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: denotes the Einstein tensor, is the usual Newton constant and the scalar field is a position dependent Newton constant.

The action for pure Einstein gravity on a spacetime manifold is

For gravity interacting with fluids or matter fields, one can add another term to the action to account for a source term , so that

and

.

Varying the total action gives the usual Einstein field equations

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 is now elevated to the status of a scalar field then must now be brought under the action integral

where is now a source for the scalar field . 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 was introduced as the inverse of a position-dependent Newton constant so that . The action for the theory is

Varying the action gives the field equations

and a scalar wave equation

Here, is the source tensor arising from the matter while is the source tensor for the field (refs.)

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

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

or

or

)

where is the source term for the scalar field and is the tensor that arises from the position dependence of within the variation of the action. The Bianchi identities and the conservation of energy and momentum demands that

In the absence of matter

Explicitly, the tensor is

or

The trace and covariant divergence are

The consistency or constraint condition also gives

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

where and . When , then the energy-momentum tensor is traceless giving

\

The tensor which solves this equation has the form

For the choice of the Brans-Dicke parameter and , this is then equivalent to the original Brans-Dicke source tensor (-) The action this tensor is

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

For the Killing vectors we require that so that and so G can vary with the radial coordinate only. The radial derivatives will be denoted and . 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

where and . The source tensor for fluid system in hydrostatic equilibrium has the usual form

with an equation of state so that . The conservation equation arising from the Bianchi identities is and leads to the condition

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

\

Here, is an energy density induced by the position-dependence of Newton’s constant, and is the associated radial pressure and

, 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)

The source tensor will be left arbitrary for the moment, but it is sufficient that it has same diagonalised form

The final contribution can come from a standard cosmological constant term or vacuum ‘dark energy’ with no spatial dependence such that with and . The total source tensor is then

so that

The modified Einstein equations can then be written as

For the metric (-) the nonvanishing components of the Einstein tensor are

so that the full field equations are

In terms of the metric functions and

The conservation of energy and momentum requires that which gives

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