I have a small subtle problem with the sign of the energy-momentum tensor for a scalar field as derived by varying the metric (s.b.). I would appreciate very much if somebody could help me on my specific issue. Let me describe the problem in more detail:

I conform to the sign convention [itex]g_{\mu \nu} = (+,-,-,-)[/itex]. The Lagranagian for a real scalar field is

[tex] \mathcal{L} = \frac{1}{2} \dot{\Phi}^2- (\nabla \Phi)^2 - V(\Phi ) = \frac{1}{2} g^{\mu \nu} \partial_\mu\Phi \;\partial_\nu\Phi- V(\Phi ).[/tex]

From Noether Theorem we find the energy-momentum tensor

[tex]T^{\mu \nu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \Phi)} \: \partial^\nu \Phi - \mathcal{L} g^{\mu \nu} = \partial^\mu \Phi \partial^\nu \Phi - \mathcal{L} g^{\mu \nu}.[/tex]

Now I want to derive this via varying the action

[tex]S = \int \mathcal{L} \sqrt{-g}\; dx^4[/tex]

in respect to [itex]g_{\mu \nu}[/itex]. In particular it holds

[tex]\delta S = \delta\int \mathcal{L} \sqrt{-g}\; dx^4 = -\frac{1}{2}\int T_{\mu \nu} \delta g^{\mu\nu} \sqrt{-g}\; dx^4.[/tex]

[itex]T_{\mu \nu}[/itex] is defined so that varying the action derived from the total Lagrangian

[tex] \mathcal{L_{\rm tot}} = \frac{1}{16\pi G} R + \mathcal{L}[/tex]

yields the Einstein field equations

[tex]G_{\mu \nu} = 8\pi G T_{\mu \nu}.[/tex]

(Note that

[tex]\delta\int\frac{1}{16\pi G} R \sqrt{-g}\; dx^4 = \int G_{\mu \nu} \delta g^{\mu \nu}\sqrt{-g}\; dx^4, [/tex]

therefore the - sign in the definition of [itex]T_{\mu \nu}[/itex].)

Now let's vary the lagrangian of the scalar field:

[tex]\delta \int \mathcal{L} \sqrt{-g}\; dx^4[/tex]

[tex] = \int \delta(\mathcal{L}) \sqrt{-g} + \mathcal{L} \delta(\sqrt{-g})\; dx^4[/tex]

[tex] = \int \delta \left( \frac{1}{2} g^{\mu \nu} \partial_\mu\Phi \;\partial_\nu\Phi- V(\Phi ) \right) \sqrt{-g} + \mathcal{L} \left(-\frac{1}{2} g_{\mu \nu} \delta g^{\mu \nu}\right) \sqrt{-g}\; dx^4[/tex]

[tex] = \frac{1}{2}\int \left( \delta g^{\mu \nu} \partial_\mu\Phi \;\partial_\nu\Phi - \mathcal{L} g_{\mu \nu} \delta g^{\mu \nu} \right) \sqrt{-g}\; dx^4[/tex]

[tex] = \frac{1}{2}\int \left(\partial_\mu\Phi \;\partial_\nu\Phi - \mathcal{L} g_{\mu \nu} \right) \delta g^{\mu \nu} \sqrt{-g}\; dx^4.[/tex]

Comparing this with the definition of the [itex]T_{\mu \nu}[/itex] yields

[tex]T_{\mu \nu} = -\partial_\mu \Phi \partial_\nu \Phi + \mathcal{L} g_{\mu \nu}[/tex]

leading to the opposite sign as derived by the Noether Theorem.

I would appreciate very much if somebody could explain why I get the sign wrong. I know this is a subtle (and possibly unimportant) issue but getting the wrong sign without understanding why gives a bad feeling. Thank you for any help!