south said:
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.
