[2003.08927] Rotating black holes in the novel $4D$ EinsteinGaussBonnet gravity
Authors:  Rahul Kumar, Sushant G. Ghosh 
Abstract:  Recently, Glavan and Lin [Phys. Rev. Lett. 124, 081301 (2020)] formulated the $4D$ novel EinsteinGaussBonnet gravity (EGB), by rescaling the GaussBonnet (GB) coupling $\alpha/(D4)$ and taking the limit $D \rightarrow 4$, which preserves the number of degrees of freedom thereby free from the Ostrogradsky instability. The theory admits spherically symmetric black holes generalizing the Schwarzschild black holes that are free form the singularity problems. We obtain the rotating counterparts of these spherically symmetric black holes and discuss their horizon properties and shadow cast. The effects of the GB coupling parameter on the shape and size of shadows are investigated in the context of recent M87* observations from the EHT. Interestingly, for a given parameter set, the apparent size of the shadow decreases and gets more distorted due to the GB coupling parameter. We find that within the finite parameter space, e.g. for $a=0.1$, $\alpha\leq 0.00394$, and within the current observational uncertainties, the rotating black holes of the $4D$ novel EGB gravity are consistent with the inferred features of M87* black hole shadow. 
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[2003.08927] Rotating black holes in the novel $4D$ EinsteinGaussBonnet gravity
This paper was commented on through Cosmo Comments together with 2003.07769 by Wei & Liu and made jointly available on this same forum. The comments can also be viewed as annotations on the paper via Hypothesis.

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Re: [2003.08927] Rotating black holes in the novel $4D$ EinsteinGaussBonnet gravity
We are thankful to the writer for sharing comments on our article.
The idea of 4D regularization of EGB gravity was originally led by Tomozawa (1107.1424), and later Cognola et al. (Phys. Rev. D 88, 024006 (2013)) reintroduced it by accounting quantum corrections due to a GB invariant within a classical Lagrangian approach. However, the interest in 4D regularized EGB gravity is recently revived by the Glavan and Lin (GL) (PRL 124, 081301 (2020)). The wellposedness of GL regularization procedure came heavily under the scrutiny and various concerns have been already raised (2004.03390, 2004.02858, 2004.09339, 2004.09214). In addition, some consistent and physically motivated alternate regularization procedures, leading to a welldefined action principle, for 4D EGB gravity were also proposed, which in contrary to the GL findings mainly belong to the special class of scalartensor theory of gravity (2004.09472, 2003.11552, 2003.12771). Considering a suitable KaluzaKlein reduction of GB theory in higher dimensions over a maximally symmetric internal space, a set of welldefined field equations are derived which interestingly admits the GL black hole solution. Another consistent regularized 4D EGB theory with the proper two dynamical degrees of freedom associated with the massless graviton, demands that the temporal diffeomorphism invariance of the theory must be broken (2005.03859). However, noteworthily none of the critics disprove the GL procedure at least for the maximally symmetric or spherically symmetric spacetimes (2004.02858, 2004.09472, 2004.03390). It is only because that this dimensional regularization procedure depends on the choice of higherdimensional spacetime metric and GL regularization procedure may not work well for spacetimes beyond the spherical symmetry, alternate regularization procedures were proposed.
We are not defending the GL regularization procedure for novel 4D EGB gravity but we would like to mention that we have analysed only the black hole solution. The main question arises: Is the black hole solution true in 4D GL’s EGB theory or other theories? It is worth mentioning that although these alternate regularization procedures are completely different in spirit, the GL static spherically symmetric black hole appears as a special solution of these alternate 4D regularized theories (2004.14738, 2003.07068, 2004.09472, 2003.11552). Furthermore, together with the fact that the GL black hole solution is also identical to those found in semiclassical Einstein's equations with conformal anomaly (JHEP 1004, 082 (2010)) and gravity theory with quantum corrections (PRD 88 024006 (2013), 1107.1424), this might indicate that 4D EGB regularized theory could indeed constitute a suitable effective action for gravity. However, the rotating generalization of static black holes in these gravity theories as well as their observational consequences have not yet been studied extensively. Therefore, in this spirit, our presented study on the rotating black holes is valid in these gravity theories as well.
The NewmanJanis algorithm (NJA) has been widely used to construct rotating black hole solutions from their nonrotating counterparts. While this algorithm was originally developed within general relativity, it has been more recently applied to nonrotating solutions in modified gravity theories (1105.3191, 1911.07520, 1302.6075, 1412.5424, 1410.4043, 1308.6631). It is true that exercising the NJA to an arbitrary nongeneral relativity spherically symmetric solution might introduces pathologies in the resulting axiallysymmetric metric. However, we used the modified NJA, owing to the AzregAinou's noncomplexification procedure, to derive the rotating EGB black hole metric, which has been successfully applied for generating imperfect fluid rotating solutions in the BoyerLindquist coordinates from spherically symmetric static solutions, and can also generate generic rotating regular black hole solutions. It may be true for our rotating black hole solution that it may not satisfy field equations and that is valid also for other rotating solution generated in modified gravity, and they likely to generate extra stresses. Therefore, we regard our rotating metric as a regularized EGB gravity black hole metric of an appropriately chosen set of field equations which are unknown but different from the EGB equations.
The main aim of our paper is to construct the shadows of rotating black hole in 4D EGB gravity and to investigate the effects of quadratic curvatures on the shape and size of shadows in the context of recent M87* observations from the EHT. Similar study was appeared in 2003.07769 couple of days ago. However, considering the GB coupling parameter related to the inverse string tension, unlike in the 2003.07769, we restrict ourselves only to the positive values of α. We used the shadow observables area A and oblateness D to characterize the size and shape of shadows and in turn to extract the values of black hole parameters. One potential advantage of using observables (A,D) over (R_s, δ_s) is that unlike radius R_s, area A is sensitive to both the black hole spin a and GB coupling parameter α, and thereby a given set of observable (A,D) uniquely determine the black hole parameters. For the M87* black hole shadow the axis ratio was found to be smaller than 4/3, which is equivalent to the circularity deviation of less than 10%. We consider the inclination angle of θ_o=π/2 and utilize the shadow circularity deviation and angular size to constrain the GB coupling parameter. We find that within the finite parameter space and current observational uncertainties, black holes of the theory are consistent with the inferred features of M87* black hole shadow.
Warm regards
Authors
The idea of 4D regularization of EGB gravity was originally led by Tomozawa (1107.1424), and later Cognola et al. (Phys. Rev. D 88, 024006 (2013)) reintroduced it by accounting quantum corrections due to a GB invariant within a classical Lagrangian approach. However, the interest in 4D regularized EGB gravity is recently revived by the Glavan and Lin (GL) (PRL 124, 081301 (2020)). The wellposedness of GL regularization procedure came heavily under the scrutiny and various concerns have been already raised (2004.03390, 2004.02858, 2004.09339, 2004.09214). In addition, some consistent and physically motivated alternate regularization procedures, leading to a welldefined action principle, for 4D EGB gravity were also proposed, which in contrary to the GL findings mainly belong to the special class of scalartensor theory of gravity (2004.09472, 2003.11552, 2003.12771). Considering a suitable KaluzaKlein reduction of GB theory in higher dimensions over a maximally symmetric internal space, a set of welldefined field equations are derived which interestingly admits the GL black hole solution. Another consistent regularized 4D EGB theory with the proper two dynamical degrees of freedom associated with the massless graviton, demands that the temporal diffeomorphism invariance of the theory must be broken (2005.03859). However, noteworthily none of the critics disprove the GL procedure at least for the maximally symmetric or spherically symmetric spacetimes (2004.02858, 2004.09472, 2004.03390). It is only because that this dimensional regularization procedure depends on the choice of higherdimensional spacetime metric and GL regularization procedure may not work well for spacetimes beyond the spherical symmetry, alternate regularization procedures were proposed.
We are not defending the GL regularization procedure for novel 4D EGB gravity but we would like to mention that we have analysed only the black hole solution. The main question arises: Is the black hole solution true in 4D GL’s EGB theory or other theories? It is worth mentioning that although these alternate regularization procedures are completely different in spirit, the GL static spherically symmetric black hole appears as a special solution of these alternate 4D regularized theories (2004.14738, 2003.07068, 2004.09472, 2003.11552). Furthermore, together with the fact that the GL black hole solution is also identical to those found in semiclassical Einstein's equations with conformal anomaly (JHEP 1004, 082 (2010)) and gravity theory with quantum corrections (PRD 88 024006 (2013), 1107.1424), this might indicate that 4D EGB regularized theory could indeed constitute a suitable effective action for gravity. However, the rotating generalization of static black holes in these gravity theories as well as their observational consequences have not yet been studied extensively. Therefore, in this spirit, our presented study on the rotating black holes is valid in these gravity theories as well.
The NewmanJanis algorithm (NJA) has been widely used to construct rotating black hole solutions from their nonrotating counterparts. While this algorithm was originally developed within general relativity, it has been more recently applied to nonrotating solutions in modified gravity theories (1105.3191, 1911.07520, 1302.6075, 1412.5424, 1410.4043, 1308.6631). It is true that exercising the NJA to an arbitrary nongeneral relativity spherically symmetric solution might introduces pathologies in the resulting axiallysymmetric metric. However, we used the modified NJA, owing to the AzregAinou's noncomplexification procedure, to derive the rotating EGB black hole metric, which has been successfully applied for generating imperfect fluid rotating solutions in the BoyerLindquist coordinates from spherically symmetric static solutions, and can also generate generic rotating regular black hole solutions. It may be true for our rotating black hole solution that it may not satisfy field equations and that is valid also for other rotating solution generated in modified gravity, and they likely to generate extra stresses. Therefore, we regard our rotating metric as a regularized EGB gravity black hole metric of an appropriately chosen set of field equations which are unknown but different from the EGB equations.
The main aim of our paper is to construct the shadows of rotating black hole in 4D EGB gravity and to investigate the effects of quadratic curvatures on the shape and size of shadows in the context of recent M87* observations from the EHT. Similar study was appeared in 2003.07769 couple of days ago. However, considering the GB coupling parameter related to the inverse string tension, unlike in the 2003.07769, we restrict ourselves only to the positive values of α. We used the shadow observables area A and oblateness D to characterize the size and shape of shadows and in turn to extract the values of black hole parameters. One potential advantage of using observables (A,D) over (R_s, δ_s) is that unlike radius R_s, area A is sensitive to both the black hole spin a and GB coupling parameter α, and thereby a given set of observable (A,D) uniquely determine the black hole parameters. For the M87* black hole shadow the axis ratio was found to be smaller than 4/3, which is equivalent to the circularity deviation of less than 10%. We consider the inclination angle of θ_o=π/2 and utilize the shadow circularity deviation and angular size to constrain the GB coupling parameter. We find that within the finite parameter space and current observational uncertainties, black holes of the theory are consistent with the inferred features of M87* black hole shadow.
Warm regards
Authors