Astronomy: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

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

where

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

where

• $G$ is the Gravitational constant
• $\mathrm{\Phi }(\mathbf{𝐱})$ is the self-gravitating potential from Newton's law of gravity
• ${\mathrm{\Phi }}_{ij}$ is the generalized version of $\mathrm{\Phi }$
• $\rho (\mathbf{𝐱})$ 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 $W$.

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

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

Chandrasekhar's Proof

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

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

Chandrasekhar tensor in terms of scalar potential

The scalar potential is defined as

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

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