### Question

If the speed of light were infinitely large, would the effects of time dilation and length contractions be observable?

Asked by: Undisclosed

### Answer

The effects of time dilation and length contractions would not exist
if there were no restriction on the speed of light in a vacuum. These
effects are a direct consequence of the Einstein's velocity addition
rule that, unlike the Galilean velocity addition rule, includes the
universal speed of light explicitly.

To make things a bit more clear, let's consider the following
situation: say, a car is parked on a street with its head lights
on. It is being observed by a person standing on the sidewalk. Since
both the car and the observer are at rest with respect to ground, the
observer concludes that the light emitted by cars head lights
propagates with some velocity V_{light}. Now allow the same car to roll
down the street with velocity V_{car}. According to the Galilean addition
rule our stationary observer should conclude that the speed of light
emitted by the car is equal to the sum of the velocities V_{light} (velocity
of light with respect to car) and V_{car} (velocity of the car with
respect to the observer), i.e. V_{galilean} = V_{light} + V_{car}. Clearly, in
this case V_{galilean} > V_{light}.

When Einstein postulated that the speed of light in the vacuum is the
same for all inertial observers, the need to reformulate the velocity
addition rule emerged (Einstein's velocity addition rule). Basically,
space-time had to be rescaled such that the speed of light in vacuum
remained constant. This rescaling process is physically manifested in
time dilation and length contraction. Thus, the Einstein's velocity
addition rule says: V_{einstein} = (V_{car} + V_{light})/[1+(V_{car}*V_{light})/c^{2}],
where c is the universal speed of light in vacuum.

Note, that if the V_{car} is small compared to c, the results of Galilean
additional rule are in close agreement with that given by Einstein's
addition rule. In other words, for speeds much less that c the
Galilean rule is a good approximation, and therefore the effects of
time dilation and length contraction are not noticeable at such
speeds.

Answered by: Max Chtangeev, B.S., Physics Grad Student, MIT