The zero mass limit of Kerr and Kerr-(anti-)de Sitter space-times: Revisited

TL;DR

This paper revisits the zero mass limit of Kerr and Kerr-(anti-)de Sitter space-times, analyzing wave equations, reflection coefficients, and Hawking radiation, finding no evidence of Hawking radiation in de Sitter geometry.

Contribution

It investigates the zero mass limit of Kerr-(anti-)de Sitter space-times, focusing on wave solutions, reflection coefficients, and Hawking radiation, providing new insights into their properties.

Findings

No Hawking radiation detected in de Sitter geometry

Polynomial solutions for radial wave equations identified at special frequencies

Reflection coefficients computed for waves near horizons

Abstract

We continue studying the zero mass limit of the Kerr- (Anti) de Sitter space-times by investigating the possibility of special values of the frequencies to have polynomial solutions for the radial wave equation, compute reflection coefficient for waves coming from and going to the cosmological horizon and see whether there is Hawking radiation for de Sitter geometry. We used the Parikh-Wilczek method to study this property and could not find Hawking radiation.

Equations

$$\displaystyle\frac{d}{dr}\bigg{(}(r^{2}+a^{2})D\frac{dF(r)}{dr}\bigg{)}+\bigg{(}\frac{3\Xi\omega^{2}}{\Lambda D}-\mu^{2}r^{2}+\frac{m^{2}a^{2}\Xi}{r^{2}+a^{2}}+\lambda\bigg{)}F(r)=0,$$

$$\displaystyle\frac{1}{\sin\theta}\frac{d}{d\theta}\bigg{(}\sin{\theta}C\frac{dG(\theta)}{d\theta}\bigg{)}+\bigg{(}\frac{3\Xi\omega^{2}}{\Lambda C}+\mu^{2}a^{2}\cos^{2}{\theta}+\frac{m^{2}\Xi}{\sin^{2}\theta}+\lambda\bigg{)}G(\theta)=0.$$

$$\displaystyle\frac{d^{2}H_{G}(x)}{dx^{2}}+\bigg{(}\frac{\delta}{x-1}+\frac{\epsilon}{x-b}+\frac{{\gamma}}{{x}}\bigg{)}{\frac{{dH_{G}(x)}}{{dx}}}+\bigg{(}\frac{{\alpha\beta x-q}}{{x(x-1)(x-b)}}\bigg{)}H_{G}(x)=0.$$

$$\displaystyle\frac{d^{2}G(u)}{du^{2}}+\bigg{(}\frac{1}{u-1}+\frac{1}{u+\frac{3}{a^{2}\Lambda}}+\frac{1}{2u}\bigg{)}\frac{dG(u)}{du}$$

$$\displaystyle+\frac{\frac{3}{a^{2}\Lambda}}{4u(u+\frac{3}{a^{2}\Lambda})(u-1)}\bigg{(}\frac{(1+\frac{3}{a^{2}\Lambda})\frac{3}{4}\omega^{2}}{\Lambda(u+\frac{3}{a^{2}\Lambda})}+\mu^{2}a^{2}u+\frac{m^{2}(1+\frac{3}{a^{2}\Lambda})}{(u-1)}+\lambda\bigg{)}G(u)=0.$$

$$G(x)=x^{1/2}(x-1)^{m/2}\big{(}x+{\frac{{3}}{{a^{2}\Lambda}}}\big{)}^{\frac{i\omega}{2}\sqrt{3/\Lambda}}H_{G}(a,q,\alpha,\beta,\gamma,\delta;x).$$

$$G(u)=(u-1)^{\pm{m/2}}\big{(}u+\frac{3}{a^{2}\Lambda}\big{)}^{\frac{{\pm}i\omega}{2}\sqrt{3/\Lambda}}H_{G}(b,q;\alpha,\beta;\gamma,\delta;u).$$

$$b$$

$$-q$$

$$\alpha$$

$$\beta$$

$$\gamma$$

$$\delta$$

$$u$$

$$G(x)=u^{1/2}(u-1)^{m/2}\big{(}u+{\frac{{3}}{{a^{2}\Lambda}}}\big{)}^{\frac{i\omega}{2}\sqrt{3/\Lambda}}H_{G}(c,Q,A,B,\Gamma,\Delta;u).$$

$$c$$

$$-Q$$

$$A$$

$$B$$

$$\Gamma$$

$$\Delta$$

$$u$$

$${\alpha }^{\prime }=\frac{1}{2}(i\omega \sqrt{3/\Lambda }+m)$$

$${\beta }^{\prime }=\frac{1}{2}(i\omega \sqrt{3/\Lambda }+m+3).$$

$$A=\frac{1}{2}(i\omega \sqrt{3/\Lambda }+m+1)$$

$$B=\frac{1}{2}(i\omega \sqrt{3/\Lambda }+m+4).$$

$$A=\frac{1}{2}(i\omega \sqrt{3/\Lambda }+m+1)=-n.$$

$$u^{1/2}(u-1)^{m/2}\big{(}u+{\frac{3}{a^{2}\Lambda}\big{)}^{-n-\frac{1+m}{2}}}H_{G}(a,Q,A,B,\Gamma,\Delta;u).$$

$$H_G(u)=D_1{H}_G(u-b)+(u-b{)}^{(\gamma +\delta -\alpha -\beta )}{D}_2{H}_G(u-b).$$

$$H_g(u)=D_1{H}_G(u+\frac{3}{a^2\Lambda })+(u+\frac{3}{a^2\Lambda }{)}^{\kappa }{D}_2{H}_G(u+\frac{3}{a^2\Lambda }),$$

$$y_i(u)=u^{1/2}(u-1{)}^{m/2}(u+\frac{3}{a^2\Lambda }{)}^{\frac{i\omega }{2}\sqrt{3/\Lambda }}{F}_i(u)$$

$$F_1(u)=H_G(-\frac{3}{a^2\Lambda },q_1;K,J;\frac{3}{2},m+1;u),$$

$$J=\frac{1}{2}((1+m+3+i\omega \sqrt{3/\Lambda }),$$

$$K=\frac{1}{2}(1+m+i\omega \sqrt{3/\Lambda }),$$

$$q_{1}=\frac{1}{4}\big{(}-2\frac{3}{a^{2}\Lambda}+3i\omega\sqrt{{3/\Lambda}}+2-(\frac{3\omega^{2}}{\Lambda}+m^{2})(1+\frac{3}{a^{2}\Lambda})-(9m+\lambda))\frac{3}{a^{2}\Lambda}\big{)}$$

$$F_2(u)=(u+\frac{3}{a^2\Lambda }{)}^{-i\omega \sqrt{\frac{3}{\Lambda }}}{H}_G(-\frac{3}{a^2\Lambda },q_2;L,E;;\frac{3}{2},m+1;u),$$

$$E=\frac{1}{2}((1+m+3-i\omega \sqrt{3/\Lambda }),$$

Full text

Kerr metric, wave equation, exact solutions

I Introduction

In a very interesting paper Gibbons and Volkov Gibbons have claimed that the mass going to zero limit of the Kerr Kerr metric has wormhole solutions. This claim does not agree with contrary claims Landau ; Visser . In an earlier paper tolga , we gave one explicit solution for a scalar particle coupled to the zero mass of the limit of both the Kerr, Kerr-dS and Kerr-AdS space times, using the wave equation given by Gibbons and Volkov Gibbons . In that work we had confirmed the result of Gibbons and Volkov by explicitly obtaining one of the wave solutions, which has a cut singularity along the z- axis for the angular equation, and at the origin for the radial equation. These solutions brought to attention whether, in this model, one can calculate the scattering at the origin and normal modes.

In this paper, we continue studying the properties of the wave equation using the radial variable $u=-\frac{r^{2}}{a^{2}}$, for a scalar particle in this metric. The interesting aspect of the metric used is that with appropriate choices of the transformation of the radial and polar angle coordinates, we get the same wave equation for both coordinates.

Actually, the wave equation has two solutions. One of the solutions validates the Gibbons-Volkov result, with a singularity at the origin and along the z- axis, which is interpreted as a wormhole in Gibbons . The other solution is smooth at the origin, and at the z- axis. As expected, there is no event horizon for the Kerr case in the zero mass limit of the black hole.

We have to report that there was an unfortunate mistake in eq.(1) of the Correction tolga1 to tolga , where the term without derivative should be divided by $4$. Fortunately this error does not change the essential result given in the solution, given in eq.(19), of tolga .

Here we complete this work by finding the possible normal modes to obtain the polynomial solutions for this equation, studying the possible reflection of waves coming from the cosmological horizon (dS) and from infinity (AdS) towards the origin. In an appendix, we study the solution at infinity, valid for the AdS solution. Since we could not produce a formula for connecting the solution expanded around infinity to the solution expanded around zero, we put this part to Appendix A. We, however, with a transformation, bring the point at infinity to the point at one, and compute the reflection coefficient. In Appendix B, we derive a connecting formula for Heun functions at zero and at an arbitrary finite point. In appendix C, we give solutions which are convergent at two singular points of the solutions for the de Sitter case, by changing the independent variable in the respective equation. The similar analysis is done in the main text for the AdS case.

II 2. Solutions and Normal Modes

As given in Gibbons , if we take the time dependence as $e^{i\omega t}$, and $\phi$ dependence as $e^{-im\phi}$, our wave equation for a scalar particle separates into two equations and read:

[TABLE]

[TABLE]

Here $D=1-{\frac{\Lambda r^{2}}{3}}$, $C=1+{\frac{{\Lambda a^{2}}}{{3}}\cos^{2}\theta}$ and $\Xi=1+{\frac{{\Lambda a^{2}}}{{3}}}$. The independent variables $\theta$ and its functions, and $r$ appear in the angular and radial equations with even powers. Thus, any solution which is valid for positive $\theta$ and $r$ are also valid for their negative values. If we make the transformations $x=\cos^{2}\theta$ and $u=-r^{2}/a^{2}$, we find that we get exactly the same expressions for both the radial and the angular equations.

One may wonder why we define $u$ between zero and minus infinity. We make the transformation for $u$ in this way, just to be able to fit to the standard Heun form Heun ; Ronveaux ; Slavyanov ; Hortacsu ; Tolga .

[TABLE]

Our eq.(1), with the transformation to $u$ reads

[TABLE]

We gave one of the possible solutions in tolga , for $x=\cos^{2}\theta$ which is written as

[TABLE]

Here $H_{G}$ a General Heun function ($H_{G}$) which is multiplied by some monomials to satisfy eq.(2). Thus, using the transformation $x=\cos^{2}\theta$ in eq.(2), we get the needed singularity on the $z$ axis in tolga . The solution without a singularity at the origin for the variable $u$ will not have the square root singularity at the origin, and will have different values for the parameters for the Heun function.

Now we give the two solutions for our Eq. (1) explicitly. One of the solutions have no singularity at the origin. Since they satisfy the same equation, the solutions of both the angular and radial equations differ only in the independent variable used. If one picks this solution, this will be true also for the angular equation, Eq.(2). The other solution has a square root singularity at the origin.

The solution without a singularity at the origin for the variable $u$ reads

[TABLE]

In this solution and in the solution with the singularity given below, $m$ and ${\frac{i\omega}{2}\sqrt{3/\Lambda}}$ may take positive or negative values. In our text, we generally choose the positive value in our expressions involving these entities. The parameters of the above solution are defined as

[TABLE]

Since the $\phi$ dependence is given as $e^{-im\phi}$, $m$ takes only unit values.

The solution with the singularity at the origin is

[TABLE]

The parameters for the equation above are defined as

[TABLE]

Both of the wave equations for $u$ above seem to have a singularity at $(u+\frac{3}{a^{2}\Lambda})=0$. We have to note that, for positive $\Lambda$, $u$ should be more than $-\frac{3}{a^{2}\Lambda}$ Gibbons . At $u=-\frac{3}{a^{2}\Lambda}$, we have the cosmological horizon. For negative $\Lambda$, there is no singularity at $\frac{3}{a^{2}\Lambda}$, no event horizon. Eq.s (4,6) are valid for $u$ from zero to minus infinity.

In order to simplify our solutions, we take the mass of the scalar field $\mu$ equal to zero. When the field is massless, we get

[TABLE]

and

[TABLE]

for the case without the singularity at the origin. When we have a singularity at the origin, the values of $A,B$ increase by one half.

[TABLE]

and

[TABLE]

We may also look for polynomial solutions for our Heun equation for the argument $u$. In Vieira ; Vieira1 , the boundary conditions to have resonant frequencies are given as to have the radial solution to be finite at the horizon, in our case at the origin, and well behaved at asymptotic infinity, which is at the cosmological horizon for the dS or at asymptotic infinity for the AdS cases. Vieira et al. state that to satisfy the condition at asymptotic infinity one needs polynomial solutions. The first requirement for this is to have either $A$ or $B$ equal to $-n$ Arscott1 , i.e. , where

[TABLE]

” These frequencies are the proper modes at which a black hole freely oscillates when excited by a perturbation… Since they are damped by the emission of gravitational waves, the corresponding eigenvalues are complex Sakalli ”. $n$ is the rank of the polynomial. Then our solution for the variable $u$ is given by

[TABLE]

where in $A$ and $B$, we replace $i\omega\sqrt{3/{\Lambda}}$ by ${\frac{-2n-m-1}{2}}$. The other parameter, $B$, is given in eq.(18) above. For the generic case, i.e. the standard form given by our eq. (3), this will be given as $\alpha+\beta+1=\gamma+\delta+\epsilon$. There is a second necessary criterion, which is the vanishing of a determinant given in ciftci ; karayer .

We have to note that, when $A=-n$, we get an imaginary value for $\omega$. Furthermore, the same $\omega$ is used in the time dependence of the solution as $e^{i\omega t}$. We can have not a propagating but a decaying wave, since $i\omega$ does not have an imaginary part.

If we, instead, take the time dependence of the solution as $e^{-i\omega t}$, as given in Gibbons , we have to take negative values for $m$ and have a condition on $n$ as ${-2n-1+|m|}$ should be greater than zero, which makes the highest power of the polynomial also depend on the value of $m$.

III 3. Scattering Coefficient at the Origin

There are two different cases in this section, the de Sitter and anti de Sitter cases. For the de Sitter case, we can use only a finite range of the $u$ coordinate, since we can not take $u<-3/(a^{2}\Lambda)$. Note that $u=-\frac{r^{2}}{a^{2}}$. We can use the connection formula derived in Appendix B for this case, derived, following the method in the paper by Dekar et al.hammann . For the anti de Sitter case, we need a connection formula relating the solution at the origin to the solution at minus infinity. As far as we know, such a connection formula for Heun functions does not exist in literature, and we could not derive one. There is, however, a trick, used by several authors Jaffe ; Philipp ; Lay ; Leaver ; Fiziev . We have to start by changing our variable $u$ to $v$, where this new variable is given by $v=u/(u-1)$. By doing this trick, we bring the singularity at minus infinity to $v=1$. Then, we use the formulae derived in hammann .

III.1 de Sitter case

When $\Lambda$ is positive, we can investigate if we have scattering at the origin, for waves coming from the cosmological horizon. Since the domain of the independent variable $u$ is finite, we do not need a new transformation to limit the domain to a finite interval. Still, we investigate this case, in Appendix C, using the trick of Jaffe ; Philipp ; Lay ; Leaver ; Fiziev , with no new results.

When $\Lambda$ is positive, we can study the scattering of the wave coming from the cosmological horizon at the origin, producing a reflected wave there. We write the wave approaching from the point $u=-\frac{3}{a^{2}\Lambda}$ to the origin. This necessitates writing $F(u)$ in terms of two functions with arguments $u=-\frac{3}{a^{2}\Lambda}$. For the generic case, this will be

[TABLE]

Here $b,\gamma,\delta,\alpha,\beta$ are the generic parameters of General Heun equation given in our eq. (3). We will apply this formalism to our example.

For the scalar field’s mass equal zero case, one can write the Heun solution expanded in powers of $u$, in terms of the two linearly independent Heun solutions expanded in powers of $(u+{\frac{{3}}{{a^{2}\Lambda}}})$:

[TABLE]

where $\kappa$ is chosen to be equal to the value to have the second solution in the Heun form Ronveaux . Dekar et al hammann managed to get the formula necessary to write the $H_{G}(z)$ solution expanded in terms of $z$ in terms of two similar functions, obtained when one expands around $1-z$. In Appendix B, we calculate the coefficients $D_{1}$ and $D_{2}$, applicable for expanding $H_{G}$ in terms of $H_{G}(u+{\frac{{3}}{{a^{2}\Lambda}}})$ for expansions for the generic Heun equation, adapted to our case.

To describe this calculation briefly, we start with our eq. (4), with a mass zero scalar field. For scattering for waves coming from the cosmological horizon, we take the solution, eq.(14), which is non analytic at the origin, and the second solution to this equation.

[TABLE]

Here $i$ will take two values, $1,2$.

For solutions expanded in a power series in terms of $u$, we have

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

The same equation has another solution due to the properties of the Heun equation hammann1 which has the same powers multiplying the $F_{2}$ term, and which is needed in the long calculation to obtain the connection result, given below. This second solution reads,

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

i.e. $q_{2}$ is the complex conjugate of $q_{1}$.

When expanded in terms of $u+\frac{3}{a^{2}\Lambda}$, we get two linearly solutions $F_{j}$ where $j$ takes two values $3,4$.

[TABLE]

where

[TABLE]

and

[TABLE]

where

[TABLE]

i.e. $q_{4}$ is complex conjugate of $q_{3}$.

As stated above, Dekar et al. hammann can calculate the constants necessary to write the General Heun function, which is a function of the variable $z$, in terms of two solutions as functions of the variables $1-z$.

[TABLE]

Here $y_{1}(z),y_{3}(1-z),y_{4}(1-z)$ are General Heun functions with their respective parameters. One can adapt their method to our case.

In Appendix B, using the method of Dekar et al hammann , we calculated $D_{1},D_{2}$ in terms of General Heun functions for the generic case. Now we give the specific values for the case we study here. Here we replaced $D_{1},D_{2}$ by $C_{1},C_{2}$, to stress that they are used not for the generic, but for our specific case. Then, recalling $A=K$, given in eq.s (24,33), and $B=J$, given in eq.s(25,32) and

[TABLE]

[TABLE]

we get

[TABLE]

[TABLE]

Using these relations, one ends up with the result

[TABLE]

When we write this equation in the simplified form

[TABLE]

we see the incoming and outgoing waves. Then the reflection coefficient from the origin reads

[TABLE]

III.2 Anti de Sitter case

For the anti de Sitter case, we start with our original equation, eq.(4), and again take the mass, $\mu$, equal to zero. To limit the domain of dependence to the interval between zero and one, we make to transformation $v=\frac{u}{u-1}$. This transformation will change all the terms in the original equation.

Our eq.(4), with the transformation to $v$ reads

[TABLE]

The new equation is not of the Heun form. Using a new s-homotopic tranformation by multiplying the dependent variable $H$ by $({v-\frac{1}{1+\frac{a^{2}\Lambda}{3}}})^{\kappa}$ and by choosing $\kappa$ accordingly, one can reduce the differential equation to the Heun form. This is necessary to obtain a three term recursion relation for the coefficients in our series solution.

At this point, we want to demonstrate the consequences of using the analytic and non analytic solutions ot the origin. We calculate the reflection coefficients for these two cases. We first study a solution which is smooth at the origin, like our eq. (6). Then we study a solution which is not analytic at the origin,like the one given in our eq. (14) and compare the results.

The analytic solution will read

[TABLE]

where

[TABLE]

The non analytic solution at the origin will read.

[TABLE]

where

[TABLE]

Then we expand our solution in an infinite series, using $\Sigma{a_{n}v^{n}}$, where $v=\frac{u}{1-u}$. We will get three term recursion relations. As stated in Leaver ; Lay , we have to concentrate on the terms multiplying the second derivative. In the ratio test $\frac{a_{n+1}}{a_{n}}$, which must be less than unity as $n$ goes to infinity, the important terms are the coefficients of the second derivative, since all others will be divided by $n^{2}$.

At the end, we find that, we have two choices. Either $\frac{a_{n+1}}{a_{n}}$ goes to unity or to $1+\frac{a^{2}\Lambda}{3}$ as $n$ goes to infinity. We choose the second choice since $\Lambda$ is negative for this case, and $v$ is between zero and minus one. We conclude that if the absolute value of $1+\frac{a^{2}\Lambda}{3}$ is less than unity, this series converges, and we can find a solution which is uniformly convergent both at zero and infinity for the variable $u$ in terms of $v$.

Dekar et al hammann managed to get the formulae necessary to write the $H_{G}(z)$ solution expanded in terms of $z$ in terms of two similar functions, obtained if one expands around $1-z$, when the interval is finite, in their equation [A.15]. Their formula reads

[TABLE]

The parameters in the above equation are same as the standard General Heun equation given in our eq.[3], except $q$ in that equation is equal to $-b$ here. We use these formulae to obtain the Reflection Coefficients for scattering at the origin for both of these cases.

For the analytic case, we get

[TABLE]

For the non analytic case, we get

[TABLE]

We see that these two are distinctly different, which may be interpreted as the existence of a worm hole at the origin for the non analytic case.

Note that the formulae both for the the reflection coefficient and for the connection equation are formal expressions. In the mathematical literature Arscott1 , technically, the solutions, we obtained, are referred as ”local solutions”. These solutions are analytic only within a circle which ranges from the point of expansion up to but excluding the next singularity. We, therefore, do not know if our solutions around $u=0$ are analytic at $u=-\frac{3}{a^{2}\Lambda}$ or u going to infinity and vice versa. We try to get rid of this problem by ensuring that the solution is finite at the ends by adjusting the parameter $\Lambda$ and $a$ in our equations Jaffe ; Philipp ; Lay ; Leaver . The details are at these references. We also give another application of this method to the dS case in Appendix C, where convergence exists for if $\frac{a^{2}\Lambda}{3}$ ,is greater than unity.

If we use polynomial solutions, then the solution will be analytic around all three singular points, $(0,-\frac{3}{a^{2}\Lambda},1)$. Furthermore, we have to find solutions with $\omega$ which satisfy the initial criterion for a polynomial solution which is either $\alpha$ or $\beta$ in the generic equation, eq.(3), equal to a negative integer. In our case, since there is a half integer difference between $\alpha$ and $\beta$, it is not possible to have both our $A$ and $B$ satisfy this criterion. For the dS case, the reflection formula given in eq.(50) may be of value only for polynomial solutions if the independent variable does not satisfy the limitations given.

Note that when $A=-n$ , we will get a decaying solution, which requires $\omega$ to take a negative numerical value, since we have ${\omega}$ as an exponent of one of the monomials multiplying the Heun solution. In general we may have only one of the coefficients $C_{i}$ finite, $i=1,2$ the other infinite, given the no reflection case.

IV 4. Conclusion and Discussion

We study the solutions of the wave equation for a scalar particle in the metric given in Gibbons for the zero mass limit of the Kerr (Anti) de Sitter geometry. We find two solutions, one of which has a square root singularity at the origin, both for the radial and polar angle equations, which may be interpreted as the presence of a worm hole. The other solution is smooth at the origin. We find that these solutions may have polynomial solutions, since they satisfy the first criterion given in Arscott1 for special values of frequency, if they also satisfy the second criterion ciftci ; karayer . We also write the reflection coefficient for waves coming from the cosmological horizon (dS) and from infinity at the origin.

In Appendix A, we calculate the shape of one of the solutions as $u$ goes to infinity, which is a finite phase, not infinite or zero. The second solution vanishes as $u$ goes to infinity.

V Acknowledgement

We are grateful to Prof. Nadir Ghazanfari for informing of the the Dekar et al. paper, giving us that paper and for important discussions. We are also grateful to Prof. Ibrahim Semiz for providing us a copy of the Leaver paper. We thank Prof. Cemsinan Deliduman for informative discussions. We also thank Prof. Tolga Birkandan for collaboration in earlier work on this subject tolga , and for calculating numerical values for some of our Heun solutions. We also thank Prof. Bekir Can Lütfüoglu for technical assistance. This work is morally supported by the Science Academy, Turkey, an NGO.

VI Appendix A

To study the behaviour of the General Heun equation, given in eq.(3), as $x$ goes to infinity, we transform the variable $x$ to $y$, where $y=\frac{1}{x}$ first on the generic Heun equation,

[TABLE]

to see how the standard equation changes. The above equation is transformed into

[TABLE]

We see that the new equation is not of the Heun form. To put it to the Heun form, we have to multiply the solution by $y^{\kappa}$ where $\kappa$ is adjusted to put the equation to the Heun form. After this is done we end up with the equation

[TABLE]

where the solution of this equation has to be multiplied by $y^{\kappa}$ to be a solution of the former equation. Here $\kappa$ may equal to $\alpha$ or to $\beta$, giving us two solutions. If $\kappa=\alpha$, $\beta$ goes to $\delta+\epsilon-\beta$. If $\kappa=\beta$, $\alpha$ goes to $\delta+\epsilon-\alpha$. We get two solutions:

[TABLE]

and

[TABLE]

If we want the behavior at infinity, we see that the equation behaves as $x^{-\alpha}$ or $x^{-\beta}$ plus terms with higher powers of $1/x$, since for $\frac{1}{x}=y=0$ the Heun solution $H_{y=0}$ is unity.

All through these calculations we used the Heun equation constraint $\alpha+\beta+1=\gamma+\delta+\epsilon$ on the parameters of the standard form in terms of the variable $x$. Actually, this is constraint does not really constrain anything and is a result of using not a single parameter, but the product of two parameters,$\alpha\times\beta$, in the standard form of the General Heun equation given in our eq.(3), in the term multiplying $x$.

To apply this result to our solution when $u$ goes to minus infinity, we first have to put our eq.(4) to the Heun form. This gives us two solutions, eq.s(6,14). Then we make the transformation $u=\frac{1}{s}$ on this transformed differential equation, eq.(14), and put the resulting equation to the Heun form to obtain the solution

[TABLE]

where $H_{G}$ is given as

[TABLE]

When $s$ equals zero, we see that there is only a constant term, a phase, equal to $e^{i\pi\frac{m}{4}}(\frac{a^{2}\Lambda}{3})^{-i\omega\sqrt{3/{\Lambda}}}$. The monomials multiplying $H_{G}$ cancel out and $H_{G}$ equals unity for $s=0$. Since the Heun function goes to unity at $s=0$, this validates our result given above. We see that the solution is smooth as $u$ goes to infinity. Here we used only the solution with $\alpha$. There is another solution which goes to zero for $s=0$, which corresponds to the solution given in eq.(63). The solution with $\beta$ has a square root singularity at $s=0$.

In the main text, we use the method of Jaffe ; Lay ; Philipp ; Leaver in addition to the Dekar et al hammann to study scattering of a wave coming from infinity.

VII Appendix B

Dekar et al could write the General Heun solution for the variable $z$ in terms of two independent solutions for the variable $1-z$ hammann as

[TABLE]

where

[TABLE]

,

[TABLE]

[TABLE]

Here

[TABLE]

Here $C_{1}$ and $C_{2}$ are constants that depend on the parameters of the $H_{G}$ function and $y_{1}$ satisfies the General Heun equation,

[TABLE]

By using a clever trick they can evaluate the constants $C_{1}$ and $C_{2}$ as

[TABLE]

and

[TABLE]

One can use the same method to write

[TABLE]

in terms of two independent solutions expanded around $z-a$.

[TABLE]

where $D_{1},D_{2}$ are functions of the parameters of the $H_{G}$ function on the left hand side of the above equation. Here

[TABLE]

and

[TABLE]

where

[TABLE]

Then we write the equality

[TABLE]

An equality similar to this is given in hammann for expansion around $1-z$ with the proper citation. Then we use the expansion given in our eq.(75) where we write $D_{i}$ with proper indices.

[TABLE]

Then we note that if $\gamma+\delta$ is greater than $\alpha+\beta$, we get

[TABLE]

, since when $z=a$, $H_{G}(1-a,q-a\alpha\beta;\alpha,\beta;1+\alpha+\beta-\gamma-\delta;\delta;0)=1$. We, then, let $\alpha$ go to $\gamma+\delta-\alpha$ and $\beta$ go to $\gamma+\delta-\beta$ in eqs. (79,80). We get

[TABLE]

and

[TABLE]

where

[TABLE]

When we compare eqs.(80) and (83), we see that

[TABLE]

Using eq.(81) and properly translation constants we get

[TABLE]

[TABLE]

We use this result, with appropriate changes in the parameters to fit to our example, in Section 3.

VIII Appendix C

Here we will find another solution for the differential equation given in our equation (4), whose wormhole solution was given in equations ( 14-21). We were not certain that for our eq. (4), a valid solution at one singularity may not diverge at another singularity.

The problem of finding a solution to a differential equation which is not singular at two consecutive singular points was solved by mathematicians long ago. One can find the method and references to original work in Lay . We had used this method to get a converging solution for the AdS case in Section 3 in the main text. Here we do the same thing for the dS case. This calculation dublicates the results given for the dS case, in the main text, since the interval for the independent variable for this case is finite, and only gives a limit on the values of $a^{2}$ and $\Lambda$. To find a solution which is uniformly convergent both at the origin and at the cosmic horizon, $u=-\frac{3}{a^{2}\Lambda}$, we transform to a new variable $z={\frac{u+{\frac{3}{a^{2}\Lambda}}}{1-u}}$. This transformation brings $u=0$ to $z={\frac{3}{a^{2}\Lambda}}$, and $u=-{\frac{3}{a^{2}\Lambda}}$ to $z=0$. We go through similar steps, as in the AdS case, transform the new differential equation to the Heun form by a s-homotopic transformation by multiplying the dependent variable by $(1+z)^{\nu}$. Then we expand our solution as a power series expansion around $z=0$. and find a series solution that converges uniformly if $\frac{a^{2}\Lambda}{3}$ is greater than unity. Thus, one can obtain a convergent solution at the both singular points of our respective domains of interest. The solution will read

[TABLE]

where

[TABLE]

We do not the study the analytic solution for the de Sitter case, thinking that the reflection coefficient will be different from the non analytic case anyway, since the solutions will be different.

Note also that when $u=1$, the remaining singular point is not in our domain of interest, since $u$ was defined as $u=-r^{2}/a^{2}$. When $u=1$, our physically interesting variable $r$ takes complex values.