Fig 1 represents the right triangle from which the time dilation factor was formulated.

Side ct:

This is the only side for which both terms are constant. The speed of light, c, is constant. Time, t, in the rest frame is constant. Therefore, judged from the rest frame the length of side ct is constant.

Side vt:

Again t is constant as judged from the rest frame. the relative velocity of the moving frame is referenced by v. therefore at relative velocity 0 side vt does not exist. For v>0 the greater the relative velocity the longer side vt is.

Side ct’:

Here c is, of course, constant and the value of t’ is, necessarily, dependent on the only other non constant value v from side vt. Time in the moving frame is t’.

The formulation is as follows:

a*a+b*b=c*c the Pythagorean theorem

c*c*t’*t’+v*v*t*t=c*c*t*t substitute values

c*c*t’*t’=c*c*t*t-v*v*t*t the long version of the time dilation equation

t’=t*sqrt(1-v*v/c*c) solved for t’- the time dilation equation

To demystify and simplify the following calculations the constants c and t are given values then the long version of the equation is simplified. c=1 and t=1

c*c*t’*t’=c*c*t*t-v*v*t*t the long version of the time dilation equation

t’*t’=1-v*v c*c=1 t*t=1 then simplify – this is the “short version” of the equation

To use this process velocity, v, must be expressed as a decimal fraction of c. .01c, .5c, .9c etc.

If there is any doubt as to the validity of this method feel free to perform the long calculation. The result is the same.

For the question of what happens if v=c:

In that case since c=1 then v=1 and as per above t=1

c*c*t’*t’=c*c*t*t-v*v*t*t the long version of the time dilation equation

1*1*t’*t’=1*1*1*1-1*1*1*1 substitute 1 for the appropriate values

t’*t’=0 simplify

t’=0 if t’=0 then side ct’ has no length and, therefore, does not exist.

For the question of what happens if t=0:

For t=0, c=1

Per the figure:

Side ct does not exist because t is 0

Side vt does not exist because t is 0

c*c*t’*t’=c*c*t*t-v*v*t*t the long version of the time dilation equation

1*1*t’*t’=1*1*0*0-v*v*0*0 substitutions

t’*t’=0

t’=0

per calculation:

Side ct’ does not exist because t’ is 0

For the question of what happens if v=0:

t’*t’=1-v*v the short version of the equation

t’*t’=1-0

t’*t’=1

t’=1

This particular calculation will be referred to later.

More examples:

For v=.866c

t’*t’=1-v*v the short version of the equation

t’*t’=1-.866*.866

t’*t’=1-.75

t’*t’=.25

t’=.5

For v=.01c

t’*t’=1-v*v the short version of the equation

t’*t’=1-.01*.01

t’*t’=1-.0001

t’*t’=.9999

t’=.99994999

The question now is what exactly do ct, vt and ct’ represent and why did Lorentz decide to use this formulation. The answer in understandable terms, diagrams and animations can be found in the Cal Tech video “The Mechanical Universe and Beyond” number 42 “The Lorentz Transformation” starting about minute 11 or so. You can Google it or follow this link:

http://www.learner.org/resources/series42.html?pop=yes&pid=611#Briefly stated: Lorentz and Einstein visualized two mirrors with a light pulse moving at the speed of light from, in this case, the bottom mirror to the top mirror in the moving frame. As seen from within the moving frame the path of the light pulse is side ct’. The distance that the mirror assembly or light clock in the moving frame traversed during time t of the rest frame is represented by side vt. The path of the light pulse from the bottom mirror to the top mirror as seen from the rest frame is represented by ct, the hypotenuse of the right triangle.

As previously stated and shown mathematically with the moving frame at relative velocity 0 side vt does not exist and therefore, as judged from the rest frame ct=ct’. At any relative velocity >0<c side vt of the right triangle exists. If side vt of the right triangle exists then side ct is the hypotenuse. The hypotenuse is the longest side of a right triangle. Since ct is the longest side of the triangle side ct’ is shorter than ct. At relative velocity 0 the length of side ct’ which is perpendicular to the direction of motion is the same as length ct which is constant as judged from the rest frame. At any relative velocity >0<c the length of ct’ is less than the length of ct as judged from the rest frame.

The Special Relativity stipulation that length perpendicular to the direction of motion is not changed as judged from the rest frame if the constant relative velocity of the moving frame is changed is disproved. At relative velocity 0, ct’=ct at relative velocities >0<c, ct’<ct.

Butchmurray