Perturbations of rotating black holes in modified theories of gravity

May 29, 2023

Pratik Wagle

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General relativity, one of the fundamental theories of modern physics, predicts the presence of black holes in the universe. Since their theoretical prediction, astronomers have attempted to detect the signals given off by these mysterious objects. Black hole signatures have been observed in the electromagnetic spectrum since the 1970s. More recently, in 2015, gravitational waves, first detected by the LIGO-Virgo-Kagra collaboration, have opened a new frontier in black hole observation.


Visualization of the ringdown phase of a black hole. Image made by Yasmine Steele for Illinois Physics.
A schematic representation of the spacetime of a ringing black hole in modified theories of gravity. The black hole at the center (represented by the black ball) is the remnant of a binary black hole merger that is emitting its last gravitational waves before settling down. Gravitational waves in general relativity are represented by the blue spirals moving away from the black hole. In theories of modified gravity, deviations from general relativity can manifest as deformations of gravitational waves, which are represented by the red spirals.

Image made by Yasmine Steele for Illinois Physics.

The gravitational waves can be detected on Earth when two black holes orbiting around each other in a binary system merge. This results in a burst of gravitational radiation that travels as ripples through spacetime, leaving behind a nascent black hole. This baby black hole, known as the remnant, emits its own gravitational waves, made up of a complex mix of frequencies, before coming to rest. In this "ringdown" phase, the black hole is shaking off the force of the merger like a bell after it has been rung. This cosmic dance of black holes enables us to extract valuable physics about their properties. The emitted signals also encode information about possible deviations from general relativity, allowing us to test it against other 'modified' theories of gravity in the quest for a new theory of gravity.


In our paper published in Physical Review X, our team has developed a new formalism that allows us to study the ringdown phase of gravitational waves. Though efforts were made in the past to apply the existing formalism to black holes in beyond general relativity theories of gravity, these were limited to non-rotating or slowly rotating black hole systems. However, observations showed that black holes spin about ten times faster than those considered in these approaches, making them inadequate. This limitation was due to the theoretical inability of existing techniques to apply to black holes beyond general relativity and the lack of a formalism explicitly developed to study alternative spacetimes within general relativity and beyond. Our new theoretical formalism solves these problems by providing a new framework for analytically studying black holes in modified theories of gravity. As a result, our framework can be applied to spinning black holes without requiring any restrictions on their spin.


Metric perturbations, or deformations of the spacetime itself, formed the backbone of previously used formalisms to study black hole perturbations in modified gravity theories. On the contrary, our new formalism defines perturbations of gravity itself, known as curvature perturbations, to describe the evolution of the perturbed rotating black holes beyond general relativity. We use the widely popular language of the null basis formalism presented by Ezra Newman and Roger Penrose, making it accessible to a wide range of gravitational physicists. Our formalism is also designed to encompass theories where additional fields may be introduced that couple to gravity and introduce new deformations. Finally, we extend our formalism to encompass higher-order deviations from general relativity, making our formalism more robust and complete.


Though our formalism is purely mathematical, the physical scenarios one can study using it are significant. One notable application is the analytical determination of quasinormal mode frequencies of rotating black holes without any approximations in the spin. This allows us to study the ringdown phase of gravitational waves and provides a way to test alternative theories of gravity by extracting multiple modes from the gravitational wave signal. The quasinormal mode frequencies of different parities oscillate and decay with the same rates in general relativity, but this is not true for modified gravity theories. Our formalism can be implemented to study this isospectrality breaking  and the effect of this property on gravitational wave signals. The success of this formalism extends beyond ringdown studies. It can also be applied to investigate extreme mass ratio inspirals, where the effects of a small object orbiting around a supermassive black hole can be studied perturbatively using gravitational waves in modified theories of gravity. These binary systems are one of the most important targets for future space-based gravitational wave detectors, such as LISA, to test General Relativity. This enables the computation of gravitational waveforms and momentum fluxes sourced by such inspirals, providing a complementary approach to previously well-established studies.


Our work, therefore, sets the foundation for a careful and rigorous treatment of perturbed black holes in theories beyond General Relativity, thus enabling an entirely new set of tests that may reveal new physics beyond our current understanding. Furthermore, by providing a novel analytical approach for studying black hole dynamics and gravitational wave signals in modified gravity theories, this work opens exciting avenues for research, with potential applications in astrophysics, cosmology, and fundamental gravitational physics.