In 2005 I started writing a paper, “The Four Cornerstones of General Relativity on which it doesn’t Rest.” Unfortunately, I never had a chance to finish it. The idea behind that unfinished article was this: there are four principles that are often described as “postulates” of General Relativity:

1. Principle of general relativity

2. Principle of general covariance

3. Equivalence principle

4. Mach principle

The truth is, however, that General Relativity is not really based on any of these “postulates” although, without a doubt, they played important heuristic roles in the development of the theory. Let’s take these “principles” one at a time.

1. Principle of general relativity. Albert Einstein developed Special Theory of Relativity in 2005. The theory is called “Special” because it only deals with motion of bodies in inertial frames of reference (IFR). In fact, the whole theory is about how to translate measurements made in one IFR to another IFR. General Theory of Relativity was supposed to generalize the relativity theory for all frames of reference, including non-inertial frames (that’s why it was called “General” Theory). Nothing could be further from the truth. Not only General Theory of Relativity does not describe mechanics of bodies in non-inertial frames of reference (NIFR), it doesn’t even define NIFR. Firstly, General Theory of Relativity does not aim to describe how measurements made in one frame of reference translate into another NIFR. In this sense, it is not really a “theory of relativity.” It is a theory of gravitation. The name, General Theory of Relativity is a misnomer, although it stuck with the theory. In developing relativity theory, Einstein made a critical error mistaking coordinate systems for reference frames. The two, however, are not the same. A frame of reference is a physical concept, which, as Einstein himself showed, plays a crucial role in physics. A coordinate system, on the other hand, is a mathematical abstraction that has no physical meaning or significance. It’s is just an arbitrary method of numbering points on the manifold. Just as renaming a street or changing an address of a building in the city doesn’t make the building move, so too changing the way one numbers or labels the points on a manifold does not change anything. In fact, we can wright all laws of physics in coordinate-free form. For example, the Einstein field equations of General Relativity can be simply written as

G = 8πT

Coordinates play no role here whatsoever. It so happens, that in the special case of an inertial frames of reference, it is possible to describe transition from one frame to another using Lorentz transformation. This is why, the use of coordinate systems in lieu of IRF, as it is traditionally done in Special Theory of Relativity, works. It would be improper to do so for non-inertial frames. Thus, the Principle of General Relativity is not at play in General Theory of Relativity.

2. Principle of general covariance. As discussed above, a coordinate system is a mathematical abstraction that has no physical meaning. The requirement that the laws of physics (including filed equations) are described in a form that is invariant with respect to coordinate transformations is a self-evident mathematical requirement, similar to the requirement of internal consistence, that should not be elevated to the status of a physical principle. As was pointed out by a German physicist Kretschmann in 1917, the demand that a theory be generally covariant does not limit the range of acceptable theories and any theory can be put in a covariant form. Today, we write laws of physics in coordinate-free form, so coordinates don’t even appear in the equations and this “principle” plays no role. See more on this subject in “General covariance and the foundations of general relativity: eight decades of dispute” by John D Norton (http://www.pitt.edu/~jdnorton/papers/decades.pdf).

The irony is that Einstein’s General Theory of Relativity is not entirely covariant, or invariant to coordinate transformation. The field equations are, but that’s not the whole story. The potential of gravitational field is described in General Theory of Relativity by the metric tensor, which is a covariant quantity. The strength of the field, however, is described by a non-covariant quantity, a metric connection (Christoffel symbols). This fact leads to a situation where the energy of gravitational field is described not by a tensor (such as energy momentum tensor of matter), but a pseudo-tensor, which is a non-covariant quantity, leading to non-conservation of gravitational energy! One can eliminate the energy in any volume of space simply by choosing Riemannian coordinates, which is unacceptable from physics’ point of view. This is known as the Energy Problem in General Relativity. (In 1979, in my Ph.D. dissertation, I proposed to resolve this problem by defining a frame of reference as a space of Affine connection and recasting Einstein’s field equation in form where gravity is described by the nonmetricity of the Affine connection, which leads to a fully covariant energy momentum tensor for the gravitational filed.) Yet another “postulate” falls off the pedestal.

3. Equivalence principle. The Equivalence principle is certainly very important to General Relativity. Moreover, putting the specific form of General Relativity aside, the equivalence principle provides the very motivation for using geometry to describe gravity (as geometrodynamics). Indeed, if the inertial mass (as the measure of inertia) of a body is equal (or proportional) to its passive gravitational mass (as the measure of the gravitational “charge”), all bodies will fall in a gravitational field with the same acceleration, as, in fact, they do. This means, gravitational effect on a body is completely independent of the mass of the body or its structure. This fact naturally leads to mathematical representation of a gravitational field as a curved space in which all bodies fall along the geodesic (shortest) lines and, therefore, have the same acceleration.

However, it would be an oversimplification to claim that the Equivalence Principle is a postulate of General Theory of Relativity. There are many formulation of this principle. Three general categories of Equivalence Principles (EP) are: Weak (Galilean) EP, Einstein EP and Strong EP.

In a nutshell, the Weak (Galilean) Equivalence Principle postulates the all test particles undergo the same acceleration in a gravitational field, independent of their mass, structure and properties. This fact has been demonstrated experimentally by Galileo in 1610 (and even before, by John Philoponus c. 6 c), Eötvös in 1908, and others with ever grater precision.

The Einstein EP adds to the weak EP a requirement that the result of any local non-gravitational experiment in a freely falling frame of reference is independent of the velocity of the frame of reference and its location in spacetime. In other words, Einstein’s version of the Equivalence Principle adds the requirement of the principle of relativity that results of experiment should not depend on a velocity or position of the observer, so that dimensionless quantities such as fine-structure constant, must have the same value everywhere every time. In fact, this version of the EP was experimentally tested by measuring fine-structure constant in 1976. Some suggest that weak Equivalence Principle implies Einstein Equivalence Principle.

The strong EP requires that the laws of gravitation are independent of velocity and location. In other words, it is a requirement that the result of any local experiment in a freely falling frame of reference is independent of the velocity of the frame of reference and its location in spacetime. The strong EP removes the limitation that the experiment be non-gravitational. It requires that gravitational constant be the same everywhere. Einstein’s General Relativity is the only theory that satisfies strong EP. Alternative theories typically satisfy Einstein’s EP, but violate strong EP. While the EP is centrally important to General Relativity, the version of the EP postulates by Einstein (and, therefore, called Einstein EP) may not be the strongest version of EP that his theory satisfy. It is the strong EP that differentiates Einstein’s General Relativity Theory from other theories of gravity, such as Brans-Dicke theory for example.

4. Mach’s principle. Mach’s principle is an imprecise hypothesis that local inertial frames of reference are determined by the large scale distribution of matter (as illustrated by the Newton’s rotating bucket of water). In other words, local inertia is caused by distant stars – “mass out there influences inertia here.” If all motion is inertial, how do we measure inertia of a body? What if there is a singular particle in the universe, how do we measure its inertia? According to Mach, the motion of such lone particle would be meaningless. Einstein was greatly influenced by Mach’s critique of Newton’s absolute space and the notion of preferred frame of reference.

Newton’s bucket argument described in his Philosophiae Naturalis Principia Mathematica, describes a rotating bucket with water. Initially, when the bucket starts spinning, the water remains still. With time, however, water begins to rotate with the bucket and forms a concave surface due to the centrifugal forces. According to Newton, this proves that so long as water is still with respect to the absolute space, no centrifugal forces are generated and only when water is rotating with respect to the absolute space it becomes subject to centrifugal (inertial) forces.

Mach, on the other hand, believed in absolute relativism. No motion, according to Mach could be considered other than with respect to something else. Thus, instead of talking about water rotating with respect to the absolute space, we should be talking either about water rotating relative to the walls of the bucket, or relative to Earth, or relative to distant stars. It is unclear if Mach had intended to formulate a law of physics. More likely, Mach only proposed a relative description of motion that does not invoke the term “space.” Einstein, who coined the term, “Mach’s principle,” understood it to mean the existence of actual physical interaction between the bodies causing large bodies, such as stars, to contribute to inertia and inertial forces. While Einstein was inspired by Mach’s principle, it is not one of the postulates of General Relativity.

So what are the postulates of General Theory of Relativity? I think, the short list includes:

1. 4-dimensional metric pseudo-Riemannian space with no torsion;

2. Einstein tensor proportional to the energy-momentum tensor of matter: G = 8πT.