I Deriving the Lorentz formula from my own example

[PeterDonis]

isn't that unit of distance (and time) larger in a curved spacetime than in a flat one?

The concept of the unit being "larger" or "smaller" doesn't make sense. The unit is the unit.

 

[JohnnyGui]

The concept of the unit being "larger" or "smaller" doesn't make sense. The unit is the unit.

Ah of course. Can one then say that a curved spacetime metric consists of more units than a non- or less-curved one? If not, then what is it that makes an object that is moving in a curved spacetime, differ in velocity from when it's moving in a flat spacetime, to the extent that one can't talk about relative velocity in a curved spacetime?

In the mean time, I'll try and search for some info on this since I think that I'm asking too many questions about this here.

 

[PeterDonis]

Can one then say that a curved spacetime metric consists of more units than a non- or less-curved one?

No, because there is no way of making the comparison. You can't use coordinates to do it, because the fact that two events in different spacetimes happen to have the same 4-tuple of coordinate numbers assigned to them has no physical meaning. And there is no other way to do it.

what is it that makes an object that is moving in a curved spacetime, differ in velocity from when it's moving in a flat spacetime

There's no way of making this comparison either, so the question you're asking doesn't make sense.

to the extent that one can't talk about relative velocity in a curved spacetime?

The reason one can't talk about relative velocity in curved spacetime is that the spacetime is curved. That is the difference between curved and flat spacetime that makes the concept of "relative velocity" inapplicable except locally in curved spacetime. There is no other possible comparison.

As for why curvature of the spacetime is the key property here, that's probably getting too involved for a PF thread, but I'll try. Consider how we actually compare velocities between distant objects in flat spacetime: we use an inertial frame that covers the entire spacetime. But why does the inertial frame cover the entire spacetime? Because we can take a whole fleet of observers, start them out all at rest relative to each other and moving inertially, and they will stay at rest relative to each other forever. So if observer A, over here, wants to know how fast some object is moving that is just passing observer B, he can just ask observer B how fast the object is moving relative to him, and assume that the object's velocity relative to observer A himself will be the same.

But in curved spacetime, if we take two observers, start them out at rest relative to each other, and have them move inertially, they will not stay at rest relative to each other. That is because spacetime curvature is the same thing as tidal gravity, and tidal gravity causes inertially moving objects that start out at rest relative to each other to not stay at rest relative to each other. So there is no longer any invariant way, in a curved spacetime, to relate the speed that something is moving relative to observer B, to a speed relative to observer A, because observers A and B themselves can no longer form a global inertial frame the way they could in flat spacetime.

 

[JohnnyGui]

As for why curvature of the spacetime is the key property here, that's probably getting too involved for a PF thread, but I'll try. Consider how we actually compare velocities between distant objects in flat spacetime: we use an inertial frame that covers the entire spacetime. But why does the inertial frame cover the entire spacetime? Because we can take a whole fleet of observers, start them out all at rest relative to each other and moving inertially, and they will stay at rest relative to each other forever. So if observer A, over here, wants to know how fast some object is moving that is just passing observer B, he can just ask observer B how fast the object is moving relative to him, and assume that the object's velocity relative to observer A himself will be the same.

But in curved spacetime, if we take two observers, start them out at rest relative to each other, and have them move inertially, they will not stay at rest relative to each other. That is because spacetime curvature is the same thing as tidal gravity, and tidal gravity causes inertially moving objects that start out at rest relative to each other to not stay at rest relative to each other. So there is no longer any invariant way, in a curved spacetime, to relate the speed that something is moving relative to observer B, to a speed relative to observer A, because observers A and B themselves can no longer form a global inertial frame the way they could in flat spacetime.

Ah, that made me understand it. Great example. And even if B wanted to relate the speed of something moving over a long distance just for himself (not for telling A), he wouldn't be able to do this because he knows that he won't be in an inertial frame over a long distance.

Can I say that, since inertial frames over long distances isn't maintainable, that curvature is causing acceleration for any observer? (non-inertial frames is caused by acceleration)

 

[PeterDonis]

Can I say that, since inertial frames over long distances isn't maintainable, that curvature is causing acceleration for any observer?

Not as you state it, because "acceleration" is not a precise term. And if we use the standard GR definition of "acceleration", which is proper acceleration (i.e., acceleration that you feel), the statement is false; tidal gravity by itself does not cause objects to feel any acceleration, any more than the Newtonian "force" of gravity does. Any acceleration that is felt is always due to some non-gravitational interaction.

 

[JohnnyGui]

Not as you state it, because "acceleration" is not a precise term. And if we use the standard GR definition of "acceleration", which is proper acceleration (i.e., acceleration that you feel), the statement is false; tidal gravity by itself does not cause objects to feel any acceleration, any more than the Newtonian "force" of gravity does. Any acceleration that is felt is always due to some non-gravitational interaction.

Oh, I thought the curvature of the universe is caused by gravity, i.e. acceleration, and thus we should feel gravity/acceleration wherever there is curvature.

Does this mean that gravity curving spacetime is something else than the universe being curved?

 

[PeterDonis]

I thought the curvature of the universe is caused by gravity

Spacetime curvature is the same thing as tidal gravity. But "gravity" is a broader term than just tidal gravity. Also, spacetime curvature is caused by the presence of stress-energy, through the Einstein Field Equation; it is not caused by gravity in any sense.

i.e. acceleration

Gravity is not the same thing as acceleration.

we should feel gravity/acceleration wherever there is curvature.

Why would you think that? Even in the Newtonian approximation it's obviously false: objects can fall in the neighborhood of the Earth and exhibit the effects of tidal gravity without feeling any acceleration at all.

Does this mean that gravity curving spacetime is something else than the universe being curved?

See above.

I'm getting very curious as to where you are getting your ideas about relativity. Have you actually studied any textbooks, such as Taylor & Wheeler's Spacetime Physics, or Carroll's online lecture notes? Or have you only read pop science books or articles?

 

[JohnnyGui]

I'm getting very curious as to where you are getting your ideas about relativity. Have you actually studied any textbooks, such as Taylor & Wheeler's Spacetime Physics, or Carroll's online lecture notes? Or have you only read pop science books or articles?

You could see me as someone who's just interested in cosmology and relativity and who is watching lectures (Yale) and reading some introductory articles here and there along with general books. You could expect that people like me (just having an interest on the subject) are a bit prone to having some too basic (or even false) knowledge on this. For example, I've always heard from multiple sources that gravity in general is the cause of bending spacetime and that it is the same as acceleration to explain time dilation in its presence. You're the first one who denied these for me, surely because you have studied its causes in much more detail.

I see you have named a few reliable sources on this. Now that I know them I can dive into them