Low-Energy String Theory Predicts Black Holes Hide a New Universe

A matching hypersurface between two space-times must obey the Israel junction conditions Israel ; Deruelle ; Durrer . The first condition is that the induced metric on the hypersurface is well-defined, that is, both sides of the hypersurface agree on the induced metric. The second condition is that the extrinsic curvature

$$K^{\mu}_{\ \nu}\,=\,q^{(\mu\lambda}\nabla_{\lambda}n_{\nu)},$$

(3)

jumps by an amount given by the surface stress tensor:

$$K^{\mu}_{\ \nu}\big{\rvert}^{+}_{-}\,=\,S^{\mu}_{\ \nu}.$$

(4)

We will now introduce coordinates on the black hole part of the manifold and on the part to the future of the S-brane. We will then apply the Israel junction conditions to determine initial metric of the universe to the future of the S-brane.

We start with the metric of a static black hole such as the Schwarzschild solution. The line element for the black hole is

$$ds^{2}\,=\,\frac{dr^{2}}{g(r)}-f(r)dx^{2}-r^{2}d\Omega^{2},$$

(5)

where $d\Omega^{2}=d\vartheta^{2}+\sin^{2}\vartheta d\phi^{2}$ is the line element of the surface of a sphere, $r$ and $x$ are the Schwarzschild $r$ and $t$ coordinates which are time-like and space-like, respectively, inside the black hole event horizon. The Schwarzschild metric is a special case with

$$f(r)\,=\,g(r)\,=\,\frac{r_{S}}{r}-1\,,$$

(6)

where $r_{S}$ is the Schwarzschild radius. Close to $r=0$, high curvatures will arise and consequently an S-brane will appear. The S-brane in these coordinates arises on a constant $r=r_{0}$ slice. It has the topology of $\mathbb{R}\times S^{2}$. The induced metric on the S-brane hypersurface is

$$q_{\mu\nu}\,=\,Diag\bigg{[}0,-f(r_{0}),-r_{0}^{2},-r_{0}^{2}\sin^{2}\vartheta\bigg{]}_{\mu\nu}$$

(7)

and the extrinsic curvature is

$$K^{\mu}_{\ \nu}=-Diag\bigg{[}0,\frac{\sqrt{g(r_{0})}f^{\prime}(r_{0})}{2f(r_{0})},\frac{\sqrt{g(r_{0})}}{r_{0}},\frac{\sqrt{g(r_{0})}}{r_{0}}\bigg{]}^{\mu}_{\ \nu}.$$

(8)

The universe after the S-brane is described by a Bianchi-type universe with a metric

$$ds^{2}=dt^{2}-a(t)^{2}dx^{2}-b(t)^{2}d\Omega^{2},$$

(9)

where $a(t)$ and $b(t)$ are scale factors. In these coordinates the S-brane arises on a constant time slice $t=t_{0}$. The induced metric on the S-brane hypersurface and the extrinsic curvature of this surface are

$$q_{\mu\nu}=Diag\bigg{[}0,-a^{2},-b^{2},-b^{2}\sin^{2}\vartheta\bigg{]}_{\mu\nu}$$

(10)

and

$$K^{\mu}_{\ \nu}=Diag\bigg{[}0,H_{a},H_{b},H_{b}\bigg{]}^{\mu}_{\ \nu},$$

(11)

where the Hubble parameters are $H_{a}=\dot{a}/a$ and $H_{b}=\dot{b}/b$.

The matching of the induced metric gives initial conditions for the scale factors

	$\displaystyle a(t_{0})$	$\displaystyle=\sqrt{f(r_{0})}\,,$		(12)
	$\displaystyle b(t_{0})$	$\displaystyle=r_{0},$		(13)

while the jump condition of the extrinsic curvature across the S-brane hypersurface (see Equation (4)) provides initial conditions for the derivatives of $a$ and $b$:

	$\displaystyle H_{a}+\frac{\sqrt{g}f^{\prime}}{2f}$	$\displaystyle=S^{x}_{\ x}=S,$		(14)
	$\displaystyle H_{b}+\frac{\sqrt{g}}{r_{0}}$	$\displaystyle=S^{\vartheta}_{\ \vartheta}=S.$		(15)

We see that a large enough S-brane tension leads to an initially expanding universe to the future of the S-brane, in the same way that an S-brane with sufficient tension can lead to a non-singular transition between a contracting and expanding universe Wang1 .

If $w_{x}=-1$, the equations simplify substantially and we can even obtain power law solutions for the scale factors. This case includes radiation and a cosmological constant.

If $w_{x}=-1$, the energy density depends only on $b$, and Equation (18) becomes a second order equation for $b(t)$ with no explicit time dependence. Therefore it can be converted to a first order equation for $\dot{b}(b)$ with the solution:

$$\dot{b}^{2}(b)=\sum_{i}\frac{\rho_{0}^{(i)}}{1-2w_{\Omega}^{(i)}}b^{-2w_{\Omega}^{(i)}}+\frac{C}{b}-1,$$

(21)

where $i$ labels different matter contents and C is an integration constant. We distinguish three cases with

1. 1.

   All matter ingredients have $0<w_{\Omega}$. For a sufficiently large $b$, the right-hand-side becomes zero and a limiting value of $b$ is reached at a finite proper time on the geodesics. $a\propto\dot{b}$ (Equation (17)) also goes to zero. This is a white hole space-time. An example of such matter content is radiation ($w_{\Omega}=1$).

2. 2.

   At least one ingredient has $w_{\Omega}<0$ but still $-1<w_{\Omega}$. At late times we get an expanding universe with scale factors growing with a power law $b(t)\propto t^{1/(1+w_{\Omega})}$.

3. 3.

   A cosmological constant $\Lambda$ is present ($w_{x}=w_{\Omega}=-1$). It will create an asymptotically de Sitter space-time with a Hubble parameter $H^{2}=\Lambda/3$.

Equation (21) cannot in general be integrated even if we neglect the ”-1” term. However, an interesting, easily solvable case is for $w_{\Omega}=-1/4$. Then we get

	$\displaystyle b(t)$	$\displaystyle=b_{0}\bigg{[}1+3H_{b}t/2+3H_{b}(H_{a}+H_{b}/2)t^{2}\bigg{]}^{2/3},$		(22)
	$\displaystyle a(t)$	$\displaystyle=\frac{a_{0}\ \dot{b}(t)}{b_{0}H_{b}},$		(23)

where $H_{a}=H_{a}(t_{0})$ and $H_{b}=H_{b}(t_{0})$ are determined from the matching conditions. This is an expanding, anisotropic universe with scale factors behaving as power laws at late times.

We have also considered some matter contents which have an equation of state close to that of a cosmological constant. We show that an ideal fluid which has an equation of state close to the cosmological constant also gives an approximately isotropic space at late times. Inflation initially smoothens the anisotropies, but they reappear once the scalar field decays.

A general scalar function $f\in L_{2}[S^{2}]$ can be decomposed in the basis of spherical harmonics:

$$f(\vartheta,\phi)=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}f_{lm}Y_{lm}(\vartheta,\phi).$$

(32)

Similarly, a rank two covariant tensor on the sphere can be decomposed using a generalization to the vector and tensor spherical harmonics which we will briefly introduce here, for a full review see for example EmanueleScript .

We will use upper case letters ($A$, $B$, …) for the spherical indexes $\vartheta$ and $\phi$ and lower case ($a$, $b$, …) for $r$ and $x$. We will denote the covariant derivative with respect to the 2-metric on the sphere by $\widehat{\nabla}_{A}$ and with respect to the full metric $g_{\mu\nu}$ by $\nabla_{\mu}$. We will from now on skip all the $lm$ indexes and sums over $l$ and $m$ for brevity.

The metric on the sphere is

$$\gamma_{AB}=\begin{bmatrix}1&0\\ 0&\sin^{2}\vartheta\end{bmatrix}.$$

(33)

A Levi-Civita tensor is obtained from the Levi-Civita symbol by multiplying it by the determinant of the metric:

$$\epsilon_{AB}=\sin\vartheta\begin{bmatrix}0&1\\ -1&0\end{bmatrix}.$$

(34)

By taking covariant derivatives on the sphere we can construct two quantities that transform as vectors on the sphere:

$$Y_{A}=\widehat{\nabla}_{A}Y\qquad S_{A}=\epsilon_{AB}\gamma^{BC}\widehat{\nabla}_{C}Y.$$

(35)

Both are eigenvectors of the parity transformation ($\vartheta\xrightarrow[]{}\pi-\vartheta$, $\phi\xrightarrow[]{}\pi+\phi$). Their eigenvalues are $(-1)^{l}$ and $-(-1)^{l}$ respectively, which is to be compared with the $(-1)^{l}$ parity of the spherical harmonic $Y_{lm}$. Quantities with the same parity as the spherical harmonic $Y_{lm}$ are called polar, those with the opposite parity axial, and thus $Y_{A}$ and $S_{A}$ are polar and axial vector harmonics respectively. They will be used to decompose the mixed part of the metric perturbation $h_{aA}$.

Polar and axial tensor spherical harmonic are, respectively:

	$\displaystyle Z_{AB}$	$\displaystyle=\widehat{\nabla}_{A}\widehat{\nabla}_{B}Y+\frac{l(l+1)}{2}Y\gamma_{AB},$		(36)
	$\displaystyle S_{AB}$	$\displaystyle=\widehat{\nabla}_{(A}S_{B)}\,.$		(37)

They will be used to decompose the angular part $h_{AB}$ of the metric perturbation.

The metric perturbation

$$h_{\mu\nu}(x)=g_{\mu\nu}(x)-\overline{g_{\mu\nu}}(x)$$

(38)

can be decomposed in the basis of generalized spherical harmonics:

	$\displaystyle h_{ab}$	$\displaystyle=Y\begin{bmatrix}-\overline{g_{00}}s_{0}&-s_{1}\\ -s_{1}&\overline{g_{11}}s_{2}\end{bmatrix}_{ab}$		(39)
	$\displaystyle h_{aA}$	$\displaystyle=v_{a}Y_{A}+h_{a}S_{A}$		(40)
	$\displaystyle h_{AB}$	$\displaystyle=-\overline{g_{22}}(t_{1}\gamma_{AB}Y+t_{2}\widehat{\nabla}_{A}\widehat{\nabla}_{B}Y+2hS_{AB}),$		(41)

where summation over $l$ and $m$ is implied. The angular dependence of the perturbations is in this way transformed to the $l$ and $m$ dependence of the coefficient functions: polar scalar ($s_{0}$, $s_{1}$, $s_{2}$), polar vector ($v_{0}$, $v_{1}$), polar tensor ($t_{1}$, $t_{2}$), axial vector ($h_{0}$, $h_{1}$) and axial tensor ($h$). All coefficient functions additionally depend on $r$ and $x$.

We can further take advantage of the static background ($x$-independence of the background) and decompose the $x$ dependence of the perturbations in Fourier modes. For example

$$h_{lm}(r,x)=\frac{1}{2\pi}\int h_{klm}(r)e^{ikx}dk.$$

(42)

We will drop the $k$ index from now on.

On the Bianchi side we replace the $r^{2}$ factor in the tensor perturbations by $b(t)^{2}$ and denote all the perturbation functions with the corresponding upper case letters.

We will now examine how the metric perturbations transform uunder a coordinate change:

$$\widetilde{x^{\mu}}=x^{\mu}+\xi^{\mu}.$$

(43)

A general perturbative coordinate change $\xi^{\mu}$ can be decomposed into scalar and vector parts:

$$\xi=Y\Xi^{a}\partial_{a}+(\Xi^{P}Y^{A}+\Xi^{A}S^{A})\partial_{A}\,,$$

(44)

where $\Xi^{0}$, $\Xi^{1}$, $\Xi^{P}$ and $\Xi^{A}$ are functions of $r$, $x$, $l$ and $m$.

The metric in the new coordinates is

$$\widetilde{g_{\mu\nu}}(\widetilde{x})=\overline{g_{\mu\nu}}(\widetilde{x})+h_{\mu\nu}-[\mathcal{L}_{\xi}\ \overline{g}]_{\mu\nu}\equiv\overline{g_{\mu\nu}}(\widetilde{x})+\widetilde{h_{\mu\nu}}\,.$$

(45)

The Lie derivative term is

$$[\mathcal{L}_{\xi}\ \overline{g}]_{\mu\nu}=\xi^{\alpha}\partial_{\alpha}\overline{g_{\mu\nu}}+2\overline{g_{\alpha(\mu}}\partial_{\nu)}\xi^{\alpha}=\overline{g_{\alpha(\mu}}\nabla_{\nu)}\xi^{\alpha}.$$

(46)

It can be calculated most conveniently by spelling out the covariant derivative and recognizing the $\widehat{\nabla}$ terms. We obtain the transformation rules for the coefficient functions (written for the Bianchi metric):

	$\displaystyle\widetilde{S_{0}}$	$\displaystyle=S_{0}-2\dot{\Xi}^{0}$		(47)
	$\displaystyle\widetilde{S_{1}}$	$\displaystyle=S_{1}+a^{2}\dot{\Xi}^{1}-\partial_{x}\Xi^{0}$
	$\displaystyle\widetilde{S_{2}}$	$\displaystyle=S_{2}+2(H_{a}\Xi^{0}+\partial_{x}\Xi^{1})$
	$\displaystyle\widetilde{V_{0}}$	$\displaystyle=V_{0}+\Xi^{0}-(2H_{b}+b^{2}\partial_{t})\Xi^{P}$
	$\displaystyle\widetilde{V_{1}}$	$\displaystyle=V_{1}-a^{2}\Xi^{1}-b^{2}\dot{\Xi}^{P}$
	$\displaystyle\widetilde{T_{1}}$	$\displaystyle=T_{1}-H_{b}\Xi^{0}$
	$\displaystyle\widetilde{T_{2}}$	$\displaystyle=T_{2}-2\Xi^{P}$
	$\displaystyle\widetilde{H_{0}}$	$\displaystyle=H_{0}-(2H_{b}+b^{2}\partial_{t})\Xi^{A}$
	$\displaystyle\widetilde{H_{1}}$	$\displaystyle=H_{1}-b^{2}\dot{\Xi}^{A}$
	$\displaystyle\widetilde{H}$	$\displaystyle=H-\Xi^{A}.$

We will fix a gauge. Setting $\widetilde{T_{2}}=0$ and $\widetilde{H}=0$ will completely fix $\Xi^{P}$ and $\Xi^{A}$ respectively. Then $\widetilde{V_{1}}=0$ will fix $\Xi^{1}$. A Regge-Wheeler gauge would be to further set $\widetilde{V_{0}}=0$ by fixing $\Xi^{0}$, we will discuss this in the next subsection. For the dipole case there exists no tensor spherical harmonic function and hence the parametrization of the metric perturbation using two spherical harmonic coefficients is redundant. For $l=0$ the Regge Wheeler gauge does not fix the gauge entirely (see Kobayashi:2014wsa ; EmanueleScript for a pedagogical exposure). We will ignore them in this work.

The S-brane arises on a hypersurface of constant Weyl or Kretschman scalar. Let $q(\bm{x})$ be a scalar which is constant on the S-brane. It is slighly perturbed in the presence of perturbations and it thus may no longer depend on $x^{0}$ only:

$$q(\textbf{x})=q_{0}(x^{0})+\delta q(\textbf{x}).$$

(48)

Therefore an S-brane is no longer a constant $x^{0}$ surface. We will therefore use a different time-like coordinate

$$\widetilde{x^{0}}=x^{0}+\Xi^{0}(\textbf{x})$$

(49)

such that the S-brane is a constant $\widetilde{x^{0}}$ hypersurface. In other words, we want:

	$\displaystyle q(\widetilde{\textbf{x}})$	$\displaystyle=q_{0}(\widetilde{x^{0}}-\Xi^{0})+\delta q(\widetilde{\textbf{x}})$		(50)
		$\displaystyle\approx q_{0}(\widetilde{x^{0}})-q_{0}^{\prime}(\widetilde{x^{0}})\Xi^{0}+\delta q$

to be a function of $\widetilde{x^{0}}$ only. Therefore we must choose (as in Deruelle ):

$$\Xi^{0}(\textbf{x})=\frac{\delta q(\textbf{x})}{q_{0}^{\prime}(x_{0})}\,,$$

(51)

at least in a neighbourhood of the S-brane. This in principle exhausts all the remaining gauge freedom. What concerns the axial perturbations, this gauge choice is the same as the Regge-Wheeler gauge, because axial perturbations do not transform with $\Xi^{0}$, but polar vector modes $V_{0}$ cannot be set to zero because we have already chosen $\Xi^{0}$.

We will first match the induced metric. The normal to the S-brane hypersurface is perturbed

$$n=\sqrt{g_{00}}dx^{0}+\sqrt{g^{00}}dx^{1}+\sqrt{g^{00}}h_{0A}dx^{A},$$

(52)

such that it remains orthogonal to all three spatial covectors $dx^{i}$. As expected, we find that the induced metric of each mode is the mode’s $h_{\mu\nu}$ with $0\mu$ and $\mu 0$ components set to 0. Matching the induced metric yields:

	$\displaystyle S_{2}(r_{0},x)$	$\displaystyle=s_{2}(t_{0},x)$		(53)
	$\displaystyle T_{1}(r_{0},x)$	$\displaystyle=t_{1}(t_{0},x)$
	$\displaystyle H_{1}(r_{0},x)$	$\displaystyle=h_{1}(t_{0},x)$

for all $x\in\mathbb{R}$.

The extrinsic curvature matching conditions are most conveniently decomposed if expressed in terms of a purely covariant extrinsic curvature tensor. Neglecting possible S-brane tension perturbations, the matching condition is:

$$\delta K_{\mu\nu}\big{\rvert}^{+}_{-}=h_{\mu\alpha}S^{\alpha}_{\ \nu}\,.$$

(54)

The extrinsic curvature perturbation can be decomposed in the spherical harmonic basis as was the metric. Due to orthogonality, matching is only between modes with the same $l$ and $m$ and between modes of the same parity. The $xA$ component of Equations (54) gives the axial matching condition

$\displaystyle\dot{H_{1}}+\sqrt{f}h_{1}^{\prime}=H_{b}H_{1}+\frac{\sqrt{f}}{r}h_{1}.$

(55)

From the $xx$ and $\theta\theta$ components we get the polar matching conditions

	$\displaystyle\dot{S_{2}}+\sqrt{f}{s_{2}}^{\prime}$	$\displaystyle=H_{a}S_{0}+\frac{f^{\prime}}{2\sqrt{f}}s_{0}$		(56)
	$\displaystyle\dot{T_{1}}+\sqrt{f}{t_{1}}^{\prime}$	$\displaystyle=H_{b}S_{0}+\frac{\sqrt{f}}{r}s_{0}\,.$

We will use the axial matching conditions in the next Section to obtain the axial power spectrum after the S-brane transition.

Assuming there is no axial energy-momentum tensor perturbation, the $R_{\vartheta\phi}$ component of the Ricci tensor perturbation shows us that $h_{1}$ and $H_{1}$ are not dynamical as they are completely fixed by $h_{0}$ and $H_{0}$:

$$h_{1}=\frac{f(fh_{0})^{\prime}}{ik}\qquad H_{1}=\frac{a(aH_{0})^{\bm{\cdot}}}{ik}\,.$$

(57)

Combining with $R_{t\phi}$ we can formulate the evolution of $h_{0}$ as a one-dimensional Schrödinger equation:

$$\psi^{\prime\prime}+V\psi=0\,,$$

(58)

where $\psi=h_{0}\ f^{3/2}/r$ for the black hole part of the space-time and $\psi=H_{0}\ a^{3/2}b^{-1}$ for the Bianchi part of the space-time. The effective potentials are

	$\displaystyle V^{(BH)}(r)$	$\displaystyle=\frac{L}{fr^{2}}+\frac{k^{2}}{f^{2}}+\frac{3f^{\prime}}{fr}+\bigg{(}\frac{f^{\prime}}{2f}\bigg{)}^{2}-\frac{f^{\prime\prime}}{2f}$		(59)
	$\displaystyle V^{(Bian)}(t)$	$\displaystyle=\frac{L}{b^{2}}+\frac{k^{2}}{a^{2}}+$		(60)
		$\displaystyle+3H_{b}(H_{a}+H_{b})+3\dot{H_{b}}-\dot{H_{a}}/2\,,$

where $L=l(l+1)$ for short.

Specifically, for the Schwarzschild black hole

$$V(R)=\frac{K^{2}R^{4}+LR(1-R)+4R-\frac{15}{4}}{R^{2}(1-R)^{2}}\,,$$

(61)

where we have changed to a dimensionless radial coordinate $R=r/r_{S}$ and a dimensionless $K=k\ r_{S}$. We call the first term the $K$-term, the second term the $L$-term and the rest the geometry-term. It is straightforward to analyze the behaviour of $\psi$ close to the singularity ($R\xrightarrow[]{}0$), close to the event horizon ($R\sim 1$) and at infinity ($R\xrightarrow[]{}\infty$):

	$\displaystyle\psi(R\xrightarrow[]{}0)$	$\displaystyle=C_{\infty}\ R^{-3/2}+C_{0}\ R^{5/2}$		(62)
	$\displaystyle\psi(R\sim 1)$	$\displaystyle=|1-R|^{1/2}$
		$\displaystyle\bigg{(}C_{+}^{H}\ e^{iK\log|1-R|}+C_{-}^{H}\ e^{-iK\log|1-R|}\bigg{)}$
	$\displaystyle\psi(R\xrightarrow[]{}\infty)$	$\displaystyle=C_{+}e^{iKR}+C_{-}e^{-iKR},\$

where the various $C$ are integration constants.

The quantity of interest is the transfer function

$$\mathcal{T}=\frac{|C_{\infty}|}{|C_{-}^{H}|},$$

(63)

which will allow us to compute the power spectrum just before the S-brane, given the initial conditions.

Some comments are in place. Close to the singularity the $R^{-3/2}$ term prevails and the perturbations diverge, faster than the background metric. Nevertheless, we will assume that $C_{\infty}$ is small enough, so the perturbations are still linear at the S-brane. The $R^{-3/2}$ term is dominant, and as we will see, $C_{\infty}$ will determine the power spectrum after the S-brane. Second, note that $\psi$ does not appear to be smooth at the horizon. This is a coordinate artefact which is resolved by using for example tortoise coordinates as in EmanueleScript . However, the smoothness requirement translates to the condition that only the incoming waves are present, so $C_{+}^{H}$ has to be zero.

Several methods for solving the Equation (61) have been proposed in the literature, the WKB approximation WKB , the Leaver series solution Leaver , the asymptotic iteration method asymptoticIterationMethod and others. We will first solve the differential equation by dividing the space-time into regions where the potential is simple, determine the scaling of the perturbations in each region separately and glue the solutions together. This will give us physical insight into the behaviour of the perturbations. However, this solution is very rough so we will complement our analysis with an analytical solution in the low $K$ limit and with the Leaver series solution to get accurate results.

We define an initial power spectrum, far away from the black hole as (see Equation (62)):

$$P_{i}(K,L)=|C_{+}|^{2}+|C_{-}|^{2}.$$

(64)

One simple choice is that this initial power spectrum originates from the quantum fluctuations in the asymptotically flat region. Quantized $\psi$ is a massless scalar field. Power spectrum is a Fourier transform of it’s correlator, yielding:

$$P_{i}(R)\propto\bigg{(}K^{2}+\frac{L}{R^{2}}\bigg{)}^{-1/2}\approx K^{-1}\,.$$

(65)

for large $R$.

For large values of $K$ it is known that the region exterior to the event horizon perfectly transmits the perturbations scattering , so the power spectrum at the horizon will be the same as the power spectrum at $r=\infty$. Small $K$ modes on the other hand are suppressed outside of the event horizon. We will in the following subsections show that they are also suppressed inside the horizon implying that they will also be suppressed on the Bianchi side of the space-time.

We divide the interior of the event horizon into regions such that we approximately understand the scaling of $\psi$ in each region. By combining these scalings we estimate the transfer function in the $K>>L$ limit and in the $K<<L$ limit.

We will here calculate the transfer function for low $K$ analytically, with very accurate results. Close to the event horizon the $L$-term is negligible and $\psi$ behaves as (see Equation (62)):

$$\psi_{H}=C_{-}^{H}\,(1-R)^{-iK+1/2}\,.$$

(72)

As $R$ decreases the $L$-term becomes ever more important while the $K$-term becomes ever more negligible. A regime-change occurs approximately at the radius $R_{1}$ at which both terms are equally important: $LR_{1}(1-R_{1})=KR_{1}^{4}$. To lowest order in a small $K$ expansion we get

$$R_{1}=1-\frac{K^{2}}{L}.$$

(73)

$1-R_{1}$ is of the order of $K^{2}$ and will turn out to be negligible for the lowest order estimate of the transfer function.

The governing Equation (58) is solvable by elementary functions in the region (0, $R_{1}$), where the $K$-term is neglected. We call the basis functions for the space of solutions $u_{l}(R)$ and $v_{l}(R)$. We can chose $u_{l}(R)$ such that it is the extension of the $R^{5/2}$ solution near the singularity (see Equation (62)) to the entire region (0, $R_{1}$). Similarly, we can chose $v_{l}(R)$ as an extension of the $R^{-3/2}$ solution.

$u_{l}(R)$ is of the form:

$$u_{l}(R)=R^{5/2}\sqrt{1-R}\ p_{l}(R)\,,$$

(74)

where $p_{l}(R)$ is a polynomial of order $l-2$ in $R$:

$$p_{l}(R)=\sum_{n=0}^{l-2}a_{n}R^{n}.$$

(75)

A recurrence relation for the coefficients of the polynomial is:

$$\frac{a_{n+1}}{a_{n}}=\frac{(n+3)(n+2)-l(l+1)}{(n+1)(n+5)},$$

(76)

By requiring that $u$ is an extension of the $R^{5/2}$ solution we fixed $p_{l}(0)=a_{0}=1$. For example, for $l=2,3,4$ one has:

	$\displaystyle p_{2}(R)$	$\displaystyle=1$		(77)
	$\displaystyle p_{3}(R)$	$\displaystyle=1-(6/5)R$
	$\displaystyle p_{4}(R)$	$\displaystyle=1-(14/5)R+(28/15)R^{2}.$

Note that for $K=0$, $u_{l}$ is the solution in the entire region $(R_{0},\,1)$ and satisfies the initial condition at the horizon. Therefore $C_{\infty}=0$. For $K\neq 0$ we will glue the solution in the region close to the horizon with the solution in the rest of the interior of the black hole, by requiring that $\psi$ and $\psi^{\prime}$ are continuous at $R_{1}$:

	$\displaystyle C_{0}u_{l}(R_{1})+C_{\infty}v_{l}(R_{1})=\psi_{H}(R_{1})$		(78)
	$\displaystyle C_{0}u_{l}^{\prime}(R_{1})+C_{\infty}v_{l}^{\prime}(R_{1})=\psi_{H}^{\prime}(R_{1}).$

Solving for $C_{\infty}$ gives

$$C_{\infty}=\frac{\psi_{H}^{\prime}u-\psi_{H}u^{\prime}}{\mathcal{W}},$$

(79)

where the Wronskian determinant is $\mathcal{W}=uv^{\prime}-u^{\prime}v$. A standard result in the theory of differential equations is that the Wronskian determinant is independent of $R$ if the governing equation has no term proportional to $\psi^{\prime}$, as is the case in the Equation (58). The Wronskian determinant can therefore be conveniently calculated in the vicinity of the singularity and equals $\mathcal{W}=-4$.

Substituting $\psi_{H}$ from Equation (72) and $u_{l}$ from Equation (74), neglecting terms beyond linear order in the small $K$ expansion, and dividing by $C_{0}$ gives the transfer function (Equation (63)):

$$\mathcal{T}=\frac{|p_{l}(R=1)|\ K}{4}.$$

(80)

The transfer function is linear in $K$. The proportionality constant $|p_{l}(1)|\,/4=|\sum_{n=0}^{l-2}a_{n}|\,/\,4$ as a function of $l$ is shown in Table 1. It decays approximately as $L^{-2}\approx l^{-4}$.

We now solve the evolution Equation (58) with the Leaver series expansion Leaver , to confirm the approximation in the previous subsection and to go beyond the lowest order in $K$. The Leaver series expansion is a very accurate method and was used extensively for calculating black hole quasi-normal mode (QNM) frequencies EmanueleScript . The idea is to use an asymptotic series expansion with an ansatz which has the correct behaviour at the borders between the regions. The ansatz for the interior of the event horizon is

$$\psi(R)=R^{-3/2}(1-R)^{-iK+1/2}\sum_{n=0}^{\infty}b_{n}(1-R)^{n}.$$

(81)

The first factor is the behaviour at the singularity, and the second factor is the behaviour at the event horizon. The transfer function is given by

$$\mathcal{T}=\frac{1}{|b_{0}|}\biggl{|}\sum_{n=0}^{\infty}b_{n}\biggr{|}.$$

(82)

Inserting the ansatz (81) in the governing Equation (58) yields a five-term recurrence relation

$$\alpha\,b_{n}+\beta\,b_{n-1}+\gamma\,b_{n-2}-4K^{2}b_{n-3}+K^{2}b_{n-4}=0\,,$$

(83)

with the prefactors:

	$\displaystyle\alpha$	$\displaystyle=n(n-2iK)$		(84)
	$\displaystyle\beta$	$\displaystyle=1+7n-2n^{2}+L-2K^{2}+iK(4n-7)$
	$\displaystyle\gamma$	$\displaystyle=6-7n+n^{2}-L+5K^{2}+iK(7-2n).$

Additionally, $b_{1}$, $b_{2}$, $b_{3}$ can be expressed by $b_{0}$. The recurrence relation can be solved numerically, and the results are shown in Figure 2.