Black hole background2/2/2024 A large set of coupling functions between the Gauss–Bonnet invariant and the scalar field was considered in order to understand better the behaviour of the scalarized solutions. A natural extension of these results was to consider the case of nonzero black hole charge. It was shown in that below a certain mass the Schwarzschild black hole background may become unstable in regions of strong curvature, and then when the scalar field backreacts to the metric, a scalarized hairy black hole emerges and it is physically favorable. In this setup, besides GR solutions with a trivial scalar field configuration, the scalarized hairy solutions for black holes and stars could also exist, which evades the no-hair theorems. Recently, the spontaneous scalarization with particular coupling function in ESGB theory is widely investigated. Various black hole solutions and compact objects in the four-dimensional ESGB theories were studied in the literature. Especially, the introduction of this coupling could lead to hairy black holes. As a special scalar–tensor theory with higher derivatives, the Einstein–scalar-Gauss–Bonnet (ESGB) gravity recently has attracted a lot of attention. A way to make this term meaningful in four-dimensional spacetimes is to consider its coupling with a scalar field. But it becomes a topological term in four-dimensional spacetime and has no dynamics in field equations when it is minimally coupled with Einstein–Hilbert action. It is known that the inclusion of such terms probably bring in the well-known ghost problem, and Gauss–Bonnet (GB) corrections is a counter-case which is ghost-free. The effects of higher-order curvature terms become more significant as we are exploring the strong field regime of gravity via detections of gravitational waves and black hole shadows. It was found that the resulting hairy black hole solutions have a regular scalar field behaviour and all the possible divergence could be hidden behind the horizon. However, such irregular behaviour of the scalar field on the horizon was avoided in asymptotically AdS/dS spacetime with a presence of a cosmological constant in the gravity theory. One of the first hairy black hole solution in an asymptotically flat spacetime was discussed in but soon the solution was argued to be unstable because the scalar field is divergent on the event horizon. When the scalar field backreacts to the background metric, one could expect that hairy black hole solutions would be generated. So physicists have proposed various modified gravitational theories which indeed provide richer framework and significantly help us further understand GR as well as our universe.Īmong them, the scalar–tensor theories which introduces a scalar field into the action attract lots of attention. Yet it is unabated that the GR theory should be generalized, and in the generalized theories extra fields or higher curvature terms are always involved in the action. Experimental progress on gravitational waves and the shadow of the M87 black hole further demonstrates the great success of Einstein’s general relativity (GR).
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