We follow closely the formalism developed by Reuter and Meyer (ref), which gives a set of modified Einstein equations for a running Newton constant.  We will be concerned with how the ultraviolet behaviour involving black hole singularities can potentially be altered by a running Newton constant. To avoid confusion the following notation is utilised: $\mathbf{G}_{\mu\nu}$ denotes the Einstein tensor, $G$ is the usual Newton constant and the scalar field $\mathcal{G}(x)$ is a position dependent Newton constant.

The action for pure Einstein gravity on a spacetime manifold $\mathbb{M}^{3+1}=\mathbb{M}^{3}\times\mathbb{R}$ is

$S_{EH}[g,G]=\frac{1}{16\pi G}{\displaystyle \int_{\mathbb{M}^{3+1}}} d^{4}x\sqrt{-g}\mathbf{R}$

For gravity interacting with fluids or matter fields, one can add another term to the action to account for a source term $\mathbf{T}_{\mu\nu}$, so that

$S_{tot}=\frac{1}{16\pi G} {\displaystyle \int_{\mathbb{M}^{3+1}}}d^{4}x\sqrt{-g}\mathbf{R}+S_{M}[g]$

and

$\mathbf{T}_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta S_{M}[g]}{\delta g_{\mu\nu}}$.

Varying the total action $\delta S_{tot}=0$ gives the usual Einstein field equations

$\mathbf{G}_{\mu\nu}=\mathbf{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\mathbf{R}= 8\pi G\mathbf{T}_{\mu\nu}$

Since G is constant, it remains part of a prefactor outside the integration over the Lagrangian and thus has a static rather than a dynamical role. If $G$ is now elevated to the status of a scalar field $\mathcal{G}(x)$ then $\mathcal{G}(x)$ must now be brought under the action integral

$S_{EH}[g,G]=\frac{1}{16\pi}{\displaystyle \int_{\mathbb{M}}^{3+1}} d^{4}x \sqrt{-g}\frac{\mathbf{R}}{\mathcal{G}(x)}+S_{M}+S_{\Phi}$

where $S_{\Phi}$ is now a source for the scalar field $\mathcal{G}(x)$. This will give rise to a scalar-tensor gravity theory that modifies the pure Einstein gravity theory.

The first such theory was the Brans-Dicke scalar-tensor theory. This theory originally sought to makes Mach’s principle compatible with a modified general relativity. The Brans-Dicke scalar field $\phi(x)$ was introduced as the inverse of a position-dependent Newton constant so that $\phi(x)=\mathcal{G}^{-1}(x)$. The action for the theory is

$S_{BD}=\frac{1}{16\pi}{\displaystyle \int} d^{4}x \sqrt{-g}(\phi\mathbf{R}-\omega\phi^{-1}\partial_{\mu}\phi \partial^{\mu}\phi)~+~S_{M}$

Varying the action gives the field equations

$\mathbf{G}_{\mu\nu}=8\pi(\phi^{-1}\mathbf{T}_{\mu\nu}+\mathbf{T}_{\mu\nu}^{BD})+\phi^{-1}(\nabla_{\mu}\nabla_{\nu}\phi-\square \phi)$

and a scalar wave equation

$\square\phi(x) = \square G^{-1}(x)=\frac{8\pi}{3+2\omega}\mathbf{T}_{\mu}^{\mu}(x)$

Here, $\mathbf{T}_{\mu\nu}$ is the source tensor arising from the matter while $\mathbf{T}_{\mu\nu}^{BD}$ is the source tensor for the field $\Phi$(refs.)

$\mathbf{T}_{\mu\nu}^{BD}=\frac{\omega}{8\pi \phi}[\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\nabla_{\alpha}\phi \nabla ^{\alpha}\phi]$

The key idea in the BD theory is that the position-dependent Newton constant is governed by a simple equation of motion, the Klein-Gordon equation. In this more general theory of a position-dependent Newton constant(-),the BD theory can arise as a special limiting case.

Varying the modified action gives

$\frac{1}{16 \pi}{\displaystyle \int_{\mathbb{M}}} (-\mathbf{R}_{\mu\nu}+\frac{1}{2}$  $g_{\mu\nu}\mathbf{R}) \mathcal{G}(x))$ $+ (- \nabla_{\mu} \nabla_{\nu} + g_{\mu\nu} \square) \mathcal{G}^{-1}(x)) \delta g^{\mu \nu}$

The field equations that arise from the action (-) will have the general form

$\mathbf{R}_{\mu\nu} - \frac{1}{2}g_{\mu\nu} \mathbf{R})\mathcal{G}(x))= \mathcal{G}(x)\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\square)$  $\mathcal{G}^{-1}(x))$

or

$\mathbf{G}_{\mu\nu}= \mathbf{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\mathbf{R} = 8\pi \mathcal{G}(x)\mathbf{T}_{\mu\nu}+ \mathbf{\Psi}_{\mu\nu}+ \mathbf{\Phi}_{\mu\nu}$

or

$\mathbf{G}_{\mu\nu}=\mathbf{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\mathbf{R}=8\pi \mathcal{G}(x)(\mathbf{T}_{\mu\nu}+ \widehat{\mathbf{\Psi}}_{\mu\nu}+\widehat{\mathbf{\Phi}}_{\mu\nu}$)

where $\mathbf{\Phi}_{\mu\nu}$ is the source term for the scalar field $\mathcal{G}(x)$ and $\mathbf{\Psi}_{\mu\nu}$ is the tensor that arises from the position dependence of $\mathcal{G}(x)$ within the variation of the action. The Bianchi identities and the conservation of energy and momentum demands that

$\nabla^{\mu}\mathbf{G}_{\mu\nu}=8\pi \nabla^{\mu} (\mathcal{G}(x)\mathbf{T}_{\mu\nu})+\nabla^{\mu}\widehat{\mathbf{\Psi}}_{\mu\nu}+\nabla^{\mu}\widehat{\mathbf{\Phi}}_{\mu\nu}=0$

In the absence of matter

$\mathbf{G}{\mu\nu}=\Psi_{\mu\nu}+\mathbf{\Phi}_{\mu\nu}$

$\nabla^{\mu}\mathbf{G}_{\mu\nu}= \nabla^{\mu}\widehat{\mathbf{\Psi}}_{\mu\nu}+\nabla^{\mu}\widehat{\mathbf{\Phi}}_{\mu\nu}=0$

Explicitly, the tensor $\mathbf{\Psi}_{\mu\nu}(x)$ is

$\widehat{\mathbf{\Psi}}_{\mu\nu}(x)=\mathcal{G}(x)(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\square)G^{-1}(x)$

or

$\widehat{\mathbf{\Psi}}_{\mu\nu}=\mathcal{G}(x)^{-2}[2\nabla_{\mu}\mathcal{G}(x)\nabla_{\nu}\mathcal{G}(x) -\mathcal{G}(x)\nabla_{\mu}\nabla_{\nu}\mathcal{G}(x)-2g_{\mu\nu} \nabla_{\mu}\mathcal{G}\nabla^{\mu}\mathcal{G}(x)-\mathcal{G}(x)\square\mathcal{G}(x)]$

The trace and covariant divergence are

$\widehat{\mathbf{\Psi}}_{\mu}^{\mu}=\frac{3}{\mathcal{G}^{3}(x)}[\mathcal{G}(x)\square \mathcal{G}(x)-2\nabla_{\mu}\nabla^{\mu}\mathcal{G}]$

$\nabla^{\mu}\widehat{\mathbf{\Psi}}_{\mu\nu}=\frac{1}{\mathcal{G}(x)}\nabla^{\mu} \mathcal{G}(x)(\mathbf{\Psi}_{\mu\nu}-\mathbf{R}_{\mu\nu}]$

The consistency or constraint condition also gives

$\frac{1}{\mathcal{G}(x)} \nabla^{\mu}\mathcal{G}(x)(\widehat{\Psi}_{\mu\nu}-R_{\mu\nu})+ \nabla^{\mu}\widehat{\Phi}_{\mu\nu}+8\pi (\nabla_{\mu}\mathcal{G}(x)\mathbf{T}^{\mu}_{\nu}=0$

However, the following alternative expression for the consistency condition also provides the link to the original BD theory:

$\frac{3}{2}\frac{\nabla_{\nu}\mathcal{G}(x)}{\mathcal{G}(x)^{2}}[\mathcal{G}(x)\square \mathcal{G}(x)-2(\nabla_{\mu}\mathcal{G}\nabla^{\mu}\mathcal{G}(x)))]+\nabla^{\mu} \widehat{\Phi}_{\mu\nu} -\frac{\nabla^{\mu}\mathcal{G}(x)}{\mathcal{G}(x)}\widehat{\Phi}_{\mu\nu}+4\pi \mathbf{T}\nabla_{\nu}\mathcal{G}(x)=0$

where $\mathbf{\Phi}_{\mu\nu}=\mathbf{\Phi}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\Phi_{\alpha}^{\alpha}$ and $\mathbf{T}=\mathbf{T}_{\mu}^{\mu}$. When $\mathbf{T}=0$, then the energy-momentum tensor is traceless giving

\$\frac{3}{2}\frac{\nabla_{\nu}\mathcal{G}}{\mathcal{G}(x)^{2}}[\mathcal{G}(x)\square \mathcal{G}(x)-2(\nabla_{\mu}\mathcal{G}(x)\nabla^{\mu}\mathcal{G}(x)))]+\nabla^{\mu}\widehat{\Phi}_{\mu\nu} -\frac{\nabla^{\mu}\mathcal{G}9x)}{\mathcal{G}}\widehat{\Phi}_{\mu\nu}=0$

The tensor $\widehat{\mathbf{\Phi}}_{\mu\nu}$ which solves this equation has the form

$\mathbf{\Phi}_{\mu\nu}^{BD}= \bigg(-\frac{3}{2}\bigg)\frac{1}{8\pi \mathcal{G}(x)^{3}}[\nabla_{\mu}\mathcal{G}(x)\nabla_{\nu}\mathcal{G}(x)-\frac{1}{2}g_{\mu\nu}\nabla_{\mu} \mathcal{G}(x)\nabla^{\mu}G(x)]$

For the choice of the Brans-Dicke parameter $\omega=\frac{3}{2}$ and $\mathbf{\Phi}=\mathcal{G}^{-1}$, this is then equivalent to the original Brans-Dicke source tensor (-) The action $S_{\Phi}$ this tensor is

$S _{\Phi}=\frac{3}{32\pi}{\displaystyle \int} d^{4}x \sqrt{g}\mathcal{G}^{-2}\nabla_{\mu}\mathcal{G}\nabla^{\nu}\mathcal{G}$

The modified Einstein equations are now formulated on a spherically symmetric metric. This is the standard static isotropic form

$ds^{2}=-X(r)dt^{2}+Y(r)dr^{2}+r^{2}(d\theta^{2}+ sin^{2}(\theta)d\varphi^{2}$

For the Killing vectors $K^{\mu}$ we require that $K^{\mu}\partial_{\mu}G(x)=0$ so that $G=G(r)$ and so G can vary with the radial coordinate $r$ only. The radial derivatives will be denoted $D_{r}=d/dr$ and $D_{rr}=d^{2}/dr^{2}$. It will be instructive to initially formulate the full generic modified field equations on this metric, and then solve for the specific case when there is no matter and no cosmological constant term. This will illucidate how the solution of the modified vacuum equations compares to the Standard Schwarzschild vacuum solution and the interior Schwarzschild solution. The full modified Einstein equations are

$\mathbf{G}_{\mu\nu}=\mathbf{R}_{\mu\nu}-\frac{1}{2} g_{\mu\nu}\mathbf{R} = -g_{\mu\nu}\Lambda$   $+ 8 \pi \mathcal{G}(x) \mathbf{T}_{\mu\nu}(x) +$ $\widehat{\Psi}_{\mu\nu}(x) + \widehat{\Phi}_{\mu\nu}(x) = \mathbf{T}tot_{\mu}^{\nu}$

where $\widehat{\mathbf{\Psi}}_{\mu\nu}=\mathbf{\Psi}_{\mu\nu}/8\pi \mathcal{G}(x)$ and $\widehat{\mathbf{\Phi}}_{\mu\nu}=\mathbf{\Phi}_{\mu\nu}/8\pi \mathcal{G}(x)$. The source tensor for fluid system in hydrostatic equilibrium has the usual form

$\mathbf{T}_{\mu\nu}=p g_{\mu\nu}+(p+\rho)\xi_{\mu}\xi^{nu}$

with an equation of state $p=p(\rho)$ so that $\mathbf{T}_{\mu\nu}=diag[-\rho(r),p(r),p(r),p(r)]$. The conservation equation arising from the Bianchi identities is $\nabla^{\mu}\mathbf{T}_{\mu\nu}=0$ and leads to the condition

$p^{\prime}(r)+\frac{X^{\prime}}{2X}(\rho+p)=0$

which is essentially the Tolman-Oppenheimer-Volkoff equation for relativistic stellar structure.

Next,the source tensor arising from the spatial dependence of Newton’s constant is expressed in terms of density and pressures, in the same form as for the matter fluid source so that

\$\mathbf{\Psi}_{\mu\nu}=\frac{\widehat{\mathbf{\Psi}}_{\mu\nu}}{8\pi \mathcal{G}}=diag[-\mathcal{D}(r),\mathcal{P}(r),\mathcal{P}_{a}(r),\mathcal{P}_{a}(r)]$

Here, $\mathcal{D}=-\mathbf{\Psi}_{t}^{t}$ is an energy density induced by the position-dependence of Newton’s constant, and $\mathcal{P}(r)=\mathbf{\Psi}_{r}^{r}$ is the associated radial pressure and
$\bar{\mathcal{P}}(r)=\mathbf{\Psi}_{\theta}^{\theta}=\mathbf{\Psi}_{\varphi}^{\varphi}$, are the angular or tangential pressures.

For the metric (-), the density and pressures are induced within the vacuum due to the running of the Newton constant are(refs)

$\mathcal{D}(r)=-\mathbf{\Psi}_{t}^{t}=\frac{1}{8\pi \mathcal{G}(r)}\bigg( \frac{2}{rY} \mathcal{G}^{-1}(r)\mathcal{G}^{\prime}(r)$

$+ \frac{Y^{\prime}}{2Y^{2}} \mathcal{G}^{-1}(r)$

$\mathcal{G}^{\prime}(r) +$  $\frac{2}{Y} \mathcal{G}(x)^{-2}(r) \mathcal{G}(x)^{\prime}(r) \mathcal{G}^{\prime}(r)$$+\frac{1}{Y}\mathcal{G}^{-1}(r)\mathcal{G}^{\prime\prime}(r)\mathcal{G}(r)\bigg)$

$\mathcal{P}(r)=\mathbf{\Psi}_{r}^{r}=\frac{1}{8\pi \mathcal{G}(r)}\bigg( \frac{X^{\prime}}{XY}\mathcal{G}^{-1}(r)\mathcal{G}^{\prime}(r)+ \frac{2}{Y}\mathcal{G}^{-1}(r)\mathcal{G}^{\prime}(r) \bigg)$

$\bar{\mathcal{P}}(r)=\mathbf{\Psi}_{\theta}^{\theta}=\mathbf{\Psi}_{\varphi}^{\varphi} =\frac{1}{8\pi \mathcal{G}(r)}\bigg(\frac{1}{rY}\mathcal{G}^{-1}(r)\mathcal{G}^{\prime}(r) -\frac{Y^{\prime}}{2Y^{2}}G^{-1}(r)\mathcal{G}^{\prime}(r)\nonumber\\[0.1in]+ \frac{X^{\prime}}{2XY}\mathcal{G}^{-1}(r)G^{\prime}(r)-\frac{2}{Y}\mathcal{G}^{-2}(r)\mathcal{G}^{\prime}(r) \mathcal{G}^{\prime}(r) -\frac{1}{Y}\mathcal{G}^{-1}(r)\mathcal{G}^{\prime\prime}(r)\bigg)$

The $\Phi_{\mu\nu}$ source tensor will be left arbitrary for the moment, but it is sufficient that it has same diagonalised form

$\mathbf{\Phi}_{\mu}^{\nu}=diag[-\mathscr{D}(r),\mathscr{P}(r),\bar{\mathcal{P}}(r),\bar{\mathcal{P}}(r)]$

The final contribution can come from a standard cosmological constant term $\Lambda$ or vacuum ‘dark energy’ with no spatial dependence such that $\mathbf{\Theta}_{\mu}^{\nu}=diag[-\rho_{v},p_{v},p_{v},p_{v}]$ with $\rho_{v}=\Lambda/8\pi \mathcal{G}$ and $p_{v}=\Lambda/8 \pi \mathcal{G}$. The total source tensor is then

$\mathbf{T}_{\mu}^{\nu}=diag[-D(r),P(r),\bar{P}(r),\bar{P}(r)]$

so that

$D(r)=\rho(r)+\rho_{v}+\mathcal{D}(r)+\mathscr{D}(r)$

$P(r)=p(r)+p_{v}+\mathcal{P}(r)+\mathscr{P}(r)$

$\bar{P}(r)=p(r)+p_{v}+\bar{\mathcal{P}}(r)+\bar{\mathscr{P}}(r)$

The modified Einstein equations can then be written as

$\mathbf{G}_{\mu\nu}=8\pi G\mathbf{T}_{\mu\nu}^{(tot)}$

For the metric (-) the nonvanishing components of the Einstein tensor are
$\mathbf{G}_{t}^{t},\mathbf{G}_{r}^{r},\mathbf{G}_{\theta}^{\theta},\mathbf{G}_{\varphi}^{\varphi}$ so that the full field equations are

$\mathbf{G}_{t}^{t}=-8\pi \mathcal{G} D(r)=-8\pi \mathcal{G}[\rho(r)+\rho_{v}+\mathcal{D}(r)+\mathscr{D}(r)]$

$\mathbf{G}_{r}^{r}=~8\pi\mathcal{G} P(r)=~8\pi G[p(r)+p_{v}+\mathcal{P}(r)+\mathscr{P}(r)]$

$\mathbf{G}_{\theta}^{\theta}=\mathbf{G}_{\varphi}^{\varphi}=8\pi P_{a}(r)=8\pi \mathcal{G}[p(r)+p_{v}+\mathcal{P}_{a}(r)+\bar{\mathcal{P}}(r)]$

In terms of the metric functions $X(r)$ and $Y(r)$

$\mathbf{G}_{t}^{t}=\frac{1}{r^{2}Y}-\frac{1}{r^{2}}-\frac{Y^{\prime}}{r Y^{2}}=-8\pi G(\rho(r)+\rho_{v}+\mathcal{D}(r)+\mathscr{D}(r))$

$\mathbf{G}_{r}^{r}=\frac{1}{r^{2}Y}-\frac{1}{r^{2}}-\frac{A^{\prime}}{r Y^{2}}=~8\pi \mathcal{G}(r)(p(r)+p_{v}+\mathcal{P}(r)+\mathscr{P}(r))$

$\mathbf{G}_{\theta}^{theta}=\mathbf{G}_{\varphi}^{\varphi}= -\frac{Y^{\prime}}{2rY^{2}}+\frac{X^{\prime}}{2r XY}-\frac{X^{\prime} Y^{\prime}}{4B^{2}X}-\frac{(X^{\prime})^{2}}{4YX^{2}}+ \frac{\partial_{rr}X^{ \prime\prime}}{2XY}=8\pi \mathcal{G} [p(r)+p_{v}+\mathcal{P}_{a}+\bar{\mathscr{P}}]$

The conservation of energy and momentum requires that $\nabla^{\mu}\mathbf{T}_{\mu\nu}=0$ which gives

$\frac{dP(r)}{dr}+ \frac{X^{\prime}}{2X}(D+P))+\frac{2}{r}(\mathcal{P}-\bar{\mathcal{P}})=0$