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

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

Calculation

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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

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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

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[to be added soon].

Volume element

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4-volume element

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

that is:

3-volume element

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A surface in space time is a mixture of space and time components.

Volume integrals in space-time

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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

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

with indices

Illustrative proof

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Applications in special relativity

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4-momentum density

Angular momentum in 4d

See also

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References

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  • 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