In special and general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the electric current density. Also known as vector current, it is used in the geometric context of four-dimensional spacetime, rather than three-dimensional space and time separately. Mathematically it is a four-vector, and is Lorentz covariant.
Analogously, it is possible to have any form of "current density", meaning the flow of a quantity per unit time per unit area. see current density for more on this quantity.
This article uses the summation convention for indices. See covariance and contravariance of vectors for background on raised and lowered indices, and raising and lowering indices on how to switch between them.
Motion of charges in spacetime
- is the charge density measured by an inertial observer O who sees the electric current moving at speed u (the magnitude of the 3-velocity);
- is “the rest charge density”, i.e., the charge density for a comoving observer (an observer moving at the speed u - with respect to the inertial observer O - along with the charges).
Qualitatively, the change in charge density (charge per unit volume) is due to the contracted volume of charge due to Lorentz contraction.
Charges (free or as a distribution) at rest will appear to remain at the same spatial position for some interval of time (as long as they're stationary). When they do move, this corresponds to changes in position, therefore the charges have velocity, and the motion of charge constitutes an electric current. This means that charge density is related to time, while current density is related to space.
The four-current unifies charge density (related to electricity) and current density (related to magnetism) in one electromagnetic entity.
In general relativity, the continuity equation is written as:
where the semi-colon represents a covariant derivative.
In general relativity, the four-current is defined as the divergence of the electromagnetic displacement, defined as
Quantum field theory
The four-current density of charge is an essential component of the Lagrangian density used in quantum electrodynamics. In 1956 Gershtein and Zeldovich considered the conserved vector current (CVC) hypothesis for electroweak interactions.
- Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd ed.). Oxford Science Publications. pp. 103–107. ISBN 978-0-19-853952-0.
- Roald K. Wangsness, Electromagnetic Fields, 2nd edition (1986), p. 518, 519
- Melvin Schwartz, Principles of Electrodynamics, Dover edition (1987), p. 122, 123
- J. D. Jackson, Classical Electrodynamics, 3rd Edition (1999), p. 554
- as [ref. 1, p519]
- Cottingham, W. Noel; Greenwood, Derek A. (2003). An introduction to the standard model of particle physics. Cambridge University Press. p. 67. ISBN 9780521588324.
- Marshak, Robert E. (1993). Conceptual foundations of modern particle physics. World Scientific Publishing Company. p. 20. ISBN 9789813103368.
- Gershtein, S. S.; Zeldovich, Y. B. (1956), Soviet Phys. JETP, 2 576.
- Thomas, Anthony W. (1996). "CVC in particle physics". arXiv:nucl-th/9609052.