What is the Lorentz transformation

Greg Bernhardt

Definition/Summary

A Lorentz transformation is the relation between the coordinates of two inertial observers who use the same event as their origin of coordinates.

A Lorentz boost is a Lorentz transformation with no rotation (so that both observers use the same coordinate-name for the direction of their relative velocity).

A combination of two Lorentz boosts of speeds u and v in the same direction is a third Lorentz boost in the same direction, of speed (u + v)/(1 + uv/c²).

A combination of two Lorentz boosts in different directions is not a Lorentz boost, but is a combination of a Lorentz boost and a spatial rotation (a rotation known as "Thomas precession") in the plane of those directions.

Equations

The standard Lorentz Transformation for a boost with velocity v in the x direction from coordinates $t,x,y,z$ to coordinates $t^{\prime},x^{\prime},y^{\prime},z^{\prime}$:

$$t^{\prime} = \gamma \left( t - \frac{vx}{c^2} \right)$$

$$x^{\prime} = \gamma (x - vt)$$

$$y^{\prime} = y\ \ \ \ z^{\prime} = z\ \ \ \ \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$

Redshift: $$c\,t^{\prime}\ +\ x^{\prime}\ =\ \sqrt{\frac{1\ -\ v/c}{1\ +\ v/c}}(c\,t^{\prime}\ +\ x^{\prime})$$

Velocity addition (one-dimensional): $$v\ =\ \frac{v_1 + v_2}{1\ +\ v_1v_2/c^2}$$

Lorentz-Fitzgerald contraction factor: $$\frac{1}{\gamma}$$

Time dilation factor: $$\frac{1}{\gamma}$$

Rapidity: $$\alpha\ \ \text{where}\ \ v/c\ =\ \tanh\alpha\ \ \text{and so}\ \ \gamma\ =\ \cosh\alpha$$

If the units of time and distance are adjusted so that c = 1, this standard Lorentz Transformation has the more symmetric form:

$$t^{\prime} = \gamma (t - vx)$$

$$x^{\prime} = \gamma (x - vt)$$

$$y^{\prime} = y\ \ \ \ z^{\prime} = z\ \ \ \ \gamma = \frac{1}{\sqrt{1 - v^2}}$$

Using rapidity:

$$t^{\prime} = \cosh\alpha\,t - \sinh\alpha\,x$$

$$x^{\prime} = \cosh\alpha\,x - \sinh\alpha\,t$$

$$t^{\prime}\ \pm\ x^{\prime}\ =\ e^{\mp \alpha}(t^{\prime}\ \pm\ x^{\prime})$$

$$\alpha\ =\ \alpha_1\ +\ \alpha_2$$

Extended explanation

Addition of speeds in the same direction:

If there are three observers, moving in the same direction, and calling it the same direction, then their relative speeds may be combined according to the formula v = (v1 + v2)/(1 + v1v1/c²).

This easier to understand if rapidities are used, with v/c = $\tanh\alpha$, v1/c = $\tanh\alpha_1$, v2/c = $\tanh\alpha_2$: then it simply says $\alpha\ =\ \alpha_1\ +\ \alpha_2$.

In other words: rapidities (in one dimension) add like ordinary numbers.

By combining speeds, we add the rapidities, and so we can make the combined rapidity as large as we like. However, rapidity is a tanh, and tanh is always less than 1 (because $\tanh\alpha\ =\ (1\,-\,e^{-2\alpha})/(1\,+\,e^{-2\alpha})$), and so the speed can never quite reach c.

Impossibility of exceeding c:

It is often said that nothing can be accelerated to the speed of light because its mass increases as it gets faster.

However, the fundamental reason is simply that "adding" speeds only adds tanh-1(speed/c), and so no amount of adding can make (speed/c) equal to (or greater than) 1.

Poincare transformations:

A Poincare transformation is the relation between the coordinates of any two inertial observers.

Lorentz transformations are Poincare transformations in which the origin (0,0,0,0) goes to itself.

Groups:

The Lorentz group is the group of Lorentz transformations, and is a subgroup of the Poincare group (of Poincare transformations).

The Lorentz boosts in one dimension are a group, which is a subgroup of the Lorentz group..

The Lorentz boosts in more than one dimension are not a group.

Every Lorentz transformation is a combination of a Lorentz boost and a spatial rotation. Every Poincare transformation is combination of a Lorentz boost a spatial rotation and a change of origin.

In other words: the Lorentz group is generated by Lorentz boosts and spatial rotations. The Poincare group is generated by Lorentz transformations and space-time translations.

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