1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Energy-momentum tensor: metric tensor or kronecker tensor appearing?

  1. Jul 2, 2011 #1
    Hi

    This might be a stupid question, so I hope you are patient with me. When I look for the definition of the energy-momentum tensor in terms of the Lagrangian density, I find two different (?) definitions:
    [tex]{T^\mu}_\nu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\partial_\nu \phi - {\delta^\mu}_\nu \mathcal{L}[/tex]
    [tex]{T^\mu}_\nu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\partial_\nu \phi - {g^\mu}_\nu \mathcal{L}[/tex]
    Which of the two is correct? Is this somehow a matter of convention or something like that? I have seen both more than once.
     
  2. jcsd
  3. Jul 2, 2011 #2
  4. Jul 3, 2011 #3
    Thanks, this makes sense. Am I right to say that
    [tex]T^{\mu\nu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\partial^\nu \phi - \delta^{\mu\nu} \mathcal{L}[/tex]
    and
    [tex]T^{\mu\nu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\partial^\nu \phi - g^{\mu\nu} \mathcal{L}[/tex]
    are the same only if one makes the convention that [tex]{\delta^\mu}_\nu[/tex] is no longer just a symbol for the kronecker delta but a tensor, namely [tex]{\delta^a}_b = g^{a c} g_{c b}[/tex] and that then, [tex]\delta^{ab} = {\delta^a}_b g^{bc} = g^{ab}[/tex] but that the two are not the same if [tex]\delta^{\mu \nu}[/tex] is understood as the kronecker delta?

    If I see things correctly, one has to look carefully if an appearing delta is just a symbol for the kronecker delta in components or really a (raised or lowered) version of the metric tensor. This is slightly confusing.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Energy-momentum tensor: metric tensor or kronecker tensor appearing?
  1. Metric Tensor of Earth (Replies: 9)

Loading...