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Schwarzschild solution and Machian variable G

  1. Jan 2, 2008 #1

    Jonathan Scott

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    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.

    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:


    Suppose that the Newtonian gravitational constant GN 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 GN 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 GN for illustrative purposes, but since GN 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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  3. Jan 2, 2008 #2


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    Jonathan, could you give links to references for the important papers you refer to?

    It is generally thought that in a gauge where atomic masses remain constant then GR includes Mach's Principle up to, but not including, the requirement that G should be determined by the presence of mass in motion in the rest of the universe.

    The Brans Dicke theory satisfies this requirement, so why do you say "as partly incorporated into Brans-Dicke theory"?

    Thank you,
  4. Jan 2, 2008 #3

    Jonathan Scott

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    A scanned copy of Sciama's paper "On the Origin of Inertia" is available through the links on this page:


    A translation of the Marcel Brillouin paper by Salvatore Antoci is available here (although the subject of this paper is not directly relevant to this thread, only the form in which he writes the Schwarzschild solution, which can in any case be easily checked without reference to this paper):


    As I understand it, the strength of the Machian component in Brans Dicke theory is effectively determined by a parameter. In order to match current experiment, this parameter has to be set to a value which makes the Machian effect extremely weak.
  5. Jan 3, 2008 #4
  6. Jan 3, 2008 #5


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    Well that might be just saying the BD approach (i.e. Mach) is wrong, or that the parameter [itex]\lambda[/itex] is very small, or that the BD theory is on the right lines but needs some further modification to bring it into concordance with observations, (which has been my approach).

    That notwithstanding, the BD theory does in fact fully include Mach's Principle.

    BTW thank you for those links to Brillouin's and Sciama's papers.

    Last edited: Jan 3, 2008
  7. Jan 3, 2008 #6

    Jonathan Scott

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    Some people think GR does too; it's all very debatable and depends on exactly what one means by "Mach's principle".

    However, I hope it should be very clear from my post at the start of the thread that what I personally mean in this case by "fully Machian" is that the strength of the gravitational interaction could be determined entirely by dimensionless ratios of masses and distances, without any predefined arbitrary constant G. What I did not previously realize is that even if this were true, for any system dominated by a single central mass, the Schwarzschild solution could still hold exactly.

    I've now found a better link to Brillouin's paper, in that the paper at the above URL is obviously just a copy of arXiv:physics/0002009v1 which can for example be found via http://arxiv.org/abs/physics/0002009v1
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