# Spacetime curvature

1. Jun 19, 2006

### Leonardo Sidis

I am reading The Universe in a Nutshell by Stephen Hawking, a book everyone here has probably read, or at least heard of. In the first chapter, he talks about how the effects of acceleration and gravity prove the curvature of spacetime. I'll give you some exact quotes from the book to make it a little easier:

"Someone inside a closed box, such as an elevator, could not tell whether the box was at rest in the Earth's gravitational field or was being accelerated by a rocket in free space."

This is an easy concept to grasp, but then he goes on to say:

"This equivalence between acceleration and gravity didn't seem to work for a round Earth, however--people on the opposite sides of the world would have to be accelerating in opposite directions but staying at a constant distance from each other. ...Einstein had the brain wave of realizing that the equivalence would work if the geometry of spacetime was curved and not flat, as had been assumed hitherto."

What I don't understand is this: We are not in a closed box, and we know that gravity is responsible for keeping us on Earth, for the whole surface of the Earth is feeling the same effect more or less (this would obviously not be the case if acceleration was the only thing responsible). I don't see the problem with the equivalence of these two things for a round Earth that Hawking is talking about. Is he saying that curved spacetime would allow the whole surface of a spherical object to accelerate outward while the diameter of the sphere remains unchanged? How could this be?

I hope I have explained myself alright. Thanks for any help.

2. Jun 21, 2006

### meemoe_uk

haven't you tied yourself in a logical knot?
First you say...
Yep
Right, because if acel was the force keeping people on the northen hemisphere on the ground, it would also mean people on the southern hemisphere would fall into the sky.
So you understand it.

But then you say...
> I don't see the problem with the equivalence of these two things [ acel and gravity ]

So then you don't understand it?!?
It's ultra simple, you weren't sposed to get confused until later on in the book.

3. Jun 21, 2006

### Garth

That's always the way!

Maybe the confusing thing about the equivalence principle is the concept that a brick sitting on the ground is actually accelerating upwards at g relative to the inertial frame.

There's obviously a force involved - the reaction to the weight of the brick as a support by the ground - but what about the kinetic energy? Is that to be measured relative to the inertial frame or to the ground?

Garth

Last edited: Jun 21, 2006
4. Jun 21, 2006

### Leonardo Sidis

Thanks for the responses! I see what both of you are saying, but my problem is that I understand that acceleration and gravity have similar effects on matter, but don't understand why acceleration also has to be affecting an object just because gravity is affecting it. In the example of the Earth, what I was saying was, gravity is responsible, but why does acceleration have to be in effect also? The way I see it is they are two seperate forces with very similar effects on matter, but just because one of them is acting on an object doesn't necessarily mean the other is as well.

Maybe this is where I am wrong? Does the fact that they are equivalent mean that when either acceleration or gravity is acting on an object, there must be an explanation for the other to be acting as well? Why is this?

So if my assumption of how I misunderstand it is correct, then how would the curvature of spacetime allow for the whole surface of the earth to be constantly accelerating outward without any change in its diameter?

Sorry, maybe I am having an anti-brain wave

Last edited: Jun 21, 2006
5. Jun 21, 2006

### Garth

There are not two forces acting on the 'brick', just the one gravitational force.

Newton interpreted this force as a real force, Einstein re-interpreted this same force and explained it as an inertial force, that is one caused by accelerating the brick relative to the freely falling frame.

Why should the inertial freely falling frame actually 'fall'? Einstein explained this as the effect of space-time being curved by the presence of the Earth's mass.

Draw a straight line from point A to point B on a piece of rubber sheet. Now distort the sheet, the line bends with it, not because it is intrinsically 'bent', because it will still be the shortest distance from A to B along the surface of the sheet, but because the sheet itself is bent.

In curved space-time two 'straight lines', geodesics' are curved or deflected towards each other. The geodesics of the brick and the centre of the Earth converge, or would do if the ground did not get in the way of the brick. The ground imparts a force on the brick, which it feels as its weight, and the brick is accelerated from the freely falling frame to that of the ground, and it sits stationary in the ground's frame of reference.

Garth

Last edited: Jun 21, 2006
6. Jun 21, 2006

### Leonardo Sidis

That helps a lot, thanks! Does Hawking assume that everyone will be able to instantly understand this? Because it doesn't seem real obvious to me the way he explained it.

7. Jun 22, 2006

### pervect

Staff Emeritus
Try another author. Whether it is because such a wide audience reads his book, the clunky system he has to use to write at all, or some other reason, Hawking has a bit of a reptuation for leaving confused readers in his wake.

8. Jun 23, 2006

### pmb_phy

Hawking is using a definition of the term "gravitational field" which is inconsistent with how Einstein defined it. Being inside a closed box means that you're in a region of spacetime in which tidal forces cannot be detected by the instruments you have in your possesion. Change the instruments to better ones and you can then measure the tidal forces.
That is not quite right. If rocket was accelerating relative to an inertial frame of reference in flat spacetime then there would be a gravitational field present. However this gravitational field is unlike that of Earth's since the tidal forces around Earth the can, in principle, be detected. In such cases the equivalence principle speaks only of small regions of spacetime where your instruments can't detect the tidal forces but for which the rockets frame has a gravitational field present.

Pete

9. Jun 23, 2006

### EvilEyeMonster

Would it be safe to assume that if you removed the Earth from the brick's path, that the brick will continue to accelerate toward the next nearest mass?

Now I'm confused. But I picture the brick "trying" to accelerate toward the center of the Earth's mass (and the earth being very slightly pulled toward the brick as well), with only the fact that the electromagnetic forces keeping them from passing through one another holding them in what appears to be a "still" state.

10. Jun 23, 2006

### Garth

The brick, tautologically, would be stationary in its own frame of reference. It would have a 4D world-line that would be a geodesic ("straight line") of the local space-time metric.

For example, if the Earth suddenly disappeared without affecting the brick the latter would continue in the former Earth's orbit around the Sun; in the Sun's frame of reference the brick would be appear to accelerate towards it.
Correct - it is the electro-magnetic forces in the atoms of the floor you are standing on, and in the atoms of your feet, that prevent you from falling through it. Otherwise your freely falling geodesic would converge onto that of the Earth's centre.

Garth

Last edited: Jun 23, 2006
11. Jul 20, 2006

### MeJennifer

Netwon thought that the apple accelerated towards his head because the Earth pulled it, Einstein thought that his head accelerated towards the apple because the earth pushed him!

12. Jul 20, 2006

### pmb_phy

Einstein didn't say that his head accelerated towards the apple. He said that one cannot distinguish between the falling apple and the accelerating head.

Pete

13. Jul 20, 2006

### MeJennifer

Well if he had said that it seems he would be wrong.

It seems to me that there is a clear difference between the frames, while the so called "falling" apple is in the inertial frame the so called "steady" Newton is in the accelerating frame.
So it appears that from a space-time perspective there is a real difference, Newton moves while the apple does not.
If I understand it correctly (and please correct me if I am wrong) the apple's worldline simply converges to the worldline that represents the center of the earth's gravity.

Last edited: Jul 21, 2006
14. Jul 21, 2006

### pmb_phy

Actually Einstein said the opposite, i.e. that the head is in a g-field even when the frame is one that is accelerating with respect to an inertial frame in flat spacetime. For that reason Einstein said that a g-field can be "produced" by a change in spacetime coordinates. So transforming to an accelerating frame 'produces' a gravitational field. In any case it is impossible to distinguish a uniform g-field and an uniformly accelerating frame, i.e. The statement Newton moves while the apple does not. can never be proved. Hence it can't be wrong. Of course you can disagree with Einstein if you'd like.
Yes. That is true.

Pete