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General Relativity
By Brad Rodgers (P1930) on Wednesday, May 17, 2000 - 02:52 am:

I have pondered this for some time - what precisely is the reasoning behind the fact that distance enlarges around a gravitational field. I have read several books on this, and none of them seem to give a very clear answer. If it is caused by gravitational pull, then why wouldn't the distance contract for an infalling object? And, if this is the case, wouldn't a light beam be pushed away from a body of mass when it approaches it from an angle?

Thanks,

Brad

By Sean Hartnoll (Sah40) on Wednesday, May 17, 2000 - 10:11 am:

okay, several points. Firstly note that 'distance' actually means spacetime distance, not spatial distance, this is once sense in which the 'rubber sheet' analogy can be misleading.

My immediate reaction is just to say that stretching how spacetime is just what happens, it comes out of the equations. I'll try to give an explanantion:

Your question is basically why does an object cause a dip as opposed to a bump in spacetime (I assuming you accept that particles and light travel on geodesics, or the shortest possible paths, so if spacetime dips, then the particle will tend to move towards the oject causing the dip, just draw a picture). In a sense this is obvious, gravity attracts objects, it doesn't repel them. So any theory of gravity must have objects moving towards the massive object.

Spacetime contration/expansion is not caused by gravitation pull, it IS ITSELF the source of gravitational pull. What GR is saying is that gravitational pull is due to the deformation of spacetime and that geodesics are no longer straight lines.

So you can ask: why does mass cause spacetime to deform in a certain way, but this is like asking, why does gravity exist at all and why does it do what it does? This is sort of question that nobody knows the answer to, and it's not clear we ever will.

Sean

By Tom Hardcastle (P2477) on Wednesday, May 17, 2000 - 05:07 pm:

I believe, although I can't quote anything at the moment, that simulations have been run in which the 'laws' of physics concerning gravity, electromagneticity, strong nuclear and weak nuclear have been altered slightly. In these cases, it turns out that fairly small changes in the way gravity (for example) works cause the universe to collapse within seconds of its formation, or prevent stars from forming, or do other things which would put a damper on your day.

This is sometimes used as an argument to say that gravity has to work like this in the universe that we observe, otherwise we couldn't exist to observe it. But this is philosophy and anyway, I don't think it's a very good argument.

By Sean Hartnoll (Sah40) on Wednesday, May 17, 2000 - 05:24 pm:

I guess it is a philosphical issue whether the fact that if the laws of gravity etc. were much different then the universe wouldnt exist actually counts as explaining why the universe is as it is. I would tend to think not because it only explains why there is something as opposed to something else, not why there is something as opposed to nothing. It doesn't give a mechanism for why things are as they are either.

Sean

By Brad Rodgers (P1930) on Wednesday, May 17, 2000 - 10:05 pm:

I am not sure I understand. I thought that everyone traveled through the same amount of spacetime. So,( using the equivalence principle), some one traveling in a high gravitational field would simply be moving faster than someone in a low gravitational field. So wouldn't someone moving through a high gravitational field measure the same absolute interval for the spacetime they traveled through as someone measuring from somewhere else. If you cannot explain this without using Reimanniann geometry, tensor analysis, etc., that's okay. I am starting to learn sphere geometry but I guess will just have to accept the consequences of GR until I have learned all of it.

By Sean Hartnoll (Sah40) on Wednesday, May 17, 2000 - 10:19 pm:

you're right that the absolute interval is the same measured from all frames. This is a basic fact that should never be forgotten.

What I was trying to say, sorry if it wasn't clear, is that it doesn't really make sense to talk about gravitional pull _causing_ the curvature of spacetime, it is rather that the curvature of spacetime (caused by mass if you like) is the _cause_ of gravitational pull.

Once you are happy with Calculus (i.e. have done a little bit about simple differential equations) then a book on GR that is relatively accessible and starts with special relativity first is "Introducing Einstein's Relativity" by D'Inverno. It presents things in a relatively straightforward way. I wouldn't worry about it quite yet, but once you are happy with Calculus it is good place to start.

Sean

By Brad Rodgers (P1930) on Thursday, May 18, 2000 - 01:24 am:

Is there a book that starts out asuming that you know all the calculations for Special Relativity(or at least most of them)? Or, in the book mentioned above, could I just skip a portion of it and go straight to the General Relativity? I have been able to do a majority of the calculations dealing with SR by myself with a small bit of guidance from a few books, but none of these books have really given any guidance as far as calculating GR.

By Sean Hartnoll (Sah40) on Thursday, May 18, 2000 - 11:51 am:

I think the book I mentioned is the easiest and it also has things very well explained. You could skip the SR section, although I think you would learn some things from it also because it introduces four vectors which are necessary for generalisation to GR.

By Carl Evans (P2080) on Friday, May 19, 2000 - 07:36 pm:

I read recently that the curvature of space, caused by the presence of a massive body, is able to 'deflect' a lightbeam passing nearby, thereby verifying relativity. I can understand that the light is following a curved path, and is not therefore being deflected. However, how can 'gravitation' effect say, a stationary body (with respect to a planet).....imagine placing a spaceship within the graviational field of a planet so that it does not have a velocity towards the planet at that instant. If it is not moving, and therefore not following a curved path, how can 'gravitation' accelerate the body if gravity is really curved space? Does this make sense?

By Sean Hartnoll (Sah40) on Friday, May 19, 2000 - 07:50 pm:

Hi Carl, I think I understand what you're saying I'll try to answer it.

It is true that you can locally eliminate gravity, for someone in the spaceship for example, the spaceship is at rest. So has curvature disappeared for them? Well no, because the manifestation of curvature is not whether a given point is accelerated but whether two nearby points are accelerated relative to each other. This is normally explained by considering an apple. If I have two nearby ants on an apple, and they start moving parallel to each other, ofter some time they will find that their paths have diverged, this is a manifestation of the curvature of an apple, the non-eculidean geometry that makes straight lines move apart or come together.

Now in the spaceship you were considering, the effect of curvature is to make the top of the spaceship accelerate at a different rate to the bottom of the spaceship and for the sides to accellerate inwards. These are called TIDAL forces (and are the reason why you would turn into spaguetti if you fell into a black hole) and they are the manifesation of curvature and are what you cannot get rid of. You can locally eliminate veloctiy and acceleration, but you cannot eliminate the relative acceleration of nearby points that causes tidal effects.

If this hasn't helped, say so and I'll try again.

Sean

By Carl Evans (P2080) on Saturday, May 20, 2000 - 12:47 pm:

Thanks Sean, but, as you probably expect, am I none the wiser. I imagine lots of people having problems with these concepts. However, it's really starting to bug me now. In your last entry you said that space is curved when there is relative acceleration between two bodies. In the case of a spaceship placed in a gravitational field but with no instant acceleration towards that planet, what has got relative acceleration with respect to the spaceship in order for 'gravity' to take hold? If it's the planet, why? As far as I can understand, the spaceship does not have a velocity towards or away from the planet or vice-versa.

Also, if Newton's First Law is to be obeyed, is it true to say that when someone jumps vertically from the ground and falls back to Earth, they are actually obeying the the First Law whilst following extreme space curvature i.e. 360 degrees? If I am totally wrong, don't laugh! I was just thinking of projectiles following straight line trajectories when you eliminate effects of gravity i.e. witnessing a cannon ball being fired when free falling.

Hope you understand what I don't.

Carl

By Sean Hartnoll (Sah40) on Saturday, May 20, 2000 - 02:31 pm:

okay, the answer to the first question is that if we agree that spaceship is an extended object and not just a point, what GR (and in fact Newtonian gravity also) is saying is precisely that you CANNOT eliminate acceleration throughout the spaceship at once, because it is in a gravity field.

I can prove this for you for a one dimensional Newtonian problem if you know something about Newtonian gravity:

Consider a point at x and a point at x+e, where e is small (i.e. two nearby points). Suppose they are in a gravitational field, with potential phi. (so the force due to gravity is -dphi/dx, I hope you are familiar with this idea of potential), so we have

d2x/dt2 = -dphi(x)/dx
d2(x+e)/dt2 = -dphi(x+e)/dx

Now use Taylor's theorem:

f(x+e) = f(x) + e f(x)/dx + small terms

So the second equation is

d2(x+e)/dt2 = -dphi(x)/dx - e d2phi(x)/dx2 + small terms

Drop the small terms and substract the equations

d2e/dt2 = - e d2phi/dx2

This is called the equation of geodesic deviation. There is an anologous result in GR which is somewhat harder to prove.

It is an equation for the relative acceleration between the points x and x+e because it says what happens to the distance between then, e.

Now, d2phi/dx2 is NOT ZERO and is not zero in any frame if we are in a gravitational field. So there is always a relative acceleration between nearby points, which is what we set out to prove.

The reason the person jumping seems a bit strange is that an initial velocity is also involved, which makes the thing a bit more complicated. It's not a stupid question at all, but I don't think I can give a satisfactory answer in brief. The first law doesn't come in because there is a gravitational field and hence a force. It is true that in some sense the person jmping is going along a geodesic (i.e. as straight as possible) in spacetime, but because there is an initial velocity this seems a it strange. Actually, the best way to look at it is by considering a frame which is instantaneously going at the same speed as the jumper. In this frame, the jumper is falling at the usual acceleration,g (can you see this?).

Hope this helps,

Sean

By Carl Evans (P2080) on Tuesday, May 23, 2000 - 12:05 am:

Sean, I sort of understand the first part where you prove that you cannot eliminate acceleration (although I'm going to have to look at the maths a bit more deeply). On the second part however, why is it that Newton's First Law cannot be considered if there is a gravitational field? I was under the impression that everything will still travel indefinitely along straight line trajectories, but in the above scenario, through curved space, the Earth "gets in the way" because space is distorted towards the centre of the earth. Is there anything correct in what I am saying?

I must say that I am studying A level maths at the moment, and so I might not understand any rigorous mathematical proof. Would you recommend that I buy that Relativity book by D'Inverno? Does the book answer some of the questions I've been asking?

Cheers, Carl

By Sean Hartnoll (Sah40) on Tuesday, May 23, 2000 - 12:17 am:

Note that the argument does not prove that you can't eliminate acceleration at a point, only that you can't eliminate the relative acceleration of two nearby points. I think you should be able to manage the argument. I'll answer any questions about it.

You have the right picture in your mind about the trajectories. I wouldn't want to invoke the first law because it sounds newtonian and there are forces. If you want to replace it by "particles with no forces other than gravity travel along geodesics (lines as straight as possible)" then that's fine.

I think the book is the most straightforward presentation of GR I know. That doesn't mean it won't require some work, but you should be able to get something out of it if you read it over a summer or something.

Sean