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. 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

$W_{ij}=-\frac{1}{2}\underset{V}{\int }\rho {\mathrm{\Phi }}_{ij}d𝐱=\underset{V}{\int }\rho x_i\frac{\partial \mathrm{\Phi }}{\partial x_j}d𝐱$

where

${\mathrm{\Phi }}_{ij}(𝐱)=G\underset{V}{\int }\rho ({𝐱}^{\prime })\frac{(x_i-{x_i}^{\prime })(x_j-{x_j}^{\prime })}{|𝐱-{𝐱}^{\prime }{|}^3}d{𝐱}^{\prime },\Rightarrow {\mathrm{\Phi }}_{ii}=\mathrm{\Phi }=G\underset{V}{\int }\frac{\rho ({𝐱}^{\prime })}{|𝐱-{𝐱}^{\prime }|}d{𝐱}^{\prime }$

where

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

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

$W=W_{ii}=-\frac{1}{2}\underset{V}{\int }\rho \mathrm{\Phi }d𝐱=\underset{V}{\int }\rho x_i\frac{\partial \mathrm{\Phi }}{\partial x_i}d𝐱$

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

Chandrasekhar's Proof

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

$\begin{array}{cc}{W}_{ij}& =-\frac{1}{2}\underset{V}{\int }\rho {\mathrm{\Phi }}_{ij}d𝐱\\ & =-\frac{1}{2}G\underset{V}{\int }\underset{V}{\int }\rho (𝐱)\rho ({𝐱}^{\prime })\frac{(x_i-{x_i}^{\prime })(x_j-{x_j}^{\prime })}{|𝐱-{𝐱}^{\prime }{|}^3}d{𝐱}^{\prime }d𝐱\\ & =-G\underset{V}{\int }\underset{V}{\int }\rho (𝐱)\rho ({𝐱}^{\prime })\frac{x_i(x_j-{x_j}^{\prime })}{|𝐱-{𝐱}^{\prime }{|}^3}d𝐱d{𝐱}^{\prime }\\ & =G\underset{V}{\int }d𝐱\rho (𝐱)x_i\frac{\partial }{\partial x_j}\underset{V}{\int }d{𝐱}^{\prime }\frac{\rho ({𝐱}^{\prime })}{|𝐱-{𝐱}^{\prime }|}\\ & =\underset{V}{\int }\rho x_i\frac{\partial \mathrm{\Phi }}{\partial x_j}d𝐱\end{array}$

Chandrasekhar tensor in terms of scalar potential

The scalar potential is defined as

$\chi (𝐱)=-G\underset{V}{\int }\rho ({𝐱}^{\prime })|𝐱-{𝐱}^{\prime }|d{𝐱}^{\prime }$

then Chandrasekhar proves that

$W_{ij}={\delta }_{ij}W+\frac{{\partial }^2\chi }{\partial x_i\partial x_j}$

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

See also

• Virial theorem
• Chandrasekhar virial equations