Proving Effects of Stress-Energy Tensor on Curvature

[VladZH]

Hi everyone. Could you help me to find the way to prove some things?
1)Changing of body velocity or reference frame don't contribute to spacetime curvature
2)On the contrary the change of body mass causes the change of curvature in local spacetime

I use the assumption that if we have the same tensor in the right part of Einstein field equation the curvature remains the same and changes othwerwise
$$R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda =
{8 \pi G \over c^4} T_{\mu \nu}$$

My suggestion is:
1) Let a body with some velocity has stress-energy tensor $T$. Then in another reference frame let the body stress-energy tensor be $T'$. As stress-energy tensor is invariant we should get the same tensor but different coordiantes when changing velocity or reference frame . Will it be the proof if I manage to find $\Lambda$ from the following equation and show that it is a linear transformation
$$T_{\mu' \nu'}'={\Lambda^{\mu}}_{\mu'} {\Lambda^{\nu}}_{\nu'} T_{\mu \nu}.$$
2) Let a body be stationary and has some mass. Its stress-energy tensor is $T$. Then we change the mass of a body and get $T'$ for its stress-energy tensor. So we should have $T \neq T'$. Can I use the previous equation here to prove this?

Can this work? Or I need to use the Riemann tensor and Richi scalar in the left part Einstein field equation?

Thank you.

 

[pervect]

Hi everyone. Could you help me to find the way to prove some things?
1)Changing of body velocity or reference frame don't contribute to spacetime curvature
2)On the contrary the change of body mass causes the change of curvature in local spacetime

I use the assumption that if we have the same tensor in the right part of Einstein field equation the curvature remains the same and changes othwerwise
$$R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda =
{8 \pi G \over c^4} T_{\mu \nu}$$

My suggestion is:
1) Let a body with some velocity has stress-energy tensor $T$. Then in another reference frame let the body stress-energy tensor be $T'$. As stress-energy tensor is invariant we should get the same tensor but different coordiantes when changing velocity or reference frame . Will it be the proof if I manage to find $\Lambda$ from the following equation and show that it is a linear transformation
$$T_{\mu' \nu'}'={\Lambda^{\mu}}_{\mu'} {\Lambda^{\nu}}_{\nu'} T_{\mu \nu}.$$

All tensors are covariant, so the Riemann tensor (which is what I assume you mean by the space-time curvature tensor) is covariant just because it's a tensor, and so by definition it's covariant. (I think you mean covariant rather than what you wrote, which is invariant).

This may or may not be what you mean when you say "Changing of body velocity or reference frame don't contribute to spacetime curvature". But I'm not quite sure what this English-language statement above means, it seems to me you need to decide for yourself if the more mathemmatical statement yous make (which are clear and unambiguous) are equivalent to the fuzzy and not-so-clear English language statements you make.

2) Let a body be stationary and has some mass. Its stress-energy tensor is $T$. Then we change the mass of a body and get $T'$ for its stress-energy tensor.

The question you need to ask yourself, and perhaps do a bit of research on, is this. Is the mass of a body a tensor? In this context, it's important to distinguish tensors from pseudotensors.

I'll give you a hint without a lengthly justification. The answer is basically "no", we don't have a "mass tensor".

 

[VladZH]

All tensors are covariant, so the Riemann tensor (which is what I assume you mean by the space-time curvature tensor) is covariant just because it's a tensor, and so by definition it's covariant. (I think you mean covariant rather than what you wrote, which is invariant).

I'm talking about covariance. I'm talking about the main property of tensors - invariance under coordinate transformation.

This may or may not be what you mean when you say "Changing of body velocity or reference frame don't contribute to spacetime curvature". But I'm not quite sure what this English-language statement above means, it seems to me you need to decide for yourself if the more mathemmatical statement yous make (which are clear and unambiguous) are equivalent to the fuzzy and not-so-clear English language statements you make.

Saying "Changing of body velocity or reference frame don't contribute to spacetime curvature" I mean the following:
Stress-energy tensor components are different momenta. They, in turn, include components of 4-vector as here
$$T^{\alpha\beta}({\bf x},t) = \gamma m v^\alpha v^\beta$$
1)The change of $v^\alpha$ leads to the change of some coordinates of the tensor $T$.
I make an assumtion here: if $T$ becomes a new tensor $T'$ then it affects the Riemann tensor in the left part of Einstein field equation. But as we know change of frame reference shouldn't change the curvature that is Riemann tensor in the left part.
2) The change of $m$ similarly affects $T$ components. So I want to show here that changing of $m$ -> changing of $T$ i. e gives us a new $T'$-> changing of $R$, i.e gives us a new $R'$

 

[VladZH]

I'm talking about covariance.

Sorry. I'm not talking about covariance

 

[Orodruin]

Sorry. I'm not talking about covariance

I'm talking about the main property of tensors - invariance under coordinate transformation.

Then you are, simply put, wrong. The main point about tensor equations is that they are covariant under coordinate transformations. If you will, tensors themselves are invariant objects without any reference to any coordinate system, but once you start looking at their components and coordinate transformations, it is all about how the components transform. And they do transform and hence are not invariant. Changing coordinate (i.e., changing reference frame) does change the components of a tensor. However, it does not change the values of invariants such as the scalar curvature.

 

[VladZH]

Then you are, simply put, wrong. The main point about tensor equations is that they are covariant under coordinate transformations. If you will, tensors themselves are invariant objects without any reference to any coordinate system, but once you start looking at their components and coordinate transformations, it is all about how the components transform. And they do transform and hence are not invariant. Changing coordinate (i.e., changing reference frame) does change the components of a tensor. However, it does not change the values of invariants such as the scalar curvature.

Ok, I see my approach is wrong. What are the approches to show that change of mass affects the curvature and change of velocity does not? How can we use Einstein field equation here?