The existence of transient mass fluctuations in objects subjected to large accelerations and rapid changes in acceleration depends upon "Mach's principle" and some peculiarities of "radiation reaction" forces. Mach's principle is the assertion that the physical origin of all inertial reaction forces is an interaction of the object with chiefly the most distant matter in the universe. (Inertial reaction forces are those things that push back on you when you push on stuff.) Radiation reaction forces are experienced by charged objects as they "launch" energy in the form of radiation when they are accelerated by external forces. (These are recoil forces, like those experienced when "launching" bullets out of a gun in your hand.) When examined, the origin of inertia and radiation reaction turn out to have some very strange consequences, notwithstanding that no "new physics" is involved. These ideas are explored in the following material.

The main document is kept free of detailed arguments and mathematics so that the conceptual argument is relatively unencumbered by technical details. It is written in interrogatory style to directly address questions you may have about this stuff. Links to more detailed discussions attended by some math are provided for several of the key ideas. I have tried to keep the exposition everywhere one that can be followed by anyone with only a modest technical background.

What is the origin of inertia?

- According to Newton (and others, to this day) inertia is an inherent property of matter that is independent of all other things in the universe. It is unaffected by the presence or absence of the other matter elsewhere in the universe.
- Inertial reaction forces are thought by some to arise from the interaction of accelerated objects with a local medium, a medium that is present everywhere. (A fashionable candidate for such a scheme in some quarters today is the putative zero point vacuum fluctuation electromagnetic field. There are compelling reasons to ignore schemes of this sort.)
- General relativity theory, for universes like ours, says that the origin of inertial reaction forces in accelerated (relative to the "fixed stars") local objects is the gravitational "field" created here chiefly by the presence of the most distant matter in the universe. (This is "Mach's principle", so named by Einstein.)

Does the gravitational "field" of general relativity theory that produces inertial reaction forces have a real existence independent of the distant matter that creates it?

- The answer to this question depends on the specific mathematics adopted for the interaction and how it's "interpreted". To answer it we need to know something about how gravity causes inertial reaction forces first.
- As shown by Dennis Sciama 45 years ago [
*Monthly Notices of the Royal Astronomical Society*,**113**, 34-42], the interaction that produces inertial reaction forces has all the earmarks of a "radiative" interaction. In particular, it is acceleration (not velocity) dependent, and its distance dependence is inverse first power (instead of inverse second) -- precisely like classical electromagnetic radiation.

But radiative interactions propagate at the speed of light. Inertial reaction forces, however, are instantaneous upon the application of "external" forces. How can a radiative interaction that propagates at the speed of light be responsible for a seemingly instantaneous interaction between a local object and the most distant matter in the universe?

- Good question.
- The answer to this question is muddied by a formal property of field equations called "gauge invariance" which makes it possible to look at things in several different, but equivalent, ways.
- Because of gauge invariance, there are several ways you can try to finesse the answer to this question, but the least artificial answer invokes "absorber" theory (first argued with considerable elaboration by J.A. Wheeler and R.P. Feynman in the 1940s). This theory says that when you push on something, it creates a disturbance in the gravitational field that propagates outward into the future. Out there in the distant future the disturbance interacts with chiefly the distant matter in the universe. It wiggles. When it wiggles it sends a gravitational disturbance backward in time (a so-called "advanced" wave). The effect of all of these "advanced" disturbances propagating backward in time is to create the inertial reaction force you experience at the instant you start to push (and cancel the advanced wave that would otherwise be created by you pushing on the object). So, in this view fields do
*not*have a real existence independent of the sources that emit and absorb them. [This and other relevant stuff is explained nicely in John Gribbin's,*Schrödinger's Kittens and the Search for Reality*(Little, Brown and Co., New York, 1995).] - Believe it or not, the other "interpretations" of the formalism -- all allegedly equivalent -- are even less physically plausible than this. (Yes, that's pretty hard to believe, but true nonetheless.) For some details see the "Origin of Inertia" link.

So what? Why does this have anything to do with radiation reaction and transient mass fluctuations?

- Wheeler-Feynman absorber theory was developed as an "action-at-a-distance" explanation for electromagnetic radiation reaction forces (based on earlier work by Dirac). In action-at-a-distance theories "fields" have no real existence apart from the interacting sources. And radiation reaction, instead of being assumed a force produced by a charge acting on itself in the process of launching radiation, is explained as a seemingly instantaneous interaction between a local accelerated charge and the distant matter in the universe (the "absorber") mediated by retarded and advanced disturbances. Fields are just book-keeping devices for the (delayed) interaction of sources. Wheeler-Feynman theory works very neatly.
- By analogy with electrodynamics, inertial reaction forces, because of their instantaneous/absorber nature, evidently are gravitational radiation reaction forces.

But electromagnetic radiation reaction forces in normal circumstances are miniscule, and gravity is a much weaker force than electromagnetism. So how can that be right?

- True. But, although gravity and electromagnetism are formally similar in the "linear approximation", there are important differences. The chief difference is that there are roughly equal amounts of positive and negative electric charge in the universe, but there are only positive masses out there. This means that the electric potential is everywhere very small or zero. The gravitational potential, on the other hand, is everywhere enormous -- approximately equal to the square of the speed of light. (By the way, they have the same dimensions too: velocity squared.) The gravitational forces responsible for inertial reaction forces have a factor of the gravitational potential in them. That's why they're so large.

Granting that inertial reaction forces may be like electromagnetic radiation reaction forces, what is it about radiation reaction that makes it of interest in the business of transient mass fluctuations?

- Characteristically, radiation reaction involves effects that go as and . (I'm using here the standard notation where each dot over a quantity indicates that the quantity is to be differentiated once with respect to time. [Differentiation with respect to time gives the rate at which the quantity is changing.] In this notation, is velocity, acceleration, and the rate at which the acceleration is changing.) These weird effects seemingly transiently violate conservation principles -- though when complete sets of "isolated" events are considered, no violations occur.
- Aside from acausal behavior (that is, the "effect" happens before the "cause", or so-called "pre-acceleration") and "runaway" solutions, the chief peculiarity in radiation reaction is that no reaction force is present during constant accelerations (since ), notwithstanding that radiation is being emitted. Since radiation carries away energy and momentum, common sense suggests that a reaction force should be exerted on the emitter; but none is. Since the radiation reaction force is zero, all the work being done by the external force must go into the motion of the particle. Where, then, without violating the conservation of energy, does the energy going into the radiation field come from? Sounds suspiciously like a free lunch, doesn't it? Some rather curious answers to this question can be found in the literature.
- In fact, energy conservation can be achieved by allowing transient changes in the restmasses of accelerating charges. But energy conservation isn't necessarily strictly true instant-by-instant through any of the intervals.
- In electrodynamics all this is regarded as peculiar behavior and usually soft-pedaled. Since energy isn't a source of the electromagnetic field, it isn't too important. In gravity, however, that's not true it turns out.

Well, what about gravity?

- While inertial reaction forces can be identified as gravitational radiation reaction forces because of their instantaneous/absorber character, in the simplest approximation (the 1953 argument of Sciama) transient mass fluctuations simply don't show up. This is a consequence of the fact that this formalism is like Maxwell's equations for electrodynamics where, similarly, radiation reaction is absent. In more sophisticated formalisms (like the PPN formalism of general relativity Ken Nordtvedt published in 1988 [
*International Journal of Theoretical Physics*,**27**, 1395-1404]), transient mass fluctuations are indeed present.

How do you get these effects (other than finding them in Nordtvedt's paper)?

- Well, the usual way to do this is to write down a "Lagrangian" with the right stuff in it. (A Lagrangian is a mathematical "function" that involves various different forms of energy.) Then you operate on it with the variational calculus to obtain the "equations of motion of the field". What's the "right stuff"? The stuff that produces field equations that look good to you and don't lead to obvious conflict with known facts. (If you don't want higher order radiative reaction effects, all you have to do is choose the right stuff. If you do, same prescription.)
- There is a less sophisticated way to get a field equation. You can treat the inertial reaction force on an accelerating test particle as due to the action of a field (with a strength equal to the force divided by the mass of the test particle). You may recall from undergrad electricity and magnetism that Maxwell's equation for the electric field is obtained by taking the "divergence" of the field (the scalar product of the gradient operator and the field). By Gauss's divergence theorem it equals the local charge density. By itself this procedure isn't "relativistically invariant" (that is, relativistically correct) because it ignores time varying effects that bring in magnetism and other stuff.
- The relativistically invariant generalization of taking the divergence of a vector field is to take the "four-divergence". This adds a term involving a time-derivative to the usual three-divergence. That term, when applied to the inertial reaction force field, produces the transient mass fluctuations. Since they depend on and , they're pretty obviously "higher order" radiation reaction effects. And like the case for electrodynamics, instantaneous seeming violations of conservation principles can be expected. Although time averaged over a proper "isolated" system no violations occur.
- In electrodynamics transient energy (mass) fluctuations aren't especially interesting. First of all, they're awfully small in almost all circumstances. Second, electric charge is the source of the fields, so energy fluctuations don't affect the field.
- In gravity the situation is different. Things involving energy are sources of the gravitational field, so fluctuations may well affect the field. Moreover, the fluctuations are multiplied by the gravitational potential, an enormous number, and and can be made large by doing things very quickly.

If you're unconvinced that transient mass fluctuations might really occur, you might ask: Is there any way to get rid of these pesky transient mass fluctuations?

- Well, yes. You can always choose a Lagrangian with the right stuff in it and . . . Or you can assume that Gauss's divergence theorem can't be generalized to four dimensions. This, however, isn't a very good idea. For example, it's the vanishing of the four-divergence of the Einstein tensor that told him (and everyone else) he had the right stuff on the left-hand side of his general relativity field equations. The other possibility is that the universally accepted relativistic generalization of Newton's second law -- the fundamental equation of motion of all dynamics -- is wrong.
- There is another possibility I should mention: maybe we can rescale the value of the gravitational potential everywhere from the square of the speed of light to zero. (Global rescalings of this sort are called "gauge transformations of the 'first kind'".) That would take care of the problem. Sad to say, this is an extremely bad idea. Gravity is known, as a matter of fact, to be non-linear. P.C. Peters wrote a really neat article about this a long time ago [
*American Journal of Physics*,**49**, 564-569 (1981)]. You can't do this sort of re-scaling in a non-linear theory.

Surely you're kidding. Can this stuff really be right?

- No "new physics" is involved. All of the techniques and procedures are standard stuff.
- Trendy sophisticates would probably say that this is all pretty crude. Yes. But it seems to work.

If you've made it this far and want more details:

- Origin of Inertia: How inertia originates in the cosmos.
- Radiation Reaction: The weird world of radiation reactions where you seem to get energy from nowhere and things move before you begin to push them.
- Transient Mass Fluctuations: Where it's all put together and we end up doing strange things.

By the way, a note on these documents: There's presently no good format for displaying documents directly in web browsers. In the future that may change, but for now all equations in these documents are displayed as numerous inline gif Images. As a result they may not line up with the rest of the text very well, and in short appear a little funky. Sorry about that, but it's the best that could be done.

Copyright © 1998, James F. Woodward. This work, whole or in part, may not be reproduced by any means for material or financial gain without the written permission of the author.