# Noether/Hilbert energy-momentum tensor

In chapter 4 of Carroll's book Spacetime and geometry he finds using the Hilbert action that the energy-momentum tensor for a scalar field is (see eq. (4.79)) $$T_{\mu\nu}^{\phi}=\nabla_\mu\phi\nabla_\nu\phi+g_{\mu\nu}\left(-\frac{1}{2}g^{\alpha\beta}\nabla_\alpha\phi\nabla_\beta\phi-V\right).\tag{4.79}$$ In the same chapter, he recalls Noether's theorem from which we have (see eq. (4.80)) $$T^{\mu\nu}=\frac{\delta\mathcal{L}}{\delta(\partial_\mu\phi)}\partial^\nu\phi-\eta^{\mu\nu}\mathcal{L}.\tag{4.80}$$

The issue is that when I use this second expression for the energy-momentum tensor and try to find it for a scalar field with Lagrangian density $$\mathcal{L}=-\frac{1}{2} \partial_\mu\phi\partial^\mu\phi-V,\tag{1.148}$$ with sign convention $$(-,+,+,+)$$, I got the same as $$T_{\mu\nu}^\phi$$ except for an overall minus sign. Maybe I just made a mistake but would appreciate any assistance on this, at least a confirmation that both definitions are equivalent in this case.

OP is right: Carroll's book has a sign mistake in eq. (4.80). The $$T^{00}$$-component should be bounded from below, and $$V$$ should appear with a plus sign in $$T^{00}$$. Eq. (4.79) satisfies this but not eq. (4.80).