2.1: The Tensor Virial Theorem

The tensor representation of the virial theorem is an attempt to restore some of the information lost in reducing the full vector Equations of motion described in Chapter I, section 1 to scalars. Although the germ of this idea can be found developing as early as 1903 in the work of Lord Rayleigh¹, it wasn't until the 1950's that Parker^{2, 3}, and Chandrasekhar and Fermi⁴ found the concept particularly helpful in dealing with the presence of magnetic fields. The concept was further expanded by Lebovitz⁵ and in a series of papers by Chandrasekhar and Lebovitz^{6, 7} during the 1960's, for the investigation of the stability of various gaseous configurations. Chandrasekhar³³ has given a fairly comprehensive recounting of his efforts on this subject after the original version of this monograph was prepared. However, the most lucid derivation is probably that presented by Chandrasekhar⁸ in 1961 and it is a simplified version of that derivation which I shall give here.

As previously mentioned the motivation for this approach is to regain some of the information lost in forming the scalar virial theorem by keeping track of certain aspects of the system associated with its spatial symmetries. If one recalls the full-blown vector Equations of motion in Chapter I, section 1, this amounts to keeping some of the component information of those Equations, but not all. In particular, it is not surprising that since system symmetries inspire this approach that the information to be kept relates to motions along orthogonal coordinate axes.

At this point, it is worth pointing out that the derivation in Chapter I, section 2, essentially originates from the Equations of motion of the system being considered. The derivations take the form of multiplying those Equations of motions by position vectors and averaging over the spatial volume. The final step involves a further average over time. That is to say that the virial theorem results from taking spatial moments of the Equations of motion and investigating their temporal behavior. (Recall that the Equations of motion themselves are moments of the Boltzmann transport Equation.) Since moment analysis of this type also yields some of the most fundamental conservation laws of physics (i.e., momentum, mass and energy), it is not surprising that the virial theorem should have the same power and generality as these laws. Indeed, it is rather satisfying to one who believes that "all that is good and beautiful in physics" can be obtained from the Boltzmann Equation that the virial theorem essentially arises from taking higher order moments of that Equation. With that in mind let us consider a collisionless pressure-free system analogous with that considered in Chapter I, section 2 and neglect viscous forces and macroscopic forces such as net rotation and magnetic fields as we shall consider them later. Under these conditions, Equation \ref{1.1.4} becomes

\[\rho \frac{\partial \mathbf{u}}{\partial \mathrm{t}}+\rho(\mathbf{u} \cdot \nabla) \mathbf{u}=-\rho \nabla \Phi=\rho \frac{\mathbf{du}}{\mathrm{dt}},\label{2.1.1}\]

which is simply the vector representation of either Equations Equation \ref{1.1.4} or Equation \ref{1.2.1}. In Chapter I, section 2, we essentially took the inner product of Equation \ref{1.1.4} with the position vector r and integrated over the volume to produce a scalar Equation. Here we propose to take the outer product of Equation \ref{1.1.1} with the position vector r producing a tensor Equation which can be regarded as a set of Equations relating the various components of the resulting tensors. Cursory dimensional arguments should persuade one that this procedure should produce relationships between the various moments of inertia of the system and energy-like tensors. Thus, our starting point is

\[\int_{\mathrm{v}} \rho \mathbf{r} \frac{\mathrm{d}\mathbf{u}}{\mathrm{dt}} \mathrm{dV}=\int_{\mathrm{v}} \rho \mathbf{r} \nabla \Phi \mathrm{dV}.\label{2.1.2}\]

Although some authors choose a slightly different convention, the term on the right hand side of Equation \ref{2.1.2} can properly be called the virial tensor. Now, as before let the potential be

\[\Phi(\mathbf{r})=\int_{\mathrm{v}^{\prime}} \rho\left(\mathbf{r}^{\prime}\right)\left(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)^{\mathrm{n}} \mathrm{dV}^{\prime} \quad \forall \mathrm{n}<0.\label{2.1.3}\]

Then following exactly the same manipulation as in Chapter I, only taking into account outer products instead of inner products with the position vectors, we get the virial tensor.^2.1

\[\int_{\mathrm{v}} \rho \mathbf{r} \nabla \Phi \mathrm{dV}=\mathrm{n}\left\{\frac{1}{2} \int_{\mathrm{v}}\int_{\mathrm{v’}} \rho(\mathbf{r}) \rho\left(\mathbf{r}^{\prime}\right)\left(\mathbf{r}-\mathbf{r}^{\prime}\right)\left(\mathbf{r}-\mathbf{r}^{\prime}\right)\left(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)^{\mathrm{n}-2} \mathrm{dV}^{\prime} \mathrm{dV}\right\}.\label{2.1.4}\]

If we define

\[\begin{aligned}
& \mathfrak{I}=\int_{\mathrm{v}}(\rho \mathbf{r} \mathbf{r}) \mathrm{dV} \\[4pt]
& \mathfrak{T}=\frac{1}{2} \int_{\mathrm{v}}(\rho \mathbf{u}\mathbf{u}) \mathrm{dV} \\[4pt]
& \mathfrak{U}=\frac{1}{2} \int_{\mathrm{v}} \int_{\mathrm{v}^{\prime}} \rho(\mathbf{r}) \rho\left(\mathbf{r}^{\prime}\right)\left(\mathbf{r}-\mathbf{r}^{\prime}\right)\left(\mathbf{r}-\mathbf{r}^{\prime}\right)\left(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)^{\mathrm{n}-2} \mathrm{dV}^{\prime} \mathrm{dV}
\end{aligned},\label{2.1.5}\]

Equation (2.1.2 becomes)

\[\frac{1}{2} \frac{\mathrm{d}^2 \mathfrak{I}}{\mathrm{dt}^2}=2 \mathfrak{T}+\mathrm{n} \mathfrak{U}.\label{2.1.6}\]

which is essentially the tensor representation of Lagrange's identity^2.2 where \(\mathfrak{I}\) is sometimes called the moment of inertia tensor, \(\mathfrak{T}\) the kinetic energy tensor and \(\mathfrak{U}\) the potential energy tensor. By eliminating additional external forces such as magnetic fields and rotation we have lost much of the power of the tensor approach. However, some insight into this power can be seen by considering in component form one term in the expansion of the virial tensor^2.1.

\[\int_\mathrm{v} \rho \frac{d}{d t}\left(\mathbf{r} \frac{d \mathbf{r}}{d t}\right) d V=\int_\mathrm{v} \rho \frac{d}{d t}\left(x_i \frac{d x_j}{d t}\right)dV.\label{2.1.7}\]

Since this tensor is clearly symmetric we find, by using the same conservation of mass arguments discussed earlier, that

\[\frac{d}{d t} \int_\mathrm{v} \rho\left(x_i \frac{d x_j}{d t}-x_j \frac{d x_i}{d t}\right) d V=0,\label{2.1.8}\]

which simply says the angular momentum about x_k is conserved. Thus the tensor virial theorem leads us to a fundamental conservation law which would not have been apparent from the scalar form derived earlier.