Trace of the stress energy tensor

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Discussion Overview

The discussion revolves around the properties of the stress-energy tensor in different contexts, specifically whether the trace of the stress-energy tensor is invariant across various types of matter, such as dust, fluids, and perfect fluids. It explores theoretical implications and mathematical formulations related to these tensors.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants propose that the trace of the stress-energy tensor is invariant under coordinate transformations.
  • Others argue that while the trace of any tensor is invariant under coordinate transformations, the traces of stress-energy tensors for different types of matter (e.g., electromagnetic radiation, dust, perfect fluid) are not necessarily the same.
  • It is noted that the trace of the stress-energy tensor for electromagnetic radiation is zero, while for dust it is nonzero, and for a perfect fluid, it is also nonzero but varies depending on the equation of state parameter.
  • One participant clarifies that dust is a specific case of a perfect fluid with an equation of state parameter \( w = 0 \), and that a perfect fluid's trace can take various values based on \( w \).
  • Another participant confirms that \( w \) represents pressure over density in the rest frame of the fluid.
  • A later reply specifies that "perfect fluid" refers to a perfect fluid with nonzero pressure, indicating \( w \neq 0 \).

Areas of Agreement / Disagreement

Participants express differing views on the invariance of the trace of the stress-energy tensor across different types of matter. While there is some agreement on the invariance under coordinate transformations, the specific values of the traces for different matter types remain contested.

Contextual Notes

The discussion includes assumptions about the equation of state parameter \( w \) and its implications for the trace of the stress-energy tensor, which may not be fully resolved. The relationship between different forms of matter and their respective stress-energy tensors is also explored without definitive conclusions.

dsaun777
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The stress energy tensor has many forms based on the type of matter you are describing, dust, fluid, perfect fluid... is it true that the trace of all of these matter situations is invariant?
 
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dsaun777 said:
The stress energy tensor has many forms based on the type of matter you are describing, dust, fluid, perfect fluid... is it true that the trace of all of these matter situations is invariant?
The trace of any tensor is invariant under coordinate transformations.
 
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dsaun777 said:
is it true that the trace of all of these matter situations is invariant?

As @Nugatory said, the trace of any tensor is invariant under coordinate transformations. However, the traces of the stress-energy tensors describing different kinds of stress-energy will not all be the same. For example, the trace of the stress-energy tensor describing electromagnetic radiation is zero; but the trace of the stress-energy tensor describing dust is nonzero. And the trace of the stress-energy tensor describing a perfect fluid is also nonzero, but different from that of dust.
 
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PeterDonis said:
And the trace of the stress-energy tensor describing a perfect fluid is also nonzero, but different from that of dust.
Dust is a particular case of a perfect fluid with ##w=0##. A perfect fluid has the trace ##\rho(1+3w)## of the stress-energy tensor, which can a priori take any value unless one starts introducing additional requirements on the equation of state parameter ##w##.
 
Orodruin said:
Dust is a particular case of a perfect fluid with ##w=0##. A perfect fluid has the trace ##\rho(1+3w)## of the stress-energy tensor, which can a priori take any value unless one starts introducing additional requirements on the equation of state parameter ##w##.
And w being pressure/ density?
 
dsaun777 said:
And w being pressure/ density?
Yes, in the rest frame of the fluid.
 
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Orodruin said:
Dust is a particular case of a perfect fluid with ##w=0##.

Yes, I should have been clearer, by "perfect fluid" I meant "perfect fluid with nonzero pressure", which would correspond to ##w \neq 0##.
 

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