General relativity and frame of reference

[south]

TL;DR Summary

Doubts about the possibility or impossibility of paradoxes

All three questions concern general relativity.

1. Does the curvature of spacetime depend on the frame of reference?

2. Does kinetic energy exist?

3. If it exists, does it contribute to the curvature of spacetime?

 

[Dale]

1. Does the curvature of spacetime depend on the frame of reference?

No. Curvature is a tensor. Its components depend on the frame of reference, but the tensor itself is a geometric object that is independent of the frame of reference.

2. Does kinetic energy exist?

KE is a frame-variant measurable quantity. Some people insist that all frame-variant quantities do not exist. Other people insist that all measurable quantities do exist.

3. If it exists, does it contribute to the curvature of spacetime?

Yes.

 

[south]

No. Curvature is a tensor. Its components depend on the frame of reference, but the tensor itself is a geometric object that is independent of the frame of reference.

KE is a frame-variant measurable quantity. Some people insist that all frame-variant quantities do not exist. Other people insist that all measurable quantities do exist.

Yes.

Hello Dale. Thank you veruy much !

 

[south]

If I admit that kinetic energy exists in GR, I fail to understand how it is reconciled with the independence of curvature from the frame of reference.
Let me explain my concern.1. If it exists, KE contributes to the curvature.2. KE depends on the frame of reference.3. Premises 1 and 2 together seem to lead me to admit that the kinetic contribution to the curvature depends on the frame of reference. And if it is the only contribution to the curvature, it seems to lead me to admit that the curvature depends entirely on the frame of reference.These are my concerns regarding the possibility of paradoxes, which surely are not possible because I am missing something.

 

[Ibix]

You need to be very careful about terminology here.

If I hover above a star and ask "what's the curvature here and now" I will always get the same answer (ignoring planets' gravity etc). If you are plunging towards the star and ask the same question then the answer will be time varying. So clearly there is a difference in the curvature at our locations depending on kinetic energy of the star. Furthermore, on a more global scale, there's an effect similar to length contraction going on which "shortens" the gravitational field in the direction of motion.

What general relativists mean by "curvature is frame invariant" is that if you pick an event everyone will agree the curvature at that event. So as you plunge towards the star and pass me hovering, we will agree the curvature at our location at that instant - because we are at the same place at the same time. And we will agree the curvature at any other event too. But if you pick different "frames of reference" then your global descriptions of curvature can be quite different. But that difference is similar to why the Earth looks different on different maps - the Earth is always the same, it's just where you draw what on the map that differs.

The way that kinetic energy enters the maths (all else being equal) leads to the same curvature at the same events, but different ideas about which events are at the same time or in the same place. So the description of how curvature varies across spacetime looks quite different, but is actually a description of the same thing.

There's quite a lot of complexity that I am glossing over here. The key takeaway is that yes kinetic energy can be a source of gravity, but not in the same way rest mass is a source of gravity. Mathematically, this is because the source of gravity in GR is a 4×4 tensor called the stress-energy tensor, not just a single quantity like in Newtonian gravity. Adding kinetic energy changes several components of the tensor in different ways, which can have far more complicated results than a simple "strength of gravity increases", which I suspect is what you are imagining.

 

[PeroK]

If I admit that kinetic energy exists in GR, I fail to understand how it is reconciled with the independence of curvature from the frame of reference.
Let me explain my concern.1. If it exists, KE contributes to the curvature.2. KE depends on the frame of reference.3. Premises 1 and 2 together seem to lead me to admit that the kinetic contribution to the curvature depends on the frame of reference. And if it is the only contribution to the curvature, it seems to lead me to admit that the curvature depends entirely on the frame of reference.These are my concerns regarding the possibility of paradoxes, which surely are not possible because I am missing something.

One thing you are missing is precise mathematics to inform your conclusions.

 

[Dale]

I fail to understand how it is reconciled with the independence of curvature from the frame of reference

In Newtonian physics, displacement is a vector. My wife asks me and my son to displace the couch 1 m towards her. The displacement is to my right and to my son’s left. Do you understand that my wife’s $(-1,0)$ displacement vector is the same vector as my $(0,1)$ vector and my son’s $(0,-1)$ vector? Even though they have different components. As I said before:

Its components depend on the frame of reference, but the tensor itself is a geometric object that is independent of the frame of reference.

 

[PeroK]

In Newtonian physics, displacement is a vector. My wife asks me and my son to displace the couch 1 m towards her. The displacement is to my right and to my son’s left.

With your wife still sitting on it?

 

[south]

Hello Ibix. Thank you for answering my question. Kind regards.

 

[south]

Thank you very much for answering my question.

 

[pervect]

If I admit that kinetic energy exists in GR, I fail to understand how it is reconciled with the independence of curvature from the frame of reference.
Let me explain my concern.1. If it exists, KE contributes to the curvature.2. KE depends on the frame of reference.3. Premises 1 and 2 together seem to lead me to admit that the kinetic contribution to the curvature depends on the frame of reference. And if it is the only contribution to the curvature, it seems to lead me to admit that the curvature depends entirely on the frame of reference.These are my concerns regarding the possibility of paradoxes, which surely are not possible because I am missing something.

I am suspecting that you are not familiar with tensors, and that's the root of your question. A detailed discussion of tensors is an A-level topic, so unless you've had graduate level courses or read graduate level textbooks, it's hardly surprising that you wouldn't be familiar with them. Of course there is a certain amount of guesswork on my part here , as I don't know your background.

The tensor that we use in GR that includes kinetic energy (actually the energy density) as a part of one piece of it is the stress-energy tensor. But it's too complicated for me to talk about in a short post, so instead I'll talk about the restricted case of a point particle, and how we might describe a point particle with a tensor, as that should be sufficient to give you some idea as to the answer to your question.

The tensor, or geometric object, that describes a point particle in special relativity is the energy-momentum four vector, and I am hoping that talking about it might give you some insight.

If you have a point particle, in any frame of reference, it has some energy E and some momentum p, a momentum that has three components. In any given frame, it has four numbers. Assuming a typical conventions, these would be the total energy E, and the momentum in the x,y, and z directions. The four numbers themselves are not frame invariant, but if you know the four numbers in any one frame, you can find them in any other frame.

How would we do this? The mass of the particle is invariant, and in special relativity it's given by solving for
$$m^2 c^2 = E^2 - |p|^2 c^2$$

Knowing m, we can use the relativistic formulas E=\gamma m c^2 and p = \gamma m v, where v is the velocity of the particle in the chosen frame.

Wiki has an article on this, https://en.wikipedia.org/wiki/Four-momentum, though it appears to differ by a scale factor from my approach.

The point is that the above information is that while the four numbers, which we call components of the tensor, change depending on our frame of reference, if we know all four numbers in one frame of reference, we can convert them to other frames of reference. We therefore regard the set of these four numbers as a "geometric object".


Note that in special relativity, the total energy is $\gamma m c^2$, the kinetic energy would be $(\gamma - 1) m c^2$. Neither the total energy nor the kinetic energy is a tensor by itself, you need the complete collection of energy and the magnitude and direction of the momentum to have the complete representation of the point particle.

The philosophical idea here is that if we have a collection of information about an object that's organized so that we can calculate the components in any frame of reference, we can think of this collection of information / object as a single entity that "exists" regardless of the frame of reference. It's rather similar to the way the we think of vectors as 'existing" in geometry, without necessarily having to specify the x,y, and z components. The philosophical point of view is that the vector exists as an object, without regard to any particular frame of reference. - the components of the object are a way of describing it given some human conventions, but the object / vector itself exists without the frame.

 

[south]

Thank you very much pervect for expanding and detailing the answer to my question. Kind regards.

 

[PeterDonis]

1. If it exists, KE contributes to the curvature.

Yes, but...

2. KE depends on the frame of reference.

Only sort of, and not in a way that supports your reasoning.

A more precise way of stating the effect of KE on curvature is: KE in the center of mass frame of an isolated object contributes to curvature. In other words, KE contributes to curvature as one aspect of an isolated system.

Here is an example: I have an isolated object, a star, which we will idealize as being perfectly spherical and non-rotating. If I measure the star's mass externally, I will find that there is a contribution to the mass from the kinetic energy of the atoms in the star, as measured in the star's center of mass frame. Internally, inside the star, that will mean that the spacetime curvature I measure will have a contribution from the kinetic energy of the atoms, again as measured in the star's center of mass frame. In more technical language, in this frame the time-time component of the star's stress-energy tensor will have a contribution from the kinetic energy of the atoms.

Now suppose I switch to a frame in which the star's center of mass is moving. I can transform the star's stress-energy tensor, which is the source of its spacetime curvature, into the new frame, and the time-time component in this new frame will now include a contribution from the star's overall kinetic energy in this new frame. But there will also be contributions to other stress-energy tensor components from the star's momentum in this new frame, and as far as generating spacetime curvature is concerned, those other contributions will cancel out the contribution from the star's overall kinetic energy in this new frame, leaving only the original contribution from the kinetic energy of the star's atoms in its center of mass frame.

if it is the only contribution to the curvature

KE is never the only contribution to curvature. It is always associated with a system that has other forms of stress-energy as well.