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Online QFT video lectures I'm aware of Sidney Coleman's 1975/76 sequence of 54 lectures on Quantum Field Theory. Are there any other high-quality QFT lecture series available online?
| Although the recordings of lectures from the Perimeter Institute has been done some years now, the Perimeter Institute launched the PSI Online program in 2018, which as of now contains three full courses (lecture videos + problem sets), two in Quantum Field Theory and one in advanced Condensed Matter physics.
An overview of PSI Online can be found in the following link:
https://www.perimeterinstitute.ca/training/psi-masters-program/psi-online
| {
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Gravitational time dilation at the earth's center I would like to know what happens with time dilation (relative to surface) at earth's center .
There is a way to calculate it?
Is time going faster at center of earth?
I've made other questions about this matter and the answers refers to:
$\Delta\Phi$ (difference in Newtonian gravitational potential between the locations) as directly related, but I think those equation can't be applied to this because were derived for the vecinity of a mass but not inside it.
Any clues? Thanks
| The rule I mentioned in another question, that the time dilation factor is $1+\Delta\Phi/c^2$, applies here. The derivation (found in various textbooks) depends only on the assumptions that fields are weak and matter is nonrelativistic, both of which are true for the Earth.
Modeling the Earth as a uniform-density sphere (not true, of course, but I don't care), we find that $g(r)=GMr/R^3$ where $R$ is the radius of the Earth. So
$$
\Delta\Phi={GM\over R^3}\int_0^Rr\,dr={GM\over 2R}.
$$
That means that
$$
{\Delta\Phi\over c^2}={GM\over 2Rc^2}={1\over 4}{R_s\over R}.
$$
Here $R_s=2GM/c^2$ is the Schwarzschild radius corresponding to the Earth's mass. Numerically, $R_s$ is about 9 mm, and $R$ is about 6400 km, so $\Delta\Phi/c^2=3\times 10^{-10}$.
The sign of the effect is that clocks tick slower when they're deeper in the potential well. That is, a clock at the Earth's surface ticks 1.0000000003 times faster than one at the center.
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Are many-worlds and the multiverse really the same thing? Are many-worlds and the multiverse really the same thing?
Not too long ago, Susskind and Bousso uploaded the article "The Multiverse Interpretation of Quantum Mechanics" with the thesis that the many-worlds interpretation and the multiverse of eternal inflation are one and the same thing. The parallel worlds of one are exactly the same thing as the parallel worlds of the other.
First, they claim decoherence can't happen over a complete description of the future light-cone of the measurement. Then, they apply that principle to eternal inflation. Without decoherence, superpositions of nucleating bubbles and metastable vacua can't decohere. According to the anthropic principle, most bubbles have no conscious observers, but an exponentially small minority do. Apply black hole complementarity to causal horizons.
Then, somehow, in a way I can't follow, they combine causal diamond worlds into a global multiverse. Then they claim decoherence is reversible.
My head is spinning. What are your opinions on this paper?
| In Figure 6 of their article, they draw a causal diamond, and divide the future null boundary into $B^+$ and $B^-$. They implicitly assumed that the Hilbert space of $B$ is the tensor product of the Hilbert space of $B^+$ with the Hilbert space of $B^-$. If $B$ were a spacelike surface, that would be true, but it's not. It's null, and null separated operators don't commute. The operation of taking the partial trace over $B^-$ to get the density matrix over $B^+$ is not correct.
The authors also can't agree upon whether time evolution between the different $\beta$ slices are unitary or not. They claim it isn't, but cite another article "The Census Taker's Hat" written by Susskind claiming the contrary. One gets the impression Bousso and Susskind aren't in agreement on this issue.
In Section 2.5, the authors forgot their own insight that distinct maximal causal diamonds are complementarity to each other. Because of this, it makes no sense to combine the local states of different overlapping causal diamonds together to get a more global description of a "superobserver" state.
| {
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What are the conditions for decoherence to be irreversible? Spin echo experiments have been able to reverse the motions of all the molecules in a gas in statistical mechanics in the manner of Loschmidt. The Fermi-Ulam-Pasta model has solutions with a single mode dispersing, only to recohere after quite some time has elapsed. Can the same thing happen for decoherence? What are the conditions fyor decoherence to be irreversible?
| An article you might be interested in: http://www.physics.arizona.edu/~cronin/Research/Lab/some%20decoherence%20refs/RBH97.pdf
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Is the wave function objective or subjective? Here is a question I am curious about.
Is the wave function objective or subjective, or is such a question meaningless?
Conventionally, subjectivity is as follows: if a quantity is subjective then it is possible for two different people to legitimately give it different values. For example, in Bayesian probability theory, probabilities are considered subjective, because two agents with access to different data will have different posteriors.
So suppose two scientists, A and B, have access to different information about the same quantum system. If A believes it has one wavefunction and B believes it has another, is one of them necessarily "right" and the other "wrong"? If so then the wavefunction is objective, but otherwise it must contain some subjective element.
| According to the Pusey-Barrett-Rudolph theorem, if scientists Alice and Bob disagree about their beliefs of the wave function, at least one of them has to be wrong. By your definition of objectivity, this makes the wave function "objective".
Note the PBR theorem doesn't apply to mixed density states, as in Wigner's friend scenarios. This kind of implies there is a conceptual difference between wave functions and density states.
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Graduate Physics Problems Books Need to brush up on my late-undergrad and early-grad physics and was wondering if anyone can recommend books or lecture notes (hard copy, or on-line) that also have solutions.
Two that I have come across are:
Princeton Problems in Physics with Solutions - Nathan Newbury
University of Chicago Graduate Problems in Physics with Solutions - Jeremiah A. Cronin
Spacetime Physics - Taylor & Wheeler (favorite book on special relativity; has a lot of problems with solutions at the back; a lot of the problems really enforce the material and discuss paradoxes)
If possible, please also provide a reason why you like the books as opposed to just listing them.
| Thinking Like a Physicist: Physics Problems for Undergrads: I love this book because it fosters a real sense of physical understanding, so it's not just mathematics, but actual physical reasoning. Plus, I found the problems challenging and interesting.
Then, there is always the MIT Open Course Ware in Phsyics, which has undergrad and graduate courses with assignments, lecture notes, tests, problems, solutions, etc.
If its qualifying exam questions and problems that you're after, a lot of universities will post examples of past versions online, its just a matter of looking at the department website hard enough.
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Do quantum states contain exponentially more information than classical states? Do quantum states contain exponentially more information than classical states? It might seem so at first sight, but what about in light of this talk?
| This is exactly the question to which I was seeking with a colleague of mine to give some sort of answer. We considered a game played by a team of two - say Alice and Bob - in which the value of a random variable $x$ is revealed to Alice only, who cannot freely communicate with Bob. Instead, she is given a quantum $n$-level system, respectively a classical $n$-state system, which she can put in possession of Bob in any state she wishes. We thought about evaluating how successfully they managed to store and recover the value of $x$ in the used system by requiring Bob to specify a value $z$ and giving a reward of value $f(x,z)$ to the team.
Now Holevo's bound in itself does not imply that the expected reward in the quantum case could not be larger than in the classical case. (One can easily give an example of 2 channel matrices such that the first one has a greater capacity, yet in a certain game - if it is played only a few times - it is better to use the second channel.)
Nevertheless, we've managed to show that whatever the probability distribution of $x$ and the reward function $f$ are, when using a quantum $n$-level system, the maximum expected reward obtainable with the best possible team strategy is equal to that obtainable with the use of a classical $n$-state system.
See the details in my article on the arXiv:
P.E. Frenkel and M. Weiner, Classical information storage in an $n$-level quantum system, arXiv:1304.5723.
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Why don't waves erase out each other when looking onto a wall? If I stand exactly in front of a colorful wall, I imagine the light waves they emit, and they receive should randomly double or erase out each other.
So as a result, I imagine I should see a weird combination of colors, or a full-black/full-white/very lightly perception of the wall, when all the light waves that the wall receives and emits cancel out each other or double each other.
Why doesn't that actually happen? Any time I look into a wall, I never see the wall "cancel out" of my perception. Same for radio waves. Shouldn't radio waves not work at all? There are so many sources where they could reflect and cancel out or annoy each other...
| If you look at the time domain and draw different waves with very different frequencies , and also draw some waves with very short peak, you can imagine that the sum of these low peak waves do not interfere that much in the final result of the bigger peak wave.
in digital communications, we say that the signal to noise ratio in that case is still big enough so that the receiver can tell what is bit 1 or 0 because it still hasnt reached the receiver decision frontier (a term used to show in a more didatic way that the receiver is influenced by the noise as if it was in a frontier where noise can push signal to 0 or 1 side).
in analog communication, that sum of waves do not ruin that much the communication and you still can watch the tv however you will see the noise on the screen, different from digital communication where it just needs to decide for the correct bit to show the image with no noise.
| {
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Understanding the cause of sidebands in Amplitude Modulation I've read it many places that Amplitude Modulation produces sidebands in the frequency domain. But as best as I can imagine it, modulating the amplitude of a fixed-frequency carrier wave just makes that "louder" or "quieter", not higher-frequency or lower-frequency. That is, I believe I could sketch, on graph paper, a path of a wave function that touches a peak or a trough exactly every 1/f increments, regardless of the "volume". Why do the sidebands appear?
| You are correct that the modulated signal touches a peak every 1/f and a trough every 1/f. But if you looked at the waveform of a carrier being amplitude modulated by a much lower frequency pure sinewave, it is intuitive that there is another frequency component. (That is the "feel" part.) The math in other posts shows you what those frequency relationships are.
I would add that commercial AM stations do not have signals which conform to these examples: The examples have zero crossings other than that which would be expected from the carrier (e.g., whenever the cosine term in the modulating signal = 0). Commercial AM stations do not allow their carriers to get modulated down to zero amplitude.
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Why doesn't a phone charge faster, rather than slower when it is in use In Physics class, we were building parallel circuits, and as more lights were attached in parallel, they got brighter (as more power was being provided to the lights, and the resistance decreases). So, when I charge a phone, why doesn't the battery charge faster when it is in use (eg. more devices powered on within the phone), than when it is not in use?
Clarification: All lights at once became brighter
| When you add components in parallel, the voltage across each component is the same. Thus, in a simple AC circuit, when you add an extra light bulb in parallel, the brightness of the light bulbs should stay the same, not increase.
This might seem wrong because there is less total resistance in the parallel circuit, and hence more total current, but the current is divided between the different branches. The result is that the brightness of a single light bulb is constant. The total brightness of the light bulbs combined increases. If anything, there might be some nominal decrease in brightness per light bulb due to the internal resistance of the battery - when you add more light bulbs in parallel, their effective resistance decreases, so the battery's internal resistance becomes more important and the bulbs are slightly dimmer. This should be a very small effect for a realistic circuit.
The analogy to the phone is a non-sequitur. Presumably it charges slower when in use because the phone is powered by the battery at all times, so using it drains the charge. I don't know for sure this is how it works, but absent more detailed information about the circuitry of the phone, knowing how a simple DC circuit with light bulbs in parallel works simply does not speak to the workings of the phone.
Edit: Two people have said that the charger supplies a fixed current, not a fixed voltage, and that this is the real explanation for why the phone charges slower when you are using it.
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Does a magnetic field do work on an intrinsic magnetic dipole? When you release a magnetic dipole in a nonuniform magnetic field, it will accelerate.
I understand that for current loops (and other such macroscopic objects) the magnetic moment comes from moving charges, and since magnetic fields do no work on charges ($F$?perpendicular to $v$) it follows that the work done on the dipole (that caused its gain in kinetic energy) must have come from somewhere other than the magnetic forces (like electric forces in the material).
However, what about a pure magnetic moment? I‘m thinking of a particle with intrinsic spin. Of course, such a thing should be treated with quantum mechanics, but shouldn't classical electrodynamics be able to accommodate a pure magnetic dipole? If so, when I release the pure dipole in a nonuniform B-field and it speeds up, what force did the work? Is it correct to say that magnetic fields DO do work, but only on pure dipoles (not on charges)? Or should we stick with "magnetic forces never do work", and the work in this case is done by some other force (what?)?
Thanks to anyone who can alleviate my confusion!
| Yes, of course that if a field - magnetic field - is able to make a bar magnet rotate or move, it is doing work. The statement that magnetic fields don't do any work only applies to point-like pure electric charges.
Magnetic moments may be visualized as objects with a forced motion of charges (solenoids have the same magnetic field as bar magnets), and if something is moving, the magnetic force is becoming a force that does work.
In terms of formulae, the magnetic force on a charge is $q\vec v\times \vec B$ which is identically perpendicular to $\vec v$ and that's why it does no work. However, forces on magnetic dipoles and more general objects don't have the form $\vec v\times$ - they're not perpendicular to $\vec v$, so they do work in general.
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Is 3+1 spacetime as privileged as is claimed? I've often heard the argument that having 3 spatial dimensions is very special. Such arguments are invariably based on certain assumptions that do not appear to be justifiable at all, at least to me. There is a summary of arguments on Wikipedia.
For example, a common argument for why >3 dimensions is too many is that the gravitational law cannot result in stable orbital motion. A common argument for <3 dimensions being too few is that one cannot have a gastrointestinal tract, or more generally, a hole that doesn't split an organism into two.
Am I being overly skeptical in thinking that while the force of gravity may not be able to hold objects in stable orbits, there most certainly exist sets of physical laws in higher dimensions which result in formation of stable structures at all scales? It may be utterly different to our universe, but who said a 4D universe must be the same as ours with one extra dimension?
Similarly, isn't it very easy to conceive of a 2D universe in which organisms can feed despite not having any holes, or not falling apart despite having them? For example, being held together by attractive forces, or allowing certain fundamental objects of a universe to interpenetrate, and thus enter a region of the body in which they become utilized. Or, conceive of a universe so incomprehensibly different to ours that feeding is unnecessary, and self-aware structures form through completely different processes.
While I realise that this is sort of a metaphysical question, is 3+1 dimensions really widely acknowledged to be particularly privileged by respected physicists?
| Science fiction writer (but also published physicist) Greg Egan has put quite a bit of work into investigating a universe with 4+0 dimensions: Orthogonal. Some of it is quite ingenious, eg. assuming a compact universe guarantees that the (modified) wave equation doesn't have exponentially growing solutions and time appears, without the -1 in the spacetime metric, as the local gradient of entropy.
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How do laser rangefinders work when the object surface is not perpendicular to the laser beam? I find the functioning of a laser rangefinder confusing.
The explanation usually goes like this: "you shine a laser beam onto the object, the laser beam gets reflected and gets back to the device and time required for that is used to calculate the distance".
Okay. But the object surface can be uneven and not perpendicular to the laser beam so only a tiny fraction of beam energy is reflected back to the device. And there's plenty of other radiation around, sunlight included.
How does a rangefinder manage to "see" that very weak reflected signal in a reliable manner?
| The amount of laser energy reflected back will be the limiting factor of its effective range. However, since the laser's radiation is of a specific wavelength, it won't be confused by extraneous radiation from ambient sources.
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Why are smaller animals stronger than larger ones, when considered relative to their body weight? I am interested in why many small animals such as ants can lift many times their own weight, yet we don't see any large animals capable of such a feat.
It has been suggested to me that this is due to physics, but I am not even sure what to search for. Could someone explain why indeed it is easy for smaller objects/lifeforms to support several times their own weight, but this is harder as objects/animals become larger?
| To lift anything, a life form on this planet needs muscles. If you want to lift heavier things you need
*
*more muscles and
*a stronger body/legs to support that additional weight.
Stronger muscles and bones need to be supplied with more oxygen, nutrion and so on which leads to needing a stronger heart, a better digestive system and so on. Or simpler: if you want to be stronger, you will get heavier and you need to support your own weight too. And the additional weight increases faster then the additional force you can get out of that.
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Experimental evidence showing the kinetic energy of an electron changes in a static non-uniform magnetic field? In a previous question, Does a magnetic field do work on an intrinsic magnetic dipole?, one highly rated answer suggested that static magnetic fields do work on intrinsic magnetic dipoles in a non-uniform magnetic field. I can visualise the change in kinetic energy of the nucleus of an atom coming from a change in the configuration of the electrons around the nucleus. But for an electron, since it's truly fundamental, I'm scratching my head over where the energy comes from to change its kinetic energy. If it does, then it really must come from the static magnetic field.
So what is the experimental evidence that shows the kinetic energy of an electron changes in a static non-uniform magnetic field?
| There is no evidence that the kinetic energy of an electron changes in a static magnetic field. You'd end up with a perpetual motion machine otherwise.
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Can the effects of gravity be broken by jumping? I was having a debate the other day with a work colleague where I explained that gravity is a weak force because it is easily broken. Then I remembered a lecture by someone, I forget who, that explained gravity is very weak because you can break its influence just by jumping or lifting a pencil, etc.
He countered that with something along the lines of 'that even though the pencil or your body is being moved away from the source of gravity it is still affected by gravity and thus it has weight'.
Is jumping a good example of gravity being a weak force?
P.S. You can probably tell, my colleague and I are not physicists but we enjoy our little debates, we just need to get our facts straight.
| I think what you heard in that lecture is this argument:
Gravitation is by far the weakest of the four interactions. Hence it is always ignored when doing particle physics. The weakness of gravity can easily be demonstrated by suspending a pin using a simple magnet (such as a refrigerator magnet). The magnet is able to hold the pin against the gravitational pull of the entire Earth.
Yet gravitation is very important for macroscopic objects and over macroscopic distances for the following reasons. Gravitation:
*
*is the only interaction that acts on all particles having mass;
*has an infinite range, like electromagnetism but unlike strong and
weak interaction
*cannot be absorbed, transformed, or shielded against;
*always attracts and never repels.
Jumping, or lifting a pencil, is in your example "breaking" the influence of gravity because the electromagnetic interactions between your feet and the ground are able to counteract the gravitational force of the entire planet, thus demonstrating that gravity is a weak force, so I'd say yes, it's a good example.
Source: http://en.wikipedia.org/wiki/Fundamental_interaction#Gravitation
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Nuclear decay rate affected by sun and quantum randomness If nuclear decay rate were affected by sun,
then emission probabilities would be subject to sun state and its influence,
so quantum randomness would depend on it,
Would it still be truly random?
One could argue that randomness would keep intact when consider the whole system..
But what about a non-isolable effect? what if there is no way to "define a system" without a sun/star influence?
If a variation source were found (for example neutrino emission) then perhaps it could be generated intentionally to control decay rates. Could we be sure that emission probabilities would never could be controled to reach 0 or 1?
Thanks for reading
| If decay rates are found could be reduced to 0, then we are thoroughly wrong in what we thought about nuclear decay, and that is unlikely to happen. The term "nuclear decay" would probably be obsolete in that case. Quantum mechanics, itself, may have answering to do given the strong form of what you are speculating about. But there is little case to believe that will happen. Quantum tunneling is extremely unlikely to be disproved or even modified in any meaningful way.
Here is a diagram from a previous question.
The gist of nuclear decay is quantum tunneling. Now, how could neutrinos from the sun affect this? Well it could be an interaction that is allowable or not allowable under the standard model, although I'm not an expert on that. Whatever it is, however, it's almost certainly not changing the fundamental nature of the nuclear decay by quantum tunneling, which is a truly quantum random process from our view of the universe.
Whatever field or neutrino flux that causes the effect (assuming it's verified) won't be fundamentally changing nuclear decay. It will just be another interaction in the book of nuclear interactions.
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Is the fine structure constant actually a constant or does its value depend on the energy scale? The value of the fine structure constant is given as
$$
\alpha = \frac{e^2}{4\pi\varepsilon_0\hbar c} = \frac{1}{137.035\,999..}
$$
It's value is only dependent on physical constants (the elementary charge $e$, speed of light $c$, Plancks constant $\hbar$), the vacuum permitivvity $\varepsilon_0$) and the mathematical constant $\pi$, which are considered to be constant under all cirumstances.
However the Wikipedia article Coupling constant states
In particular, at low energies, α ≈ 1/137, whereas at the scale of the Z boson, about 90 GeV, one measures α ≈ 1/127.
I don't understand how this can be possible, except that one of the physical constants above or even $\pi$ are actually not constant, but dependent on the energy scale. But that seems nonsense.
So what do physicists mean when they say that the fine structure constant $\alpha$ increases with energy? Can you perhaps reformulate the quoted sentence above so that it makes more sense?
| Do not worry, it is a constant.
There is a sloppiness and misunderstanding in physics when the energy dependence of the cross sections is wrongly attributed to the "fundamental constant" whereas it is a cross section feature.
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Calculating diffraction-limited resolution for a lens setup Supposed a lens arrangement is prepared where light from an object is collimated, focused and recollimated etc. before entering a CCD array. Given that we can calculate the diffraction-limited resolution for each lens in the system, how do we measure the diffraction limited resolution for the whole setup?
| In the spot diagrams of optic designs the ray aberrations are always compared to the size of diffraction limited spot.
In order to do this you have to find the smallest aperture in your system. This can be a deliberately placed mechanical aperture but it can also be the circumference of a lens.
I copied the following spot diagram from
http://www.astronomy.net/articles/17/.
The black ring is the diffraction limited spot size (3.3 $\mu$m if you convert from inch) the red dots are the spot with ray aberrations. You can see a strong Coma with $1\mu$m spread in x.
I read a nice discussion about this topic a while ago but can't remember which book it was in. I tried to implement it in a raytracer. It involved an iteration over all apertures in the optical train. One had to trace all apertures into the image and chose which one produced the smallest opening angle.
I think any decent book on lens design should contain a chapter on the topic. See for example chapter 6 in Warren J. Smith: "Modern optical engineering: the design of optical systems".
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significance of maxima and minima of time varying kinetic energy of a system Consider a system of particles where the kinetic energy of the system is varying with time. I'd like to know the significance (or meaning) of the time derivative of the kinetic energy being zero at a point. What is the significance of time instances where the kinetic energy has maxima and minima ?
| As Anna pointed out in her comment, it really depends on the system. In general, there's not much you can say except that the minimum of kinetic energy corresponds to the minimum speed and the maximum of kinetic energy corresponds to the maximum speed.
There are many systems in which the minimum speed (and thus the minimum kinetic energy) is zero, and in those systems, finding the times at which the kinetic energy is a minimum tells you when the object stops.
Mathematically, if you actually take the derivative of kinetic energy with respect to time, you get
$$\frac{\mathrm{d}K}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t}\biggl[\frac{1}{2}mv^2\biggr] = m\vec{v}\cdot\vec{a}$$
(using the nonrelativistic expression for $K$). There are three ways this can be equal to zero:
*
*$v = 0$: the object is not moving. This corresponds to a minimum of kinetic energy.
*$a = 0$: the object is not accelerating (and by $F = ma$ is also experiencing no net force). This corresponds to either constant velocity motion, or a maximum of kinetic energy, in a simple harmonic oscillator for example.
*$\vec{v}\perp\vec{a}$: the object is experiencing pure centripetal acceleration, which means it's moving with a momentarily constant radius of curvature. This can be either a minimum or a maximum of kinetic energy, in an orbit for example.
Using the relativistic expression, you get
$$\frac{\mathrm{d}K}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t}\bigl[(\gamma - 1)mc^2\bigr] = \frac{m\vec{v}\cdot\vec{a}}{\bigl(1-\frac{v^2}{c^2}\bigr)^{\frac{3}{2}}}$$
from which the same conclusions follow.
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What do we consider "Perpetual Motion" I know this is a bad question to most serious Physics but I have a question about what is considered “Perpetual motion.” The Foucault pendulum in the UN consists of sphere that passes directly over a raised metal ring at the centre that contains an electromagnet, which induces a current in the copper inside the ball. This supplies the necessary energy to overcome friction and air resistance and keeps it swinging uniformly. Now the swing of the pendulum is induced but the 36h 45m clockwise shift generated bay the earths rotation is perpetual or as long as the earth rotates. Is this assumption correct?
Does a generator that works on tidal movements not fall in to the same assumption?
| From Wikipedia: "Perpetual motion describes hypothetical machines that operate or produce useful work indefinitely and, more generally, hypothetical machines that produce more work or energy than they consume, whether they might operate indefinitely or not."
The key is not moving indefinitely, but DOING WORK. People didn't build perpetual motion machines to observe inertial movement, but try to power factories, transportation, etc without any energy source.
| {
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what cools bottle of water faster: ice or snow Imagine you have a pile of snow and a pile of ice shards. You put a soda bottle which has a room temperature into both piles. Which bottle is going to cool down faster?
| This depends on contact area between bottle & ice/snow, and consistency of snow.
If there is not much air in the snow, it should have bigger contact area with bottle, and thus heat will be transferred faster.
Ice will contact with the bottle mainly at shards edges, so contact area is small.
PS. Adding water will change everything, as contact area would be maximized in both cases.
| {
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Length of a curve in D dimensional euclidean space In a book I am reading on special relativity, the infinitesimal line element is defined as $dl^2=\delta_{ij}dx^idx^j$ (Einstein summation convention) where $\delta_{ij}$ is the euclidean metric. Next, if we have some curve C between two points $P_1$ and $P_2$ in this space then the length of the curve is given as $\Delta L = \int_{P_1}^{P_2}dl$
I am having trouble deriving the next statement, which I quote:
A curve in D-dimensional Euclidean
space can be described as a subspace
of the D-dimensional spce where the D
co-ordinates $x^i$ are given by single
valued functions of some parameter
$t$, in which case the length of the
curve from $P_1=x(t_1)$ to
$P_2=x(t_2)$ can be written
$$\Delta L = \int_{t_1}^{t_2}\sqrt{\delta_{ij}
\dot{x}^i \dot{x}^j} dt \qquad
\mbox{where}\; \dot{x}^i\equiv
\frac{dx^i}{dt}$$
| Well, $\mathrm{d}l$ represents an infinitesimal length along the curve. $\mathrm{d}t$ also represents an infinitesimal length along the curve, although if the parametrizations $t$ and $l$ are different, the two infinitesimal lengths are not going to be the same. You can write the identity $\mathrm{d}l = \frac{\mathrm{d}l}{\mathrm{d}t}\mathrm{d}t$ and substitute in the definition of $\mathrm{d}l$:
$$\frac{\mathrm{d}l}{\mathrm{d}t}\mathrm{d}t = \frac{\sqrt{\delta_{ij}\mathrm{d}x^i\mathrm{d}x^j}}{\mathrm{d}t}\mathrm{d}t$$
Now, since $\mathrm{d}x^{i(j)}$ and $\mathrm{d}t$ are positive lengths, you can do a little algebraic manipulation:
$$\frac{\mathrm{d}l}{\mathrm{d}t}\mathrm{d}t = \sqrt{\frac{\delta_{ij}\mathrm{d}x^i\mathrm{d}x^j}{\mathrm{d}t^2}}\mathrm{d}t = \sqrt{\delta_{ij}\dot x^i\dot x^j}\mathrm{d}t$$
This is not 100% mathematically rigorous, but in physics we think of derivatives and differentials as the limit of finite lengths and ratios, so it makes physical sense at least. And as long as your parametrization isn't singular, the math should hold up. (If you do have a singular parametrization, then I believe the result you're asking about still holds, although you need to resort to more precise math to prove it.)
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(Almost) double light speed Let's say we have $2$ particles facing each other and each traveling (almost) at speed of light.
Let's say I'm sitting on #$1$ particle so in my point of view #$2$ particle's speed is (almost) $c+c=2c$, double light speed? Please say why I am incorrect :)
EDIT: About sitting me is just example, so in point of view of #1 particle, the second one moves at (almost) $c+c=2c$ speed?
| You are both correct and wrong.
If - sitting on one photon - you would measure the velocity of the approaching photon, the figure received would be exactly c.
However, if two photons separated by the distance of 1 light-year are sent toward each other, they will meet after exactly six months and exactly in the middle of this distance i.e. 1/2 lightyear. Go figure what the relative speed of these photons was :)
| {
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How to find the principal point in an image? I need to find the principal point in an image. Its a point where the principal axis intersects the image sensor.
Due to misalignment this point is not at the center of image always(or image sensor). I need to precisely determine its location using any of the optical methods available(if any). Any suggestions are greatly appreciated.
More information: The camera is giving me a live feed over a screen and I am able to store, analyze any part of it in real time or later.
| To do this you would need to take an image of some sort of reference object, like a regular grid pattern. You would also have to know that your imaging system has some measurable field aberrations which are radially symmetric, such as distortion. Then you could do some simple image processing to locate the center of the distortion pattern.
In the case of a regular grid pattern, you would plot the positions of the grid lines in the image. With third order distortion, the grid lines should be spaced quadratically (ie. with the space between each grid line increasing linearly). The grid spacing will reach a max (or min) at the "principal point." (I use scare quotes around "principal point" because the principal point has a specific meaning in geometrical optics, and it is not the meaning used in the question here.)
| {
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How is thermodynamic entropy defined? What is its relationship to information entropy? I read that thermodynamic entropy is a measure of the number of microenergy states. What is the derivation for $S=k\log N$, where $k$ is Boltzmann constant, $N$ number of microenergy states.
How is the logarithmic measure justified?
Does thermodynamic entropy have anything to do with information entropy (defined by Shannon) used in information theory?
| I think that the best way to justify the logarithm is that you want entropy to be an extensive quantity -- that is, if you have two non-interacting systems A and B, you want the entropy of the combined system to be
$$
S_{AB}=S_A+S_B.
$$
If the two systems have $N_A,N_B$ states each, then the combined system has $N_AN_B$ states. So to get additivity in the entropy, you need to take the log.
You might wonder why it's so important that the entropy be extensive (i.e., additive). That's partly just history. Before people had worked out the microscopic basis for entropy, they'd worked out a lot of the theory on macroscopic thermodynamic grounds alone, and the quantity that they'd defined as entropy was additive.
Also, the number of states available to a macroscopic system tends to be absurdly, exponentially large, so if you don't take logarithms it's very inconvenient: who wants to be constantly dealing with numbers like $10^{10^{20}}$?
| {
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Clebsch-Gordan Identity I'm trying to take advantage of a particular identity for the sum of the product of three Clebsch-Gordan coefficients, however, the present form of my equation is slightly different. Is there a symmetry relation that will allow me to change:
$\sum_{\alpha\beta\delta}C_{a\alpha b\beta}^{c\gamma}C_{d\delta b\beta}^{e\epsilon}C_{d\delta f\phi}^{a\alpha}$
Into:
$\sum_{\alpha\beta\delta}C_{a\alpha b\beta}^{c\gamma}C_{d\delta b\beta}^{e\epsilon}C_{a\alpha f\phi}^{d\delta}$
Notice I need to swap $j_2m_2$ with $jm$ in the last Clebsh-Gordan coefficient. Does anyone know a way to do this?
Note: My notation follows that of Varshalovich, $C_{j_1 m_1 j_2 m_2}^{jm}$
| Notice that $C^{22}_{1111}=1$ but $C^{11}_{2211}=0$. I don't think that this is true unless $a=d$ and the sums over $\alpha$ and $\delta$ have the same range.
| {
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How to calculate concentration of vapor at the surface of a water drop I'm reading a paper that examines the evaporation rates of water. In the final formula, it has the following constant:
$c_s - c_\infty $ where $c_s$ is the concentration of the vapor at the sphere surface and $c_\infty$ is the concentration of the vapor at infinity.
I'm fairly confident in how to derive $c_\infty$:
1) Calculate water vapor pressure $p_s = 610.78 e^{\frac{17.2694 T}{T+238.3}}$
2) Actual vapor pressure is then: $ p=(Relative Humidity)p_s$
3) Using ideal gas law gives concentration at infinity: $c_\infty = \frac{(Molar Mass of Water)p}{RT}$
This generally looks like it gives me answers consistent with the paper's values. But, if I screwed up, please let me know.
My problem is with $c_s$. It's weird, because I recognize how easy it should be to get this, but just can't. I've asked some of the other grad students here and they basically all are befuddled. I think this is the physicist's version of a "tip of my tongue" experience. So, can anyone give me a link or method by which I can get this?
Thanks.
| So, it turns out that the answer is incredibly simple. Which is what I suspected given the mechanism: Question seems super easy, but can't seem to get answer of the top of my head. So I go searching for an answer, but because the solution is so obvious, no book has it explicitly. Ask people, again, no one seems to have it in mind, but can use complicated formulas based on it using already derived numbers. Ladeeda.
All the paper is looking for is the difference between actual vapor pressure at the drop as opposed to vapor pressure in the room. So all one needs to do is do steps 1 and 3 above and skip 2. Then multiply the result by (1-H) where H is the relative humidity. Good Times!
| {
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Batman spotlight in the sky I have noticed that obstructing a spotlight typically results in a blurred shadow unlike the crisp batman symbol in the comics of batman. Is there a way to create a spotlight with a crisp batman symbol?
| You need to have a focusing lens built into your projector. After that, it's just like any other movie projector sort of thing, you can focus it provided you have an object to project it onto (like a flat cloud). Physically, the clarity of the image is limited by two or three things.
(1) The size of your optics (the large diameter lens) gives a limitation in that it's impossible for a lens of, say, 2 meters, to focus light in a direction more accurately than the wavelength of light divided by that diameter. For the 550 nm that the human eye adapts to at night, this is about 550nm/2m = 0.000000275 radians = 0.00002 degrees. This is probably not going to be a problem.
(2) There's also a limit based on the size of your light source (the bright thing from which the light emits, for example, the filament of a clear glass light bulb). This is a thermodynamic limit; you can't focus your light source so as to produce a temperature hotter than the light source. In practice, this means that if your light source has a size of 1cm, then you can't focus the beam from it to a size smaller than 1cm. By the way, this restriction is the limiting presence in the number of pixels displayed by a projection system. High intensity (small size high light output) sources are expensive and so projector bulbs are sometimes almost as expensive as the rest of the projector.
(3) If the image is viewed off-axis, a non-flat projection screen (or a translucent screen such as a cloud) will make the image blurred. But this won't be such a big deal from near where the image is projected.
Neither of these restrictions will significantly degrade your image. Good hunting!
| {
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If you view the Earth from far enough away can you observe its past? From my understanding of light, you are always looking into the past based on how much time it takes the light to reach you from what you are observing.
For example when you see a star burn out, if the star was 5 light years away then the star actually burnt out 5 years ago.
So I am 27 years old, if I was 27 light years away from Earth and had a telescope strong enough to view Earth, could I theoretically view myself being born?
| Yes, you can. And you do not even need to leave the Earth to do it.
You are always viewing things in the past, just as you are always hearing things in the past. If you see someone do something, who is 30 meters away, you are seeing what happened $(30\;\mathrm{m})/(3\times10^8\;\mathrm{m}/\mathrm{s}) = 0.1\;\mu\mathrm{s}$
in the past.
If you had a mirror on the moon (about 238K miles away), you could see about 2.5 seconds into earth's past. If that mirror was on Pluto, you could see about 13.4 hours into Earth's past.
If you are relying on hearing, you hear an event at 30 m away about 0.1 s after it occurs. That is why runners often watch the starting pistol at an event, because they can see a more recent picture of the past than they can hear.
To more directly answer the intent of your question: Yes, if you could magically be transported 27 lightyears away, or had a mirror strategically placed 13.5 lightyears away, you could see yourself being born.
| {
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Is spacetime simply connected? As I've stated in a prior question of mine, I am a mathematician with very little knowledge of Physics, and I ask here things I'm curious about/things that will help me learn.
This falls into the category of things I'm curious about. Have people considered whether spacetime is simply connected? Similarly, one can ask if it contractible, what its Betti numbers are, its Euler characteristic and so forth. What would be the physical significance of it being non-simply-connected?
| I suppose there are many aspects to look at this from, anna v mentioned how Calabi-Yao manifolds in string theory (might?) have lots of holes, I'll approach the question from a purely General Relativity perspective as far as global topology.
Solutions in the Einstein Equations themselves do not reveal anything about global topology except in very specific cases (most notably in 2 (spacial dimensions) + 1 (time dimension) where the theory becomes completely topological). A metric by itself doesn't necessarily place limits on the topology of a manifold.
Beyond this, there is one theorem of general relativity, called the Topological Censorship Hypothesis that essentially states that any topological deviation from simply connected will quickly collapse, resulting in a simply connected surface. This work assumes an asymptotically flat space-time, which is generally the accepted model (as shown by supernova redshift research and things of that nature).
Another aspect of this question is the universe is usually considered homogenous and isotropic in all directions, topological defects would mean this wouldn't be true. Although that really isn't a convincing answer per say...
| {
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Calculating time for a fully charged UPS I have a UPS of 1000 Volts connected with 2 batteries each of 150 Amp. How much time it will take to consume the whole UPS (after fully charged) when a device of 1Amp is getting electricity form that UPS.
Please also explain me the calculation.
| There are some missing data in your question.
*
*What voltage does the batteries have, I'm going to assume 12V since it is common.
*Battery capacity, you typed it as 150A but I guess it was 150Ah. Please note that a normal car battery on a car like a VW Golf has approx 60Ah so 150Ah is a quite big battery.
*Output power, you state that you have 1000V (Volts) out, but I guess you are talking about 1000W (Watts). And that should be maximum power.
*Output voltage should be either 230V (or 110V). I will assume 230V.
The first thing is that a device connected to the UPS and is using 1A, that is 1A at 230V or $230V * 1A = 230W.$
Then we go inside the UPS and have a look. We still has the 230W but since it is now a 12V system, we need to reverse the conversion into amps again and get something like $\frac{230W}{12V}=19A$.
And then we divide the battery capacity with this current and get something like,
$\frac{(150*2)Ah}{19A}=15h$.
Please note that those 15 hours is the best value you can expect to see, but since there is losses in the conversion from 12V to 230V we probably loose some 10-20% of the energy, and that translate directly into a shorter time.
Let's say 80% of 15h would be approx 12h.
Then I must add that Fortunato does have a point. Batteries degrade over time and can't hold the same charge, so make sure you have some margin and check/service your UPS on yearly basis.
| {
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How come an anti-reflective coating makes glass *more* transparent? The book I'm reading about optics says that an anti-reflective film applied on glass* makes the glass more transparent, because the air→film and film→glass reflected waves (originated from a paraxial incoming wave) interfere destructively with each other, resulting on virtually no reflected light; therefore the "extra" light that would normally get reflected, gets transmitted instead (to honor the principle of conservation of energy, I suppose?).
However, this answer states that "Superposition is the principle that the amplitudes due to two waves incident on the same point in space at the same time can be naively added together, but the waves do not affect each other."
So, how does this fit into this picture? If the reflected waves actually continue happily travelling back, where does the extra transmitted light come from?
* the film is described as (1) having an intermediate index of refraction between those of air and glass, so that both the air-film and film-glass reflections are "hard", i.e., produce a 180º inversion in the phase of the incoming wave, and (2) having a depth of 1/4 of the wavelength of the wave in the film, so that the film-glass reflection travels half its wavelength back and meets the air-film reflection in the opposite phase, thus cancelling it.
| The wave reflected from the air-film interface continues happily traveling back, as you say, but so does the wave reflected from the film-glass interface. Since they are the same frequency but in antiphase, they interfere destructively as long as they keep going. Since the superposition principle states that they do not affect each other in any way, they keep on going as long as they like.
However, there is no energy transported backwards in the reflected waves, because the energy is proportional to the square of the total electric field. It is the energy that is conserved, not the electric field, so all the energy (if not absorbed) is transmitted.
| {
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Is there any way to increase a rubber-bands lifetime? Rubber-bands are simple, yet very useful. Old rubber bands(5 years?) get brittle? Why is that?
| If a rubber band is usable for 5 years, it's a very good one.
There is not much you can do to prevent brittleness other than storing them in a dark and cool place. Light and oxygen are the most dangerous enemy of rubber.
The biggest influence on lifespan is the manufacturer, because they can choose
to use a persistent (expensive) type of rubber (eg Kalrez®, ask for prizes!)
Of course there are less extreme rubbers in prize.
Another way to extend life is to change the color; adding carbon black prolonges the life of rubber substantially (Protection from light) and by adding antioxidants.
The background of embrittlement of polybutadiene (polyisoprene) based
rubbers is crosslinking by radical chain reactions with oxygen.
Cheap urethane rubbers die from similar reactions, but those usually
become sticky, even liquid by this processes.
| {
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What happens to light in a perfect reflective sphere? Let's say you have the ability to shine some light into a perfectly round sphere and the sphere's interior surface was perfectly smooth and reflective and there was no way for the light to escape.
If you could observe the inside of the sphere, what would you observe? A glow? And would temperature affect the outcome?
Seems silly, it's just something I've always thought about but never spent enough time (until now) to actually find an answer.
| Just another perspective:
Since the sphere is non-ergodic, your observation depends on your and the source locations inside the sphere. For ergodic shapes (ellipsoid, etc), you will see an evenly lit world.
| {
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Simplest interferometer I want to build simplest interferometer which should be able to measure movements down to fraction of wavelength.
What is the simplest scheme for that, and what are the requirements for a laser?
I have a bunch of laser diode-based ones, and I guess they might be not coherent enough...
Are green DPSS ones any better?
| Well, one problem you are likely to encounter is that your setup will likely be vibrating with amplitude on this order.. do you have a floating optical table?
Coherence is probably not that important. At a minimum you will need a beamsplitter, two mirrors and a diode or some other way to measure the interference pattern. A lens or two to magnify the pattern will also be helpful.
| {
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Software for simulating 3D Newtonian dynamics of simple geometric objects (with force fields) I'm looking for something short of a molecular dynamics package, where I can build up simple geometric shapes with flexible linkages/etc and simulate the consequences of electrostatic repulsion between surfaces. Something, say, that would let me simulate billiards in a magnetic field.
Does anyone have good recommendations for a 3D Newtonian dynamics software package (free or not) with these sort of capabilities?
| I've not tried it myself, but you may want to check out the Elmer multiphysics package, which looks like it can do time dependendent 3D finite element mechanics + electrostatics.
Here's a nice pic they have of a beam being deflected by electrostatic attraction:
It's F.O.S.S. They have windows binaries and it's in the ubuntu repo.
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Do eyeballs exhibit chromatic aberration? Fairly straightforward question. If not, why not?
I suspect that if they do, it is not perceived due to the regions of highest dispersion being in one's region of lowest visual acuity.
| Yes, it does. We don't see it because our brain automagically 'correct it' because it always see the same aberration from the childhood.
Our eye focuses on 'green' wavelength as it's its peak sensitivity, so red and especially violet lines are usually slightly out-of-focus.
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'Getting in' to research physics? I'm going to be choosing a university course soon, and I want to go into a branch of physics. A dream job for me would be to work in research, however, I do realise that this isn't for everyone and is difficult to reach. So what is the best way to go about achieving this aim? What things can I do which will help me?
| I'd say study hard, but also learn about the fundamentals of experiment design, learn how to write proper scientific papers, and learn how publish your work. These things aren't (in my experience) typically taught in the undergraduate or even graduate level coursework, but are absolutely crucial to "success" in a research environment.
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Solve spring system I am not physics in training so if anything is unclear please request clarification.
I have a set of objects to be placed on a line, linked with springs of known lengths (L) and stiffnesses (K). I want to solve the equilibrium positions of each object (x).
o---------o--------------o
x0 x1 x2
|~~~~L1~~~|~~~~~~L2~~~~~~|
|~~~~~~~~~~L3~~~~~~~~~~~~|
I am following this Wikipedia entry - I believe the idea is to set the forces on each object zero and then solve linear equations.
However, my problem is a bit more complicated. My objects have known lengths, so the problem becomes:
ooooo------ooo------------oooo
x0 x1 x2
|~~L1~~| |~~~~~~L2~~~~|
|~~~~~~~~~L3~~~~~~~~~~|
I would also forbid the objects occupy the same place. Can someone please help me derive a solution here? Thanks so much.
| I'm assuming you want to solve such problems automatically on a computer. If you want to solve them individually by hand, the best method will obviously be different.
Without the constraint that the objects can't pass through each other, the problem is an unconstrained quadratic optimization problem, which is solved simply by taking the gradient of that quadratic function and setting it to zero (which is a linear equation).
With the constraint, the problem becomes a quadratic programming problem. The potential energy matrix Q will indeed be positive definite (because if you make add a sufficiently large multiple of any perturbation to the coordinates, the potential energy will eventually go up), so it's also an example of convex programming.
There are many libraries available to solve convex quadratic programming problems for you with well-known algorithms. That Wikipedia article gives the examples of OpenOpt, qp-numpy, CVXOPT...
So to solve your problem you just convert the given spring and object lengths and spring stiffnesses into the format of a quadratic programming problem, call an appropriate library function to solve it, and then interpret the solution as the desired equilibrium positions.
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Roughly how many atoms thick is the layer of graphite left by a pencil writing on paper? Actually I can't expand much as the question pretty much explains the query. I would be interested in the method of estimating an answer as well as a potential way to measure it experimentally. Thanks.
P.s. I'm not sure what tags to include for this one - for those that can, feel free to edit them as you see fit.
| Although I don't know anything about this, using some rough estimates I think I can get the right order of magnitude:
*
*Volume of graphite in a pencil: $10 cm$ cylinder of $1 mm$ thick = $0.314 mm^3$ (error: ~factor 2)
*Maximum surface a pencil can write: $50 km$ $\times$ $1$ mm = $10 m^2$ (error: ~factor 5)
*Thickness of the graphite layer: Volume / Surf. Area = $31.4$ nanometers
*Size of a (carbon) atom: $0.22 nm$ (error: 10%)
*Thickness of the layer: $31.4 nm /0.22 nm$ = $142$ carbon atoms
so I'd say 'about a 100 atoms' (or at least more than 10 and less than 1000). I might have been a bit conservative with my error estimates but this seems reasonable.
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Why is 55-60 MPH optimal for gas mileage of a passenger car? My driver's education teacher back in high school said 55 MPH is optimal for gas mileage of a passenger car. Just last week, I read an article in a magazine saying 60 MPH is optimal. These numbers are pretty close, so there's some validity in the statement. What's the physics explanation for this 55-60 MPH sweet spot?
| It's really an engineering question. Cars are designed for efficiency nowadays; especially in Europe the taxes are directly related to the MPG ratings on standardized test cycles. But even in the US, there are standard test cycles for fuel effiency. Those test cycles typically have large segments driven at ~55 mph.
Now an engineer that's optimizing for fuel efficiency has to make tradeoffs. Adding more gear ratios can improve engine efficiency, but increases costs. His marketing department really doesn't ask for efficiency at 75 mph; that buys them nothing. So there's generally little commercial need for a 6th gear to optimize fuel efficiency at those speeds (You do see 6th gears more commonly on German cars - no speed limit there)
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Could a human run horizontally inside a Wall of Death? A popular circus stunt is for a motorcycle rider to ride inside a bowl shaped depression called a "Wall of Death." The rider goes higher and higher up the wall until they are actually horizontal. I wonder if a human could do the same.
| The speed you need (explained http://physics.mut.ac.th/PhysicsMagic/wall.htm)
$$
V^2 > R g/u
$$
$v$ = the velocity (m/s)
$R$ = Radius of the pit (m)
$g$ = acceleration of gravity (9.8m/s^2)
$u$ = coeff of static friction
So if u is 1 (the maximum possible), and you are an olympic athlete that can run 100m in 10s
Then you apparently could with a radius of about 10m
$$
R = uV^2/g = 10^2/9.8 = 10m
$$
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Books that every layman should read To add to Books that every physicist should read:
A list of popular physics books for people who aren't necessarily interested in technical physics.
(see also Book recommendations)
| I can recommend the following physics books. They are all somewhat different, but collectively they provide a good picture of where ideas in physics are today and how modern theories developed.
The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius, Graham Farmelo
Quantum Enigma, Bruce Rosenblum and Fred Kuttner
The Infinity Puzzle, Frank Close
About Time, Adam Frank
Knocking on Heaven's Door, Lisa Randall
The Grand Design, Stephen Hawking and Leonard Mlodinow
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Is the earth expanding? I recently saw this video on youtube:
http://www.youtube.com/watch?v=oJfBSc6e7QQ
and I don't know what to make of it. It seems as if the theory has enough evidence to be correct but where would all the water have appeared from? Would that much water have appeared over 60 million years? Also what would cause it to expand. The video suggests that since the time of dinosaurs the earths size has doubled in volume, how much of this is and can be true?
[could someone please tag this, I don't know what category this should come under]
| Although considered a viable alternative hypothesis in the past, the expanding earth hypothesis is now generally considered to be obsolete, given the overwhelming evidence in support of plate tectonics. The following paper uses geologic evidence and a classical physical analysis of the earth-moon system to show there is no evidence that the earth's radius has significantly changed for at least the last ~620 Ma.
Williams, G.E. (2000), "Geological constraints on the Precambrian history of the Earth’s rotation and the moon’s orbit" (PDF), Reviews of Geophysics 38 (1): 37–59
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Creation of matter in the Big Bang I appreciate your patience to my neophyte question. I am working on my dissertation in philosophy (which has nothing or little to do with physics) about the "problem of naming." Briefly what I am arguing is that when we name something, we stop it from being anything or everything else. It is a phenomenological question and has a lot to do with language as an object.
My question for you is that, is it true that all matter was somehow formed in the Big Bang or in those famous three minutes following? I think I understand that helium and hydrogen were formed and are they then to be considered the basis of all matter today? A friend said to me a long time ago that we are made of the same atoms that were present at the Big Bang; could this possibly be true? (And how wonderful if it is...)
| The closer to the Big Bang you look, the more fundamental are the particles that constitute the universe (quarks, leptons, etc.) - go far back enough and you reach the point where only a field description makes sense because of the curvature of the universe.
But after the 3rd minute and up to around the 20th minute, the universe had cooled down enough for nuclei to form and yet was hot enough for fusion to occur. This was the period of nucleosynthesis when the first elements in the periodic table were forged. Most of the nuclei formed were hydrogen and helium and their various isotopes but there were also trace amounts of lithium and beryllium.
This is confirmed today in the precise ratio of elements observed in the universe with the significant abundance of hydrogen and helium, and is one of the best numerical confirmations of the Big Bang theory.
Rocky planets and humans and what not however require higher elements like carbon, nitrogen, oxygen and silicon to form. These were not formed in the Big Bang like H and He. They were forged at the centres of stars and dispersed in violent supernovae explosions that took place millions to billions of years after the Big Bang.
That still is pretty wonderful - that every atom in our body has its origins in either stardust or the Big Bang.
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Smoothed particle hydrodynamics in cosmological N-body simulations What is the role of smoothed particle hydrodynamics (SPH) in cosmological N-body simulations like the Millenium Run (performed with Gadget-2)?
| To answer your specific question: absolutely none.
The Millenium run is a "dark matter-only" simulation. In this sort of simulation gas physics is taken to play a negligible role. All the gas (and stars, indeed all "baryonic matter" as it's called in the jargon) is removed and replaced with additional dark matter. The extra dark matter is added just to keep the density of the Universe the same as it would have been with the gas included. On some scales, the behavior of the dark matter is not very sensitive to the processes of "baryon physics", so this process ends up giving a simulation output that is a decent approximation to the large-scale structure of the Universe. It's also much cheaper in terms of computing time to run a dark matter-only simulation than one that includes gas physics.
Smoothed particle hydrodynamics is, as the name implies, a technique to solve the equations of hydrodynamics, so it is only used in simulations that model gas (stars are usually treated as a collisionless fluid, or if resolution is sufficient, as individual point masses, and so are treated separately). In SPH, the gas is discretized as a collection of Lagrangian "particles". Each particle has a position and velocity, and carries fluid properties (e.g. temperature, pressure, etc.) as well. To obtain the properties of the fluid at an arbitrary point in space, the appropriate property is calculated as a kernel-smoothed average over nearby particles. Typically the nearest particles carry the most weight, and particles beyond a certain distance don't contribute at all. These "smoothed" values are the values used in solving the equations of hydrodynamics (Navier-Stokes equations, or some simplification thereof), which yields the forces acting on each particle and the energy to be exchanged between particles, allowing integration of the system.
Recent cosmological simulations have begun to include hydrodynamics. The two currently most relevant are EAGLE, which uses SPH (modified P-Gadget-3 code), and Illustris, which uses an entirely different technique called a moving mesh (AREPO code) to solve the hydrodynamics equations.
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How much solar energy hits a rooftop in Israel? I was asked to today about the available plans (in Israel) to install photo-voltaic receptors on the roof and sell the energy. This is a legitimate plan and there are several companies that do this. The claim is that the initial investment is repaid after roughly 5 years which sounds too optimistic to me. I want to carry the calculation myself and check the results. Where can I find some data about the light flux that hit the earth in a typical year?
note
I'm not sure, since this is not my specific field of expertise, about the commonly used term.
| For the middle east, typically around 2000 KWh/m^2/year
A good place to start is wiki page for insolation (technical term for sunlight arriving)
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What does the quantification of causes and effect look like, for clouds in offshore wind turbine wakes? At Horns Rev windfarm off the coast of Denmark, sometimes in winter, clouds appears in the wake of the turbines. I've only seen photos of the phenomenon when the wind direction is exactly aligned with the grid layout - that is, it's blowing directly from a turbine to its closest neighbour. That may be because it's most picturesque then (and thus most likely to be photographed); or it may be that there's something going in the fluid dynamics that requires that alignment for the phenomenon to occur.
I guess there are several things at work here: that wake losses are highest when wind is exactly aligned with one axis of the turbine grid; that air temperatures vary with height above water; that the temperature is low enough to be close enough to form fog anyway (and in the photo, it looks like they're a layer of mist just above the sea's surface); that the turbine's wake is mixing air from different altitudes
A study started in early 2011 at Lawrence Livermore National Laboratory on turbine wakes, following on from a study by DONG energy on wakes at Horns Rev (787 kB pdf here)
I'm wondering if it's possible to predict when the phenomenon in the photo here might occur. So my question is - what's the specific formulation of what's going on, here: what does the quantification of causes and effect look like?
| Here is a paper published in Energies, freely accessible:
http://www.mdpi.com/1996-1073/6/2/696
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In dimensional analysis, why the dimensionless constant is usually of order 1? Usually in all discussions and arguments of scaling or solving problems using dimensional analysis, the dimensionless constant is indeterminate but it is usually assumed that it is of order 1.
*
*What does "of order 1" mean? 0.1-10?
*Is there any way, qualitative or quantitative, to see why the dimensionless constant is of order 1?
*Are there exceptions to that? I mean cases where the dimensionless constant is very far from 1? Could you give some examples? Can such exceptions be figured out from dimensional analysis alone?
| Notable exception: Reynolds number.
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Why does the weighing balance restore when tilted and released I'm talking about a Weighing Balance shown in the figure:
Press & Hold on onside of the horizontal beam and then release it. It makes some oscillations and comes back to equilibrium like shown in the figure.
Both the pans are of equal equal masses. When the horizontal beam is tilted by an angle using external force, the torque due to these pan weights are equal in magnitude & opposite in direction. Then why does it come back to it position? What's making it to come back?
| It behaves this way because that's how it was built. By adjusting the mass distribution, we could make a scale that flops to one side, is roughly balanced at all angles, etc. However, those scales would not be useful, so the scale isn't built that way.
It might be assumed from the left/right symmetry of the picture that the system cannot decide which way to go, and so is at an equilibrium point. This equilibrium will be stable if a small perturbation (rotating the beam a small angle) raises the center of mass. It will be unstable if a small perturbation lowers the center of mass.
Beyond that, it is difficult to say how the center of mass moves simply by looking at your picture because we do not completely understand the mass distribution and the location of the pivot point.
When finding the center of mass, we can ignore any stationary pieces because we are only interested in the change of the height of the center of mass. Additionally, if the pans hang freely down, it appears as if one will rise by the same amount the other falls, and thus they will not change the height of their center of mass when considered jointly. They can also be ignored.
Let's assume the rest of the scale rotates rigidly. In that case, the center of mass of the rigid portion we're considering will be constrained to a circle with its center at the pivot point. If the center of mass is exactly at the bottom of the circle, we have a stable equilibrium. Otherwise, it is unstable.
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Must the action be a Lorentz scalar? Page 580, Chapter 12 in Jackson's 3rd edition text carries the statement:
From the first postulate of special relativity the action integral must be a Lorentz scalar because the equations of motion are determined by the extemum condition, $\delta A = 0$
Certainly the extremeum condition must be an invariant for the equation of motion between $t_1$ and $t_2$, whereas I don't see how the action integral must be a Lorentz scalar. Using basic classical mechanics as a guide, the action for a free particle isn't a Galilean scalar but still gives the correct equations of motion.
| First, observe that although the non-relativistic Lagrangian is not invariant. It changes by a total derivative, thus the equations of motions remain invariant.
The reason of the difference between the Lorentzian and the Galilean cases is that the group action of the Lorentz group on the classical variables (positions and momenta) is a by means of a true representation, while in the case of the Galilean group the representation is projective.
In the Language of geometric quantization, $exp(i \frac{S}{\hbar})$, where $S$ is the action is a section in $L \otimes \bar{L}$, where $L$ is the prequantization line bundle and $\bar{L}$ its dual. In other words, the action needs not be a scalar, only an exprssion of the form:
$\bar{\psi}(t_2)exp(i \frac{S(t_1, t_2)}{\hbar})\psi(t_1)$, where $\psi(t)$ is the wavefunction at time $t$ and $S(t_1, t_2)$ is the classical action between $t_1$ and $t_2$.
The reason that the representation in the Galilean case is projective is related to the nontriviality of the cohomology group $H^2(G, U(1))$ in the Galilean case in contrast to the Lorentz case.
I have given a more detailed answer on a very similar subject in my answer to Anirbit: Poincare group vs Galilean group and in the comments therein.
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Does the wavelength always decrease in a medium? I was studying a GRE Physics Test problem where optical light with a wavelength of 500 nm travels through a gas with refractive index $n$.
If we look at the equations for wave motion and index of refraction
$$c=\lambda_0\nu\quad\text{(in vacuum)}$$
$$v = \lambda\nu\quad\text{(in medium)}$$
$$n = c/v$$
we see that, if the frequency is constant, the wavelength decreases in the medium compared to vacuum. Is this a consistent property at all frequencies and for all mediums with refractive index real and greater than 1?
Are there dielectrics which change the frequency (still for n > 1), and is there an example of that?
| The answer to your first question
Is this a consistent property at all frequencies and for all mediums with refractive index real and greater than 1?
is Yes.
Not only in optics but in other wave mechanics too.
Refraction of light is the most commonly observed phenomenon, but any type of wave can refract when it interacts with a medium, for example when sound waves pass from one medium into another or when water waves move into water of a different depth
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Why did my liquid soda freeze once I pulled it out of the fridge and opened it? This isn't a duplicate to "Why did my liquid soda freeze once I pulled it out of the fridge?". My question is why soda froze after it was opened. Opening a can or bottle seems to have a larger effect than just jostling it.
Is it because of the disturbance noted in the previous question? Is it related to the pressure decrease? Is it because of the release of some CO2 when it was opened?
| Farenheit calibrated his thermometer to have 0 degrees be the freezing point of salt water. Fresh water freezes at 32F. Water with stuff dissolved in it freezes at a lower temperature than water with less stuff dissolved in it.
Another hypothesis for why soda freezes after opening is that when you open it some CO2 outgasses and less CO2 is dissolved in the water. This will raise the temperature at which the soda freezes.
It is possible that the soda was cooled to 27F, and that the freezing point of soda with lots of CO2 is 25F so it stays a liquid. When you open it, enough CO2 escapes to change the freezing point to 30F and some of the soda freezes.
A test would be to see if the soda gets colder or warmer when it is opened. In the example above, the soda would warm up from 27F to 30F when opened. My guess is that that a soda that freezes when opened will get slightly warmer, but that a soda that does not freeze when opened will get slightly cooler - so slightly that I doubt I could measure it.
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With potential $V(x)= ax^6$ the quantized energy level $E$ depends on which power of $n$? A particle in one dimension moves under the influence of a potential $V(x)= ax^6$, where $a$ is a real constant. For large $n$, what is the form of the dependence of the energy $E$ on $n$?
| You might not be aware, but the quantum number "n" has a classical interpretation as the action variable "J". The action variable measures the area in phase space of the classical orbit,
$$J = \oint p {dx\over dt} dt$$
And the correspondence between J and n was known before quantum theory was developed. It is easy to work out the orbit shape in phase space for any value of J, and figure out what the dependence between E and J is, classically. This would solve your problem in the correspondence limit, in the limit of large n.
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Physical interpretation of describing mass in units of length I'm working in Taylor and Wheeler's "Exploring Black Holes" and on p.2-14 they use two honorary constants: Newton's constant divided by the speed of light squared e.g. $G/c^2$ as a term to convert mass measured in $kg$ to distance.
Without doing the arithmetic here, the "length" of the Earth is 0.444 cm; and of the sun is 1.477 km. To what do these distances correspond? What is their physical significance, generally?
| They represent the scale on which general relativisic effects dominate physics related to bodies of that mass.
For instance if you were to create a (un-rotating, uncharged) black hole of 1 Earth mass it's event horizon would have a radius of about $9\text{ mm} = 2 * M_\text{Earth}$ in those units.
For scales much, much larger than the "length" of the mass, general relativity may be neglected. For intermediate scale in comes in as corrections on order of $\frac{l}{L}$ where $l$ is the mass in the scaled units and $L$ is the length scale of the problem.
This is similar to what particle physicists do by setting $c = \hbar = 1\text{ (dimensionless)}$ energy scales and length scales become inter-changeable.
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How can it be that the beginning universe had a high temperature and a low entropy at the same time? The Big Bang theory assumes that our universe started from a very/infinitely dense and extremely/infinitely hot state. But on the other side, it is often claimed that our universe must have been started in a state with very low or even zero entropy.
Now the third law of thermodynamic states that if the entropy of a system approaches a minimum, it's temperature approaches absolut zero.
So how can it be that the beginning universe had a high temperature and a low entropy at the same time? Wouldn't such a state be in contradiction to the third law of thermodynamics?
| Let me show you that there is no contradiction by pointing out e.g. that for ordinary expansion periods (that is away from first order phase transitions, decouplings...) the total entropy is actually constant in time while the universe is getting bigger and cooler. Or, going back in time, the universe is getting hotter while S is kept constant. How is this adiabatic expansion possible? Well, the space is expanding but the space of particles' momenta is redshifting too, and the net result is a constant phase space volume. Since S measure this volume, the resulting entropy remains constant.
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Collision of Phobos Mars has two moons: Phobos and Deimos. Both are irregular and are believed to have been captured from the nearby asteroid belt.
Phobos always shows the same face to Mars because of tidal forces exerted by the planet on its satellite. These same forces causes Phobos to drift increasingly closer to Mars, a situation that will cause their collision in about 50 to 100 million years.
How I can calculate, given appropriate data, the estimated time at which Phobos will collide with Mars?
| Dynamical models over a likely timescale (say $10^6$ to $10^9$ years) would have significant error bars as mentioned above, and therefore one off predictions about individual moons have little validity.
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Experimental evidence for parallel universes/multiverses My idea of physics is that it is a collection of mathematical laws relating observables. And that one can perform alot of mathematical derivations on these laws to produce new laws between observables. My question is how does one translate a mathematical equation into 'there exist other universes like ours'?
How does one derive that there exist other universes, what phenomena do they explain?
Which observables suggest other universes?
| There is no evidence that another universe exists. There are various physical theories that allow for the existence of parallel universes, and as far as I know there are no widely accepted theories that prohibit their existence outright, so strictly speaking, I suppose it's possible. But it's unlikely that we would ever be able to detect them if they do exist.
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Why Do Hurricane Balls Spin So Fast? I was wondering if anyone could offer an explanation as to why the balls described in this video spin so fast.
Here's the setup: Two metal balls are wielded together. When spun with air, they acquire a massive amount of rpm.
| Look at around 0:34 in the video. I want to point out something relevant to the question here. The end of the tube is narrowed. That is, in technical terms, a nozzle. Nozzles are extremely common in engineering and they work as a form of mechanical leverage just like a lever. I should also note that the straw itself is already a form of nozzle and allows (I think) a more potent blow than would otherwise be possible with the lips.
A human mouth has limitations. The most accurate way to frame this would be to say that one's mouth can only produce a given flow rate, $\dot{m}$, at a certain pressure above atmosphere, $\Delta P$. Combined, these give an energy rate, or power, that can be produced by the mouth.
The comment by Steve Melvin does apply - that the balls can not move faster than the fluid that is passing by it. However, the small outlet of the straw he uses is a way to make a tradeoff, getting high fluid velocity by sacrificing volume of flow. This would, in fact, be rather more difficult to do as accurately and gracefully with mechanical forces. This type of easy conversion ability of forms of fluid mechanical work is a major reason that hydraulics is such a useful science.
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Could the Schrödinger equation be nonlinear? Is there any specific reasons why so few consider the possibility that there might be something underlying the Schrödinger equation which is nonlinear? For instance, can't quantum gravity (QG) be nonlinear like general relativity (GR)?
| In addition to the classic Weinberg paper cited above, there's this shorter version, and then follow ups by Peres 1989 on how it violates the 2nd law, by Gisin on how it allows superluminal communications, and by Polchinski on how it would allow for an 'Everett' phone.
More recently, there's this mathematical argument against nonlinear QM by Kapustin.
| {
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Why is the string theory graviton spin-2? In string theory, the first excited level of the bosonic string can be decomposed into irreducible representations of the transverse rotation group, $SO(D-2)$. We then claim that the symmetric traceless part (i.e. the 35 rep) is the spin-2 graviton - but isn't the label "spin-2" intrinsically 3+1 dimensional? I.e. it labels the representation under the little group $SU(2)$?
| I don't know anything about string theory, but the graviton has been described as a spin-2 particle well before string theory. In his "Lectures on Gravitation" Feynman explains why the graviton must be integer spin, then explains why it can't be 0 or 1, then proceeds to attempt to build the quantum field theory of a spin-2 graviton, simply because it's the next simplest thing that could work. (It doesn't, Feynman abandoned this line of research.)
| {
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Why does gravity forbid local observables? I heard in a conference that gravity forbids to construct local gauge invariants like $\mathrm{Tr}\left\{−\frac{1}{4} F_{μν}^{a}F_{a}^{μν}\right\}$ and only allows non-local gauge invariant quantities like Wilson Loops: $\mathrm{Tr}\mathcal{P}\exp\left[\oint_{\gamma} A^{a}dx_{a}\right]$. Could someone explain me where does it come from?
| General coordinate invariance lets you arbitrarily set the values of the metric and it's first derivative at any one point-- Fermi coordinates . Since you can do this, constructions like the maxwell term you describe above will be necessarily coordinate-dependent, and thus, not local observables.
| {
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Meaning of dimension in dimensional analysis I was wondering what dimension can mean in physics?
I know it can mean the dimension of the space and time.
But there is dimensional analysis. How is this dimension related to and different from the previous one? How is it related to and different from units (e.g. kilometer)?
| A dimension (in dimensional analysis) is defined by the transformation law of an object under changes in scale. If I have an object which is twice as big, it has 4 times the surface area and 8 times the volume, so the surface area has dimension of length-squared, and the volume has dimension of length-cubed. Dimensional analysis is applied whenever you have a quantity where there is a scale that you can change.
There are two different notions of dimension of space which took a while to be disentangled. The topological dimension is defined inductively by the cutting properties of the space. If a space can be cut in two by a point, it's 1 dimensional. If it can be cut in two by a 1 dimensional shape, it's two dimensional. This type of definition requires care for wild shapes, but it produces an integer dimension of the space.
The scaling dimension, or fractal dimension, is defined differently, in terms of distances on the space. The scaling dimension counts the number of boxes of size A required to cover the space, and sees how this goes up as A gets small. The exponent is the scaling dimension.
| {
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How and why will the Milky Way collide with the Andromeda galaxy? Hubble's law says that the universe is expanding. How come the Milky Way and the Andromeda galaxies are on a collision course? How will they end up colliding with each other?
| The large-scale structure of the Universe is expanding. However, gravity still works, and it's especially powerful if the distance is small. E.g., the Earth is still pulling your body closer to it, even though the Universe is expanding. The Earth and the Moon still attract each other, even though the Universe is expanding. Our whole galaxy is still held together by gravity, even though... You get it.
M31 (a.k.a. the Andromeda Galaxy) is "only" 2.5 million light years away, which is pretty close, by cosmic standards. At that scale, for large objects such as galaxies, gravity overrides cosmic expansion. There is a net positive attraction between us and M31. Our galaxies are being pulled together by gravity.
As to what exactly will happen when they collide, take a look at this simulation:
https://www.youtube.com/watch?v=4disyKG7XtU
Or just download the software yourself, it's free:
http://universesandbox.com/
Only galaxies which are very far away from each other are being pulled apart by the cosmic expansion faster than gravity could pull them together.
| {
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Paramagnet: Negative specific heat? for a simple paramagnet ($N$ magnetic moments with values
$-\mu m_i$ and $m_i = -s, ..., s$) in an external magnetic field $B$, I have computed the Gibbs partition function and thus the Gibbs free energy $G(B,T)$ and obtained
a simple approximation for small $B$ fields, which has the basic form
$$G(B,T) = N k T a - N b B^2/T$$
for suitable constants $a$ and $b$. From this, I would like to compute the heat capacities for both constant $B$ and constant magnetization, $M$. I can obtain the magnetization from
$$M = -\frac{\partial G}{\partial B} = 2 N b B/T$$
and from this the total energy via
$$E = -MB$$
For the heat capacity at constant field, I would then write this in terms of $B$ and $T$ alone, i.e.
$$E = - 2 N b B^2 / T$$
so
$$C_B = 2 N b B^2 / T^2$$
this makes sense for small $B$ and large $T$ as we have a finite-state system that achieves saturation: Eventually, there simply are no more available states to add more energy to the system.
Now for the heat capacity at constant magnetization. My idea was to just rewrite $E$ again, this time as a function of $M$ and $T$ alone, which gives
$$E = -MB = -\frac{M^2 T}{2Nb}.$$
This time, obviously
$$\frac{\partial E}{\partial T} = -\frac{M^2}{2Nb}$$
so I have a negative heat capacity.
Now, I have a hard time visualizing this. I increase the temperature and thereby
lower the energy? Is this because increasing the temperature increases
spin fluctuations so they are not as aligned anymore and, in absence of a kinetic energy term, these fluctuations don't carry energy themselves?
Or have I gone wrong at some part in my derivation?
| It is the constant magnetization assumption which causes this. If you increase the temperature at constant $B$, then the magnetization would reduce, so to keep magnetization constant, you have to increase the magnetic field. The energy, $E=-MB$, so as you increase $B$ at constant $M$ you are making energy more negative.
Does that make sense?
| {
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Why are color values stored as Red, Green, Blue? I learned in elementary school that you could get green by mixing blue with yellow.
However with LEDs, TFTs, etc. you always have RGB (red, green, blue) values?
Why is that? From what you learned in elementary yellow would be the 'natural' choice instead of green.
| When you mix colors using Watercolors, then they mix as "Subtractive Colors". However, Light itself mixes as "Additive Colors".
Even though it might seem strange why the inherently same thing works so differently, it makes sense if you think about Watercolors, etc. as absorbing everything but that specific color.
| {
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Naïve relativistic schrodinger equation
Possible Duplicate:
Why are higher order Lagrangians called 'non-local'?
Bjorken and Drell presents the equation:
$$i\hbar\frac{d\psi}{dt}=H\psi=\sqrt{p^2 c^2+m^2 c^4}\psi=\sqrt{-\hbar^2 c^2 \nabla^2+m^2 c^4}\psi$$
The squareroot can be expanded to obtain an equation with all powers of the derivative operator. What do they mean when they say this leads to a non-local theory?
And is this equation incorrect or just impractical?
| To see why the theory is nonlocal, consider the effect of the derivative operator... I like to put things on a lattice, so I will: $\psi_i=\psi(x_i)$, then the derivatives (in 1D, for simplicity) become
$$\nabla^2 \psi_i \propto (\psi_{i-1}-2\psi_i + \psi_{i+1})$$
Now, you can see what happens as you continue to apply derivatives (as you must, in the expansion of the square root) -- For high order derivatives, the time-derivative of $\psi$ at a lattice site will depend on the instantaneous spatial values of the whole field!
| {
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How would you store heat? Um .. naive question perhaps but if somebody wanted to store heat, how would they go about it? Can heat be stored?
I'm told that decomposing kitchen waste in a closed vessel results in a rise in temperature on the body of the vessel. I'm just wondering whether it could be stored for later use.
| If you want to store heat in a battery-like device, you could use the heat to power a turbine, generate electrical energy, and store it as chemical energy in a battery. This is extremely inefficient, but I think this is most analogous to what you are asking.
You could also find a high-energy chemical reaction in equilibrium. This would store some of the heat as chemical energy, but would have to be kept at the same temperature or the chemical mixture would start producing heat.
Really, though, the best way is probably dmckee's "warm stuff + insulated container".
| {
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Graphene space elevator possible? I just read this story on MIT working on industrial scale, km^2 sheet production of graphene.
A quick check of Wikipedia on graphene and Wikipedia on space elevator tells me
Measurements have shown that graphene has a breaking strength 200 times greater than steel, with a tensile strength of 130 GPa (19,000,000 psi)
and
The largest holdup to Edwards' proposed design is the technological limit of the tether material. His calculations call for a fiber composed of epoxy-bonded carbon nanotubes with a minimal tensile strength of 130 GPa (19 million psi) (including a safety factor of 2)
Does this mean we may soon actually have the material for a space elevator?
| @lurscher of course I understand it's from GEO, the fact that GEO is the net zero apparent acceleration point is the reason it would be "unfurled" from GEO. If your point behind the stages is that it could be carried up in segments, then yes, no one ever argued otherwise. The only thing your $k^N$ mathematics shows is that it could theoretically be made with any material, regardless of its specific strength. This is true for any compression structure as well. There is still a practical problem if the approach results in needing trillions of tons of material.
| {
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Why is it important that Hamilton's equations have the four symplectic properties and what do they mean? The symplectic properties are:
*
*time invariance
*conservation of energy
*the element of phase space volume is invariant to coordinate transformations
*the volume the phase space element is invariant with respect to time
I'm most inerested in what 3 and 4 mean and why they are important.
| I think the properties 3 and 4 are important because in these way the probability
distribution in phase space is conserved and the information is also conserved.
Some systems dont have this property, these are chaotic. In these systems the volume
in phase space could increase until filling the complete phase space in a fractal
way.
| {
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What does it take to understand Maxwell's equations? Assume I want to learn math and physics enough to reach a level where I understand Maxwell's equations (The terms and reasoning in the equations I.e. why they "work"). What would I have to learn in order to have the tools I need to make sense out of it?
I'm kindof looking for a road map which I can use to get started and know what to focus on. The fields are pretty big so some pruning would be very helpful (if possible)
| I learned undergraduate electrostatics from Wangsness' Electromagnetic Fields. The text is one of the clearest and most comprehensible I've seen at this level and I highly recommend it. The first chapter covers only vector calculus and provides an excellent basis for the necessary math. We got to Maxwell's equations at the end of the first semester so you should expect to spend at least half a year to get to that level.
| {
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Is an electron/proton gun possible? In the 1944 SF story “Off the Beam” by George O. Smith, an electron gun is constructed along the length of a spaceship. In order to avoid being constrained by a net charge imbalance, it is built to also fire the same number of protons in the other direction, dissipating the mass of the “cathode”.
With current knowledge, is this plausible? That is,
*
*Can a practical (i.e., not built with unobtanium insulators) electrostatic device like an electron gun separate and accelerate electrons and protons in this manner?
*Can it actually disassemble solid matter? If so, how does the composition of the cathode affect the difficulty?
| An example of a real-life electron/proton beam is the Neutral Beam Injector used in magnetically confined fusion devices. They're used to inject faster particles into the fusion plasma, raising the overall temperature. However an ordinary (charged) proton/deuteron beam will be bent away by the strong magnetic fields surrounding the plasma, so the beam has to be made neutral first. This is done by sending the ions through a gas where they can pick up electrons, then selecting only the neutral ones with another magnetic field.
| {
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Open quantum systems and measuring devices The Copenhagen interpretation by Niels Bohr insists that quantum systems do not exist independently of the measuring apparatus but only comes into being by the process of measurement itself. It is only through the apparatus that anything can be said about the system. By necessity, the apparatus has to be outside the system. An open quantum system. Can quantum mechanics be applied to closed systems where the measuring apparatus is itself part of the system? Can a measuring apparatus measure itself and bring itself into existence?
| Q: Can quantum mechanics be applied to closed systems where the measuring apparatus is itself part of the system? | {
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How does one calculate the force applied on an object by a magnetic field? I've tried very hard to find an answer to this question, and every path leads me to an abstract discussion of fundamental forces. Therefore, I will propose two very specific scenarios and see if they yield the result that I am looking for.
Scenario One
Let's say that I have a vertical tube exactly 1 inch in diameter that is completely incapable of holding an electromagnetic charge and has a frictionless surface. Resting inside this tube is a steel ball also exactly 1 inch in diameter. If a cylindrical magnet, also exactly 1 inch in diameter is slowly lowered into the tube, how does one determine the exact point at which the force being applied to the steel ball by the magnet will cause the ball to overcome gravity and rise toward the magnet? Is there even any way to determine this? What further information would I need?
Scenario Two
I have the same tube from above with the cylindrical magnet resting on the bottom of the tube, north pole facing upward. Suspended by a weightless string in the tube is an identical magnet, north pole facing downward. If the bottom magnet is slowly raised, how does one calculate the exact point at which the suspended magnet will begin to move upward? Is this calculation possible? What further information would I need for this calculation?
Extra Question
How are weight capacities on magnets calculated? I.e. if a whitepaper says that a magnet is capable of lifting 25 pounds, how is the correct size magnet calculated?
| I am consider Scenario One:
For given ball and magnet the magnitude of the interaction force between them, depending on the distance $x$, follows the formula:
$$F=\frac{const}{x^7}$$ where $const$ depends on the ball's radius, on the magnetic dipole moment of the magnet and on the magnetic permeability(expected to be constant) of the material of the ball.
You can determine $const$ by measuring the distance $x$ under gravity(with known weight of the magnet).
A derivation of the formula for $F$ requires the vector calculus. I guess it is not particularly familiar with you.
| {
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Why do ships lean to the outside, but boats lean to the inside of a turn? Small vessels generally lean into a turn, whereas big vessels lean out.
Why do ships lean to the outside, but boats lean to the inside of a turn?
For example, a boat leaning into a turn:
Image Source.
And a ship leaning out:
Image source
| The answer to the question is; leaning out or leaning in is a result of the vessel's rudder's position relative to the keel. If the rudder is ABOVE the keel, the vessel will lean OUT; if the rudder is below the keel, the vessel will lean IN. ("Rudder"- the method or mechanism by which the flow of water under the vessel is diverted to cause a change direction of the vessel. This method or mechanism might be a movable plate, or a movable jet of water generated by a propellor.)
| {
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Mathematical Universe Hypothesis What is the current "consensus" on Max Tegmark's Mathematical Universe Hypothesis (MUH) which claims every concievable mathematical structure exists, including infinite different Universes etc.
I realize it's more metaphysics than physics and that it is not falsifiable, yet a lot of people seem to be taking a liking to it, so is it something that is plausible ?
I've yet to hear any very good objections to it other than "it's crazy", but are there any real technical problems with it?
| According to the metaphysical Church-Turing thesis, all that exists has to be Turing computable. There should also be a unity oneness to all that exists.
So, dare I say, maybe all that exists is ONE universal computer dovetailing over all possible programs a la Schmidhuber, or ONE quantum computer running a superposition of all possible quantum programs.
| {
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can radiocarbon dating be used on living things? I have been going through a wiki article about worlds oldest living creature. As a matter of fact its a plant, a shrub to be precise.
Wiki says that the plant age was determined by carbon dating. But what i know is that carbon dating can only be done for dead tissues i.e which no longer assimilates atmospheric carbon onto itself.
Can someone clarify this??
regards,
| When trees grow, they add successive layers to a central core. After only a year or two, this middle core becomes established and stops growing. The living part of a tree is mostly in the bark and layer immediately below that (as well as leaves and roots). This is why woodpeckers don't kill trees but girdling one will. As a consequence, it is possible to take a core from a tree (without killing it) and perform carbon dating on the inner portions.
| {
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How can a conductor be grounded yet there are induced charges on it? A classic example for the method of images is the following, quoted from Griffiths's Introduction to Electrodynamics, page 121:
Suppose a point charge $q$ is held a distance $d$ above an infinite grounded conducting plane. Question: What is the potential in the region above the plane?
Griffiths continued on solving the example using the method of images setting V=0 on the plane as one of the boundary conditions saying "since the conducting plane is grounded".
Now, of course there will be an induced surface charge density. My question is, how can this be since the plane is grounded?
Does the word grounded have different meanings? sometimes it means not charged and the others it means the potential there is 0?
| Think of the ground plane as being an infinitely big electrical conductor, initially uncharged.
We bring in a (say positive) point charge close to the surface of the ground plane. An negative image charge is induced on the surface near the point charge, and since the net charge on the ground plane is zero, an opposite positive charge is pushed off "to infinity".
Now suppose we bring a positively charged conductor up to the ground plane, and then connect it electrically. The positive charge in our conductor gets discharged and cancels the negative induced surface charge, but the positive charge that had been pushed off to infinity is still out there and gets redistributed. Since it is a finite charge distributed over an infinite body, there is effectively zero charge density.
| {
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Realistic projectile motion I am working on a project involving a simulation of the motion of a projectile (in 3D) aimed at a moving target. The way projectile motion is analyzed in most introductory physics books is not accurate enough for this project. I would like to know what other influences on the motion of a projectile, including air resistance and spin, I need to take into account. What is a good book on this subject?
| If you have time to read a book, I strongly recommend Richard Feynman Lecture on Physics:
http://en.wikipedia.org/wiki/The_Feynman_Lectures_on_Physics
It is everything you want from a book:
*
*fun to read
*never boring
*it will change the way you see the world
*everyone can read it
In your case, I would focus on the first volume, Mechanics I, particularly from chapter 8.
| {
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Could a ship equipped with Alcubierre drive theoretically escape from a black hole? Could a ship equipped with Alcubierre drive theoretically escape from a black hole?
Also, could it reach parts of the universe that are receding faster than the speed of light from us?
| In my understanding, which is based on simple facts and admittedly not a big understanding of how black holes would work with regard to this...
A warp bubble would compress space in front of the ship and expand space behind the ship. This should effectively move the black hole away from the ship, descreasing the effort needed to escape. It also means that if you had fallen past the event horizon, you might find yourself outside it again as the warping of space moves the black hole, and thus the event horizon away. At this point, I suppose it would come to this: is the speed of the warp bubble a factor in your escape velocity, or does the bubble only serve to buy you more time, and you need to have a velocity within the bubble to actually escape? I'm not the one to answer that lol
| {
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Problem with an electricity / thermodynamics assignment I've been trying to figure this one out for a while on my own, so I'd like to ask for your help if you could offer some.
The task states:
A heater made out of a wire with a diameter $R = 0.2\text{ mm}$, length $4\pi\text{ m}$ and electrical resistivity of $0.5\times 10^{-6}\ \Omega\;\mathrm{m}$ is connected to a voltage source of $220\text{ V}$, sinked in the water.
Which mass of water will it heat up from $20^{\circ}\mathrm{C}$ to $50^{\circ}\mathrm{C}$ in the time of 10 minutes? (C of water = $4200\ \mathrm{J\;kg}/\mathrm{K}$)
I know I have the electrical properties of the wire and the thermodynamic properties of the water, but I don't know how to proceed from there. We've been studying electricity and I am not really aware how I can connect it with thermodynamics?
| find m from this equation,
$$
mC_p\Delta T = \frac{V^2}{R}t
$$
where,$$R=\frac{\rho l}{\pi r^2}$$
| {
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The visibility of air For pilots of gliders or sailplanes, the 'thermal' is the most important phenomena of the air. A thermal is classically described as an upward flow of air caused by ground level heating of air that rises in bubbles or a connected stream of warmed air. Given sufficient velocity of the rising air, a gliding craft, bird or even trash and debris can be lifted thousands of feet. It can be also noted that 'dust devils' can result from especially vigorous flows and that other even stronger phenomena like tornadoes and cyclonic storms are related.
But in the absence of markers like dust, the air flow is generally invisible until possibly the flow reaches an altitude where the water vapor contained, condenses to form a cloud. And again if the flow is strong enough and contains enough water, a thunder storm is possible.
So, the question. Given all of the above is it possible to see the mass of air that makes up the thermal? Is there anything in the difference between the thermal and the surrounding air that could be detected and presented graphically?
| Air of different temperature and pressure has different refractive index - but the difference is very small.
If you think of putting a piece of glass into water and trying to see it. The difference between glass (1.5) and water (1.33) is pretty large, air at different temperatures has refractive index differences of parts-per-million
Astronomers at least measure the refractive index profile to design adaptive optics systems, but it takes a lot of equipment (lasers and large telescopes) - I don't know if meteorologists do it much.
| {
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Is the cooling rate of a (very) cold object, sitting next to an AC higher or lower? In more detail:
If i have two soda cans, both are cooled to exactly 4 degrees celsius,
And i put one in a 25 degrees room, and the other next to an AC vent set to 16 degrees.
After three minutes, which one should be colder than the other and why?
Edit: To clarify - if I have a cold soda can, should I place it near the AC vent or not (if I like my drink cold)? Which location will cause faster heating?
| The air around the soda can effects how fast it transfers heat. If the temperature difference is low it will change gradually because the actual energy it takes to transfer energy is constant. waters specific heat is 4181.3 J/(kg·K), water has the second highest specific heat capacity. This means it takes a certain amount of energy to transfer. The lower the temperature the slower the energy transfered is.
The can in the cold air should remain colder from the exprience from labs i've done with it in the past.
| {
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Transmission and reflection What is the transmission amplitude of a wavefunction $\phi(x)=e^{ikx}(\tanh x -ik)$? I would have thought that it is $\tanh x -ik$ since this is the factor associated with the forward travelling $e^{ikx}$ but then since the reflection coefficient is $0$, we have that the reflection probability is $0$, but $|\tanh x-ik|^2$ is dependent on $x$ so not identically $=1$? Where have I gone wrong?
| The clue here is that the coefficients of $\mathrm e^{\mathrm ikx}$ and $\mathrm e^{-\mathrm ikx}$ are to be evaluated at infinity. The terms and concepts you're using apply to a situation where we have an incoming wave proportional to $\mathrm e^{\mathrm ikx}$ for $x\to-\infty$. It interacts with a system around the origin and is partially reflected, transmitted and/or absorbed. The reflected component is an outgoing wave proportional to $\mathrm e^{-\mathrm ikx}$ for $x\to-\infty$, and the transmitted component is an outgoing wave proportional to $\mathrm e^{\mathrm ikx}$ for $x\to\infty$.
In your case, $\tanh x\to\pm1$ for $x\to\pm\infty$. Thus the incoming wave has amplitude $-1-\mathrm ik$, and the transmitted wave has amplitude $+1-\mathrm ik$, and the transmission coefficient is
$$\frac{+1-\mathrm ik}{-1-\mathrm ik}=\frac{\left(1-\mathrm ik\right)^2}{-1-k^2}=-\left(\frac{1-\mathrm ik}{|1-\mathrm ik|}\right)^2=\exp\left(\mathrm i\left(\pi-2\arctan k\right)\right)\;.$$
This has magnitude $1$, so probability is conserved, as there are no reflected or absorbed components. The interaction merely shifts the phase of the wave by a $k$-dependent angle $\pi-2\arctan k$.
| {
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Dependence of Friction on Area Is friction really independent of area? The friction force, $f_s = \mu_s N$. The equation says that friction only depends on the normal force, which is $ N = W = mg$, and nature of sliding surface, due to $\mu_S$.
Now, less inflated tires experiences more friction compared to well inflated tire. Can someone give clear explanation, why friction does not depend on area, as the textbooks says?
| The increased 'resistance' of an underinflated tyre is due to mechanical deformation, friction is independent of area as suggested. The simplest explanation for me is that: as area increases the applied force per unit area decreases, but there is more contact surface to resist motion.
Added as per Zass' suggestion below:
$$\rm{Friction}= \rm{Material\ Coefficient} \times \rm{Pressure} \times \rm{Contact Area}$$
Where the material coefficient is a measure of the 'grippiness' of the material, the pressure applied to the surface and the area of the surfaces in contact. So we can see the area in the pressure term cancels with the third term.
This is not to be confused with traction, where spreading the motive force over a larger area can help.
| {
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Are specific heat and thermal conductivity related? Are there any logical relationship between specific heat capacity and thermal conductivity ?
I was wondering about this when I was reading an article on whether to choose cast iron or aluminium vessels for kitchen.
Aluminium has more thermal conductivity and specific heat than iron ( source ).
This must mean more energy is required to raise an unit of aluminium than iron yet aluminium conducts heat better than cast iron.
Does it mean that aluminium also retains heat better ?
How does mass of the vessel affect the heat retention?
| For metals there is a connection between the thermal conductivity and electric conductivity (Wiedemann–Franz law).
However specific heat is not directly related. This is because electric and thermal conductivity are due to the electrons, however the specific heat is mostly due to the ion vibrations (phonons).
Despite "classical" intuition electrons contribute almost nothing for specific heat in metals. Electrons in a typical metal behave close to an ideal fermion gas, in a very deep quantum range (typical Fermi temperature is about 40K).
| {
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Is decoherence due to coarse graining or coupling with the environment? In the literature, sometimes one reads that decoherence is due to the coupling of the system to the external environment, and sometimes one reads that it is due to coarse graining over the microscopic degrees of freedom. Are these two different cases of decoherence, or is one more fundamental than the other?
| The more conventional way is to describe decoherence as being due to the "coupling to the environmental degrees of freedom" that are traced over. However, the "environmental degrees of freedom" may also include geometrically internal degrees of freedom of a physical system such as a cat – unmeasurably complicated correlations in the properties of the individual atoms.
Because the degrees of freedom you may track – e.g. whether or not a cat is alive – become entangled with many degrees of freedom you can't track – e.g. a jungle of phonons propagating through the cat – you may derive that the density matrix for the latter becomes quickly diagonalized. In this picture, the phonons' degrees of freedom would be considered "environment" by those who say that decoherence is due to the coupling to the environment. The other group wouldn't call it the environment. Instead, it would refer to coarse-graining in which all microstates of a cat which are alive, regardless of the state of the phonons etc., are clumped together. One would derive the appearance of the non-pure density matrix via another approximation but the qualitative outcome would be the same: decoherence.
| {
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How do we perceive hotness or coldness of an object? Some objects, especially metallic ones, feel cold on touching and others like wood, etc. feel warm on touching. Both are exposed to same environment and are in their stable state, so some kind of equilibrium must be being reached. What is this equilibrium?
And how do we perceive hotness or coldness of an object? Does skin have some kind of heat sensors, etc., which transmit signals to brain? Like, how do the eye transmit/convey an image formed on the retina to brain?
| To the first part of your question: What is this equilibrium? At room temperature the metallic or wooden object is in equilibrium with it's environment. When you touch the object this equilibrium is disturbed. As the skin on your fingers will be quite a bit above room temperature when you touch the object heat will flow from your finger to the object.
The amount of heat depends on the contact area, force with which you touch the object etc. Your finger will get colder, as the heat is flowing into the object. An equilibrium is reached again, when this heat flow does not change over time. This can mean that the object has the same temperature as your skin or that the heat is going into the environment at a constant rate through the object.The cold or warm feel is depending on the amount of heat that flows out of your finger into the object and is sensed by the actual temperature change of your finger. The biological mechanism has been described by Vineet Menon nicely.
| {
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why evaluate at lambda = 0 I am trying to understand Herbert Goldstein's introduction to 4-vectors. He describes a 1-D curve in spacetime $ P_(\lambda) $ then he says a 4 vector is defined as the tangent vector to this curve $$ v = \biggr ( \frac {dP} {d\lambda}\biggr)_{\lambda =0} $$
why is $ \lambda $=0? what does that have to do with anything? I have been staring at this for like 20 minutes I still don't understand what he is talking about... it's giving me problems because i need to understand this part later because it is relevant to how tensors transform
also he says $ \lambda $ is a measure of a length along the curve... i don't really follow that point either... i though $ \lambda $ could be any parameter like proper time etc.
any help on this??
|
he says a 4 vector is defined as the tangent vector to this curve
That is not true in general. A four-vector is not always defined as the tangent vector to a curve. In the book they are computing a tangent vector to a curve in 3+1D spacetime; the tangent vector is just one example of a four-vector.
In particular, the formula given tells you how to compute the tangent vector at a specific point $\mathcal{A}$. Since the curve runs from $\mathcal{A}$ to $\mathcal{B}$ and is parametrized by $\lambda \in [0,1]$, $\lambda = 0$ is the value which corresponds to the point $\mathcal{A}$. So if you're going to define the tangent vector at $\mathcal{A}$, you need to set $\lambda = 0$.
| {
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Is there any anti-gravity material? I want to know if there is any anti-gravity material. I am thinking of making flying vehicles which are made up of anti-gravity material so that they will not experience any gravity on them and can easily take off and be more fuel efficient. Is there any such thing? Or any workaround?
| Anti-gravity is impossible, as it would let you build a perpetual motion system, as follows.
Assume we have a system in which we can capture the kinetic energy of a falling mass. For example, a ball that falls onto a scooped wheel to drive it. Take the ball and move a sheet of anti-grav material under it. As the ball now no longer feels the earth's gravity, we can push it up above the wheel without using any energy. Now remove the anti-grav sheet. The ball will fall onto the wheel and thus generate energy.
This violates the law of conservation of energy, i.e. the first law of thermodynamics. Hence it is impossible, and hence anti-gavity cannot exist.
| {
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Should annealed disorder be characterized by the average of the partition function? Most of the literature says that for a quenched average over disorder, an average over the log of the partition function must be taken:
\begin{equation}
\langle \log Z \rangle,
\end{equation}
while for the annealed average, it's
\begin{equation}
\langle Z \rangle.
\end{equation}
But a while ago, I came across a book that said that the annealed average is not $\langle Z \rangle$, though I don't remember what it said should be calculated instead.
Does anyone know which book this is, or what they might want to calculate instead of $\langle Z \rangle$ for the annealed average?
| I've found the book I was looking for: Ma - Modern Theory of Critical Phenomena (~ p366).
It says that rather the difference between a quenched and annealed average is the probability distribution it should be averaged over. Both types of average (if I understand it correctly) should be taken over the free energy:
\begin{equation}
-T \langle \ln Z \rangle
\end{equation}
but with different meanings of $\langle \cdots \rangle$ for the quenched and annealed averages.
It says that $-T \ln \langle Z \rangle$ should not be used.
| {
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"timestamp": "2023-03-29T00:00:00",
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About the Ether Theory acceptance Why was the Ether Theory refused by Modern Physics? If you please explain me, I just wanted to understand it more.
| It's not entirely true that ether theory was refused by modern physics, but seen as superfluous and over-complicated compared to special relativity's notion of time and space. There's a version called Lorentz ether theory which postulates that it's not possible to detect the absolute ether, and gives the same results as predicted by special relativity. Hence it's not possible to experimentally determine the difference between the two theories.
But whereas LET is constructed in an hoc way by adding time dilation and Lorenz contraction into the theory to make the ether undetectable, these follow from the two postulates of SR. So the sheer elegance and simplicity in favour of SR is why it's taken more seriously than LET.
| {
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Why is beta negative decay more common than beta positive? In simple terms, why is beta negative decay more common than beta positive?
I know it's something to do with occuring inside/outside the nucleus - but I can't find a simple, easy to understand explanation!
| Beta-minus decay occurs in nuclei with an excess of neutrons, while beta-plus decay takes place in neutron-deficit nuclei. A lot of natural background radiation on Earth is due to fission or alpha-decay of heavy radioactive elements. The remains of fission or alpha-decay are neutron-rich nuclei, so beta-minus decay is more common on Earth.
Whereas on stars beta-plus decay is typical, because neutron-deficit nuclei are produced in nuclear fusion.
| {
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Can there be black light? I mean is it possible to devise a machine that outputs darkness? I understand there are various colours that light can have. But i was wondering why there is no 'black' light. What is the logical explanation for this? I mean I am expecting an answer that goes beyond mentioning the spectrum details. All I could think of was a machine as powerful as a blackhole; it could bend the light so hard that all we would see is darkness. But is there any other way?
P.S. I am a programmer and didn't study much Physics beyond high school. This question is not a goof. I am not asking this question for fun. I seriously have this curiosity.
| In the final boss room of Peper Mario: The Thousand Year Door, the boss turns regular candles into dark candles. I don't think it's possible to have a light sources that's like those candles except that it actually makes the room darker. One way you could have that type of light source is to have a source of particles that annihilate with photons travelling in the same direction without producing anything in the process. Probably no such particles exist. A cold black object can however appear to be emitting coldness and you can actually feel a warm shadow from that source when you hide behind something. That's because the total amount of radiation it's reflecting or emitting is less than the rest of the surfaces in the room. Hiding behind something from the point of view of the cold black object warms you up only because it makes you receive radiation from the thing you're hiding behind.
| {
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Quality factor for a quantum oscillator? I've been reading papers about nanomechanical oscillators, and the concept of quality factor often pops up. I understand to some extent about Q factor in classical sense, but since nanomechanic oscillators are often treated quantumly, what does Q factor mean then?
| There is no difference in interpretation. In both classical and quantum oscillators, if they have any dampening, the Q factor is higher the lower the dampening is. In quantum mechanics, it is common to relate the dampening to the half-life, but as far as I can tell, there is no further difference.
EDIT: the definition of Q is still the same, $\Delta f / f$ (though now that I re-read the wiki link you gave, I see that this is true only for high Q). If your system is treated quantumly, you just need to calculate the decay of the state you are interested in, or the linewidth of the energy level.
| {
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How long does it take for expanding space to double in size? I have been reading about Hubble's constant and trying to make 'sense' of the theory of the expanding Universe. Is is possible that space in the universe expands uniformly? If so, absent of other forces (ie gravity), how long does it take for the distance between any to dimensionless points in the universe to double in length?
I've tried to work the math as follows:
$\frac{74.2 \text{ km}}{\text {s Mpc}}\times\frac{\text{Mpc}}{3,261,564\text{ ly}}\times\frac{\text{ly}}{9,460,730,472,581 \text{ km}}\times\frac{31,557,600\text{ s}}{\text{yr}} = \frac{1}{13,177,793,645 \text{ yr}}$
Using the continuously compounding interest formula
$FV= Pe^{rt}$
$2 = 1\times e^{(1/13,177,793,646)t}$
$\ln(2) = \frac{1}{13,177,793,646}\times t$
$t = 9,134,150,511 \text{ yr}$
So it would take 9 billion years for the distance between any two points in space to double in length?
If this is so, when two points in 3D space double in distance apart, the space itself increased by $2^3 = 8$ so the time it would take for space itself to double in size would be
$t = 1,141,768,813 \text{ yr}$
Since the Universe is only about 15 billion years old and started from a singularity of volume $0$,
I would have to assume that the rate of expansion of space isn't constant over time?
Does the time for the distance between two points to double in length vary based on the original distance between those to points?
| The way to answer your question is to take the Friedman equation and put in the components you want to. In the Standard Model of Cosmology you'd put in radiation, matter and lamdba. You then solve the equation for the scale factor $a(t)$. (This can be automated with a program like Mathematica.) You'll get an explicit $a(t)$ function that you can plot and see how the Universe expands.
A very nice pedagogical introduction is in Barbara Ryden's "Introduction to Cosmology", there's a PDF version online:
http://www.astro.caltech.edu/~george/ay21/readings/Ryden_IntroCosmo.pdf
Pg. 119 Figure 6.5 is just what you're looking for.
| {
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Red shift and time distortion Superman throws a light emitting object away from himself fast enough to notice a red-shift. The object passes through a region in which time runs more slowly. From Superman's perspective, does the red-shift change as the object slows or is the light's appearance unaffected?
| The object doesn't slow in this case, it speeds up. The redshift is increased. The clock-slowing-down factor is more traditionally called the gravitational potential, and where clocks are slow, this is close to a massive object. In the extreme limit that the light-emitting-object is approaching a black hole, time stops relative to Superman, so that Superman sees the object infinitely redshifted.
| {
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"timestamp": "2023-03-29T00:00:00",
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Theoretical basis for black hole evaporation What is the basis for black hole evaporation?
I understand that Hawking-radiation is emitted at the event horizon, a theoretical result originating in General Relativity and Quantum Field Theory, but it seems to me that additionaly one has to assert an integral conservation law for mass/energy, ie. for a sphere surrounding the black hole.
Does such a conservation law hold for the simplest case of a Schwarzschild metric?
I am grateful for any related classic paper references.
EDIT: The usual heuristic for understanding Hawking-radiation is: virtual pair, one falls in, one goes out; the ones going out are called Hawking-radiation. But what about the ones going in? Naively, it seems there should also be Hawking-radiation going inward, which would actually increase the black hole's mass.
| The Hawking radiation is calculated by the methods of "quantum field theory on curved backgrounds". One may show that the final state contains the thermal radiation (corresponding to the right, Hawking temperature) for any field that propagates on the black hole background.
Quantum field theory on a curved background tells us many more microscopic details than just the existence of the radiation, its temperature, and the fact that the black hole is shrinking. But yes, of course, the total mass/energy conservation law holds. By Noether's theorem, it holds for any physical system whose laws are invariant under translations in time. Emmy Noether proved it in general, for any system: but of course, if one works with a specific field theory such as "the Klein-Gordon quantum field theory on a curved background", one may see what the mass/energy conservation means in much more detail.
The key paper for the Hawking radiation is obviously Hawking's 1975 paper, Particle creation by black holes:
http://scholar.google.com/scholar?hl=en&q=hawking+particle+creation&btnG=Search&as_sdt=0%2C5&as_ylo=&as_vis=0
I think it contains everything needed to answer basic questions such as yours.
| {
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Relativistic transformation of the wave packet length Let us suppose we have an excited atom at rest. It has a certain mean lifetime $\tau_0$. If we wait sufficiently long time $t>>\tau_0$, we will find a deactivated atom and a (spherical) electromagnetic wave function of photon with about $\tau_0\cdot c$ long layer with non zero probability to find a photon within. Something like a fast expanding probability "ring" with a $\tau_0\cdot c$ width of the ring.
Now, let us consider this system in a moving reference frame. It seems to me that this width $\tau_0\cdot c$ is relativistic invariant: it is a difference between two "fronts" of electromagnetic wave rather than a length of a material body subjected to the Lorentz contraction. Is it correct? In other words, whether this picture relativistic invariant?
| No, the probabilistic density isn't Lorentz-invariant. First of all, the probabilistic density isn't a Lorentz scalar: it is a time component of a 4-vector whose spatial components encode the probability current.
Second, it is obviously not true that the ring will be equally thick in all directions. While it's true that the photons are moving by the speed of light in any inertial system, they're moving from a different center because the atom itself is moving (from the moving frame's viewpoint).
So the ring from a moving reference frame's viewpoint will look like one of the black or purple squeezed annuli on the picture below:
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