Stress energy tensor for fields

[pervect]

In the case of swarms of particles, the stress energy tensor can be derived by considering the flow of energy and momentum "carried" by the particles along their worldlines.

Is there a way to interpret the field definition of the stress energy tensor from Wald, p455 E.1.26

$$T_{ab} \propto \frac{\delta S_M}{\delta g^{ab}}$$
Where $S_M$ is the action of the "matter" field.

as being due to the "flow" of momentum? If so, how exactly?

 

[samalkhaiat]

I am not sure what exactly your question? Do you want to "derive" the Hilbert definition for T from some physical principle? My answer in this case is No, no such derivation exist. It is one of most annoying definitions in theoretical physics.
Or, Do you want to interpret the field SET in terms of some physical processes?

 

[Bill_K]

It is one of most annoying definitions in theoretical physics.

T^μν = 2 δS/δg_μνexpresses concisely the fact that T^μν is the source of the gravitational field, and automatically yields a stress-energy tensor that is symmetric and conserved.

Do you find its electromagnetic analog J^μ = δS/δA_μ equally annoying?

 

[pervect]

I'm trying to build off definitions like the following.

The stress-energy tensor is symmetric and defined so that $T^{uv}$ is the flux of momentem $p^{u}$ across a surface of constant $x^{v}$

Baez has a similar approach. Schutz is referened, I don't have that textbook alas.

In the case of a gas , or a swarm of particles, for the 1space + 1 time case, this can be illustrated neatly by a space-time diagram of the particles, as per the attachment. The particles are assumed to not interact at all (no fields).

I wanted to make the definition more general though. I was hoping to say a few words about how the idea generalized from the simplistic "swarm of particles" to the more general cases, such as fields. I don't see anyway to do this at the moment, however.

I was thinking of making a small FAQ on the topic, we get enough quesitons about it.

The attached diagram should give some insight into the approach I'm taking. There are a few more diagrams, another for "flow in the x direction", and some illustrations of how you can compute the flow in an arbitrary direction (say the t' direction of a boosted observer) knowing the flow in the t and x directions.

 

[Mentz114]

If the velocities in a Lagrangian are replaced by the gradients $\frac{\partial \phi}{\partial x^\mu}$of a field $\phi$ it's easier to see the connection between the field Lagrangian and the SET than it is with the mechanical description. See Itzykson&Zuber, page 22.

 

[samalkhaiat]

Do you find its electromagnetic analog J^μ = δS/δA_μ equally annoying?

No, I don't because I can derive it (not just define it) within the canonical formalism. This is

exactly the reason why Hilbert definition of SET is annoying: There is no room in the

canonical formalism which allows you to derive the expression
$$\frac{ \delta S_{ m } }{ \delta g^{ \mu \nu } } = \frac{ 1 }{ 2 } \sqrt{ - g } \ T_{ \mu \nu }.$$

If you know how to derive it, I would love to see how.