Finding Stress-Energy Tensor: General Rules & Variational Principle

[davidge]

I wonder if there is a "general rule", a kind of "algorithm" for finding the components of the Stress-Energy tensor in for particular cases.
For the Einstein tensor, just by knowing the metric, one can find the components of it. What about the Stress-Energy tensor?

One way I thought of (and that is the way I have been encountering in books on GR) is using what we know about the scenario... to make approximations and then deriving the components.

Another way I thought of is by using variational principle. Would this require knowing the field that describes the matter in the situation we are considering, right? Is knowing the field too hard?

 

[haushofer]

What do you mean exactly?How about the functional derivative of the matter action wrt the field?

 

[PeterDonis]

For the Einstein tensor, just by knowing the metric, one can find the components of it. What about the Stress-Energy tensor?

If you know the Einstein tensor, you know the stress-energy tensor, since the Einstein Field Equation equates the two.

 

[davidge]

What do you mean exactly?How about the functional derivative of the matter action wrt the field?

Excuse me if I'm not fully capable of expressing myself, this is due my lack of knowledge in English.

If you know the Einstein tensor, you know the stress-energy tensor, since the Einstein Field Equation equates the two.

Ok, but how does one know it in a situation where there are no information about the metric (and thus about the Einstein tensor)?

 

[PeterDonis]

how does one know it in a situation where there are no information about the metric

You have to know something about what kind of matter, energy, fields, etc. are present. For example, if you know a perfect fluid is present, you use the stress-energy tensor for a perfect fluid. (That's how the standard FRW solutions in cosmology are derived.)

 

[sweet springs]

I wonder if there is a "general rule", a kind of "algorithm" for finding the components of the Stress-Energy tensor in for particular cases. For the Einstein tensor, just by knowing the metric, one can find the components of it. What about the Stress-Energy tensor?

How about
$$\begin{align*} T^{\mu \nu} \ =\ \rho c^2\ \left( \begin{array}{cccc} u^{0} u^{0} & u^{0} u^{1} & u^{0} u^{2} & u^{0} u^{3} \\ u^{1} u^{0} & u^{1} u^{1} & u^{1} u^{2} & u^{1} u^{3} \\ u^{2} u^{0} & u^{2} u^{1} & u^{2} u^{2} & u^{2} u^{3} \\ u^{3} u^{0} & u^{3} u^{1} & u^{3} u^{2} & u^{3} u^{3} \\ \end{array} \right) \end{align*}$$
?

 

[haushofer]

That's the EM-tensor for dust.

 

[davidge]

You have to know something about what kind of matter, energy, fields, etc. are present

As I asked in the opening post, is it hard to know the exact field equations that describe the matter?

 

[PeterDonis]

How about

As @haushofer pointed out, this is not a "general rule" for constructing a stress-energy tensor. It is a particular case: $T_{\mu \nu} = \rho u_\mu u_\nu$, which describes a perfect fluid with zero pressure, rest energy density $\rho$, and 4-velocity $u_\mu$.

 

[PeterDonis]

is it hard to know the exact field equations that describe the matter?

Depends on the kind of matter and what approximations you want to use. The most general method is to find the Lagrangian that describes the matter (or whatever you want to call it--we don't usually call electromagnetic fields "matter", for example, but we know a Lagrangian for them), and then use that to derive field equations, equations of motion, stress-energy tensor, or whatever else you need. But there is no single way to find the Lagrangian; you just have to figure it out for each individual case you're interested in.

 

[davidge]

Depends on the kind of matter and what approximations you want to use. The most general method is to find the Lagrangian that describes the matter (or whatever you want to call it--we don't usually call electromagnetic fields "matter", for example, but we know a Lagrangian for them), and then use that to derive field equations, equations of motion, stress-energy tensor, or whatever else you need. But there is no single way to find the Lagrangian; you just have to figure it out for each individual case you're interested in.

Ah, cool.