List of relativistic equations

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A list of relativistic equations:

Contents

List of Equations

Lorentz Transformation

The Lorentz transformation:

Time dilation:

Length contraction:

Velocity Subtraction:

Four-Vectors and Matrices

The matrix version of the Lorentz transformation:

The metric tensor:

The space time interval:

Four velocity:

Four-momentum:

Energy in Relativity

Energy and the four-momentum's time term:

Kinetic energy:

Other useful energy-momentum relations:

Four-force:

Doppler Shift

General Doppler shift:

Doppler shift for emitter and observer moving right towards each other (or directly away):

Doppler shift for emitter and observer moving in a direction perpendicular to the line connecting them:

Derivation of the equations of Special Relativity

To derive the equations of special relativity, one must start with two postulates:

From these two postulates, all of special relativity follows.

Time Dilation

Now, with the above postulates in mind, imagine you are in a train, a car, a bus, or any other vehicle, moving with a velocity v with respect to someone standing on the ground as your vehicle goes by. Inside your vehicle, you shine a light upwards to a mirror on the ceiling, where the light bounces back down. If the height of the mirror is h, and the speed of light c, then the time it takes for the light to go up and come back down is:

However, to the observer on the ground, the situation is very different. Because the train is moving by the observer on the ground, the light beam appears to move diagonally instead of straight up and down. To visualize this, picture the light being emitted at one point, then having the vehicle move until the light hits the mirror at the top of the vehicle, and then having the train move still more until the light beam returns to the bottom of the vehicle. The light beam will have appeared to have moved diagonally upward with the train, and then diagonally downward. This path will help form two-right sided triangles, with the height as one of the sides, and the two straight parts of the path being the respective hypotenuses:

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