Stress Energy Components: 3 or 4 Densities?

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SUMMARY

The discussion centers on the representation of the stress-energy tensor, specifically whether it should be expressed as three densities or four densities. The consensus is that the stress-energy tensor is correctly represented as four-densities, contradicting common approximations that use three densities. The participants emphasize the importance of maintaining time as an explicit variable in integrals to accurately capture momentum, reinforcing the necessity of proper sourcing in scientific discourse.

PREREQUISITES
  • Understanding of the stress-energy tensor in general relativity
  • Familiarity with the concept of densities in physics
  • Knowledge of integral calculus, particularly in the context of multi-dimensional integrals
  • Basic grasp of the relationship between space-time and momentum
NEXT STEPS
  • Research the mathematical formulation of the stress-energy tensor in general relativity
  • Study the implications of four-dimensional densities in physics
  • Explore the role of time in integrals related to momentum
  • Review academic sources on the spin tensor and its applications
USEFUL FOR

Physicists, students of general relativity, and researchers interested in the mathematical foundations of energy-momentum relations will benefit from this discussion.

Ryan Rankin
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I usually come across expressions for the Stress energy tensor showing them as three densities (normally over a space-like slice.
In wikipedia's article on the spin tensor, they clearly (it does not appear to be an error) both write about and express the components of the stress energy tensor as four-densities. Is the three density just an approximation then?
For example:
06db8c0990a297671e457be2a9e4ba13.png

I usually come across this as an integral over three-space.
 
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Wikipedia is wrong. In fact the statement itself makes no sense whatsoever. You cannot get momentum at time t if you have integrated t out!

Also, please note that it is customary (and required!) to link to your sources. Others should not have to go through the process of finding something you talk about to verify what you have said.
 

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