The only reason behind the existence of the Special Theory of Relativity is the experimental fact that the speed of light is independent of any motion of the source or receiver. This is because the usual concept of speed is inconsistent with the invariance of c unless one redefines length and time units accordingly.
In his paper 'On the Electrodynamics of Moving Bodies' (published in 1905), Einstein derives his theory by considering the propagation of a light signal in two different reference frames, essentially in the following way:
he first examines the propagation the light signal needs to travel a given distance x in the system of the source, which is given by
(1) x(t) = ct ,
where t is the time and c is the speed of light. Here the speed of light is
(2) dx(t)/dt = c ,
as required.
Then he considers the propagation in a system moving with speed v relative to the source, i.e.
(3) x'(t) = ct-vt =(c-v)t .
Here the speed is obviously
(4) dx'(t)/dt = c-v ,
which contradicts the requirement that it should have the same value c in all reference frames.
In order to remedy the situation, Einstein therefore concludes that the space and time coordinates have to be re-scaled (dependent on the velocity), which leads to the well known formulae identical to the Lorentz Transformation. This re-definition of the original length and time units in order to be able apply the usual addition of velocities (Eq.(3)) is clearly invalid (believing that it is correct is like believing that one could fit a car into a 1 meter wide parking space just by measuring the car with a correspondingly expanded ruler unit).
In addition to this argument regarding the consistency of the physical definitions (which is further illustrated by the thought experiment in the box below), it can anyway be shown that Einstein's derivation of the Lorentz Transformation is mathematically inconsistent (see
here for more).
Obviously, the propagation of light signals has to be described by a completely different concept of 'speed' than the usual one applied by Einstein:
the invariance of c can only mean that the time T for a light signal to travel from the source to the observer does not depend on the velocity of either of them but only on their distance x(t=0) at the time of the emission , i.e.
(5) T = x(t=0)/c .
The mental picture of light as an entity travelling independently through space is therefore wrong; if one wants c to be truly invariant, there can only be a moment of emission and a moment of detection, and the 'speed' of light is simply given by the difference between the two and the distance at the instant of emission. One might ask how the light signal 'knows' what this distance is, but this would be a metaphysical question like the one how the earth knows how to react to the gravitational pull of the sun. It is just a law of nature.
This circumstance could for instance be of relevance for the apparent
anomalous acceleration (slowdown) of the Pioneer 10 and 11 spacecrafts, as the traditional way of calculating the travel time of the communication signal underestimates the corresponding distance approximately by the observed mismatch (it is obvious that for the speeds involved (10s of km/sec) this problem will only become apparent for large distances, as observed).
In order to illustrate the above theoretical argument regarding the correct interpretation of the invariance of the speed of light, consider the following thought experiment:
An observer in a car is moving away from a ground based observer (GBO) with velocity v. Assume that both carry synchronized clocks (these could be started for instance through a contact when the car sets off from the GBO). Assume also that the car and the GBO are connected by two ropes, one fixed to the car and reeling off at the GBO, the other fixed to the GBO and reeling off at the car. Both ropes carry markings which indicate the length of rope that has reeled off (i.e. the distance between both). When the reel at the GBO has reached the marking x 1, a contact triggers a light signal to be sent after the car. The clock at the GBO at this moment reads T 1=x 1/v . Since obviously the same amount of rope has reeled off at the car as well, the marking at the latter does at this moment also read x 1 and since the rope has reeled off with the same speed, the car-clock does also shows T 1'=T 1=x 1/v. Now, the observer in the car knows (by agreement) that at this moment the GBO (who is receding with velocity v relatively to him) sends a light signal to him which should reach him at his time T 2'=T 1'+x 1/c = T 1 +x 1/c as the speed of light does not depend on the relative velocity of the GBO. For the observer in the car the signal has therefore taken an interval T2'-T1'=x1/c to reach him. During this time the rope has reeled off by a further x 2'-x 1'=v*(T 2-T 1')=x 1*v/c and reads therefore the distance x2'=x1*(1+v/c). At this moment (when the light signal reaches the car) the car-reel blocks and stops the relative motion of the two. Now, for the GBO the same amount of rope must have reeled off, so he knows that the light signal reached the car when the latter was at a distance x2=x2'=x1*(1+v/c) and since for him the rope also reeled off with velocity v, his clock also shows the time T 2=x 2/v = x 1*(1/v+1/c)=T 1+x 1/c , i.e. T2-T1=x1/c.
The travel time of the light signal is therefore velocity independent and hence identical in either reference frame, in contrast to the result that Special Relativity would yield.
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A concluding word regarding the invariance of c: this is generally displayed merely as an experimental fact, but if one accepts that light is an electromagnetic wave, it is indeed a theoretical necessity: a lightwave can not possibly require a physical carrier medium because otherwise it could not propagate through empty space; on the other hand, without an 'ether', the speed of light has to be constant with regard to source and receiver or otherwise it would be completely undefined. The point is in fact that a light wave needs no carrying medium as it carries itself (to be more precise, according to Maxwell's Equations, the electric wave carries the magnetic wave and vice versa; it is somewhat ironic that Maxwell himself did obviously not realize this as he believed in the ether theory and a positive outcome of the Michelson-Morley experiment). On the other hand, experiments like the Sagnac effect, which apparently contradict the invariance of c and suggest the presence of an absolute reference frame, could well be explained by the presence of the earth's magnetic field. One should expect that with a sufficiently compensated magnetic field these experiments also give a negative result i.e. the conclusions on this page should then apply.
Of course, for the addition of velocities of material objects, the speed of light is not relevant at all and velocities are added in the usual way, i.e. the relative speed between two objects can be larger than c.
NOTE: Because the usual addition of velocities does not apply to light, the notion of a 'speed' or 'velocity' of light should therefore strictly speaking not be used at all. With regard to light there can only be distances and light travel times (at least as the propagation in a field free vacuum is concerned).