Solution: Interval Invariance Implies Universal Speed of Light 

Return to current sectionDefining Spacetime.


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. 



Return to current sectionDefining Spacetime.


© David Hestenes 2005, 2014