*** TIME DILATION AND PARADOXES ***

Morning brain teasers are fun!


Time Dilation

One of the most enthralling aspects of Relativity is its new understanding of time. The term "time dilation" might evoke images of Salvadore Dali's timepieces hanging on twigs, however, time dilation is all but surrealistic. As stated earlier, if the speed of light is constant, time cannot be constant. In fact, it doesn't make sense to speak of time as being constant or absolute, when we think of it as one dimension of spacetime. Special Relativity states that time is measured according to the relative velocity of the reference frame it is measured in. Despite of the simplicity of this statement, the relativistic connection between time and space are hard to fathom. There are numerous ways to illustrate this:

The four dimensions of spacetime.

In Relativity the world has four dimensions: three space dimensions and one dimension that is not exactly time but related to time. In fact, it is time multiplied by the square root of -1. Say, you move through one space dimension from point A to point B. When you move to another space coordinate, you automatically cause your position on the time coordinate to change, even if you don't notice. This causes time to elapse. Of course, you are always traveling through time, but when you travel through space you travel through time by less than you expect. Consider the following example:

Time dilation; the twin paradox.

There are two twin brothers. On their thirtieth birthday, one of the brothers goes on a space journey in a superfast rocket that travels at 99% of the speed of light. The space traveller stays on his journey for precisely one year, whereupon he returns to Earth on his 31st birthday. On Earth, however, seven years have elapsed, so his twin brother is 37 years old at the time of his arrival. This is due to the fact that time is stretched by factor 7 at approx. 99% of the speed of light, which means that in the space traveller’s reference frame, one year is equivalent to seven years on earth. Yet, time appears to have passed normally to both brothers, i.e. both still need five minutes to shave each morning in their respective reference frame.

Time in the moving system will be observed by a stationary observer to be running slower by the factor t':

:form:

As it can be seen from the above function, the effect of time dilation is negligible for common speeds, such as that of a car or even a jet plane, but it increases dramatically when one gets close to the speed of light. Very close to c, time virtually stands still for the outside observer.

More here [link to www.thebigview.com]

Train and Tunnel Paradox

Suppose that a train robber decides to stop a train inside tunnel. The proper length of the train is 60 m, while the proper length of the tunnel is 50 m. The train is traveling at 4/5 the speed of light. According to proper lengths, the train would not fit inside the tunnel. But the robber plans to use relativity to his advantage. The length of the moving train in the rest frame of the tunnel, and of the robber, is 36 m. The robber computes this and decides to trap the train inside the tunnel, since, in his frame, the train should fit. From the point of view of the train's engineer, however, the tunnel is only 30 m long, just half the length of the train. The engineer knows that his 60 m train will not fit completely into the tunnel. The robber thinks that the train will fit, whereas the engineer is sure it will not. But either the train will fit, or it will not; it cannot do both. Who is correct?

:Train paradox:

Solution here [link to www.astro.virginia.edu]