Defining Spacetime

Previous section: Special Relativity with Geometric Algebra.

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Einstein (1905) recognized that Newtonian mechanics is inconsistent with Electromagnetic Theory, and he traced the difficulty to the Newtonian concept of time. He resolved this problem by adopting two principles:

          1. Principle of Relativity. Einstein adopted this principle from Newtonian theory, but raised
              its status from a mere corollary to a basic principle.


          2.
Invariance of the speed of light. Einstein assumed that the speed of light c has the
              same value in all inertial systems.


Minkowski (1908) incorporated these principles into a new conceptual fusion of space and time that can be defined with GA by the following assumptions:


          1.
In a given inertial system, the time t and place x is of an event is represented as a single
              point X
= ct + x in a 4-dimensional space called spacetime (see spacetime maps below).


          2.
The spacetime displacement 

                        X X2 − X1 (t2 − t1(x2 − x1= c+ x

              between events X1 and X2 has an invariant magnitude |Xcalled the interval or
              proper distance between the events and given by 

                        X X̃  =  ε |X|2  =  (ct)2 (x)2 ,

              where X̃ = ct x, and the signature  ε of the interval has the value 1, 0 or –1, and,
              respectively, the interval is said to be
timelike, lightlike, or spacelike


          3.
Invariance of the interval means that

                       X̃  =  ε |X|2  =  ε |X'|2  =  X' X̃'

              where X' X'2 − X'1 ct' x' is the same interval represented in some other inertial
             
system.


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

               (Click here for a solution to this problem.)


Footnote: Because of the importance of c in this theory and the mountain of experimental data
                supporting the theory, its value is now defined to be  
c  299,792,458 m/s. The
                second is defined in terms of the period of radiation emitted from a certain energy
                level transition in 133Cs at absolute zero. Then, with the given value for 
c, the
                meter is defined as the length traveled by light in (1/299,792,458) seconds ≈ 3.34 ns.
                The combined quantity
ct shows up naturally in these equations, is used in the
                spacetime diagrams on following pages, and has the units of meters. It is a useful
                practice to think of
ct as representing meters of time or time meters. We say one
                time meter (
ct = 1) corresponds to a time of about 3.34 ns. 




Previous sectionSpecial Relativity with Geometric Algebra.

Next section: Spacetime Maps.


© David Hestenes 2005, 2014