2.6: Summary

In this chapter we have continued the development of the virial theorem as it appears in more contemporary usage. The tensor virial theorem is a more general form of Lagrange's identity which when averaged over time provides rather general expressions for the coordinated behavior of some energy like tensors of the system. Further insight into the nature of this process is discovered in the second section where we find that taking higher order spatial moments of the Equation of motion is equivalent to recovering the information selectively lost in the classical derivation of the virial theorem. In principle this approach could be used in a prescription for the complete solution of the Equations of motion. However, it seems likely that in practice it would be more difficult than implementing a direct numerical solution of the original Equations themselves. The importance of the method lies in the fact that such moment expressions for stable systems are normally rapidly convergent. Thus, the largest amount of information can be recovered with the least effort.

In the next two sections, we considered the effects of a relativity principle on the development of the virial theorem. We found that large velocities require large gravitational fields to keep them in check and thus one might argue that a separate discussion of the effects of special relativity is not warranted. However, there are at least two dynamically stable systems for which this is not true (i.e., pure radiation spheres which approximate some models of super massive stars and white dwarfs where low-mass, high velocity electrons, were kept in check by the high-mass, low velocity nucleons). In addition, Lagrange's identity is applicable to systems which are not in equilibrium and hence may be relativistic. For this reason, we have developed the special and general relativistic versions of the theorem separately and will return in the last chapter to discuss some specific applications of them. I have attempted throughout the chapter to emphasize the similarity of the derivation of the virial theorem and particularly in section 3, the perceptive reader may have noticed that the derivation is equivalent to carrying out

\[\int_{\mathrm{v}} \mathrm{r} \cdot \square \cdot \mathfrak{J} \mathrm{dV}=0,\label{2.6.1}\]

where r is a four vector in the Lorentz metric. However, we ignored all contributions from the time-like part. These would have been of the form

\[\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d t}} \int_\mathrm{v} t \mathfrak{J}_{00} \mathrm{d V}=\int_\mathrm{v} \left(\mathrm{t} \sum_\mathrm{j} \frac{\partial \mathfrak{J}_{0 j}}{\partial \mathrm{x}_\mathrm{j}}\right) \mathrm{d V},\label{2.6.2}\]

which after appropriate application of the divergence theorem, becomes

\[\frac{\mathrm{d}}{\mathrm{dt}}\left(\int_{\mathrm{v}} \varepsilon \mathrm{dV}\right)=\frac{\mathrm{dE}}{\mathrm{dt}}=-\int_{\mathrm{s}} \frac{\rho \mathrm{c}^2}{\gamma} \frac{\mathrm{d} \mathbf{x}}{\mathrm{dt}} \cdot \mathrm{d} \mathbf{S}.\label{2.6.3}\]

Because of the linear independence of the time and space coordinates, this is not a new result but rather an expression of the conservation of energy. Essentially it states that the time rate of change of the total energy of the system equals the momentum flux across the system boundary multiplied by c².

When the metric is no longer the 'flat' Lorentz metric as in the case in section 4, things are no longer simple. It is this loss of simplicity which caused me to stray from the more rigorous approaches of other sections. Thus rather than stress the manipulative complexity of the post-Newtonian approximation, I have attempted to provide physical motivation from the existence of terms arising in the Equations of motion that results from the non-linear nature of general relativity. The derivation presented in this section follows exactly the prescription of earlier sections, but for simplicity I presented the development of the scalar version of the virial theorem only. Sticking with the post-Newtonian approximation avoided some difficult problems of uniqueness and interpretation.

However, I remain convinced that a general formulation of Lagrange's identity and the virial theorem which is compatible with the field Equation of Einstein exists and its formulation would be most rewarding^†.

†

Since the Pachart edition of this effort was written, there has been some significant progress in this area. Bonazzola³⁰, formulated the virial theorem for the spherical, stationary case in full general relativity. He notes that in the non-stationary, non-spherical case, the existence of gravitational radiation destroys any strict equivalence between the general relativistic and Newtonian cases so that there can be no unique formulation of the virial theorem. Thus any formulation of a virial-like relation will depend on the specific nature of the configuration. This approach was extended by Vilain31 who applied a similar formulation of a general relativistic virial theorem to the stability of perfect fluid spheres.

For most stellar astrophysical applications the post-Newtonian result is probably sufficient. In the last section of this chapter some of the powers of the virial theorem to deal with difficult situations became apparent. The results which have been generalized to include additional effects are not the result of any new physical concepts. Rather, they are the result of the specific identification of the physical contributions to the system made by such attributes as magnetic fields and macroscopic motion. Although I included only magnetic fields throughout most of the discussion, the inclusion of electric fields in the Maxwell field tensor makes it clear how to proceed should they be present. Lastly we looked again at velocity dependent forces not so much with an eye to their effect on the virial theorem, but rather with a view to their persistent presence in the variational form of Lagrange's identity. The presence of all these complicating aspects is included only to make their interplay explicit. The basic theorem must hold; all the rest is done to glean more insight.