2.3: Special Relativity and the Virial Theorem

So far we have considered only the virial theorem that one obtains from the Newtonian Equations of motion. Since there are systems such as white dwarfs, wherein the dynamic pressure balancing gravity is supplied by particles whose energies are very much larger than their rest energy, it is appropriate that we investigate the extent to which we shall have to modify the virial theorem to include the effects of special relativity. For systems in equilibrium, the virial theorem says \(2T = Ω\). One might say that it requires a potential energy equal to 2T to confine the motions of particles having a total kinetic energy T. As particles approach the velocity of light the kinetic energy increases without bound. One may interpret this as resulting from an unbounded increase of the particle's mass. This increase will also affect the gravitational potential energy, but the effect is quadratic. Thus we might expect in a relativistic system that a potential energy less than 2T would be required to maintain equilibrium. This appears to be the conclusion arrived at by Chandrasekhar when, while investigating the internal energy of white dwarfs he concludes that as the system becomes more relativistically degenerate, \(T\) approaches \(Ω\) and this "must be the statement of the virial theorem for material particles moving with very nearly the velocity of light."¹⁴

This is indeed the case and is the asymptotic limit represented by a photon gas or polytrope of index \(n = 3\) (see Collins³²). In order to obtain the somewhat more general result of a relativistic form of Lagrange's identity, we shall turn to the discussion of relativistic mechanics of Landau and Lifshitz¹⁵ mentioned in Chapter I. As most discussions in field mechanics generally start from a somewhat different prospective, let us examine the correspondence with the starting point of the Equations of motion adopted in the earlier sections. Generally most expositions of field mechanics start with the statement that

\[\square \cdot \mathfrak{J}=0,\label{2.3.1}\]

where \(\mathfrak{J}\) is the Maxwell stress-energy tensor and \(\square\) is known as the four-gradient operator. This is equivalent to saying that there exists a volume in space-time sufficiently large so that outside that volume the stress-energy tensor is zero. This equivalence is made obvious by applying Gauss' divergence theorem so that

\[\int_{\mathrm{R}} \square \cdot \mathfrak{J} \mathrm{dR}=\int_{\mathrm{S}} \mathfrak{J} \cdot \mathrm{d} \mathbf{s}=0.\label{2.3.2}\]

In short, Equation \ref{2.3.1} is a conservation law. We have already seen that the fundamental conservation laws of physics are derivable from the Boltzmann transport Equation as are the Equations of motion. Indeed, the operation of taking moments is quite similar in both cases. Thus, both starting points are equivalent as they have their origin in a common concept.

Although the conceptual development for this derivation is inspired by Landau and Lifshitz the subscript notation will be largely that employed by Misner, Thorne, and Wheeler¹⁶. Tempting as it is to use the coordinate free geometry of these authors, the concept of taking moments at this point is most easily understood within the context of a coordinate representation so for the moment we will keep that approach. In a Lorentz coordinate system, Landau and Lifshitz give components of the 4-velocity of a particle as

\[\mathrm{u}_\alpha=\frac{\mathrm{dx}_\alpha}{\mathrm{ds}} . \quad \alpha=0 \cdots 3.\label{2.3.3}\]

where \(\mathrm{ds} / \mathrm{dt}=\mathrm{c}\left(1-\mathrm{v}^2 / \mathrm{c}^2\right)^{1 / 2}=\gamma \mathrm{c}\). Note this is a somewhat unconventional definition of \(\gamma\).

The components of the energy-momentum tensor are

\[\begin{aligned}
\mathfrak{J}_{\alpha \beta}&=\rho \mathrm{cu}_\alpha \mathrm{u}_\beta(\mathrm{ds} / \mathrm{dt}), \\[4pt]
\text{and specifically}\quad \mathfrak{J}_{0 \mathrm{j}}&=\mathrm{i} \rho \mathrm{c}^2 \mathrm{u}_{\mathrm{j}},
\end{aligned}\label{2.3.4}\]

which are clearly symmetric in α and β. Since \(\sum_{\alpha=0}^3 \mathrm{u}_\alpha^2=-1\), the trace of Equation \ref{2.3.4} is

\[\sum_{\alpha=0}^3 \mathfrak{J}_{\alpha \alpha}=-\rho c^2 \gamma.\label{2.3.5}\]

In terms of the three-dimensional components the conservation law expressed by Equation \ref{2.3.1} can be written as

\[\frac{1}{\mathrm{ic}} \frac{\partial \mathfrak{J}_{\mathrm{j} 0}}{\partial \mathrm{t}}+\sum_{\mathrm{k}=1}^3 \frac{\partial \mathfrak{J}_{\mathrm{jk}}}{\partial \mathrm{x}_{\mathrm{k}}}=\mathrm{c} \frac{\partial\left(\rho \mathrm{u}_{\mathrm{j}}\right)}{\partial \mathrm{t}}+\sum_{\mathrm{k}=1}^3 \frac{\partial \mathfrak{J}_{\mathrm{jk}}}{\partial \mathrm{x}_{\mathrm{k}}}=0 .\label{2.3.6}\]

Substituting for the components of \(\mathrm{j}\) and repeating the algebr of earlier derivations we have^2.3

\[\sum_{\mathrm{j}=1}^3 \frac{\mathrm{d}}{\mathrm{dt}} \int_{\mathrm{v}} \frac{\rho}{2} \frac{\mathrm{d}}{\mathrm{dt}}\left(\mathrm{x}_{\mathrm{j}} \mathrm{x}_{\mathrm{j}} / \gamma\right) \mathrm{dV}=\Omega+\mathrm{T}+\int_{\mathrm{v}} \gamma \tau \mathrm{dV},\label{2.3.7}\]

where \(\tau\) is the kinetic energy density of relativistic particles¹⁷. Again, using the conservation of mass arguments in Chapter I, this becomes

\[\sum_{\mathrm{j}=1}^3 \frac{1}{2} \frac{\mathrm{d}^2}{\mathrm{dt}^2} \int_{\mathrm{V}}\left(\rho \mathrm{x}_{\mathrm{j}} \mathrm{x}_{\mathrm{j}} / \gamma\right) \mathrm{dV}=\Omega+\mathrm{T}+\int_{\mathrm{V}} \gamma \tau \mathrm{dV}.\label{2.3.8}\]

where, if we define the volume integral on the left to be the relativistic moment of inertia, we can write

\[\frac{1}{2} \frac{\mathrm{d}^2}{\mathrm{dt}^2}\left(\mathrm{I}_{\mathrm{r}}\right)=\Omega+\mathrm{T}+\int_{\mathrm{v}} \gamma \tau \mathrm{dV}.\label{2.3.9}\]

In the low velocity limit \(\gamma \rightarrow 1\) so that \(\mathrm{I}_\mathrm{r} \rightarrow \mathrm{I}\) and we recover the ordinary Lagrange's identity. In the relativistic limit as \(\gamma \rightarrow 0\) we recover for stable systems the Chandrasekhar result that \(T + Ω = 0\) (i.e., the total energy E = 0). Thus, it is fair to say that Equation \ref{2.3.9} is an expression of the Lagrange's identity including the effects of special relativity.

It is worth noting by analogy with section 1 that the tensor relativistic theorem can be derived by taking the outer or 'tensor' product of the space-like position vector with Equation \ref{2.3.6} and integrating over the volume. Following the same steps that lead to Equation (N.2.3.3), we get

\[\mathrm{c} \int_{\mathrm{v}} \mathrm{x}_{\mathrm{i}} \frac{\partial\left(\rho \mathrm{u}_{\mathrm{j}}\right)}{\partial \mathrm{t}} \mathrm{dV}-\int_{\mathrm{v}} \mathfrak{J}_{\mathrm{ij}} \mathrm{dV}=0.\label{2.3.10}\]

Since \(\mathfrak{J}_{\mathrm{ij}}\) is symmetric, we can add Equation \ref{2.3.10} to its counterpart with the indices interchanged, and get

\[\mathrm{c \int_\mathrm{v}\left(x_i \frac{\partial\left(\rho u_j\right)}{\partial t}+x_j \frac{\partial\left(\rho u_i\right.}{\partial t}\right) d V-2 \int_\mathrm{v} \mathfrak{J}_{i j} d V=0},\label{2.3.11}\]

or finally

\[\mathrm{\frac{1}{2} \frac{d^2}{d t^2}\left(\int_\mathrm{v}\left[\rho\left(x_j x_i\right) / \gamma\right) d V=\int_\mathrm{v} \mathfrak{J}_{i j} d V\right.}.\label{2.3.12}\]

The integral on the left can be viewed on the relativistic moment of inertia tensor while the right hand side is just the volume integral of the components of the energy momentum tensor. Following the prescription used to generate Equation \ref{2.3.11}, but subtracting the transpose of Equation \ref{2.3.11} from itself, yields

\[\mathrm{\int_{\mathrm{v}}\left(\mathrm{x}_{\mathrm{i}} \frac{\partial\left(\rho \mathrm{u}_{\mathrm{j}}\right)}{\partial \mathrm{t}}-\mathrm{x}_{\mathrm{j}} \frac{\partial\left(\rho \mathrm{u}_{\mathrm{i}}\right.}{\partial \mathrm{t}}\right) \mathrm{dV}=0},\label{2.3.13}\]

which becomes by application of Leibniz’s law

\[\frac{\mathrm{d}}{\mathrm{dt}}\left[\int_{\mathrm{v}} \frac{\rho}{\gamma}\left(\mathrm{x}_{\mathrm{i}} \frac{\partial\left(\rho \mathrm{u}_{\mathrm{j}}\right)}{\partial \mathrm{t}}+\mathrm{x}_{\mathrm{j}} \frac{\partial\left(\rho \mathrm{u}_{\mathrm{i}}\right.}{\partial \mathrm{t}}\right) \mathrm{dV}\right]=0,\label{2.3.14}\]

and is the relativistic form of the expression for the conservation of angular momentum obtained in section 1, Equation \ref{2.1.8}.