General Tensor contraction: Trace of Energy-Momentum Tensor (Einstein metric)

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Discussion Overview

The discussion revolves around the trace of the energy-momentum tensor in the context of general relativity, specifically examining the mathematical formulation and contraction of tensors. Participants explore the implications of the energy density and pressure in the fluid approximation of the tensor.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents the energy-momentum tensor as \( T_{ij} = \rho u_i u_j - p h_{ij} \) and attempts to compute its trace, leading to a discrepancy with the expected result \( \rho - 3p \).
  • Another participant asks for clarification on the contraction of indices, specifically \( g^{ij} g_{ij} \) and \( g^{ij} u_i u_j \).
  • A response indicates that \( g^{ij} u_i u_j = 1 \), while the contraction \( g^{ij} g_{ij} \) is questioned, with suggestions that it could be \( g \) or \( 0 \).
  • Further clarification is provided that \( g_{ij} g^{ij} = \delta_{ii} = 4 \), prompting a discussion about the dimensionality of tensors and the notation used.
  • Participants express uncertainty about the dimensionality of tensors, with one noting that tensors can exist in different dimensions and questioning how a specific dimension was inferred in this context.

Areas of Agreement / Disagreement

Participants show some agreement on the mathematical properties of tensor contractions, but there is uncertainty regarding the dimensionality of the tensors and the implications of the calculations presented. The discussion remains unresolved regarding the correct interpretation of the trace and the dimensionality of the tensors involved.

Contextual Notes

There are limitations in the assumptions made about the dimensionality of the space and the definitions of the tensors involved. The discussion reflects varying levels of familiarity with tensor calculus and general relativity concepts.

tetris11
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Okay so I have:

Eqn1) Tij=\rhouiuj-phij = \rhouiuj-p(gij-uiuj)

Where Tij is the energy-momentum tensor, being approximated as a fluid with \rho as the energy density and p as the pressure in the medium.


My problem:
Eqn2) Trace(T) = Tii = gijTij = \rho-3p

My attempt:

Tr(T) = Tii = gij[\rhouiuj-p(gij-uiuj)]
= [\rhogijuiuj-pgijgij+pgijuiuj)]
= \rhou - p + pu

which doesn't equal rho-3p (eqn2) as required, so I've done something wrong.
I think I've contracted incorrectly but I don't know why... please help?
 
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What are the following?

g^{i j} g_{i j} = ?

g^{i j} u_i u_j = ?
 
Well,
g^{i j} u_i u_j = 1

g^{i j} g_{i j} = ??
uh... g? or 0?

Might need to help me out here, maths isn't my first language...
 
Cheers man, that actually makes complete sense - but just for the record:

gij gij = δii = n, where n is number of dimensions?

I'm just wondering how you knew it was four without knowing how dimensions it was.
Tensors aren't all 4-d, right?
 
tetris11 said:
Cheers man, that actually makes complete sense - but just for the record:

gij gij = δii = n, where n is number of dimensions?

I'm just wondering how you knew it was four without knowing how dimensions it was.
Tensors aren't all 4-d, right?

Well this is the 'Special & General Relativity' board so it was probably just a good guess?
 

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