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. 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 Chandrasekhar proves that
Setting we get , taking Laplacian again, we get .