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Question: Stress energy tensor of a minimally coupled scalar field is $T_{\mu\nu} = \partial_\mu\phi\partial_\nu\phi - \left[\frac{1}{2}(\nabla\phi)^2+V(\phi)\right]g_{\mu\nu}$.

Derive the scalar field equations of motion from the conservation law $\nabla_\mu T^{\mu\nu}=0$.

Attempt:

Before starting, I was trying to figure out what the solution should look like. We are dealing with a scalar field, so my guess was we should arrive at some relativistic differential equation for a wave propagating at the speed of light. I would expect the D'Alembertian operator to appear. Note that this is what I have deduced from reading around a bit. I could be wrong in my assumptions.

I first need to contract $T_{\mu\nu}$ to get its inverse.

$$T^{\alpha\beta} = g^{\beta\nu}g^{\alpha\mu}T_{\mu\nu}. (1)$$

$$\therefore \nabla_\alpha T^{\alpha\beta} = \nabla_\alpha (g^{\beta\nu}g^{\alpha\mu}T_{\mu\nu}) = g^{\beta\nu}g^{\alpha\mu}\nabla_{\alpha}T_{\mu\nu}. (2)$$

The first two terms of (2) are zero since by definition the covariant derivative of the metric tensor is zero.

$$\nabla_{\alpha}T_{\mu\nu} = \nabla_\alpha(\partial_\mu\phi\partial_\nu\phi) - \left[\frac{1}{2}\nabla_{\alpha}(\nabla\phi)^2+ \nabla_{\alpha}V(\phi)\right]g_{\mu\nu}. (3)$$

So we have the an equation

$$g^{\beta\nu}g^{\alpha\mu}\left(\nabla_\alpha(\partial_\mu\phi\partial_\nu\phi) - \left[\frac{1}{2}\nabla_{\alpha}(\nabla\phi)^2+ \nabla_{\alpha}V(\phi)\right]g_{\mu\nu}\right) = 0 .\space (4)$$

Am I on the right track? The D'Alembertian is defined as $\Box = g^{\mu\nu}\partial_v\partial_\mu$. I might be able to recover this if I play around a bit, then get some second order equation.

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  • $\begingroup$ Have you tried using the product rule to expand the derivatives in equation (4), to see if anything cancels? $\endgroup$ Oct 31, 2021 at 20:25
  • $\begingroup$ @bsafaria see my answer here physics.stackexchange.com/questions/512424/… $\endgroup$
    – Noone
    Oct 31, 2021 at 20:43
  • $\begingroup$ @ApolloRa I'll take a look at it, thank you. $\endgroup$
    – bsafaria
    Oct 31, 2021 at 20:51

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