Postulates of special relativity
Albert Einstein derived the theory of special relativity in 1905, from principles now called the postulates of special relativity. Einstein's formulation is said to only require two postulates, though his derivation implies a few more assumptions.
The idea that special relativity depended only on two postulates, both of which seemed to follow from the theory and experiment of the day, was one of the most compelling arguments for the correctness of the theory
Postulates of special relativity
1. First postulate2. Second postulate
The two-postulate basis for special relativity is the one historically used by Einstein, and it is sometimes the starting point today. As Einstein himself later acknowledged, the derivation of the Lorentz transformation tacitly makes use of some additional assumptions, including spatial homogeneity, isotropy, and memorylessness. Hermann Minkowski also implicitly used both postulates when he introduced the Minkowski space formulation, even though he showed that c can be seen as a space-time constant, and the identification with the speed of light is derived from optics.
Alternative derivations of special relativity
Historically, Hendrik Lorentz and Henri Poincaré derived the Lorentz transformation from Maxwell's equations, which served to explain the negative result of all aether drift measurements. By that the luminiferous aether becomes undetectable in agreement with what Poincaré called the principle of relativity. A more modern example of deriving the Lorentz transformation from electrodynamics, was given by Richard Feynman.George Francis FitzGerald already made an argument similar to Einstein's in 1889, in response to the Michelson–Morley experiment seeming to show both postulates to be true. He wrote that a length contraction is "almost the only hypothesis that can reconcile" the apparent contradictions. Lorentz independently came to similar conclusions, and later wrote "the chief difference being that Einstein simply postulates what we have deduced".
Following these derivations, many alternative derivations have been proposed, based on various sets of assumptions. It has often been argued
that a formula equivalent to the Lorentz transformation, up to a non-negative free parameter, follows from just the relativity postulate itself, without first postulating the universal light speed. These formulations rely on the aforementioned various assumptions such as isotropy. The numerical value of the parameter in these transformations can then be determined by experiment, just as the numerical values of the parameter pair c and the Vacuum permittivity are left to be determined by experiment even when using Einstein's original postulates. Experiment rules out the validity of the Galilean transformations. When the numerical values in both Einstein's and other approaches have been found then these different approaches result in the same theory.
Insufficiency of the two standard postulates
Einstein's 1905 derivation is not complete. A break in Einstein's logic occurs where, after having established "the law of the constancy of the speed of light" for empty space, he invokes the law in situations where space is no longer empty. For the derivation to apply to physical objects requires an additional postulate or "bridging hypothesis", that the geometry derived for empty space also applies when a space is populated. This would be equivalent to stating that we know that the introduction of matter into a region, and its relative motion, have no effect on lightbeam geometry.Such a statement would be problematic, as Einstein rejected the notion that a process such as light-propagation could be immune to other factors
Including this "bridge" as an explicit third postulate might also have damaged the theory's credibility, as refractive index and the Fizeau effect would have suggested that the presence and behaviour of matter does seem to influence light-propagation, contra the theory. If this bridging hypothesis had been stated as a third postulate, it could have been claimed that the third postulate were falsified by the experimental evidence.
The 1905 system as "null theory"
Without a "bridging hypothesis" as a third postulate, the 1905 derivation is open to the criticism that its derived relationships may only apply in vacuo, that is, in the absence of matter.The controversial suggestion that the 1905 theory, derived by assuming empty space, might only apply to empty space, appears in Edwin F. Taylor and John Archibald Wheeler's book "Spacetime Physics".
A similar suggestion that the reduction of GR geometry to SR's flat spacetime over small regions may be "unphysical" was acknowledged but rejected by Einstein in 1914.
Einstein revisited the problem in 1919
A further argument for unphysicality can be gleaned from Einstein's solution to the "hole problem" under general relativity, in which Einstein rejects the physicality of coordinate-system relationships in truly empty space.
Alternative relativistic models
Einstein's special theory is not the only theory that combines a form of light speed constancy with the relativity principle. A theory along the lines of that proposed by Heinrich Hertz allows for light to be fully dragged by all objects, giving local c-constancy for all physical observers. The logical possibility of a Hertzian theory shows that Einstein's two standard postulates are not sufficient to allow us to arrive uniquely at the solution of special relativity.Einstein agreed that the Hertz theory was logically consistent, but dismissed it on the grounds of a poor agreement with the Fizeau result, leaving special relativity as the only remaining option. Given that SR was similarly unable to reproduce the Fizeau result without introducing additional auxiliary rules, this was perhaps not a fair comparison.
Mathematical formulation of the postulates
In the rigorous mathematical formulation of special relativity, we suppose that the universe exists on a four-dimensional spacetime M. Individual points in spacetime are known as events; physical objects in spacetime are described by worldlines or worldsheets. The worldline or worldsheet only describes the motion of the object; the object may also have several other physical characteristics such as energy-momentum, mass, charge, etc.In addition to events and physical objects, there are a class of inertial frames of reference. Each inertial frame of reference provides a coordinate system for events in the spacetime M. Furthermore, this frame of reference also gives coordinates to all other physical characteristics of objects in the spacetime; for instance, it will provide coordinates for the momentum and energy of an object, coordinates for an electromagnetic field, and so forth.
We assume that given any two inertial frames of reference, there exists a coordinate transformation that converts the coordinates from one frame of reference to the coordinates in another frame of reference. This transformation not only provides a conversion for spacetime coordinates, but will also provide a conversion for all other physical coordinates, such as a conversion law for momentum and energy, etc.
We also assume that the universe obeys a number of physical laws. Mathematically, each physical law can be expressed with respect to the coordinates given by an inertial frame of reference by a mathematical equation which relates the various coordinates of the various objects in the spacetime. A typical example is Maxwell's equations. Another is Newton's first law.
1. First Postulate
2. Second Postulate
Informally, the Second Postulate asserts that objects travelling at speed c in one reference frame will necessarily travel at speed c in all reference frames. This postulate is a subset of the postulates that underlie Maxwell's equations in the interpretation given to them in the context of special relativity. However, Maxwell's equations rely on several other postulates, some of which are now known to be false.
The second postulate can be used to imply a stronger version of itself, namely that the spacetime interval is invariant under changes of inertial reference frame. In the above notation, this means that
for any two events A, B. This can in turn be used to deduce the transformation laws between reference frames; see Lorentz transformation.
The postulates of special relativity can be expressed very succinctly using the mathematical language of pseudo-Riemannian manifolds. The second postulate is then an assertion that the four-dimensional spacetime M is a pseudo-Riemannian manifold equipped with a metric g of signature, which is given by the Minkowski metric when measured in each inertial reference frame. This metric is viewed as one of the physical quantities of the theory; thus it transforms in a certain manner when the frame of reference is changed, and it can be legitimately used in describing the laws of physics. The first postulate is an assertion that the laws of physics are invariant when represented in any frame of reference for which g is given by the Minkowski metric. One advantage of this formulation is that it is now easy to compare special relativity with general relativity, in which the same two postulates hold but the assumption that the metric is required to be Minkowski is dropped.
The theory of Galilean relativity is the limiting case of special relativity in the limit . In this theory, the first postulate remains unchanged, but the second postulate is modified to:
The physical theory given by classical mechanics, and Newtonian gravity is consistent with Galilean relativity, but not special relativity. Conversely, Maxwell's equations are not consistent with Galilean relativity unless one postulates the existence of a physical aether. In a number of cases, the laws of physics in special relativity can be deduced by combining the postulates of special relativity with the hypothesis that the laws of special relativity approach the laws of classical mechanics in the non-relativistic limit.