# Formulae

Here are some formulae occurring in relativistic optics. They are meant for reference only, and we have given little explanation. Further discussion of them may be found in the paper Visualising special relativity which is part of this site.

The formulae presented below use the following symbols:

### Where:

v = velocity of observer.
c = the speed of light (~300 000 km/s).
γ = the gamma factor
α' = the angle measured by a moving observer between an incoming light ray and the velocity vector.
α = the original angle.
ν' = frequency measured by a moving observer.
ν = original frequency.
I' = intensity measured by a moving observer.
I = original intensity.

## The Gamma Factor

The gamma factor is fundamental part of relativistic formulae. It gets the name 'factor' because some relativistic effects increase linearly in proportion to gamma (e.g. length contraction and time dilation). Often it is more useful to look at the gamma factor rather than the speed. For example, there are large differences in relativistic effects between 0.99c ( γ = 7) and 0.999c ( γ = 22). While the speed change doesn't necessarily suggest that, the gamma factor does.

## Angular Compression

This formula rotates all incoming light rays towards the velocity vector. In the case of the rollercoaster, and most of our movies, this is in the direction we're looking towards. This effect increases almost linearly with the gamma factor.

Further discussion of this formula may be found in the paper Visualising special relativity which is part of this site.

## The Doppler Effect

This is a similar formula to that for classical doppler shifting, apart from the relativistic gamma factor, due to time dilation, which causes blue shifting.

The numerator of the fraction depends on the angle alpha and hence is directionally dependent. This effect will increase linearly as v/c increases, (this explains the comparisons page). The cos α will be maximized when α = 0, so the most up doppler shifting (towards blue) occurs in line with the velocity vector. When α = π/2, cos α goes to zero, so there is no doppler shift al all perpendicular to the velocity vector. Finally when α = π, cos α is minimized and negative, hence the most down doppler shifting (towards red), occurs in the opposite direction to the velocity vector.

Further discussion of this formula may be found in the paper Visualising special relativity which is part of this site.

## The Intensity Effect

This form of this equation is similar to the doppler effect equation, so see above for a discussion of its behavior.

Further discussion of this formula may be found in the paper Visualising special relativity which is part of this site.