<aside> 🧾

MC-286 · E0 analytic derivation + E1 symbolic witness

Certificate SHA-256: 392403dba17bad14afa4ff04890baef21b110e08b37bfdbc128d1e8e3300e73d

</aside>

Theorem

Let ((M,g)) be a four-dimensional Lorentzian spacetime, (psi) matter fields, and (C) a real scalar continuity field. Declare

$$ S[g,C,\psi]=\int_M \sqrt{-g}\left[\frac{R-2\Lambda}{16\pi G}+\mathcal L_C+\mathcal L_m\right]d^4x $$

with

$$ \mathcal L_C=-\frac12\nabla_\mu C\nabla^\mu C-V(C). $$

Then stationary variation gives

$$ G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G\left(T_{\mu\nu}^{(m)}+T_{\mu\nu}^{(C)}\right), $$

$$ T_{\mu\nu}^{(C)}=\nabla_\mu C\nabla_\nu C-g_{\mu\nu}\left[\frac12\nabla_\alpha C\nabla^\alpha C+V(C)\right], $$

and

$$ \Box_g C-V'(C)=0. $$

The contracted Bianchi identity implies total covariant conservation. Directly,

$$ \nabla^\mu T_{\mu\nu}^{(C)}=\left(\Box_g C-V'(C)\right)\nabla_\nu C, $$

so the scalar sector is conserved on shell.

Proof

1. Vary the Einstein-Hilbert plus cosmological term. Up to the standard boundary contribution,

$$ \delta\left[\sqrt{-g}(R-2\Lambda)\right]=\sqrt{-g}(G_{\mu\nu}+\Lambda g_{\mu\nu})\delta g^{\mu\nu}+\text{boundary}. $$

1. Use (deltasqrt{-g}=-(1/2)sqrt{-g}g_{munu}delta g^{munu}) and vary (g^{alphabeta}nabla_alpha Cnabla_beta C). This yields the stated (T_{munu}^{(C)}). Adding the matter variation and setting (delta S/delta g^{munu}=0) gives the Einstein equation with total stress-energy.
2. Vary (C) with compactly supported (delta C):

$$ \delta S_C=\int\sqrt{-g}\left[-\nabla^\mu C\nabla_\mu(\delta C)-V'(C)\delta C\right]d^4x. $$