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Mach scale

Mach's principle

In theoretical physics, particularly in discussions of Theories of gravitation, Mach's principle (or Mach's conjecture) is the name given by Einstein to a vague hypothesis first supported by the physicist and philosopher Ernst Mach. The broad notion is that "mass there influences inertia here". This concept was a guiding factor in Einstein's development of the general theory of relativity. In many respects, this is a true statement in the general theory. However, because this principle is so vague, many distinct statements can be (and have been) made which would qualify as a Mach principle.

The basic idea also appears before Mach's time, in the writings of George Berkeley. The book Absolute or Relative Motion? (1896) by Benedict Friedländer and his brother Immanuel contained ideas similar to Mach's principle.

A very general statement of Mach's principle is "Local physical laws are determined by the large-scale structure of the universe."

Mach's principle

There is a fundamental issue in Relativity theory. If all motion is relative, how can we measure the inertia of a body? We must measure the inertia with respect to something else. But what if we imagine a particle completely on its own in the universe? We might hope to still have some notion of its state of rotation. Mach's principle is sometimes interpreted as the statement that such a particle's state of motion has no meaning in that case.

In Mach's words, the principle is embodied as follows:

"[The] investigator must feel the need of... knowledge of the immediate connections, say, of the masses of the universe. There will hover before him as an ideal insight into the principles of the whole matter, from which accelerated and inertial motions will result in the same way."
Albert Einstein seemed to view Mach's principle as something along the lines of:
"...inertia originates in a kind of interaction between bodies..."
In this sense, at least some Mach principles are related to philosophical holism. Mach's suggestion can be taken as the injunction that gravitation theories should be relational theories. Einstein brought the principle into mainstream physics while working on general relativity. Indeed it was Einstein who first coined the phrase Mach's principle. There is much debate as to whether Mach really intended to suggest a new physical law since he never states it explicitly.

The writing in which Einstein found inspiration from Mach was "The Science of Mechanics", where the philosopher criticized Newton's idea of absolute space, in particular the argument that Newton gave sustaining the existence of an advantaged reference system: what is commonly called "Newton's bucket argument".

In his Philosophiae Naturalis Principia Mathematica, Newton tried to demonstrate that one can always decide if one is rotating with respect to the absolute space, measuring the apparent forces that arise only when an absolute rotation is performed. If a bucket is filled with water, and made to rotate, initially the water remains still, but then, gradually, the walls of the vessel communicate their motion to the water, making it curve and climb up the borders of the bucket, because of the centrifugal forces produced by the rotation. Newton says that this thought experiment demonstrates that the centrifugal forces arise only when the water is in rotation with respect to the absolute space (represented here by the reference frame solidal with the earth, or better, the distant stars); instead, when the bucket was rotating with respect to the water no centrifugal forces were produced, this indicating that the latter was still with respect to the absolute space.

Mach, in his book, says that the bucket experiment only demonstrates that when the water is in rotation with respect to the bucket no centrifugal forces are produced, and that we cannot know how the water would behave if in the experiment the bucket's walls were increased in depth and width until they became leagues big. In Mach's idea this concept of absolute motion should be substituted with a total relativism in which every motion, uniform or accelerated, has sense only in reference to other bodies (i.e., one cannot simply say that the water is rotating, but must specify if it's rotating with respect to the vessel or to the earth). In this view, the apparent forces that seem to permit discrimination between relative and "absolute" motions should only be considered as an effect of the particular asymmetry that there is in our reference system between the bodies which we consider in motion, that are small (like buckets), and the bodies that we believe are still (the earth and distant stars), that are overwhelmingly bigger and heavier than the former. This same thought had been expressed by the philosopher George Berkeley in his De Motu. It is then not clear, in the passages from Mach just mentioned, if the philosopher intended to formulate a new kind of physical action between heavy bodies. This physical mechanism should determine the inertia of bodies, in a way that the heavy and distant bodies of our universe should contribute the most to the inertial forces. More likely, Mach only suggested a mere "redescription of motion in space as experiences that do not invoke the term space". What is certain is that Einstein interpreted Mach's passage in the former way, originating a long-lasting debate.

Most physicists believe Mach's principle physical theory that would explain a mechanism by which the stars can have such an effect. Although Einstein was intrigued and inspired by Mach's principle, Einstein's formulation of the principle is not a fundamental assumption of general relativity. There have been attempts to formulate a theory which is more fully Machian, such as Brans-Dicke theory, but most physicists argue that none have been fully successful.

Mach's principle in modern General Relativity

Einstein—before completing his development of the general theory of relativity—found an effect which he interpreted as being evidence of Mach's principle. We assume a fixed background for conceptual simplicity, construct a large spherical shell of mass, and set it spinning in that background. The reference frame in the interior of this shell will precess with respect to the fixed background. This effect is known as the Lense-Thirring effect. Einstein was so satisfied with this manifestation of Mach's principle that he wrote a letter to Mach expressing this:
"it... turns out that inertia originates in a kind of interaction between bodies, quite in the sense of your considerations on Newton's pail experiment... If one rotates [a heavy shell of matter] relative to the fixed stars about an axis going through its center, a Coriolis force arises in the interior of the shell; that is, the plane of a Foucault pendulum is dragged around (with a practically unmeasurably small angular velocity)."

The Lense-Thirring effect certainly satisfies the very basic and broad notion that "matter there influences inertia here The plane of the pendulum would not be dragged around if the shell of matter were not present, or if it were not spinning. As for the statement that "inertia originates in a kind of interaction between bodies", this too could be interpreted as true in the context of the effect.

More fundamental to the problem, however, is the very existence of a fixed background, which Einstein describes as "the fixed stars". Modern relativists see the imprints of Mach's principle in the Initial-Value Problem. Essentially, we need to separate spacetime into slices of constant time. When we do this, Einstein's equations can be decomposed into one set of equations which must be satisfied on each slice, and another set which describes how to move between slices. The equations for an individual slice are elliptic partial differential equations. In general, this means that only part of the geometry of the slice can be given by the scientist, while the geometry everywhere else will then be dictated by Einstein's equations on the slice.

In the context of an asymptotically flat spacetime, the boundary conditions are given at infinity. Heuristically, the boundary conditions for an asymptotically flat universe define a frame with respect to which inertia has meaning. By performing a Lorentz transformation on the distant universe, of course, this inertia can also be transformed.

See also

References

  • Sciama, D. W. (1971). Modern Cosmology. Cambridge: Cambridge University Press. OCLC 6931707. Dennis Sciama helped renew interest in Mach's principle with his writings in (among other places) this textbook.
  • A Look at the Abandoned Contributions to Cosmology of Dirac, Sciama and Dicke (arxiv:0708.3518)
  • Graneau, Peter; Graneau, Neal (2006). In the Grip of the Distant Universe - The Science of Inertia. Massachusetts: World Scientific. ISBN 981-256-754-2. A revival of Mach's principle, based on a Newtonian paradigm, to explain inertia forces.
  • arxiv.org, a machian model of Dark energy

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