# How does the curvature of spacetime affect space?

1. Apr 19, 2015

### RisingSun361

If the fabric of the universe is made of both space and time, and curving spacetime affects time, then I'm guessing it also affects space. I'm aware that an object shortens in length as it approaches the speed of light. But in the case of gravity, is space relative like time? Does an object on Earth have a different length than it would if placed on the Sun or a neutron star?

2. Apr 19, 2015

### phinds

It is unfortunate that you are "aware" of that because it is not true.

It IS true that length appears contracted to a remote observer but it does not change locally (that is, in its own frame of reference).

You, for example, are traveling right now as you read this at .99999c relative to a particle in the CERN accelerator. Do you feel any shorter?

3. Apr 19, 2015

### Staff: Mentor

in general, yes, it can. But you have to be careful about what that means. See below.

If you mean, can we split up spacetime into space and time in different ways in curved spacetime as well as in flat spacetime, yes, we can.

The problem here is defining what "different length" means. In flat spacetime, you can compare the lengths of objects at different spatial locations. But in curved spacetime, you can't do that; more precisely, you can't do it in a unique way. So if I have two objects of identical construction, one on Earth and one on a neutron star, there is no unique way of comparing their lengths to see if they are "the same". The only comparison we have is a comparison of local measurements at each location: for example, taking two rulers of identical construction and putting one on Earth next to the first object, and the other on the neutron star next to the second object. If both of these measurements give the same results (which they will if the objects are of identical construction), then the objects have the same length.

What this illustrates is that curvature of space does not show up as objects having different lengths. It shows up as the large-scale geometry of space being non-Euclidean. For example, if we form a triangle in space around the Earth by having three rockets "hover" in space above the Earth and measuring the distance between them, the sum of the angles of this triangle won't be exactly 180 degrees. And if we did the same experiment around a neutron star, placing the rockets so as to make the side lengths of the triangle the same as when we did it around the Earth, we would find the sum of the angles deviating from 180 degrees by a larger amount, indicating that the neutron star, with its larger mass, produces more space curvature.

4. Apr 19, 2015

### pervect

Staff Emeritus
Curved space-time can be expressed as a rather complex mathematical object known as the Riemann curvature tensor. Choosing an observer (or more formally, choosing the basis) one might described a specific instance of curved space-time as being 'curved space" according to one observer (the time components of the tensor are zero), or "curved time" by another observer (the space components of the tensor are zero). So curved space-time could be either.

Curved space can be described as the sum of the angles of a triangle being other than 180 degrees. This particular definition doesn't generalize well to curved space-time. Unfortunately I'm not aware of any definition of curved space-time that doesnt involve the ideas of "parallel transport", and I don't want to try to get into those ideas in detail. I will say that it's basically equivalent to say that the sum of the angles of a triangle not being 180 degrees is equivalent to the idea that if you parallel-transport a vector around a closed curve (like a triangle), the vector always comes back unrotated if there is no curvature, while it will in general come back rotated if curvature is present.

The Lorentz contraction happens in what we call "flat space time", the space-time of special relativity -a space-time that is not curved. But there are some important lessons that can be learned from SR that are an aid to understanding GR and curvature, even though SR itself won't tell you much about curvature (being "flat"). The way I would describe Lorentz contraction is that length just isn't a fundamental, observer-independent quantity. Length depends on the observer. It turns out that there is a fundamental observer independent quantity, which is known as the "Lorentz interval". There's a rather simple formula - the square of the Lorentz interval is equal to the difference of the square of the distance, minus the square of the time difference. Trying to describe the physics in observer-dependent terms like length quickly gets very confusing. The more abstract approach, of describing physics in terms of observer independent quantities, is more productive and clearer in the long run, though it is initially challnging becuase the fundamental ideas are abstractions, rather than the familiar (but observer-dependent) ideas of "length" and "time".

5. Apr 19, 2015

### Ibix

I think it is correct to say that, if one were to drill a hole through the Earth and measure its "depth", D, and the Earth's circumference, C, then $C\neq\pi D$. But, if you take your tape measure into deep space and construct a circle with it then $C'=\pi D'$. However there is no non-arbitrary sense in which we can say that it was C, D, or a bit of both that changed. And it has to be a fairly specific set of choices of measurements for it to be possible even to get that much.

6. Apr 19, 2015

### Ibix

A better way to put that:

You drill a hole through the Earth and fit a rod to it, then build a ring around the equator. Transport rod and ring to deep space and you will find that the rod does not fit the ring, even when you account for enginering issues like thermal expansion. There's no way to say that it was the ring or the rod or both that changed. And, in fact, we don't think either changed (the rod will have length D and the ring will have circumference C before and after the move). Instead we think that the "shape" of space-time is different near massive objects.

I think that's better...

7. Apr 19, 2015

### A.T.

You are still confused about remote/local vs. different reference frames. The length of an object is contracted even locally, in a frame where the object moves.

8. Apr 19, 2015

### A.T.

9. Apr 19, 2015

### phinds

Hm ... why am I not correct in saying that if that were true then all objects would have an infinite number of different lengths since there are an infinite number of reference frames in which each object is moving at different speeds for each reference frame. I DO remember probably having this conversation before and you are right in that if the foregoing is wrong, then I AM still confused. Thanks for your patience in helping me w/ this.

10. Apr 19, 2015

### A.T.

Even in Newtonian mechanics each object can have an infinite number of different possible positions and velocities in all those infinitely many reference frames. Relativity adds length to those frame dependent quantities.

The problem is your use of the words "remote" and "local". Usually they refer to "distant" and "close". But you use them as "in relative motion" and "at relative rest".

11. Apr 19, 2015

### phinds

Well of course, but the length of the object itself never changes. I thought that's what we were talking about and that was my point.

Yes, that's what I do and I see your point. I guess that's sloppy terminology. I'll try to remember to change my use. I did indeed mean that from the frame of reference in which the object is in motion, it appears to have a different length than what it has in its own frame of reference, in which it is not in motion.

Thanks

12. Apr 19, 2015

### A.T.

That "appears" is also a very unfortunate word choice. It simply has a different length in different frames.

13. Apr 19, 2015

### wabbit

It does have a proper length though (if that expression is unauthorized, rest length then or whatever is the correct name), the one measured in any frame in which it is at rest. This distinguishes it from its length in non-rest frames, so even if apparent length isnt authorized terminology I find it understandable - and quite natural.

14. Apr 19, 2015

### phinds

You're right of course, I just have a hard time expressing it that way because I always think it implies that the proper length has changed (even though when I stop to think about it, I know that's not what is meant).