Toy Universes

We continue our notes on Mach’s Principle (MP) and its various interpretations. Today we will briefly point out a sixth interpretation of MP of our list of 21, namely;

Mach Principle 6: “As induction of inertial forces by accelerated masses by analogy with electromagnetic induction.”

There is a formal analogy between the equations of electrodynamics, described by Maxwell’s equations, and the weak-field limit and slow motion approximation of Einstein’s General Relativity equations (GR). Such description is referred to as “gravitomagnetism“.

One can show that the angular momentum of a stationary mass-energy current, in the above-mentioned limit of GR, plays a similar role as the magnetic dipole moment of an equally stationary charge current. Therefore, similarly to electromagnetic induction, the orbital plane of a test body is dragged along the sense of rotation of a massive central body, the so-called “frame-dragging” phenomenon.

So the question is: how far such a result, intrinsically manifested in the equations of GR in the weak-field, slow motion limit, could be extended to a stronger principle? That is essentially MP #6.

The interested reader is referred to Ciufolini’s review article on Nature (subscription required), and his article in [Barb95] , page 386. A freely available paper of interest is also his arxiv review.

We continue our notes on Mach’s Principle (MP) and its various interpretations. Today we will briefly point out a fifth interpretation of MP of our list of 21, namely;

Mach Principle 5: “As the generation of inertial forces in any body accelerated with respect to distant masses”.

You can tell when a body is accelerated by the emergence of the so-called inertial forces acting on the body, which do not exist if the body is at rest or moving at constant velocity(*). The statement concerns the question of whether such inertial forces arise from the motion of the body with respect with distant bodies.

It is well-known that Newton’s bucket experiment shows that inertial forces inducing the concavity of the surface of water in the rotating bucket arise independently of whether the water is rotating with respect to the bucket. In other words, it indicates that inertial forces seem not to be related to motion relative to other bodies.

Mach, however, pointed out that we do not know whether the concavity of the surface of the water would arise if in the experiment the bucket’s walls were increased many orders in width! (A brilliant observation, by the way). In other words, that experiment may rule out the effect of nearby masses, but not necessarily the integrated effect of distant masses in the Universe. This is the point of MP # 5.

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(*) …with respect to what… ? – you would certainly claim to add some reference against which you establish that a body is at rest or moving in constant velocity. According to Newton, the reference is, of course, the Absolute Space. In a Machian view, velocity can only make sense if measured relatively to other bodies, so one could say that the average motion of all bodies in the Universe makes such a reference frame. Today, one would use the Cosmic Microwave Background Radiation (CMBR) “rest frame”, that is, if a body does not move through the average CMBR, then it is “at rest”, and accordingly for a constant velocity movement. In any case, General Covariance extends the notion of inertial frames of reference, so that what is important is that the physical laws have the same mathematical form under arbitrary coordinate transformations. More on that later.