General Relativity - Machian Viewpoint

The Machian interpretation of general relativity holds that inertia, and the distinction between accelerated and unaccelerated motion, are not properties an object possesses on its own but are relations to the distribution of all other matter in the universe. On this view, general relativity is best understood not merely as a theory of curved spacetime geometry treated as a freestanding entity, but as a partial, imperfect realization of Ernst Mach's critique of Newtonian absolute space - a theory in which, ideally, mass and motion everywhere are mutually determining and nothing is left over to act as an unexplained backdrop. Holders of this viewpoint include philosophers of physics, historians of science, and a minority of working relativists who think the geometric reading of General Relativity common in textbooks obscures the relational core of the theory and leaves residual absolute elements that Einstein himself wanted to eliminate.

The starting point is Mach's reanalysis of Newton's rotating bucket experiment. Newton had argued that because the water's surface curves upward when the bucket spins, even when bucket and water are not in relative motion to each other, the curvature must be caused by rotation relative to absolute space itself, not relative motion among bodies. Mach rejected this. In his *Science of Mechanics*, Mach argued that the water's behavior could only ever be tested, in practice, against the rest of the matter in the universe - the fixed stars, the distant galaxies - and that invoking an undetectable absolute space to explain the effect was explanatorily idle. He proposed instead that the water curves because it is rotating relative to the total mass distribution of the universe, and that if one could somehow hold the water fixed and instead rotate the entire remaining mass of the universe around it, the same centrifugal effect ought to appear. Inertia, on this view, is not an intrinsic property of bodies but a relational effect of all matter on all other matter.

Einstein read Mach's *Science of Mechanics* as a student and later credited it as a decisive influence on his thinking, alongside the equivalence principle, in the years leading up to general relativity. Einstein coined the term “Mach's principle” for the general inspiration he found in Mach's work on mechanics, even though Mach himself never formulated the principle succinctly. Machians hold that Einstein's original ambition was for general relativity to be a fully relational theory: that the metric field at any point should be wholly determined by the distribution of mass-energy, with no residual geometric structure left unexplained by matter. In Einstein's own words, the gravitational field is completely determined by the masses of bodies, and because mass and energy are identical, the field is conditioned and determined by the energy tensor. Machians regard this ambition as the proper lens for interpreting the theory's structure and significance, even where the finished mathematics falls short of it.

Proponents of this viewpoint are typically candid that strict formulations of Mach's principle are not satisfied by general relativity as Einstein actually completed it, and they treat this as a problem internal to the theory's foundations rather than evidence against the Machian project itself. They point to several well-known anti-Machian features:

• Minkowski spacetime - flat, empty spacetime with no matter at all - nonetheless supports a perfectly well-defined inertial structure, geodesics, and centrifugal effects, which is precisely what Mach's principle says should be impossible in an empty universe.
• The Gödel solution describes a rotating universe with no privileged non-rotating frame, which Machians read as a sign that the theory permits exactly the kind of “absolute rotation” Mach wanted to banish. (1)
• Mach's principle in essence advocates shunning the unobservable absolute space and time of Newton in favor of the directly observable background matter of the universe, which determines geometry and inertia, yet Einstein's finished theory contains several anti-Machian features despite his expectation that general covariance would automatically secure a Machian result. (2)
• Some Machians further note an in-principle tension between Mach's principle and the equivalence principle: the former is global, tying local inertia to the energy-momentum content of the whole universe, while the latter is strictly local, since tidal effects that distinguish gravity from inertia are only negligible in a small region.

Rather than treating these results as decisive counterevidence, holders of this viewpoint argue that they show general relativity to be a Machian theory only in spirit and only partially in execution - that Einstein built a theory strongly inspired by Mach without ever fully cashing out the principle in the field equations, and that the anti-Machian solutions are artifacts of boundary conditions and idealizations (infinite empty space, eternal rotating dust) that would not arise in any physically realistic, matter-filled universe.

A more technical strand of the viewpoint, pursued mostly by physicists rather than philosophers, attempts to show that ordinary general relativity can be re-expressed in variables that make its Machian content explicit, or that a modified theory closer to Mach's intent can reproduce GR's successful predictions while discarding its absolute residues. Proposed routes include:

• Reformulating the metric as a product of a scalar density tied to cosmological structure and a unit-determinant tensor density carrying the gravitational degrees of freedom, so that the “Machian” cosmological content and the local gravitational dynamics are cleanly separated within the same formalism.
• Brans-Dicke theory and related scalar-tensor extensions, which promote Newton's gravitational constant to a dynamical scalar field set by the surrounding mass distribution, a move read by many Machians as a more faithful realization of Mach's intent than Einstein's field equations.
• Shape dynamics and other relational reformulations that dispense with absolute spacetime scale and attempt to derive inertial structure purely from the instantaneous configuration of matter.

Advocates of these programs do not all agree on which formalism is correct, but they share the conviction that the relational core Mach identified is real physics that a sufficiently complete theory must capture explicitly, rather than an artifact of nineteenth-century positivist scruples that modern field theory has simply outgrown.

Machians frequently point to frame-dragging, the Lense-Thirring effect, as the clearest empirical case where general relativity behaves in a recognizably Machian way: a rotating mass measurably drags the local inertial frames of nearby space along with it, meaning that “what counts as not rotating” near a spinning body is itself influenced by that body's rotation. This was confirmed in low-earth orbit by Gravity Probe B and by analysis of LAGEOS satellite data. Machians treat frame-dragging as evidence that inertia is not fixed once and for all by an absolute background but is genuinely shaped, to a measurable if small degree, by the surrounding mass distribution - exactly the kind of effect Mach's principle predicts and pure Newtonian absolute space rules out.

Mach's critique of Newtonian mechanics appeared in his 1883 *Die Mechanik in ihrer Entwicklung*, written decades before general relativity existed, as part of a broader empiricist and anti-metaphysical program: Mach was suspicious of any theoretical posit, such as absolute space, that could not in principle be cashed out in observable relations between bodies. Einstein encountered this critique as a student in Zurich and later said it had shaken his faith in mechanics as a final, unrevisable foundation for physics.

Through the 1910s, as Einstein developed general relativity, he treated Mach's principle as a guiding heuristic and a desideratum the finished theory ought to satisfy, distinct from the equivalence principle and general covariance but woven together with them in his own exposition. After completing the theory in 1915, Einstein continued for some years to argue, notably in correspondence with Willem de Sitter, that general relativity vindicated Mach's principle. De Sitter's own cosmological solution - essentially empty of matter yet possessing well-defined geometric structure - was an early sign of trouble for this claim.

By the 1920s and 1930s, as more exact solutions were studied, it became clear that general relativity permits matter-free or nearly matter-free solutions with well-defined inertial structure, undermining a strict reading of Mach's principle. Einstein himself eventually repudiated Mach's principle later in life, after giving up his attempts to formulate general relativity in a fully Machian sense. This did not end the viewpoint; rather, it split it. Some Machians concluded the principle needed reformulation rather than abandonment, leading to the scalar-tensor and topological reformulations pursued later in the century. Others held that Einstein's repudiation was a loss of nerve rather than a refutation, and that the relational reading remains the correct philosophical frame for the theory regardless of whether the field equations fully realize it.

Later surveys have catalogued as many as ten distinct formulations of Mach's principle, none of which is unambiguously satisfied by general relativity, which Machians take as evidence that the project is conceptually rich and unfinished rather than a single claim that has been cleanly falsified.

• Ernst Mach (1838-1916) - Austrian physicist and philosopher of science whose critique of Newtonian absolute space, developed on broadly empiricist grounds, is the foundational source for the entire viewpoint.
• Albert Einstein (1879-1955) - coined the term “Mach's principle” and treated it as a central motivation for general relativity during its development, though he later abandoned the hope that his finished theory fully realized it.
• Dennis W. Sciama (1926-1999) - British cosmologist who proposed an explicitly Machian theory of inertia in the 1950s, arguing that the inertial mass of a body arises from its gravitational interaction with all the other matter in the universe.
• Carl H. Brans and Robert H. Dicke - developed Brans-Dicke theory in 1961, a scalar-tensor alternative to general relativity motivated substantially by a desire to incorporate Mach's principle more fully than Einstein's field equations do. (3)
• Julian Barbour - contemporary physicist and philosopher associated with shape dynamics and relational approaches to mechanics and gravity, who has written extensively defending a Machian, relationist reading of physics against absolute space and time.
• Herbert Pfister - physicist known for technical work cataloguing and clarifying the various distinct formulations of Mach's principle and assessing which are and are not satisfied by general relativity.

Holders of this viewpoint disagree on several points:

• Reform versus reinterpretation. Some Machians think general relativity itself, properly reformulated in the right variables, already is a Machian theory and that the appearance of anti-Machian solutions like Minkowski space or the Gödel universe is a mathematical artifact rather than physical content. Others think Einstein's actual field equations are irreducibly non-Machian and that a genuinely Machian theory of gravity, such as Brans-Dicke theory or a successor, must replace them.
• How strong a thesis is at stake. There is disagreement over whether Mach's principle should be read as a strict metaphysical claim - that absolute space and inertia without matter are strictly impossible - or as a weaker heuristic and explanatory preference, satisfied if matter substantially influences inertia even where it does not wholly determine it. Frame-dragging is treated by some as decisive vindication of even the strong thesis and by others as merely confirming the weak thesis, leaving the strong thesis still unsettled.
• Whether the empty-universe solutions are physically relevant. Some Machians argue that solutions like Minkowski spacetime or de Sitter space are mathematical idealizations with no matter at all and therefore tell us nothing about the physically realized universe, which is never actually empty; others regard dismissing a theory's own solutions as ad hoc and think a fully adequate theory should not admit such solutions even as idealizations.
• Relation to logical positivism. Some proponents present Machian relativism as continuous with Mach's original verificationist epistemology, in which absolute space is rejected because it is unobservable in principle. Others, wary of resting physics on a now largely abandoned epistemological program, defend the relational content of the view on physical and mathematical grounds alone, independent of whether strict verificationism is correct.

• General Relativity - Main Topic
• General Relativity - Standard Geometric Interpretation Viewpoint - the contrasting viewpoint treating spacetime curvature as a geometric structure in its own right, not reducible to relations among matter
• Geocentrism - Machian Relativist Viewpoint - a distinct viewpoint, concerning the foundations of astronomy rather than the foundations of inertia, which draws on Machian relationism about motion to argue that no reference frame, including a geocentric one, is privileged by general relativity
• General Relativity - History
• general-relativity-machian-interpretation-viewpoint-debate
• mach-principle-formulations-controversy
• brans-dicke-theory

1. Ernst Mach, *Die Mechanik in ihrer Entwicklung historisch-kritisch dargestellt* (Leipzig: F. A. Brockhaus, 1883); translated by Thomas J. McCormack as *The Science of Mechanics: A Critical and Historical Account of Its Development* (Chicago: Open Court, 1893).
2. Hervé Dahan and Lawrence M. Krauss, "Emergent Physics on Mach's Principle and the Rotating Vacuum," accessed via arXiv, https://arxiv.org/abs/1506.00882.
3. Carl H. Brans and Robert H. Dicke, "Mach's Principle and a Relativistic Theory of Gravitation," *Physical Review* 124, no. 3 (1961): 925-935.
4. D. W. Sciama, "On the Origin of Inertia," *Monthly Notices of the Royal Astronomical Society* 113, no. 1 (1953): 34-42.
5. Julian Barbour and Herbert Pfister, eds., *Mach's Principle: From Newton's Bucket to Quantum Gravity*, Einstein Studies, vol. 6 (Boston: Birkhäuser, 1995).
6. Albert Einstein to Willem de Sitter, correspondence excerpted in John Norton, "Mach's Principle Before Einstein," in Barbour and Pfister, *Mach's Principle*.
7. "Einstein and Beyond: A Critical Perspective on General Relativity," arXiv preprint, https://arxiv.org/abs/1605.09236.
8. C. W. F. Everitt et al., "Gravity Probe B: Final Results of a Space Experiment to Test General Relativity," *Physical Review Letters* 106, no. 22 (2011): 221101.