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.^[1][2][3] 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

${\displaystyle W_{ij}=-{\frac {1}{2}}\int _{V}\rho \Phi _{ij}d\mathbf {x} =\int _{V}\rho x_{i}{\frac {\partial \Phi }{\partial x_{j}}}d\mathbf {x} }$

where

${\displaystyle \Phi _{ij}(\mathbf {x} )=G\int _{V}\rho (\mathbf {x'} ){\frac {(x_{i}-x_{i}')(x_{j}-x_{j}')}{|\mathbf {x} -\mathbf {x'} |^{3}}}d\mathbf {x'} ,\quad \Rightarrow \quad \Phi _{ii}=\Phi =G\int _{V}{\frac {\rho (\mathbf {x'} )}{|\mathbf {x} -\mathbf {x'} |}}d\mathbf {x'} }$

where

• ${\displaystyle G}$ is the Gravitational constant
• ${\displaystyle \Phi (\mathbf {x} )}$ is the self-gravitating potential from Newton's law of gravity
• ${\displaystyle \Phi _{ij}}$ is the generalized version of ${\displaystyle \Phi }$
• ${\displaystyle \rho (\mathbf {x} )}$ is the matter density distribution
• ${\displaystyle V}$ is the volume of the body

It is evident that ${\displaystyle W_{ij}}$ is a symmetric tensor from its definition. The trace of the Chandrasekhar tensor ${\displaystyle W_{ij}}$ is nothing but the potential energy ${\displaystyle W}$.

${\displaystyle W=W_{ii}=-{\frac {1}{2}}\int _{V}\rho \Phi d\mathbf {x} =\int _{V}\rho x_{i}{\frac {\partial \Phi }{\partial x_{i}}}d\mathbf {x} }$

Hence Chandrasekhar tensor can be viewed as the generalization of potential energy.^[4]

Chandrasekhar's Proof

Consider a matter of volume ${\displaystyle V}$ with density ${\displaystyle \rho (\mathbf {x} )}$. Thus

${\displaystyle {\begin{aligned}W_{ij}&=-{\frac {1}{2}}\int _{V}\rho \Phi _{ij}d\mathbf {x} \\&=-{\frac {1}{2}}G\int _{V}\int _{V}\rho (\mathbf {x} )\rho (\mathbf {x'} ){\frac {(x_{i}-x_{i}')(x_{j}-x_{j}')}{|\mathbf {x} -\mathbf {x'} |^{3}}}d\mathbf {x'} d\mathbf {x} \\&=-G\int _{V}\int _{V}\rho (\mathbf {x} )\rho (\mathbf {x'} ){\frac {x_{i}(x_{j}-x_{j}')}{|\mathbf {x} -\mathbf {x'} |^{3}}}d\mathbf {x} d\mathbf {x'} \\&=G\int _{V}d\mathbf {x} \rho (\mathbf {x} )x_{i}{\frac {\partial }{\partial x_{j}}}\int _{V}d\mathbf {x'} {\frac {\rho (\mathbf {x'} )}{|\mathbf {x} -\mathbf {x'} |}}\\&=\int _{V}\rho x_{i}{\frac {\partial \Phi }{\partial x_{j}}}d\mathbf {x} \end{aligned}}}$

Chandrasekhar tensor in terms of scalar potential

The scalar potential is defined as

${\displaystyle \chi (\mathbf {x} )=-G\int _{V}\rho (\mathbf {x'} )|\mathbf {x} -\mathbf {x'} |d\mathbf {x'} }$

then Chandrasekhar^[5] proves that

${\displaystyle W_{ij}=\delta _{ij}W+{\frac {\partial ^{2}\chi }{\partial x_{i}\partial x_{j}}}}$

Setting ${\displaystyle i=j}$ we get ${\displaystyle \nabla ^{2}\chi =-2W}$, taking Laplacian again, we get ${\displaystyle \nabla ^{4}\chi =8\pi G\rho }$.

See also

• Virial theorem
• Chandrasekhar virial equations