If one rewrites the Schwarzschild solution in terms of a radial coordinate R = r - 2GM where r is the Schwarzschild radial coordinate (as for example is done by Marcel Brillouin in his 1923 paper where he explains why he considers that r = 2GM is effectively the origin), then all of the factors involving the gravitational constant G are then in exactly the right form to support a fully Machian variable form of G, as in Dennis Sciama's illustrative theory (in his paper "On the Origin of Inertia") and as partly incorporated into Brans-Dicke theory.(adsbygoogle = window.adsbygoogle || []).push({});

In terms of this R coordinate, the metric is as follows (with explicit G factors):

[tex]ds^2 = -\frac{1}{(1 + 2Gm/R)} \, dt^2 \: + \: (1 + 2Gm/R) \, dR^2 \: + \: (1 + 2Gm/R)^2 \, R^2 \, (d \theta ^2 + \sin^2 \theta d \phi^2)[/tex]

It is possible for the Schwarzschild solution to give exactly the correct experimental results for any single central object yet for the true value of G to vary exactly as predicted by Machian theories. This is because all references to the gravitational constant consist of powers of the following expression, as explained below:

[tex]\frac{1}{(1+2Gm/R)}[/tex]

Suppose that the Newtonian gravitational constant G_{N}is actually a variable which exactly satisfies the following Machian relationship (a special case of the Whitrow-Randall relation) when the sum is taken for all masses in the universe:

[tex]G_N \Sigma \frac{m_i}{R_i} = 1/2}[/tex]

Let G be the value which G_{N}would have at a point which is distant from the local mass but not far enough away to affect the other terms. This is a constant for local calculation purposes. This can be written as follows, where the sum is assumed to include the local mass, which is then subtracted out again:

[tex]G = \frac{1}{2 (\Sigma m_i/R_i - m/R)}[/tex]

We then find that the interesting factor 1/(1+2Gm/R) is then simply equivalent to the following, without any G in sight (nor indeed any unit of mass or distance), where the sums again include the local mass:

[tex]\frac{\Sigma m_i/R_i - m/R}{\Sigma m_i/R_i}[/tex]

That is, this factor is simply an abbreviated expression for the ratio of the sum of m/R for every mass in the universe as seen from a point far away from the local mass m and at distance R from that mass.

The same expression can also be written in terms of G_{N}for illustrative purposes, but since G_{N}is a variable, this form is not helpful for calculation purposes:

[tex]1 - 2 G_N\, m/R[/tex]

I believe that this result means that the standard experimental evidence that GR is completely correct for single central masses, based on the Schwarzschild solution, does not even begin to rule out the possibility that G varies in a Machian way.

However, experiments relating to the variation of G with time and location (taking into account possible effects on space, time and other units) could of course be used to narrow down the Machian possibilities.

Does this result appear to be correct, and if so, is it a known result?

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# Schwarzschild solution and Machian variable G

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