### [2003.07769] Testing the nature of Gauss-Bonnet gravity by four-dimensional rotating black hole shadow

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**May 04 2020**This paper was commented on through

In the last two months, there has been significant interest in the so-called "4D novel Einstein-Gauss-Bonnet gravity" (4DnEGBg). It is known that the most general D-dimensional metric, diffeomorphism-invariant theory with only two degrees of freedom and second-order equations of motion, is described by the Lovelock Lagrangian. In 4D this reduces to GR, whereas in 5D the first beyond-GR term is the Gauss-Bonnet invariant. In 4D, the Gauss-Bonnet invariant yields a topological action, or in other words it is a total derivative, which does not contribute to the equations of motion. One way of seeing this is to explicitly compute the variation of the Gauss-Bonnet invariant with respect to the metric, which contains a factor proportional to $(D-4)$ and, hence, vanishes identically in $D=4$. Recently, Glavan and Lin proposed in 1905.03601 to rescale the Gauss-Bonnet coupling by dividing by the offending $(D-4)$ factor. This, which one could think of as a dimensional regularization procedure, yields apparently non-trivial equations of motion in 4D, challenging the special role attributed to GR in 4D by Lovelock's theorem, and has spurred significant interest in the community, with a large number of follow-up works.

Two such works are 2003.07769 by Wei & Liu and 2003.08927 by Kumar & Ghosh, who compute the shadows of rotating black holes (BHs) in 4DnEGBg. They then compare these shadows against the shadow of M87* detected by the Event Horizon Telescope (EHT) and use this comparison to constrain the Gauss-Bonnet coupling $\alpha$. Given the strong interest in both 4DnEGBg and BH shadows following the EHT detection, these papers are very timely and are already making a strong impact in the field, as shown by the large number of citations. However, in all of this there is an elephant in the room, which is whether 4DnEGBg is a sensible theory to begin with.

In fact, a serious concern was raised in 2004.03390, where it was shown that the most general $D$-dimensional Einstein-Gauss-Bonnet theory does not admit a well-defined 4D limit. The reason is that the Gauss-Bonnet tensor in $D$ dimensions can generically be split into a tensor which carries information about the number of dimensions and another Lanczos-Bach tensor which does not care about $D$. The former has a well-defined $D \to 4$ limit, whereas the latter does not. Naively, the 4DnEGBg is recovered by setting the Lanczos-Bach tensor to zero, but doing so results in a violation of the Bianchi identities in 4D. Another option is to have a discontinuity in the Gauss-Bonnet tensor, but this option is pathological, too. A more heuristic way of explaining this problem is that in order to define 4DnEGBg in 4D, one first needs to work in $D>4$ and dispose of $(D-4)$ coordinates before taking the $D \to 4$ limit. However, in general one cannot dispose of $(D-4)$ coordinates freely since there is no canonical prescription for this procedure. This is quite different from compactification or dimensional reduction where one does not really dispose of the extra coordinates, but changes their sizes. The net result is that 4DnEGBg is only well-defined on $D>4$-dimensional spacetimes with a high degree of symmetry. This is the reason why Glavan and Lin managed to make sense of 4DnEGBg in 1905.03601 on FLRW or static spherically-symmetric spacetimes, but probably this cannot be done for a general spacetime. A similar concern was raised by Ai in 2004.02858.

The problem with the two papers in question is that they study rotating BH solutions by applying the Newman-Janis algorithm to the static spherically-symmetric BH solutions studied by Glavan and Lin. The resulting space-time no longer has the high degree of symmetry of the original static spherically-symmetric space-time, but is now only axially symmetric. Therefore, one might worry that the theory is ill-defined on this background. In fact, one problem appears in the paper by Wei & Liu, who find just above Eq. (33) that the 4DnEGBg equations are only satisfied if the polar angle $\theta$ satisfies $\theta=\pi/2$, for which the solution is maximally symmetric. In fact, they write "we should introduce some other fields in the GB action, which is very different from the case of general relativity. So the solution obtained here is not the GB vacuum solution and some matter fields should be included in order to be consistent with the gravitational field equations". This is actually a huge problem. It essentially amounts to the statement I wrote above that 4DnEGBg does not in general satisfy the Bianchi identity and one needs to add additional tensor fields to ensure the latter. Then, the key question is whether the rotating BH solutions found by Wei & Liu and Kumar & Ghosh even make sense? While Wei & Liu explicitly point out this issue, I do not see it mentioned in Kumar & Ghosh.

Besides this key issue, it would be interesting to better understand the differences between Wei & Liu and Kumar & Ghosh. The two papers appeared within two days from one another and basically seem to study shadows of the same BH solutions. For instance, the domain of validity of the Gauss-Bonnet coupling the two papers study is different. Could the authors of each paper compare their choice against the other?

Furthermore, the procedure used to compare their shadows against the shadow of M87* is different: Wei & Liu fits for a maximum of 10% offset in the major diameter, whereas Kumar & Ghosh fits for a maximum of 10% circularity deviation. Could the authors of each paper compare their choice against the other?

Finally, Wei & Liu also consider negative values for the Gauss-Bonnet coupling $\alpha$. Does this make sense, if we remember that this coupling should be the inverse string tension, if we interpret Einstein-Gauss-Bonnet gravity as arising from heterotic string theory, and hence should be positive-definite?

*Cosmo Comments*together with 2003.08927 by Kumar & Ghosh and made jointly available below. The following comments can also be viewed as annotations on the paper via Hypothesis.In the last two months, there has been significant interest in the so-called "4D novel Einstein-Gauss-Bonnet gravity" (4DnEGBg). It is known that the most general D-dimensional metric, diffeomorphism-invariant theory with only two degrees of freedom and second-order equations of motion, is described by the Lovelock Lagrangian. In 4D this reduces to GR, whereas in 5D the first beyond-GR term is the Gauss-Bonnet invariant. In 4D, the Gauss-Bonnet invariant yields a topological action, or in other words it is a total derivative, which does not contribute to the equations of motion. One way of seeing this is to explicitly compute the variation of the Gauss-Bonnet invariant with respect to the metric, which contains a factor proportional to $(D-4)$ and, hence, vanishes identically in $D=4$. Recently, Glavan and Lin proposed in 1905.03601 to rescale the Gauss-Bonnet coupling by dividing by the offending $(D-4)$ factor. This, which one could think of as a dimensional regularization procedure, yields apparently non-trivial equations of motion in 4D, challenging the special role attributed to GR in 4D by Lovelock's theorem, and has spurred significant interest in the community, with a large number of follow-up works.

Two such works are 2003.07769 by Wei & Liu and 2003.08927 by Kumar & Ghosh, who compute the shadows of rotating black holes (BHs) in 4DnEGBg. They then compare these shadows against the shadow of M87* detected by the Event Horizon Telescope (EHT) and use this comparison to constrain the Gauss-Bonnet coupling $\alpha$. Given the strong interest in both 4DnEGBg and BH shadows following the EHT detection, these papers are very timely and are already making a strong impact in the field, as shown by the large number of citations. However, in all of this there is an elephant in the room, which is whether 4DnEGBg is a sensible theory to begin with.

In fact, a serious concern was raised in 2004.03390, where it was shown that the most general $D$-dimensional Einstein-Gauss-Bonnet theory does not admit a well-defined 4D limit. The reason is that the Gauss-Bonnet tensor in $D$ dimensions can generically be split into a tensor which carries information about the number of dimensions and another Lanczos-Bach tensor which does not care about $D$. The former has a well-defined $D \to 4$ limit, whereas the latter does not. Naively, the 4DnEGBg is recovered by setting the Lanczos-Bach tensor to zero, but doing so results in a violation of the Bianchi identities in 4D. Another option is to have a discontinuity in the Gauss-Bonnet tensor, but this option is pathological, too. A more heuristic way of explaining this problem is that in order to define 4DnEGBg in 4D, one first needs to work in $D>4$ and dispose of $(D-4)$ coordinates before taking the $D \to 4$ limit. However, in general one cannot dispose of $(D-4)$ coordinates freely since there is no canonical prescription for this procedure. This is quite different from compactification or dimensional reduction where one does not really dispose of the extra coordinates, but changes their sizes. The net result is that 4DnEGBg is only well-defined on $D>4$-dimensional spacetimes with a high degree of symmetry. This is the reason why Glavan and Lin managed to make sense of 4DnEGBg in 1905.03601 on FLRW or static spherically-symmetric spacetimes, but probably this cannot be done for a general spacetime. A similar concern was raised by Ai in 2004.02858.

The problem with the two papers in question is that they study rotating BH solutions by applying the Newman-Janis algorithm to the static spherically-symmetric BH solutions studied by Glavan and Lin. The resulting space-time no longer has the high degree of symmetry of the original static spherically-symmetric space-time, but is now only axially symmetric. Therefore, one might worry that the theory is ill-defined on this background. In fact, one problem appears in the paper by Wei & Liu, who find just above Eq. (33) that the 4DnEGBg equations are only satisfied if the polar angle $\theta$ satisfies $\theta=\pi/2$, for which the solution is maximally symmetric. In fact, they write "we should introduce some other fields in the GB action, which is very different from the case of general relativity. So the solution obtained here is not the GB vacuum solution and some matter fields should be included in order to be consistent with the gravitational field equations". This is actually a huge problem. It essentially amounts to the statement I wrote above that 4DnEGBg does not in general satisfy the Bianchi identity and one needs to add additional tensor fields to ensure the latter. Then, the key question is whether the rotating BH solutions found by Wei & Liu and Kumar & Ghosh even make sense? While Wei & Liu explicitly point out this issue, I do not see it mentioned in Kumar & Ghosh.

Besides this key issue, it would be interesting to better understand the differences between Wei & Liu and Kumar & Ghosh. The two papers appeared within two days from one another and basically seem to study shadows of the same BH solutions. For instance, the domain of validity of the Gauss-Bonnet coupling the two papers study is different. Could the authors of each paper compare their choice against the other?

Furthermore, the procedure used to compare their shadows against the shadow of M87* is different: Wei & Liu fits for a maximum of 10% offset in the major diameter, whereas Kumar & Ghosh fits for a maximum of 10% circularity deviation. Could the authors of each paper compare their choice against the other?

Finally, Wei & Liu also consider negative values for the Gauss-Bonnet coupling $\alpha$. Does this make sense, if we remember that this coupling should be the inverse string tension, if we interpret Einstein-Gauss-Bonnet gravity as arising from heterotic string theory, and hence should be positive-definite?

*[These comments were shared with us by a member of the cosmology community. They do not necessarily reflect the opinion of the Cosmo Comments team.]*