Ineffective Higher Derivative Black Hole Hair
Inspired by possibility that the Schwarzschild black hole may not be the unique spherically symmetric vacuum solution to generalisations of general relativity, we consider black holes in pure fourth order higher derivative gravity treated as an effective theory. Such solutions may be of interest in addressing the issue of higher derivative hair or during the later stages of black hole evaporation. Non-Schwarzschild solutions have been studied but we have put earlier results on a firmer footing by finding a systematic asymptotic expansion for the black holes and matching them with known numerical solutions obtained by integrating out from the near horizon region. These asymptotic expansions can be cast in the form of trans-series expansions which we conjecture will be a generic feature of non-Schwarzschild higher derivative black holes. Excitingly we find a new branch of solutions with lower free energy than the Schwarzschild solution, but as found in earlier work, solutions only seem to exist for black holes with large curvatures meaning that one should not generically neglect even higher derivative corrections. This suggests that one effectively recovers the non-hair theorems in this context.
Birkoff’s theorem jebsen1921general ; birkhoff1923relativity ; ANDP:ANDP19233771804 ; eiesland1925group implies that the unique spherically symmetric, asymptotically flat, vacuum solution of general relativity is the Schwarzschild black hole. Despite apparently having a large entropy, this black hole is completely characterised by just one parameter – namely its mass. Essentially taking ; together with spherical symmetry and asymptotic flatness; gives
Recently, it was shown in Lu:2015cqa that in fourth order gravity, the stronger assumption of a static spherically asymptotically flat solution leads to a weaker restriction that the Ricci scalar, , but not the Ricci tensor, , is zero111It is not hard to see that in general relativity vacuum solutions necessarily have .222For some interesting earlier work on Birkoff’s theorem in higher derivative gravity see Oliva:2011xu ; Oliva:2012zs . This in turn opens up the exciting possibility that there are so-called non-Schwarzschild black holes. For reasons we will review later, the fourth order equations of motion are numerically unstable (or “stiff”) and one can only integrate them for a while before they diverge. In Lu:2015cqa it was found that by tuning the initial conditions near the horizon one can successively integrate out further and further approaching asymptotically flat space. This is convincing evidence for the existence of such solutions. We have put these results on a firmer footing by matching a systematic asymptotic expansion to the numerical solutions. While it was know that these solutions where characterised by just two near horizon parameters, from the asymptotic form of the solution, we explicitly see that in addition to the mass, the solutions are characterised by the size of a massive Yukawa mode associated with the higher derivative terms. Armed with the asymptotic form of the solution we where also able to find a new branch of solutions that were previously missed.
We are interested in studying these solutions for three reasons. Firstly, the fact that Birkhoff’s theorem is weakened once we add higher derivatives, suggests the possibility of higher derivative hair which could play a role in understanding black hole entropy. Although one finds that these solutions are only characterised by one additional parameter, it does suggest that even higher order terms in the Lagrangian could lead to more hair. Secondly, as a Schwarzschild black hole shrinks by Hawking evaporation, treating gravity as an effective theory, one expects that at some length scale, higher derivative terms may become important. This means that non-Schwarzschild black holes may play some role in the late time evolution of black holes. Finally, we would like to study asymptotically AdS non-Schwarzschild black holes which could have interesting holographic interpretations. This paper serves as a warm up for that project.
The laws of black hole thermodynamics bh1 ; haw generalise once one adds higher derivative terms Wald:1993nt ; Jacobson:1993vj ; Iyer:1994ys ; Jacobson:1994qe . In particular the Bekenstein-Hawking entropy, given by
where is the horizon area (in Plank units), is replaced by the Wald entropy which, for a spherical solution, is given by Sen:2007qy
with the relevant action. We take the non-Schwarzschild black hole’s entropy to given by the Wald entropy.
Although we will discuss it in detail later (see Figures 14, 13, 12 and 11 for detailed plots), it is worth mentioning some features of the thermodynamics of the non-Schwarzschild black holes. Unlike the Schwarzschild black hole, the horizon radius decreases as these black holes get colder. At a fixed horizon size, one has both a Schwarzschild and a non-Schwarzschild solution except for one critical point where the solutions coincide. Below this critical size, the non-Schwarzschild black hole is colder than the Schwarzschild solution, while above the critical size it is hotter than the Schwarzschild solution. We will call the two sets of solutions the cold and hot branches respectively333Only the hot branch was found in Lu:2015cqa .. Like the Schwarzschild black hole their entropy decreases as one increase the temperature. However, on the cold branch, the black hole’s entropy seems to approach a finite value as the temperature approaches zero. Bizarrely on the hot branch, as one increases the temperature, the entropy reaches zero at some point and then becomes negative as the temperature increases further. We can not see how this makes any physical sense. Finally it is interesting to note that, at a fixed temperature, both branches have lower entropy than the Schwarzschild solution. Intriguingly, also at fixed temperature, the cold branch has a lower free energy, while the hot branch has a higher free energy, than the corresponding Schwarzschild one.
In addition to troubling thermodynamics, both branches of the non-Schwarzschild solutions are highly curved which would not generically be consistent with neglecting even higher order curvature terms in the action. One could insist on truncating the theory rather than treating it as an effective theory but this will come at the cost of introducing ghosts stelle1978classical . These unphysical excitations could be responsible for some of the strange thermodynamic features of the solutions. This suggests that the only physically reasonable black hole one can take is the Schwarzschild solution eliminating the prospect of higher derivative hair at the first non-trivial order. We have called this class of hair “ineffective” since it disappears if one insists on the self-consistency of the low energy effective description444In principle, there could be special situations in which it makes sense to consider a truncated theory in which case our objections to this hair falls way. Unfortunately it does not seem likely that such “special” hair could generically play a role in understanding black hole entropy..
On a technical note, the asymptotic expansion we find take the form of a trans-series. We conjecture that this will be a generic feature and may prove to be a useful tool for studying the asymptotics of solutions to higher derivative theories in general.
In section 2 we review results on non-Schwarzschild black holes. Then in section 3, we show how to systematically find the asymptotic form of the metric followed by a discussion of the thermodynamics one extracts by matching the numerical and asymptotic solutions in section 4. In section 5 we present our conclusions and prospects for future work. Finally some technical aspects are left to Appendix A and Appendix B which cover further details of the asymptotic expansion and our numerical results respectively.
2 Non-Schwarzschild black holes
We review some of the results of Lu:2015cqa ; Lu:2015psa on non-Schwarzschild black holes which will be useful for us (with some added details, and slight changes in notation and emphasis). We consider a general pure gravitational action of the form
where is the Weyl tensor, and are dimensionless constants555One can add a Gauss-Bonnet term which, in four dimensions, does not affect the equations of motion, but can shift the Wald entropy.. is the Planck scale and is some length scale associated with the curvature scale of our solution at the horizon – for a Schwarzschild black hole . Up to , the equations of motion, , which follow from (2.1) are Lu:2015cqa :
where is the covariant derivative. One can check that vacuum solutions of general relativity satisfy (2.3) so that in particular the Schwarzschild blackhole is a solution666The non-trivial result one has to use is the Bianchi identity for the Weyl Tensor hawkingellis , , so that ..
In stelle1978classical , it was shown that when one studies linear fluctuations of (2.3) about flat space, in addition to the usual massless spin-2 graviton mode, there is are massive spin-0 (with ) and spin-2 (with ) modes. The massive modes will introduce Yukawa like behaviour, , in the asymptotic form of the metric. This means that, to obtain asymptotically flat solutions, one needs to finely tune conditions in the bulk of the spacetime to avoid the growing Yukawa mode. Furthermore, the growing mode makes the numerical integration of the equations tricky since rounding errors can inadvertently excite the growing mode leading to so-called stiff equations which easily blow up777Of course such Yukawa modes could also have implications for the stability of such solutions against perturbations which we do not consider at this stage.. Also notice that one needs to take and positive to avoid tachyon-like modes.
In Lu:2015cqa it was shown that any asymptotically flat black hole solution to (2.3) must have but not necessarily . As is well known jebsen1921general ; birkhoff1923relativity ; ANDP:ANDP19233771804 ; eiesland1925group , spherical symmetry and implies that the only black hole solution is the Schwarzschild one. With the weaker requirement that , there is the possibility of other solutions. Given the complexity of (2.3), it is challenging to find analytic solutions but one can resort to numerics and perturbative expansions. The matching of a near horizon expansion with numerical integration was studied in Lu:2015cqa – we have further studied an asymptotic expansion about flat space which compliments this earlier work.
Following on from the discussion in the previous paragraph, we will only be interested in solutions with from now on. This means that the value of will not effect the equation of motion so we will ignore it888 Alvarez-Gaume:2015rwa investigates non-Schwarzschild higher derivative black holes with . They are working in quadratic gravity which does not start with the usual Einstein-Hilbert term . Since the equations of motion are different in that case, our argument does not apply.. Furthermore, once we neglect the terms involving , it is not hard to see that the trace of the equations of motion (2.3) leads to .
2.1 Near horizon expansion
We now consider the near horizon form of the metric which give the boundary conditions for numerical integration of the equations of motion. Given that we can neglect the terms involving , the only length scale in (2.3) is 999We put in a factor of into the definition of for later convenience. . It is convenient to scale out this dependence by defining a dimensionless radial coordinate . We take the spherically symmetric ansatz
With this ansatz, the trace of the equations of motion, 101010As mentioned in the beginning of this section, the trace of the equations of motion, reduces to in this case., gives
where, , denotes derivatives with respect to , and the equation of motion, , gives
To study black hole solutions, one takes the following expansion about the horizon (at some ) :
Note that the constant can be shifted by a trivial rescaling of and consequently does not entering into the equations of motion (although does need to be tuned if one demands that asymptotes to at the end). One can show that corresponds to the Schwarzschild solution so that the parameter
gives an indication of the degree to which our solution deviates from the Schwarzschild case.
and from (1.3), one finds the Wald entropy:
where as usual the Bekenstein-Hawking entropy, , is one quarter the horizon area. It will turn out, for a given mass, the Schwarzschild and non-Schwarzschild black holes have a different horizon area (see Figure 12). This means that when determining entropy as a function of mass, both terms in (2.10) lead to shifts in the entropy.
Systematically solving (2.5)-(2.6) for and at each order in , one finds that there are two free parameters which can be taken to be and . For instance at second order one finds111111Given the system’s numerical instability, we solved the equations up to order to accurately set up the initial conditions.
3 Asymptotic expansion
To give more solid evidence for non-Schwarzschild black holes we will find a systematic asymptotic expansion which we can match to the near horizon expansion discussed in the previous section. It appears that, in principle, this expansion is extendable to any order – albeit at the cost of having to evaluate some integrals beyond second order numerically.
Generalising stelle1978classical , it turns out to be more convenient to rewrite the metric as follows
where as before . We then consider an expansion about flat space
together with the two following linear combinations of the equations of motion (2.3):
Recall from the previous section, that (3.4) implies in this context.
3.1 First order expansion
These two equations can be decoupled and easily integrated using new variables. Letting
one finds that (3.6) can be written
Solving (3.13) we find121212Multiplying (3.13) by one finds that can be written as a total derivative . This is easily integrated leading to a second order linear equation which can then be integrated to give (3.15).
which can in turn be integrated:
As discussed in Lu:2015psa , we expect our solution to have 4 free parameters and indeed, substituting (3.15) and (3.17) back into the equations of motions shows that we must have . Furthermore, requiring that our solution asymptotes to flat space means one must eliminate the growing Yukawa mode and we set . The constant corresponds to a trivial rescaling of the time coordinate and requiring sets 131313The constant, , essentially maps to the constant, , of the near horizon expansion in some non-trivial way.. So finally the first order solution for the metric components is
which is a special case of results in stelle1978classical . Given the boundary conditions we have imposed, at first order, there are two independent parameters in this expansion, and . The parameter is related to the mass, while is related to the strength of the Yukawa mode . Recalling that, , and letting
we see that we can consider an expansion in and of the form
with and dimensionless.
To get a feel for the parameter , we note that the effective Newtonian potential seen by a particle far away from the black hole is
In practice, since the equations are linear at each order of , one can also use the expansion (3.2) and easily read off powers of and from (3.20). Notice that near the horizon (at ) one might roughly expect
even for small black holes. Similarly, for the other part of ,
which is, a priori, difficult to estimate. Given (3.25) and (3.26) are not necessarily small near the horizon, one should not generically expect the expansion, (3.2), to be valid there. On the other hand, if and fall off fast enough, the expansion should become good some distance from the horizon as these terms become small. In fact, we will find that as increases, our expansion becomes worse and worse very near the horizon but rapidly converges as one moves just a small distance away from the horizon (measured in units of ).
Finally, we have a technical aside. We find it convenient to introduce a third parameter in the asymptotic expansion, , so that
As discussed earlier, one is free to scale as long as one scales time to compensate. We would like to work in units for which give , but one does not initially know how the near horizon expression for , (2.7), should be scaled to achieve this. For convenience, when setting the boundary conditions for our numerical integration, we fix in (2.7). Fitting a value for to our numerical results then allows us to determine how we should rescale to ensure that . Scaling , and using , gives the modified form of (2.9) we will be using
3.2 Second order expansion
Now, expanding the equations of motion (3.4) to second order we find
and once again is given by (3.14). Notice that (3.33) has the same form as (3.13) except that there is now a source. We can solve (3.33) by integrating the source over an appropriate Green’s function, leading to
The terms involving are just solutions to the homogeneous equation that appeared at first order – without loss of generality we can set them to zero. In a similar fashion, we do not specify the lower limits of the integrations since any constant pieces can be absorbed into the homogeneous solution. Performing the integrals (and discarding the homogeneous part) we find
where is the Exponential Integral defined by
To get some intuition for the terms involving Ei appearing in (3.40), note that for large ,
whereas, for small ,
The integrations in (3.46) produce two constants of integration which we neglected. One of the constants can be absorbed into the mass and the other corresponds to a rescaling of the time coordinate so that no new degrees of freedom are introduced at this order. For more details of the calculation see Appendix A.
3.3 Higher order expansion
At higher orders, we can make the substitutions
and it would appear that the equations to order , take the form
for some and . As at second order, we find a solution of the form
We note that, as for the second order case, no new degrees of freedom are introduced at each order. Unfortunately, in practise, the integrals become increasingly difficult to evaluate due to non-trivial integrals involving Ei and Meijer-G functions, although in principle, it seems one could evaluate up to arbitrary order using numerical integration. More details, including the order solutions, are in Appendix A.
3.4 Features of the asymptotic expansion
We would like to investigate some features of the asymptotic expansion. When all the dust has settled, we find the expression for the metric components to second order in and is
Notice that with , it is easy to check that we recover the Schwartz child solution (up to ):
At large , we find the slightly simpler expressions
We note that the expansions (3.63) and (3.67) look like transseries (see trans for a nice introduction to transseries). Perhaps starting with an appropriate transseries ansatz would be an easier way to approach this problem which we hope to look at in the future. Given the appearance of Yukawa modes at first order, it is perhaps not surprising that we end up with a transseries expansion & and we conjecture that this feature will be a common characteristic of the asymptotics of other higher derivative black holes.
Figure 1 is a log plot of , , and as a function of . These are terms in (3.23) with coefficients , and that do not involve the mass . It is clear from the plot that as increases, successive terms become exponentially smaller at higher orders in the expansion. There is similar behaviour when one compares other higher order terms in the expansion to the lower order term which effectively act as sources for them. This is good evidence that the expansion will converge at least at for sufficiently large – we leave a more rigorous examination of convergence for possible future work. The plots in Figure 1 all approach straight lines indicating exponential fall off on a log plot.
Figure 2-Figure 4 are plots of different terms at the same order in the asymptotic expansion. What is clear is that for , the terms , which correspond to the pure Schwarzschild black hole, dominate for large . Far from the horizon, the non-Schwarzschild terms become negligible. The non-Schwarzschild terms also become linear at large once again indicating exponential fall off.
From Figure 1, one see that for small , higher order terms in the expansion are larger than the lower order terms. This suggests that the expansion breaks down at some point. We know that the expansion for the Schwarzschild black hole only breaks down at the horizon in these coordinates (which would be at ). We calculate the asymptotic expansion up to third order and our numerical investigations show that as becomes large, the asymptotic solution does not match well with the near horizon expansion. Consequently, we will need numerical integration to match the near horizon solution with the asymptotic expansion which is what we cover in the next section.
4 Matching and Thermodynamics
In this section we study the non-Schwarzschild thermodynamics, as well as the validity of our asymptotic expansion. At the most basic level, we need to find the relationship between the mass of the solution, which can be extracted asymptotically, and, temperature and entropy, which are encoded in the horizon data.
As discussed in section 2.1, the near horizon solution is characterised by two parameters and with the former encoding the deviation for the Schwarzschild solution and the latter encoding the position of the horizon. As shown in Lu:2015cqa , for a given , aside from the Schwarzschild solution, which corresponds to , there is one additional asymptotically flat solution with 141414There is a special point at where the Schwarzschild solution is the only solution. In Lu:2015cqa , the solutions with where found.. As discussed below, for a given , one has to tune to obtain a regular solution. In section 3 we showed that the asymptotic solution is characterised by two parameters, the mass , and strength of the Yukawa excitation, (see (3.24)). The Schwarzschild solution corresponds to . We will discuss below how for a given non-Schwarzschild solution, one can extract and by matching the near horizon and asymptotic expansions numerically.
We start with a similar procedure to the one outlined in Lu:2015cqa . For a fixed , one uses the near horizon expansion to set the boundary conditions for numerical integration of (2.5) and (2.6), starting a little bit away from the horizon. One then has to find so that the positive Yukawa mode, which would cause the solution to blow up, is not excited.
Adopting a shooting method, we scan over a range of guesses for , successively tuning it so that the integration can extend further and further – this process is shown in Figure 5.151515We used Mathematica for the numerical integration with the stiffness switching method, successively increasing the precision and accuracy goals as we fine tuned . It was claimed in Lu:2015cqa , that they where able to extend the numerics integration as far as . While this is good evidence for the existence of non-Schwarzschild solutions, there is no guarantee that one will eventually hit some sort of barrier. We found that one does not have to integrate too far before the asymptotic expansion, which also involves only three parameters, , and , matches very well with numerical integration161616We used a non-linear fit to match the numerical results with the asymptotic expansion.. This gives even stronger evidence for the non-Schwarzschild solutions.
4.1 Convergence and Asymptotics
Before going on to consider the thermodynamics, we would like to discuss some investigations of the near horizon and asymptotic expansions, as well as our matching procedure.
In Figure 6 we show a numerical plot for and , together with the near-horizon expansion at various orders. It is clear from the plot that the near horizon expansion breaks down pretty quickly, namely before , even at high order. We see that one does not greatly extend the expansion’s domain of validity each time one increases the order of the expansion. We found this behaviour was universal, over all values of investigated, with the near horizon expansion, even up to , not extending far.
Given the computing power at our disposal, we did not extend our numerical integration much beyond 40-50 . As mentioned, and as can be seen from Figure 1 - Figure 4, terms in our asymptotic expansion involving fall of exponentially fast. This means that our solution is soon very well approximated by the Schwarzschild making it relatively easy to find the mass, , and . Using a non-linear fitting procedure we extract the parameters by finding the best fit between our numerical and asymptotic solutions (over some appropriately chosen range of ).
In Figure 7 we superimpose our fit of the and asymptotic expansions onto Figure 6 171717The asymptotic expansions where fitted to the numerical results over the range .. We see that there is a good match even close to the horizon. Furthermore, although it is not too easy so see on this graph, the plot looks like a better fit. Unfortunately, we also found that the asymptotic expansion is generally not good in the limited domain of validity of the near horizon expansion.
This is evident in Figure 8 where we’ve plotted numerical results for together with fits for the asymptotic expansions. We see that the expansion diverges more from the near horizon solutions than the expansion but on the other hand the expansion converges more rapidly to the numerical results as increases. We find similar results for other values of with the divergence between the asymptotic and near horizon expansion increasing as increases (where is obtained from our best fit procedure). This behaviour is not too surprising if we look back at Figures 4, 3, 2 and 1, where terms which appear with a coefficient of can be very large for small but fall off very fast. Putting this all together strongly suggests that expansion (3.23) becomes good for sufficiently large. On the other hand, this means that in general, to find the relationship between near horizon and asymptotic data some numerical integration is required. Lastly, we would like to note that while the values obtained for and by fitting the or asymptotic expansions to the numerical results only differ by 1% or less, the values obtained for can have much larger variation. For example, for the case shown in Figure 8, fitting the second order expansion gave while fitting the third expansion gave . Once again, we can understand this from the fact that terms involving fall off very quickly leaving an essentially Schwarzschild like solution.
While we can see from Figures 8 and 7 that the numerical and asymptotic results agree well beyond a few multiples of , to get a better feel for our results we now switch to log-log plots to make small differences at large more visible.
The plot in Figure 9 has the same as Figure 7 but it makes the difference between our fits and the numerical result more visible. The dotted vertical line in the plot corresponds to the point at which we set the boundary conditions for the numerical integration using the near horizon result. It is clear that the expansions are good fits for the numerical data and that increasing the order of the expansion improves the fit. In Figure 10, we wanted to check the effect of changing the range of over which we fit the asymptotic expansion. We shifted the starting point, , for the fit, keeping the end point fixed. In this case, the best fit was obtained at . The different values we took for had very little effect on the best fit for (and ) but changed considerably. Ideally, one should try to optimise the value of to obtain the best fit since, a priori, one does not know at what radius one can really start to trust the asymptotic expansion. We did not bother doing that since it had little effect on the mass extracted from the solution which is what we where primarily interested in.
Having investigated how well our asymptotic expansion fits the numerical results, we can happily go ahead and investigate the thermodynamics. Since we found that using the expansion did not alter our results for the mass (and ) very much, we just used the expansion up to to find the parameters of the solutions – these results are summarised in Appendix B.
Firstly, we consider Figure 11 which is a plot of temperature as a function of (recall that corresponds to the length scale of the horizon in units of ). The solid line denotes the non-Schwarzschild solution while the dashed line corresponds to the Schwarzschild one. As mentioned in the introduction, above a certain critical point (already found in Lu:2015cqa ) the non-Schwarzschild solution is hotter than the Schwarzschild one whereas below it the non-Schwarzschild one is colder. We call these two branches the hot (red line) and cold branches (blue line) respectively. Notice that, unlike the Schwarzschild case, the temperature of the non-Schwarzschild decreases as decreases. Extrapolating the cold branch to smaller , it would appear as if one approaches a zero temperature solution – unfortunately the numerical ingratiation becomes increasingly unstable and we where not able to investigate this limit.
Now, looking at Figure 12, which is a plot of mass as a function of , we start to see some more peculiar features of these non-Schwarzschild solutions. Once again the the dashed line corresponds is the Schwarzschild black hole with the hot and cold non-Schwarzschild branches represented by solid red and blue lines respectively. In contrast to the Schwarzschild solution, the mass of the non-Schwarzschild solutions increases as decreases. On the cold branch the mass seems to saturate at around as the temperature approaches zero. On the hot branch, the mass actually becomes negative at some point, continuing to become smaller as increases without any apparent lower bound. For Schwarzschild black holes, a negative mass is associated with a naked singularity but the negative mass non-Schwarzschild do not seem to violate cosmic censorship.