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Because MOND revolves around acceleration, which is so much in the heart of inertia, one is directed, with the above imagery in mind, to consider that inertia itself, not just MOND, is a derived concept reflecting the interactions of bodies with some agent in the background. The idea, which is as old as Newton's second law, is the basic premise of the Mach's principle. The great sense that this idea makes has lead many to attempt its implementation. The agent responsible for inertia had been taken to be the totality of matter in the Universe.

Arguably, an even better candidate for the inertia-producing agent, which I have been considering since the early 1990s, in the hope of understanding MOND's origin, is the vacuum. The vacuum is known to be implicated in producing or modifying inertia; for example, through mass renormalization effects, and through its contribution to the free Maxwell action in the form of the Euler-Heisenberg action [Itzykson & Zuber 1980]. Another type of vacuum contributions to inertia have been discussed by [Jaekel & Reynaud 1993]. But, it remains moot whether the vacuum can be fully responsible for inertia.

The vacuum is thought to be Lorentz invariant, and so indifferent to motion with constant speed. But acceleration is another matter. As shown by Unruh in the 1970s, an accelerated body is alive to its acceleration with respect to the vacuum, since it finds itself immersed in a telltale radiation, a transmogrification of the vacuum that reflects his accelerated motion. For an observer on a constant-acceleration (a) trajectory this radiation is thermal, with T = alpha a, where alpha ident hbar / 2pi kc. The effect has been also calculated approximately for highly relativistic circular motions; the spectrum is then not exactly thermal. In general, it is expected that the effect is non-local; i.e., depends on the full trajectory.

Unruh's result shows that the vacuum can serve as an inertial frame. But this is only the first step. The remaining big question is how exactly the vacuum might endow bodies with inertia. At any rate, what we want is the full MOND law of inertia, with the transition occurring at accelerations of order a0 that is related to cosmology. We then have to examine the vacuum in the context of cosmology. How it affects, and is being affected by, cosmology. One possible way in which cosmology might enter is through the Gibbons-Hawking effect, whereby even inertial observers in an expanding universe find themselves embedded in a palpable radiation field that is an incarnation of the vacuum. The problem has been solved for de Sitter Universe, which is characterized by a single constant: the cosmological constant, Lambda, which is also the square of the (time independent) Hubble constant. In this case the spectrum is also thermal with a temperature T = alpha c(Lambda / 3)1/2.

In the context of MOND it is interesting to know what sort of radiation an observer sees, who is accelerated in a non-trivial universe: If the Unruh temperature is related to inertia, then it might be revealing to learn how this temperature is affected by cosmology. This can be gotten for the case of a constant-acceleration observer in a de Sitter Universe. For this case the radiation is thermal with a temperature T = alpha(a2 + c2 Lambda / 3)1/2 [Deser & Levin 1997]. Inertia, which is related to the departure of the trajectory from that of an inertial observer, who in de Sitter space sees a temperature alpha c(Lambda / 3)1/2, might be proportional to the temperature difference

Equation 12       (12)

and this behaves exactly as MOND inertia should: it is proportional to a for a > > a0 ident 2c(Lambda / 3)1/2, and to a2 / a0 for a < < a0; and, we reproduce the connection of a0 with cosmology. Of course, in the modified-inertia paradigm this would reflect on a "linear", constant-acceleration motion, while circular trajectories will probably behave differently. But the emergence of an expression a-la MOND in this connection with the vacuum is very interesting.

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