A better Einstein-Hilbert action formula?

Does spatial Ricci curvature work as the energy density?

In the text below we are looking for solutions for a static setup. We do not try to handle a dynamic configuration yet.

We have to find an action formula where positive pressure is able to cause positive Ricci curvature in the spatial metric.

The integral of the local spacetime volume over

       R_s + L_M

is one candidate. There R_s denotes the Ricci curvature of the spatial metric.

There is an obvious drawback in R_s. If we have a round mass on the sheet, then R_s is negative around the mass, because the rubber somewhat resembles a saddle surface there. Having a negative energy density somewhere does not sound good. Also, R_s might not be a realistic measure of the deformation energy.

How to determine the temporal metric?

How to restrict the temporal metric? One can make the action integral as small as one wishes by slowing down the local flow of time. Einstein was able to restrict the slowdown of time by including the temporal metric in the calculation of the Ricci curvature.

If the lagrangian would be of a rubber sheet, then the vertical position z plays the role of the slowdown of the local time: the smaller the z, the slower the local flow of time.

In the rubber model, z is not an independent variable but is determined by the spatial metric (= the stretching of the rubber sheet). In the variation of the action integral, z is not among the parameters we can vary. Only the spatial metric parameters (usually g_11, g_22, g_33) take part in the variation.

A (non-equivalent) alternative to z is to use the thickness of the rubber sheet to measure the pace of time. The volume of rubber stays constant unless a very large pressure is applied to it. The pace of time would be determined by the spatial stretching of the rubber. However, if we have a long straight steel bar resting on a rubber sheet, then the tension of the rubber is almost constant close to the steel bar. The depression in the rubber is deeper close to the bar. Obviously, the thickness of the rubber is not a good measure for the pace of time.

The natural coordinate system C and the elastic energy density E(g)

If we paint a coordinate grid in the unstretched rubber, we get a global spatial coordinate system C. We can define the spatial metric with respect to those global coordinates. Then the metric tells how much the rubber sheet is stretched at that point and we can calculate the deformation energy. The deformation energy of a single weight on the rubber sheet is spread throughout space, while the term R in the Einstein-Hilbert action is nonzero only inside the mass.

Let us denote the elastic energy density of the rubber sheet by

       E(g),

where g is the spatial metric in the global coordinate system C. Then the action is the integral on

       E(g) + L_M

over the spacetime volume.

The temporal metric is determined by the spatial metric. If the action is completely analogous to a real newtonian mechanics rubber sheet model, then we know that energy is conserved.

Let

       sqrt(g_11) = 1 + s.

Then the elastic energy density due to the stretch s is something like

       k s^2 / (1 + s).

Modify the Einstein-Hilbert action so that also positive pressure contributes besides L_M?

The lagrangian would then be something like

        R + L_M + L'_M,

where L'_M means the "derivative" of L_M when the spatial metric is varied. Positive pressure would act like a positive mass density.

A problem with ad hoc lagrangians like this is that they probably do not conserve energy and allow a perpetuum mobile.

Let us check if someone has investigated a gravitational "source" of the type above.

Earlier alternative theories to general relativity

https://en.wikipedia.org/wiki/Alternatives_to_general_relativity

At a first glance, none of the theories listed in Wikipedia explicitly specifies positive pressure of a source of gravity. Pressure acting as a source could be a consequence of their definitions, though.

None of the theories listed in Wikipedia explicitly claims to be a rubber sheet model. How do they show that energy is conserved, then?

https://arxiv.org/abs/1603.07655

The model of Tenev and Horstemeyer is equivalent to general relativity. They have a very stiff and thin rubber plane whose elastic energy is in the bending of the plane.