Gravitational Field of a Spherical Perfect Fluid

For the case of the Kerr black hole we confirmed our previous result - quantum particles do not cross the black hole horizon. We have also came up with the criteria for the particle to be reflected from the Kerr black hole, which is that its azimuthal quantum number should be zero. Next, we have shown that the ordinary Schwarzschild metric does not correspond to a spherically distributed normal ideal fluid and to model gravitational field of an isolated realistic source, including black holes, one needs to use more general metric with the extra metric function. The minimal radius, where the ordinary Schwarzschild metric still can be used, is found.


Introduction
Spacetime singularities are an inevitable feature for many of the solutions of the Einstein equations [1]. The most important solution of this type is the Schwarzschild metric: where we introduced the notation: In the used natural system of units (c = = G = 1) the metric function A(r), which determines the Schwarzschild horizon, has the dimension of the length, while another metric function B(r) is dimensionless. The special case of (1), with B(r) = 0, corresponds to the ordinary Schwarzschild metric, which is often used to describe Black Holes (BHs).
The only way to explore gravitational field is the observations of paths of particles. Currently the most accepted criterion of geometrical singularities is based on the investigations of geodesic incompleteness [2,3]. Based on the analysis of the classical geodesic equations, it was found that BHs are characterized by surrounding event horizons, which screen the naked singularities at the center [4][5][6]. To avoid the problems with this central singularity in 1939 paper Einstein demonstrates that there is a lower limit for the orbital radius of a classical particle in a circular orbit around the Schwarzschild sphere, and concludes that particles will not cross the horizon and reach the center of the Schwarzschild solution [7]. However, this result obtained using the classical Hamilton-Jacobi (geodesic) equation, appears to be wrong. It is known that in the case of so called coordinate singularities (for which the curvature invariants are finite), by specific singular coordinate transformations the horizon can be "repaired" and cannot prevent particles to reach the central singularity [4][5][6].
For the ordinary Schwarzschild metric (B = 0 and A = 2M in (1)) the result in the direction of Einstein's hopes, but for quantum particles, was obtained in our previous papers [8,9]. While the singular coordinate transformations (which are necessary to hide the horizon singularity) do not cause problems on the level of the classical geodesic equations (3) (which contain the first derivatives of the coordinate functions), they usually lead to the appearance of delta-functions at the horizon, r = 2M, in the equations of quantum particles (which contain the derivatives of the generalized momenta), via Euler-Lagrange equations, i.e. the second derivatives of the wavefunction φ [8,9]. This fact restricts singular coordinate transformations, like introduced by Kruskal-Szekeres, Eddington-Finkelstein, Lemaître, or Gullstrand-Painlevé, which give delta-functions in the second derivatives. Indeed, for the case of the Schwarzschild BH, these coordinates contain either the factor √ 2M − r, or ln |2M −r|, the second derivatives of which lead to the appearance of delta-functions at r = 2M.
Using physically boundary conditions at a BH horizon, in our previous paper we have found the real-valued exponentially time-dependent solutions (with the complex phases) of equations for quantum particles in Schwarzschild's coordinates (without performing singular coordinate transformations) [8,9]. This means that particles after all do not enter the horizon, but are absorbed and some are reflected by it. The consequence of this observation is that the minimal radius of any isolated body is its Schwarzschild radius, which can potentially solve the main BH mysteries.
Note that similar real-valued solution in Schwarzschild coordinates was obtained in [10], where it was nevertheless assumed that classical geodesics are extendable across the horizon, while the complex phase was connected with the creation of particles by the gravitational field of BHs. However to regularize the singularity of the particles propagator at the horizon (which shows that classical particles are stopped from entering the BH), the singular point was removed by the introduction of the infinitesimal surrounding integration contours in the definition of the new radial coordinate.
In our opinion the argument against our conclusion (that the horizon is special place for quantum particles), that for a large BH the Kretschmann invariant, diverges only when r → 0, while at the horizon is small, and thus, gravitational effects should be small there, does not works. The smallness of curvature invariants does not mean that curvature tensor, i.e. gravitational field, is also small. For example, the three from six non-zero independent components of the mixed Riemann tensor, diverge at r = 2M as well. The equations of motion of a system of classical particles in the quadruple approximation, unlike the Hamilton-Jacobi equation (3), contain derivatives of the 4momentum and the Riemann tensor itself [11], where J αβγδ is the quadruple moment of the source, S αβ is the spin tensor and u ν is the 4-velocity. Therefore, already on the classical level the particles will feel the spin and quadruple forces, F µ , which for the Schwarzschild case diverges at the horizon r = 2M. Note that the validity of the expression (5) at r = 2M is obtained from the assumption of the type 0/0 = 1. Same is right for the determinant of Schwarzschild's metric, where the product of the components of metric tensor, g tt · g rr , is ill-defined at r = 2M. In general g tt and g rr are independent functions and the cancelation of their zeros for the spherically symmetric case is accidental only, since follows from the validity of the vacuum Einstein equations. However, exact spherical symmetry and true vacuums are rarely, if ever, observed.

Klein-Gordon equation in Kerr spacetime
Macroscopic BHs are formed from collapsing stars and they should have an angular momentum. Therefore, it is important to demonstrate that quantum particles do not cross the horizon for Kerr BHs as well. Let us consider a quantum particle close to the Kerr BH event horizon and demonstrate that there is a damping of the wave function, like in the Schwarzschild case [8,9].
Due to the facts that General Relativity does not care about spin, for considering the motion of quantum particles it is sufficient to study the Klein-Gordon equation, which in the field of uncharged Kerr BH of the mass M and with the angular parameter a can be written in the form [12]: where µ is the particle mass and ∆ = r 2 − 2Mr + a 2 . Let us suppose that in the Kerr spacetime it is possible to separate variables in the wavefunction in the form: Unlike the common case here we have chosen the real-valued time dependence, e −ωt , since in the limit a → 0 the Kerr solution reduces to the Schwarzschild one, which was investigated in our previous papers [8,9]. This leads to the fact that the mixed term in (8) (which contains ∂ 2 /∂t∂φ and is absent in the Schwarzschild case) becomes complex and to eliminate it we need to set the azimuthal quantum number, m, to zero. Substitution of (9) into (8) gives the following system of equations: where C is the separation constant. Changing in (11) the radial variable, r, by the physical variable, ∆, and taking m = 0 (to make the equation real), we find, Near the horizon, ∆ → 0, this equation takes the form: where we had introduced the parameter, .
The regular solution to the equation (13), indicates that as a particle moves toward the horizon, ∆ → 0, its radial wavefunction tends to zero, R(∆) → 0, as for the Schwarzschild case [8,9], i.e. quantum particles do not cross the horizon. In difference with the Schwarzschild metric, due to the coupling of the particles and BH's angular momentums, the reflection by the BH is possible only for the infalling particles with the zero azimuthal quantum number, m = 0. For completeness let us show that another component of the wavefunction (9), S(θ), stays finite at the horizon. The change of the variable, cos θ ≡ x, brings the equation (10) to the form: which has the solution in Legendre polynomials, i.e. S(θ) is a finite function.

Metric of spherical ideal fluid
It seems natural to replace any singular point in Riemann spacetime by a matter source, e.g. nobody worries about central singularity of the Coulomb potential. The basic idea behind General Relativity is that geometry does not exist separately from matter and any physical vacuum solutions of the Einstein equations should be associated with some realistic matter source. This should also be true for the Schwarzschild metric (1), for which the Einstein gravitation field equations can be written in the form: where primes denote derivatives with the respect to the radial coordinate r. From the first equation of this system one can find The function M(r) here is always positive quantity (the energy density, T t t , is assumed to be positive) and is a measure of the amount of a mass located within a surface of the radius r.
Since most isolated bodies, large enough to support a strong gravitational field, are constructed of material in a state of high fluidity, the most natural assumption for the other components of the energy-momentum tensor in (17) would be the equation of state of a perfect fluid, where p represents the pressure, which usually is assumed to be also a positive quantity. Then from the Einstein equations (17) one obtains a single second order non-linear differential equation for the metric functions A(r) and B(r): One trivial solution to this equation, corresponds to the ordinary Schwarzschild case, where the integration constants are usually used to switch between the exterior (C 2 = 0) and interior (C 1 = 0) Schwarzschild metrics. It is believed that the ordinary Schwarzschild metric (22) corresponds to the gravitational field of a realistic spherically symmetric mass, including BHs. However, for an ideal fluid (19), for which B ′ (r) equals to zero in whole space-time, from (17) one finds Then: a). If p is positive, the energy density appears to be negative; b). The negative pressure violates the Strong Energy Condition; c). The case p = 0 corresponds to the sourceless situation, This means that the ordinary Schwarzschild metric (22) does not correspond to a spherically distributed normal perfect fluid. In addition, we think that it is misleading to consider (22) as the solution of the free Einstein equations in the whole spacetime, due to the presence of the central singularity, where some kind of matter source should be placed. So to study gravitational field of a spherically symmetric isolated ideal fluid and behaviors of geodesics in the fields of realistic BHs, we need to use the more general metric (1), which approaches the ordinary Schwarzschild form (with B ′ = A ′ = 0) only far from the source. Let us find radii where the mass function A(r) = 2M(r) can be considered as a constant, and thus the ordinary Schwarzschild metric (22) is valid. In the approximation A ′ → 0 the equation (20) reduces to the form: This equation contains two special points. One of them corresponds to the Schwarzschild horizon, r = 2M, while the second one, at r = 5M, is located beyond the Schwarzschild sphere. From (25) it is clear that the assumption A ′ , B ′ → 0 is valid only when Already in the region 2M < r < 5M , one cannot assume that the functions M(r) and B(r) are constant, i.e. the pressure terms (19) are negligible. This means that a spherical perfect fluid source, which is described by the metric (1), cannot be regarded as isolated in the volume of the radius ∼ 5M. This also implies that realistic Schwarzschild BHs are not described by the vacuum metric (22) and to explore behaviors of particles close to a BH horizon one needs to use the general spherically symmetric solution (1). Note that the situation that close to a BH horizon properties of geodesics are changing, i.e. Schwarzschild BHs acquire an effective 'atmospheres' [13], reminds the 'firewall' conjecture [14,15], and can be modeled by the introduction of an effective cosmological constant in this region [9].

Conclusions
To conclude, in this paper for the case of the Kerr BH we had confirmed our previous result, that quantum particles do not cross the BH horizon. This fact demonstrates that one should be careful with physical interpretation of the results obtained using singular coordinate transformations. We have also came up with the criteria for the particle to be reflected from the Kerr BH, which is that its azimuthal quantum number should be zero. The fact that particle cannot cross the horizon means that minimal radius of any isolated body is its Schwarzschild radius, which potentially can solve the main BH mysteries.
Next, we have shown that the ordinary Schwarzschild metric does not correspond to a spherically distributed normal perfect fluid and to model gravitational field of an isolated realistic source, including BHs, one need to use more general metric with the extra metric function. It is found that already in the region 2M < r < 5M beyond a BH's Schwarzschild horizon one cannot assume that the mass parameter M is a constant.