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


 * $$W_{ij} = -\frac{1}{2} \int_V \rho \Phi_{ij} d\mathbf{x} =\int_V \rho x_i \frac{\partial \Phi}{\partial x_j} d\mathbf{x}$$

where
 * $$\Phi_{ij}(\mathbf{x}) = G \int_V \rho(\mathbf{x'}) \frac{(x_i-x_i')(x_j-x_j')}{|\mathbf{x}-\mathbf{x'}|^3} d\mathbf{x'}, \quad \Rightarrow \quad \Phi_{ii} = \Phi = G \int_V \frac{\rho(\mathbf{x'})}{|\mathbf{x}-\mathbf{x'}|} d\mathbf{x'} $$

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


 * $$W= W_{ii} = -\frac{1}{2} \int_V \rho \Phi d\mathbf{x} = \int_V \rho x_i \frac{\partial \Phi}{\partial x_i} d\mathbf{x}$$

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

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



\begin{align} 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{align} $$

Chandrasekhar tensor in terms of scalar potential
The scalar potential is defined as


 * $$\chi(\mathbf{x}) = -G \int_V \rho(\mathbf{x'}) |\mathbf{x}-\mathbf{x'}|d\mathbf{x'}$$

then Chandrasekhar proves that


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

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