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

This article discusses 4d volume applied to special and general relativity - see four-dimensional space and Minkowski space for other fuller descriptions.

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:

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

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].

Volume element

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

See also

• tesseract
• light cone

References

• Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
• T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, p. 67, ISBN 978-1107-602601