General Relativity

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JamesCFraser
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General Relativity

Postby JamesCFraser » Wed Apr 09, 2008 9:25 pm UTC

I'm not a physicist, so please don't be mad at me if this question is poorly worded or in and of itself stupid.

My question probably doesn't require a thorough grasp of precisely what General Relativity is, so I imagine any physicist worth his salt could answer it. Anyway, my question is thus:

Does General Relativity encapsulate all of the principles of Special Relativity, or is it the case that General Relativity is only relevant to gravity? (I know this is a poorly phrased question, and it also demonstrates my ignorance...)

The reason I'm asking this is that I said you would use equations from General Relativity to answer what speed a particle would be traveling at if you were given its mass and kinetic energy (in the case where Newtonian equations would give you a speed greater than the speed of light), to which a know-it-all friend rather rudely interrupted, called me stupid, and said you would use Special Relativity.

I was under the impression that General Relativity contained Special Relativity, but the Wikipedia article seems to focus on the implications of General Relativity when gravity is involved.


I'm rather hoping I was right, as feeding said know-it-all's ego is a bad idea. I don't mind about being wrong, and, quite frankly, could do with some physics education, but I'm rather hoping his response wasn't accurate.

Thanks for your time.

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Re: General Relativity

Postby antonfire » Wed Apr 09, 2008 9:35 pm UTC

You are right in that General relativity encapsulates special relativity, but he is right in that the equation you're looking for is from special relativity.

To make an analogy, you wouldn't say that, F=ma is an equation from special relativity, would you?

But, honestly, who cares what you call it?


Edit: made wording clearer.
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Re: General Relativity

Postby Yakk » Wed Apr 09, 2008 9:37 pm UTC

Yes, you could use General Relativity. But that would be overkill.

Special Relativity contains all of the equations you need to work out the solution to that problem, and is lots easier to work with. High-school algebra difficulty vs mid to upper-year university calculus and analysis.

Imagine if you said "we should use a gun to take the air out of the tires", and someone responded "silly, you deflate tires by using the valve". Sure, a gun can deflate a tire -- but that isn't a good way to do it. :)
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Re: General Relativity

Postby JamesCFraser » Wed Apr 09, 2008 9:39 pm UTC

antonfire wrote:You are right in that General relativity encapsulates special relativity, but he is right in that the equation you're looking for is from special relativity. You wouldn't say that, F=ma is an equation from special relativity, would you?

But, honestly, who cares what you call it?


Well, it doesn't really matter. The point is that he is very insolent to those around him because he is very insecure about his own level of intelligence (for one, he got rejected from his first choice of university). However, despite the fact I sympathise with him, I don't appreciate the way he now treats everyone as if he is the most intelligent person we could ever hope to meet and we are all idiots. Showing that he's not always right may help to stop him from behaving like this, I hope.

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Re: General Relativity

Postby taby » Wed Apr 09, 2008 9:42 pm UTC

Sure, the concept of the 4-momentum vector was introduced to clarify the meaning of Special Relativity, but it also plays an incredibly crucial role in General Relativity. The source of gravitation is a 4x4 matrix (2nd rank tensor), where the top row and leftmost column just happen to be this very same 4-momentum vector!

I think that if your friend actually knew some of the basics behind General Relativity, they wouldn't have objected. Indeed, it's called General Relativity because it handles rectilinear motion just as well as Special Relativity (plus a whole lot more). In fact, the only important differentiation in regard to relativity in this case is not Special vs. General, but Galilean vs. Einsteinian.

Sure you can say that Special Relativity is a simpler method for handling this situation, but then I must go one step further and say that plain old Newtonian mechanics is even better when the object's relative speed is much less than the speed of light.
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Re: General Relativity

Postby antonfire » Wed Apr 09, 2008 9:50 pm UTC

I still say that, while the 4-momentum is important in GR, it comes from SR, so it's not quite right to say that you'd be using "an equation from general relativity" to solve the problem. It's like saying that you're using "an idea from quantum mechanics" when you're doing something with a Hamiltonian.

Calling it an equation from GR bothers me.
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Re: General Relativity

Postby taby » Wed Apr 09, 2008 9:53 pm UTC

antonfire wrote:I still say that, while the 4-momentum is important in GR, it comes from SR, so it's not quite right to say that you'd be using "an equation from general relativity" to solve the problem. It's like saying that you're using "an idea from quantum mechanics" when you're doing something with a Hamiltonian.

Calling it an equation from GR bothers me.


I completely understand. This is also why I mention that if the speed is low, calling this an equation from SR bothers me just as equally. It all depends on the problem's initial conditions.

In physical reality however, everything discussed here does boil down to being a problem solvable only by GR, since there is no velocity larger than 0 where relativistic corrections to momentum do not occur (even if the corrections may be minute), and there is no such thing as a truly rectilinear path (even if the accelerations may be minute).

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Re: General Relativity

Postby Yakk » Wed Apr 09, 2008 10:05 pm UTC

Oh, by that standard, we don't have physics that we know solves your problem.

Nobody has figured out how to make QM and General Relativity consistent with each other in every case.

You can get them to produce predictions that match our precision of measurement in many cases, but the same is true of using Newtonian Mechanics at medium mass/medium force/medium energy/medium time models.

The math in both GR and SR reduce to Newtonian Mechanics up to a margin of error smaller than we can detect over the domain that Newtonian Mechanics matches observations up to our margin of observation error, so they do have a strictly larger domain of validity. :)
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Re: General Relativity

Postby Scigatt » Thu Apr 10, 2008 1:29 am UTC

antonfire wrote:I still say that, while the 4-momentum is important in GR, it comes from SR, so it's not quite right to say that you'd be using "an equation from general relativity" to solve the problem. It's like saying that you're using "an idea from quantum mechanics" when you're doing something with a Hamiltonian.

Calling it an equation from GR bothers me.


What about, say, using the equivalence principle to solve an HS physics problem?

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Re: General Relativity

Postby taby » Thu Apr 10, 2008 1:35 am UTC

Yakk wrote:Oh, by that standard, we don't have physics that we know solves your problem.

Nobody has figured out how to make QM and General Relativity consistent with each other in every case.

You can get them to produce predictions that match our precision of measurement in many cases, but the same is true of using Newtonian Mechanics at medium mass/medium force/medium energy/medium time models.

The math in both GR and SR reduce to Newtonian Mechanics up to a margin of error smaller than we can detect over the domain that Newtonian Mechanics matches observations up to our margin of observation error, so they do have a strictly larger domain of validity. :)


Surely you realize that I was being ridiculous on purpose, just to prove the point that being so nitpicky is silly. :)

And besides, I was the one who was suggesting Newtonian mechanics in the first place! :P

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Re: General Relativity

Postby Socal Swimmer » Thu Apr 10, 2008 4:04 am UTC

On a more simple note, Special relativity deals with inertial reference frames (no acceleration, what you are doing) whereas general relativity deals with accelerated reference frames (which is the same as a gravitational field). Apparently GR contains SR (I didn't know this) but it seems that SR would be much easier to use.

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Re: General Relativity

Postby JamesCFraser » Thu Apr 10, 2008 7:07 am UTC

Socal Swimmer wrote:On a more simple note, Special relativity deals with inertial reference frames (no acceleration, what you are doing) whereas general relativity deals with accelerated reference frames (which is the same as a gravitational field). Apparently GR contains SR (I didn't know this) but it seems that SR would be much easier to use.

The douche was technically wrong, but (annoyingly) right.


Actually, in retrospect, the situation was in a gravitational field (it was a cosmic ray arriving at Earth), but the effect of that would be negligible.

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Re: General Relativity

Postby Socal Swimmer » Thu Apr 10, 2008 7:14 am UTC

JamesCFraser wrote:
Socal Swimmer wrote:On a more simple note, Special relativity deals with inertial reference frames (no acceleration, what you are doing) whereas general relativity deals with accelerated reference frames (which is the same as a gravitational field). Apparently GR contains SR (I didn't know this) but it seems that SR would be much easier to use.

The douche was technically wrong, but (annoyingly) right.


Actually, in retrospect, the situation was in a gravitational field (it was a cosmic ray arriving at Earth), but the effect of that would be negligible.


while we are in retrospect, are we attempting to determine the velocity with respect to the earth, or to an inertial reference frame?
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Re: General Relativity

Postby JamesCFraser » Thu Apr 10, 2008 10:54 am UTC

Socal Swimmer wrote:
JamesCFraser wrote:
Socal Swimmer wrote:On a more simple note, Special relativity deals with inertial reference frames (no acceleration, what you are doing) whereas general relativity deals with accelerated reference frames (which is the same as a gravitational field). Apparently GR contains SR (I didn't know this) but it seems that SR would be much easier to use.

The douche was technically wrong, but (annoyingly) right.


Actually, in retrospect, the situation was in a gravitational field (it was a cosmic ray arriving at Earth), but the effect of that would be negligible.


while we are in retrospect, are we attempting to determine the velocity with respect to the earth, or to an inertial reference frame?


An inertial reference frame

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Re: General Relativity

Postby Socal Swimmer » Thu Apr 10, 2008 7:12 pm UTC

JamesCFraser wrote:
Socal Swimmer wrote:
JamesCFraser wrote:
Socal Swimmer wrote:On a more simple note, Special relativity deals with inertial reference frames (no acceleration, what you are doing) whereas general relativity deals with accelerated reference frames (which is the same as a gravitational field). Apparently GR contains SR (I didn't know this) but it seems that SR would be much easier to use.

The douche was technically wrong, but (annoyingly) right.


Actually, in retrospect, the situation was in a gravitational field (it was a cosmic ray arriving at Earth), but the effect of that would be negligible.


while we are in retrospect, are we attempting to determine the velocity with respect to the earth, or to an inertial reference frame?


An inertial reference frame


then gravity is irrelevant. So the effect of gravity is 0 (not just negligible).

based on the equivalence principle, an inertial frame = 0 gravity.
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Re: General Relativity

Postby Klotz » Fri Apr 18, 2008 3:16 am UTC

Special relativity is general relativity in Minkowski space (aka flat space).

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Re: General Relativity

Postby Snoof » Sat Apr 19, 2008 7:36 am UTC

Socal Swimmer wrote:On a more simple note, Special relativity deals with inertial reference frames (no acceleration, what you are doing) whereas general relativity deals with accelerated reference frames (which is the same as a gravitational field).


SR can handle acceleration just fine - it just can't handle accelerating reference frames. There's a difference. So while it makes no sense to speak of the Centre Of Mass frame of an object which is accelerating (relative to some inertial frame), you can still solve problems with that accelerating object.
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General Relativity

Postby Tac-Tics » Fri Aug 22, 2008 6:46 pm UTC

Special Relativity is easy cheese. However, I'm having trouble understanding the implications of general relativity. I've tried reading through the material on Wikipedia (The Introduction to General Relativity), but it's not all clear and hardly concise. I don't have any really specific questions, but I know there are people on this forum who can help me pick up on the main ideas.

So far, this is what I understand. Einstein first got his inspiration for the general theory when he realized that an accelerating system can still be treated like a frame of reference. Just as you can't know your absolute velocity, you cannot know your absolute acceleration -- you have to measure it against something. And so, if you are accelerating at the same speed as another object, say, a moon, it looks like you and the moon you're measuring against are both at rest. And thus, special relativity applies in your system. And this is called the equivalence principle.

I also understand why gravity fields must bend light. Suppose you are in an elevator in deep space with a laser pointer mounted on the wall, facing the wall opposite it. You can use a straight edge to determine exactly where the laser will hit the opposite wall. Now, because of the equivalence principle above, you can use the same methods to determine where the laser will hit, even if the elevator is accelerating due to a gravity field. However, to an observer outside of the elevator, say standing on a planet below towards which the elevator is falling, he will see that the path of the light is curving inside the elevator -- noting that the height the elevator when the laser is fired is greater than the height of the laser when the light beam hits the opposite wall.

I hope I'm understanding that much correctly.

My guess is that the path of the light, relative to the guy on the planet's surface, would be a hyperbola. Just like a ball in free-fall. But unlike a ball, you have to keep its speed constant along the path, so the faster the elevator falls, the slower time moves for the people inside.

Now this is about where my understanding breaks down (if I'm not mistaken elsewhere). I know there is a red/blue shit that occurs in these situations, but I'm not sure exactly how they crop up. The wikipedia article also explains a "tidal effect" which they do a poor job of explaining. At one point, looking over the article a few days ago, I had a faint hint as to where the whole geometric interpretation comes from, but I lost it shortly afterwards.

So anyone able to give me a better picture of what's going on, I'd love to hear your explanations!

Unlike in special relativity, where the


EDIT: Looking back at my post, I may be mistake about the relative nature of acceleration. Hopefully this is a good point where someone can step in and explain to me what's really going on :-)

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Re: General Relativity

Postby hitokiriilh » Fri Aug 22, 2008 9:15 pm UTC

Relativity is pretty ridiculous. It's my favorite physical theory, but it's sometimes difficult to understand how the geometric formulation of physics in GR relates to how we perceive reality. In particular, space-time is a manifold and can be treated geometrically as a static structure on which only the set of points matter and not the parameterization of trajectories or this fleeting thing called 'time'. Consequently, it's rather perplexing to understand why we see the geometry of space-time as a continuous ordered set of states that change with time and not as a static geometric structure (basically: why do we experience the phenomenon of time elapse?). I feel like I'm digressing though...

Back to the point. You know how each physical theory answers a fundamental question with a set of equations? Newton's 2nd law answers "If we apply a force to an object, how can we figure out how it will move?" Maxwell's equations answer "Given a 4-current distribution, what will the field strength tensor be?" (this of course being the manifestly relativistic formulation of E&M). Special relativity answers "What set of coordinate transformations leave the matrix representation of the Minkowski metric invariant?" with the Lorentz transformations (more generally, the entire Poisson group leaves it invariant - these include parity transformations, time reversal transformation, translations, etc...but the latter is trivial and the former two aren't really physical when thinking in term of relating two reference frames). Well, in all of these previous theories assume space-time is flat. I think it's a rather naive assumption. General relativity aims to fix this. It answers the question "What is the metric on space-time?" with Einstein's field equations.

General relativity was motivated by three principles: Mach's principle (which you summarized nicely), the equivalence of gravitational mass and inertial mass (which makes an acceleration indistinguishable from a gravitational field), and the strong equivalence principle (equivalence of all inertial frames). In order to make absolute acceleration a thing of the past, absolutely everything would have to gravitate and respond similarly to gravitational interactions. In Newtonian gravity, light wouldn't have been bent by earth's gravitational field. The person falling was decidedly in the accelerated frame; the observers on the surface of earth were decidedly inertial observers. General relativity turns this on its head. Any frame that's in free fall is an inertial frame, and consequently the person falling in the elevator would see the light go straight; any frame that *locally* sees the effects of gravitational 'forces' is being accelerated. So since you're sitting on a chair reading this message and see the effect of gravity holding your computer flush against the table or lap you're actually in an accelerating frame. Now drop a pen - neglecting air resistance it's in an inertial frame until it hits the floor.

So inertial frames move along geodesics (curved space equivalent of a straight line) in space-time. Geodesics are determined by the metric. A metric tells us how to locally measure distance and angles on a manifold. Force deviate an object off of its geodesic

Let's say you have two geodesics right next to each other than start off parallel. In general they'll eventually diverge. If you hook two point masses together via a spring and send them along these initially parallel geodesics, the spring will stretch. The force that stretches the spring is the so-called tidal force you heard about. You can can of think of it in terms of Newtonian gravity: since the field goes as [imath]\frac{1}{r^2}[/imath], if you drop a spring with one end at a smaller distance from the center of the gravitating object than the other end, the spring will stretch and the force that stretches the spring is a tidal force. It's aptly named because this effect is responsible for tides.

Blue/redshifting occurs basically for the same reason the doppler effect happens. If you accelerate towards a light source, you'll go progressively faster and the light will get bluer and bluer. Since an acceleration is equivalent to a gravitational field, if you sit in a gravitational field and look at a source of light that's sitting out in flat space far away, the light will get blueshifted by the time it makes its way to you. Similarly, if the light source is sitting in a gravitational field and you're in flat space far away, the light will be redshifted as this is equivalent to the source accelerating away from you. Mind-blowing, but you REALLY have to abandon all Newtonian and Gallilean intuition when working with general relativity.

Edit: You're right in the whole relative acceleration thing. That's exactly Mach's principle and it's built into the field equations. If everything in the universe accelerated in the same direction at the same rate the metric would be indistinuishable from a metric in a universe in which nothing accelerated.

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Re: General Relativity

Postby Goemon » Fri Aug 22, 2008 10:37 pm UTC

I think I'd be a little careful about saying "acceleration is relative".

In the sense described above (everything in the universe experiences uniform acceleration), it may be true, but in general it isn't.

Two spaceships sit next to each other in otherwise empty (flat) space. Suddenly each notices that the other is moving away, faster and faster. Which one is accelerating? Is there any way to tell? Or is the physics the same no matter which one we consider to be accelerating?

The answer is that it makes a difference. Everyone in the universe, no matter where they are or how they're moving (including passengers in the two spaceships) can instantly point at spaceship "A" and declare "THAT one is accelerating!" There's no question about it. In this sense, acceleration is NOT relative. It's easy for anyone and everyone to verify absolutely which spaceship is "really" accelerating.

And I think it's a bit difficult to summarize everything that General Relativity means in a paragraph or two...

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Re: General Relativity

Postby hitokiriilh » Sat Aug 23, 2008 12:32 am UTC

Goemon wrote:I think I'd be a little careful about saying "acceleration is relative".

In the sense described above (everything in the universe experiences uniform acceleration), it may be true, but in general it isn't.

Two spaceships sit next to each other in otherwise empty (flat) space. Suddenly each notices that the other is moving away, faster and faster. Which one is accelerating? Is there any way to tell? Or is the physics the same no matter which one we consider to be accelerating?

The answer is that it makes a difference. Everyone in the universe, no matter where they are or how they're moving (including passengers in the two spaceships) can instantly point at spaceship "A" and declare "THAT one is accelerating!" There's no question about it. In this sense, acceleration is NOT relative. It's easy for anyone and everyone to verify absolutely which spaceship is "really" accelerating.

And I think it's a bit difficult to summarize everything that General Relativity means in a paragraph or two...


I thought that at first too, but the relative acceleration thing can be answered by nothing more than looking at the metric. In the examples you gave, the metric was implicitly assumed to be flat and the objects were considered test objects to which the metric barely couples. Mach's principle is kind of built into the field equations. If everything in the universe were accelerating in the same direction at the same rate, the metric would indeed be different - coupling according - and you wouldn't be able to do the 'release a ball and see if it moves away from you' experiment because the ball would be accelerating too. In such a universe, this acceleration would be entirely invisible. In fact, recall that a Newtonian acceleration is equivalent to a global gravitational field. Seeing everything in the universe uniformly accelerating is equivalent to global gravitational field, which can always be eliminated by a coordinate transformation and is entirely equivalent to flat space. Jerk and acceleration gradients, however, are another matter.

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Re: General Relativity

Postby Goemon » Sat Aug 23, 2008 4:23 am UTC

I'm not denying that IF everything in the universe accelerates at the same uniform rate, then it's like it's not accelerating at all. We can't tell whether it is or not; the result is flat spacetime. But that's beside the point. The point is that we CAN tell whether it's spaceship A or spaceship B that REALLY IS accelerating with respect to "the universe."

Sorry if I'm quibbling, but I've had lots of arguments with people who claim that the twin paradox implies neither twin can be younger than the other because the situation is perfectly symmetrical. Each twin can claim the other dashed off and came back; there's no difference whatsoever between the twins. To which I always answer acceleration isn't relative! Whether it's Mach's principal that causes it or not, the simple fact is that we know for certain when somebody or something accelerates; there's nothing relative about it.

So I don't think we're really disagreeing about anything - I just don't want anyone to get the impression that "acceleration is relative" in the same way velocity is relative.
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Re: General Relativity

Postby hitokiriilh » Sat Aug 23, 2008 4:44 am UTC

Naaaahhh, not quibbling at all. =) You're absolutely right - we CAN tell if A or B is acc. with respect to the universe. And no, we're not disagreeing. Sorry if it appeared so. >.<

Oh, and you should punch the twin paradox people...in the face...hard...maybe with some brass knuckles. I've been in that position before and it just demonstrates such a lack of understanding of SR. It's pretty easy to solve the paradox just by drawing world lines and talking about simultaneity.

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Re: General Relativity

Postby gmalivuk » Sat Aug 23, 2008 3:35 pm UTC

In addition to the fact that a whole bunch of threads already discuss relativity, there was already one with the exact same topic title, with which this newer thread has now been merged.

Seriously, people, if you're going to post about something like relativity, please use the handy search function.
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