Editing Mach's principle (section)

== Mach's principle in general relativity ==
{{more citations needed section|date=November 2015}}
Because intuitive notions of distance and time no longer apply, what exactly is meant by "Mach's principle" in general relativity is even less clear than in Newtonian physics and at least 21 formulations of Mach's principle are possible, some being considered more strongly Machian than others.<ref name=":0" />{{rp|530}}  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 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 [[Precession|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:
{{quote|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).<ref name=Einstein/>}}

The Lense–Thirring effect certainly satisfies the very basic and broad notion that "matter there influences inertia here".<ref>{{cite journal |author=Bondi, Hermann |author2= Samuel, Joseph |name-list-style=amp |title=The Lense–Thirring Effect and Mach's Principle |journal= Physics Letters A |volume=228 |issue=3 |pages=121 |arxiv=gr-qc/9607009 |date= July 4, 1996 |doi= 10.1016/S0375-9601(97)00117-5 |bibcode= 1997PhLA..228..121B |s2cid= 15625102 }}  A useful review explaining the multiplicity of "Mach principles" which have been invoked in the research literature (and elsewhere).</ref>  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]] 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 equation]]s.  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.{{clarify|date=June 2017}}

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{{clarification needed|date=November 2021}}.

A stronger form of Mach's principle applies in [[Wheeler–Mach–Einstein spacetime]]s, which require spacetime to be spatially [[Compact space|compact]] and [[Globally hyperbolic manifold|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]]).<ref name=":0" />{{rp|188–207}}

There have been other attempts to formulate a theory that is more fully Machian, such as the [[Brans–Dicke theory]] and 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".<ref name=":0" />{{rp|106}}

However, Einstein was convinced that a valid theory of gravity would necessarily have to include the relativity of inertia:
{{quote|text=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:
# The principle of relativity as expressed by general covariance.
# The principle of equivalence.
# Mach's principle (the first time this term entered the literature): … that the ''g<sub>µν</sub>'' are completely determined by the mass of bodies, more generally by ''T<sub>µν</sub>''. 
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.
 |author=[[Abraham Pais]]
 |source=in ''Subtle is the Lord: the Science and the Life of Albert Einstein'' (Oxford University Press, 2005), pp. 287–288.
}}