Mach's Principle and the Accelerating Universe

[e2m2a]

With the current astronomical data indicating the universe is expanding at an accelerated rate, does this not invalidate the Mach's Principle? My understanding is when Einstein tried to incorporate this principle into his general theory, he assumed the universe had to be bounded and closed. He believed, as Mach did, the vicinity of other mass contributes to the inertial properties of matter. But, if the universe is not closed and expanding, and if the mean mass density of the universe is decreasing, does this not work against the condition of proximity of mass to determine the inertial properties of matter?

 

[Jonathan Scott]

With the current astronomical data indicating the universe is expanding at an accelerated rate, does this not invalidate the Mach's Principle? My understanding is when Einstein tried to incorporate this principle into his general theory, he assumed the universe had to be bounded and closed. He believed, as Mach did, the vicinity of other mass contributes to the inertial properties of matter. But, if the universe is not closed and expanding, and if the mean mass density of the universe is decreasing, does this not work against the condition of proximity of mass to determine the inertial properties of matter?

Although GR loosely fits with Mach's Principle, at the detailed level (as Einstein later realized) it already conflicts with it, in that for example the gravitational constant G is assumed to be a universal constant in GR, but Mach's principle would predict an effective value of G which depends on the distribution of matter in the universe and hence varies slightly with location and perhaps also with time.

Observations within the solar system suggest that any variation of G with time is extremely small, and this is considered evidence against alternative theories which satisfy Mach's Principle. However, if one considers that such a theory can also affect relative masses and distances, this does not rule out the possibility that the variation in the local value happens to be too small to be observed during the current era.

As we only have one universe to experiment on, and the gravitational potential everywhere is effectively dominated by distant masses, it is difficult to arrange a significantly different distribution of mass in order to test Mach's principle.

We cannot at present measure G with any accuracy outside the solar system, and the effects which are currently ascribed to dark matter and dark energy mean that our knowledge of gravitational effects on a larger scale is even more approximate.

This means that although GR is currently our best available theoretical model, experiment has not totally eliminated the possibility that some other theory which incorporates Mach's principle might eventually prove to be more accurate than GR.

 

[e2m2a]

If inertia here is caused by acceleration of an object with respect to distant masses out there, how do Machian relativists account for the instantaneous, immediate effect locally of inertia without violating causality under the speed of light restriction? With an expanding universe this seems to be even more of a problem.

 

[Jonathan Scott]

If inertia here is caused by acceleration of an object with respect to distant masses out there, how do Machian relativists account for the instantaneous, immediate effect locally of inertia without violating causality under the speed of light restriction? With an expanding universe this seems to be even more of a problem.

I've heard that odd argument before, but I don't know where it comes from. There's no need for the influence to be instantaneous any more than there is with other gravity theories. The main thing that happens in the Machian case is that the gravitational effect of a local mass is not absolute but rather in a sense relative to the effect the rest of the masses in the universe, as seen now (that is, as they would have been when light currently being received started out from them). As a first approximation, the most significant experimental difference from GR would be that the effective value of G would vary a little with location close to a large enough mass. However, the detailed differences depend on the theory, and it is possible for that variation to vanish to first order if the theory differs from GR in other ways as well.

One common feature of many Machian theories is that they partly or totally satisfy the Whitrow-Randall relation, or a more general variant of it:

Sum(GM/Rc²) = k

where the sum is for all masses in the universe and their distances from any observation point, and k is 1 for the simple Whitrow-Randall case or a small numeric constant which depends on the theory. With this relation, G becomes an abbreviation for k/Sum(M/Rc²) for all masses in the universe as seen from that point.

One way of looking at this is that if the whole universe were rotating very very slowly round one then according to Mach's theory the expected total effect of the frame-dragging should be to cause that rotation to be canceled out. This is sometimes called the "sum for inertia".

This relation means that the time-dilation factor (1-Gm/rc²) in a typical relativistic gravitational potential of a local central mass m at distance r is no longer simple to work with because according to the Whitrow-Randall relation, G varies with the distance from the local mass. However, if you redefine G to exclude the local mass from the sum you find that it is now effectively constant (in that it only references distant masses) but that the time-dilation factor changes to 1/(1+Gm/rc²).

This means that although G varies with location, the variation due to the local mass can be factored out, to first order, by slightly changing the form of the expression for the potential, so it then works as if there were a constant G due to all other masses.
The formulae in the previous paragraph are only exact for k=1. For k=1/n, the general form becomes 1/(1+nGm/rc²)^(1/n).

This type of variable G cannot simply be plugged into existing GR solutions, because they would no longer exactly satisfy the Einstein equations and because the different form of the potential factor has a detectable effect on second-order terms, specifically the PPN beta parameter, which has been experimentally verified using the precession of Mercury's perihelion and Lunar Laser Ranging. Any Machian theory which allows G to vary in this way can therefore only be viable if it includes field equations which give rise to the correct second-order terms.