2.4: General Relativity and the Virial Theorem

The development of quantum theory and the formulation of the general theory of relativity probably represent the two most significant advances in physical science in the twentieth century. In light of the general nature and wide applicability of the virial theorem it is surprising that little attempt was made during that time to formulate it within the context of general relativity. Perhaps this was a result of the lack of physical phenomena requiring general relativity for their description or possibly the direction of mathematical development undertaken for theory itself. For the last twenty years, there has been a concerted effort on the part of relativists to seek coordinate-free descriptions of general relativity in order to emphasize the connection between the fundamental geometrical properties of space and the description of associated physical phenomena. Although this has undoubtedly been profitable for the development of general relativity, it has drawn attention away from that technique in theoretical physics known as 'moment analysis'. This technique produces results which are in principle coordinate independent but usually utilize some specific coordinate frames for the purpose of calculation.

Another point of difficulty consists of the nature of the theory itself. General relativity, like so many successful theories, is a field theory and is thus concerned with functions defined at a point. Virtually every version of the virial theorem emphasizes its global nature.^† That is, some sort of symmetrical volume is integrated or summed over to produce the appropriate physical parameters. This difference becomes a serious problem when one attempts to assign a physical operational interpretation to the quantities represented by the spatial integrals. The problem of operational definition of macroscopic properties in general relativity has plagued the theory since its formulation. Although continuous progress has been made, there does not exist any completely general formulation of the virial theorem within the framework of general relativity at this time. This certainly is not to say that such a formulation cannot be made. Indeed, what we have seen so far should convince even the greatest skeptic that such a formulation does exist since its origin is basically that of a conservation law (see footnote at the end of the chapter). Even the general theory of relativity recognizes conservation laws although their form is often altered. Let us take a closer look at the origin of some of these problems. This can be done by taking into account in a self-consistent manner in the Einstein field Equations all terms of order 1/c². This is the approach adopted by Einstein, Infield, and Hoffman¹⁹ in their approach to the relativistic n-body problem and successfully applied by Chandrasekhar²⁰ to hydrodynamics. Although more efficient approximation techniques exist for the calculation of higher order relativistic phenomena such as gravitational radiation, this time honored approach is adequate for calculating the first order (i.e. c²) terms commonly known as the post-Newtonian terms.^†† During the first half of 1960s, in studying the hydrodynamics of various fluid bodies, Chandrasekhar developed the virial theorem to an extremely sophisticated level. The most comprehensive recognition of this work can be found in his excellent book on the subject⁹. One of the highlights not dealt with in the book are his efforts to include the first-order effects of general relativity. In an impressive and lengthy paper during 1965, Chandrasekhar developed the post-Newtonian Equations of hydrodynamics including a formulation of the virial theorem²⁰. It is largely this effort which we shall summarize here. One of the fundamental difficulties with the general theory of relativity is its non-linearity. The physical properties of matter are represented by the geometry of space and in that turn determines the geometry of space. It is this non-linearity that causes so much difficulty with approximation theory and with which the Einstein, Infield, Hoffman theory (EIH) deals directly.

†

It is worth noting that in order to 'test' any field theory against observation, it is necessary to compare integrals of the field quantities with the observed quantities. Even something as elementary as density is always "observed" by comparing some mass to some volume. It is impossible in principle to measure anything at a point. This obvious statement causes no trouble as long as we are dealing with concepts well within the range of our experience where we can expect our intuition to behave properly. However, beyond this comfortable realm, we are liable to attribute physical significance and testability to quantities which are in principle untestable. The result is to restrict the range of possibility for a theory unnecessarily.

††

For a beautifully concise and complete summary of the post-Newtonian approximation, see Misner, Thorne, and Wheeler, Gravitation (1973), w. H. Freeman & Co., San Francisco, Chapter 39.

The basic approach assumes that the metric tensor can be written as being perturbed slightly from the flat-space or Euclidean metric so that

\[\mathfrak{g}_{\alpha \beta}=\mathfrak{g}_{\alpha \beta}^{(\mathfrak{O})}+\mathfrak{h}_{\alpha \beta},\label{2.4.1}\]

where the \(\mathfrak{h}_{\alpha \beta}\) are small terms of the order of c^-2 or smaller. This enables one to determine the elements of the energy-momentum tensor up to terms of the order of c^-2 from its definition so that

\[\mathfrak{J}_{\alpha \beta}=(\varepsilon+\mathrm{P}) \mathrm{u}_\alpha \mathrm{u}_\beta-\mathrm{P}\mathfrak{g}_{\alpha \beta}.\label{2.4.2}\]

The Einstein field Equations can be written in terms of the Ricci tensor and the energy-momentum tensor as

\[\Re_{\alpha \beta}=-\frac{8 \pi \mathrm{G}}{\mathrm{c}^4}\left(\mathfrak{J}_{\alpha \beta}-\frac{1}{2} \mathrm{J}\mathfrak{g}_{\alpha \beta}\right),\label{2.4.3}\]

where J is just the trace \(\mathfrak{J}_{\alpha \beta}\). The Ricci tensor is essentially a geometric tensor and contains information relating to the metric alone.

The EIH approximation provides a prescription for solving the field Equations in various powers of 1/c² given the information concerning \(\mathfrak{J}_{\alpha \beta}\) and \(\mathfrak{g}_{\alpha \beta}\). In general the procedure determines the metric coefficients to one higher order than was originally specified. This procedure can be repeated but there remain some unsolved problems as to convergence of the scheme in general. At any point one may use the perturbed metric and the prescription for obtaining the Equations of motion to generate a set of perturbed Equations of motion. The relativistic prescription that free particles follow geodesic paths is logically equivalent to stating the four-space divergence of the energy-momentum tensor is zero. That is

\[\square \cdot \mathfrak{J}_{\alpha \beta}=0.\label{2.4.4}\]

Indeed it is this condition that in the flat-space metric yields of the Euler-Lagrange Equations of hydrodynamic flow. It was this condition that we needed in section 3 to obtain a form of the virial theorem appropriate for special relativity. Unfortunately the process of taking the divergence looses one order of approximation and thus it is not possible to go directly from the perturbed metric to the Equations of motion and maintain the same level of accuracy. One must first pass through the field Equations and the EIH approximation scheme. In order to follow this prescription one must first start with an approximation to the metric \(\mathfrak{g}_{\alpha\beta}\). Here it is traditional to invoke the principle of equivalence that requires that²¹

\[\mathfrak{h}_{\alpha \beta}=-\frac{2 \Psi}{c^2} \delta_{\alpha \beta}+\mathbf{O}\left(<1 / c^3\right).\label{2.4.5}\]

With this as a starting point one may proceed with the approximation scheme and obtain the Equations of motion. Like many approximation schemes the mathematical manipulations are formidable and physical insight easily lost. However, progress is rapid in a field such as this and what was an original research effort by Chandrasekhar in 1965 becomes a 'homework' problem for Misner, Thorne and Wheeler¹⁶ in 1972 (e.g. M.T.W. exercise 39.13.) Although it is contrary to the spirit of this book to quote results without derivation, I find in this case I must. We have laid neither a sufficient mathematical framework nor developed the general theory of relativity sufficiently to present the derivation in detail without consuming excessive space. Instead, consider the types of effects we might expect general relativity to introduce and see if these can be identified in the resulting Equations of motion.

First, energy is matter and therefore its motion must be followed in the Equations of motion as well as that of matter. This is really a consequence of special relativity but insofar as this 'added' mass affects the metric, we should find its effects present. The distortion of space also changes or at least complicates what is meant by a volume and thus it is useful to define a density ρ* which obeys a continuity Equation

\[\begin{aligned}
& \frac{\partial \rho^*}{\partial \mathrm{t}}+\nabla \cdot \left(\rho^* \mathbf{u}\right)=0, \\[4pt]
\text{where }& \rho^*=\rho_0\left[1+\left(\mathrm{u}^2+3 \Psi\right) / \mathrm{c}^2\right] .
\end{aligned}\label{2.4.6}\]

For purposes of simplification, Chandrasekhar finds it convenient to define a slightly altered form of the density which explicitly contains the internal energy of the gas and has a slightly altered continuity Equation

\[\begin{aligned}
&\frac{\partial \sigma}{\partial \mathrm{t}}+\nabla \cdot \left[\sigma \mathbf{u}+\frac{1}{\mathrm{c}^2}\left(\rho_0 \frac{\partial \Psi}{\partial \mathrm{t}}-\frac{\partial \rho_0}{\partial \mathrm{t}}\right)\right]=0,\\[4pt]
\text{where} \quad&\sigma=\rho_0\left[1+\left(\mathrm{u}^2+2 \Psi+\Pi+\mathrm{P} / \rho_0\right) / \mathrm{c}^2\right] .
\end{aligned}\label{2.4.7}\]

Here \(\Pi/\mathrm{c}^2\) is the internal energy of the gas and P is the local pressure. It is worth nothing that \(\rho_o\) is the density one would find in the absence of general relativity but where relativistic effects are important it is essentially a non-observable quantity since one can not devise a test for measuring it.

The general theory of relativity is a non-linear theory and thus we should expect terms to appear which reflect this non-linearity. They will be of different types. Firstly one should expect effects of the Newtonian potential Ψ affecting the metric directly which in turn modifies Ψ. These terms are indeed present but Misner, Thorne and Wheeler show that they can be represented by direct integrals over the mass distribution²². Secondly, since the matter and the metric are inexorably tied together, motion of matter will 'drag' the metric which will introduce velocity dependent terms in the 'potentials' used to represent those terms.

Both these effects can indeed be represented by 'potentials' but not just the Newtonian potential. Thus, various authors introduce various kinds of potentials to account for these non-linear terms. With this in mind the Equations of motion as derived by Chandrasekhar become²³

\[\frac{\partial(\sigma \mathbf{u})}{\partial \mathrm{t}}-\rho_0 \nabla \Psi+\nabla\left[\left(1+2 \Psi / \mathrm{c}^2\right) \mathrm{P}\right]+\frac{\rho_0}{\mathrm{c}^2}\left\{\frac{\mathrm{d}}{\mathrm{dt}}\left[4 \mathbf{u} \Psi-\frac{7}{2} \mathbf{Y}-\frac{1}{2} \nabla(|\mathbf{Y}|)\right]+4 \mathbf{u} \cdot (\nabla \mathbf{Y})-\frac{1}{2}(\mathbf{u} \cdot \nabla)\left[\mathbf{Y}-\frac{1}{2} \nabla(|\mathbf{Y}|)\right]\right\}+\frac{1}{2 \pi \mathrm{Gc}^2}\left[\nabla^2 \Phi \nabla \Psi+\nabla^2 \Psi \nabla \Phi\right]=0.\label{2.4.8}\]

Here, the various potentials which we have introduced can be defined by the fact that they satisfy a Poisson's Equation of the form

\[\begin{aligned}
\nabla^2 \Psi & =-4 \pi \mathrm{G} \rho_0 \\[4pt]
\nabla^2 \Phi & =-4 \pi \mathrm{G} \rho_0 \varphi \\[4pt]
\nabla^2 \mathbf{Y} &=-4 \pi \mathrm{G} \rho_0 \mathrm{u}\\[4pt]
\text { where } \varphi&=\mathrm{u}^2+\Psi+\frac{1}{2} \Pi+\frac{3}{2}\left(\mathrm{P} / \rho_0\right)
\end{aligned}\label{2.4.9}\]

Y is a vector potential whose source is the same as that of the Newtonian potential weighted by the local velocity field. Similarly, Φ is a scalar potential whose source is again that of the Newtonian potential but weighted by a function ϕ related to the total internal energy field.

Expansive as the Equations of motion are we may still derive some comprehension for the meaning of the various terms in Equation \ref{2.4.8}. The first two terms are basically Newtonian, indeed neglecting the contribution to the mass from energy \(\sigma=\rho_{\mathrm{o}}\) and they are identical to the first term of the Newtonian-Euler-Lagrange Equations of hydrodynamic flow. The first part of the third term is just the pressure gradient and thus also to be expected on Newtonian grounds alone. The remaining contribution to the pressure gradient results from the space curvature introduced by the presence of the matter and is perhaps the most likely relativistic correction to be expected. The remaining tensors are the non-linear interaction terms alluded to earlier. The lengthy expression in braces contains the effects of the 'dragging' of the metric by the matter and the induced velocity dependent terms. The last term represents the direct effect of the matter-energy potentials on the metrics and this effect, in turn, is propagated to the potentials themselves.

Having obtained the Equations of motion for the system, the procedure for obtaining the general relativistic form of Lagranges' identity is the same as we have used repeatedly in earlier sections. For simplicity, we shall compute the scalar version of Lagranges' identity by taking the inner product of Equation \ref{2.4.8} with the position vector r. We should expect this procedure to yield terms similar to the classical derivation but with differences introduced by differences between ρ_o , ρ*, and σ. In addition we shall take our volumes large enough so that volume integrals of divergence vanish. In this regard it is worth noting that if volume contains the entire system, then by the divergence theorem

\[\int_{\mathrm{v}} \nabla \mathrm{AdV}=\frac{1}{3} \int_{\mathrm{v}} \nabla \cdot (\mathbb{1} \mathrm{A}) \mathrm{dV}=0 .\label{2.4.10}\]

Thus, by integrating the Equations of motion over the volume yields

\[\frac{\mathrm{d}}{\mathrm{dt}} \int_{\mathrm{v}}\left\{\sigma \mathbf{u}+\frac{\rho_0}{\mathrm{c}^2}\left[4 \mathbf{u} \Psi-\frac{7}{2} \mathbf{Y}-\nabla(|\mathbf{Y}|)\right]\right\} \mathrm{dV}=0 \equiv \frac{\mathrm{d}}{\mathrm{dt}} \int_{\mathrm{v}} \mathbf{K} \mathrm{dV},\label{2.4.11}\]

after noting that remaining integrals in the braces { }of Equation \ref{2.4.8} can be integrated by parts to zero. Equation \ref{2.4.11} is a statement of conservation of linear momentum. This is a useful result for simplifying Equation \ref{2.4.8}. Now we are prepared to write down Lagrange's identity by letting the integral of Equation \ref{2.4.11} be the local linear momentum density K and taking the scalar product of r with Equation \ref{2.4.8}.

\[\mathbf{r} \cdot \frac{\mathrm{d} \mathbf{K}}{\mathrm{dt}} -\rho_0 \cdot \nabla\left[\Psi+\mathrm{P}\left(1+2 \Psi / \mathrm{c}^2\right)\right]+\frac{4 \rho}{\mathrm{c}^2} \mathbf{r} \cdot [\mathbf{u} \cdot (\nabla \mathbf{Y})]-\frac{1}{2} \mathbf{r} \cdot (\mathbf{u} \cdot \nabla)[\mathbf{Y}-\nabla(|\mathbf{Y}|)]
-\frac{2 \rho}{\mathrm{c}^2}(\varphi \mathbf{r} \cdot \nabla \Psi+\mathbf{r} \cdot \nabla \Phi)=0\label{2.4.12}\]

After multiple integration by parts and liberal use of the divergence theorem, this becomes

\[\frac{\mathrm{d}}{\mathrm{dt}} \int_{\mathrm{V}}(\mathbf{r} \cdot \mathbf{K}) \mathrm{dV}=2 \mathrm{T}+\Omega+3 \int_{\mathrm{V}}\left(1+2 \Psi / \mathrm{c}^2\right) \mathrm{PdV}+\frac{1}{\mathrm{c}^2}\left[4 \mathrm{W}+\langle\Phi\rangle-\frac{7}{4} \mathrm{Y}-\frac{1}{4} \mathrm{Z}\right],\label{2.4.13}\]

Where

\[\begin{aligned}
\mathrm{T} & =\frac{1}{2} \int_{\mathrm{v}} \sigma \mathrm{u}^2 \mathrm{dV} \\[4pt]
\Omega & =-\frac{1}{2} \int_{\mathrm{v}} \rho_0 \Psi \mathrm{dV} \\[4pt]
\mathrm{W} & =\int_{\mathrm{v}} \rho_0 \mathrm{u}^2 \Psi \mathrm{dV} \\[4pt]
<\Phi> & =\int_{\mathrm{v}} \rho_0 \varphi \Psi \mathrm{dV} \\[4pt]
\mathrm{Y} & =\int_{\mathrm{v}} \rho_0 \mathbf{u} \cdot \mathbf{Y}\mathrm{ d V}
\end{aligned},\label{2.4.14}\]

and

\[\mathrm{\left.Z=\int_{v}\int_{v^{\prime}}\left\{\rho_0 \rho_0^{\prime}\left(\left[\mathbf{u} \cdot \left(\mathbf{r}-\mathbf{r}^{\prime}\right)\right] [ \mathbf{u}^{\prime} \cdot \left(\mathbf{r}-\mathbf{r}^{\prime}\right)\right] /\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^3\right)\right\} d V^{\prime} d V}.\nonumber\]

This can be made somewhat more familiar if we re-write the left hand side of Equation \ref{2.4.13} so that

\[\mathrm{\frac{1}{2} \frac{d^2}{d t^2}\left(\int_v \sigma r^2 d V\right) +\frac{1}{c^2} \frac{d}{d t} \int_v \rho_0\left[4 \Psi(\mathbf{r} \cdot \mathbf{u})-\frac{7}{2} \mathbf{r} \cdot \mathbf{Y}-r \cdot \nabla(|\mathbf{Y}|)\right] d V=2 T+\Omega+3 \int_v\left(1+2 \Psi / c^2\right) P d V +\frac{1}{c^2}\left[4 \mathrm{W}+<\Phi>-\frac{7}{2} \mathrm{Y}-\frac{1}{4} \mathrm{Z}\right]}\label{2.4.15}\]

The first term on the left hand side of Equation \ref{2.4.15} is just \(\frac{1}{2} \mathrm{d}^2 \mathrm{I} / \mathrm{dt}^2\) in the Newtonian limit. The second term arises from the correction to the metric resulting from the potential and the 'dragging' of the metric due to internal motion. The first two terms on the right are just what one would expect in the Newtonian limit while the next term can be related to the total internal energy.

This term contains a relativistic correction resulting directly from the change in metric due to the presence of matter. The remaining terms are all energy like and the first two (W and <Φ>) represent relativistic corrections arising from the change in the potential caused by the metric modification by the potential itself. The last two involve metric dragging.

We have gone to some length to show the problems injected into the virial theorem by the non-linear aspects of general relativity. Writing Lagrange's identity as in Equation \ref{2.4.15} emphasizes the origin of the various terms - whether they are Newtonian or Relativistic. Although terms to this order should be sufficient to describe most phenomena in stellar astrophysics, we can ask if higher order terms or other metric theories of gravity provide any significant corrections to the virial theorem. The Einstein, Infield Hoffman approximation has been iterated up to 2 1/2 times^{24, 25} in a true tour-de-force by Chandrasekhar and co-workers looking for additional effects. At the 2 1/2 level of approximation, radiation reaction terms appear which could be significant for non-spherical collapsed objects which exist over long periods of time. Using a parameterized version of the post-Newtonian approximation, Ni²⁶ has developed a set of hydrodynamic Equations which must hold in nearly all metric theories of gravity and that depend on the values (near unity) of a set of dimensionless parameters. This latter effort is useful for relating various terms in the Equations to the fundamental assumptions made by different theories.

Perhaps the most obvious lesson to be learned from the EIH approach to this problem is that continued application of the theory is not the way to approach the general results. However, of some consequence is the result that conservation laws for energy, momentum, and angular momentum exist and are subject to an operational interpretation at all levels of approximation. Thus it seems reasonable to conjecture that these laws as well as the virial theorem remained well posed in the general theory.