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in classical general relativity a causal particle inside the event horizon is inevitably pulled towards the center of the black hole , at least until it reaches the cauchy horizon representing a barrier to predictability @xcite . in particular , no classically relativistic particle , be it of positive or vanishing rest - mass , can escape from the black hole interior . however , hawking @xcite has shown that photons in effect can leave the black hole if a quantized photon field in the curved spacetime is supposed , cf . @xcite .    in general , quantization of spin - weighted waves in a curved spacetime yields the notion of spin particles in a gravitational field . as such , spin - weighted waves are the basis for semiclassical quantum gravity . their properties and behavior in the outer space of kerr black holes have been extensively studied @xcite , and could in part even be extended to charged kerr - newman black holes @xcite . by the symmetries of these spacetimes the equations turn out to be separable in special coordinate frames , thus being mathematically tractable to a certain extent .    in the framework of classical general relativity the examination of waves in the outer space of a black hole seemed sufficient , since the event horizon acts like a perfect semi - permeable membrane , letting in any form of energy and matter but allowing none to get out .    in the present paper we extend these considerations to the region beyond the event horizon . topologically , this means nothing particular , since the spacetime curvature remains finite in this domain and the event horizon is nothing more than a coordinate singularity . as expected , the event horizon causes a singular repulsive potential in the respective wave equations . however , more detailed examination shows the remarkable , and to our knowledge yet unmentioned , property of the potential barrier to be singular enough to prevent nearly all kinds of spin - weighted waves from tunneling through it  unless electromagnetic waves . our analysis below shows that in fact a photon inside an uncharged rotating black hole may tunnel through the event horizon to the outside region , but a graviton , a neutrino or an electron may not . if the black hole is non - rotating , no particle at all can tunnel out of it . in case of a rotating and electrically charged black hole it remains still unknown what kind of particle may tunnel out , it is only sure that electrons and neutrinos _ may not_.    the present paper is organized as follows . in section [ sec - kn ] we introduce the basic notation and properties of kerr - newman spacetimes . in section [ sec - spin - waves ] we analyze the equations for massless waves in kerr geometry and present a proof that only photons of certain discrete frequencies may tunnel out of a rotating but non - extremal kerr black hole . the case of extremal rotation is considered in section [ sec - rotation ] , the mathematically not completely tractable case of massless waves in a charged reissner - nordstrm spacetime in section [ sec - rn ] , and the dirac equation in a general kerr - newman spacetime in section [ sec - dirac ] . finally , in section [ sec - discussion ] we sum up the results and discuss them . suppose a kerr - newman black hole with the three real parameters @xmath0 , @xmath1 , and @xmath2 . they are related to the mass @xmath3 ( in kg ) , the angular momentum @xmath4 ( in kg m@xmath5 s@xmath6 ) and the electrical charge @xmath7 ( in kg@xmath8 m@xmath9 s@xmath6 ) by the relations @xmath10 here @xmath11 is the gravitational constant , and @xmath12 the speed of light . the non - vanishing contravariant components @xmath13 of the metric tensor in boyer - lindquist coordinates @xmath14 ,  , @xmath15 , @xmath16 , @xmath17 , @xmath18 are then @xmath19   \displaystyle      g^{t\varphi } = { ( 2mr - q^2 ) a \over \rho \bar\rho                \delta},\quad      g^{\varphi\varphi }       = - { \delta - a^2 \sin^2\theta \over \rho \bar\rho                \delta                    \sin^2\theta } ,   \end{array}\ ] ] ( @xmath20 ) , where @xmath21 with the _ event horizon _ @xmath22 and the _ cauchy horizon _ @xmath23 given by @xmath24 only the points of the set @xmath25 have infinite riemann curvature and thus locate the curvature singularity of the spacetime . for @xmath26 , it in fact forms a ring of radius @xmath27 in the equatorial plane @xmath28 , cf .  @xcite . because the _ cosmic censorship hypothesis _ forbids naked singularities , the event horizon @xmath22 must necessarily exist . thus the square root of ( [ horizonte ] ) must be real , i.e. @xmath29 let be @xmath30 , and define @xmath31 @xmath32 \{0 , @xmath33 , @xmath34 , @xmath35 } as the _ spin - weight_. then the sourcefree perturbation equations for scalar ( @xmath31 @xmath36 0 ) , two - component neutrino ( @xmath31 @xmath36 @xmath33 ) , electromagnetic ( @xmath31 @xmath36 @xmath34 ) , and gravitational fields ( @xmath31 @xmath36 @xmath35 ) are given by wave equations which by the the symmetries of the kerr spacetime , viz . stationarity and axialsymmetry , admit the separable solutions @xmath37 with the constants @xmath38 and @xmath39 @xmath32 @xmath40 . here @xmath41 obeys the radial equation @xmath42   \displaystyle            + 4 { \mathrm{i } } s\omega r - \lambda \bigg ) r_s = 0 , \label{radial equation}\end{aligned}\ ] ] and @xmath43 solves the angular equation @xmath44   \displaystyle          - { ( m + s \cos\theta)^2 \over \sin^2\theta }          - s(s-1 ) + a \bigg ) s_s = 0 , \label{angular equation}\end{aligned}\ ] ] with @xmath45 and @xmath46 , cf . @xcite the constants @xmath47 , @xmath48 @xmath32 @xmath49 are separation constants obtaining some discrete values depending on the boundary conditions of @xmath43 . ( for details see @xcite ,  2.1 . ) to be more explicit , @xmath47 @xmath36 @xmath50 @xmath36 @xmath51 @xmath52 @xmath53 @xmath54 @xmath55 , where @xmath51 @xmath36 @xmath56 @xmath54 @xmath57 @xmath52 @xmath58 , with @xmath59 @xmath60 @xmath61 and @xmath59 @xmath32 @xmath62 for integer spin @xmath31 , @xmath59 half - odd integer for @xmath63 @xmath36 1/2 , cf . @xcite eq . ( 3.10 ) . [ satz - potential ] let be given the operator @xmath64 @xmath65 @xmath66 @xmath65 , @xmath67 with @xmath68 with the functional transformation @xmath69 thes radial equation ( [ radial equation ] ) is equivalent to @xmath70 @xmath36 @xmath71 .    _ _ first we see that with the functions @xmath72 given by the transformation @xmath73 ( i.e. , @xmath74 @xmath36 @xmath75 and @xmath76 @xmath36 @xmath77 ) the radial differential equation ( [ radial equation ] ) is equivalent to @xmath78 , with the differential operator @xmath79 where @xmath80 and @xmath81 ( cf . @xcite , eqs . ( 3 ) , ( 96 ) , ( 97 ) in chapter 8) . with @xmath82 we see that @xmath83 since d@xmath84/d@xmath16 @xmath36 @xmath85 , we have @xmath86 by elementary means ( e.g.  @xcite , p.  323 ) we deduce from this result that , by the transformation @xmath87 the equation @xmath78 is equivalent to @xmath88 . [ lemma - tunneling ] let @xmath89 be a solution of the differential equation @xmath70 @xmath36 0 . then the asymptotic behavior of @xmath89 is given by @xmath90              \phantom{\scriptstyle r \to \infty}\\[-1.5ex ]              \longrightarrow\\[-1.5ex ]              { \scriptstyle r \to \infty } \\[-1.5ex ]          \end{array }          a_{\mathrm{in}}\ , r^s\ , { \mathrm{e}}^{-{\mathrm{i } } \omega r }          + a_{\mathrm{out}}\ , r^{-s}\ , { \mathrm{e}}^{{\mathrm{i } } \omega r},\ ] ] @xmath90              \phantom{\scriptstyle r \to m}\\[-1.5ex ]              \longrightarrow\\[-1.5ex ]              { \scriptstyle r \to m } \\[-1.5ex ]          \end{array }          b_{\mathrm{in}}\ , { \mathrm{e}}^{-{\mathrm{i } } k r }          + b_{\mathrm{out}}\ , { \mathrm{e}}^{{\mathrm{i } } k r }      \label{u - asymptotic}\ ] ] with the complex amplitudes @xmath91 , @xmath92 , @xmath93 , @xmath94 @xmath32 @xmath95 and the complex exponent @xmath96 given by ( cf . figure [ fig - sqrt ] ) @xmath97 ^ 2 \over ( m^2-a^2)^2 }           + { |s| - 1 - \lambda - 4 { \mathrm{i } } s \omega m \over m^2 - a^2 } } \!. \label{k - def}\ ] ]    ( 0,0 )   of a complex number @xmath98 in the complex plane . if @xmath98 @xmath36 @xmath99 e@xmath100 , then @xmath101 @xmath36 @xmath102 e@xmath103 [ fig - sqrt],title="fig:",width=188 ]    ( 48,36 ) ( 20.5,26.5)(0,0)i ( 33,13)(0,0)1 ( 2,26)(0,0)@xmath98 ( 28,30)(0,0)@xmath101 ( 16 , 3)(0,0)@xmath104 ( 20.5,19.5)(0,0)@xmath105 ( 31,23)(0,0)@xmath106    _ proof . _ first we consider the behavior for @xmath16 @xmath66 @xmath107 . in this case , @xmath108 @xmath36 @xmath109 @xmath52 @xmath110 with @xmath109 @xmath36 @xmath111 @xmath54 @xmath112 . thus asymptotically @xmath70 @xmath66 @xmath113 @xmath52 @xmath114 , which is solved by the first limit ( [ u - asymptotic ] ) as is seen by direct computing .    on the other hand , we calculate by @xmath115 @xmath36 @xmath116 @xmath54 @xmath117 that @xmath108 @xmath66 @xmath118 as @xmath16 @xmath66 @xmath0 . hence , @xmath70 @xmath66 @xmath119 @xmath52 @xmath120 , which is solved by the second limit in ( [ u - asymptotic ] ) . note that the complex conjugate of @xmath121 in ( [ a - s - def ] ) equals the radial operator @xmath122 of the waves of spin @xmath123 , @xmath124 @xmath36 @xmath122 . in particular , @xmath125 @xmath36 @xmath126 @xmath36 0 , and thus @xmath127 the advantage of the transformation in theorem [ satz - potential ] is the fact that the wronskian of two solutions of the differential equation @xmath70 @xmath36 0 is constant . in particular , the behavior of two _ independent _ solutions at the boundaries of the domain of definition determines the reflexion and transmission coefficients of the potential @xmath128 , cf .  @xcite . by the peeling theorem of newman and penrose @xcite , for a null field of spin weight @xmath31 the ingoing and outgoing solutions differ in magnitude by the factor @xmath129 . accordingly , the solutions @xmath41 show the asymptotic behavior @xmath130 this is in accordance with the asymptotic behavior of @xmath89 , using the transformation ( [ u - s - def ] ) and @xmath84 @xmath36 @xmath131 .    [ satz - tunneling ] the only massless particles that can tunnel out of a non - extremal kerr black hole are photons with the discrete frequencies @xmath132 their wave numbers @xmath133 at @xmath16 @xmath36 @xmath0 are given by @xmath134 depending on the angular quantum number @xmath39 @xmath32 @xmath40 . the amplitudes of a photon wave with the asymptotic behavior ( [ u - asymptotic ] ) is determined by the conservation law @xmath135    _ proof . _ a kerr black hole with mass @xmath0 and rotation parameter @xmath1 is non - extremal if and only if @xmath136 . for @xmath137 @xmath36 @xmath138 as in ( [ omega - m ] ) we compute straightforwardly the functions @xmath139 @xmath36 @xmath140 as @xmath141 inserting @xmath138 into the expression for @xmath96 in ( [ k - def ] ) yields @xmath142 ^ 2            + { |s|-1 - \lambda - 4 { \mathrm{i } } s \omega_m m \over m^2 - a^2 } .\ ] ] by theorem [ satz - potential ] a massless particle of spin - weight @xmath31 is given as a wave function @xmath89 satisfying @xmath70 @xmath36 0 . since the wkb approximation is applicable if @xmath108 is varying slowly enough , @xmath143 we may use it for a solution @xmath144 in a region close enough to the event horizon at @xmath16 @xmath36 @xmath145 and far enough from the turning points @xmath108 @xmath36 0 , see @xcite  8.2 , say @xmath146 @xmath147 @xmath16 @xmath147 @xmath148 . in this regime , the transmission coefficient @xmath149 is given by @xmath150 \label{transmission - coefficient}\ ] ] the integral diverges if @xmath108 has a pole at @xmath16 @xmath36 @xmath145 of order not smaller than 2 , but it attains a finite value for a pole of order 1 since @xmath151 there are three terms in @xmath108 which may result in a pole of order 2 at @xmath16 @xmath36 @xmath145 , namely @xmath152 the third term , the imaginary part , vanishes ( in the non - extremal case , where @xmath145 @xmath153 @xmath0 ) if and only if @xmath154 @xmath36 0 or @xmath31 @xmath36 @xmath71 . the first two terms , the real part , vanish at @xmath16 @xmath36 @xmath145 if and only if @xmath155 @xmath36 @xmath156 @xmath54 @xmath157 @xmath158 . since @xmath108 is never purely real and negative , a pole both in the real part as well as in the imaginary part of @xmath108 is also a pole of re@xmath159 , cf . figure [ fig - sqrt ] . thus in ( [ q - s - poles ] ) the real and the imaginary terms of @xmath108 resulting in a pole of order 2 have to vanish simultaneously at @xmath16 @xmath36 @xmath145 , i.e.  @xmath154 @xmath36 0 and @xmath63 @xmath36 1 . this means that the integral @xmath160 exists only for photons with frequencies @xmath161 . in particular , with ( [ k - m ] ) we obtain @xmath162 @xmath36 @xmath163 . this leads to the functions @xmath164 @xmath165 now let @xmath89 be a solution of @xmath70 @xmath36 0 with the asymptotic behavior ( [ u - asymptotic ] ) . then @xmath166 solves @xmath167 @xmath36 0 and is therefore a wave of spin @xmath123 , too . by @xmath168 @xmath36 @xmath121 it follows that @xmath169 @xmath36 0 , i.e.   @xmath170 is a wave with spin @xmath171 . hence the wronskian of the two solutions of @xmath172 @xmath36 0 is given asymptotically by @xmath173      &      \begin{array}{c } \\[-4.25ex ]          \phantom{\scriptstyle r \to \infty}\\[-1.5ex ]          \longrightarrow\\[-1.5ex ]          { \scriptstyle r \to \infty } \\[-1.5ex ]      \end{array }      &      2 { \mathrm{i}}\omega \ ,       ( |a_{\mathrm{out}}|^2 - |a_{\mathrm{in}}|^2 ) ,      \nonumber \\ *      \left[u_s , u_{-s}^*\right ]      &      \begin{array}{c } \\[-4.25ex ]          \phantom{\scriptstyle r \to m}\\[-1.5ex ]          \longrightarrow\\[-1.5ex ]          { \scriptstyle r \to m } \\[-1.5ex ]      \end{array }      &      2 { \mathrm{i } } \ { \mathrm{re}}\ , k\      ( |b_{\mathrm{out}}|^2 - |b_{\mathrm{in}}|^2 ) , \label{[u , u]-asymptotic}\end{aligned}\ ] ] since the wronskian is constant for an equation of the form @xmath174 @xmath36 0 , the assertion follows . an extremal kerr black hole is given by the maximally possible angular momentum @xmath27 @xmath36 @xmath0 . in this case , we have @xmath145 @xmath36 @xmath175 @xmath36 @xmath0 , and the functions @xmath84 and @xmath140 in ( [ rho - delta ] , [ k - def ] ) simplify to @xmath140 @xmath36 @xmath176 @xmath54 @xmath117 , @xmath84 @xmath36 @xmath158 . according to ( [ q - s - def ] ) this yields @xmath177 now @xmath108 has a pole of order 4 at @xmath178 . to apply a similar consideration as in the proof of theorem [ satz - tunneling ] , we have to arrange the nominators to obtain a zero of order 3 . for convenience , we again consider the imaginary and the real part of @xmath108 separately . first we note that @xmath115 must vanish to avoid a pole of order 3 , i.e.  @xmath115 @xmath36 0 , or equivalently @xmath179 with this , @xmath140 simplifies to @xmath140 @xmath36 @xmath180 , and therefore @xmath181 i.e. , the imaginary part of @xmath108 has a pole of order 1 . regarding the real part , we insert our derived expression for @xmath140 to obtain @xmath182 at @xmath16 @xmath36 @xmath0 , the nominator vanishes if and only if @xmath47 @xmath36 @xmath183 @xmath52 @xmath184 , or with ( [ omega - m - extremal ] ) , @xmath185 therefore , massless spin waves only can tunnel out of an extremal kerr black hole under a restrictive condition on the separation constant @xmath47 . since @xmath184 , is integer for integer spin @xmath31 , @xmath39 has to be even to guarantee that @xmath186 and thus @xmath47 is an integer . ( remember that @xmath47 has to be integer for integer spin @xmath31 . ) on the other hand , @xmath39 also has to be even such that @xmath47 is a square of an odd - half integer for @xmath63 @xmath36 @xmath187 . ( note : @xmath184 @xmath36 @xmath188 , and @xmath47 @xmath36 @xmath189 ; since @xmath190 @xmath36 0 or 1 mod 4 , @xmath191 @xmath36 1 or 3 mod 4 ; this is only a square number if @xmath191 @xmath36 1 mod 4 , because any square number @xmath192 @xmath36 0 or 1 mod 4 . ) for @xmath1 @xmath36 0 and 0 @xmath147 @xmath193 @xmath194 @xmath195 , the kerr - newman spacetime represents a static charged black hole , the reissner - nordstrm black hole . a general master equation like teukolsky s equation governing massless spin waves in the reissner - nordstrm geometry and leading to radial and angular equations like ( [ radial equation ] ) , ( [ angular equation ] ) could not be derived to date . nonetheless , the basic equations governing electromagnetic and gravitational waves ( @xmath63 @xmath36 1 , 2 ) can be transformed to @xmath196 @xmath36 0 , where @xmath197 @xmath36 @xmath198 @xmath52 @xmath199 @xmath52 @xmath200 with @xmath201 and @xmath202 . \label{g - s - rn}\ ] ] here the separation constant @xmath192 is given by @xmath203 @xmath36 @xmath204 , the indices @xmath205 and @xmath206 take the two values @xmath207 @xmath36 1,2 , but @xmath208 , and the constants @xmath209 are given by @xmath210 these relations can be readily deduced from equations ( 244 ) and ( 245 ) in @xcite ,  5 p.  244 .    by the same procedure as in the proof of theorem [ satz - potential ] the spin - weighted wave equations can be transformed to @xmath211 @xmath36 0 with @xmath121 @xmath36 d@xmath212d@xmath213 @xmath52 @xmath108 and @xmath108 @xmath36 @xmath200 @xmath54 @xmath214 @xmath54 @xmath215 . but since @xmath216 @xmath36 @xmath217 @xmath52 @xmath218 , @xmath108 has a pole of order not smaller than 2 as long as @xmath219 . hence for waves of spin - weight @xmath63 @xmath36 1 or 2 no tunneling out of a reissner - nordstrm black hole can occur for a particle of mass @xmath220 and electrical charge @xmath221 let be given the constant @xmath222 @xmath36 @xmath223 , such that @xmath224 is the compton wavelength . then it is described by the dirac equation in a general spacetime , cf . @xcite eq . ( 16 ) ( for @xmath225 ) . in the kerr - newman geometry it can be reduced to the complex radial differential equation @xmath226 @xmath36 0 , where @xmath197 @xmath36 @xmath198 @xmath52 @xmath199 @xmath52 @xmath200 with @xmath227 and @xmath228 ( note that @xmath72 is related to the usual radial function @xmath41 by ( [ radial transformation ] ) ) , and the angular equation @xmath229 , with @xmath230 here @xmath231 and @xmath232 . the constants @xmath47 , @xmath48 @xmath32 @xmath49 are separation constants obtaining discrete values depending on the boundary conditions of @xmath43 . for details see @xcite , for the kerr case ( @xmath30 ) see @xcite ,  104 eqs . ( 123 ) , ( 124 ) .    to analyze the potential , a transformation analogous to the proof of theorem [ satz - potential ] has to be done , where now by @xmath233 , @xmath234 cf . ( [ g-s ] ) . this yields a complex potential @xmath108 differing from ( [ q - s - def ] ) only by terms depending on @xmath222 . as long as @xmath47 @xmath235 @xmath236 , these terms are of order @xmath237 , and the proof of theorem [ satz - potential ] in essence can be applied . hence no tunneling of electrons out of a kerr - newman black hole can occur , unless for the case of extremal rotation . if @xmath47 @xmath36 @xmath236 the situation changes , since now the terms by which @xmath108 differs from the massless case , are of order @xmath238 . first we notice that @xmath47 @xmath52 @xmath239 @xmath36 @xmath240 . therefore , with @xmath84 @xmath36 @xmath241 , @xmath242 @xmath36 1 , @xmath243 @xmath36 @xmath244 , we obtain @xmath245 with these equations the terms deciding the occurence of a pole of order 2 at the horizon @xmath16 @xmath36 @xmath145 are determined by @xmath246 to avoid a pole of order 2 , the imaginary part has to vanish . this implies @xmath247 ( even in the extremal case @xmath0 @xmath36 @xmath248 @xmath52 @xmath193 , for then @xmath84 @xmath36 @xmath158 and @xmath0 @xmath36 @xmath175 ) , or @xmath249 the real part @xmath250 @xmath251 re@xmath108 @xmath36 @xmath252 then vanishes if and only if @xmath145 @xmath36 @xmath175 @xmath36 @xmath0 , which is equivalent to @xmath195 @xmath36 @xmath248 @xmath52 @xmath193 . this is the extremal case , and we see that then @xmath140 @xmath36 @xmath253 @xmath254 , i.e.  @xmath255 or @xmath256 therefore , @xmath108 has a pole of order 2 at the event horizon , such that no tunneling may occur for @xmath47 @xmath36 @xmath236 . in the present paper we consider properties of spin - weighted waves in various kerr - newman spacetimes . we analyze the singular potential barriers of the respective wave equations emerging at the event horizon . it turns out that the potential barriers are repulsive enough to prevent spin waves with spin @xmath63 @xmath36 0 , @xmath187 , 2 from tunneling out of the black hole , but weak enough to admit the tunneling of photons in case of non - extremal rotation .    in particular , if @xmath1 denotes the angular momentum of a kerr - newman spacetime and @xmath2 its electrical charge , then the following table shows the occurence of tunneling ( ` @xmath257 ' ) , its impossibility ( `  ' ) and the present ignorance ( ` ? ' ) with respect to the various kinds of spins @xmath31 . @xmath258      spin $ |s|$ & $ 0 $ & $ \frac12 $ &    $ 1 $    &    $ 2 $ \\[.5ex ] \hline      $ a=0 $ &    --      &     --      &    --     &    --     \\      $ 0<|a|<m$ , $ q=0 $            &    --      &     --      & $ \times$ &    --     \\      $ |a|=m$ , $ q=0 $           & $ \times$ & $ \times$   & $ \times$ & $ \times$ \\      $ 0<|a| , |q|<m$            &    --      &     --      &     ? &     ?     \\      \hline \hline \end{tabular}% } \ ] ] the case @xmath259 , i.e.  charged rotating kerr - newman black holes , still remains unknown . the crucial role that rotation plays in the known cases suggests that photons may tunnel out of _ any _ rotating black hole . a very strong hint supporting this conjecture is the analysis of one of the authors @xcite , where any electromagnetic wave _ in the outer space _ of a rotating kerr - newman black hole can gain some energy from the hole . however , we still have no such hints for gravitons . for the case of extremal rotation , @xmath260 , any particle can tunnel out of the black hole . because of the cosmic censorship hypothesis ( [ ccc ] ) , @xmath2 vanishes necessarily in this case . there is only the restriction on the separation constant @xmath47 given by ( [ lambda - sm - extremal ] ) to be an even integer . whether such an integer exists at all depends on the sturm - liouville equation for @xmath43 , ( [ angular equation ] ) or ( [ angular dirac ] ) , respectively . mathematically , in case of @xmath30 or @xmath261 a necessary and sufficient condition for the occurence of the tunnel phenomenon is the vanishing of @xmath262 , the factor of the first derivative in the radial wave equations , and of the function @xmath140 , cf . equations ( [ g - s ] ) , ( [ k - def ] ) for massless waves of spin @xmath31 , and ( [ g - s - kn ] ) , ( [ k - def - kn ] ) for spin-@xmath187 waves of mass @xmath220 @xmath36 @xmath263 . this condition yields a potential with an order-1 pole at the event horizon @xmath16 @xmath36 @xmath145 , see figure [ fig - pot ] .     the real and imaginary part of @xmath108 for @xmath264 , @xmath30 , @xmath265 , @xmath266 , @xmath161 ; also plotted is the constant @xmath267 . the tics on the horizontal @xmath16-axis are in units of @xmath0 . note the poles at @xmath145 @xmath36 @xmath268 and @xmath175 @xmath36 @xmath269 . ] remarkably , the real part of a potential with a pole of order one is attractive from the outer space of the black hole and repulsive from the interior , but the imaginary part tends to @xmath270 from the outside and to @xmath271 from the interior . a wave packet inside the black hole with nonvanishing outgoing amplitude @xmath94 then reaches the outer space with nonvanishing amplitude @xmath92 ; if , e.g. , @xmath94 is known at @xmath178 and @xmath272 , the outside amplitude is given by ( [ conservation - law ] ) with @xmath273 .    where does this all lead us to ? notably , the tunnel phenomenon studied in this paper is a semiclassical phenomenon . this implies that the black hole gravitational field has to be strong enough compared to the typical energy of the wave field such that it does not perturb the curvature essentially , i.e.  such that it is a test field . the tunnel phenomenon seems to be related to another semiclassical phenomenon , the hawking radiation . but whereas the hawking radiation is a black hole thermodynamical effect occuring in particular for static black holes , the tunneling only exists in case of rotation . as such it resembles the classical superradiance effect , but this affects only , and all , particles with integer spin . a complete theory of quantum gravity must include the tunnel phenomenon  and should explain it . especially the puzzles concerning gravitons and photons in charged rotating black holes should be solved . to date , the most promising candidate for a theory of quantum gravity is m - theory . it has made strong progress in the last few years , not only with respect to black hole phenomena . one of the most remarkable aspects in this context certainly is the deduction of the notion of entropy of a black hole @xcite . however , to our knowledge the analysis of rotating black holes , be they uncharged or charged , in m - theory is still missing .    to conclude , rotation in general relativity over and over again reveals surprising phenomena , a fact which is not only demonstrated by @xcite , @xcite , or @xcite . succeeding in integrating rotation in m - theory would give deep insight .
the potentials of spin - weighted wave equations in various kerr - newman black holes are analyzed . 
 they all form singular potential barriers at the event horizon . applying the wkb approximation it is shown that no particle can tunnel out of the interior of a static black hole . 
 however , photons inside a non - extremely rotating kerr black hole may tunnel out into the outer space , whereas neutrinos , electrons , and gravitons may not . 
 if the rotation is extremal , any particle may tunnel out , under restrictive conditions . 
 it is unknown whether photons and gravitons may tunnel out if the black hole is charged and rotating .