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.

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 k^{2}T.

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+k^{2})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 (k^{2}-1)c
T/2.

The velocity of B relative to A is then:

From this, all of special relativity follows.

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