Is Self-Adjusting Isotropy a Valid Concept in Gravitational Theory?

[Jonathan Scott]

One well-known effect of a gravitational source in GR is that it bends space (which has the effect of doubling the deflection of a light ray passing the sun compared with Newtonian gravity).

One way of thinking of this bending is that it maps a tangential plane as seen by a local observer into a sort of rounded cone (which however is extremely close to flat for weak gravitational fields) within an overall flat coordinate system describing the region. Considering only curvature of space, the angle from flatness all round the cone is Gm/rc^2 radians, where m is the mass of the source and r is the distance of the closest point from the center of the source mass (so the total deflection angle of a spatial line is 2Gm/rc^2, half of that for a light beam, assuming I've remembered that formula correctly).

This means that from the point of view of an observer at that point, when the source is approached, the fixed stars anywhere near that plane are effectively shifted towards the part of the sky away from the source, by that tiny angle, effectively adjusting the apparent distribution of the rest of the source masses in the universe to make up for the unevenness caused by the local mass.

If we assume the Whitrow-Randall-Sciama relation holds approximately, so the sum of GM/Rc^2 for every mass in the universe gives some simple constant n of order 1, and that the overall shift effect also involves a simple constant k to represent the average effect on the fixed stars over the whole sphere, then applying the conical shift to the fixed stars gives an effect of redistributing an amount of potential source equal to (nk Gm/rc^2) in the opposite direction, tending to cancel out the "uneven" effect on the potential of the local mass. [It strikes me that it would be very neat if nk = 1, giving an exact balance, but I doubt that this could work in GR].

This effect effectively adds up linearly for any combination of sources in any direction, and seems to imply that the average directions of distant sources will effectively be automatically adjusted so that on average sources (weighted according to potential ) appear to be more isotropically distributed. I find this very interesting, and quite Machian in its "self-adjusting" nature.

Does this appear to be a valid conclusion? Any other thoughts on the implications?

 

[mfb]

If we assume the Whitrow-Randall-Sciama relation holds approximately, so the sum of GM/Rc^2 for every mass in the universe gives some simple constant n of order 1

That mass is not in one direction - it has a nearly uniform distribution, so deflections should cancel to a very, very good approximation.
If I understand your constant k correctly, it would be extremely small.

In addition, the shift is not present for the source causing the shift itself, and not correlated to other sources - if you look at light coming from a specific age (like the CMB), the fluctuations there are not canceled by matter somewhere else.

It looks a bit arbitrary to consider "all mass in a specific direction in the sky", where there could be some cancellation effect.

 

[Jonathan Scott]

That mass is not in one direction - it has a nearly uniform distribution, so deflections should cancel to a very, very good approximation.
If I understand your constant k correctly, it would be extremely small.

In addition, the shift is not present for the source causing the shift itself, and not correlated to other sources - if you look at light coming from a specific age (like the CMB), the fluctuations there are not canceled by matter somewhere else.

It looks a bit arbitrary to consider "all mass in a specific direction in the sky", where there could be some cancellation effect.

OK, I need to explain this better.

There are two sums involved - a directional (vector) sum of the potential terms due to every source mass, which is expected to cancel to around zero, and the total scalar potential, which is of order 1, which I've called n.

The angular shift from a local source causes the "fixed stars" vector contribution to be shifted. The effect on stars around the tangential direction is that their contribution to the vector sum is modified by an amount equal to their share of the scalar potential multiplied by Gm/rc^2 for the local source, in the direction perpendicular to the tangential plane and away from the local source. The effect on stars further away from the tangential direction is more difficult to define as it depends on how the radial component of distance is defined, but I am expecting "k" to be a significant fraction, say 1/2, even if distances in the perpendicular distance are unchanged.

What I'm saying overall is that the effect of curved space means that the directional sum of potential terms is less affected by a local mass than one might expect, and if nk were equal to 1 it might not vary at all, and might in fact be zero everywhere (although I think that would require some change from GR towards a more Machian theory to work exactly).