Lorentz Factor

It is usually defined
where
:

\beta = rac{u}{c}

is the velocity in terms of the Speed Of Light ,

u

$au$

c

For large γ:

v \approx (1-rac {1} {2} \gamma ^{-2})c

PROOF

First of all, one must realize that for every observer, light travels at the ''same'' speed of light (which is why the speed of light is represented as a constant (

c

)). Imagine two observers: the first, observer

A

, traveling at a constant speed

v

with respect to a second Inertial Reference Frame in which observer

B

is stationary.

A

points a laser “upward” (perpendicular to the direction of travel). From

B

's perspective, the light is traveling at an angle. After a period of time

t_B

A

has traveled (from

B

's perspective) a distance

d = v t_B

; the light had traveled (also from

B

perspective) a distance

d = c t_B

at an angle. The upward component of the path

d_t

of the light can be solved by the Pythagorean Theorem .

d_t = \sqrt{(c t  _B)^2 - (v t_B)^2}

Factoring out

ct_B

gives us,

d_t = c t\sqrt{1 - {\left(rac{v}{c}
ight)}^2}

This distance is the same distance that

A

sees the light travel. Because the light must travel at

c

A

's time,

t_A

, will be equal to

rac{d_u}{c}

. Therefore

t_A = rac{c t_B \sqrt{1 - {\left(rac{v}{c}
ight)}^2}}{c}

which simplifies to

t_A = t_B\sqrt{1 - {\left(rac{v}{c}
ight)}^2}

Information About

Lorentz Factor