# ⓘ Machs principle. In theoretical physics, particularly in discussions of gravitation theories, Machs principle is the name given by Einstein to an imprecise hypo ..

## ⓘ Machs principle

In theoretical physics, particularly in discussions of gravitation theories, Machs principle is the name given by Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach. The idea is that the existence of absolute rotation is determined by the large-scale distribution of matter, as exemplified by this anecdote:

You are standing in a field looking at the stars. Your arms are resting freely at your side, and you see that the distant stars are not moving. Now start spinning. The stars are whirling around you and your arms are pulled away from your body. Why should your arms be pulled away when the stars are whirling? Why should they be dangling freely when the stars dont move?

Machs principle says that this is not a coincidence - that there is a physical law that relates the motion of the distant stars to the local inertial frame. If you see all the stars whirling around you, Mach suggests that there is some physical law which would make it so you would feel a centrifugal force. There are a number of rival formulations of the principle. It is often stated in vague ways, like "mass out there influences inertia here". A very general statement of Machs principle is "local physical laws are determined by the large-scale structure of the universe".

This concept was a guiding factor in Einsteins development of the general theory of relativity. Einstein realized that the overall distribution of matter would determine the metric tensor, which tells you which frame is rotationally stationary. Frame-dragging and conservation of gravitational angular momentum makes this into a true statement in the general theory in certain solutions. But because the principle is so vague, many distinct statements can be and have been made that would qualify as a Mach principle, and some of these are false. The Godel rotating universe is a solution of the field equations that is designed to disobey Machs principle in the worst possible way. In this example, the distant stars seem to be revolving faster and faster as one moves further away. This example doesnt completely settle the question, because it has closed timelike curves.

## 1. History

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

## 2. Einsteins use of the 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 motion. Machs principle is sometimes interpreted as the statement that such a particles state of motion has no meaning in that case.

In Machs words, the principle is embodied as follows:

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 Machs principle as something along the lines of:

.inertia originates in a kind of interaction between bodies.

In this sense, at least some of Machs principles are related to philosophical holism. Machs 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 Machs 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 Newtons idea of absolute space, in particular the argument that Newton gave sustaining the existence of an advantaged reference system: what is commonly called "Newtons 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.This thought experiment demonstrates that the centrifugal forces arise only when the water is in rotation with respect to the absolute space 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 buckets walls were increased in depth and width until they became leagues big. In Machs 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. 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 Machs passage in the former way, originating a long-lasting debate.

Most physicists believe Machs principle was never developed into a quantitative physical theory that would explain a mechanism by which the stars can have such an effect. It was never made clear by Mach himself exactly what his principle was. Although Einstein was intrigued and inspired by Machs principle, Einsteins formulation of the principle is not a fundamental assumption of general relativity.

## 3. Machs principle in general relativity

Because intuitive notions of distance and time no longer apply, what exactly is meant by "Machs principle" in general relativity is even less clear than in Newtonian physics and at least 21 formulations of Machs principle are possible, some being considered more strongly Machian than others. A relatively weak formulation is the assertion that the motion of matter in one place should affect which frames are inertial in another.

Einstein, before completing his development of the general theory of relativity, found an effect which he interpreted as being evidence of Machs 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 Machs 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 Newtons pail experiment. If one rotates 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, Machs 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. Machs principle may therefore have a future – but not without the quantum theory.

## 4. Variations in the statement of the principle

The broad notion that "mass there influences inertia here" has been expressed in several forms. Hermann Bondi and Joseph Samuel have listed eleven distinct statements that can be called Mach principles, labelled Mach0 through Mach10. Though their list is not necessarily exhaustive, it does give a flavor for the variety possible.

• Mach3: Local inertial frames are affected by the cosmic motion and distribution of matter.
• Mach9: The theory contains no absolute elements.
• Mach7: If you take away all matter, there is no more space.
• Mach8: Ω = def 4 π ρ G T 2 {\displaystyle \Omega \ {\stackrel {\text{def}}{=}}\ 4\pi \rho GT^{2}} is a definite number, of order unity, where ρ {\displaystyle \rho } is the mean density of matter in the universe, and T {\displaystyle T} is the Hubble time.
• Mach5: The total energy, angular and linear momentum of the universe are zero.
• Mach0: The universe, as represented by the average motion of distant galaxies, does not appear to rotate relative to local inertial frames.
• Mach6: Inertial mass is affected by the global distribution of matter.
• Mach10: Overall rigid rotations and translations of a system are unobservable.
• Mach2: An isolated body in otherwise empty space has no inertia.
• Mach1: Newton’s gravitational constant G is a dynamical field.
• Mach4: The universe is spatially closed.

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