Mach's principle

• 1. Introduction

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  Mach's principle asks one of the strangest questions in all of physics: why do your arms fly outward when you spin? Not because of anything touching them, but because of the distant stars. That is the claim, at least, and it is strange enough that Albert Einstein named a principle after it and spent years debating whether his own theory of gravity satisfied it.

  The physicist and philosopher Ernst Mach put the idea into print in his 1883 book The Science of Mechanics. He was challenging Newton's conception of absolute space, a conception Newton himself had defended with a famous bucket of water. From that challenge came something that refuses to stay settled: a question about whether the universe's matter determines, at every point, what counts as truly rotating and what counts as truly still.

  Why did Einstein call it a principle if no one can agree on what it says? And why, after more than a century, does the origin of inertia remain, in the words of one physicist who knew the problem well, "the most obscure subject in the theory of particles and fields"?

• 2. Newton's Bucket and Berkeley's Doubt

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  In his Philosophiae Naturalis Principia Mathematica, Isaac Newton argued that you can always tell whether you are truly rotating. Fill a bucket with water and set it spinning. At first the water stays flat while the bucket walls move around it. Then the water picks up the rotation, curves upward along the sides, and climbs the walls. Newton read that climbing as proof of rotation relative to absolute space, not rotation relative to the bucket. When the bucket turned and the water stayed still, the water was flat. When the water turned too, the water curved. The bucket's motion relative to the water was the same in both cases; the result was not. Something else, Newton said, must explain the difference.

  Mach's counterargument, published in 1883, was precise in its skepticism. He pointed out that Newton's experiment only demonstrates that no centrifugal forces appear when the water rotates relative to the bucket. It says nothing about what would happen if the bucket's walls were enlarged to the scale of leagues, until the bucket itself became the dominant mass in the experiment. In Mach's view, absolute motion should be replaced by a total relativism in which every motion, uniform or accelerated, has meaning only with respect to other bodies.

  The philosopher George Berkeley had expressed this same thought earlier in his De Motu, which Mach's argument echoes. After Mach, the book Absolute or Relative Motion?, published in 1896 by Benedict Friedlaender and his brother Immanuel, carried similar ideas further. Whether Mach intended a new physical law or merely a redescription of motion without invoking the term space is, as the source notes, genuinely unclear.

• 3. Einstein Coins the Name

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  Einstein was the one who gave the principle its name, and the naming carries significance: it was Einstein who first used the phrase "Mach's principle" in print. He encountered the key passage in The Science of Mechanics and read it as a call for gravitation theories to be relational theories, where inertia originates in interaction between bodies rather than in any background absolute space.

  In 1918, Einstein listed three principles on which a satisfactory theory of gravitation should rest. The first was the principle of relativity as expressed by general covariance. The second was the principle of equivalence. The third was Mach's principle, which he stated as the requirement that the gravitational metric tensor be completely determined by the mass of bodies, more generally by the energy-momentum tensor. He placed all three on equal footing.

  By 1922, Einstein noted that other physicists were content to proceed without that third criterion. He responded that such contentedness would appear incomprehensible to a later generation. Yet Einstein himself eventually acknowledged, in a candid assessment recorded in the source, that Mach's principle had not brought physics decisively farther, and that inertia's origin remains obscure. He closed with a conditional: Mach's principle may have a future, but not without the quantum theory.

• 4. The Lense-Thirring Effect

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  Before completing the general theory of relativity, Einstein found what he believed was evidence that Mach's principle was physically real. He imagined a large spherical shell of mass set spinning against a fixed background. Inside the shell, the reference frame precesses relative to that background. This is the Lense-Thirring effect, and Einstein was moved enough by it to write directly to Mach.

  In that letter, quoted in the source, Einstein described how, if a heavy shell of matter is rotated relative to the fixed stars about an axis through its center, a Coriolis force arises inside the shell. A Foucault pendulum inside would have its plane dragged around, though with an angular velocity Einstein described as practically unmeasurably small. He framed it explicitly as confirmation of Mach's thoughts on Newton's bucket.

  The effect does satisfy the broad claim that matter there influences inertia here: remove the spinning shell or stop it from spinning, and the pendulum's plane would not be dragged. But the Lense-Thirring effect depends on a fixed background, which Einstein called the fixed stars. Modern relativists see the deeper imprint of Mach's principle not in this effect but in the initial-value problem of general relativity, where elliptic partial differential equations on each time-slice mean that only part of the geometry can be freely specified; the rest is dictated by Einstein's equations.

• 5. Counting Formulations

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  Hermann Bondi and Joseph Samuel catalogued the problem's scope by listing eleven distinct statements that can each be called a Mach principle. They labelled them Mach0 through Mach10, taking inspiration from the Mach number used in aerodynamics. Their list is not exhaustive; the source notes at least 21 formulations are possible when general relativity is the setting, some considered more strongly Machian than others.

  A relatively weak formulation holds only that matter in one place should affect which frames are inertial in another. Stronger formulations require spacetime to be spatially compact and globally hyperbolic, as in Wheeler-Mach-Einstein spacetimes, where the distribution of matter and field energy-momentum at a Cauchy surface determines the inertial frame at every point.

  In 1953, the Cambridge University physicist Dennis W. Sciama attempted to give the principle quantitative form by adding an acceleration-dependent term to the Newtonian gravitation equation. Sciama's term involved the distance between particles, the gravitational constant, the relative acceleration, and the speed of light in vacuum. He called the effect inertial induction. Other attempts at a more fully Machian theory include the Brans-Dicke theory and the Hoyle-Narlikar theory of gravity, though most physicists argue that none has been fully successful.

• 6. The 1993 Tubingen Poll

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  At an exit poll of experts held in Tubingen in 1993, physicists were asked two questions about general relativity and Mach's principle. To the direct question "Is general relativity perfectly Machian?", three respondents said yes and 22 said no. The result was decisive.

  The second question was softer: "Is general relativity with appropriate boundary conditions of closure of some kind very Machian?" There the vote shifted: 14 yes against 7 no. The boundary conditions matter. An asymptotically flat spacetime defines, through conditions given at infinity, a frame with respect to which inertia has meaning; a Lorentz transformation applied to the distant universe transforms that inertia. Closure changes the picture.

  The Godel rotating universe stands as the sharpest counterexample. It is a solution of Einstein's field equations constructed precisely to disobey Mach's principle as thoroughly as possible. In that universe, distant stars appear to revolve faster and faster the further away they are. The source notes that this example does not completely settle the question of physical relevance because the Godel universe contains closed timelike curves, which create their own interpretive complications. The debate has never been cleanly resolved, and the principle remains a vague but persistent presence at the edge of gravitational theory.