Towards a formalism for mapping the spacetimes of massive compact objects: Bumpy black holes and their orbits

TL;DR

The paper tackles the problem of testing whether observed massive compact objects are Kerr black holes by mapping their spacetimes through the accumulated orbital phase of nearby bodies. It develops a formalism of bumpy black holes—spacetimes that are almost black holes but possess controlled deformations in their multipole structure—and analyzes equatorial orbits within a Weyl-based perturbative framework, focusing on how a bumpiness parameter imprints on periapsis precession both in the weak and strong field. The key contributions are (i) a concrete construction of axisymmetric bumpy black holes via first-order metric perturbations around Schwarzschild, (ii) explicit weak-field and strong-field calculations for three perturbation scenarios (polar point masses, an equatorial ring, and a pure quadrupole), and (iii) demonstration that bumpiness can be read off from orbital precession and is measurable with coherent phase-tracking observations, enabling stringent tests of the no-hair hypothesis. This work lays the foundation for using bumpy black holes as a practical, strong-field probe to validate GR's black hole solutions and motivates future generalizations to Kerr spacetimes and inclined orbits, with implications for gravitational-wave and X-ray timing tests.

Abstract

Observations have established that extremely compact, massive objects are common in the universe. It is generally accepted that these objects are black holes. As observations improve, it becomes possible to test this hypothesis in ever greater detail. In particular, it is or will be possible to measure the properties of orbits deep in the strong field of a black hole candidate (using x-ray timing or with gravitational-waves) and to test whether they have the characteristics of black hole orbits in general relativity. Such measurements can be used to map the spacetime of a massive compact object, testing whether the object's multipoles satisfy the strict constraints of the black hole hypothesis. Such a test requires that we compare against objects with the ``wrong'' multipole structure. In this paper, we present tools for constructing bumpy black holes: objects that are almost black holes, but that have some multipoles with the wrong value. The spacetimes which we present are good deep into the strong field of the object -- we do not use a large r expansion, except to make contact with weak field intuition. Also, our spacetimes reduce to the black hole spacetimes of general relativity when the ``bumpiness'' is set to zero. We propose bumpy black holes as the foundation for a null experiment: if black hole candidates are the black holes of general relativity, their bumpiness should be zero. By comparing orbits in a bumpy spacetime with those of an astrophysical source, observations should be able to test this hypothesis, stringently testing whether they are the black holes of general relativity. (Abridged)

Figures (3)

• Figure 1: Periapsis precession in the strong field of a Schwarzschild black hole: the "extra" azimuth accumulated over a single orbit with parameters $(p,\varepsilon)$. The solid curve is for eccentricity $\varepsilon = 0.7$; the dashed one is for $\varepsilon = 0.1$. As we go to the weak field, $\delta\phi$ approaches zero --- orbits approach closed ellipses. The degree of precession is large in the strong field: the limiting value $\sim 12$ radians corresponds to almost 2 extra revolutions around the symmetry axis.
• Figure 2: Periapsis precession in the strong field of a bumpy black hole: the "extra" azimuth accumulated due to the hole's bumpiness as a function of $(p,\varepsilon)$. We have put $\mu = 0.01$ and $b = 2.5$. By the linearity of all relevant equations in mass, the degree of precession scales proportional to $\mu$. The solid curve is for $\varepsilon = 0.7$; the dashed one is for $\varepsilon = 0.1$. The dotted curve shows the weak-field prediction. Both strong-field results asymptotically approach the weak-field prediction for large $p$.
• Figure 3: Periapsis precession in the strong field of a bumpy black hole: the "extra" azimuth accumulated due to the hole's bumpiness as a function of $(p,\varepsilon)$. This plot is identical to Fig. \ref{['fig:b2.5']} except that we have put $b = 1.5M$.