Momentum

The momentum $$\mathbf P$$ is a bivector quantity with the following ten components.

$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$

The inertia tensor $$\mathcal I$$ is given by

$$\mathcal I = {\Large\sum} \gamma m{\left[\begin{array}{cccc|ccc|ccc} -1 & 0 & 0 & 0 & 0 & 0 & 0 & -x & -y & -z \\ 0 & 1 & 0 & 0 & 0 & z & -y & -ct & 0 & 0 \\ 0 & 0 & 1 & 0 & -z & 0 & x & 0 & -ct & 0 \\ 0 & 0 & 0 & 1 & y & -x & 0 & 0 & 0 & -ct \\ \hline 0 & 0 & -z & y & y^2 + z^2 & -xy & -zx & 0 & ctz & -cty \\ 0 & z & 0 & -x & -xy & z^2 + x^2 & -yz & -ctz & 0 & ctx \\ 0 & -y & x & 0 & -zx & -yz & x^2 + y^2 & cty & -ctx & 0 \\ \hline -x & -ct & 0 & 0 & 0 & -ctz & cty & c^2t^2 - x^2 & -xy & -zx \\ -y & 0 & -ct & 0 & ctz & 0 & -ctx & -xy & c^2t^2 - y^2 & -yz \\ -z & 0 & 0 & -ct & -cty & ctx & 0 & -zx & -yz & c^2t^2 - z^2 \end{array}\right]}$$ .

As in classical mechanics, the inertia tensor $$\mathcal I$$ has units of mass × length².

See Also

• Velocity