RELATIVITY QUESTIONS! (and other common queries)

[PM 2Ring]

[1] I kid.

* giggles *

http://en.wikipedia.org/wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function#The_case_s.3D1

[Mostlynormal]

So I have a question about Black Holes.

While reading a book on physics (which included a lot of science history) I came across something that basically said that scientists expected black holes to have different shapes depending on what made the black hole, and then were surprised by some result or another that said nonrotating ones had to be spherical and rotating ones bulged perpendicular to the axis of rotation. This confused me for two reasons:

1) Why did scientists expect the black hole to have different shapes? I'm aware of (though I don't fully understand) the problems caused by black holes being indistiguishable except for rotation, which messed with entropy (they may have resolved this later some other way by now, like I said, I don't fully understand it). However, my intuition tells me that gravity concentrated at a single point of infinite density would be completely uniform, and therefore, the event horizon, which was by definition the boundary at which spacetime was so warped that light could not escape, would be completely spherical. So what led scientists to beleive this wouldn't be the case?

2) Also my intutition is apparently wrong anyway for rotating black holes. So is there an explanation, in not too technical terms, as to why a black hole's event horizon would "bulge" due to rotation?

[doogly]

As for 1, I have no idea what that book is going on about. Do they cite any particular people with particular ideas?

As for 2, everything bulges when it spins. Look at the earth, it is a little thicker at the equator. An event horizon is a very different thing than a hunk of rock, but at leas you can get a sense that this is reasonable.

[starslayer]

So I have a question about Black Holes.

While reading a book on physics (which included a lot of science history) I came across something that basically said that scientists expected black holes to have different shapes depending on what made the black hole, and then were surprised by some result or another that said nonrotating ones had to be spherical and rotating ones bulged perpendicular to the axis of rotation. This confused me for two reasons:

1) Why did scientists expect the black hole to have different shapes? I'm aware of (though I don't fully understand) the problems caused by black holes being indistiguishable except for rotation, which messed with entropy (they may have resolved this later some other way by now, like I said, I don't fully understand it). However, my intuition tells me that gravity concentrated at a single point of infinite density would be completely uniform, and therefore, the event horizon, which was by definition the boundary at which spacetime was so warped that light could not escape, would be completely spherical. So what led scientists to beleive this wouldn't be the case?

2) Also my intutition is apparently wrong anyway for rotating black holes. So is there an explanation, in not too technical terms, as to why a black hole's event horizon would "bulge" due to rotation?

Is the book you read "Black Holes and Time Warps" by Kip Thorne, by any chance?

Anyway, to your first question, in the early days of black hole and GR research (note that they weren't even called black holes then), no one was willing to accept a point mass with infinite density. Physicists today still aren't, really, and for good reason: it doesn't make any damned sense. What people were willing to accept was that the star (in the West, black holes were called "collapsed stars" before John Wheeler coined the term "black hole") collapsed inward beyond the Schwarzschild radius, but did not become a point mass. Even today, no one believes it actually becomes a point mass. So, the thinking was that if you, say, had a star with a mountain on it, as it shrank, the mountain would become larger and larger relative to the star. You would naively expect it to leave some imprint on the horizon as the star collapses further inwards, since the mountain is ostensibly still there, just completely hidden from view. But relativists in the 1960s were able to show that this isn't what happens at all - once the star shrinks beyond its Schwarzschild radius, GR says that complete collapse is inevitable and the resulting black hole will be perfectly spherical, assuming the star wasn't rotating. As with most weird results, especially those that completely go against physicists' intuition, people were slow to accept this.

For your second, not really, none that I can think of. If you're feeling brave, have a look at the Wiki page for the Kerr metric. That page at least has the mathematical reason for it. I suppose you could sort of intuitively think about as centrifugal force pushing the horizon out, but that explanation doesn't sit well with me since we're dealing with spacetime curvature.

[Mostlynormal]

Thanks for the explanation. The Kerr Metric goes a bit above my head but thanks for trying.

[yurell]

If I have a box full of photons with the energy equivalent of 1kg of matter, does the box behave like a 1kg box (presuming that the inside is made of perfect mirrors, of course)? Is there any way for an observer to tell it apart from a 1kg box with no photons?

[doogly]

Measure the pressure on the walls, and check how it reacts to heat.

[yurell]

How would you measure the pressure on the walls without opening the box? But measuring the box's reaction to heat is a great idea.
Gravitationally it would look like a 1kg box wouldn't it? Would it behave like a 1kg box when a force is applied? Would the moment of inertia be different?

[PM 2Ring]

How would you measure the pressure on the walls without opening the box? But measuring the box's reaction to heat is a great idea.

Well, you just need to equip the box walls with some sort of strain gauge that can be read from the outside.

Gravitationally it would look like a 1kg box wouldn't it? Would it behave like a 1kg box when a force is applied? Would the moment of inertia be different?

Yes, yes, no.

Of course, it'd be rather difficult to actually perform this experiment without a source of perfectly-reflecting unobtainium.

[phlip]

If you apply a force to, say, the left side of the box, to accelerate it to the right... then there will be a higher pressure from the photons on the left side of the box than on the right, which will mean you have to push harder. I haven't run the numbers, but I wouldn't be surprised if the amount you have to push harder is 1kg * a.

Once the box is in motion to the right, all the photons moving from left to right will be slightly blue-shifted, while all the photons moving from right to left will be slightly red-shifted (from your perspective). But the blue-shifted photons will, I think, have more energy than their previous levels by more than the red-shifted ones have less... so it's a net increase, which is where the energy goes from having to push harder on the box.

Rotation is more complicated, but I believe ultimately similar.

[Goemon]

Let's see... 1kg of photons have a momentum E/c = mc = c. If they're trapped in a box of height h, and all happen to be moving vertically, they'd bounce back and forth with a round trip time of 2h/c. That exerts a total force dp/dt = c^2/h on the top of the box (momentum transfer = 2c for a reflection) . At a height of 100km, the photon pressure would support a cube of water half a kilometer on a side.

With two perfect mirrors and 500 grams of anti-matter, you could set a city permanently floating 100km high...

[Eebster the Great]

Let's see... 1kg of photons have a momentum E/c = mc = c. If they're trapped in a box of height h, and all happen to be moving vertically, they'd bounce back and forth with a round trip time of 2h/c. That exerts a total force dp/dt = c^2/h on the top of the box (momentum transfer = 2c for a reflection) . At a height of 100km, the photon pressure would support a cube of water half a kilometer on a side.

With two perfect mirrors and 500 grams of anti-matter, you could set a city permanently floating 100km high...

Why are the photons only exerting pressure on one side of the box?

[Goemon]

and all happento be moving vertically

The aforementioned force on the top of the box, and of course the same plus 1kg worth of force on the bottom.

[SU3SU2U1]

How would you measure the pressure on the walls without opening the box?

Squeeze the box.

[Slothrop]

I probably won't be able to describe this in any understandable way, but here it goes.

Why aren't there a 90 degrees phase shift between the electric and magnetic field oscillations in (?) a photon? I would have guessed that the electric field would be at its strongest where the change in the magnetic field is the fastest.

Figures like this (http://en.wikipedia.org/wiki/File:Light-wave.svg) indicate that I'm wrong.

[Robert'); DROP TABLE *;]

I probably won't be able to describe this in any understandable way, but here it goes.

Why aren't there a 90 degrees phase shift between the electric and magnetic field oscillations in (?) a photon? I would have guessed that the electric field would be at its strongest where the change in the magnetic field is the fastest.

Figures like this (http://en.wikipedia.org/wiki/File:Light-wave.svg) indicate that I'm wrong.

Having misread this question and invented a completely wrong answer for it, I'd like to know this too. I would've also thought that the weakest point in the electric field would correspond to the peak in the other. (Not both peaks occurring simultaneously, as the diagram shows)

[starslayer]

Presuming you're up on your vector math, construct a generic electric field wave, then take its curl and integrate with respect to time. You'll recover a magnetic field wave that is in phase and perpendicular to the electric one. Specifically, you'll get the cross product of the wave propagation vector and the electric field. Wiki has it written out, as usual (it's at the bottom of the article).

Your mistakes were in not remembering that Faraday's Law says that the time derivative of the magnetic field is related to the curl of the electric field, not the field itself. To wit:

\nabla \times \bf{E} = \frac{-\partial \bf{B}}{\partial t}

[Slothrop]

Cheers, starslayer. I definitely need to brush up my vector math, but that wiki seems to go through it straight enough.

[starslayer]

Another way you can think of it is seeing that the curl is essentially a set of differences of spatial derivatives of a vector field. Therefore, Faraday's Law basically relates how the magnetic field is changing in time to how the electric field is changing in space. Ampere-Maxwell does this same thing for the magnetic field (it takes the same form as Faraday's Law in absence of currents).

Let's start with the electric field again, like Wiki does. Around the crest or trough of an oscillation, the electric field isn't changing much in space or time, and those derivatives are close to zero. Therefore, the curl will be small. Since curl(E) is small, then the rate of change of B with respect to time is also small. This means that B also has to be at a crest or trough. Therefore, the electric and magnetic fields must be in phase.

Similarly, we can look at where the electric field is zero. There, it is changing the fastest, so its derivatives and curl are also at their largest. This means that the magnetic field must also be changing its quickest there, and presuming it is an oscillating field, its strength must be zero/at its minimum as well. Again, we come to the conclusion that the electric and magnetic waves are in phase.

[Robert'); DROP TABLE *;]

Minor question about black holes now.

According to my algebra, whenever anything falls into a black hole and "hits" the event horizon, the gravitational potential energy it loses is equal to exactly half of its rest mass in "deep space," (where GPE=0 by convention) regardless of how big the black hole is, what's falling in, or whether the in-falling object is massive or not. Is this actually significant, beyond being a nice round number? Would it make sense to treat it as a "floor" of warped space-time, where GPE=0 is the "ceiling?"

[doogly]

No, that seems about right. What you're doing is saying the potential energy is 0 at infinity, and at the horizon it is GmM/r, but with r=2GM. Energy is now m/2. Yay.

This seems quite reasonable. And it's not problematically naive to be mixing the GR and Newtonian ways of thinking so long as your black hole is large enough that the horizon is out pretty far. Also note that we have ignored any talk of angular momentum. That does get a bit fuzzy. This summarizes the full GR picture pretty nicely:
http://en.wikipedia.org/wiki/Two-body_p ... relativity

[Taikand]

Am I the only person that believes quantum physicists should stop doing philosophical interpretations and they should have powerful mathematical models before throwing around hypotheses.Looking around the Internet for information about the experiments usually lead me to some pop-science explanations. I'd like to find a source where I can learn quantum physics without analogies but that are nonetheless understandable. I really want to learn but what I find is either too dumbed down or too complex. IF anyone has knowledge of any courses then please send them to me.

[starslayer]

They do have powerful mathematical models; that's kind of what quantum mechanics and QFT and the Standard Model are all about. What it all actually means is another question, which is where the philosophy comes in. I also doubt you're going to find that much middle ground; even an undergraduate level treatment usually either involves a ton of handwaving with some philosophical type arguments (find a textbook on "modern physics" for that), or is going to require knowledge of differential equations and linear algebra (even the simplest start with the Schrodinger equation is going to use both).

[eSOANEM]

Am I the only person that believes quantum physicists should stop doing philosophical interpretations

Lots of physics is built on what were originally purely philosophical grounds. The equivalence principle as used by Einstein to form GR is an excellent example of this. I've also seen it said that the Schroedinger equation is an example of this (although I tend to disagree as it is, with only a few, easily justifiable even at the time, assumptions, deducible from the de broglie relation, Planck's relation and classical mechanics).

Such philosophy is often the driving force behind new theories (as an example of this, Penrose followed a line of reasoning where, because both collapse and gravity don't play nicely with the Schroedinger equation, the two must be related. Following this, he comes up with a theory with objective collapse caused by gravitational self-interaction and even designs a practical experiment to test it (although it needs to be done in microgravity so probably won't be done any time soon)).

Furthermore, in order to actually obtain predictions from QM, you need some philosophical interpretation of what the wave function is and at what point probabilities can be observed (when collapse occurs, or even whether it does at all). Otherwise, all you're doing is evolving some function with no way of relating it to the universe.

Looking around the Internet for information about the experiments usually lead me to some pop-science explanations. I'd like to find a source where I can learn quantum physics without analogies but that are nonetheless understandable. I really want to learn but what I find is either too dumbed down or too complex. IF anyone has knowledge of any courses then please send them to me.

You're going to need a pretty good knowledge of differential equations in order to do anything other than calculate the number of photons emitted by a megawatt red laser (which isn't really "proper" quantum mechanics).

Assuming you're happy with that, the first thing to do would be to play with the time independent schroedinger equation in one dimension for various potential wells.

The easiest wells to start with would be infinite square wells. These have a potential of 0 inside the well (which has some width) and infinity elsewhere.

See if you can derive discrete energy levels and then what those levels are.

Then try a finite square potential well, this doesn't go to infinity outside the well, but to some finite value (although it must be greater than the total energy of the particle if you want to get a normalisable wavefunction (you do)).

From these, it should be fairly easy to derive the confinement energy (the ground state of an infinite square well of a given width) and from this, and coulomb's law provide an argument for the value of the bohr radius from the pov of energetics. You can then do the same thing with a muon instead of an electron to show that they bring the nuclei much closer together (to approximately their compton wavelength IIRC) and so could, if muons were long-lived enough, be used to catalyse fusion.

Then you could have a look at a quantum harmonic oscillator (a partical in a quadratic potential).

etc.



You can do quite a lot of playing with just the Schroedinger equation in one dimension, to do more, you'll need a lot more maths which, from the sounds of it, is the limiting factor.



A couple of notes about wavefunctions:

A physically realisable wavefunction must be normalisable. This, roughly, means the integral of its modulus squared over all of space must have some finite value.

In order to recover probabilities, that finite value should be one (it should be a normalised wavefunction), this can be done by dividing by the integral of the modulus squared over all space.

Once you have a normalised wavefunction, the probability of being in a certain part of whatever space you're in (if you're working with the time-independent schroedinger equation, this will be position space) is given by the integral of the modulus squared within that region.

The wavefunction should, for continuous potential wells, be continuous and have a continuous slope (this means that the momentum cannot arbitrarily jump) and, the curvature should be continuous too (the energy cannot arbitrarily jump). Furthermore, the slope should be continuous unless there is an infinite jump in the potential well (because the well is a function of position, in order to conserve momentum, it cannot change all over the place).



Oh, and all of this stuff I've just described, that's all well before any SR or anything gets involved, so before any QFT. It gets a lot more complicated and, if you want an easy route through it, there isn't really one which doesn't require a lot of maths to be learnt.

[doogly]

Add to this the reality that Schrodinger didn't actually understand quantum mechanics. His interpretation was flawed, because he did not grok Hilbert space. You, too, after a first formal undergrad course, would even then be likely to not grok Hilbert space. It's generally quite hard.

There are a lot of things that go on in physics that are exciting. Relativity, optics, solid state, the particle zoo, radiation, classical chaos and dynamical systems... lots lots more. The most conceptually demanding, philosophically rewarding, and mathematically intricate is definitely quantum mechanics. It is the highest hanging fruit, and there is no shortcut to it. I don't mean to be discouraging, just to rank difficulties here. If you compare it in particular to the other "modern" pillar, relativity, it is vastly more unintuitive.

[Yakk]

Hilbert Space?

Complete Inner Product Spaces?

A vector space with a linear, conjugate symmetric positive definite binary operation whose Cauchy sequences in the derived norm converge in the space?

A group of elements called vectors and a field of scalars with a two additional functions -- scalars x vector to vector function that is field-linear in group vector addition, and a vector x vector to field function that is linear in the first element, conjugate symmetric between the two arguments, and positive definite when called on the same argument twice, and you can build a norm from calling it twice on the same argument in which Cauchy sequences converge?

What is there to grok?

[doogly]

The fact that this is the space where states \psi live. Many people like to write down a psi(x) a la Schro and think the story is done here, that a state ontologically *is* some map from your configuration space manifold to the complex numbers.

[Twelfthroot]

A quick question to verify my understanding of relativity: if I'm not mistaken, for any object which I measure at rest with respect to myself to be n meters long, I can find a frame in which I measure that object to be x < n meters long for any arbitrary x as long as I move sufficiently fast (or move the object sufficiently fast) in the direction of measurement. Is there any way for me to observe the object to be longer than n meters?

If my understanding is correct, there is no inertial reference frame in which the object can appear longer than it appears to me when we're at rest wrt each other. However, I haven't yet explored the consequences of general relativity. Is this possible when taking acceleration/gravity into account?

[eSOANEM]

I can't give a definite answer, but, because space is locally flat, this should (I'm almost certain) be true for short lengths and probably generally (excepting cases in very strong fields) too.

[thoughtfully]

The "Lorentz Contraction" is strictly a contraction, as you thought. If L₀ is the length in the rest frame, the length (along the direction of motion) is L' = L₀/γ, where γ = (1-v²/c²)^-1/2.
Sorry, radicals refuse to render right. Say it five times fast!

I don't believe it's drastically different in accelerating frames or high gravitational fields, but I bet somebody more familiar with those will have something to say about it.

[starslayer]

It isn't. With strong gravitational fields, you can get the object to appear longer due to lensing, but that isn't a coordinate effect like Lorentz contraction is.

[Goemon]

If you're deep in a gravitational field, you'll perceive objects far above to be longer than expected - that is, a stick that you measure to be a meter long when it's next to you will extend across > 1m of "coordinate" distance when it's far above you (using coordinates normalized to you). Acceleration in flat space, on the other hand, doesn't affect length measurements.

[Taikand]

Am I the only person that believes quantum physicists should stop doing philosophical interpretations

You're going to need a pretty good knowledge of differential equations in order to do anything other than calculate the number of photons emitted by a megawatt red laser (which isn't really "proper" quantum mechanics).
[...]
Oh, and all of this stuff I've just described, that's all well before any SR or anything gets involved, so before any QFT. It gets a lot more complicated and, if you want an easy route through it, there isn't really one which doesn't require a lot of maths to be learnt.

Believe, I dislike QM so much that I won't stop before understanding and either accept or prove false.What I hate most is when people present the experiments they say things like "we're going to mark a photon" or "we're deleting the marker on the photon" and I can't find any source on how exactly it works like.That's one of the reasons why I find it hard to accept their experimental results, because the exact nature of the processes they use may be better answers than saying that "photons make a pact when you're not looking at them".

[eSOANEM]

The problem is that the people presenting experiments like that are generally doing so to a broadly lay audience. In this case, they have to try to come up with some explanation people will understand, but these explanations are not the physics; they are not correct, they are tools designed to help people gain some acceptance of the experimental results/the phenomenon.

The actual physics is the maths and there's no way round that. If you want to get around all the dodgy explanations, you're going to have to do a lot of work learning all the necessary maths (the language in which the physics is expressed) and then the physics (the equations themselves).

As for "mark a photon", the only times I can see people saying that would be in cases where the photons are emitted one at a time (such as the single-photon double-slit experiment). In this case, no such marking takes place; it is purely a way of helping people visualise the location of the photon at a given point in time (in as much as one exists).

[Taikand]

The actual physics is the maths and there's no way round that. If you want to get around all the dodgy explanations, you're going to have to do a lot of work learning all the necessary maths (the language in which the physics is expressed) and then the physics (the equations themselves).

Exactly!We should tell people:"Screw cats in a box, learn some math!".Those explanations only make me hate QM even more because we get people that say things like "My thoughts can bend reality, that's why you must always think positive.IT'S SCIENCE!"
And, I want to understand the math, unfortunately I have to learn for my finals now so I can't really do any extra-curricular work. Can't wait to get to college.

[Dopefish]

Unfortunately, an awful lot of people are very put off from math, which can lead to people being put off from physics because they're 'bad at math'. I think both physics and math are awesome for their own sake, but there's surely plenty of people out there who like physics because they find it interesting (because they've heard of all the [pop sci interpretations of] cool stuff quantum and such can do), and only eventually get around to learning the math because they're interested in the physics, even if they dislike the math itself.

At high school/first year level, it's largely a matter of marketing the material, rather then teaching it unfortunately. You can get the gist of a lot of physics by drawing on intuition and analogies, but those tend to be a bit misleading when you get into the realm of quantum mechanics and relativity, so those areas in particular people tend to be particularly confused in when they discover that their pop sci knowledge of isn't a sufficient replacement for hard math.

[gmalivuk]

Those explanations only make me hate QM even more because we get people that say things like "My thoughts can bend reality, that's why you must always think positive.IT'S SCIENCE!"

Do you really believe people would stop saying stupid things about science if they stopped hearing explanations of it in lay terms? People *still* believe ridiculous nonsense about magnets, despite the fact that a bright child who isn't very good at math could quite easily learn enough about magnetism to understand why such nonsense doesn't work.

[eSOANEM]

At high school/first year level, it's largely a matter of marketing the material, rather then teaching it unfortunately.

This. Furthermore, this is necessarily so (irritating as that is). Without such explanations, fewer people would bother with physics at high level and many of those who do would find it harder to learn the necessary maths because they'd have no intuition behind it.

[Robert'); DROP TABLE *;]

Assuming that a) time dilation is such that things falling into a black hole take an effectively infinitely long time to fall past the EH when seen from arbitrarily far away, and b) a combination of Hawking radiation and cooling CMBR mean that every black hole will eventually evaporate, I have a couple of questions:
1) How can a black hole form in the first place? It would seem that as two very dense masses approach each other, the time dilation they experience will increase, and so they will never actually combine into a singularity in finite time.
2) How can something fall past the EH? It must surely take a larger amount of time than the black hole will actually exist for, (AFAIK) since the time dilation near the EH increases without bound.

[PM 2Ring]

Assuming that a) time dilation is such that things falling into a black hole take an effectively infinitely long time to fall past the EH when seen from arbitrarily far away, and b) a combination of Hawking radiation and cooling CMBR mean that every black hole will eventually evaporate, I have a couple of questions:
1) How can a black hole form in the first place? It would seem that as two very dense masses approach each other, the time dilation they experience will increase, and so they will never actually combine into a singularity in finite time.
2) How can something fall past the EH? It must surely take a larger amount of time than the black hole will actually exist for, (AFAIK) since the time dilation near the EH increases without bound.

Well, as you point out, the extreme time dilation is in the frame of the observer at infinity; material in freefall near the EH will cross the EH rather rapidly according to its own clock. Still, I think it's important to recognize that black hole formation takes a very long time in the frame of distant observers, and that for such observers black holes may effectively evaporate before they finish forming, assuming Hawking radiation is real.

Note that we still only have indirect evidence for the existence of black holes, via gravitational lensing and the activity of accretion disks. We may never have any observational evidence for Hawking radiation, since it's so weak, and it will be a long time before the CMBR is cool enough to allow even stellar mass black holes to begin to evaporate via Hawking radiation.

FWIW, the basic Schwarzschild black hole is a rather artificial entity; it arises as a somewhat simple solution to the Einstein field equations. In this solution, the Schwarzschild black hole is the only object in its universe, and it's eternal. Obviously, the mathematics that describe the dynamics of real black hole formation are a bit more complicated than the simple static Schwarzschild solution.