# The Lorentz metric

## The Lorentz metric

### The Lorentz metric

For lorentz:

\((\delta v )^TM\delta v = \delta t ^ 2 - \delta x^2-\delta y^2 - \delta z^2\)

\(Action = \int \sqrt {\delta t ^ 2 - \delta x^2-\delta y^2 - \delta z^2}\)

\(Action = \int \sqrt {1 - \dot x^2-\dot y^2 - \dot z^2}\delta t\)

\(Action = \int \sqrt {1-v^2}\delta t\)

### The Lorentz metric with \(c\)

For lorentz with c

\((\delta v )^TM\delta v = \delta c^2t ^ 2 - \delta x^2-\delta y^2 - \delta z^2\) p \(Action = \int \sqrt {\delta c^2 t ^ 2 - \delta x^2-\delta y^2 - \delta z^2}\)

\(Action = \int \sqrt {1 - \dfrac{\dot x^2}{c^2}-\dfrac{\dot y^2}{c^2} - \dfrac{\dot z^2}{c^2}}c\delta t\)

\(Action = \int \sqrt {1 - \dfrac{v^2}{c^2}}c\delta t\)

Because \(c\) is constant, we can simplify to:

\(Action = \int \sqrt {1 - \dfrac{\dot x^2}{c^2}-\dfrac{\dot y^2}{c^2} - \dfrac{\dot z^2}{c^2}}\delta t\)

\(Action = \int \sqrt {1 - \dfrac{v^2}{c^2}}\delta t\)

### Lorentz rotations

### Lorentz boosts

### The Lorentz group

The Lorentz group consists of the Lorentz rotations and the Lorentz boosts.

### The PoincarĂ© group

### Group contraction from Lorentz to Euclid

### Spacetime interval

### Proper time