User:F=q(E+v^B)/4-volume



In special and general relativity - the 4-volume is the content of a hyperparallelepiped in 4d Minkowski spacetime.

Calculation
The volume of a hyperparallelepiped with vector edges A, in the time direction and B, C, D in the spatial directions, is given by:


 * $$\Omega = {}^\star(\mathbf{A}\wedge\mathbf{B}\wedge\mathbf{C}\wedge\mathbf{D})$$

where the orientation is so that time t points towards the future, and the vectors in this order form a right-hand tetrad. The basis 4-form is:


 * $$\mathbf{e}_0\wedge\mathbf{e}_1\wedge\mathbf{e}_2\wedge\mathbf{e}_3 $$

where e0 points to the future, and e1, e2, e3 point in increasing spatial directions, these form a right-handed triad.

In tensor index notation (including the summation convention), it can be calculated using the Levi-civita symbol, equivalently as a determinant:


 * $$\begin{align}

\Omega & = \epsilon_{\alpha\beta\gamma\delta} A^\alpha B^\beta C^\gamma D^\delta \\ & = \begin{vmatrix} A^0 & A^1 & A^3 & A^3 \\ B^0 & B^1 & B^3 & B^3 \\ C^0 & C^1 & C^3 & C^3 \\ D^0 & D^1 & D^3 & D^3 \\ \end{vmatrix} \end{align}$$

The boundary of the hyperparallelepiped
Just as the boundary of a 3d parallelepiped is a net of parallelograms; the boundary of a 4-volume tesseract is a net of 3d paralleleipipeds.

Diagrammatic interpretation
[to be added soon].

4-volume element
The components of the vectors for the 4-volume element are:


 * $$A^\alpha = (dt,0,0,0),\ B^\alpha= (0,dx,0,0), \ C^\alpha = (0,0,dy,0), \ D^\alpha = (0,0,0,dz) $$

that is:


 * $$d^4 \Omega = dt \wedge dx \wedge dy \wedge dz $$

3-volume element
A surface in space time is a mixture of space and time components.


 * $$d^3 \Sigma_\mu = \epsilon_{\mu|\alpha\beta\gamma|} dx^\alpha \wedge dx^\beta \wedge dx^\gamma $$

Volume integrals in space-time
Surface and volume integrals in spacetime are over all the space and time components mixed, not simply integrals over space then time or vice versa.

Gauss' theorem in flat spacetime
The generalization of the divergence theorem (also called Gauss' theorem) in index-freen notation is:


 * $$\int_\mathcal{V} (\nabla \cdot \boldsymbol{\mathsf{T}}) d^4 \Omega = \oint_{\partial \mathcal{V}} \boldsymbol{\mathsf{T}} \cdot d \boldsymbol{\Sigma} $$

with indices


 * $$\int_\mathcal{V} \frac{\partial}{\partial x^\gamma}T_{\alpha\beta\gamma} d^4 \Omega = \oint_{\partial \mathcal{V}} T_{\alpha\beta\gamma} d^3 \Sigma $$

Applications in special relativity
4-momentum density

Angular momentum in 4d