Principle of Equivalence

In 1907, Einstein introduced the principle of equivalence. He proposed a thought experiment involving an accelerating box and an occupant inside who could not distinguish between acceleration and a gravitational field. This was due to the equivalence of inertial mass (the resistance of objects to motion changes) and gravitational mass (the mass that governs gravitational attraction).

The equivalence of gravitational and accelerated motion became a cornerstone of his theory.

Mach's Principle

Einstein also embraced Mach's principle. Ernst Mach’s 1893 assertion concerning inertial motion challenged Isaac Newton’s notion of an absolute frame of accelerated motion.

In his Principia (1687), Newton illustrated that a spinning bucket of water would exhibit curvature on its surface, suggesting a distinction between accelerated and non-accelerated motion, unlike the inertial motion described in Einstein’s special theory. If true, this posed a significant obstacle to developing a relativity theory for accelerated motion.

Mach contended that the bucket’s motion was relative to all matter in the universe. As the bucket spun, any reference frame at rest relative to it, like an ant on the rim, would observe the stars and planets whirling around it.

Mach’s idea implied that in a universe devoid of matter, the bucket would not show any surface curvature. Similarly, asserting that the universe as a whole rotates would be meaningless without something to reference.

In 1912, Einstein claimed to have developed a theory demonstrating Mach’s principle with a force interaction causing the bucket’s curvature. He proposed in 1913 that

> "the entire inertia of a point mass is an interaction with the presence of all the remaining masses and based on a kind of interaction with them."

He termed this concept the hypothesis of the relativity of inertia, positing that the collective motion of galaxies defines an inertial state.

General Covariance

In 1912, Einstein also moved beyond his earlier mathematical education, aided by his friend Marcel Grossmann, to adopt the advanced formalism of Ricci, Levi-Civita, Riemann, and Gauss. This framework allowed for the development of physical theories within arbitrary and even unspecified coordinate systems. Building on Minkowski’s geometry, Einstein and Grossmann began constructing a theory that was generally covariant.

Utilizing general covariance, they jointly published a theory on accelerated motion’s relativity, integrating Minkowski’s coordinate systems and then transforming them into an accelerating frame. Einstein illustrated how the metric—defining distances in Minkowski space—reflected the parameters of the accelerating motion.

By this point, most of Einstein's theory of relativity was formulated, but all his geometries remained flat, lacking inherent curvature. The theory needed non-flat spacetimes to account for gravity. However, their formulation did not align with Newton’s, leading them to propose a non-generally covariant gravity theory instead.

For the next three years, Einstein grappled intensely with reconciling gravity and general covariance. He published several papers attempting to assert that physical theories could not be generally covariant, effectively contradicting his previous collaboration with Grossmann.

Yet, by 1915, these challenges dissipated, culminating in the publication of his renowned review article on general relativity in 1916, which continues to be celebrated today. The resolution of these issues largely stemmed from Einstein's abandonment of certain principles he had initially believed were essential to his theory but were not.

Einstein Refutes Mach’s Principle

While popular narratives of relativity adhered to Einstein’s early perspective, his views on the principles outlined in 1918 began to evolve almost immediately.

The first to be discarded was Mach’s principle, which Einstein started distancing himself from as early as 1919. By 1924, he equated Mach’s initial, Newtonian interpretation of inertia with "action at a distance." His modified version encountered problems, as the definition of inertial motion relied on a "makeshift" quantity—the stress energy tensor representing matter and energy.

By 1954, shortly before his death, he stated, “In my opinion, we ought not to speak about Mach’s principle any more.”

Ultimately, the introduction of Mach’s principle was meant to address challenges in his earlier theories, and its abandonment did not hinder his final formulation. His ultimate theory of relativity encompassed Mach’s principle by being a field theory that described the interaction between spacetime curvature and matter. Consequently, matter influences space and time, and vice versa. There is no necessity for the metric field to be entirely dictated by matter distribution (which might not hold true considering the need for a cosmological constant) since matter distribution is influenced by the metric and vice versa.

Thus, while Newton assigned an absolute frame of reference to the accelerated motion of his bucket and Mach proposed that the frame was defined by the distribution of bodies in the universe through some action at a distance, Einstein’s mature understanding recognized that the distribution of bodies contributes to the very shape of space and time. The bucket experiences accelerated motion not directly because of surrounding bodies but because it exists within a spacetime shaped by all observable matter and energy fields.

General Covariance Refuted

Although Einstein discarded Mach’s principle—which was never integrated into technical explanations of general relativity—he regarded general covariance as an essential feature of his theory.

However, as early as 1917, Kretschmann pointed out that general covariance was physically vacuous. He argued that

> "any physical theory can be made to comply with any arbitrary relativity postulate, even the most general one, without altering any of its content subject to empirical testing."

Thus, Einstein’s assertion that general covariance is a fundamental principle of his theory is devoid of physical meaning.

In 1918, Einstein reluctantly acknowledged Kretschmann’s observation:

> "I believe Herr Kretschmann’s argument to be correct, but the innovation proposed by him is not commendable."

In this statement, he invoked Ockham’s razor, suggesting that while it is possible to make any theory generally covariant, many, like Newton’s, would be excessively complex to the point of impracticality. Thus, he emphasized the "heuristic force" of the principle in favor of adopting his theory.

This marked a significant retreat from a foundational principle to one that was merely a practical guideline. Can one imagine Newton's laws being regarded solely as heuristic? They are meant to delineate the correct physical laws rather than simply indicate them.

It took only five years for Cartan (1923) to devise a generally covariant mathematical framework for Newton’s laws, further undermining Einstein’s Ockham’s razor argument.

Today, most theories are articulated in a generally covariant format developed in the 1960s and 70s, completely devoid of predetermined coordinates. The primary distinction between Einstein’s theory and these contemporary models is that the metric and manifold (geometry) of spacetime is not predefined.

Equivalence Principle Redefined

In contemporary discourse, Mach’s principle and general covariance are rarely mentioned in experimental tests. However, the equivalence principle is frequently examined, often leading to headlines proclaiming, "Einstein is right again!" But is this entirely accurate? The answer is both yes and no.

The equivalence principle in use today differs from the one Einstein endorsed in 1918 and maintained throughout his lifetime.

Despite the mathematical sophistication he gained, Einstein never relinquished his concept of the accelerating elevator thought experiment as the foundation for the equivalence principle. In 1922, he presented the idea of one coordinate system, K, that is not uniformly accelerating, and another, K’, that is:

> "there is nothing to prevent our conceiving this gravitational field as real; that is, the conception that K is ‘at rest’ and a gravitational field is present can be considered equivalent to the notion that only K is an ‘allowable’ system of coordinates and that no gravitational field is present."

However, physicists almost universally accepted the equivalence principle as something different, encapsulated by contemporary physicist Wolfgang Pauli’s words:

> "For every infinitely small world region (i.e., a world region that is so small that the space- and time-variation of gravity can be neglected in it), there always exists a coordinate system, K, in which gravitation has no influence on the motion of particles or any other physical process."

Should one inquire about the equivalence principle, physicists may reference Einstein’s example, yet when asked for a mathematical definition, they will provide Pauli’s version. The issue lies in the fact that these interpretations are not synonymous. Pauli’s version is often referred to as "local Lorentz covariance," which is the one subjected to experimental validation.

Einstein opposed this interpretation, advocating for a weaker equivalence principle applicable to uniform acceleration, rather than arbitrary gravitational fields, positioning it between the Lorentz covariance of special relativity and the general covariance of general relativity. In his own words from 1916:

> "The requirement of general covariance of equations encompasses that of the principle of equivalence as a quite special case."

In essence, Einstein contended that general covariance superseded the equivalence principle, but unfortunately, general covariance is too broad to effectively delineate the theory.

Does General Relativity Generalize Relativity? (Hint: No)

Another issue with general covariance is that, in addition to being physically vacuous, it failed to generalize the principle of relativity that Einstein believed it would.

By the 1960s, numerous scientists asserted that "general relativity" was misnamed; it was merely Einstein’s Theory of Gravity. Far from generalizing relativity, Einstein’s theory contradicted it by asserting that spacetime possesses an absolute geometric shape, thus limiting relativity to flat geometries only.

In 1959, Fock, a critic of general relativity, summarized Einstein’s misunderstanding of his own theory:

> "The fact that the theory of gravitation, a theory of such amazing depth, beauty, and coherence, was not correctly understood by its author should not surprise us. We should also not be surprised at the gaps in logic, and even errors, which the author permitted himself when he derived the basic equations of the theory."

Years later, in 1974, Fock remarked on general relativity:

> "[G]eneral relativity cannot be physical, and physical relativity cannot be general."

This sentiment echoed the growing consensus from the 1960s onward that Einstein’s Theory of Gravity was not a generalization of relativity but rather a refutation, akin to how special relativity countered Newton’s laws.

This realization gained traction with the emergence of group theory in the 1950s as the standard for defining all physical theories. Group theory serves to define symmetries, allowing for transformations of mathematical objects within a theory without altering their essence. Lorentz symmetry was one of the first to be identified.

So, what is the symmetry group of general relativity?

It is not general covariance, which is merely a symmetry of coordinate systems. Rather, it is the identity group—a trivial group containing one transformation from an entity to itself, adjusted by a constant factor.

Kretschmann was the first to make this observation in 1917 (in the same paper referenced earlier). Through extensive mathematical analysis, he concluded that:

> "Einstein’s theory satisfies no relativity principle at all... it is a completely absolute theory."

In Einstein's theory, once the physical parameters of the system are established, the metric representing the gravitational field is fixed. The only variability lies in changing coordinates. Since coordinate transformations are not physical, this implies a lack of true freedom.

Thus, what Einstein misinterpreted as general relativity is, in fact, devoid of relativity—merely a freedom to alter coordinates.

Part of the reason for this misconception arises from the transition from special to "general" relativity, where time measurements through clocks and distance measurements through rods, which hold significance in the coordinate frames of special relativity, lose their physical relevance. In Einstein's gravitational theory, coordinate systems hold no relation to measurements. Instead, measurements can only be derived from collapsing coordinate system-dependent vectors and tensors into scalar (numerical) values that are coordinate invariant. Consequently, the Lorentz covariance of special relativity is an outcome of attributing physical significance to coordinate frames, which is unfeasible in the general theory.

What Are the Principles of Einstein’s Theory?

While general covariance and generalized relativity seem ineffective in defining relativity, and the identity group fails to serve as a meaningful symmetry group, many expositions rely on the one principle that Einstein ultimately rejected: infinitesimal Lorentz covariance (the equivalence principle).

However, this principle is also vulnerable to critique since coordinate transformations should not negate the presence of a gravitational field if it exists (which is defined by a non-vanishing curvature in the spacetime geometry). This may explain Einstein's aversion to the concept.

These phenomena can be observed since all objects, regardless of size, experience non-zero tidal forces in the presence of a gravitational field. Even the tiniest raindrop encounters them. Tidal forces represent differentials in gravitational pull on an object, and one cannot entirely eliminate them from the theory. Thus, the equivalence principle, which is frequently validated through experiments, is only applicable because tidal forces are sufficiently small to be disregarded.

For instance, astronauts and their spacecraft in free fall are never in a genuine zero-g state; rather, the tidal forces exerted by Earth are negligible. While one can choose a coordinate system that suggests local Lorentz covariance, this is unphysical.

Another potential foundational principle is the inherent freedom of the manifold itself, implying that no absolutes exist within spacetime, as the manifold is dynamic. This would invalidate generally covariant Newtonian theory since it contains an absolute space and time. This may have been Einstein's true aim: the eradication of any absolutes.

This concept is compelling, yet it does not imply a lack of rules, as Riemann certainly held assumptions about curved geometries similar to Euclid's regarding flat ones.

A third possibility is that the theory encompasses no non-trivial symmetries (i.e., it exclusively contains the identity group). Thus, the basic principles could be summarized as: spacetime is geometric, Einstein's field equations govern its relationship to matter, and gravity manifests as curvature. This formulation is empirically useful, albeit lacking in explanatory depth.

When merging the aforementioned ideas, Einstein’s equations can be easily derived from a fundamental assertion of a differential form on a spacetime geometry. Consequently, they embody a complete lack of freedom, meaning that the theory is physically constrained by itself. Thus, spacetime is solely governed by the principles of geometric manifolds.

Where Do We Go From Here?

Einstein’s gravitational theory has demonstrated remarkable success, yet most physicists believe it serves as an effective theory, a lower-order approximation of the true gravity theory and possibly a more comprehensive understanding of the universe. Various approaches to formulating a quantum theory of gravity have focused on one symmetry group or another, but given the absence of non-trivial groups in Einstein’s own framework, is this the correct path? Or does it make more sense to eliminate the symmetries from other forces? Only time will reveal the answer.

Norton, John D. “General covariance and the foundations of general relativity: eight decades of dispute.” Reports on Progress in Physics 56.7 (1993): 791.