Curvature-squared counterterm variation tensors

Let $(M,g)$ be a smooth Lorentzian four-manifold of signature $(-,+,+,+)$. Use its unique torsion-free, metric-compatible connection $\mathrm{\nabla }$ and the curvature convention $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$. Define $R^a{}_{bcd}$ in each chart by $R({\mathrm{\partial }}_c,{\mathrm{\partial }}_d){\mathrm{\partial }}_b=R^a{}_{bcd}{\mathrm{\partial }}_a$, and sum repeated indices; $g^{ab}$ denotes the inverse matrix to $g_{ab}$. Then set $R_{abcd}=g_{ae}{R}^e{}_{bcd}$, ${\mathrm{Ric}}_{bd}=g^{ac}{R}_{abcd}$, $S=g^{bd}{\mathrm{Ric}}_{bd}$, and $G_{ab}={\mathrm{Ric}}_{ab}-\frac{1}{2}S{g}_{ab}$ using this convention. In a chart whose coordinates have units of length, use the positive metric volume density $d{\mathrm{v}\mathrm{o}\mathrm{l}}_g=\mathrm{\mid }\det (g_{ab}){\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\mid }d{x}^{0}d{x}^{1}d{x}^{2}d{x}^3\mathrm{\mid }\mathrm{.}$

A compactly supported inverse-metric variation is a smooth family of smooth Lorentzian metrics $g_{ϵ}$ for $ϵ$ in an open interval about $0$, with $g_0=g$, for which $h^{ab}:={{\mathrm{\partial }}_{ϵ}{g}_{ϵ}^{ab}\mid }_{ϵ=0}$ is a compactly supported smooth symmetric contravariant two-tensor. Every such $h^{ab}$ is admissible for a sufficiently small parameter interval because Lorentzian signature is open and $h^{ab}$ has compact support. For a smooth metric $q$, write the curvature-squared densities as ${\mathcal{L}}_{R^2}(q):=S[q{]}^{2}d{\mathrm{v}\mathrm{o}\mathrm{l}}_q$ and ${\mathcal{L}}_{{\mathrm{Ric}}^2}(q):={\mathrm{Ric}}_{cd}[q]{\mathrm{Ric}}^{cd}[q]d{\mathrm{v}\mathrm{o}\mathrm{l}}_q$. Define the first variations by integrating their pointwise density derivatives at $ϵ=0$, not by subtracting possibly infinite total actions: $\delta I_{R^2}[g;h]:={\int }_M{{\mathrm{\partial }}_{ϵ}{\mathcal{L}}_{R^2}(g_{ϵ})\mid }_0,\delta I_{{\mathrm{Ric}}^2}[g;h]:={\int }_M{{\mathrm{\partial }}_{ϵ}{\mathcal{L}}_{{\mathrm{Ric}}^2}(g_{ϵ})\mid }_0\mathrm{.}$ These derivatives are smooth compactly supported densities determined only by $g$ and $h$, even if $g_{ϵ}-g$ is not compactly supported for nonzero $ϵ$. Indeed, in a chart each density coefficient is a smooth function of the inverse metric and its coordinate derivatives through order two on the open set of nondegenerate metrics. The parameter chain rule therefore makes its derivative at zero a finite linear combination, with smooth background coefficients, of $h^{ab}$ and its coordinate derivatives through order two. Every such term vanishes outside $\mathrm{supp}h$. Smooth joint dependence allows the parameter and coordinate derivatives to commute, so no higher-order parameter terms of the family enter. This proves both compact support and independence of the chosen family with the same first inverse-metric derivative; integration of these compact densities is finite. In particular $h=0$ gives zero first variations. No finiteness of the actions at nonzero parameter, or compact support of their differences, is assumed or concluded.

For a particular model instance, supply smooth symmetric covariant two-tensors $H_{ab}^{(R^2)}$ and $H_{ab}^{({\mathrm{Ric}}^2)}$ that satisfy, for every compactly supported inverse-metric variation, $\delta I_{R^2}[g;h]={\int }_M{{\mathrm{\partial }}_{ϵ}{\mathcal{L}}_{R^2}(g_{ϵ})\mid }_0={\int }_M{H}_{ab}^{(R^2)}{h}^{ab}d{\mathrm{v}\mathrm{o}\mathrm{l}}_g,$ $\delta I_{{\mathrm{Ric}}^2}[g;h]={\int }_M{{\mathrm{\partial }}_{ϵ}{\mathcal{L}}_{{\mathrm{Ric}}^2}(g_{ϵ})\mid }_0={\int }_M{H}_{ab}^{({\mathrm{Ric}}^2)}{h}^{ab}d{\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$ The displayed positive signs, inverse-metric variation, and curvature convention fix the counterterm sign and normalization. In length coordinates both tensors have units ${\mathrm{m}}^{-4}$. This Definition specifies the selected tensors as model inputs and does not assert their existence for every possible background or any solution of a gravitational field equation.

The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

Covariant field families and variations

The time orientation in The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors can be supplied locally; no global time-orientability assumption is needed here. At each point choose a timelike tangent vector and extend its chart components constantly near that point. Smoothness of $g$ makes this vector field timelike on a sufficiently small open neighborhood, where its cone selects a time orientation. Apply the supplier on each such neighborhood. Its unique torsion-free, metric-compatible connections agree on overlaps: each restricted connection satisfies those same two conditions for the restricted metric, regardless of which timelike cone was selected. They therefore define one global smooth connection. The displayed curvature operator, lowering and contractions then agree on overlaps as well, since they use only that connection and $g$, not the local cone selections. Thus the supplier's smooth tensor definitions apply to every background in this Definition, including backgrounds with no global time orientation. This local argument makes no simultaneous choice of a cone at every point.