# The factor of 1/2 in the Einstein-Hilbert action

• pellman
In summary, the factor of 1/2 in the Einstein-Hilbert action is arbitrary and does not affect the theory, but it is kept to maintain consistency with long-held conventions and to account for the factor in the Hamiltonian.

#### pellman

Why the factor of 1/2 in the Einstein-Hilbert action?

The Einstein field equation is

$$\frac{1}{\kappa}\left(R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}\right)=T_{\mu\nu}$$

where $\kappa=\frac{c^4}{8\pi G}$

This can be derived from extremizing the action

$$S=\int{(\mathcal{L}_{G}+\mathcal{L}_{matter})d^4x}$$

with respect to variation in the metric. Typically the gravitational Lagrangian is given as

$$\mathcal{L}_{G}=\frac{c^4}{16\pi G}R\sqrt{-g}=\frac{1}{2\kappa}R\sqrt{-g}$$

I have seen this 1/2kappa scaling of the Einstein-Hilbert action frequently in the literature. The matter Lagrangian must then be scaled in such a way that

$$\delta\mathcal{L}_{matter}=-\frac{1}{2}T_{\mu\nu}\delta g^{\mu\nu}$$

Any idea why the 1/2s? Why not just $\mathcal{L}_{G}=\frac{1}{\kappa}R\sqrt{-g}$ and $\delta\mathcal{L}_{matter}=-T_{\mu\nu}\delta g^{\mu\nu}$ ?

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The factor is arbitrary and doesn't affect the theory but note that any factor present in the Lagrangian also shows in the Hamiltonian. Dropping that factor of 1/2 would ultimately force you to drop the factor of 1/2 in the kinetic energy. So, to keep every thing consistent with long held conventions, that 1/2 factor is kept.

• 1 person