Abstract
Einstein’s introduction of Mach’s principle, shortly after the publication of the General Relativity Theory and his remarks of 1921 are considered, especially his prediction that “spectator matter” should alter the masses of nearby objects. Difficulties of implementing the principle in cosmological terms are mentioned. Sciama’s vector gravity theory is laid out, noting the role of the vector potential in the production of inertial reaction forces. From Sciama’s work it follows that “critical cosmic matter density” and cosmic scale spatial flatness with their concomitant condition that the total scalar gravitational potential \( \varphi \) is equal to the square of the speed of light is required if inertial reaction forces are solely due to gravity. The Wilkinson Microwave Anisotropy Probe results, which show space at cosmic scale to be flat, are mentioned in this connection. Carl Brans’ argument about the role of spectator matter and Ken Nordtvedt’s comments on gravitomagnetism are then discussed. Radiation reaction, gravity waves, and the instantaneity of inertial reaction forces are then investigated. The relationalist and physical interpretations of Mach’s principle are mentioned, and the chapter concludes with the statement of the Mach-Einstein-Sciama laws of inertia.
So strongly did Einstein believe at that time in the relativity of inertia that in 1918 he stated as being on an equal footing three principles on which a satisfactory theory of gravitation should rest:
- 1.
The principle of relativity as expressed by general covariance.
- 2.
The principle of equivalence.
- 3.
Mach’s principle (the first time this term entered the literature):…that the \( {{g}_{{\mu \nu }}} \) are completely determined by the mass of bodies, more generally by \( {{T}_{{\mu \nu }}} \).
In 1922, Einstein noted that others were satisfied to proceed without this [third] criterion and added, “This contentedness will appear incomprehensible to a later generation however.”
….It must be said that, as far as I can see, to this day Mach’s principle has not brought physics decisively farther. It must also be said that the origin of inertia is and remains the most obscure subject in the theory of particles and fields. Mach’s principle may therefore have a future – but not without the quantum theory.
--Abraham Pais, Subtle is the Lord: the Science and the Life of Albert Einstein, pp. 287–288.
(Quoted by permission of Oxford University Press, Oxford, 1982)
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Notes
- 1.
Initially conceived of by George Pugh and Leonard Schiff in the 1960s, Gravity Probe B was a collection of high precision gyroscopes flown in a satellite in polar orbit intended to detect the dragging of spacetime caused by the rotation of Earth. The project, which flew several years ago, spanned decades and cost nearly a billion dollars. One noted relativist, queried by the press on the launch of the satellite, was reported to have remarked, “never was so much spent to learn so little.” The history of this project is yet to be written. But it will doubtless prove fascinating.
- 2.
Nowadays in some quarters so-called SI units are used. They make the magnitudes of many things normally encountered in field theory unintuitively large or small. I use the traditional Gaussian units of field theory because there was a good reason why they were adopted decades ago by those who work in this area.
- 3.
Newton is routinely credited with the discovery of the inverse square law of universal gravitation. But his contemporary Robert Hooke claimed to have independently discovered the inverse square law before Newton made public his claim. Newton refused the presidency of the Royal Society until shortly after Hooke’s death. Shortly thereafter, the Royal Society moved to new quarters, and Hooke’s papers from the 1680s were lost in the move. Whether Hooke actually discovered the inverse square nature of gravity, absent his papers, is a matter of conjecture. It seems unlikely, though, that he discovered the universal nature of the interaction.
- 4.
Actually, the “missing mass” problem was first identified in the 1930s by Fritz Zwicky by applying the “virial theorem” to clusters of galaxies. The virial theorem says that on average, the kinetic and potential energies of galaxies in clusters should be the same. So, by measuring the motions of galaxies in a cluster, you can estimate the mass of the cluster. It leads to galaxy cluster mass estimates 10–100 times greater than the light emitted suggests is present. Only later was it extended to encompass cosmology, too.
- 5.
Or Einstein’s vector approximation equation for the force exerted by spectator matter that is accelerating on other local objects.
- 6.
The Nordtvedt effect proposes that gravitational potential energies do contribute to the mass-energy of things and predicts (small) deviations from the predictions of GRT that would follow. Such effects have not been observed.
- 7.
He also predicted that the masses of things should vary as they are accelerated, an effect of the sort that we’ll be looking at in the next chapter.
- 8.
Nordtvedt considered only a rigid sphere of uniform density of modest dimensions. He did not extend the argument to the case where the sphere is the entire universe, as did Sciama.
- 9.
See: J. Sultana and D. Kazanas, arXiv:1104.1306v1 (astro-ph.CO, later published in the Journal of Modern Physics D). They find that the “Sciama” force is one quarter of that needed for an exact inertial reaction force. The factor of 4 discrepancy arises from the fact that Sultana and Kazanas simply assumed the “Sciama” force without deriving it from GRT, and Sciama’s calculation is not exactly equivalent to a general relativistic calculation like Nordtvedt’s. The difference is the factor of 4 that when multiplied times their result returns 1 almost exactly.
- 10.
The field strength of gravitational radiation depends on the frequency at which it is emitted. Gravitational waves, all other things held constant, depend on the fifth power of the emission frequency. This strong frequency dependence has led some to speculate that very high frequency gravitational waves might be used for propulsive purposes. Since the momenergy in gravity waves produced by human scale sources is so hopelessly minute, even allowing for unrealistically high frequency sources, gravity waves hold out no promise of practical scale effects.
- 11.
The technical term for such an acceleration is a “jerk.”
- 12.
Self energy in electrodynamics arises because the parts of an electric charge repel the other parts of the charge, and work must be done to compress the parts into a compact structure. The energy expended to affect the assembly is stored in the field of the charge. When the electron was discovered by J. J. Thomson in 1897, it was not long until H. A. Lorentz and others suggested that the electron’s mass might be nothing more than the energy stored in its electric field (divided by c 2). They used this conjecture to calculate the so-called “classical electron radius” that turns out to be about 10−13 cm. But should you assume that the size of the electron is zero, the energy of assembly turns out to be infinite.
- 13.
The “amplitude” (for an oscillatory field) or “field strength” (the magnitude of the scalar potential or field vector) is not the same as the “intensity” of the field. The intensity is proportional to the square of the field strength. So, a field whose strength decreases as 1/r has an intensity that decreases as 1/r 2, as does electromagnetic radiation (light), for example. When the intensity decreases at this rate, some energy just barely makes it to “asymptotic infinity.” If the intensity decreases faster than 1/r 2, as it does for any field whose strength decreases more quickly than 1/r, then no freely propagating energy makes it to asymptotic infinity.
- 14.
As Faraday discovered in the early 1840s when Newton’s “third letter to Bentley” was first published. Hitherto, Newton’s true views on action-at-a-distance were not generally known. After reading Newton’s letter, it is said that Faraday became positively boorish regaling everyone with the news that Newton rejected action-at-a-distance.
- 15.
Two exceptions to this rule should be noted. First, a spherical object whose parts are undergoing a uniform radial acceleration does not radiate as the quadrupole moment is and remains zero. While such an expansion changes the radial tension in the field, it produces no “kink” in the field of the sort shown in Fig. 2.1. Second, there are those who hope to find a way to couple an object directly to the distant matter in the universe and produce accelerations without the need for an anchoring local mass. Such speculations are sometimes referred to as “field effect” propulsion. Hope springs eternal.
- 16.
When I read it as a grad student in the 1960s, I thought he was nuts. But Feynman knew what he was doing. Frank Wilczek recounts (in The Lightness of Being, pp. 83–84) a conversation with Feynman in 1982 about fields: “… He had hoped that by formulating his theory directly in terms of paths of particles in space-time – Feynman graphs – he would avoid the field concept and construct something essentially new. For a while, he thought he had. Why did he want to get rid of fields? ‘I had a slogan, … The vacuum doesn’t weigh anything [dramatic pause] because nothing’s there! …’” Feynman initially thought that his path integral approach captured the chief feature of the action at a distance theory: no freely propagating radiation in spacetime.
- 17.
Paul Davies, author of many popular books on physics, however, recounts in his About Time that it was attendance at one of Hoyle’s lectures on this topic that set him on his early research career.
- 18.
In this connection, Paul Davies relates an apposite story: “… I ventured: “What is the origin of the random phase assumption?” To my astonishment and dismay, [David] Bohm merely shrugged and muttered: “Who knows?”
“But you can’t make much progress in physics without making that assumption,” I protested.
“In my opinion,” replied Bohm, “progress in science is usually made by dropping assumptions!”
This seemed like a humiliating put-down at the time, but I have always remembered these words of David Bohm. History shows he is right. …
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Addenda
Addenda
Addendum #1: On the Origin of Inertia Article
“On the Origin of Inertia” by D.W. Sciama, Monthly Notices of the Royal Astronomical Society, vol. 113, pp. 34–42. Reprinted under Wiley’s fair dealing policy.
Addendum #2: Brans on Gravitational Energy Localization
Reprinted excerpt with permission from C.H. Brans, “Mach’s principle and the Locally Measured Gravitational Constant in General Relativity,” Physical Review, vol. 125, pp. 388–396 (1962). Copyright 1962 by the American Physical Society.
Addenda #3: Excerpt from Nordtvedt
Ken Nordtvedt, “Existence of the Gravitomagnetic Interaction,” International Journal of Theoretical Physics, vol. 27, pp. 1395–1404. Reprinted with permission of Springer Verlag.
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© 2013 James F. Woodward
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Woodward, J.F. (2013). Mach’s Principle. In: Making Starships and Stargates. Springer Praxis Books(). Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5623-0_2
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