Mathisson–Papapetrou–Dixon equations

Mathisson–Papapetrou–Dixon equations

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]

\frac{D}{ds}\left(m u^\lambda + u_\mu \frac{DS^{\lambda\mu}}{ds} \right) = -\frac{1}{2}u^\pi S^{\rho\sigma} R^\lambda{}_{\pi\rho\sigma}

\frac{DS^{\mu\nu}}{ds} + u^\mu u_\sigma \frac{DS^{\nu\sigma}}{ds} - u^\nu u_\sigma \frac{DS^{\mu\sigma}}{ds} = 0

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).

Contents

  • Mathisson–Papapetrou equations 1
  • Papapetrou–Dixon equations 2
  • See also 3
  • References 4
    • Notes 4.1
    • Selected papers 4.2

Mathisson–Papapetrou equations

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

\frac{D}{ds}m u^\lambda = -\frac{1}{2}u^\pi S^{\rho\sigma} R^\lambda{}_{\pi\rho\sigma}

\frac{DS^{\mu\nu}}{ds} + u^\mu u_\sigma \frac{DS^{\nu\sigma}}{ds} - u^\nu u_\sigma \frac{DS^{\mu\sigma}}{ds} = 0

using the same symbols as above.

Papapetrou–Dixon equations

See also

References

Notes

  1. ^
  2. ^
  3. ^
  4. ^
  5. ^
  6. ^
  7. ^

Selected papers