## Solution: Interval Invariance Implies Universal Speed of Light

Problem: Prove that spacetime interval invariance implies that the speed of light has the same constant value
in all inertial systems.

Solution:  Consider event X1 to be a burst of light from a particular place and event X2 to be the sensing of that burst
at another place. If the speed of light in this inertial system is
c, the invariance of the interval between
these two events means that

(ct)2 − (x)2  =  (ct)2 − (ct)2  =  0

should also be zero when calculated in another inertial system for these same two events. In that
other system, let's say the speed of light is
c'. Then:

(ct')2 − (x')2  =  (ct')2 − (c't')2  =  0 .

Dividing out
t' leaves c = c'. The first c is part of the definition of interval and may be thought of as a
conversion factor relating units of time and length, the second,
c', is the speed of  light measured in an
arbitrary inertial system, and they must be the same to maintain interval invariance.