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Mach's principle


Mach's Principle is essentially Ernst Mach's C19th statement of the general principle of relativity combined with a suggestion as to how it might be implemented.

Definition of Mach's Principle

Arguments about "the principle" are complicated by the fact that Mach does not seem to have ever stated or defined it as such. Mach did produce the basic argument for the general principle of relativity, and argued that we could treat rotation as being purely relative, as long as we allowed the existence of gravitational fields of a non-traditional nature. Mach's argument has been characterised as "mass there dictates inertia here" . When we forcibly accelerate a body, and it resists due to its inertia, this inertia is a function of the body's mass interacting with the mass of the background universe. Since this remote interaction is immediate, it has to be an interaction of the existing fields of the two sets of masses, which brings in the equivalence principle, the assumed equivalence of inertial and gravitational mass.

Mach's argument was published and popularised in his book "The Science of Mechanics" (""), but since the book went through at least fifteen different editions, with some of the wording changing between editions, and not all editions being translated into English, it is difficult to choose a definitive version of Mach's statement of the argument.

Results of Mach's Principle

The main criticism of Mach's Principle was that it was an essentially "empty" and "retrospective" concept that generated no obvious new physical predictions – to blame rotational forces on the existence of an external shell of rotating matter was all very well, but we could not conduct experiments by changing the spin-rate of the universe, or by creating other cosmological-scale hollow shells and spinning them to produce measurable internal forces.

However, the use of topological arguments lets us extend Mach's principle from the case of inward-facing hollow shells to outward-facing massed bodies:

  • We notice: A rotating system feels an outward force, perpendicular to its rotation axis
    • MP explains: A rotating hollow shell generates an outward force within it, which can be expressed as a gravitational field
      • Topology says: A rotating star exerts in inward force, roughly proportional to the square of the velocity times mass, in other words, to its kinetic energy.
        • GR says: a rotating star shows a stronger gravitational attraction, due to the additional gravitational mass associated with its additional localised energy.
  • We notice: a missile thrown from a rotating system appears to deflect sideways, in the same direction as the moving background stars.
    • MP explains: A rotating hollow shell exerts a rotational force on its contents, acting in the direction of rotation.
      • Topology says: A rotating star exerts a sideways dragging effect on light and matter around it, acting in the direction of rotation.
        • GR says: A rotating star causes the surrounding region of spacetime to be twisted, so that light and matter deflect in the direction of rotation.
  • We notice: If we undergo forced acceleration, geeforces push us against the accelerating surface
    • MP explains: An accelerated hollow shell exerts a force on its contents, acting in the direction of acceleration.
      • Topology says: A forcibly-accelerated mass creates an associated dragging force in the region around it.
        • GR says: A forcibly-accelerated mass creates a spacetime distortion that encourages nearby matter and light to defect in the direction of acceleration.

To these four "standard" Machian GR effects, Einstein added a fifth in ~1921:

  • If inertia is a mass-field (gravitational) interaction, then increasing the density of the surrounding field ought to increase the inertia of a test body.
    • Hence, the inertia of a body shoudl be greater in a more intense gravitational filed.
      • Hence, a clock should tick more slowly in a more intense gravitational field.

Rather than making no testable predictions, Mach's principle arguably predicts from C19th logic the existence of gravitational time-dilation (Einstein 1911, Pound-Snider 1959), the dragging of inertial frames around a rotating body (Gravity Probe B, 19xxx)), and anticipates E=mc^2 (association of mass with localised energy, 1905).


Although Einstein considered these aspects of Mach's Principle to be fully supported in GR1916, the downgrading of the GPoR in 1960 to avoid conflicts with special relativity (GR1960) led to Mach's principle being presented as an old idea that had outlived its usefulness, and was no longer a supported principle under the new 1960s version of GR. towards the end of his career, Einstein also seemed to consider MP as having made all the predictions that it was capable of making, and was rejecting a fully observerspace approach, perhaps partly because of his unhappiness with the Copenhagen interpretation of quantum mechanics.

See also: