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A new study led by STRUCTURES YRC member Pedro G.S. Fernandes has presented stationary and axially symmetric black hole solutions to Einstein's field equations of General Relativity incorporating gravitational effects of the black hole's astrophysical environment.
Black holes are a central prediction of Einstein’s theory of general relativity. In their simplest form, they were theoretically described already in 1916 by physicist and astronomer Karl Schwarzschild, who discovered an exact vacuum solution of Einstein's field equations describing a static, spherically symmetric, non-rotating black hole. In contrast, astrophysical black holes are expected to spin, since they form from the collapse of rotating stars or from mergers of compact objects carrying angular momentum. An appropriate description for such objects is given by the Kerr metric, a stationary, axially symmetric vacuum solution discovered by Roy Kerr in 1963.
Most commonly, stellar or galactic black holes are modelled using the Kerr solution, while neglecting the gravitational influence of their surroundings. While this description has been extraordinarily successful, in realistic contexts, black holes are not isolated objects, but generally thought to be embedded in complex matter-rich environments. Aside from accretion disks, these consist of the galaxy's dark matter halo, which contributes small background gravitational effects that are typically neglected.
In a recent study by Pedro G. S. Fernandes et al., the authors went beyond this idealized description by constructing rotating black hole solutions that explicitly incorporate such a surrounding matter distribution. Generalizing Kerr's vacuum solution, their model utilizes an anisotropic fluid to source a stationary, axially symmetric spacetime geometry. The particular shapes of these spacetimes incorporate both the black hole – characterized by its mass and its spin – and its astrophysical environment. The gravitational influence of the latter, modelled in the framework of the so-called Einstein cluster model, results in additional functions compared to the Kerr metric. These functions determine how the dark matter halo alters the radial and angular geometry of spacetime, as well as frame-dragging effects and they need to be solved numerically.
The researchers computed the physical properties of these solutions and studied the gravitational impact of the surrounding matter on characteristic observables. They found that characteristic orbits of matter and light around the black hole are shifted and that the apparent size of the black hole shadow – a key observational property – can become substantially larger than predicted by the Kerr model. Moreover, these effects grow with increasing spin of the black hole and weaken when the surrounding halo of matter is more diffuse. Interestingly, the study also shows that black holes surrounded by matter can spin faster than the maximum allowed for an isolated black hole in vacuum, which is called the Kerr limit. In vacuum, exceeding this limit would remove the event horizon and violate the so-called cosmic censorship hypothesis. In the presence of surrounding matter, however, the bound is shifted to higher spin values.
The results indicate that environmental effects need not always be merely negligible corrections, but can play an important role in the interpretation of high-precision studies of black holes. As black hole imaging continues to improve in accuracy, properly accounting for astrophysical environments will be essential to avoid misinterpreting observational signatures as genuine deviations from general relativity itself. Similarly, environmental effects may have implications for future gravitational-wave observations.
The findings were published in Physical Review Letters.
Original Publication:
Fernandes, P. G. S. and Cardoso, V., “Spinning Black Holes in Astrophysical Environments”, Physical Review Letters, vol. 135, no. 21, Art. no. 211403, APS, 2025. doi:10.1103/9shv-5d21.