Mach's principle

# Mach's principle

Discussion

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In theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

, particularly in discussions of gravitation theories, Mach's principle (or Mach's conjecture) is the name given by Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

to an imprecise hypothesis often credited to the physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

and philosopher Ernst Mach.

The idea is that the local motion of a rotating reference frame 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 don't move?

Mach's 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 were whirling around you, Mach suggests that there is some physical law which would make it so you would feel a centrifugal force
Centrifugal force
Centrifugal force can generally be any force directed outward relative to some origin. More particularly, in classical mechanics, the centrifugal force is an outward force which arises when describing the motion of objects in a rotating reference frame...

. There are a number of rival formulations of the principle
Mach principle
In theoretical physics, particularly in discussions of gravitation theories, a Mach principle is any of a class of principles which are more specific statements of Mach's principle....

. It is often stated in vague ways, like "mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

out there influences inertia
Inertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

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

This concept was a guiding factor in Einstein's development of the general theory of relativity. Einstein realized that the overall distribution of matter would determine the metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

, 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 which would qualify as a Mach principle, and some of these are false. The Gödel rotating universe
Gödel metric
The Gödel metric is an exact solution of the Einstein field equations in which the stress-energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles, and the second associated with a nonzero cosmological constant...

is a solution of the field equations which is designed to disobey Mach's principle in the worst possible way. In this example, the distant stars seem to be rotating faster and faster as one moves further away. This example doesn't completely settle the question, because it has closed timelike curves.

## History

The basic idea also appears before Mach's time, in the writings of George Berkeley
George Berkeley
George Berkeley , also known as Bishop Berkeley , was an Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism"...

. The book Absolute or Relative Motion? (1896) by Benedict Friedländer and his brother Immanuel contained ideas similar to Mach's principle.

## Einstein's 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 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:
Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

seemed to view Mach's principle as something along the lines of:
In this sense, at least some of Mach principles are related to philosophical holism
Holism
Holism is the idea that all the properties of a given system cannot be determined or explained by its component parts alone...

. Mach's suggestion can be taken as the injunction that gravitation theories should be relational theories
Relational theory
In physics and philosophy, a relational theory is a framework to understand reality or a physical system in such a way that the positions and other properties of objects are only meaningful relative to other objects...

. Einstein brought the principle into mainstream physics while working on general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

. 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
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

'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
Bucket argument
Isaac Newton's rotating bucket argument was designed to demonstrate that true rotational motion cannot be defined as the relative rotation of the body with respect to the immediately surrounding bodies...

".

In his Philosophiae Naturalis Principia Mathematica
Philosophiae Naturalis Principia Mathematica
Philosophiæ Naturalis Principia Mathematica, Latin for "Mathematical Principles of Natural Philosophy", often referred to as simply the Principia, is a work in three books by Sir Isaac Newton, first published 5 July 1687. Newton also published two further editions, in 1713 and 1726...

, 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
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...

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
George Berkeley
George Berkeley , also known as Bishop Berkeley , was an Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism"...

in his De Motu
De Motu (Berkeley's essay)
De Motu is an essay written by George Berkeley and published in 1721. Its full title is De Motu or The Principle and Nature of Motion and the Cause of the Communication of Motions. The essay was unsuccessfully submitted for a prize that had been offered by the Royal Academy of Sciences at...

. 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 was never developed into a quantitative 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
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

. 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.

## 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
Frame-dragging
Einstein's general theory of relativity predicts that non-static, stationary mass-energy distributions affect spacetime in a peculiar way giving rise to a phenomenon usually known as frame-dragging...

. Einstein was so satisfied with this manifestation of Mach's principle that he wrote a letter to Mach expressing this:
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 humans seem to wish to separate spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

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 describe 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
Asymptotically flat spacetime
An asymptotically flat spacetime is a Lorentzian manifold in which, roughly speaking, the curvature vanishes at large distances from some region, so that at large distances, the geometry becomes indistinguishable from that of Minkowski 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
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...

on the distant universe, of course, this inertia can also be transformed.

• Woodward effect
Woodward effect
The Woodward effect is a hypothesis proposed by James F. Woodward, a physicist at California State University, Fullerton, that energy-storing ions experience transient mass fluctuations when accelerated. While some have expressed doubt about this hypothesis, no respected theoreticians have yet...

• Mach principle
Mach principle
In theoretical physics, particularly in discussions of gravitation theories, a Mach principle is any of a class of principles which are more specific statements of Mach's principle....

• Inertial frame
• Absolute rotation
• Absolute space
• Luminiferous aether
Luminiferous aether
In the late 19th century, luminiferous aether or ether, meaning light-bearing aether, was the term used to describe a medium for the propagation of light....

• Newton's bucket
• Frame-dragging
Frame-dragging
Einstein's general theory of relativity predicts that non-static, stationary mass-energy distributions affect spacetime in a peculiar way giving rise to a phenomenon usually known as frame-dragging...

• Brans–Dicke theory
• General relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

• Bucket argument
Bucket argument
Isaac Newton's rotating bucket argument was designed to demonstrate that true rotational motion cannot be defined as the relative rotation of the body with respect to the immediately surrounding bodies...

• Wheeler–Feynman absorber theory