Horizon curvature and spacetime structure influences on black hole scalarization

Experimental progress on gravitational waves GW1 ; GW2 ; GW3 and the shadow of the $M87$ black hole EHT further demonstrates the great success of Einstein’s general relativity (GR). 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 Nojiri:2006ri ; Clifton:2011jh ; MG3 . 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.

Among them, the scalar-tensor theories which introduces a scalar field into the action attract lots of attention Fujii . When the scalar field backreacts to the background metric, one could expect that hairy black hole solutions would be generated. One of the first hairy black hole solution in an asymptotically flat spacetime was discussed in BBMB but soon the solution was argued to be unstable because the scalar field is divergent on the event horizon bronnikov . 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. 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 Martinez:1996gn ; Banados:1992wn ; Martinez:2004nb ; Zloshchastiev:2004ny ; Martinez:2002ru ; Dotti:2007cp ; Torii:2001pg ; Winstanley:2002jt ; Martinez:2006an ; Kolyvaris:2009pc ; Charmousis:2014zaa ; Khodadi:2020jij .

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 is known that the inclusion of such terms probably bring in the well-known ghost problem stelle , and Gauss-Bonnet (GB) corrections is a counter-case which is ghost-free. 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. A way to make this term meaningful in four-dimensional spacetimes is to consider its coupling with a scalar field stringT . As a special scalar-tensor theory with higher derivatives Horndeski ; Deffayet:2009mn , the Einstein-scalar-Gauss-Bonnet (ESGB) gravity recently has attracted a lot of attention. Especially, the introduction of this coupling could lead to hairy black holes. Various black hole solutions and compact objects in the four-dimensional ESGB theories were studied in the literature Mignemi_1993 ; Kanti_1996 ; Torrii_1996 ; Ayzenberg_2014 ; Kleihaus_2011 ; Kleihaus_2016a .

Recently, the spontaneous scalarization with particular coupling function in ESGB theory is widely investigated. 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 Bekenstein ; Antoniou_2018 ; Antoniou:2017hxj . It was shown in Doneva:2017bvd 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. A natural extension of these results was to consider the case of nonzero black hole charge. 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 Doneva:2018rou -Fernandes:2019rez .

Various generalizations of the spontaneous scalarization procedure were followed in the literature. The scalarization due to a coupling of a scalar field to Ricci scalar was studied in Herdeiro:2019yjy and in the presence of Chern-Simons invariant was studied in Brihaye:2018bgc . The generalized study of scalarized black hole solutions and compact objects in asymptotical flat spacetime of ESGB theories was discussed in Silva:2017uqg -Hunter:2020wkd . The spontaneous scalarization of asymptotically AdS/dS black holes in ESGB theory with a negative/positive cosmological constant was extended in Bakopoulos:2018nui ; Brihaye:2019gla ; Bakopoulos:2019tvc ; Bakopoulos:2020dfg ; Lin:2020asf . Especially, the connections of asymptotically AdS black holes scalarization with holographic phase transitions in the dual boundary theory was studied in Brihaye:2019dck ; Guo:2020sdu . Recently the spontaneous scalarization in f(R) gravity theories in the presence of a scalar field minimally coupled to gravity with a self-interacting potential was discussed in Tang:2020sjs . Also in ESGB theory the spin-induced black hole spontaneous scalarization, which is the outcome of linear tachyonic instability triggered by rapid rotation, was explored in Collodel:2019kkx ; Dima:2020yac ; Doneva:2020kfv ; Herdeiro:2020wei ; Berti:2020kgk ; Zhang:2020pko .

The scalarization in asymptotical flat, AdS and dS black holes in ESGB gravity theory was studied, however detailed information on how different spacetime structures influence the scalarization process is not disclosed. An interesting question is how the cosmological constant leaves its imprint on the black hole scalarization and in which kind of spacetime the scalar field could be formed more easily. Besides, the studies of spontaneous scalarization in the existed literatures were focused on the black hole with spherical horizon, but as known in AdS spacetime, one can have AdS black holes with different topologies and it is interesting to study how these topologies of the black hole horizon affect the scalarization procedure¹¹1Note that in Motohashi:2018mql , the authors discussed the influence of the topology in the matter configuration on the spontaneous scalarization. They found that comparing to the spherical distribution, the planar symmetric matter distribution makes the spontaneous scalarization easier to occur.. Thus, the aim of this work is to probe the signal of scalarization in various black holes with different horizon curvature and try to understand its effects on the scalarization processes. To this end, we first consider the test scalar field perturbation and analyze its effective potential in various black hole backgrounds. Then we study the tachyonic instabilities of the scalar field near the horizon by solving the perturbation equation. We find that the scalar hair around the black hole with spherical horizon is easier to form in AdS spacetime than in flat case. While in dS case, the regular condition near cosmological horizon leads to divergence at large distance, so that the test field approximation breaks down. Moreover, our study on the effect of horizon curvatures indicates that the scalar hair around AdS black hole with toroidal horizon could be easier to be formed than that with hyperbolic horizon, but more difficult than that with spherical horizon. Both the effective potential analysis and the test field solution conclude the above observations.

The work is organized as follows. We write down the general equation of motion of a test scalar field in the background of four dimensional black holes with different topologies in ESGB gravity theory in section II. In section III, we study the signal of scalarization of a black hole with positive curvature in different spacetimes. Then we investigate the signal of a black hole scalarization with negative curvature and zero curvature in section IV and section V respectively. The last section is devoted to conclusions and discussions.

We consider four-dimensional ESGB theories with the action

where $\Lambda$ is the cosmological constant, $\lambda$ is the coupling constant between Gauss-Bonnet term and scalar field with the dimension of length and $\nabla_{\mu}$ denotes the covariant derivative. The scalar field coupling function $f(\Phi)$ only depends on the neutral scalar field $\Phi$, and the Gauss-Bonnet term is given by

where $R,R_{\mu\nu}$ and $R_{\mu\nu\alpha\beta}$ are the Ricci scalar, Ricci and Riemann tensors, respectively. The Einstein equation and Klein-Gordon equation can be derived from the above action as

where the energy-momentum tensor $\Gamma_{\mu\nu}$ is given by

with $\Psi_{\mu}=\lambda^{2}\frac{df(\Phi)}{d\Phi}\nabla_{\mu}\Phi$.

It is obvious that different forms of the coupling function $f(\Phi)$ shall give different properties of the ESGB theory. As addressed in Doneva:2017bvd , $f(\Phi)$ could satisfy conditions $\frac{df(\Phi)}{d\Phi}\mid_{\Phi=0}=0$ and $\frac{d^{2}f(\Phi)}{d\Phi^{2}}\mid_{\Phi=0}=b^{2}>0$, where $b$ is a constant, in order to admit Schwarzschild black hole as a background solution and further explore the black hole scalarization in the ESGB theory. Moreover, one usually assumes that the scalar field vanishes at infinity and normalizes the constant $b$ to unity. Thus, in this work, we shall choose the form of the coupling function as $f(\Phi)=\frac{1}{2}(1-e^{-\Phi^{2}})$ which also satisfies $f(\Phi=0)=0$.

When the scalar field vanishes, i.e., $\Phi=0$, the gravity theory admits black hole solutions, whose metric takes the general form,

where $M$ is the integral constant related the black hole mass, and $k$ determines the curvature of two dimensional geometry near the horizon with the line element $d\overrightarrow{x}_{k,2}^{2}$ given by

where $k=1,0$ and $-1$ correspond to black hole horizon with spherical, planar and hyperbolic topology, respectively. Then the corresponding GB term can be easily computed as

In order to explore the (in)stability of the background black holes, we consider a small fluctuation in the form $\delta\Phi=e^{-i\omega t}\phi(r)Y(x_{1},x_{2})$ where $Y(x_{1},x_{2})$ satisfies Laplace-Beltrami equation $\Box_{2}Y(x_{1},x_{2})=-AY(x_{1},x_{2})$. We consider that $Y(x_{1},x_{2})$ is spherical harmonics $Y_{\ell,m}$ for positive curvature so that $A=\ell(\ell+1)$; For zero curvature $Y(x_{1},x_{2})$ is plane wave case as $e^{i\mathcal{K}(x+y)}$ so that $A=\mathcal{K}^{2}$; and for negative curvature $Y(x_{1},x_{2})$ is spherical harmonics $Y_{-\frac{1}{2}\pm i\zeta,\mathfrak{m}}$ and $A=\frac{1}{4}+\zeta^{2}$ Koutsoumbas:2008pw ; Wang:2001tk . Then we use the tortoise coordinate $dr_{*}=\frac{dr}{g(r)}$, and bring the Klein-Gordon equation into a Schrodinger-like form by reforming $\phi(r)=\frac{\varphi(r)}{r}$

where the effective potential takes the form

Noted that in the Schrodinger-like form (9) and (10) of the perturbation, we have only kept the leading term of the expansion $\frac{df(\Phi)}{d\Phi}\simeq\Phi(1-\Phi^{2}+\frac{1}{2}\Phi^{4}+\mathcal{O}(\Phi^{2}))$ in the potential function.

We then analyze the signal of scalarization of the system by computing the effective potential with the lowest mode $(\ell=\mathcal{K}=\zeta=0)$. For simplification, we could focus on the static perturbation with $\omega=0$, thereby we shall solve the radial perturbed equation for $\phi(r)$,

and see how the scalar field with the lowest mode emerges as the Gauss-Bonnet coupling increases. To control the variables of the black holes and compare the results, we shall set $r_{h}=1$ in various cases, namely, we fix the size of the black holes such that the black hole mass is solved via $g(r=r_{h})=g(r_{h}=1)=0$ in the following discussion. We would numerically employ the spectral method to solve the differential equations.

Since the existence of above black hole with different curvatures is dependent on the type of asymptotic background, i.e., the sign of $\Lambda$, we would discuss the signal of scalarization of various black holes in terms of the curvature. Moreover, we shall rewrite $\Lambda=\pm 3/L^{2}$ where $L$ is the curvature AdS radius. As it was discussed in Guo:2020sdu ; Kolyvaris:2013zfa because the scalar field is coupled to GB invariant, the effective mass of the scalar field receives a contribution from this coupling. Then for various values of the coupling, the effective mass of the scalar field becomes tachyonic outside the background black hole and the fluctuations are unstable destabilizing in this way the background metric.

In this section, we focus on the background solution with positive curvature, i.e., $k=1$ in (6), in which case there exist asymptotical flat $(\Lambda=0)$, AdS $(\Lambda<0)$ and dS $(\Lambda>0)$ black holes as possible backgrounds. Consequently, we shall separately study the signal of scalarization in all these asymptotical backgrounds, and then do comparisons on the emergence of non-trivial scalar field which is introduced by the Gauss-Bonnet coupling in these different spacetime backgrounds.

In this section we will study the signal of scalarization if the background metric is a topological black hole (TBH). For the action

the presence of a negative cosmological constant allows the existence of black holes with a topology ${R}^{2}\times\Sigma$, where $\Sigma$ is a two-dimensional manifold of constant curvature. The simplest solution of this kind, when $\Sigma$ has negative constant curvature, reads

where $d\sigma^{2}_{2}$ is the line element of $\Sigma$ which is just (6) with $k=-1$, $\Lambda=-3L^{-2}$ and $M=2G\mu$. The line element $d\sigma^{2}_{2}$ is locally isomorphic to the hyperbolic manifold $H^{2}$ and $\Sigma$ must be of the form

where $\Gamma$ is a freely acting discrete subgroup (i.e., without fixed points).

The configurations (24) are asymptotically local AdS spacetimes and these black holes are known as TBHs mann -Cai:2001dz . It has been shown in Gibbons:2002pq that the massless configurations where $\Sigma$ has negative constant curvature are stable under gravitational perturbations and the stability of the topological black holes was discussed in Birmingham:2007yv ; Wang:2001tk . In a series of papers Koutsoumbas:2006xj ; Koutsoumbas:2008pw ; Koutsoumbas:2008yq it was shown that there is evidence that a vacuum TBH goes over to a hairy configuration, the MTZ black hole Martinez:2004nb , through a second-order phase transition. Therefore it is interesting to see if in the presence of the coupling between scalar field and the GB term, there is evidence of TBH scalarization.

In this case, the GB term (8) is evaluated as ${\cal R}^{2}_{GB}=\frac{4}{r^{2}}(g^{\prime}(r)^{2}+(g(r)+1)g^{\prime\prime}(r))$. To fulfill $g(r_{h})=0$, the black hole mass should satisfy $M=r_{h}^{3}/L^{2}-r_{h}$. The positivity of black hole mass forces $0<L<r_{h}$, while the positivity of Hawking temperature requires $L<\sqrt{3}r_{h}$. Thus for $r_{h}\leq L<\sqrt{3}r_{h}$, the topological black hole have a positive Hawking temperature, but the black hole mass could be zero or negative, and the stability of the these massless and negative mass black hole has been investigated in Birmingham:2007yv . It is noted that the topological black holes have interesting structures in their own right, here for security we focus on topological black hole with positive mass, such that we have $0<L<1$ as we have set $r_{h}=1$. Then the effective potential is shown in Fig. 5. Without the GB coupling (see the left plot), the effective potential is always positive, so there is no signal of tachyonic instability to induce the scalarization, so that the TBH keeps stable. With $\lambda=0.3$ (see the middle plot), the potential well becomes deeper as $L$ decreases . When $L$ is small enough, the potential well would be deep enough to allow the tachyonic instability to happen. In the right plot with fixed $L=0.5$, as $\lambda$ increases, the potential well becomes deeper. We note that the rule for negative curvature in AdS case extracted from Fig. 5 is qualitatively the same as that in the positive curvature case (see Fig. 3). But with the same parameters, the potential well for $k=-1$ is shallower than that in the case with $k=1$. This means that the onset of spontaneous scalarization is easier to happen for AdS black hole with positive curvature than that with negative curvature.

Again, to further fix the parameters for the threshold of the nontrivial scalar field, we need to call back the perturbation equation (11). Near the event horizon, the scalar field satisfies

While in asymptotical infinity, the behavior is also (17). We shall also set $\phi_{+}=0$ in the numeric.

The critical threshold parameters in $(\lambda,L)$ space is shown on the left of Fig. 6 where in the green shadow region the scalar field in nontrivial. As $L$ increases, the critical coupling also increases which is similar as that in the cases with $k=1$. The curve tends to be divergent as $L\rightarrow 1$ because there is no black hole solution when $L\geq 1$ in our setup. By comparing with left of Fig. 6 and Fig. 4, we find that the scalar hair is more difficult to form in the negative curvature than in the positive case. This is consistent with the conclusion from the comparison of the potential function.

We also present the scalar hair near the horizon and the scalar charge at infinity as the function of $\lambda$ for different values of $L$ in the middle and right of Fig. 6, respectively. With the increase of $L$, the threshold value $\lambda$ becomes larger, which matches the observation on the left of Fig. 6. Moreover, as $L$ becomes larger, the curves become lower and lower rapidly and the value of $\phi_{h}(\phi_{-})$ decreases. Especially, when $L$ goes closely to the unity, $\phi_{h}$ becomes very small, which indicates that the scalarization of the background black hole becomes very difficult. This behaviour is consistent with that in positive curvature at small coupling (see the middle plot of Fig. 4) as well as that in zero curvature shown in next section.

We have observed that scalarizations in hyperbolic AdS black hole backgrounds depend on two factors, the coupling strength between the scalar field and the GB term and the cosmological constant through the radius $L$. For the chosen $L$, once the coupling parameter $\lambda$ increases, the effective potential in Fig. 5 becomes deeper which can ignite tachyonic instability and start the hair formation more easily. When we have big enough $L$, in Fig. 6 we see that the required critical $\lambda$ to start the scalarization becomes bigger. In the middle panel of Fig. 6, we find that for bigger $L$, the formed scalar hair is weaker even we have stronger coupling strength $\lambda$. The growth of $L$ counteracts the effect of the $\lambda$ and hinders the scalarization process.

AdS black holes with a flat space ($k=0$) is described as toroidal black hole spacetimes. Thus the corresponding redshift function in the metric (6) is $g(r)=-\frac{M}{r}+\frac{r^{2}}{L^{2}}$ ($M=\frac{1}{L^{2}}$ as $r_{h}=1$) and the GB term is ${\cal R}^{2}_{GB}=\frac{4}{r^{2}}(g^{\prime}(r)^{2}+g(r)g^{\prime\prime}(r))$.

We plot the effective potential in Fig. 7. The potential function with $\lambda=0$ does not show negative well which insures the stability of the original toroidal hole. When the coupling between the scalar field and GB term is turned on, for the chosen AdS radius $L$, the potential well near the horizon appears and becomes deeper as $\lambda$ increases. For the same coupling strength $\lambda$, similar potential well behavior will appear when we decrease the AdS radius $L$. The deep potential well can bound the scalar field near the horizon and lead to the instability of the original toroidal black hole. This property is similar to those observed in the case with positive curvature (Fig. 3) and negative curvature (Fig. 5). Through careful comparisons, we find that for the same parameters, the potential well in zero curvature case is deeper than that in the negative curvature case, while shallower than that in the positive curvature background. This indicates that in ESGB theory, the horizon curvature influences the scalarization. For AdS black hole with toroidal horizon the scalar hair can be formed more easily than the hyperbolic case, but if compared with the spherical AdS black hole, the scalarization process in toroidal AdS black hole is still more difficult.

To see the signal of the scalar field growing near the horizon, we solve the perturbation equation (11) with the boundary condition near the horizon

The behavior is also (17) in asymptotical infinity, and we set $\phi_{+}=0$ as usual in the numeric.

The phase diagram in parameters $(L,\lambda)$ space is shown in the left panel of Fig. 8 where below the blue line, the scalar field perturbation will trivially die out, but above this line the black hole scalarization will happen. The values on this line are critical threshold values to ignite the scalarization. When the AdS radius $L\rightarrow\infty$, we see that the threshold coupling $\lambda$ linearly approaches infinity, which means that the toroidal AdS black hole cannot be scalarized in such limits. In the limit $L\rightarrow\infty$, the background returns to the flat spacetime and there is no black hole with the zero curvature.

With some fixed $L$, we study the scalar field near the horizon and the scalar charge at infinity as the function of $\lambda$, and the results are shown in the middle and right of Fig. 8, respectively. Note that for large GB coupling, the probe limit may be not reasonable and backreaction should be involved. But in some sense our result for small GB coupling is reliable and we see the signal of scalarization from the scalar field perturbation. In Guo:2020sdu , we constructed the scalarized hairy black hole with planar horizon in AdS space by considering the full backreaction. There we have set $L=1$ and the threshold value is $\lambda\simeq 0.64$ which is consistent with that in Fig. 8 via careful comparison.

In this paper, we discussed the signal of scalarization for black holes with different curvature topologies in various spacetimes. We considered the scalar field as a probe to the general relativity solution in Einstein-scalar-Gauss-Bonnet theory. Our discussion is based on the analysis of its effective potential and the onset of nontrivial solution to the perturbed scalar equation.

We studied the signal of black hole scalarization with positive curvature in asymptotical flat, AdS and dS spacetime. We first analyzed the validity of the probe approximation in various spacetimes. We found that in flat and AdS cases, a regular condition of test scalar field near event horizon could correspond to convergent (vanishing) condition at the asymptotical infinity, meaning the test field approximation could be valid; while in dS case, the regular condition near cosmological horizon leads to divergence at large distance, so that the test field approximation breaks down. We then compared the effect of asymptotical flat and AdS spacetime spacetime structures on the scalarization process. Without the GB coupling $\lambda$, the effective potential of the scalar field are alway positive and no well emerges. As we increased the coupling, the negative potential well would show and becomes deeper which may lead to tachyonic instability. With the same parameters, the potential well for AdS black hole is deeper than that for Schwarzschild black hole, meaning that the scalar hair around the spherical AdS black hole may be easier to form. We solved the perturbed scalar field equation in the parameter space $(\lambda,L)$, and obtained the critical line above which the non trivial scalar hair can be found. Fig. 4 shows that the scalar cloud around AdS black hole could form easier than that around the Schwarzschild black hole. Moreover, in the limit $L\rightarrow\infty$, the threshold coupling in AdS case approaches the value in Schwarzschild black hole from below as shown in the left of Fig. 4.

We also investigated the signal of AdS black hole scalarization process with different curvatures of the horizon. For AdS black hole with positive $(k=1)$, negative $(k=-1)$ and zero $(k=0)$ curvatures, the properties of effective potential is qualitatively the same (see Fig. 3, Fig. 5 and Fig. 7, respectively), that’s to say, no negative potential well appears with $\lambda=0$ for instability. The potential well near horizon appears and becomes deep as $\lambda$ (or $L$) increases (decreases), which could bound the scalar field near the horizon and lead to instability. However, the potential well for zero curvature is deeper than that for negative curvature but shallower than that for positive curvature. Notice that the effect of horizon curvature on the potential could be roughly analytically evaluated from the expression (10), i.e, $V(r)=g(r)\left(\frac{g^{\prime}(r)}{r}+\frac{A}{r^{2}}+m^{2}-\frac{\lambda^{2}}{2}{\cal R}^{2}_{GB}\right)$. With fixed $L$ and $\lambda$, we can diagnose the potential as follows. (i) $m^{2}=-2/L^{2}+12\lambda^{2}/L^{4}$ for different horizon curvatures is the same in our setup. (ii) For the lowest mode, the topological term $A$ satisfies $A(k=1)=A(k=0)=0<A(k=-1)=1/4$ . (iii) Fixing horizon radius implies that the black hole mass satisfies $M(k=1)>M(k=0)>M(k=-1)$, then it is easy to obtain that the group term $T=\frac{g^{\prime}(r)}{r}-\frac{\lambda^{2}}{2}{\cal}R^{2}_{GB}$ in the potential satisfies $T(k=1)<T(k=0)<T(k=-1)<0$ for fixed $r$ near horizon. (iv) The overall factor $g(r)$ fulfills $g(k=1)>g(k=0)>g(k=-1)>0$ for fixed $r$ outside the horizon. Combining (i)-(iv), it is not difficult to reduce that the effect of horizon curvature on the negative potential well is $V(k=1)<V(k=0)<V(k=-1)$ near horizon. It is noted that only the angular eigenvalues term $A$ mentioned in point (ii) is truly the topological effect, and the terms mentioned in points (iii-iv) are actually geometrical effect. Thus, in this sense, the effect from the horizon curvature on the potential as well as the scalarization is reflected by the interplay between the topology and geometry.

The behavior of potential indicates that in ESGB theory, the scalar hair around AdS black hole with toroidal horizon may be easier to be formed than that with hyperbolic horizon but more difficult than that with spherical horizon. Moreover, we collected the critical threshold curves for different curvatures in Fig. 9 where the blue dot represents the threshold value of Schwarzschild black hole in the limit $L\rightarrow\infty$. It is obvious that for the AdS black hole with fixed $L$, the scalar hair is the easiest to be formed around spherical horizon, then around the toroidal horizon and it is the most difficult to be bounded near hyperbolic horizon.

We shall present two comments on Fig. 9. $i)$ In the limit $L\rightarrow 0$, the threshold coupling in all three cases is very small in which case the scalarization could easily realized. Moreover, we know from the effective potential that the GB coupling should be involved for the tachyonic instability and it further leads to the spontaneous scalarization. So we may argue that the threshold value would be tiny as $L\rightarrow 0$, or the scalarization could cut off at certain tiny $L$. This direction with large $\Lambda$ deserves further study. $ii)$ In the limit $L\rightarrow\infty$, the threshold coupling for zero or negative curvature are divergent, suggesting that the scalar field can not be bounded in these cases. This is understandable because as $L\rightarrow\infty$, the spacetime become flat and only Schwarzschild black hole with positive curvature exists as background in probe limit. Moreover, it is interesting to involve the full backreaction of the scalar field into the geometry and verify our results.