physics - Postulates of special relativity

Postulates of special relativity

1. First postulate (principle of relativity)

Observation of physical phenomena by more than one inertial observer must result in agreement between the observers as to the nature of reality. Or, the nature of the universe must not change for an observer if their inertial state changes.

Every physical theory should look the same mathematically to every inertial observer.

To state that simply, no property of the universe will change if the observer is in motion. The laws of the universe are the same regardless of inertial frame of reference.

2. Second postulate (invariance of c)

The speed of light in vacuum, commonly denoted c, is the same to all inertial observers, is the same in all directions, and does not depend on the velocity of the object emitting the light. When combined with the First Postulate, this Second Postulate is equivalent to stating that light does not require any medium (such as "aether") in which to propagate.

The second postulate is an assertion that the invariant speed of the universe is a finite speed c, as opposed to Galilean relativity, in which the invariant speed is infinite.

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 co-ordinate system $(x 1, x 2, x 3, t)$ for events in the spacetime M. Furthermore, this frame of reference also gives co-ordinates to all other physical characteristics of objects in the spacetime, for instance it will provide co-ordinates $(p 1, p 2, p 3, E)$ for the momentum and energy of an object, co-ordinates $(E 1, E 2, E 3, B 1, B 2, B 3)$ 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 co-ordinates from one frame of reference to the co-ordinates in another frame of reference. This transformation not only provides a conversion for spacetime co-ordinates $(x 1, x 2, x 3, t)$ , but will also provide a conversion for all other physical co-ordinates, 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 co-ordinates given by an inertial frame of reference by a mathematical equation (for instance, a differential equation) which relates the various co-ordinates 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)

Every physical law is invariant under inertial co-ordinate 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 $0 < c < \infty$ with the following property. If A, B are two events which have co-ordinates

(x 1, x 2, x 3, t)

and

(y 1, y 2, y 3, s)

in one inertial frame

F

, and have co-ordinates

(x' 1, x' 2, x' 3, t')

and

(y' 1, y' 2, y' 3, s')

in another inertial frame

F'

, then

$\sqrt{(x_1-y_1)^2 + (x_2-y_2)^2 + (x_3-y_3)^2} = c(s-t)$ if and only if $\sqrt{(x'_1-y'_1)^2 + (x'_2-y'_2)^2 + (x'_3-y'_3)^2} = c(s'-t')$ .

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. It turns out that the Second Postulate can be mathematically deduced from the First Postulate and Maxwell's equations, in which case c is given by $c = 1/\sqrt{\mu_0 \epsilon_0}$ , where $μ 0$ and $ε 0$ are the permeability and permittivity of vacuum respectively. Since Maxwell's equations govern the propagation of electromagnetic radiation such as light, it is thus common practice to refer to c as the speed of light, and one can interpret the Second Postulate as nothing more than an assertion that electrodynamics as described by Maxwell's equations is indeed correct, in contrast with the earlier theory of Galilean relativity which was in contradiction to Maxwell's equations (unless one postulated an aether). However, it is worth noting that the formulation of the Second Postulate as given above does not actually require the existence of electromagnetic radiation or Maxwell's equations.

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 co-ordinates

(x 1, x 2, x 3, t)

and

(y 1, y 2, y 3, s)

in one inertial frame

F

, and have co-ordinates

(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 $E = m c 2$ ) 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.

Categories: Special relativity