Einstein Tensor and Stress Energy Tensor of Scalar Field

[Phinrich]

TL;DR Summary

What is the connection, if any, between the Einstein Tensor and the Stress-Energy Tensor of a Scalar Field ?

Hi All.

Given that we may write

And that the Stress-Energy Tensor of a Scalar Field may be written as;

These two Equations seem to have a similar form.

Is this what would be expected or is it just coincidence?

Thanks in advance

 

[Phinrich]

OK I think this is self evident. All we are saying is that the right side of the first Equation = the right side of the second equation. Nothing sinister.

Give me the prize for Raspberry of the Month

 

[Phinrich]

Thats not even the Einstein Tensor that's the Ricci Tensor.

 

[PeterDonis]

Summary: What is the connection, if any, between the Einstein Tensor and the Stress-Energy Tensor of a Scalar Field ?

The fact that the Einstein tensor equals the Stress-Energy tensor (times a constant that depends on your choice of units) according to the Einstein Field Equation. (This is true for any stress-energy tensor, not just the stress-energy tensor of a scalar field.)

However, as has been pointed out, $R_{\mu \nu}$ is not the Einstein tensor, it's the Ricci tensor. The Einstein tensor is

$$
G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R^\alpha{}_\alpha
$$

And the Einstein Field Equation is just $G_{\mu \nu} = 8 \pi T_{\mu \nu}$ (in natural units where $G = c = 1$). The equation in the Ricci tensor that you derived is obtained by "trace reversing" the Einstein Field Equation--taking the trace of the EFE to obtain $- R^\alpha{}_\alpha = 8 \pi T^\alpha{}_\alpha$ and then rearranging terms to put the trace on the RHS.

The specific form of $T_{\mu \nu}$ depends on the kind of matter and energy that are present.

the Stress-Energy Tensor of a Scalar Field may be written as

Where are you getting that from? It doesn't look like the SET of a scalar field that I'm familiar with.

 

[Phinrich]

http://www.math.uct.ac.za/sites/def...rs/Prof_George_Ellis/Overview/GR Lectures.pdf
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