© Jonathan Scott 1995, 1998.
This document is based on one posted by e-mail to several people in January 1995. Since then (partly as a result of feedback), I've found that the main line of reasoning is much more arbitrary and less rigorous than it seemed at the time, but it is still very plausible, especially as I have also found since then that it seems to be closely related to the theory of Yilmaz. I've also recently found "Gravitation and Inertia" by Ciufolini & Wheeler very useful in helping to resolve some questions arising from my studies in this area.
I am currently interested in a theory of gravity derived by Dennis Sciama from Mach's Principle in the 1950s [1]. In its original form, it is more of an illustration than a complete theory, and appears to give a PPN beta factor which disagrees with experiment. However, I recently noted that the treatment of self-energy in GR and similar theories seems to contain an inconsistency, and when an additional correction is made for this effect, it is Sciama's theory which gives the correct PPN beta. I am therefore interested in finding out more about whether my self-energy correction is plausible and whether the existing experiments (especially the binary pulsars) could be used to distinguish between my modified version of Sciama's theory and GR. Unfortunately, my skills in tensors and curved space are very limited, and I have so far only been able to handle the mathematical details using isotropic coordinates for a central body.
As I am a computer systems software programmer rather than a physicist or mathematician, I have no current contact with relevant people except via the internet. (I recently studied in my spare time for a few years with the UK Open University and obtained a first class honours degree in Physics and Mathematics subjects, but the courses were not at a high enough level to handle GR in detail, and studying from home did not bring me into very much contact with any real experts). I have received some very useful advice via the sci.physics newsgroup, including the references to gr-qc articles and the paper by Sciama. I have a copy of Misner, Thorne and Wheeler "Gravitation" and I have recently referred to a library copy of Will's "Theory and Experiment in Gravitational Physics" although as I am no longer a student I couldn't borrow it, so I guess I'll have to buy my own copy (of the 1993 revised edition) soon. However, I don't have the skills to evaluate my theory directly using the formulations in those books, nor to work out the exact modification necessary to Einstein's field equations to make them match my theory, although I have some ideas based on analogy.
Here's a summary of my theory. I would appreciate any feedback on whether you find it comprehensible, plausible and usable, whether you have seen anything similar before and whether you can help me work out whether it predicts anything which can be tested by existing or future experiment.
The theory was not deliberately developed to be different from GR but was rather derived from the requirement for any gravitational theory to be self-consistent when describing the effects of any number of masses up to and including the whole universe, as seen by observers who themselves might be at different potentials. (I did however start on this path on the suspicion that GR did not appear to be self-consistent in this sense). The basic principle is that the relative rates of clocks at rest at different gravitational potentials must combine correctly, so A/B times B/C = A/C, and the relative rate for any pair of clocks must be independent of the location or potential of the observer. One obvious Newtonian solution to this problem would be for the clock rate to vary as the exponential of the Newtonian potential difference, but it appeared possible that there were other solutions.
On Christmas Day 1986, I found that consistency appeared to require that the relative rates of two clocks at different potentials was a function of the linear sums of "mc2/r" for everything in the universe as seen from those two points, where the definition of what was meant by "mc2/r" must be locally consistent with the Newtonian definition but for remote masses was simply defined as whatever made this rule consistent.
Although the expression exp(-sum(Gm/rc2)) has this property, Special Relativity approximations to gravity require that the potential should vary as a four-vector under small local transformations, and I could not make this work.
A much more successful idea was that this function was simply 1/sum(mc2/r), corresponding to G being equal to c2/sum(m/r). The vector version then simply replaces mc2 with the four-vector energy-momentum and the transformation rules and conservation rules are trivial. Inertia and the shape of space then become functions of the distribution of matter in the universe, explicitly satisfying Mach's principle and making G a property of the current state of the universe rather than an arbitrary constant.
However, some people on the sci.physics Usenet newsgroup pointed out that when you plug this expression into the PPN formalism, it does not predict the correct perihelion advance for Mercury. I played around with modified versions of this idea for a while but none of them were quite so neat.
After mentioning this old idea again some years later (in 1995) on one of the newsgroups, I was referred to a theory by Sciama, described in his paper "On the Origin of Inertia" [1], which derives a specific requirement for G/c2 to be effectively equal to 1/sum(m/r) for all masses in the universe, matching the original form of my own theory, in order for inertia to be precisely accounted for by the gravitational effect of the entire universe. He used a special-relativistic flat space approximation, but with this encouragement I decided to resume work on this form of my theory and to try to work out the detailed solution for the central mass case including relativistic and higher order effects.
I then discovered that in the coordinate system I had used, m was not actually a constant, for reasons explained below, and once I had the appropriate correction for this term, I had enough detail to test the theory using the PPN approximation.
The result was quite simple, in that the dt2 factor in the metric in isotropic coordinates for a central spherically symmetrical mass becomes exp(-2Gm/rc2) and the dx2, dy2 and dz2 factor becomes approximately exp(2Gm/rc2). The dt factor is determined exactly by the basic theory but the dx factor is based on the assumption that Einstein's field equations for empty space are correct at least to first order.
The way this result was obtained is described in the following sections.
The multiplicative gravitational potential at one point relative to a reference point using Sciama's theory (in the scalar approximation) is:
sum(m/r) for universe as seen from reference point -------------------------------------------------- sum(m/r) for universe as seen from observation point
Split this up into a sum for all distant masses, abbreviated to M/R, and the term for the local central mass, m/r, and assume the reference point to be a long way from the local central mass. The potential ratio is then:
M/R 1 --------- or ----------------- M/R + m/r 1 + 1/(M/R) * m/r
Let G stand for = 1/c2 * M/R (where this excludes the local mass so it is effectively constant for the central case).
The multiplicative potential can then be written as:
1 ----------- 1 + Gm/rc2
This is then the exact potential factor for total mass-energy mc2 distributed with spherical symmetry within a sphere of radius r.
(The use of c in this expression is not quite as simple as it seems, since the effective values of both c and G vary with potential in isotropic coordinates. I actually split this as (G/c4) (mc2/r) where G/c4 does not depend on observer potential provided G and c are observed at the same place, and similarly mc2/r does not vary with observer potential, at least to first order. When I refer to "mass" but include the c2 factor, this is more accurately described as the rest energy).
If this potential factor is plugged into the PPN formalism, it gives the wrong value for beta. This is where I originally gave up.
However, I later noticed that it appears to be incorrect to treat m as a constant in this formulation, or any similar Post-Newtonian approximation, for reasons explained in the next section.
If the mass m was built up from smaller masses, the value m here would not necessarily match the sum of the constituent masses, and it also might vary slightly with r if the gravitational field had non-zero energy density. In order to determine the details, I considered the following model.
If an observer was sitting at distance r from the central mass and saw some additional mass being brought in from infinity, the clocks of that observer would be running slower than at the original location of the mass by a factor 1/(1 + Gm/rc2) so the observer would see the energy of the incoming incremental mass to be (1 + Gm/rc2) times its original rest energy, effectively because of the kinetic energy gained falling in towards the existing central mass.
This means that if the central mass were built up from nothing, a local observer would see additional energy (mc2 integral of Gm/rc2 dm). This gives a self-energy term 1/2 Gm2/r by which the energy of the central object exceeds the sum of the energies of the incoming masses. (Note that this is related to the radius of the central object but only to the distance of the not observer. Once energy has passed into the sphere containing the observer, its radial distribution does not affect the field at the observer, as is shown by Birkhoff's theorem).
Taking the self-energy term into account in the central mass, the local observer would also see a second order central energy term given by the integral of the effect of the first order self-energy term, and so on.
The series for the effective total energy in terms of the sum m of the masses which formed the body is then as follows:
mc2 (1 + 1/2 (Gm/rc2) + 1/3! (Gm/rc2)2 + 1/4! (Gm/rc)^3...)
When this series is substituted for mc2 in the potential factor expression 1/(1 + Gm/rc2), a minor miracle occurs and we get exp(-Gm/rc2). In this form, m is now a constant and the PPN formalism can be used to analyze the predictions. The most sensitive test is that this expression has the PPN beta factor equal to 1 as in General Relativity. I have also calculated Mercury's perihelion advance directly using this formula in isotropic coordinates and an Euler-Lagrange form of the geodesic equations, and the results come out exactly the same as for GR, matching the experimental results.
However, the dt2 factor exp(-2Gm/rc2) can never become zero or negative regardless of how large the mass or how small the radius, so this metric cannot give rise to black holes.
Another property of this particular expression for the potential is that the gravitational force in the field of a central mass in isotropic coordinates is identical to the Newtonian form GmM/r2, where m is the linear measure of mass. (Unfortunately, I don't think this type of simplification applies in many-body situations. I was originally hoping that that the exponential form would extend to multiple masses, but I now suspect that it only works when they are at the same location. In one sense this is a good thing, as otherwise it would look so similar to Ni's theory that it would probably suffer from the same defects).
The derivation of the self-energy above also leads to an unexpected conclusion about the location of that energy.
According to a distant observer, the nearby observer's clocks now run slow by a factor of exp(-Gm/rc2). This means that according to the distant observer, the nearby observer's calculation of the energy is wrong by this factor. When the above series for the locally observed mass is multiplied by this factor, it changes the sign of alternate terms, giving:
mc2 (1 - 1/2 (Gm/rc2) + 1/3! (Gm/rc2)2 - 1/4! (Gm/rc)^3...)
This means that the amount of energy within the sphere depends on r, the size of the spherical shell. When r is very large, the total energy is mc2, showing that no energy was lost in building up the body. When r decreases, a small part of the energy is no longer enclosed, showing that a positive energy density exists in free space.
The energy density of the field is of course obtained by differentiating the energy expression with respect to r and dividing by the surface area of the sphere (4 pi r2) giving a first-order gravitational energy density of Gm2/(8 pi r4) or equivalently g2/(8 pi G) where g is the magnitude of the local acceleration. This is exactly analogous to the Maxwell electromagnetic energy density e0E2/2. The gravitational force constant has the opposite sign from the electric force constant (since "like charges" repel but all masses attract) but because of the clock factor the energy of the field is positive when seen by the distant observer, whose view is most "objective" in this case.
The above derivation of the energy density can be done using the first-order term only to show that it apparently applies to any theory which can be described in isotropic coordinates and can be approximated by the Newtonian potential factor (1 - Gm/rc2).
This implies a non-zero energy density in supposedly "empty" space, which means that if Einstein's field equations for empty space are correct, they will nevertheless give the wrong answer at some level of accuracy near massive bodies because a small amount of gravitational self-energy lies in the field outside the body itself!
It is obvious that the apparent distribution of this energy density depends on the relative acceleration and rotation of the frame of reference, and a free fall observer does not see any gravitational forces acting (except that the universe tends to rotate a tiny bit) so the transformation rules for gravitational energy under changes of reference frame are unlike those for conventional particles. However, as far as I can see, the field energy model works nicely (at least as a first order approximation) for multiple masses as well, giving global energy conservation as seen by a distant observer.
For purposes of analyzing distant events, an important part of this theory is its red-shift predictions. The gravitational time dilation factor can either be expressed as 1/(1 + Gm/rc2) where m is the total effective mass including the self-energy of the field up to radius r, or as exp(-Gm/rc2) where m is the linear measure of the rest mass which went to form the body. Assuming this expression is equal to 1/(1 + z) where z is the red shift, the first form gives z = Gm/rc2 where m includes the self-energy up to radius r, and the second form gives z = exp(Gm/rc2) - 1 where m is the linear measure of mass.
Note that as r increases, the two forms of mass converge, so at a sufficient distance from the object, the total energy including any gravitational fields near to the object is equal to the energy of the mass from which the object was originally constructed, regardless of the final density or structure of the central object.
If you followed it this far, thanks! Even if there's some obvious flaw, I hope that the new approach may have helped exercise your mind.
I wish I had more time to work on the details of this theory, but being a computer programmer rather than a professional physicist I find it difficult to get feedback and to know when I'm making elementary mistakes, and I certainly can't afford to give up the "day job" long enough to spend much time at the local university library trying to find the answers. Anyway, please let me know if you have any feedback or any further questions.
[1] On the Origin of Inertia. D.W. Sciama. MNRAS Vol.113 No.1 p.34 1953
If you are interested in discussing the above ideas any further, please feel free to contact me by e-mail: jonathan_scott@attglobal.net.