부산대학교 도서관

This paper raises a natural question concerning the motion of {\it extended} bodies in general relativity: How can we make sense of the concept that material bodies in relativity follow geodesic curves? More specifically, consider an exact solution of Einstein's equation. Let us assert that we can `single out' a body within this solution. This body is required, particularly, to satisfy two properties: It must be small (i.e., we should be able to localize it) and of small mass (i.e., it should not distort the space-time too much). A standard example would be to single out the earth in a sun-earth 2-body system. \par Two existing approaches to address this problem are mentioned, which this paper aims to combine: One represents the body by a stress-energy distribution, the other by a family of smooth tensor fields that are conserved and satisfy an energy condition. The overall strategy may be summarized as follows: consider a background metric $g_{ab}$ and a curve $\gamma$ on the space-time. In addition, let us assume to have a family of smooth (stress-energy) tensor fields $T^{ab}$ that are conserved and satisfy an energy condition. Moreover, we assume the fields $T^{ab}$ to be supported on arbitrarily small neighborhoods of $\gamma$. In case of matter interaction (e.g., charged bodies), some extra assumptions need to be made to account for this interaction. The aim is to show, under suitable conditions, that with such a configuration the curve $\gamma$ must be a geodesic. \par In Section 2 of the paper, the authors discuss the limit of a point-like particle. This limit is instructive since the path of the body, in this case, actually is described by a single curve. Methodologically, this description is done in terms of distributions (reviewed in Appendix A of the paper, along with a review of tensor densities in Appendix B). Let us fix the following terminology: Curly brackets, $\{\}$, denote the action of a distribution $\ssf{T}^{ab}$ on a test field $\ssf{x}_{ab}$ (i.e., a smooth tensor density of weight 1 and with compact support). Let us furthermore introduce the following energy conditions: A symmetric tensor field $T^{ab}$ satisfies the {\it dominant energy condition} if $T(u,v)\geq0$ for any timelike or null vectors $u, v$. A symmetric tensor field $t_{ab}$ satisfies the {\it dual energy condition} if $t_{ab}T^{ab}\geq0$ for every $T^{ab}$ that satisfies the dominant energy condition. Finally, a symmetric distribution $\ssf{T}^{ab}$ is said to satisfy the {\it energy condition} if $\ssf{T}\{\ssf{x}\}\geq0$ for every test field $\ssf{x}_{ab}$ that satisfies the dual energy condition everywhere. \par Section 3 of the paper is devoted to extended bodies and constitutes the main section of the paper. In line with the general strategy, consider a collection of intermediate space-times starting from an exact solution with an extended object and reducing size and/or mass of the body. The following notion of {\it tracking} is introduced: Let $C$ be a collection of space-times and $\gamma$ a curve. We say that $C$ tracks $\gamma$ if the following holds: Let $\ssf{x}_{ab}$ be an arbitrary test field (satisfying the dual energy condition in a neighborhood of $\gamma$ and generic at some point of $\gamma$). Then there is an element $T^{ab}$ in $C$ such that $\ssf{T}\{\ssf{x}\}>0$. \par The two main theorems of the paper address two major cases, namely, the case without matter interaction, and charged particles. \par Theorem 3. Let $\gamma$ be a timelike curve in the space-time $(M, g_{ab})$. Let $C$ be a collection of fields $T^{ab}$ that each satisfy the dominant energy condition. Let $C$ track $\gamma$ and let each $T^{ab}$ be conserved. Then there exists a sequence of $T^{ab}$ (each a positive multiple of some element of $C$) that converges to $u^au^b\delta_\gamma$ in the sense of distributions ($u$ is a vector field required to be the unit tangent on $\gamma$). \par Roughly speaking, Theorem 3 says that, under conservation, a family of stress-energies that (suitably) collapses onto $\gamma$ includes a sequence converging to this distribution of a point particle. Next, since each $T^{ab}$ in the collection is conserved, conservation holds for the limiting distribution too. From this it can be concluded that $\gamma$ indeed must be a geodesic. \par A number of properties and limitations of Theorem 3 are discussed in the paper. We are not going to discuss these in detail here, as more importantly, the paper then sets out to make generalizations. In particular, Theorem 3 is extended to bodies carrying charge. Such bodies are described by a pair of fields, containing the stress-energy $T^{ab}$ and a charge-current $J^a$. Moreover, on the space-time $(M,g_{ab})$ we get an electromagnetic field $F_{ab}$. In the same spirit as before, we now require a collection of pairs $(T^{ab},J^a)$ that tracks $\gamma$. The force law $\nabla_bT^{ab}=F^a_bJ^b$ and the conservation law $\nabla_aJ^a=0$ need to be satisfied, and in addition the charge-mass ratio must be bounded by a non-negative number $\kappa$. \par Theorem 4. Let a collection $C$ (suitably) track the curve $\gamma$. Then there is a number $\kappa'$ with $|\kappa'|\leq\kappa$ and a sequence of pairs $(T^{ab},J^a)$, each multiples of elements in $C$, that converges to $(u^au^b\delta_\gamma,\kappa'u^a\delta_\gamma)$. Further generalizations are also discussed in the paper but appear to be more involved than the case of the electromagnetic interaction. Section 4 is dedicated to some examples for globally hyperbolic spaces (hyperbolicity is only a technical assumption ensuring enough interesting solutions). Specifically, for Maxwell's equations, it is confirmed that the collection of all solutions tracks every null geodesic. Similar results are derived for the Klein-Gordon equation and the charged Klein-Gordon field. In the latter case, geodesics are replaced by Lorentz-force curves. Some related questions are touched upon in Section 5, such as how tachyonic bodies are prohibited through tracking. In summary, this paper provides an inspiring perspective on the relativistic motion of extended bodies, both for conserved stress-energy and in presence of electromagnetic charges. A range of examples makes the subject more tangible, while available approaches are also briefly discussed and compared.