Postulates of special relativity

In physics, Albert Einstein's 1905 theory of special relativity is derived from first principles now called the postulates of special relativity. Einstein's formulation only uses two postulates, though his derivation implies a few more assumptions.

Postulates of special relativity

1. First postulate (principle of relativity)

The laws of physics take the same form in all inertial frames of reference.

2. Second postulate (invariance of c)

As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body. Or: the speed of light in free space has the same value c in all inertial frames of reference.

The two-postulate basis for special relativity is the one historically used by Einstein, and it remains 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. Also Hermann Minkowski 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 (1892–1905) 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 (see History of Lorentz transformations and Lorentz ether theory). A more modern example of deriving the Lorentz transformation from electrodynamics (without using the historical aether concept at all), was given by Richard Feynman.

Following Einstein's original derivation and the group theoretical presentation by Minkowski, many alternative derivations have been proposed, based on various sets of assumptions. It has often been argued (such as by Vladimir Ignatowski in 1910, or Philipp Frank and Hermann Rothe in 1911, and many others in subsequent years) that a formula equivalent to the Lorentz transformation, up to a nonnegative 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.

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 (if the object is a point particle) or worldsheets (if the object is larger than a point). 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 $(x_{1},x_{2},x_{3},t)$ 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 $(p_{1},p_{2},p_{3},E)$ for the momentum and energy of an object, coordinates $(E_{1},E_{2},E_{3},B_{1},B_{2},B_{3})$ 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 $(x_{1},x_{2},x_{3},t)$, but will also provide a conversion for all other physical coordinates, such as a conversion law for momentum and energy $(p_{1},p_{2},p_{3},E)$, etc. (In practice, these conversion laws can be efficiently handled using the mathematics of tensors.)

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 (for instance, a differential 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 (Principle of relativity)

Under transitions between inertial reference frames, the equations of all fundamental laws of physics stay form-invariant, while all the numerical constants entering these equations preserve their values. Thus, if a fundamental physical law is expressed with a mathematical equation in one inertial frame, it must be expressed by an identical equation in any other inertial frame, provided both frames are parameterised with charts of the same type. (The caveat on charts is relaxed, if we employ connections to write the law in a covariant form.)

2. Second Postulate (Invariance of c)

There exists an absolute constant $0<c<\infty$ with the following property. If A, B are two events which have coordinates $(x_{1},x_{2},x_{3},t)$ and $(y_{1},y_{2},y_{3},s)$ in one inertial frame $F$, and have coordinates $(x'_{1},x'_{2},x'_{3},t')$ and $(y'_{1},y'_{2},y'_{3},s')$ in another inertial frame $F'$, then

${\sqrt {(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}+(x_{3}-y_{3})^{2}}}=c(s-t)\quad$ if and only if $\quad {\sqrt {(x'_{1}-y'_{1})^{2}+(x'_{2}-y'_{2})^{2}+(x'_{3}-y'_{3})^{2}}}=c(s'-t')$.

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 (e.g., Maxwell's equations cannot account for the quantum attributes of electromagnetic radiation).

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

$c^{2}(s-t)^{2}-(x_{1}-y_{1})^{2}-(x_{2}-y_{2})^{2}-(x_{3}-y_{3})^{2}$

$=c^{2}(s'-t')^{2}-(x'_{1}-y'_{1})^{2}-(x'_{2}-y'_{2})^{2}-(x'_{3}-y'_{3})^{2}$

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 (1,3), 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 $c\to \infty$ (which is sometimes referred to as the non-relativistic limit). In this theory, the first postulate remains unchanged, but the second postulate is modified to:

If A, B are two events which have coordinates $(x_{1},x_{2},x_{3},t)$ and $(y_{1},y_{2},y_{3},s)$ in one inertial frame $F$, and have coordinates $(x'_{1},x'_{2},x'_{3},t')$ and $(y'_{1},y'_{2},y'_{3},s')$ in another inertial frame $F'$, then $s-t=s'-t'$. Furthermore, if $s-t=s'-t'=0$, then

$\quad {\sqrt {(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}+(x_{3}-y_{3})^{2}}}$

$={\sqrt {(x'_{1}-y'_{1})^{2}+(x'_{2}-y'_{2})^{2}+(x'_{3}-y'_{3})^{2}}}$.

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 surprising number of cases, the laws of physics in special relativity (such as the famous equation $E=mc^{2}$) 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.