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
[edit]The volume of a hyperparallelepiped with vector edges A, in the time direction and B, C, D in the spatial directions, is given by:
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:
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:
The boundary of the hyperparallelepiped
[edit]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
[edit][to be added soon].
Volume element
[edit]4-volume element
[edit]The components of the vectors for the 4-volume element are:
that is:
3-volume element
[edit]A surface in space time is a mixture of space and time components.
Volume integrals in space-time
[edit]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
[edit]The generalization of the divergence theorem (also called Gauss' theorem) in index-freen notation is:
with indices
Illustrative proof
[edit]Applications in special relativity
[edit]4-momentum density
Angular momentum in 4d
See also
[edit]References
[edit]- 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