Help with Stress Tensor: Local vs Other Observers

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Discussion Overview

The discussion revolves around the properties of the stress-energy tensor, particularly in the context of local versus non-local observers. Participants explore the implications of Lorentz transformations on the components of the tensor and the invariance of certain scalar quantities derived from it.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant describes the stress-energy tensor as T_mn = r0 U^m U^n, noting that for a local observer, only T_00 is non-zero, while for other observers, many components may be non-zero and larger than those of the local observer.
  • Another participant asserts that while the components of the tensor change under boosts, the scalar length remains unchanged, suggesting that invariants derived from the tensor do not change.
  • A participant questions how the scalar or length can remain the same if the components for other observers appear larger, proposing that the metric tensor might adjust the components to maintain invariance.
  • One participant acknowledges the non-invariance of energy under Lorentz transformations, suggesting that this affects the stress-energy tensor's behavior.
  • Another participant clarifies that tensors are covariant and their components change under coordinate transformations, which raises questions about the invariance of T_mn.
  • A participant references the electromagnetic force tensor, noting that while its components change, certain invariants remain constant, and expresses uncertainty about the invariants of the stress-energy tensor.

Areas of Agreement / Disagreement

Participants express differing views on the invariance of the stress-energy tensor under coordinate transformations, with some asserting that it is covariant while others question the implications of this on its components and invariants. The discussion remains unresolved regarding the nature of invariance and the behavior of the tensor across different observers.

Contextual Notes

Participants discuss the implications of Lorentz boosts on the stress-energy tensor and its components, highlighting the complexity of energy invariance and the role of the metric tensor in transforming components.

Jitu18
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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.
 
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If you boost a 4-vector, the vector changes, but the scalar length is unchanged.
So it is with Tmn, the components change but scalars made from it do not.
 
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.
 
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.
 
Then r u saying that T_mn will not remain invariant under cordinate transformation. If so then how come it became tensor?
 
Consider the EM force tensor Fmn. 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 ).
 
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.
 

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