Chandrasekhar potential energy tensor

In astrophysics, Chandrasekhar potential energy tensor provides the gravitational potential of a body due to its own gravity created by the distribution of matter across the body, named after the Indian American astrophysicist Subrahmanyan Chandrasekhar.Chandrasekhar, S; Lebovitz NR (1962). "The Potentials and the Superpotentials of Homogeneous Ellipsoids" (PDF). Ap. J. 136: 1037–1047. . . Retrieved March 24, 2012.Chandrasekhar, S; Fermi E (1953). "Problems of Gravitational Stability in the Presence of a Magnetic Field" (PDF). Ap. J. 118: 116. . . Retrieved March 24, 2012.Chandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 9. New Haven: Yale University Press, 1969. The Chandrasekhar tensor is a generalization of potential energy in other words, the trace of the Chandrasekhar tensor provides the potential energy of the body.

Definition

The Chandrasekhar potential energy tensor is defined as

where

where

is the Gravitational constant is the self-gravitating potential from Newton's law of gravity is the generalized version of is the matter density distribution is the volume of the body

It is evident that is a symmetric tensor from its definition. The trace of the Chandrasekhar tensor is nothing but the potential energy .

Hence Chandrasekhar tensor can be viewed as the generalization of potential energy.

Chandrasekhar's Proof

Consider a matter of volume with density . Thus

Chandrasekhar tensor in terms of scalar potential

The scalar potential is defined as

then ChandrasekharChandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 9. New Haven: Yale University Press, 1969. proves that

Setting we get , taking Laplacian again, we get .

See also

Virial theorem Chandrasekhar virial equations

References