The Energy-Momentum Tensor

[agostino981]

I am a bit confused here.

In the Einstein Field Equation, there is a tensor called stress-energy tensor in wikipedia and energy-momentum tensor in some books or papers which is $$T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta(\mathcal{L} \sqrt{-g})}{\delta g_{\mu\nu}}$$

Is it equivalent to the energy-momentum tensor I came across in QFT?

$$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial( \partial_\mu\phi_{a})}\partial^\nu\phi_a -g^{\mu\nu}\mathcal{L}$$

Thanks in advance.

 

[samalkhaiat]

I am a bit confused here.

In the Einstein Field Equation, there is a tensor called stress-energy tensor in wikipedia and energy-momentum tensor in some books or papers which is $$T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta(\mathcal{L} \sqrt{-g})}{\delta g_{\mu\nu}}$$

This is the general definition of the SYMMETRICAL energy-momentum tensor.

$$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial( \partial_\mu\phi_{a})}\partial^\nu\phi_a -g^{\mu\nu}\mathcal{L}$$
Thanks in advance.

This is the CANONICAL energy-momentum tensor. For scalar fields, the two are identical. For other fields they differ by a total divergence. They are equivalent in the sense that both leads to the same energy-momentum 4-vector
$$P^{ \mu } = \int d^{ 3 } x T^{ 0 \mu } ( x )$$

 

[stevendaryl]

I am a bit confused here.

In the Einstein Field Equation, there is a tensor called stress-energy tensor in wikipedia and energy-momentum tensor in some books or papers which is $$T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta(\mathcal{L} \sqrt{-g})}{\delta g_{\mu\nu}}$$

Is it equivalent to the energy-momentum tensor I came across in QFT?

$$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial( \partial_\mu\phi_{a})}\partial^\nu\phi_a -g^{\mu\nu}\mathcal{L}$$

Thanks in advance.

They are not, in general, the same. However, in QFT, the stress-energy tensor is not unique, because you can add additional terms to it that have no effect on conservation laws. There is a procedure for tweaking the canonical stress-energy tensor to get a modified tensor, the Belinfante–Rosenfeld stress–energy tensor, that (according to Wikipedia, at least) agrees with the Hilbert stress-energy tensor used by General Relativity:
http://en.wikipedia.org/wiki/Belinfante–Rosenfeld_stress–energy_tensor

 

[pervect]

Wald has a good discussion of this, and shows that the first form arises naturalliy from formulating GR as a Lagrangian theory.

 

[agostino981]

Thanks! That clears things up.