Phase Transitions and Stability of Eguchi-Hanson-AdS Solitons

As we have just seen, the Eguchi-Hanson-AdS₅ soliton is characterized by a single integer $p$. For several reasons, it will be fruitful to consider these solutions for large values of $p$.

Beginning with the Eguchi-Hanson-AdS₅ metric (3), we perform the following transformations

$\displaystyle t$

$\displaystyle=\frac{2\tau}{p}\,,\quad r=az\,,\quad\theta=\frac{4\rho}{p}\,,\quad\varphi=\psi-\frac{p}{2}\phi\,,$

(8)

and then take the limit $p\to\infty$. The result is

$$ds^{2}=-z^{2}d\tau^{2}+\frac{\ell^{2}dz^{2}}{z^{2}f(z)}+\frac{\ell^{2}z^{2}f(z)}{4}d\varphi^{2}+\ell^{2}z^{2}\left[d\rho^{2}+\rho^{2}d\phi^{2}\right]+\mathcal{O}\left(\frac{1}{p}\right)$$

(9)

where

$$f(z)=1-\frac{1}{z^{4}}\,.$$

(10)

One can then convert the polar coordinates on the $\mathbb{R}^{2}$ to standard Cartesian coordinates, giving

$$ds^{2}=-z^{2}d\tau^{2}+\frac{\ell^{2}dz^{2}}{z^{2}f(z)}+\frac{\ell^{2}z^{2}f(z)}{4}d\varphi^{2}+\ell^{2}z^{2}\left[dx^{2}+dy^{2}\right]+\mathcal{O}\left(\frac{1}{p}\right)\,.$$

(11)

In the strict $p\to\infty$ limit, this is an AdS₅ soliton belonging to the class first reported in Horowitz:1998ha , in coordinates such that the location of the bubble is at $z=1$. One can easily check that the coordinate $\varphi$ is periodic with period $2\pi$. Strictly speaking, there are several topologically distinct solitons that can be obtained from the same local metric (11) depending on identifications performed on the auxiliary flat directions $(x,y)$ — see, e.g., Page:2002qc . In this case, the soliton corresponding to the large $p$ limit of Eguchi-Hanson-AdS has no identifications on these coordinates, i.e. the spatial part of the boundary metric is $\mathbb{S}^{1}\times\mathbb{R}^{2}$. This is the same configuration first considered in Horowitz:1998ha , and henceforth we will refer to this case as “the” AdS soliton. To the best of our knowledge, this connection between the Eguchi-Hanson-AdS₅ soliton and the AdS soliton has not been previously reported on¹¹1The connection between the Eguchi-Hanson and AdS soliton geometries could be inferred from the results for lensed CFT partition functions Shaghoulian:2016gol . We are grateful to Edgar Shaghoulian who, after this work was completed, brought this reference to our attention. .

Regularity of the solution requires that $p\geq 3$ is an integer. Therefore, the $p\to\infty$ limit may be most cautiously considered as a ‘formal’ limit. Nonetheless, it is difficult to overstate the utility of this result. As we will see in the subsequent sections, many of the quantities of interest cannot be evaluated exactly for the Eguchi-Hanson-AdS₅ soliton, but the asymptotics of these quantities can be effectively captured by the corresponding quantities for the AdS soliton. Said another way, Eguchi-Hanson-AdS₅ solitons for large values of $p$ behave in a manner similar to the AdS soliton.

While it is well-known that black holes satisfy a first law and Smarr relation, similar relationships can be found for solitons and soliton-black hole configurations. This was rigorously demonstrated in the asymptotically flat case in Kunduri:2013vka , and extended to a particular example of an asymptotically globally AdS soliton in Andrews:2019hvq . Here we apply these considerations to the Eguchi-Hanson-AdS₅ soliton. While the mass was calculated in the original manuscript Clarkson:2006zk , the notion of thermodynamic volume Kastor:2009wy — which proves crucial for deriving the Smarr relation and first law in this case — was at that point not developed. We note also that considerations of extended thermodynamics have been previously carried out for Eguchi-Hanson-dS soliton in Mbarek:2016mep .

Let $\xi=\partial_{t}$ be the stationary Killing field. It has zero divergence so $d\star\xi=0$, where $\star$ is the Hodge dual. It follows that one can write the closed four-form $\star\xi=-d\star\varpi$ for some locally defined 2-form $\varpi$, or equivalently

$$\xi=\star d\star\varpi.$$

(12)

On the other hand a basic identity is

$$d\star d\xi=2\star\text{Ric}(\xi)=\frac{8}{\ell^{2}}d\star\varpi$$

(13)

where the second equality follows from the Einstein equation $R_{ab}=-\tfrac{4}{\ell^{2}}g_{ab}$ and (12). This means we have the conservation equation

$$d\star\left[d\xi-\frac{8}{\ell^{2}}\varpi\right]=0$$

(14)

which we will integrate over a spatial hypersurface $t=$constant. If we introduce the basis

	$\displaystyle e^{0}$	$\displaystyle=\sqrt{g}dt,\qquad e^{1}=\frac{dr}{\sqrt{fg}},\qquad e^{2}=r\sqrt{f}\left(d\psi+\frac{\cos\theta}{2}d\phi\right),$		(15)
	$\displaystyle e^{3}$	$\displaystyle=\frac{r}{2}d\theta,\qquad e^{4}=\frac{r}{2}\sin\theta d\phi$

and assume $\varpi$ takes the form

$$\varpi=A(r)e^{0}\wedge e^{1}+B(r)e^{1}\wedge e^{2}$$

(16)

then a calculation gives

$$A(r)=\frac{1}{\sqrt{f}}\left(\frac{r}{4}+\frac{C_{1}}{r^{3}}\right),\qquad B(r)=\frac{C_{2}}{r^{2}\sqrt{g}}$$

(17)

so that

$$\varpi=\frac{1}{f}\left(\frac{r}{4}+\frac{C_{1}}{r^{3}}\right)dt\wedge dr+\frac{C_{2}}{rg}dr\wedge\left(d\psi+\frac{\cos\theta}{2}d\phi\right).$$

(18)

whose Hodge dual is

$$\star\varpi=-\frac{1}{4}\left(\frac{r^{4}}{4}+C_{1}\right)\sin\theta d\psi\wedge d\theta\wedge d\phi+\frac{C_{2}}{4}\sin\theta dt\wedge d\theta\wedge d\phi.$$

(19)

Note that the degeneracy of $\partial/\partial\psi$ at the ‘centre’ $r=a$ implies that $\star\varpi$ is not well defined there unless $C_{1}$ is chosen to be $C_{1}=-a^{4}/4$. However, as is typical — and as we will see below — regularity is not the correct prescription for fixing the parameter $C_{1}$.

Next, we integrate the closed form defined by (14) over a hypersurface $\Sigma$ defined as a surface of constant time, $t=$ constant, over the region $R_{0}\leq r\leq\infty$ with $R_{0}>a$. On this region $\varpi$ is well-defined and we can apply Stokes’ theorem. The identity gives

$$0=\int_{\Sigma}d\left[\star\left(d\xi-\frac{8}{\ell^{2}}\varpi\right)\right]=\int_{\partial_{\infty}\Sigma}\star\left(d\xi-\frac{8}{\ell^{2}}\varpi\right)-\int_{\partial\Sigma_{R_{0}}}\star\left(d\xi-\frac{8}{\ell^{2}}\varpi\right)$$

(20)

where $\partial_{\infty}\Sigma$ and $\partial\Sigma_{R_{0}}$ represent the asymptotic and inner boundaries respectively.

Let us first focus on the contribution at conformal infinity. A calculation shows that as $r\to\infty$,

$$\star d\xi=\left(-\frac{r^{4}}{2\ell^{2}}+\frac{a^{4}}{2\ell^{2}}+O(1/r^{2})\right)\sin\theta d\psi\wedge d\theta\wedge d\phi\,,$$

(21)

and we note that the divergent term is precisely cancelled by the corresponding divergent term of $\star\varpi$ when the two terms are combined as in (20). We identify a renormalized Komar mass as Kastor:2009wy

$$M_{\rm Komar}:=-\frac{3}{32\pi}\int_{\partial_{\infty}\Sigma}\star\left(d\xi-\frac{8}{\ell^{2}}\varpi\right)=-\frac{3\pi}{8p\ell^{2}}\left(a^{4}+4C_{1}\right)\,.$$

(22)

The fact that the free parameter $C_{1}$ appears in the Komar mass can be understood as an ambiguity in the ground state energy. We will now fix this ambiguity.

We can ensure the integral over the asymptotic boundary evaluates to the mass by choosing $C_{1}$ appropriately. To calculate this we use the Ashtekar-Magnon procedure Ashtekar:1984zz which is well-defined in this setting. The relevant component of the Weyl tensor is

$$C^{t}_{~{}rtr}=-\frac{a^{4}}{(r^{2}+\ell^{2})(r^{4}-a^{4})}=-\frac{a^{4}}{r^{6}}+O(r^{-8})$$

(23)

as $r\to\infty$. Setting $\Omega=\ell/r$ and defining the conformal metric $\bar{g}_{ab}=\Omega^{2}g_{ab}$ with $\Omega=0$ as $r\to\infty$, the Ashtekar-Magnon mass is then defined as

$$Q[\partial_{t}]=\frac{\ell}{16\pi}\int_{\partial M}\bar{\mathcal{E}}^{a}_{~{}b}(\partial_{t})^{b}dS_{t}$$

(24)

where $dS_{t}$ is a constant time slice of the conformal boundary, which has a round lens space metric of radius $\ell$. The quantity $\bar{\mathcal{E}}^{a}_{~{}b}$ is the electric part of the Weyl tensor

$$\bar{\mathcal{E}}^{a}_{~{}b}=\frac{\ell^{2}}{\Omega^{2}}\bar{g}^{cd}\bar{g}^{ef}n_{d}n_{f}C^{a}_{~{}cbe},$$

(25)

with unit spacelike normal $n=d\Omega$.

Noting that $\bar{g}^{rr}=r^{4}\ell^{2}$, it is then a straightforward matter to obtain the relevant component

$$\bar{\mathcal{E}}^{t}_{~{}t}=\frac{r^{6}}{\ell^{6}}C^{t}_{rtr}=-\frac{a^{4}}{\ell^{6}}$$

(26)

yielding

$$M:=Q[\partial_{t}]=\frac{\ell}{16\pi}\int_{\partial M}\left(-\frac{a^{4}}{\ell^{6}}\right)\ell^{3}\frac{\sin\theta}{4}d\psi d\theta d\phi=-\frac{\pi a^{4}}{8\ell^{2}p}$$

(27)

so the mass is negative, a fact already observed in Clarkson:2006zk — c. f. eq.(10) of that work. Comparing the above with the result of the Komar integration implies $C_{1}=-a^{4}/6$.

Once the Komar mass has been computed, the thermodynamic volume is identified by evaluating the integral of the Killing potential over the inner boundary, and taking the limit $R_{0}\to a$. This gives:

$$V=-\int_{\partial\Sigma_{R}}\star\varpi=\frac{\pi^{2}}{2p}\left(a^{4}+4C_{1}\right)=\frac{\pi^{2}a^{4}}{6p}\,.$$

(28)

using the choice $C_{1}=-a^{4}/6$. For some black hole solutions the thermodynamic volume can be interpreted as the volume of a Euclidean ball of radius $a$ that is removed from the spatial hypersurface Kubiznak:2016qmn . This interpretation is not available in this case because there is no ‘ball’ in Euclidean space for which a lens space is its boundary. In this case, the thermodynamic volume $V$ has a topological origin: it arises not due to the presence of an horizon, but instead because the choice of constant $C_{1}$ that leads to the correct mass leads to a Killing potential that is not regular at the location of the bubble.

Noting that thermodynamic pressure is Kubiznak:2016qmn

$$P:=-\frac{\Lambda}{8\pi}=\frac{3}{4\pi\ell^{2}}$$

(29)

then we have

$$M=-PV$$

(30)

which is the Smarr formula for this system. Using the regularity condition (6) and the mass (27) we have

$$dM=-\frac{\pi}{4p}\left(\frac{p^{2}}{4}-1\right)^{2}\ell d\ell$$

(31)

Alternatively, using (28), (29), and the regularity condition (6), we have

$$VdP=-\frac{\pi}{4p}\left(\frac{p^{2}}{4}-1\right)^{2}\ell d\ell$$

(32)

and consequently

$$dM=VdP$$

(33)

which is the expression of the first law for the Eguchi-Hanson-AdS₅ soliton.

The fact that the regularity condition is required for the validity of the first law is consistent with previous studies of (extended) mechanics of smooth geometries Kunduri:2013vka ; Gunasekaran:2016nep ; Andrews:2019hvq . It is worth remarking that in some cases, e.g., for accelerating black holes or spacetimes containing Misner strings, it is possible to formulate the Smarr relation and first law without requiring the regularity condition to hold Appels:2016uha ; Bordo:2019tyh . It may be interesting to better understand when, exactly, regularity of the geometry is crucial for formulating a sensible first law and Smarr relation.

By sending $t\to i\tau$ the Eguchi-Hanson-AdS₅ solution (3) may be analytically continued to produce a Riemannian (positive signature) Einstein metric:

$$ds^{2}=g(r)d\tau^{2}+r^{2}f(r)\left[d\psi+\frac{\cos\theta}{2}d\phi\right]^{2}+\frac{dr^{2}}{f(r)g(r)}+\frac{r^{2}}{4}d\Omega^{2}_{2}.$$

(34)

The geometry is smooth and complete with an $S^{2}$-bolt at $r=a$ provided the regularity condition (7) is imposed. We now periodically identify the $\tau$-coordinate as $\tau\sim\tau+\beta$ so that it parameterizes an $S^{1}$. The vector field $\partial_{\tau}$ is nowhere vanishing since $g(r)>0$ and therefore this $S^{1}$ does not degenerate. In particular there is no condition on $\beta$. The underlying manifold will therefore be $S^{1}\times T^{*}S^{2}$ (the latter factor being the cotangent bundle of $S^{2}$). The metric is conformally compact with conformal boundary $S^{1}\times L(p,1)$ equipped with the conformal boundary metric

$$\gamma=d\tau^{2}+\ell^{2}\left[\left(d\psi+\frac{\cos\theta}{2}d\phi\right)^{2}+\frac{d\Omega_{2}^{2}}{4}\right].$$

(35)

The metric on the boundary $L(p,1)$ is the round metric.

We may easily produce a Riemannian Einstein metric with the same conformal boundary by taking appropriate angular identifications of the Euclidean Schwarzschild-AdS₅ metric to obtain an Einstein metric on $\mathbb{R}^{2}\times L(p,1)$:

$\displaystyle ds^{2}$

$\displaystyle=U(r)d\tau^{2}+U(r)^{-1}dr^{2}+r^{2}\left[\left(d\psi+\frac{\cos\theta}{2}d\phi\right)^{2}+\frac{d\Omega_{2}^{2}}{4}\right]$

(36)

where $U(r)=1-\mu/r^{2}+r^{2}/\ell^{2}$. We take, as above, $\psi\sim\psi+2\pi/p$ and $\theta\in(0,\pi)$, $\phi\in(0,2\pi)$ and $r>r_{+}$ where $r_{+}$ is the largest root of $U(r)$. As is well known, regularity at $r=r_{+}$, the largest root of $U(r)$, requires that the angle $\tau$ must be identified as $\tau\sim\tau+\beta$ with

$$\beta=\frac{2\pi\ell^{2}r_{+}}{2r_{+}^{2}+\ell^{2}}.$$

(37)

Thus for fixed temperature $T=\beta^{-1}$ there are two possible black holes

$$r_{+}=\frac{\pi\ell^{2}\pm\ell\sqrt{\pi^{2}\ell^{2}-2\beta^{2}}}{2\beta}$$

(38)

provided $T>T_{min}$ where

$$T_{min}=\frac{\sqrt{2}}{\pi\ell}.$$

(39)

Note that rather than closing smoothly to an $S^{1}\times S^{2}$ ‘bolt’ as in Euclidean Eguchi-Hanson-AdS₅, the above space has a $L(p,1)$ bolt.

We now follow the standard procedure Emparan:1999pm ; Balasubramanian:1999re to compare the finite Euclidean on-shell actions for these two possible infilling metrics for fixed temperature $T=\beta^{-1}$. The renormalized Euclidean action is

$$I=-\frac{1}{16\pi G}\left[\int_{M}\left(R_{g}+\frac{12}{\ell^{2}}\right)d\text{Vol}(g)+2\int_{\partial M}\left(\text{Tr}K-\frac{3}{\ell}-\frac{\ell R_{h}}{4}\right)d\text{Vol}(h)\right]$$

(40)

where $h$ is the metric induced on a hypersurface $r=R$ and $R_{g},R_{h}$ are the respective scalar curvatures of the metrics $g$ and $h$.

For Eguchi-Hanson-AdS₅ we have

$$I_{EH}=\frac{\beta\ell^{2}\text{Vol}(L(p,1))}{16\pi G}\left[\frac{3}{4}-\left(\frac{p^{2}}{4}-1\right)^{2}\right]$$

(41)

where $\text{Vol}(L(p,1))=2\pi^{2}/p$ is the volume of the boundary $L(p,1)$. For the Euclidean black hole metric (36) a computation gives

$$I_{BH}=\frac{\beta\ell^{2}\text{Vol}(L(p,1))}{16\pi G}\left[\frac{3}{4}+\frac{r_{+}^{2}}{\ell^{2}}\left(1-\frac{r_{+}^{2}}{\ell^{2}}\right)\right].$$

(42)

With the actions at hand, simple calculations reveal that the situation is analogous to the Hawking-Page transition Hawking:1982dh . At low temperature the Eguchi-Hanson-AdS₅ soliton has the least action and dominates the canonical ensemble, whereas at sufficiently large temperatures, it is a large black hole that dominates. When the actions are equal there is a transition analogous to the Hawking-Page transition. The temperature at which the phase transition occurs can be shown to be

$$T_{\rm EHB}=\frac{1}{2\pi\ell}\frac{4+\sqrt{p^{4}-8p^{2}+20}}{\sqrt{2+\sqrt{p^{4}-8p^{2}+20}}}\,.$$

(43)

At large values of $p$ this has the asymptotic form

$$T_{\rm EHB}=\frac{p}{2\pi\ell}\left[1+\frac{1}{p^{2}}+\frac{5}{2p^{4}}+\cdots\right]\,.$$

(44)

In the strict $p\to\infty$ limit, this phase transition is related to that which occurs between the toroidal AdS black hole and the AdS soliton Surya:2001vj .

Let us begin with a consideration of time-like geodesics, i.e. $M\neq 0$. To illustrate some similarities/differences with time-like geodesics in AdS, we will restrict attention here to radial time-like geodesics. After defining $\mathcal{E}=E/M$ and rescaling the affine parameter accordingly, we arrive at the equation

$$\dot{r}^{2}=f(r)\left[\mathcal{E}^{2}-g(r)\right]\,.$$

(58)

We immediately see that the large$-r$ turning point of the motion is exactly the same as it is for AdS²²2Of course, the radial coordinate for the soliton is not the same as the radial coordinate for pure AdS. These differences disappear at sufficiently large $r$, as can be confirmed by putting the metric into Fefferman-Graham form. What we mean here is that the functional form of $r_{\rm max}$ is identical.:

$$r_{\rm max}=\ell\sqrt{\mathcal{E}^{2}-1}\,.$$

(59)

This result is sensible — the space is asymptotically locally AdS and so at large enough distances the radial motion should approach that of AdS. There is a further constraint to consider since the only physically relevant cases are those for which $r_{\rm max}\geq a$. This in turn enforces that the energy must be larger than a given threshold,

$$\mathcal{E}\geq\mathcal{E}_{\rm min}=\frac{p}{2}\,,\quad\text{where}\quad\frac{a}{\ell}=\sqrt{\frac{p^{2}}{4}-1}\,.$$

(60)

A second turning point arises due to the presence of the bubble, this is at $r=a$, where $f(r)=0$. The motion of a massive particle is therefore oscillatory, bouncing back and forth on the interval $a\leq r\leq r_{\rm max}$.

The presence of the bubble has implications for the motion at smaller values of $r$. To highlight this, consider the acceleration

$$\ddot{r}=-\frac{r}{\ell^{2}}-\frac{a^{4}}{r^{3}\ell^{2}}+\frac{2a^{4}(\mathcal{E}^{2}-1)}{r^{5}}$$

(61)

where the first term on the right-hand side is the acceleration term present in pure AdS. This makes manifest the well-known fact that the motion of time-like geodesics in AdS is periodic with period $2\pi\ell$. Further, note that the sign of the acceleration in pure AdS is always negative — a stone tossed in AdS will always be returned to sender.

The additional acceleration terms for $a\neq 0$ have interesting consequences. The last term in the above always makes a positive contribution, since $\mathcal{E}\geq\mathcal{E}_{\rm min}>1$. Close to the bubble this positive term actually dominates, leading to a region where the acceleration is positive, indicating a repulsion.

It is a simple matter to prove this. First, note that sufficiently large $r$, the leading AdS term will always dominate, meaning that $\ddot{r}<0$ at large $r$. Next, let us consider the possibility of solutions to the equation $\ddot{r}=0$. Define a new variable $r=\ell(x+\alpha)$ where $\alpha=a/\ell$. Then, the equation $\ddot{r}=0$ becomes equivalent to the polynomial equation³³3While an explicit solution for $x$ in this equation can be obtained, it is sufficiently complicatedly that it is not beneficial to present it here.

$$x^{6}+6\alpha x^{5}+15\alpha^{2}x^{4}+20\alpha^{3}x^{3}+16\alpha^{4}x^{2}+8\alpha^{5}x+2\alpha^{4}\left(1-\mathcal{E}^{2}+\alpha^{2}\right)=0\,.$$

(62)

Every term in this polynomial is manifestly positive except for the very last one. It will vanish for

$$\mathcal{E}=\mathcal{E}_{\rm min}=\frac{p}{2}\,.$$

(63)

For $\mathcal{E}>\mathcal{E}_{\rm min}$ the last term is negative. In this case, applying Descartes’ rule of signs tells us that there will be a single positive value of $x$ where the above polynomial has a zero. There are no zeros for positive $x$ under other circumstances. Undoing our substitutions, $x>0$ implies $r>a$. Thus, we have concluded that there is exactly one zero for the acceleration for $r>a$. Since we know from the above analysis that $\ddot{r}<0$ for sufficiently large $r$, we then conclude that in a neighbourhood of the bubble the acceleration is positive for particles with $\mathcal{E}>p/2$. Since this corresponds to the minimum possible energy, it follows that all massive particles feel a repulsion in the vicinity of the bubble, except those for which $\mathcal{E}$ is exactly $\mathcal{E}_{\rm min}$ as these particles just sit at the bubble without motion.

The extent of this repulsion is bounded and approaches a constant, as can be seen by expanding the acceleration in the vicinity of the bubble:

$$\ddot{r}=\frac{(4\mathcal{E}^{2}-p^{2})}{\ell\sqrt{p^{2}-4}}+\mathcal{O}(r-a)\,.$$

(64)

The origin of this effect is the negative mass of the bubble. To see this, it is instructive to re-write the expression for the acceleration in terms of the mass given in (27). It is a simple matter to show that it then takes the form,

$$\ddot{r}=-\frac{r}{\ell^{2}}-\frac{8pM}{\pi}\frac{\left(2r_{\rm max}^{2}-r^{2}\right)}{r^{5}}\,,$$

(65)

where $r_{\rm max}$ was introduced in eq. (59). Since $r\leq r_{\rm max}$ we see directly that if it were possible to have positive mass, then the acceleration would always be attractive. However, since the mass is necessarily negative, there is a competition between the confining potential of AdS and the negative mass repulsion of the bubble. This leads to a thin layer in the vicinity of the bubble where massive particles find themselves accelerated away from the bubble.

Finally, it is important to emphasize that the repulsion does not result in a situation where a (positive energy) particle ‘hovers’ at some fixed position $r>a$. When $\mathcal{E}>\mathcal{E}_{\rm min}$ the velocity is necessarily non-zero at the point where the acceleration vanishes. The only case when it is possible for $\dot{r}=\ddot{r}=0$ simultaneously is when $\mathcal{E}=\mathcal{E}_{\rm min}$, corresponding to a particle at the location of the bubble.

The (radial) motion of massive particles in Eguchi-Hanson-AdS₅ is therefore qualitatively similar to the motion of massive particles in AdS. The motion is forever oscillatory, with particles confined in some layer surrounding the bubble, the thickness of which depends on the energy of the particle. Unsurprisingly, this is suggestive that Eguchi-Hanson-AdS₅ may suffer from a similar non-linear instability as global AdS, however, here without the possibility of forming horizons Dold:2017hwr .

Consider, now, null geodesics. These are obtained by setting $M=0$ in (56), yielding

$$\dot{r}^{2}=4f(r)-\frac{16g(r)\eta^{2}}{r^{2}}-\frac{16\hat{C}}{r^{2}}f(r)g(r)$$

(66)

where $\eta:=p_{\psi}/E$ and $\hat{C}=C/E^{2}$ and we have performed an appropriate rescaling of the affine parameter. For convenience, we will work with dimensionless parameters by scaling out $\ell$ (this is equivalent to fixing $\ell=1$). Doing so produces

$$\dot{r}^{2}=V(x)=\frac{4}{x^{3}}P(x),\qquad P(x)=c_{3}x^{3}+c_{2}x^{2}+c_{1}x+c_{0}.$$

(67)

where $x=r^{2}$ and

$$c_{3}=1-4\hat{C}-4\eta^{2},\quad c_{2}=-4(\hat{C}+\eta^{2}),\quad c_{1}=-a^{4}(1-4\hat{C}),\quad c_{0}=4a^{4}\hat{C}.$$

(68)

Trajectories are only allowed in regions where the effective potential $V(r)>0$ with turning points at the zeroes, whereas regions with $V(r)<0$ are forbidden. Stable trapping will occur if there exist $x_{1},x_{2}$ such that $0<a^{2}\leq x_{1}<x_{2}$ with $V(x_{i})=0$, $V(x)>0$ for $x\in(x_{1},x_{2})$, and $V(x)<0$ in neighbourhoods to the left of $x_{1}$ and right of $x_{2}$ (see (Gunasekaran:2020pue, , Fig 2)). This translates into similar conditions on the cubic $P(x)$. First note that $\hat{C}\geq 0$ by definition. Observe that

$$P(0)=4a^{4}\hat{C}\geq 0,\qquad P(a^{2})=-4a^{4}(1+a^{2})\eta^{2}\leq 0.$$

(69)

We now consider several distinct cases.

In AdS, a light ray sent from a given point completes a round-trip to infinity and back in finite time coordinate time. Taking this point to be the orign, we have

$$T_{\rm AdS}=2\int_{r=0}^{r=\infty}\frac{\dot{t}}{\dot{r}}=2\int_{0}^{\infty}\frac{1}{g(r)}=\pi\ell\,.$$

(78)

This in turn defines a fundamental frequency naturally associated with AdS:

$$\omega_{\rm AdS}\ell=\frac{2\pi\ell}{T_{\rm AdS}}=2\,.$$

(79)

This is relevant because the fundamental frequency matches the spacing for the overtones of scalar normal modes in AdS. Since we will study scalar normal modes in the next section, it is relevant to understand if a similar effect occurs for the soliton.

The computation for the light-crossing time in the soliton proceeds along the same lines. The integral to evaluate is now

$$T_{\rm p}=2\int_{a}^{\infty}\frac{1}{g(r)\sqrt{f(r)}}dr\,,$$

(80)

where here the subscript $p$ refers to the integer defining regularity of the geometry. This integral actually has a closed form expression,

	$\displaystyle\frac{T_{p}}{\ell}$	$\displaystyle=\frac{1}{48\sqrt{2\pi}(p^{2}-4)^{3/2}}\left\{48(p^{2}-4)^{2}\Gamma\left(\frac{3}{4}\right)^{2}\left[{}_{2}F_{1}\left(-\frac{1}{4},1,\frac{1}{4},\frac{16}{(p^{2}-4)^{2}}\right)-1\right]\right.$
		$\displaystyle\left.\,+\Gamma\left(\frac{1}{4}\right)^{2}\left[48(p^{2}-4)+\frac{256}{p\sqrt{p^{2}-8}}{}_{2}F_{1}\left(\frac{1}{2},\frac{3}{4},\frac{7}{4},\frac{16}{(p^{2}-4)^{2}}\right)\right]\right\}\,.$		(81)

Expanding this in different limits is more useful. First, for $p\to 2$, the expansion is

$$\frac{T_{p}}{\ell}=\pi-\sqrt{\frac{2}{\pi}}\Gamma\left[\frac{3}{4}\right]^{2}\sqrt{p-2}-\frac{4\pi\Gamma\left[1/4\right]+3\sqrt{2}\Gamma[3/4]^{3}}{24\sqrt{\pi}\Gamma[3/4]}(p-2)^{3/2}+\frac{\pi}{2}(p-2)^{2}+\mathcal{O}(p-2)^{5/2}\,.$$

(82)

The other limit of interest where this can be expanded is $p\to\infty$. In this case the expansion reads,

$$\frac{T_{p}}{\ell}=\frac{\Gamma[1/4]^{2}}{\sqrt{2\pi}p}+\sqrt{\frac{2}{\pi}}\frac{\Gamma[1/4]^{2}-8\Gamma[3/4]^{2}}{p^{3}}+\frac{3\left(17\Gamma[-3/4]^{2}-256\Gamma[3/4]^{2}\right)}{8\sqrt{2\pi}p^{5}}+\mathcal{O}(p^{-6})\,.$$

(83)