# 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:

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

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

### The Lorentz group

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