Classical mechanics (and Newtonian gravity) is based on the principles of absolute space, absolute time, and absolute motion, including acceleration. According to Newton, absolute time and space respectively are “universal” (the same for all observers) or independent aspects of objective reality, and do not depend upon physical events, but are a “stage” setting within which physical phenomena occur. These ideas were originally opposed by Leibniz, Berkeley and Huygens, arguing that as observers, we can only epistemically access relative notions of space, time, and motion. For instance, a distance only makes sense between two physical entities, and time between two events.
Newton’s justification of absolute space and motion is portraited in his rotating bucket of water experiment at the beginning of his 1687 masterpiece Principia, which can be summarized as follows:
A bucket with water is set spinning around its axis and at first, the walls of the bucket rotate relative to the stationary water while the surface of the water remains flat as prior to the spinning. After the water starts to rotate as well, it rises towards the walls and its shape is no longer flat when the spinning bucket and water are at rest relative to each other. Interactions between water and walls (due to their relative velocities) cannot explain the shape of the water, since prior to spinning, when the shape of the water was flat, they were also at rest relative to each other. Newton assumed this experiment could be used to measure rotation with respect to an absolute space, “without relation to anything external, which remains always similar and immovable”, which produced the inertial forces that made the water rise.
Leibniz’s relativistic space, time and motion ideas were further developed by Ernst Mach in his 1883 The Science of Mechanics (which inspired Einstein to develop GR, who originally coined the term “Mach’s principle”), in which he criticizes Newton’s conclusion of his bucket of water experiment stating that “No one is competent to say how the experiment would turn out if the sides of the vessel increased in thickness and mass till they were ultimately several leagues thick”.
According to Mach, the relative rotation of the water with respect to the bucket produces no noticeable centrifugal forces, and such forces are instead produced by its relative rotation with respect to Earth and the other celestial bodies. In this way, the relative motion between the several leagues thick bucket and the water could produce noticeable centrifugal forces in the water, and perhaps their non-relative motion could reduce these forces when both rotate with respect to the rest of the universe. In contrast to Newton, who attempted to explain the physical effects of inertia through a sort of resistance to motion within an unobservable absolute space with no physical properties, Mach conceived inertia as an interaction that required other external bodies to manifest.
Newton did not suspect that the change in the water’s shape could be due to rotation relative to the rest of the universe because his action at a distance gravitational force from his infinite and homogeneous universe acting on the water cancels out according to Newton’s shell theorem and it is independent of velocity or acceleration. But no local phenomena can ever be isolated from the rest of the universe, which certainly reaches the water even if the sum of these forces is zero.
According to Mach, and in opposition to Newton’s conclusions, a spinning bucket of water in an empty universe would not change the water’s shape (one could not detect relative motion in an empty universe, it would be undefinable), and if the rest of the universe was set spinning while the bucket was at rest, the water surface would curve, as both the spinning of the bucket or the spinning of the universe are indistinguishable systems without an absolute space.
Moreover, there exists an exact coincidence between the local measurement of the angular velocity of the Earth through Foucault’s pendulum and the cosmological measurement through the apparent movement of distant stars and galaxies, which Newtonian gravity and mechanics cannot explain because it does not causally connect both measurements (and consider the determination of inertial frames by the fixed stars and Foucault pendulum a coincidence), but constitutes the basic idea of Mach’s principle: “The universe, as represented by the average motion of distant galaxies, does not appear to rotate relative to local inertial frames.”
According to Mach, and known as Mach’s principle, the inertia of a body is not an independent and intrinsic property of matter (unlike in Newtonian mechanics and GR, where a particle in an empty universe has inertial properties), but rather the result of the action of the universe as a whole. Mach suggested that the fixed background distribution of matter in the universe must exert the inertial forces on a local accelerating body. In this way, the reference system with respect to which the universe is at rest or in uniform and rectilinear motion is a true inertial reference system. In Ernst Mach words, “I have remained to the present day the only one who insists upon referring the law of inertia to the Earth, and in the case of motions of great spatial and temporal extent to the fixed stars.” Hence, inertial frames should be defined with respect to this rest frame, and local physical laws must be determined by the large-scale structure of the universe.
According to Newton, forces are viewed as isolated causes, and in absence of such causes, the law of inertia dictates motion. But Mach argued that singular and isolated causes cannot exist independently, and that inertial motion is not the result of an absence of causes, but the interaction with the causally connected rest of the universe.
A frame linearly accelerated relative to an inertial frame in the Minkowskian spacetime of Special Relativity is locally identical to a frame at rest in a gravitational field. Einstein’s equivalence principle assumes that inertial and gravitational masses are equal, and the metric tensor in GR determines the inertial mass of a body. Gravity is the only long-distance force that cannot be screened. Moreover, there always exists a reference frame in which inertial forces vanish, just like for the case of gravitational forces in free fall. These equivalences indicate that both inertial and gravitational forces are of the same nature. Following Mach, in any two-body interaction, the influence of all other matter inside their causal spacetime should be taken into account. Inertia is thus a form of gravitational induction, appearing when a body is accelerated with respect to the rest of the universe, subjected to retarded action. It seems that to describe, for example, the Earth’s motion around the Sun, only local masses and distances are required, but the Machian perspective implies that the universe’s action is already taken into account in the Newtonian laws. The only way this can be true is through the Newtonian gravitational constant, which together with the choice of an inertial absolute space frame, both constitute the two arbitrarily choices of Newton.
General Relativity, although featuring some Machian effects such as frame dragging, is not fully Machian, since the solutions of local dynamics do not depend of the mass-energy distribution far away. For instance, in the Schwarzschild solution, space is asymptotically flat at infinity. Another example is that a mass in an empty universe (flat Minkowski space) does have a non-vanishing inertial mass. This could not be the case if inertia were completely determined by the gravitational interaction with other masses.
Is Mach’s principle the way to extend gravity beyond General Relativity towards a more fundamental theory? How can the gravitational constant be derived from Mach’s principle? We will explore the theories of modified gravity and modified inertia based on Mach’s principle in a future post.
References:
MOND as a Transformation Between Non-inertial Reference Frames Via Sciama’s Interpretation of Mach’s Principle, International Journal of Theoretical Physics, Volume 63, article number 271, (2024)
Mach’s principle critique to Newton’s bucket of water
Jonathan Fay’s Mach’s principle