The smeared-horizon observer of a black hole

A class of observers is introduced that interpolate smoothly between the Schwarzschild observer, stable at spatial infinity, and the Kerr-Schild observer, who falls into a black hole. For these observers the passing of the event and inner horizon takes a finite time, which diverges logarithmically when the interpolation parameter $\sigma$ goes to zero. In the field theoretic approach to gravitation, the behavior at the horizons becomes regular, making the mass of the metric well defined.


The outgoing smeared-horizon observer 1 Introduction
The most amazing property of black holes is their event horizon, which even led Einstein initially to be skeptical about the Schwarzschild metric.An observer at spatial "infinity" will never live the day to observe something falling into a black hole.The related infinite redshift at the event horizon is the ultimate limit of the redshift known from sirenes on cars that move away from us.
On the other hand, there are the Painlevé-Gullstrand and Kerr-Schild metrics, which do not exhibit a horizon for the observers related to them.It is the aim of the present paper to introduce a class of intermediate observers, the "smeared horizon observers", which interpolate between the Schwarzschild and Kerr-Schild metrics by a free parameter σ.Some related properties are discussed.
In section 2 we introduce the metric of the smeared-horizon observers and consider properties like its eigenvectors and their behaviors in the black hole interior, and analyze outgoing and ingoing spherical mass shells.In section 3 we connect to recent exact solutions for the black hole interior and its role for the smeared-horizon observer.In section 4 we connect to a recent class of exact solutions for smooth, cored black hole interiors.In section 5 we show that the field theoretic approach to gravitation, connected to the Landau-Lifshitz pseudo tensor for the gravitational field, becomes well defined at the would-be horizons and hence everywhere, so that allows to properly define the mass of the metric.

Generalized Schwarzschild and Reissner-Nordström metric
For smooth functions N (r) and S(r) we consider the generalization of the Schwarzschild metric in spherical coordinates r µ = (t, r, θ, φ), µ = 0, 1, 2, 3, with dΩ = (dθ, sin θdφ).The Schwarzschild metric [1] is described by N (r) = 1, S(r) = 2GM/r; the Reissner Nordström metric [2,3] by . The latter has an event (e) horizon and an inner (i) horizon, where S(R e,i ) = 1 ( S = 0).They are located at In our units = c = 1 and µ 0 = 4π, the Planck mass is m P = 1/ √ G.It holds that 0 < S < 1 in the outer space, so that S < 0 there.Inside a core bounded by the inner horizon R i , one has S < 1.In the Schwarzschild metric, the inner horizon coincides with the origin, but this is a special case.In the Reissner Nordström metric, S → −∞ for r → 0.
We have recently proposed a class of exact solutions where S is regular with S ∼ r 2 for r → 0. The latter property implies the presence of a finite core bounded by an inner horizon R i .As in the Reissner Nordström metric, the region between the inner and event horizons, termed the mantle, is a standard vacuum, described by the Reissner Nordström metric with S > 1.In these models one has N (r) = 1 for r > R i and 0 < N ≤ 1 for r < R i .

The Painlevé-Gullstrand observer
De één heeft meer in zijn mars dan de ander 1

Dutch expression
One of the mysteries of black holes is that in its interior, from the event horizon to the inner horizon, the roles of r and t are interchanged.The reversed role of r and t in its interior is counterintuitive, and so is the infinite redshift for signals from the horizon to a stationary observer at spatial infinity.
The first step to investigate the issue was made by Painlevé in 1921 [4], and Gullstrand in 1922 [5], actually intended to question the Schwarzschild solution.For completeness, we recall their approach.In the Schwarzschild metric they introduce a new time coordinate by setting dt S = dt pg − (2GM/r) dr/(1 − 2GM/r), so as to obtain This metric is regular except for r → 0, so that the observer does not notice an event horizon.In integral form one has t S = t pg −2GM {2y−log[(y+1)/(y−1)]}, where y = r/2GM .CONTENTS

The ingoing Kerr-Schild observer
In this work we follow the approach of Kerr-Schild [6] to generate new metrics.We start by allowing in the Kerr-Schild metric a dilated time dt → N (r)dt ks , The one-form dk = k µ dr µ involves a null vector k µ , viz.g µν ks k µ k ν = 0.The standard cases and their connection to t S are dk = k µ dr µ = N dt ks + dr, dt S = dt ks + S N S dr, (iks), (2.5) where iks stands for an ingoing Kerr-Schild observer and oks for an outgoing one.Unlike (2.1), this is regular also at S = 1, S = 0.For the iks case it reads The inverse iks metric is coded in Indeed, k µ is a null vector, k µ k µ = N 2 g 00 + 2N g 01 + g 11 = 0.The situation for the oks is obtained by setting dr → −dr and ∂ r → −∂ r .

The ingoing smeared-horizon observer (sho)
The present work proposes a new class of observers, to be called "smearedhorizon observers" (sho); the ingoing ones are for some σ ≥ 0 defined by S dr. (2.9) The latter form interpolates between Schwarzschild's stationary observer at infinity (σ = 0) and the ingoing PG observer (σ → ∞).This observer still falls in, but slower than the latter.We shall take σ constant, though it can actually be a function of r.
The line element takes the form The inverse metric is coded in At any finite σ, g µν and g µν are regular, notably at the would-be horizon(s) where S = 0. Taking σ small exposes Schwarzschild's event horizon at an arbitrary precision.

Eigenbasis of the ingoing smeared-horizon metric
The eigenvalues of g ish µν in (2.10) are, next to λ 2 = −r 2 and λ 3 = −r 2 s 2 θ , with L defined by Outside the BH core, one has N = 1 and S = 2GM/r − GQ 2 /r 2 , so that L → GM/r at large r.In the limit σ → 0 one has sinh 2L → 1 2 (N 2 S2 − 1)/N S, which diverges at S = 0 and changes sign there; at small σ, these divergences are rounded.In that case, L changes sign at S = σ 2 + O(σ 4 ), which codes two locations in the mantle, one beyond R i and the other below R e .
For σ → ∞ one describes the Kerr-Schild observer; with N ≤ 1 and S ≥ 0, L and E remain positive, making t act as timelike coordinate also in the interior.
For small σ there is a transition region which regularizes the singularity at the event horizon.The regime S < 0 applies to the outer space and also to the BH core; in both cases, the prefactor 1/σ 2 in sinh E makes E ≫ 1 and hence dξ 0 ∼ dt and dξ 1 ∼ dr as usual.In the mantle, the opposite case S > 1 involves sinh E < 0 and E ≪ −1, so that there is the switched connection between t and r, dξ 0 ∼ −dr and dξ 1 ∼ dt, known from the interior of the Schwarzschild BH.The width of the transition region is ∆ S ∼ σ 2 .For small σ one can analyze the transition by coding r in a parameter λ such that In the absence of surface layers, N = 1 at the event and inner horizon, whence r reads where w = ± √ M 2 − Q 2 , with the + (−) sign at the event (inner) horizon.Starting inside the mantle near the event horizon and going outwards, λ increases from negative to positive values, with the event horizon S = 0 located at λ c = ln( √ 2 + 1).Related behavior occurs around the inner horizon.

Outgoing shells in the frame of the ingoing sho
An outgoing spherical shell for a massless field involves dξ 0 = dξ 1 , that is,

.20)
This vanishes at the inner and event horizons where E = L.With N = 1 outside the core, the large r behaviors L → GM/2r and sinh 2E → (1 + σ 2 )r/2GM σ 2 , imply dr/dt → 1 − 2GM σ 2 /(1 + σ 2 )r, an outgoing motion.Eq. (2.15) shows that the crossing E = L occurs at the would-be horizons R i,e where S = 1.Setting t e = −1/S ′ (R e ), the relation leads for r > R e to r − R e ∼ exp(t/2t e ) and for r < R e to R e − r ∼ exp(t/2t e ).The passing of the event horizon occurs for t → −∞.Near the inner horizon we set t i = 1/S ′ (R i ) and obtain likewise r − R i ∼ exp(−t/2t i ) and for r < R i to R i − r ∼ exp(−t/2t i ); the passing occurs for t → +∞.In the mantle, t decreases when r increases.Let us apply this to the Schwarzschild metric (S = 2GM/r and N = 1), employ units 2GM → 1, so that t e → 1, and define r = r − 1.One has It keeps the Schwarzschild singularity t ∼ | log r| for the time to go from a point 0 < r − 1 ≪ 1 just outside of the event horizon to a location well away.Including a charge Q, the adimensional units express S = 2GM/r−GQ 2 /r 2 as S = 1/r − q 2 /4r 2 with q = m P Q/M .It yields where r e,i = 1 2 (1 ± 1 − q 2 ) denote the event and inner horizons, respectively, and the r j are the 4 complex roots of S2 + σ 2 = 0. Since they arise from S = ±iσ, they take the explicit forms It is seen that t keeps logarithmic divergences for signals emitted close to r e,i .As above for q = 0, the effect of a finite σ is to double their prefactor.In other words, for σ → 0, ∆t absorbs half of the logarithms of the first line in (2.23).But at finite σ, ∆t itself is regular for all real r.
In conclusion, in the description by the ingoing sho, it takes infinite time for the shell to emerge from the BH.This stems with the popular statement "nothing can escape from a black hole".CONTENTS

Ingoing shells described by the ingoing sho
An ingoing spherical shell for a massless field involves dξ 0 = −dξ 1 , that is, which is finite for E = L, that is, at the horizons.Integrating this for the Schwarzschild metric yields

.27)
This remains finite as t ∼ log 1/σ for r → 1, so the ingoing massless shells are observed to go quickly through the event horizon.For the charged case, it involves ∆t of eq.(2.23), (2.28) Compared to the outgoing case (2.23), next to r → −r, the explicit logarithms disappear, while the root sum is maintained.
In conclusion, for the incoming sho, incoming shells only need a finite time to pass the event horizon: for that process, no event horizon is noticed.

The outgoing smeared-horizon observer
So far we considered the ingoing PG observer.The outgoing PG observer relates formally to the Schwarzschild observer by r-reversal in going from (2.5) to (2.6), so that dt S = dt ks − S N S dr (3.29) Correspondingly, there is a sign change in the shift term in (2.9).
The reversed role of ingoing and outgoing motion corresponds to timereversal, and turns the black hole into a so-called white hole.In the exterior, dr/dt > 0 leads to time delay dt S /dt > 1.In the Schwarzschild interior, increasing time-like coordinate r corresponds to increasing time.
With dr → −dr and fixing the overall sign such that e 0 → (1, 0, 0, 0) and e 1 → (0, 1, 0, 0) for large r, the eigenvectors ( In the exterior, one has N = 1 and E ≫ 1 for small σ, so that dξ 0 = e −L dt and dξ 1 = e L dr, as usual; in the mantle E ≪ −1 so that dξ 0 = e −L dr and dξ 1 = −e L dt; and in the core dξ 0 = √ N e −L dt and dξ 1 = √ N e L dr.Outgoing shells are described by dξ 1 = dξ 0 , so that and ingoing ones by dξ 1 = −dξ 0 , so that Apart from an overall sign, this coincides with (2.20).Hence ingoing shells, described by the outgoing sho, take infinite time to pass the horizon.Outgoing shells, on the other hand, only take a finite time, since (3.33) is regular.

Exact solutions for the black hole interior
A class of exact solutions for the BH metric, which is regular everywhere in the core, was proposed recently.

The stress energy tensor for the sho
For general N, S and σ, the Einstein tensor has the nontrivial elements with G 2 2 = G 3 3 due to spherical symmetry.In the Schwarzschild case σ = 0, G µ ν is diagonal; a value σ > 0 does not modify these elements, but creates the G 0 1 element provided N ′ = 0.This represents a radial energy current for the smeared-horizon observer falling in onto the static energy distribution.So Q(r) is the enclosed charge.The related energy density is Notice that the connections (4.44) and (4.45) are just as in special relativity.
Assuming that ρ λ , ρ q , ρ ϑ and p ϑ vanish in the mantle, our task is to provide their physical meaning in the core r ≤ R i , for suitable functions N , S.

A class of exact solutions
We recently presented an exact solution for a charged black hole core r ≤ R i [7].
Here we recall it and then consider it for the smeared horizon observer.
The basic motivation is that in the stellar collapse, electrons are more easily ejected than protons, so that the black hole is positively charged.Since the Coulomb force is much stronger than the Newton force, the fraction of surplus charge needs only be of order m N /m P ∼ 10 −19 .The binding energy of the nucleons is released when their density is large enough.The rest mass of the up and down quarks and the electrons makes up only 1% of the energy; when it is neglected, the problem allows an exact solution with N = 1, corresponding to a vanishing matter temperature and neglect of rest masses.
Consider a core charge Q c be distributed as with F q = 1 for x ≥ 1.This generates an electrostatic energy density The solution for S(r) which goes from 0 at r = 0 to 1 at R i , is given by with 0 ≤ x ≤ 1.The integrals are well behaved when the charge density ρ q is finite at r = 0, so that F q (y) ∼ y 3 for y → 0. The solution rests on the insight that the zero point energy density of the quantum vacuum can act as a zero point battery or zero point storage, and locally absorb the energy density For continuity with the vacuum, this must vanish at R i , which fixes q i .It holds that s which should be non-negative at this first crossing of S = 1 starting from S = 0 at r = 0. Continuity With s i between 0 and 2, Q c /M c ranges from 1 2 √ 3 to 1, that is to say, from quite charged to maximally charged.
It has been put forward that surface charge layers may be present on the outer side of the inner and event horizons [7].
The same idea of a nonuniform vacuum energy combined with electric fields has been applied to dark matter [8].

The exact solutions for the smeared-horizon observer
The connections (2.9), (3.30) lead to the transformation from r µ S to r µ sh given by ∂r µ S /∂r ν sh = δ µ ν + αδ µ 0 δ 1 ν , with the inverse ∂r µ sh /∂r ν S = δ µ ν − αδ µ 0 δ 1 ν , where α = ± σ 2 S/(σ 2 + S2 )N S. The Einstein tensor transforms as G µ sh ν = (∂r µ sh /∂r μ S )G μν (∂r ν S /∂r ν sh ), which coincides with the diagonal G µ ν tensor of the Schwarzschild case, and contains an extra term The class of exact solutions of section 4.2 involves N (r) = 1 and hence G 0 0 = G 1 1 , so that the G 0 1 term does not show up.But when ρ ϑ and p ϑ are nontrivial, so is G 0 1 .This relates already to the situation where thermal matter still involves a termperature T = 0, but the rest masses of the up and down quarks and electrons are accounted for.They bring deviations at the percent level, as shown in a numerical approach [7].

Einstein gravity as a field in flat space time
The aim of the present section is to consider the mass of the regularized metrics of previous section.When the stress energy tensor T µν is integrated over space, the conservation T µν ; ν = 0 does not lead to a conserved quantity.This led Landau and Lifshitz to derive their pseudo tensor τ µν representing the energy momentum tensor of the gravitational field itself [9].However, it does not transform as a tensor.It has been pointed out that it becomes a proper object provided it is evaluated in Cartesian coordinates and transformed from there [10,11].This approach is based on an underlying Minkowski space time, in which Noether's theorem assures a properly stress energy tensor.

General approach
Gravitation can be described as a field (a "pudding") in a standard Minkowski space, so that fits naturally with the matter fields in the standard model of particle physics.We consider Cartesian and spherical coordinates, dσ 2 = η µν dx µ dx ν = γ µν dr µ dr ν , x µ = (t, x, y, z), r µ = (t, r, θ, φ); and, for some metric g µν , the Riemann metric ds 2 = g µν dr µ dr ν . (5.52) With g = det(g µν ) and γ = det(γ µν ), the combinations act as tensor fields in flat space time.This also applies to the mantle of the BH, which for the Schwarzschild metric is the full interior, even though the role of physical time is played there by the radial parameter r.
One can define the "acceleration tensor"[11] where the column denotes covariant differentiation in flat space with its Christoffel coefficients γ µ νρ vanishing for Cartesian coordinates.The Einstein equations take the form (5.55) Θ µν is conserved in Minkowski space, Θ µν :ν = 0; this condition coincides with T µν ; ν = 0, the conservation of T µν in Riemann space.By eliminating T µν with use of the Einstein equations G µν = 8πGT µν , one gets This expresses t µν in terms of the metric alone.It can be verified that the second order derivatives drop out, together with certain first order derivatives, so that there remains only a bilinear expression in first order derivatives, The fact that all second order derivatives could be collected in A µν arrives from absorbing the square-root factors in k µν = g/γ g µν .
In this field theoretic approach in Minkowski space, it is natural to identify t µν as the stress energy tensor of the gravitational field.In Cartesian coordinates it coincides with the Landau Lifshitz pseudo tensor.This clarifies its role: the Landau Lifshitz approach is correct in Cartesian coordinates; from them one can transform the results to any other coordinate systems.In the above approach, this is guaranteed by the covariant derivatives in flat space.The material stress energy tensor in Minkowski space is (g/γ)T µν .
In the above cases, the determinants of the metrics are γ = −r 4 s 2 θ and g = −N 2 r 4 s 2 θ , which implyies k µν = N g µν .The mass (energy) of the metric is For the Friedman metric in cosmology, it follows [11] that Θ 00 = 0, so that, loosely speaking, "it costs no energy to create a universe".This is the ultimate free lunch, more ultimate than the effect of the cosmological constant (dark energy, inflation) alone, for which the energy cost is known to be compensated by the gain of work.Indeed, Θ 00 = 0 holds also in the radiation and matter phases.

The mass experienced by the smeared-horizon observer
General relativity allows various identifications of mass, in particular the farfield mass experienced by a Newtonian observer.But a complete theory must deal with the near field and behaviour in the interior, and show that all is well there -if it is.As we demonstrate now, this goal is reached for our class of smooth exact solutions observed by the smeared horizon observer.For the metric (2.10), Θ 00 is equal for the ish and osh, reading .59) independent of N .It is seen that σ regularizes the 1/ S2 term, that is, the poles at the horizons.The other non-trivial elements of Θ µ ν , which is diagonal, are independent of σ, The Einstein equations yield for the material (non-gravitational) energy density (5.61) The gravitational energy density t 00 = Θ 00 /N 2 − T 00 is proportional to 1/N 2 and reads (5.62) The definition of Θ µν imposes that Θ 00 is a total derivative.Indeed,

.63)
It leads to the integral The origin r = 0 did not contribute, since S ∼ r 2 in our regularized approach.Actually, the r → 0 value of (5.64) vanishes even in the Schwarzschild case S = 2GM/r and the RN case S = 2GM/r − GQ 2 /r 2 .The important finding is that the quadratic divergencies of (5.63) in the σ = 0 case, at the S = 0 locations of the inner and event horizons, are regulated by any finite σ, so that there is no longer a nasty "integration across the poles".
With regularized S(r) → 0 at r → 0, the mass is E = P 0 (∞); since S = 2GM/r − GQ 2 /r 2 for r > R e , this yields E = M for all σ. (5.65) For the Schwarzschild metric, the mass M is in the field theoretic approach determined by the gravitational field alone, since T 00 = 0 implies that its energy density equals t 00 = Θ 00 , while the singularity at r → 0 is put under the rug.In terms of r = r/2GM it reads when regularized by σ t 00 = 2r − 1 − 3(2 + σ 2 )r 2 + 2(2 + 3σ 2 + 2σ 4 )r 3 − (1 + 3σ 2 + 2σ 4 )r 4 (8πGr 2 ) [(r − 1) 2 + σ 2 r2 ] 3 (5.66)CONTENTS That the integral is regular and finite, yielding M as it should be, cannot hide that Θ 00 and t 00 have an 1/r 2 divergence at the origin, pointing at singular behavior of the Schwarzschild metric.Equations (5.63), (5.61), (5.62) and (5.66) show that the smeared horizon observer encounters for small σ a smeared, non-sharp horizon, and only a true one when the limit σ → 0 is taken first.The mass is thus well defined and takes the far field value M for any finite value of σ.

Outlook
The class of smeared horizon observers which are introduced in this papers allow for a complete and singularity-free description of black holes.On the one hand, the exact solutions of ref. [7] have been carried over without effort to these new observers; on the other hand, their singularities at the inner and event horizons in the field theoretic description of gravitation, get smeared, so that the energy density is finite everywhere and the mass of the black hole well defined.
These smeared horizon observers keep some peculiarities: for the ingoing observer, infalling shells cross the horizon in a finite time, but outgoing shells need infinite time.For the outgoing observer, outgoing shells emerge in finite time, but infalling ones need an infinite time.
This puts forward the possibility to see matter falling into the core of a black hole in a finite time by an ingoing smeared horizon observer; when this shell is next repelled by exerting some force on it, and made to go outwards, and, likewise, the observer is modified into an outgoing smeared horizon observer, he is capable to see the shell emerging in a finite time.A perhaps simpler setup is to investigate, for this type of observation, the geodesic motion of a point particle which enters the black hole mantle and next the core, turns around the origin, and goes out again.
Another open question is the form of Hawking radiation for smearedhorizon observers.