Toy Universes

Gravity inside our minds

Mach’s Principle Number 4

with 7 comments

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

Mach Principle 4: “As a mechanical interaction of masses, in particular through as interactive Machian ‘kinetic energy’”.

That particular interpretation is not only interesting by itself, but also for its apparent historical origin. The above-mentioned ‘Machian kinetic energy’ is explained in the article by John D. Norton on pages 32-33 of [Barb95], from which my brief exposition is here based.

The idea emerges as a Machian hypothesis by its own essentially because it arises from a quote by Einstein in a 1913 paper, where he discussed “the hypothesis of the relativity of inertia” and cites an “ingenious pamphlet of the Viennese mathematician W. Hofmann”.

Essentially, Hofmann (1904) argued that the standard kinetic energy L of a body of mass m (L ={1 \over 2} m v^2) was unsatisfactory, due to an intrinsic asymmetry: the kinetic energy of m with respect to M would not be the same as that of M with respect to m. This consideration would be a necessary outcome of the assumption of the relativity of inertia.

Therefore, in order to have a symmetric law for the kinetic energy, Hofmann proposed that

L = k M m f(r) v^2,

where k is a constant and f an undetermined function. The idea was: since the mass m measured of a given particle should arise as a contribution of all masses in the Universe according to that law, then by an integration of all masses, that formula would have to converge to the usual formula for the kinetic energy.

Further work along that line was carried out by H. Reissner in 1914-15, but he cited Mach for the modified kinetic energy law, and Hofmann was not mentioned.

(Well, I was unsuccessful to find any further online references for W. Hofmann, either a bio or link to his “pamphlet”…)

For historical details, curiosities and references, the reader is referred to Norton’s article.

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Written by Christine

January 27, 2012 at 5:20 PM

Posted in Mach's Principle

7 Responses

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  1. You likely have run across the notes in “Mach’s Principle: From Newtons Bucket to Quantum Gravity”. There is discussion of three people concerning the “Machian kinetic energy” idea, Hofmann, Reissner, and Schrodinger. Schrodinger’s paper is from 1925, does not reference Reissner or Hofmann, and uses a different f(r) than Reissner, so it could be that all three of them devised the notion independently.

    Bill Davidson

    January 30, 2012 at 3:15 AM

  2. Yes, thank you for pointing that out. Indeed, it seems likely that this concept was conceived independently by the three of them. Do you know of any subsequent (recent) work along those lines?

    By the way, the discussion sections of that book are extremely interesting and one can only wonder how much had been missed not having attended it…



    January 30, 2012 at 7:54 AM

    • I don’t know of any further work along those lines, just what was mentioned in those notes. Maybe Schrodinger got about as far as was possible using simple Newtonian ideas.

      One might argue that the Hoyle-Narlikar theory fits the bill since their gravitational action principle contains an “inter-particle” product.


      Bill Davidson

      January 30, 2012 at 6:24 PM

    • Oops, I see now that your reference “Barb95″ was that book I mentioned earlier, so of course you know all about it. I should have read the whole blog thread before commenting at all.

      Bill Davidson

      January 30, 2012 at 7:02 PM

      • No problem, it’s fine. You have enriched what I have written – admittedly just a brief summary anyway. Thanks,



        January 30, 2012 at 7:19 PM

      • Thanks. I enjoyed you blog. I found it through a Google Alert for “Mach’s Principle”. It is sure a fun topic and is not usually discussed well on the internet. It is especially a treat to read your reporting of the people and history.

        If you really want me to be less terse I’d be happy to do so, but all I can say is what I think and I have no particular qualifications. Please correct me when you know I’m in error:

        Going further with your well-stated Mach’s Principle 4, formalizing the approach seems to end up with inertial mass that varies in time and place and is even anisotropic. Schrodinger called it “radial” and “tangential” inertial mass. What is cool, though, about Schrodinger’s 1925 paper is that he can easily deduce the precession of Mercury’s perihelion using his simple model Machian universe, and with a lot less fancy machinery than is needed to do it by general relativity (GR).

        To me his success points out that there are really two alternate ways to envision the working of gravity; (1) as an effect directly on local space-time as done in GR, or (2) as an effect by one mass on another mass within a “flat” background. GR was the favored approach and won the verdict of history. But if you want to take the other route you can if only you try hard enough, and that is what Hoyle and Narlikar attempted. Their section on conformal invariance (6.2) in their book “Action At A Distance in Physics and Cosmology” is really interesting. It indicates you have to make a hard choice between retaining the geodesic equation on the one hand and conformal invariance on the other.

        So with GR you may already have a theory that is fully Machian in the sense of Mach’s Principle 4, but you can’t easily prove it.

        Bill Davidson

        January 30, 2012 at 9:07 PM

  3. Thank you for the additional information, that’s interesting and useful! I will have to take a closer look at that paper by Schrödinger as well as the discussion in Hoyle and Narlikar’s book.

    You make an interesting point. It seems a really difficult question and I wonder how it could be addressed. The only hand waving idea that occurs me now in order to conciliate both approaches would be by showing that the local metric (given in GR) could be consistently written as an integral expression involving the masses in an otherwise flat spacetime. But I’m certainly being naive here. I must read the material that you mention to penetrate further into the arguments.




    January 30, 2012 at 10:24 PM

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