In the latter part of the 1800's it was well known that Newtonian mechanics and Maxwell's 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" independent of the speed of source and 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 and modifications to the experiment were unsuccessful. Albert Einstein's approach was to modify the way we do mechanics.

Einstein introduced two postulates. First, physical laws are the same for all observers. This would not conflict with Newton's first law, and is incorporated in Newtonian mechanics. But second, the speed of light is the same for all observers, as in Maxwell's theory, is in direct conflict with Newtonian mechanics.

The "problem" becomes apparent if we consider a special kind of clock, the light pulse clock which uses a pulse of light bouncing between two mirrors spaced h (say- half a metre) apart. We say h metres of time have elapsed each time the pulse is reflected back and forth, Dt' = 2h/c. Note the prime refers to the "clock frame", where the pulse begins and ends in the same place. Another observer will call this the "moving clock".
If the clock is moving past another observer at speed V, the time it takes to go from one mirror to another is ... longer! The path is the hypotenuse of the right triangle h,½VDt, ½cDt' and we have

where g² = c²/(c²-V²)    g = 1/(1-V²/c²)^½ = 1/(1-b²)^½   and  b=V/c.
(Notice also that sinq=b, the aberration formula sometimes seen incorrectly as tanq=V/c.)

The elapsed time as measured by the observer who sees the measurements at different positions is a factor (1-b²)^-½ larger than that measured by the observer who makes the measurements at the same place. This is the time dilation (sometimes called dilitation) effect.

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 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 2km and at sea level. Since muons have a half-life of 2.2ms, the average distance traveled, ignoring time dilation, would be 660m. Almost none of the muons should reach sea level. In fact, a large percentage of the muons do, in disagreement with classical mechanics. On the other hand, taking time dilation into account an observer on the earth finds the muon's lifetime is about 20ms, so the average distance traveled before decay is 6km. Thus, most of the muons reach the earth. From the point of view of the muon, the distance traveled is, using Lorentz contraction, about 220m, and the required time is only 0.73ms, considerably less than the 2.2ms 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. 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

wherea(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)^½ and ½(a-1/a) = (V/c)(1-V²/c²)^½.

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 transformation equations first found by Lorentz for E&M:

or since we prefer to measure time t in metres, ct:

with b=V/c and g=1/(1-b²)^½ .....(12)

The Relativity of Simultaneity
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.
The events are A=(ct_c,x_a, 0, 0), and B:=(ct_c,x_b,0,0). The components of these two events, as measured by S', are given by the Lorentz transformation equations (y=z=0):

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 ¹t'_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 |cDt'| = |V/c|L_o < L_o  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 (11a) and (11b), this implies that x²+y²+z²-c²t² or, if you prefer, c²t²-x²-y²-z² 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.

The inverse equations are obtained by switching the primes and changing the sign of b. Summarizing......(18):

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.