# Mathisson–Papapetrou–Dixon equations

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In physics, specifically general relativity, the **Mathisson–Papapetrou–Dixon equations** describe the motion of a spinning massive object, moving in a gravitational field. Other equations with similar names and mathematical forms are the **Mathisson-Papapetrou equations** and **Papapetrou-Dixon equations**. All three sets of equations describe the same physics.

They are named for M. Mathisson,^{[1]} W. G. Dixon,^{[2]} and A. Papapetrou.^{[3]}

Throughout, this article uses the natural units *c* = *G* = 1, and tensor index notation.

For a particle of mass *m*, the **Mathisson–Papapetrou–Dixon equations** are:^{[4]}^{[5]}

where: *u* is the four velocity (1st order tensor), *S* the spin tensor (2nd order), *R* the Riemann curvature tensor (4th order), and the capital "*D*" indicates the covariant derivative with respect to the particle's proper time *s* (an affine parameter).

## Mathisson–Papapetrou equations

For a particle of mass *m*, the **Mathisson–Papapetrou equations** are:^{[6]}^{[7]}

using the same symbols as above.

## Papapetrou–Dixon equations

## See also

- Introduction to the mathematics of general relativity
- Geodesic equation
- Pauli–Lubanski pseudovector
- Test particle
- Relativistic angular momentum
- Center of mass (relativistic)