Postulates of special relativity
1. First postulate (
principle of relativityIn physics, the principle of relativity is the requirement that the equations, describing the laws of physics, have the same form in all admissible frames of reference....
)
- The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion.
2. Second postulate (invariance of
c)
- As measured in an 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.
Alternate Derivations of Special Relativity
The two-postulate basis for special relativity outlined above is the one historically used by Einstein, and it remains the starting point today. However
Hendrik LorentzHendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...
and
Henri PoincaréJules Henri Poincaré was a French mathematician and theoretical physicist, and a philosopher of science...
derived their version of the theory from
Maxwell's equationsMaxwell's equations are a set of four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined to show that light is an electromagnetic wave...
and the principle of relativity. The
Minkowski spaceIn physics and mathematics, Minkowski space is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated. In this setting the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for...
formulation is also used. As Einstein himself later acknowledged, the derivation tacitly makes use of some additional assumptions, including spatial homogeneity, isotropy, and memorylessness. Following Einstein's original derivation, many alternative derivations have been proposed, based on various sets of assumptions. It has often been claimed (such as by Ignatowsky in 1910 and many others in subsequent years) that special relativity follows from just the relativity postulate itself. This claim can be misleading because actually these formulations rely on the aforementioned various assumptions such as isotropy. Nevertheless the Lorentz transformations, up to a nonnegative free parameter, can be derived without first postulating the universal lightspeed. 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 permittivity of free space 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.
See also
Special relativity (alternative formulations)As formulated by Albert Einstein in 1905, the theory of special relativity was based on two main postulates:# The principle of relativity — The form of a physical law is the same in any inertial frame....
.
Mathematical formulation of the postulates
In the rigorous mathematical formulation of special relativity, we suppose that the universe exists on a four-dimensional
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...
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
worldsheetIn string theory, the worldsheet is a two-dimensional manifold which describes the embedding of the string in spacetime. It is a direct generalization of the familiar worldline of a particle in special and general relativity....
s (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
energyIn physics, energy is a scalar physical quantity that describes the amount of work that can be performed by a force, an attribute of objects and systems that is subject to a conservation law...
,
momentumIn classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section "modern definitions of momentum" on this page...
,
massIn physics, mass commonly refers to any of three properties of matter, which have been shown experimentally to be equivalent: inertial mass, active gravitational mass and passive gravitational mass...
,
chargeElectric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields...
, etc.
In addition to events and physical objects, there are a class of
inertial frames of referenceIn physics, an inertial frame of reference is a member of the subset of reference frames with the property that every physical law takes the same form in each such frame. In contrast, in the set of non-inertial frames the laws of physics are frame-dependent, and the usual physical forces must be...
. Each inertial frame of reference provides a
coordinate systemIn mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. This concept is part of the theory of manifolds. "Scalars" in many cases means real numbers, but, depending on context, can mean complex...
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 fieldThe electromagnetic field is a physical field produced by electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. Light is the electromagnetic field in a certain frequency range...
, 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. (In practice, these conversion laws can be efficiently handled using the mathematics of
tensorTensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another...
s).
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 equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
) which relates the various coordinates of the various objects in the spacetime. A typical example is
Maxwell's equationsMaxwell's equations are a set of four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined to show that light is an electromagnetic wave...
. Another is
Newton's first lawNewton's laws of motion are three physical laws that form the basis for classical mechanics. They are:# In the absence of force, a body either is at rest or moves in a straight line with constant speed....
.
1. First Postulate (
Principle of relativityIn physics, the principle of relativity is the requirement that the equations, describing the laws of physics, have the same form in all admissible frames of reference....
)
- Every physical law is invariant under inertial coordinate transformations. Thus, if an object in spacetime obeys the mathematical equations describing a physical law in one inertial frame of reference, it must necessarily obey the same equations when using any other inertial frame of reference.
2. Second Postulate (Invariance of
c)
- There exists an absolute constant with the following property. If A, B are two events which have coordinates and in one inertial frame , and have coordinates and in another inertial frame , then
- if and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional combined with its reverse ; hence the name...
.
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
invariantInvariant and invariance may have several meanings, among which are:* Invariant , an expression whose value doesn't change during program execution* In computer science, a type in overriding that is neither covariant nor contravariant...
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 transformationIn physics, the Lorentz transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frame of reference. It reflects the surprising fact that observers moving at different velocities report...
.
The postulates of special relativity can be expressed very succinctly using the mathematical language of
pseudo-Riemannian manifoldIn differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
s. 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 relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
, 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 (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 and in one inertial frame , and have coordinates and in another inertial frame , then . Furthermore, if , then
- .
The physical theory given by
classical mechanicsIn the fields of physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies geometrically distributed within a certain...
, 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 ) 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.