Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The Energy-Momentum Tensor

  1. Dec 31, 2012 #1
    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.
  2. jcsd
  3. Dec 31, 2012 #2


    User Avatar
    Science Advisor

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

    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
    [tex]P^{ \mu } = \int d^{ 3 } x T^{ 0 \mu } ( x )[/tex]
  4. Dec 31, 2012 #3


    User Avatar
    Staff Emeritus
    Science Advisor

    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:
  5. Dec 31, 2012 #4


    User Avatar
    Staff Emeritus
    Science Advisor

    Wald has a good discussion of this, and shows that the first form arises naturalliy from formulating GR as a Lagrangian theory.
  6. Jan 1, 2013 #5
    Thanks! That clears things up.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook