“Mathisson–Papapetrou–Dixon equations”的意思、由来-开放百科全书

In physics, specifically general relativity, the Mathisson–Papapetrou–Dixon equations describe the motion of a massive spinning body 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.

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

Matthisson-Papapetrou–Dixon equations

the vector

is the four-velocity of some reference point

in the body, and the skew-symmetric tensor

is the angular momentum

of the body about this point. In the time-slice integrals we are asuming that the body is compact enough that we can use flat coordinates within the body where the energy-momentum tensor

is non-zero.

As they stand, there are only ten equations to determine thirteen quantities. These quantities are the six components of

, the four components of

and the three independent components of

. The equations must therefore be supplimented by three additional constraints which serve to determine which point in the body has velocity

. Matthison and Pirani originally chose to impose the condition

which, although involving four components, contains only three constraints because

is identically zero. This condition, however, leads to mysterious "helical motions"

^[4]. The Tulczyjew-Dixon condition

is perhaps more physically reasonable as it selects the reference point

to be the body's center of mass in the frame in which its momentum is

Accepting the Tulczyjew-Dixon condition

, we can manipulate the second of the MPD equations into the form

This is a form of Fermi-Walker transport of the spin tensor along the trajectory - but one preserving orthogonality to the momentum vector

rather than to the tangent vector

. Dixon calls this M-transport.

See also

References

Notes

1. ^{{cite news |author=M. Mathisson|title=Neue Mechanik materieller Systeme |journal=Acta Physica Polonica |volume=6 |year=1937 |pages=163–209 |url=http://inspirehep.net/record/48323/citations}}
2. ^{{cite journal |title=Dynamics of Extended Bodies in General Relativity. I. Momentum and Angular Momentum |author=W. G. Dixon |url=http://rspa.royalsocietypublishing.org/content/314/1519/499.full.pdf+html |year=1970 |doi=10.1098/rspa.1970.0020 |journal=Proc. R. Soc. Lond. A |volume=314 |issue=1519 |bibcode=1970RSPSA.314..499D |pages=499–527}}
3. ^{{cite journal |author=A. Papapetrou |title=Spinning Test-Particles in General Relativity. I |url=http://rspa.royalsocietypublishing.org/content/209/1097/248.full.pdf+html |year=1951 |doi=10.1098/rspa.1951.0200 |journal=Proc. R. Soc. Lond. A |volume=209 |issue=1097 |bibcode=1951RSPSA.209..248P |pages=248–258}}
4. ^{{cite journal|title=Mathisson's helical motions demystified|journal=AIP Conf. Proc.|volume=1458 |pages=367–370 |author1=L. F. O. Costa |author2=J. Natário |author3=M. Zilhão |year=2012|arxiv=1206.7093|doi=10.1063/1.4734436|series=AIP Conference Proceedings }}

Matthisson-Papapetrou–Dixon equations

See also

References

Notes

Selected papers