**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 *= *X _{2} *−

*X*=

_{1}*c*(

*t*−

_{2}*t*) + (

_{1}**x**

_{2}

**−**

**x**

_{1}) =

*c*

*∆*

*t*+

*∆*

**x**

between events *X*_{1} and *X*_{2} has an **invariant magnitude** |*∆**X**| *called the interval or* proper distance*** **between the events and given by

*∆**X **∆**X̃ =* *ε* *|**∆**X**|*^{2} = (*c**∆**t*)^{2} − (*∆***x**)^{2} ,

where *∆**X̃ *= *c**∆**t *− *∆***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

*∆**X **∆**X̃ =* *ε* *|**∆**X**|*^{2} = *ε* *|**∆**X'**|*^{2} = *∆**X' **∆**X̃'*

where *∆**X' *= *X' _{2} *−

*X'*=

_{1}*c*

*∆*

*t'*+

*∆*

**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 ^{133}Cs 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

**. We say one**

*time meters*time meter (

*ct*= 1) corresponds to a time of about 3.34 ns.

__Previous section__**: ****Special Relativity with Geometric Algebra****.**

**Next section: ****Spacetime Maps****.**