Question

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

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 Vlight. Now allow the same car to roll down the street with velocity Vcar. 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 Vlight (velocity of light with respect to car) and Vcar (velocity of the car with respect to the observer), i.e. Vgalilean = Vlight + Vcar. Clearly, in this case Vgalilean > Vlight.

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: Veinstein = (Vcar + Vlight)/[1+(Vcar*Vlight)/c2], where c is the universal speed of light in vacuum.

Note, that if the Vcar 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.

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Science Quote

'If one wishes to obtain a definite answer from Nature one must attack the question from a more general and less selfish point of view.'

Max Planck
(1858-1947)