Help with Stress Tensor: Local vs Other Observers

[Jitu18]

Well lately i have in mess for this. The problem is about the stress energy tensor. Well we know that
T_mn = r0 U^m U^n
where r0 is mass density and U is proper velocity. Ok now consider the local observer. For him except for U^0 other U^m will be jero. So for local observer.
T_00 = r0 c^2
other component of this tensor will be zero. Surely for other observer almost all the component maybe nonzero. And T_00 component for other observer will be
T_00 = r0 c^2 (dt/dTou)^2
here tou is proper time. And it will be greater than T_00 of local observer. So how can the T tensor remain invariant. Sure its trace can't be the same. For other observer T_00 it self is bigger let alone the other nonzero component. Please help me here. Its bugging me a lot. If u dnt understand something about my writing than tell me.

 

[Mentz114]

If you boost a 4-vector, the vector changes, but the scalar length is unchanged.
So it is with T_mn, the components change but scalars made from it do not.

 

[Jitu18]

Yes i know it. But that's the problem. Local observer has only one nonzero component while other observer's same tensor have lot of nonzero component. some of those component may be bigger than the local observer's only componet. So will the scalar or length will be same? Other observer's T tensor's magnitude sure seems much bigger. So how will the mixed tensor's diagonal terms sum will be same. Or is it that the metric tensor will be such that when converting those contravariant tensor to mix tensor the other observer's component will be such drastically reduced so that their sum will be the same as the local observer's only component.

 

[Mentz114]

I was overlooking the fact that energy is not Lorentz invariant. If you boost T then the kinetic energy will increase and the gravitational effect should be different.

 

[Jitu18]

Then r u saying that T_mn will not remain invariant under cordinate transformation. If so then how come it became tensor?

 

[Mentz114]

Consider the EM force tensor F^mn. Under Lorentz boost the components of the tensor change, but F²=-(E²-B²) remains the same. Only scalars formed from tensors are invariant. I don't know what the invariants of T are ( or if it has any ).

 

[Ich]

Then r u saying that T_mn will not remain invariant under cordinate transformation. If so then how come it became tensor?

Tensors are covariant, not invariant. Their components change under coordinate transformations.