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Einstein on General Relativity: Does a Pendulum’s Swing Prove the Earth’s Rotation, or that the Universe is Revolving Around the Earth
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“According to the general theory of relativity, one must consider that the Foucault pendulum adjusts itself to be rotation-free with respect to the total mass of the universe. This mutual influence should however not be interpreted as action at a distance: themasses define the gμν-Feld in space, and this field defines the inertial behavior of the mass of the Foucault pendulum.

Albert Einstein. Typed Letter Signed, “A. Einstein”, with handwritten annotation, “gµν-Feld” [i.e., the gravitational metric field]. To Axel Frey Samsoie. Berlin, February 10, 1921. 1p, in German.

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A highly desirable letter From Einstein’s Nobel Prize year, 1921. Einstein points out the interdependence of gravity and motion to re-affirm the relativity of all motion and the absence of privileged reference frames.

Samsoie had questioned: “when we ascertain a rotation of the Earth by means of Foucault’s pendulum, is the rotation a motion with respect to the Sun, the stars, or to something else?” In other words, if all frames of reference are valid, couldn’t a Foucault pendulum “prove” that the earth is stationary and that the whole universe rotates around it?

Einstein’s brief answer may be his most complete explanation of this key General Relativity question on the possibility of a “frame dragging” effect due to the earth’s rotation. He draws on Mach’s principle, in a way that evokes John Wheeler’s famous description of the essence of General Relativity: “Spacetime tells matter how to move; matter tells spacetime how to curve.”

Following Newton’s laws of motion, the pendulum is actually moving in a “straight line” as defined by the curvature of spacetime. The nonlinear motion that the pendulum appears to sustain from our perspective is an illusion tracing of the rotation of the earth beneath it.

Mach’s Principle (or conjecture) demonstrates the interdependence of gravity and motion. The statement that the pendulum is “rotation-free with respect to the total mass of the universe” is Einstein signaling that the universe is not rotating around a still earth, but in fact it is the earth that is rotating. However, Einstein appears to have changed his own position on the issue and even on Mach’s Principle, more than once over the course of his career, so it is notable to have him here on record—some five years after the publication of his epochal Grundlage statement (Einstein’s greatest and most substantial work, ‘Die Grundlage der allgemeine Relativitätstheorie’ or ‘The Foundations of the General Theory of Relativity’)—affirming its relativity.

Einstein had referenced Foucault’s pendulum in an important 1912 thought experiment relating to Mach’s Principle, which led Einstein to posit both the relativity of inertial mass and the Einstein-Lense-Thirring frame-dragging effect. (Cf. Einstein’s historic 1913 letter to Mach.)

Though some attention was paid in the first decade of General Relativity to the question of whether the theory could account for the rotation of a gyroscope, surprisingly little attention was paid to the question of a Foucault pendulum—hence Samsoie’s question to Einstein.

Samsoie’s original letter, on which Einstein drafted his response, is in the Einstein archives in Jerusalem. Einstein originally thought to begin his answer by stating: “From the point of view of the Lorentzian Theory [i.e., Special Relativity], the answer to your question was that Foucault's experiment indicated the rotation against the ether. One can keep this statement from the standpoint of General Relativity.”

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Excerpted from Wikipedia
“Mach's principle” is the name given by Einsteinto an imprecise hypothesis he credited to the physicistand philosopherErnst Mach, which explained how rotating objects, such as gyroscopes and spinning celestial bodies, maintain a frame of reference. The idea is that the existence of absolute rotation(the distinction of local inertial framesvs. rotating reference frames) is determined by the large-scale distribution of matter, as exemplified by this anecdote:[2]

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?...

This 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, which tells you which frame is rotationally stationary. Frame-draggingand conservation of gravitational angular momentummakes this into a true statement in the general theory in certain solutions. But the principle is vague; many distinct statements have been made that would qualify as a Mach principle, though some of them are false.

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. Mach's principle is sometimes interpreted as the statement that such a particle's state of motion has no meaning in that case.

Mach's suggestion can be taken as the injunction that gravitation theories should be relational theories. Einstein found inspiration from Mach’s "The Science of Mechanics," which criticized Newton's idea of absolute space, in particular Newton’s bucket argument" sustaining the existence of an advantaged reference system.

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 experimentdemonstrates that the centrifugal forces arise only when the water is in rotation with respect to the absolute space (represented here by the earth's reference frame, 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 said 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. He believed the 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, discrimination between relative and "absolute" motions is an effect of the particular asymmetry of our reference system. Bodies which we consider in motion are small (like buckets), and bodies that we believe are still (the earth and distant stars) that are overwhelmingly bigger and heavier than the former.

Mach was likely only suggested a "re-description of motion in space as experiences that do not invoke the term space".  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.

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 forcearises in the interior of the shell; that is, the plane of a Foucault pendulumis dragged around (with a practically unmeasurably small angular velocity).[6]

The Lense–Thirring effect certainly satisfies the very basic and broad notion that "matter there influences inertia here".[8]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 spacetimeinto slices of constant time. When we do this, Einstein's equationscan 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.[clarification needed]

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

A stronger form of Mach's principle applies in Wheeler–Mach–Einstein spacetimes, which require spacetime to be spatially compactand globally hyperbolic. In such universes Mach's principle can be stated as the distribution of matter and field energy-momentum (and possibly other information) at a particular moment in the universe determines the inertial frame at each point in the universe (where "a particular moment in the universe" refers to a chosen Cauchy surface).[7]: 188–207 

There have been other attempts to formulate a theory that is more fully Machian, such as the Brans–Dicke theoryand the Hoyle–Narlikar theory of gravity, but most physicists argue that none have been fully successful. At an exit poll of experts, held in Tübingen in 1993, when asked the question "Is general relativity perfectly Machian?", 3 respondents replied "yes", and 22 replied "no". To the question "Is general relativity with appropriate boundary conditions of closure of some kind very Machian?" the result was 14 "yes" and 7 "no".[7]: 106 

However, Einstein was convinced that a valid theory of gravity would necessarily have to include the relativity of inertia. So strongly did Einstein believe at that time in the relativity of inertia that in 1918 he stated as being on an equal footing three principles on which a satisfactory theory of gravitation should rest:

1.     The principle of relativity as expressed by general covariance.

2.     The principle of equivalence.

3.     Mach's principle (the first time this term entered the literature): … that the gµν are completely determined by the mass of bodies, more generally by Tµν.

In 1922, Einstein noted that others were satisfied to proceed without this [third] 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, Mach's 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. Mach's principle may therefore have a future – but not without the quantum theory.

— Abraham Pais, in Subtle is the Lord: the Science and the Life of Albert Einstein (Oxford University Press, 2005), pp. 287–288.

Variations in the statement of the principle
The broad notion that "mass there influences inertia here" has been expressed in several forms. Hermann Bondiand Joseph Samuel have listed eleven distinct statements that can be called Mach principles[10], some of which are:  

  • Mach1: Newton's gravitational constantG is a dynamical field.
  • Mach2: An isolated body in otherwise empty space has no inertia.
  • Mach4: The universe is spatially closed.
  • Mach5: The total energy, angular and linear momentum of the universe are zero.
  • Mach7: If you take away all matter, there is no more space.
  • Mach9: The theory contains no absolute elements.

Hans Christian Von Bayer, The Fermi Solution: Essays on Science, Courier Dover Publications (2001), ISBN 0-486-41707-7, page 79.

Steven, Weinberg (1972). Gravitation and Cosmology. USA: Wiley. pp. 17. ISBN 978-0-471-92567-5.

Stephen W. Hawking & George Francis Rayner Ellis (1973). The Large Scale Structure of Space–Time. Cambridge University Press. p. 1. ISBN 978-0-521-09906-6.

G. Berkeley (1726). The Principles of Human Knowledge.See para. 111–117, 1710.

Mach, Ernst (1960). The Science of Mechanics; a Critical and Historical Account of its Development. LaSalle, IL: Open Court Pub. Co. LCCN 60010179.This is a reprint of the English translation by Thomas H. MCormack (first published in 1906) with a new introduction by Karl Menger

A. Einstein, letter to Ernst Mach, Zurich, 25 June 1913, in Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 978-0-7167-0344-0.

Julian B. Barbour; Herbert Pfister, eds. (1995). Mach's principle: from Newton's bucket to quantum gravity. Volume 6 of Einstein Studies. Boston: Birkhäuser. ISBN 978-3-7643-3823-7.

Bondi, Hermann & Samuel, Joseph (July 4, 1996). "The Lense–Thirring Effect and Mach's Principle". Physics Letters A. 228 (3): 121. arXiv:gr-qc/9607009. Bibcode:1997PhLA..228..121B. doi:10.1016/S0375-9601(97)00117-5. S2CID 15625102.

Sciama, D. W. (1953-02-01). "On the Origin of Inertia". Monthly Notices of the Royal Astronomical Society. 113 (1): 34–42. Bibcode:1953MNRAS.113...34S. doi:10.1093/mnras/113.1.34. ISSN 0035-8711.