## Wednesday, 12 December 2012

### The Twins' Paradox of Einstein's Relativity

Don't worry about the math: all the graph shows is that as one approaches relativistic speeds i.e. near light speed, time in your reference frame, a rocket ship, starts to slow down in relation to the time on the Earth. Unfortunately your rocket ship starts to gain an exponential increase in mass and energy required for the velocity.

What remains probably, in popular imagination, the most bizarre consequence of Einstein's Special Relativity is the paradox of the identical twins Jack and Jill. At their birth Jill goes on a space voyage across the Milky Way galaxy at close to the speed of light. When she returns to the earth several years later, she is a young woman in her twenties but her twin, Jack is an old man about a hundred years old.

How does Special Relativity explain this bizarre effect?

Special Relativity predicts the Jack and Jill paradox because it assumes that the twins' biological clock run like atomic clocks. Atomic clocks do not keep absolute Newtonian time but run at different rates in different worldliness. The twins age at different rates, therefore, because the worldliness of Jack and Jill are not the same. Rather, time, as measured on Jill's worldline, is significantly shorter than on Jack's.

What is a worldline and why does it affect the running of biological and atomic clocks?

Herman Minkowski in 1907 proposed how to represent spacetime as a four-dimensional continuum structured geometrically with respect to the behavior of photons. In Minkowski's spacetime geometry, all spacetime curves are categorized into four major types, namely, timelike, spacelike, lightlike and null curves. This classification has important consequences. All massive particles(i.e. particles with mass greater than zero) cannot attain to the speed of light, thus all events in describing their behavior in spacetime is with reference to a path traced by timelike curves or in special cases by spacelike curves. Massless particles, however, can attain to the speed of light consequently their trajectory in spacetime is referred to null curves.

The term "worldline" was used by Minkowski in reference to null or timelike curves. A "worldline" therefore tracks the path in spacetime of a massless particle such as a photon or a massive particle of subatomic dimensions. Now, the length measured along a worldline which is a timelike curve is said to be "proper time." Atomic clocks keep proper time along their timelike worldliness. Worldlines are formally defined only for pointlike particles, but in applying the worldline metric to atomic clocks, it is assumed that the volume of space occupied by the clock is negligible for the purpose of the analysis.

Atomic clocks go slower at close to the speed of light relative to an inertial frame because at close to the speed of light the length of the interval along timelike curves is stretched. This phenomenon is termed "time dilation." All processes, therefore, slow down at speeds close to the speed of light.

It is important to remember that Special Relativity deals with uniform motion and that at close to the speed of light, the space traveler in uniform motion is not only unaware of his motion but is also unaware of the slowing down of his biological process relative to her twin on earth. If one were to send Jill on a space trip, in a space ship approaching the speed of light(discounting her initial acceleration described separately by General Relativity) to a star 50 light years away, she will make the return trip in 100 years of earth time, but time for her aboard the starship would be only 20 years.

A time dilation effect similar to the effect at uniform motion close to the speed of light is observed in a gravitational field. Gravitational potential stretches out the length of the interval along timelike spacetime curves and thus biological and atomic clocks move slower in a gravitational field.

Calling all Guest Bloggers! Could you write a great post like this?