On defining astronomically meaningful Reference Frames in General Relativity

TL;DR

Extends astronomical reference frames from the post-Newtonian regime to exact GR by analyzing shearfree observer congruences and their asymptotic properties. The approach characterizes when a spacetime admits a coordinate system adapted to a congruence with vanishing shear and asymptotically zero vorticity and acceleration, leading to a metric structure $ds^{2}=-e^{2\Phi}[dt-\mathcal{A}_{i}dx^{i}]^{2}+h_{ij}dx^{i}dx^{j}$ with $h_{\alpha\beta}=f\chi_{\alpha\beta}$ (conformal rigidity). Key contributions include explicit conditions for frames anchored to distant inertial objects, a clear metric form for shearfree congruences, and a rigorous separation of ZAMOs from true astronomical frames in the presence of frame dragging. The results apply to stationary spacetimes (e.g., Kerr) and certain non-asymptotically flat cases, providing a principled framework to avoid misinterpretations in astrophysical modeling and to extend IAU-style references to exact GR.

Abstract

In a recent paper we discussed when it is possible to define reference frames nonrotating with respect to distant inertial reference objects (extension of the IAU reference systems to exact general relativity), and how to construct them. We briefly review the construction, illustrating it with further examples, and caution against the recent misuse of zero angular momentum observers (ZAMOs).

Figures (2)

• Figure 1: An astronomically meaningful reference frame: coordinate system adapted to a shearfree congruence of observers $\mathcal{O}(u)$ with asymptotically vanishing acceleration and vorticity. (a) Connecting vectors ($X^{\alpha}$ and $\partial_{i}$) have fixed direction in the orthonormal triad $\{{\bf e}_{\hat{\imath}}\}$, yielding a grid of points everywhere at fixed directions with respect to each other, and anchored to inertial frames at infinity. (b) 3D space representation of such a grid. In the special case of conformally stationary spacetimes (e.g. the Kerr metric) this materializes in that light rays from remote sources arrive at fixed directions in such a frame.
• Figure 2: Examples of congruences in flat spacetime: (a) rigidly rotating congruence; (b) congruence associated to Milne's universe GriffithsPodolsky2009 (purely expanding: $a^{\alpha}=\omega^{\alpha}=\sigma_{\mu \nu}=0$); (c) free or irrotational vortex (e.g., a whirlpool)---shearing congruence. The plots depict the spatial velocities $dx^{i}/dt=u^{i}/u^{0}$. Red dots show the evolution of an initial squared array of observers: in (a) and (b) the squared shape is preserved along the congruence, reflecting the preservation of angles between observers. In (c), by contrast, the shape is completely distorted as it evolves along the congruence.