The Postulates of Special Relativity

The theory of special relativity can be derived formally from a small number of postulates. The fundamental postulates of special relativity can be expressed in various ways; you may find different versions of them in different books.

The first two postulates below are assertions about the structure of spacetime, while the last two postulates form the heart of special relativity.

Statement: “Space and time form a 4-dimensional continuum”.

Watch this 4D hypercube rotate (46K GIF movie).

The postulate that spacetime forms a 4-dimensional continuum is a generalization of the classical Galilean concept that space and time form separate 3 and 1 dimensional continua. The postulate of a 4-dimensional spacetime continuum is retained in general relativity.

Physicists widely believe that this postulate must ultimately break down, that space and time are quantized over extremely small intervals of space and time, the Planck length \(\sqrt{G \hbar / c^3} \approx 10^{-35}\) meters, and the Planck time \(\sqrt{G \hbar / c^5} \approx 10^{-43}\) seconds, where \(G\) is Newton's gravitational constant, \(\hbar \equiv h / ( 2\pi )\) is Planck's constant divided by 2 pi, and \(c\) is the speed of light.

Statement: “There exist global spacetime frames with respect to which unaccelerated objects move in straight lines at constant velocity”.

A spacetime frame is a system of coordinates for labelling space and time. Four coordinates are needed, because spacetime is 4-dimensional.

A frame in which unaccelerated objects move in straight lines at constant velocity is called an inertial frame. One can easily think of non-inertial frames: a rotating frame, an accelerating frame, or simply a frame with some bizarre Dahlian labelling of coordinates.

A globally inertial frame is an inertial frame that covers all of space and time.

The postulate that globally inertial frames exist is carried over from classical mechanics (Newton's first law of motion).

Implicit in the assumption of the existence of globally inertial frames is the assumption that the geometry of spacetime is flat, the geometry of Euclid. In general relativity, this postulate is replaced by the weaker postulate that local (not global) inertial frames exist. A locally inertial frame is one which is inertial in a ‘small neighbourhood’ of a spacetime point. In general relativity, spacetime can be curved.

Statement: “The speed of light \(c\) is a universal constant, the same in any inertial frame”.

This postulate is the nub of special relativity, and much of the content of these pages is concerned with exploring its paradoxical consequences, starting with the next page, The Paradox of Special Relativity. Amongst other strange consequences, the postulate implies that the time dimension behaves in many ways as if it were an imaginary spatial dimension. Watch this 4D spacetime hypercube rotate (47K GIF movie).

Measuring speed requires being able to measure intervals of both space and time: speed is distance travelled divided by time elapsed. Inertial frames constitute a special class of spacetime coordinate systems; it is with respect to distance and time intervals in these special frames that the speed of light is asserted to be constant.

In general relativity, arbitrarily weird coordinate systems are allowed, and light need move neither in straight lines nor at constant velocity with respect to bizarre coordinates (why should it, if the labelling of space and time is totally arbitrary?). However, general relativity asserts the existence of locally inertial frames, and the speed of light is a universal constant in those frames.

In 1983, the General Conference on Weights and Measures officially defined the speed of light to be \[ c = \mbox{299,792,458 meters/second ,} \] and the meter, instead of being a primary measure, became a secondary quantity, defined in terms of the second and the speed of light.

Statement: “The laws of physics are the same in any inertial frame, regardless of position or velocity”.

Physically, this means that there is no absolute spacetime, no absolute frame of reference with respect to which position and velocity are defined. Only relative positions and velocities between objects are meaningful.

It is to be noted that the Principle of Special Relativity does not imply the constancy of the speed of light, although the postulates are consistent with each other. Moreover the constancy of the speed of light does not imply the Principle of Special Relativity, although for Einstein the former appears to have been the inspiration for the latter.