In the early 1900's it was already recognized that Newtonian mechanics and electromagnetic theory were not consistent. In Newtonian mechanics, two observers moving relative to each other find different velocities for a moving body, the difference being their relative velocity. In electromagnetic theory, radiation (in vacuum) always moves with the "speed of light" irrespective of the observer. Most of the attempts to remove this discrepancy involved modifications to electromagnetic theory. The "ether" concept supposed a single reference frame in which electromagnetic theory was valid, but attempts to find this frame, such as the Michelson-Morley experiment, were unsuccessful. Albert Einstein's approach was to modify Newtonian mechanics. Since the speed of light (about 3x10⁸ m/s) is so large, the well known results in Newtonian mechanics are not invalidated; rather the theory is extended.

Einstein introduced two postulates. Postulate I: physical laws are the same for all observers. Postulate II: the speed of light is the same for all observers. For a single observer it might appear that no changes are necessary. This is true for kinematic variables, such as time, position, velocity, and acceleration. But Postulate I does require changes in the definitions of dynamical variables, such as momentum, energy, and force, since laws (e.g. momentum conservation) must be valid in all frames if they are valid in one. To see how the formulation of special relativity comes about, we start by considering two situations.

The first supposes an observer in a vehicle sending a pulse of light from the floor to a mirror directly above on the ceiling and back to the floor. If the speed of light is c and the height of the vehicle is h, then the total time for the pulse to travel up and down is Dt_o = 2h/c (the subscript zero on t recognizes that the measurements by this observer occur at the same place). An observer beside the road, however, believes that the two measurements (times of emission and reception of the light pulse) are separated by a distance VDt, where V is the speed of the vehicle and Dt is the light pulse transit time as measured by this observer. This transit time is Dt = 2[h² + (VDt/2)²]^1/2/c, since the speed of light must be c. Now

so that

The elapsed time as measured by the observer who sees the measurements at different positions is a factor [1 - V²/c²]^-1/2 larger than that measured by the observer who makes the measurements at the same place. This is the time dilation effect. At this point it is useful to define g = [1 - V²/c²]^-1/2, so Dt = gDt_o. Note that for V = 300 m/s (about 600 mph; a high speed by everyday standards) g = 1 + 5x10^-13 » 1. Even for V = 3x10⁷ m/s, g = 1.00504 is still very close to unity. However, if V = 0.999c, then g = 22.37 and, for V even closer to c, g may be made as large as desired. Values of V which are equal to or greater than c in magnitude are not permitted in the theory and, as will be seen later, cannot be attained. Also note that g³1, and that g is an even function of V, i.e. g(-V) = g(V).

The new concept, inherent in time dilation, is that observers have individual time frames, just as they have individual space frames based on their own location. The relations between the coordinates of two observers must involve both space and time.

The second experiment supposes that one observer is holding a rod, length L_o (according to this relatively stationary observer), and another observer, moving with speed V relative to the first, measures the length of the rod by the time it takes to pass by. Since the second observer makes these measurements at the same place, this time may be denoted by Dt_o and the length is L = VDt_o. The observer holding the rod measures this time as Dt = gDt_o so that L_o = VDt = VgDt_o = gL. The relation between the two length measurements is L = L_o/g. This is the Lorentz contraction effect (note L £ L_o).

Experimental verification of time dilation was provided by observations on muons produced in the upper atmosphere by cosmic rays. The intensity of muons with speeds of 0.994c was measured at a height of 2000 m and at sea level. Since muons have an average lifetime of Dt_o = 2.2 ms, the average distance traveled, ignoring time dilation, would be (3x10⁸)(2.2x10^-6) = 660 m, so almost none of the muons should reach sea level. Actually a large percentage of the muons do so, in disagreement with classical Newtonian mechanics. On the other hand, taking time dilation into account with g = 9.14, an observer on the earth finds the muon's lifetime to be about 20 ms, so the average distance traveled before decay is 6000 m. Thus a reasonable fraction of the muons reach the earth. From the point of view of the muon, the distance traveled is, using Lorentz contraction, 2000/9.14 = 220 m, and the required time is only 220/3x10⁸ = 0.73 ms. This is considerably less than the 2.2 ms lifetime.

The Lorentz Transformation

The primary concern of special relativity is the determination of the relations between kinematic variables for two different observers. Suppose that two observers, S and S', with relative speed V in the x direction (V for S' relative to S and, by symmetry, - V for S relative to S'), pass at a common origin at a common time zero (this simplifies the algebra without changing the physics). The question now is: How are the coordinates of an event as measured by S': (x', y', z', t'), related to the coordinates of the same event as measured by S: (x, y, z, t); i.e. what are the equations for each of x', y', z', and t', as functions of the set of variables (x, y, z, t)?

Since the new feature in the theory is the constancy of the speed of light, consider a pulse of light starting at the common origin at the common time zero and moving in the x direction. Observer S writes the equation of motion for this pulse as x = ct, and S' writes it as x' = ct'. Thus x - ct = 0 implies x' - ct' = 0, and vice versa.

Whatever equations relate (x' ,y' ,z' ,t') to (x, y, z, t) must embody this condition. A suitable relation, applicable to all x, t, x', and t', is

where a(V) is a function of the velocity, V, of S' relative to S. Since the velocity of S relative to S' is - V, the inverse relation must be

Next, consider what happens to equation (4) if the direction of the x axis is reversed so that x ® - x,
x' ® - x', and V ® - V:

or, rearranging,

A similar equation could have been written down by considering a light pulse traveling backwards along the x axis, but it would not have had 1/a(V), explicitly.

The exact form for a is obtained by using more information about the relative velocity. Consider the equation of motion for the origin of the S' frame: according to S' it is x' = 0, and, according to S, it is x = Vt. Substituting these into equations (3) and (6) yields

and

Changing the sign on both sides of equation (7) and dividing by equation (8) gives

or

The positive sign for the square root is adopted to agree with the case V = 0. Also, to ensure that both a and a^-1 make sense, the condition - c < V < c is imposed on V (the physical reason for this condition appears later). Now, equations (3) and (6), along with equation (10), determine x' and t' in terms of x and t. Using the combinations (x' - ct') and (x + ct) seems somewhat indirect, since the aim is to get equations for x' and t', but these combinations are the natural consequence of Postulate II. In this regard, it is useful to note that ½(a +1/a) = (1 - V²/c²)^-1/2 and  ½(a - 1/a) = (V/c)(1 - V²/c²)^-1/2

It might appear that equation (3) is too restrictive. Why not a relation such as (x' - ct') = a(V)tan(x - ct)? However, any such relation, in which the rate of change of the primed variables with respect to the unprimed variables is position and time dependent, has an unfortunate consequence: If S finds that Newton's First Law is satisfied then S' finds that Newton's First Law is not satisfied. But this violates Postulate I. The linear relation assumed in equation (3) is the only one possible.

The space dimensions perpendicular to the relative motion can not distinguish V from - V, i.e. S and S' must be equivalent, so y' can not be greater in magnitude than y and y can not be greater in magnitude than y'. Technically this provides two possible solutions for y': y' = y and y' = - y. But when V = 0, y' = y, and, as V increases continuously, no discontinuous change can occur. So y' = y for all V. Similarly, z' = z.

The result is the set of Lorentz transformation equations:

with (as before)

To understand the role of the Lorentz transformation equations it is useful to reconsider Lorentz contraction. Suppose that a rod lies along the x axis and is at rest in the frame of S', so that it has a velocity V in the x direction according to S. Now suppose observer S determines the length of the rod by measuring two events: the position, x_a, of the back of the rod at time t_c, and the position, x_b, of the front of the rod at the same time t_c. Formally, the events are A: (x_a, 0, 0, t_c), and B: (x_b, 0, 0, t_c). The components of these two events, as measured by S', are given by the Lorentz transformation equations:

and

When the equations for x_a' and x_b' are combined, the terms involving t_c cancel to yield

i.e. L = L_o/g, as before.

However, a new feature is now apparent: t_b' ¹ t_a'. The events that S regards as simultaneous are not simultaneous according to S'. In general, the concept of simultaneity depends on the observer. The time difference according to S' is

Note that |Dt'| = |V/c|(L_o/c) < L_o/c, since |V| < c. So, according to S', no light pulse, nor any slower signal, can travel between event A and event B. This condition is obviously true for S.

Another useful result is obtained by combining equations (3) and (6) so that the a factors cancel:

In conjunction with equations (11b) and (11c), this implies that x² + y² + z² - c²t² is the same for all observers, i.e. it is invariant.

Assuming that both observers are at the origin at time zero is convenient (it simplifies the algebra) but is not necessary. More generally one may deal with space and time differences. Also it is nice to use the combinations b = V/c and ct. This yields

The inverse equations are obtained by switching the primes and changing the sign of b. Further, Dx² + Dy² + Dz² - c²Dt² is invariant.

One interesting consequence of the Lorentz transformation equations is the "addition" of velocities equation. Suppose a particle moves in the x direction with velocity v' = Dx'/Dt' as measured by S'. Then the velocity as measured by S is

If v' = c, then this gives v = c. If |v'| < c, then |v| < c.

Momentum and energy are very useful concepts in Newtonian mechanics, indeed, conservation of momentum and energy are central themes in collision calculations, etc. In special relativity a new feature is added. If one observer determines that momentum and energy are conserved, then so must all other observers (Postulate I). I.e. if Sp_xi is the same before and after a collision then this must also hold for Sp'_xi (here particles are specified by a subscript i and S denotes summation over all particles). If one tries to define p_x º mv_x = mDx/Dt then

and, when one sums over a set of particles, the p_x in the denominator makes it impossible to establish momentum conservation in both reference frames. One would like something where only the numerator changes with the frame. The best candidate is p_x = mDx/Dt_o since Dt_o (the time interval as determined in a frame moving with the particle) is the same for all observers. But Dt_o = Dt/g_o, so this is equivalent to p_x = g_omv_x, where g_o = [1 - v²/c²]^-1/2 (g_o is used here to emphasize that the transformation is from the reference frame in which the particle is stationary). Now

where p_o = m(cDt)/Dt_o = g_omc. Thus, provided that p_o is also a conserved quantity (see below), momentum conservation can also be established in the second frame:

Since both the sums on the right hand side of this equation are the same before and after the collision, this also holds for the sum on the left hand side, i.e. the x component of the total momentum is conserved in the second reference frame. The other components (including p_o) follow likewise.

What is this new quantity p_o? In the non-relativistic limit p_o = g_omc » mc, and, since mass is a conserved quantity in non-relativistic collisions, p_o appears to be a conserved quantity. But the non-relativistic limit usually assumes that c becomes infinite, which would not be appropriate here. A more careful analysis is needed. When v/c is small, the Binomial Theorem gives the approximation

so that

where K is the non-relativistic kinetic energy. This suggests that the kinetic energy in special relativity should be defined as K = (g_o-1)mc² and that mass and kinetic energy conservation should apply in collisions (although only conservation of p_oc = mc² + K is needed).

At this point Mother Nature has her say. Neither mass nor kinetic energy are conserved in collisions! But mc² + K is conserved. So conservation of momentum and energy are still valid if the particle's energy is defined as E = g_omc².

The non-conservation of mass and kinetic energy appears as a conversion between these two. Since E/c² has the units of mass and E is usually measured in units of eV or MeV, it is convenient to introduce a new mass unit: MeV/c². Likewise, a new momentum unit is introduced: MeV/c. An important conversion process is the proton cycle in the Sun. There, four protons and two electrons are converted into a helium-4 nucleus, six photons, and two neutrinos. The proton mass is 938.272 MeV/c², the electron mass is 0.511 MeV/c², the helium-4 mass is 3727.380 MeV/c², and the photons and neutrinos have no mass. The change in mass is [4(938.272) + 2(0.511) - 3727.380] MeV/c² = 26.73 MeV/c², so 26.73 MeV of kinetic energy is released in each such conversion.

To recapitulate, in special relativity the momentum of a particle of mass m and velocity v is defined as

and its energy is defined as

with

The transformation equations for the momentum and energy are

Choosing b = p_xc/E makes p'_x = 0 and, if the x direction is chosen to be the direction of p, then the S' frame has p' = 0. When p and E are the total momentum and total energy, respectively, for a system of particles, the Lorentz transformation with b = pc/E yields S' as the center of momentum frame with zero total momentum.

Both total momentum and total energy are conserved in collisions and the transformation equations ensure that these conservation laws hold for all observers. Note that, just as position and time have to be associated, so do momentum and energy.

For a single particle

so that

and

Equations (30) and (31) are the heart of the relativistic kinematics of elementary particles. Note that v, b, and g do not appear. The equation for E allows the energy of a known particle (known mass) to be calculated when its momentum has been measured. The equation for m allows the mass of a particle to be obtained (and the particle identified) if E and p have been determined for it by, say, using energy and momentum conservation in a collision where all but one of the particles are known.

If needed, the velocity of a particle can be obtained as

Finally, it should be noted that the energy definition, E = g_omc², gives v in terms of E as

so that no matter how much energy is given to a particle, its speed is always less than c.

Particle Decay

As a practical example, consider the identification of a particle by the observation of its decay products. Suppose an unknown particle decays to produce a proton, mass m₁ = 938.3 MeV/c², and a pion, mass
m₂ = 139.6 MeV/c², and the momenta of the proton and pion are measured to be p₁ = (852.9i) MeV/c and p₂ = (203.2i + 116.3j) MeV/c, respectively. What is the mass of the unknown particle?

The momentum and energy of the unknown particle must be related to those of the proton and pion by momentum and energy conservation. Thus the total momentum, i.e. the momentum of the unknown particle, is

To obtain the total energy, the individual energies of the proton and pion must first be calculated using their masses and momenta. These are

for the proton, and

for the pion. Hence the total energy, i.e. the energy of the unknown particle, is

Now that the momentum and energy are available, the mass may be calculated:

The mass of the unknown particle is 1115.6 MeV/c², which identifies it as a lambda particle.

This exposition has necessarily been limited in extent. Topics such as the construction of coordinate systems, acceleration, the relativistic description of waves, the relativistic Doppler effect, the angular momentum tensor, force, the relativistic description of electromagnetic theory, and the electromagnetic energy tensor, have not been addressed. A full presentation may be found in Introduction to Special Relativity by Wolfgang Rindler (Oxford University Press, Oxford, 1991).

Unless otherwise indicated, the symbol c denotes the speed of light and has a value of 3x10⁸ m/s. Also
b = velocity/c and g = (1-b²)^-1/2.