# Mach's Principle and Equivalence

1. Dec 14, 2007

### yogi

In free space far removed from significant particulate matter, inertial reaction of a test mass will be isotropic. But if Mach's Principle is the root cause of inertia, then a nearby massive object should modify the inertia of such a test mass so that its reactance to acceleration will be directionally dependent, that is, the test mass will exhibit less reactionary force when accelerated toward the massive object and a greater reaction when accelerated in the opposite direction. Convention has it that the inertia of the test remains isotropic but the directional reactance is due to the G field of the nearby massive body. Why is it incorrect to argue that the nearby mass is actually modifying the inertia of the test mass a la Mach since equivalence precludes the distinquishing of G forces from Inertial forces?

Last edited: Dec 14, 2007
2. Dec 14, 2007

### Garth

Mach's principle does not necessarily involve anisotropic inertia.

In tensor scalar theory, such as the Brans Dicke theory, a scalar field $\phi$, coupled covariantly to the matter in motion in the rest of the universe, determines local inertial mass, that is the value of the particle mass, not a direction of any anisotropic inertial forces.

$$\Box^2 \phi = 4\pi\lambda T^{\sigma}_{\sigma}$$

As rest mass, as measured by comparison with a standard mass, is constant, this variation reveals itself as a variation in G (= $\phi^{-1}$) instead. (As only GM can be measured in any Cavendish-type experiment)

Garth

Last edited: Dec 15, 2007
3. Dec 14, 2007

### yogi

Garth - thanks for the reply - but if cosmic density were not uniform on the large scale, for example if there were an imbalance caused by a large galactic concentration somewhere in the universe - then how does one know that Mach's principle will not lead to a directional difference in the inertia of local particles?

4. Dec 15, 2007

### Garth

It might, but it depends on the type of Machian gravitational theory being proposed.

If the Equivalence Principle holds then the presence of masses in the rest of the universe determines the inertial frame of reference. Once in an inertial frame the laws of motion are completely unaffected by the presence of those masses apart from tidal forces.

The Brans Dicke theory includes the scalar field in such a way that the Equivalence Principle still holds:

$${T_{M}}^{\mu}_{\nu}_{;\mu} = 0$$.

So in the Brans Dicke theory inertia is isotropic in a freely falling frame. As I said, Mach's Principle does not necessarily involve anisotropic inertia.

Garth

Last edited: Dec 15, 2007