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Postulates of special relativity

Postulates of special relativity

1. First postulate (principle of relativity)

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)

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 one most commonly used today. However, 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 sometimes been claimed (such as by Ignatowsky in 1910 and many others in subsequent years) that special relativity follows from just the relativity postulate itself, but this claim has been thoroughly investigated and is known to be incorrect, as is obvious from the fact that Galilean relativity (for example) also satisfies the relativity postulate, but it is not equivalent to special relativity.

Also, it is well known that the assumptions of homogeneity, isotropy, memorylessness, reciprocity, and relativity are sufficient to imply the relationship between inertial coordinate systems up to a constant, but they are not sufficient to establish whether that constant is positive, negative, or zero. This is what distinguishes special relativity from Galilean (and Euclidean) relativity. In every purported derivation of special relativity that claims not to make use of some postulate in addition to the postulate of relativity, the derivation proceeds only to the point of requiring that inertial coordinate systems are related by either Euclidean, Galilean, or Lorentz transformations. Some other assumption is required to single out which of these forms is applicable. The resulting structures of spacetime under these three assumptions are profoundly different. In particular, the Minkowski metric of special relativity (based on the Lorentz transformation) is not positive-definite, so it does not satisfy fundamental properties of a metric space (such as the triangle inequality). Some authors contend that no additional postulate or assumption (beyond relativity) is needed because the decision as to whether the characteristic constant is positive, negative, or zero can simply be determined experimentally. Indeed Einstein's original paper emphasizes the empirical basis of the postulates. The relativity postulate is also empirically based. A postulate is not eliminated merely by pointing out that it is empirically based..

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 $\left(x_1,x_2,x_3,t\right)$ 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 $\left(p_1,p_2,p_3,E\right)$ for the momentum and energy of an object, coordinates $\left(E_1,E_2,E_3,B_1,B_2,B_3\right)$ 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 $\left(x_1,x_2,x_3,t\right)$, but will also provide a conversion for all other physical coordinates, such as a conversion law for momentum and energy $\left(p_1,p_2,p_3,E\right)$, 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)

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 $0 < c < infty$ with the following property. If A, B are two events which have coordinates $\left(x_1,x_2,x_3,t\right)$ and $\left(y_1,y_2,y_3,s\right)$ in one inertial frame $F$, and have coordinates $\left(x\text{'}_1,x\text{'}_2,x\text{'}_3,t\text{'}\right)$ and $\left(y\text{'}_1,y\text{'}_2,y\text{'}_3,s\text{'}\right)$ in another inertial frame $F\text{'}$, then
$sqrt\left\{\left(x_1-y_1\right)^2 + \left(x_2-y_2\right)^2 + \left(x_3-y_3\right)^2\right\} = c\left(s-t\right)$ if and only if $sqrt\left\{\left(x\text{'}_1-y\text{'}_1\right)^2 + \left(x\text{'}_2-y\text{'}_2\right)^2 + \left(x\text{'}_3-y\text{'}_3\right)^2\right\} = c\left(s\text{'}-t\text{'}\right)$.

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

$c^2 \left(s-t\right)^2 - \left(x_1-y_1\right)^2 - \left(x_2-y_2\right)^2 - \left(x_3-y_3\right)^2$
$= c^2 \left(s\text{'}-t\text{'}\right)^2 - \left(x\text{'}_1-y\text{'}_1\right)^2 - \left(x\text{'}_2-y\text{'}_2\right)^2 - \left(x\text{'}_3-y\text{'}_3\right)^2$
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 $\left(x_1,x_2,x_3,t\right)$ and $\left(y_1,y_2,y_3,s\right)$ in one inertial frame $F$, and have coordinates $\left(x\text{'}_1,x\text{'}_2,x\text{'}_3,t\text{'}\right)$ and $\left(y\text{'}_1,y\text{'}_2,y\text{'}_3,s\text{'}\right)$ in another inertial frame $F\text{'}$, then $s-t = s\text{'}-t\text{'}$. Furthermore, if $s-t=s\text{'}-t\text{'}=0$, then
$quad sqrt\left\{\left(x_1-y_1\right)^2 + \left(x_2-y_2\right)^2 + \left(x_3-y_3\right)^2\right\}$
$= sqrt\left\{\left(x\text{'}_1-y\text{'}_1\right)^2 + \left(x\text{'}_2-y\text{'}_2\right)^2 + \left(x\text{'}_3-y\text{'}_3\right)^2\right\}$.

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.

Notes

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