Interpreting Stress-Energy Tensor in General Relativity

[paweld]

What is the interpretation of stress energy-tensor in general relativity.
According to General Relativity by Wald $T^{\mu\nu} u_\mu u_\nu$
is energy density (chapter 4.2) where $u_\mu$ is observer 4-veliocity.
But for isolated particle $T^{\mu\nu} = \gamma m V_\mu V_\nu$
(http://en.wikipedia.org/wiki/Stress_energy_tensor#Isolated_particle") in particle reference frame
we obtain density of energy: $m / \gamma$ and not $m$
(because $V_\mu^2=1/\gamma^2$ - it's not 4-veliocity but simpliy veliocity).

 

[Polyrhythmic]

The stress-energy tensor contains information about energy density, momentum density and flow of momentum(force). The Einstein field equations show that it is related to the curvature of spacetime.

 

[Entropia]

The stress-energy tensor in general relativity is a mathematical representation of the distribution of energy and momentum in spacetime. It describes the density and flux of energy and momentum at each point in spacetime, and how they affect the curvature of spacetime.

One interpretation of the stress-energy tensor is that it represents the sources of gravity in the universe. In Einstein's field equations, the stress-energy tensor is on the right hand side, indicating that it is the source of the curvature of spacetime. This means that the distribution of matter and energy in the universe is what determines the structure of spacetime.

Another interpretation is that the stress-energy tensor represents the local energy-momentum content of a given region of spacetime. This means that it describes the energy and momentum that are present at a specific point in spacetime, as observed by an observer with a specific 4-velocity.

The stress-energy tensor also has a special interpretation in the case of an isolated particle. In this case, it represents the energy density of the particle as observed in its own reference frame. This is because the 4-velocity of the particle, which is used to calculate the stress-energy tensor, is normalized to be 1 in its own reference frame. This interpretation is important in understanding the behavior of particles in curved spacetime, such as in the case of black holes.

In summary, the stress-energy tensor in general relativity has multiple interpretations, including representing the sources of gravity, the local energy-momentum content of a region of spacetime, and the energy density of an isolated particle. It is a fundamental concept in understanding the relationship between matter and spacetime in the theory of general relativity.