Minkowski Spacetime
This section requires some knowledge of calculus, and possibly vectors
as well. If you do not understand integration, try our page on Bondi
k-calculus
Minkowski spacetime was developed shortly after Einstein published his
paper on the theory of special relativity. It extends the 'standard' equations
of relativity to arbitrary paths through space, and is best suited to regions
of space devoid of large amounts of matter and energy.
In space, Pythagoras' theorem states that if a line has components along the x,y, and z axis of dx,dy, and dz respectively, then the total length of the line, ds, is given by
In Minkowski spacetime, this equation is changed slightly to
ds2 = c2 dt2 - ( dx2 + dy2 + dz2 )
where dt is the component of the line along the time axis.
Now suppose A is a nonaccelerating observer travelling along the time axis. (So their position is constant for all time) Suppose that B travels at a constant velocity, v , along the x-axis with respect to A.
The velocity, as every novice physics student knows, is v = dx/dt where dx and dt are changes in position and time. Thus, using the distance defined above,
So the length of B's path is a constant multiple of A's length as assumed in Bondi's k-calculus! The key to using Minkowski spacetime is knowing (without proving here) that the time experienced by an observer is related to the length of its path in spacetime. Actually,
which gives the standard equation of relativity:
dt = dT / (1-v2/c2)1/2
