Geometric optics in general relativity using bilocal operators

We consider the standard problem of observational astronomy, i.e., the observations of light emission from a distant region of spacetime in general relativity. The goal is to describe the changes between the measurements of the light performed by a sample of observers slightly displaced with respect to each other and moving with different 4-velocities and 4-accelerations. In our approach, all results of observations can be expressed as functions of the kinematic variables, describing the motions of the observers and the emitting bodies with respect to their local inertial frames, and four linear bilocal geodesic operators, describing the influence of the spacetime geometry on light propagation. The operators are functionals of the curvature tensor along the line of sight. The results are based on the assumption that the regions of emissions and observations are sufficiently small so that the spacetime curvature effects are negligible within each of them, although they are significant for the light propagation between them. The new formulation provides a uniform approach to optical phenomena in curved spacetimes and, as an application, we discuss the problem of a fully relativistic definition of the parallax and position drifts (or proper motions). We then use the results to construct combinations of observables which are completely insensitive to the motion of both the observer and the emitter. These combinations by construction probe the spacetime geometry between the observation and emission regions and in our formalism we may express them as functionals of the Riemann tensor along the line of sight. For short distances one of these combinations depends only on the matter content along the line of sight. This opens up the possibility to measure the matter content of a spacetime in a tomographylike manner irrespective of the motions of the emitter and the observer.

Figure 1

Flat, distant regions $N_{\mathcal{E}}$ and $N_{\mathcal{O}}$ connected by a family of null geodesics. The fiducial null geodesic ${\gamma }_0$ connects events $\mathcal{O}$ and $\mathcal{E}$. We consider other geodesics lying entirely within a narrow tube around ${\gamma }_0$. The geodesics may be parametrized by a pair of vectors in the tangent space $T_{\mathcal{O}}\mathcal{M}$ giving the initial displacement ($\delta x_{\mathcal{O}}$) and the initial direction deviation ($\mathrm{\Delta }{l}_{\mathcal{O}}$) with respect to ${\gamma }_0$. Similar parametrization $(\delta x_{\mathcal{E}},\mathrm{\Delta }{l}_{\mathcal{E}})$ can also be used in $N_{\mathcal{E}}$, at point $\lambda ={\lambda }_{\mathcal{E}}$ along the displaced geodesic.

Figure 2

In the parallel light rays approximation, we assume that the regions $N_{\mathcal{O}}$ and $N_{\mathcal{E}}$ are small enough and at the same time sufficiently separated that we can treat the light rays of null geodesics passing through them as parallel when discussing the relation of the direction deviation vector at $\mathcal{O}$ to the displacement vectors at both ends, see the figure on the left-hand side. This amounts to neglecting all possible perspective distortions of three-dimensional images and allows us to consider only flat, two-dimensional projections to the screen space, see the figure on the right-hand side.

Figure 3

Geometry of the quotient spaces ${\mathcal{Q}}_{\mathcal{O}}$, ${\mathcal{P}}_{\mathcal{O}}$ and their counterparts on the other side. Elements of ${\mathcal{Q}}_{\mathcal{O}}$ correspond to the vectors in $T_{\mathcal{O}}\mathcal{M}$, or points in $N_{\mathcal{O}}$, identified if they are separated by a multiple of $l_{\mathcal{O}}$. Geometrically it is the space of null straight lines in $N_{\mathcal{O}}$ parallel to ${\gamma }_0$. The two-dimensional subspace ${\mathcal{P}}_{\mathcal{O}}$ corresponds to null straight lines parallel to ${\gamma }_0$ which additionally lie on the null hyperplane orthogonal to $l_{\mathcal{O}}$.

Figure 4

Null tangent vectors $l_{\mathcal{O}}$ at $\mathcal{O}$ and $l_{\mathcal{E}}$ at $\mathcal{E}$ define two corresponding foliations by null hypersurfaces orthogonal to $l_{\mathcal{O}}$ and $l_{\mathcal{E}}$ respectively. In the flat light cones approximation, only the pairs of points lying on the corresponding leaves can be connected by a null geodesic. This requirement restricts the events in $N_{\mathcal{E}}$ from which an observer in $N_{\mathcal{O}}$ can register signals at a given instant of time.

Figure 5

A view of the projection of the null hypersurfaces containing $\mathcal{E}$ and $\mathcal{O}$. The angle at which the observer sees another point in an extended emitter’s cross section is determined by $\delta x_{\mathcal{E}}$, the Jacobi operator and the observer’s 4-velocity.

Figure 6

A pointlike luminous object at $\mathcal{E}$ and two other pointlike luminous objects nearby, forming a triangle on the cross section plane in the direction of the propagation of light. The angular diameter distance between $\mathcal{O}$ and $\mathcal{E}$ is obtained as the square root of the ratio between the area of the triangle on the cross section plane and the stereographic angle occupied by the triangle with vertices at the apparent positions of the 3 objects at the observer’s sky.

Figure 7

A view of the projection of the null hypersurfaces containing $\mathcal{E}$ and $\mathcal{O}$. The angle at which the displaced observer sees the point source at $\mathcal{E}$ as compared to the one at $\mathcal{O}$, i.e., the classical parallax, is determined by the Jacobi operator, the emitter-observer asymmetry operator, and the observer’s 4-velocity.

Figure 8

If the perpendicular part of the emitter-observer asymmetry operator $m_{\perp }$ vanishes, it is possible to draw a thin and long parallelogram made of two, initially parallel null geodesics $\gamma$ and ${\gamma }^{\prime }$, the displacement vector $\delta x_{\mathcal{O}}$ and its parallel transport $-\delta {\widehat{x}}_{\mathcal{O}}$. ${\gamma }_0$ plays the role of its diagonal. In this case determining the parallax angle for an observer displaced by $\delta x_{\mathcal{O}}$ is equivalent to the problem of the angular size of an object extending from $\mathcal{E}$ to $-\delta {\widehat{x}}_{\mathcal{O}}$.

Figure 9

A single, pointlike luminous source at $\mathcal{E}$ observed by an observer $o_1$ at $\mathcal{O}$ as well as two comoving observers $o_2$ and $o_3$ displaced with respect to $o_1$, performing their measurements on the same null hypersurface. The projection of their positions to the screen space perpendicular to $l_{\mathcal{O}}$ yields a triangle $T_2$ of area $A_{\mathcal{O}}$. On the other hand, superimposing the registered positions of the sources by the three observers yields a triangle $T_1$ on the “combined” celestial sphere, whose solid angle area is denoted by ${\mathrm{\Omega }}_{\mathcal{O}}$.

Figure 10

The observer and the emitter in gravitationally bound systems, undergoing short-time orbital motions around their free-falling barycenters.

Figure 11

The standard way to describe the position of a point mass $\delta x_{\mathcal{O}}$ is to decompose the position vector into the tangent and perpendicular components with respect to the barycenter 4-velocity $U_{\mathcal{O}}$ (on the left). Here we use a modified decomposition: $\delta x_{\mathcal{O}}$ decomposed into the part parallel to $U_{\mathcal{O}}$ and $\sigma$ orthogonal to $l_{\mathcal{O}}$. The decompositions differ by the tilt of the 3-plane containing the second component.