# Bondi K-Calculus

In simple terms, Bondi k-calculus is a method of deriving the effects of Einstein's theory of special relativity which requires only basic mathematics, and yet gives all the appropriate results.

Now before all of the 'scientists' click to the next page and dismiss this as a derivation without substance, let me present a few details. The method was created by Sir Hermann Bondi, who not only wrote numerous papers on the theory of relativity, but was also given a professorship at Cambridge University. The derivation presented here is taught in undergraduate and graduate level physics courses around the world. Unfortunately it is rarely taught in High Schools, where its simplicity would benefit all students struggling with the traditional lorentz transformation derivation of relativity.

#### Lesson 1

As is done in most derivations, let us limit space to a single dimension to spare having to draw four-dimensional figures on a two dimensional screen. In the diagram below, let the vertical axis represent time and the horizontal axis represent space. Then a curve in the diagram represents a point in space which is 'moving'.(As time progresses, the point changes its position). If the curve is a straight line, the point moves at a constant velocity. Let A and B be two such lines, (representing observers or spaceships, or whatever seems appropriate) which cross at some point. Each observer carries a clock and at the instant they meet, both reset their clocks to 0. As soon as they cross, A begins shining a flashlight at B for a period of time T.

PROPOSITION - If A emits light for 1 second, B receives the light for k seconds. ( This proposition is not a huge leap of faith. It seems reasonable to have a linear relationship). Then B receives the light from A for a time kT. Now it seems natural to assume that in such a situation, with nothing else to refer to, A and B should not be able to tell which one of them is moving. Thus if B sends light for one second, A should receive it for k seconds. Thus suppose that while A is sending light, B is reflecting the light. Then A would receive the reflection for a time k(kT), or k2T. Now A wonders when B stopped receiving the signal. B claims that the signal ends at time kT. But the speed of light is constant, so the time at which the signal ended is halfway between when A ended the signal and when the signal A receives ended. And this gives that the time A thinks B stopped receiving the signal as (1+k2)T/2. Hence the two observers see this event occuring at two different times. This is called TIME DILATION.

Similarly, A might wonder how far away B is when the event occurs. Since the speed of light is constant, ( denote it by c), the distance is given by the speed times the length of time light travels between the observers. A simple calculation gives the distance as (k2-1)c T/2.

The velocity of B relative to A is then:

v / c =(k - 1/k) / ( k + 1/k )

From this, all of special relativity follows.

Lesson 2

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