Through a Black Hole Singularity

Flavio Mercati1 and David Sloan2

1Departamento de Fısica, Universidad de Burgos, E-09001 Burgos, Spain;2Department of Physics, Lancaster University, Lancaster UK.

(Dated: September 23, 2021)

We show that the Kantowski–Sachs model of a Schwarzschild black hole interior can be slightlygeneralized in order to accommodate spatial metrics of different orientations, and in this formulationthe equations of motion admit a variable redefinition that makes the system regular at the singularity.This system will then traverse the singularity in a deterministic way (information will be conservedthrough it), and evolve into a time-reversed and orientation-flipped Schwarzschild white hole interior.

The interior of the Schwarzschild metric can be un-derstood as a special case of the Kantowski–Sachs classof spacetimes [1]. These are homogeneous cosmologicalmodels with an S2×R spatial topology, and a spacetimemetric of the form:

dτ2 = −N(σ)2dσ2 +A(σ)2dρ2 +B(σ)2dΩ2 . (1)

The ordinary Schwarzschild metric is found as the par-ticular case

N =(2Mσ − 1

)−1/2, A =

(2Mσ − 1

)1/2, B = σ , (2)

if we call σ = r and ρ = t. This, of course, is only validwhen r < 2M , i.e. the region inside the event horizon,where the r coordinate is timelike and the t coordinateis spacelike.

The manipulation we described allows us to under-stand the Schwarzschild singularity in a similar man-ner to the Big Bang, and to translate progress in theunderstanding of homogeneous cosmological singularitiesinto advancement in the physics of black holes. A re-cent approach [2–4]1 allowed us to prove an existenceand uniqueness theorem for the solutions of Einstein’sequations at and beyond the singularity of homogeneouscosmologies. By extending the configuration space of thetheory to include the information about the orientationof spatial slices, this result allows us to prove that to eachand every collapsing solution ending up in a singularity,there corresponds one and only one expanding solutionthat evolves away from the singularity with opposite ori-entation. The singularity is a degenerate hypersurfacewhich cannot support a nonzero volume because it is ef-fectively one- or two-dimensional. At this hypersurface,the spatial orientation flips.

Thanks to the device described at the beginning, theseinsights on homogeneous cosmologies can be applied toblack hole singularities described as Kantowski–Sachsspacetimes.

1 See also [5] for a closely-related approach, and [6] for a possiblequantum origin for the quiescence mechanism.

THE EMPTY KANTOWSKI–SACHS MODEL

After imposing the ansatz (1), the Einstein–Hilbert La-grangian reads

L = 12κ

∫d3x√−gR = 4πNλ

κ

(A− A(B)2+2BAB

N2

)+ K ,

(3)where κ = 8πGc−4 and λ =

∫ r2r1dρ is the width of a fidu-

cial interval of radii over which we integrate (by homo-geneity, the metric outside this interval will be identical

to the one inside). K = 4πλκN

(AB2 + 2BAB

)appears as

a total derivative, and is therefore a boundary term thatcan be removed (it is minus the Gibbons–Hawking–Yorkterm [7, 8]).

In terms of the canonical momenta PA = ∂L/∂A,PB = ∂L/∂B, we can write the total Hamiltonian H =PAA+ PBB − L as:

H =N

ν2

(P 2AA− 2PAPBB

4B2− ν4A

), (4)

where ν =√

4πλκ . With the following canonical transfor-

mation:

A =e− x√

2

ν, B =

ex+y√

2

ν,

PA = −√

2νex√2 (px − py) , PB =

√2νe− x+y√

2 py ,

(5)

the Hamiltonian takes the simple form:

H =N ν

2e− x+2y√

2

(p2x − p2y − 2e

√2y). (6)

We are free to choose the lapse function N , and the

obvious choice is N = 1ν e

x+2y√2 , which simplifies the pref-

actor and gives us the elementary Hamiltonian

H =1

2

(p2x − p2y

)− e√2y . (7)

This Hamiltonian makes px a conserved quantity, andthe Hamiltonian constraint H ≈ 0 imposes that p2x =

p2y + 2e√2y. This is the Hamiltonian of a one-dimension

arX

iv:2

109.

1075

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[gr

-qc]

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Sep

2021

2

-15 -10 -5 5 10 15x

-15

-10

-5

5

y

2 4 6 8 10

15

10

-5

5

y

e 2 y

FIG. 1. Examples of solutions of the Kantowski–Sachs systemin the x − y plane, with a plot of the potential on the side.The curves asymptotes to the singularity on the left, and thehorizon on the right. The gradient on the background repre-sents the values of the volume degree of freedom v (darker =larger), and the black dots represent the points of maximumvolume.

nonrelativistic point particle with potential 2e√2y and

energy p2x. The general solution to Hamilton’s equationsis:

x = px(s− s1) , py = k tanh

(k(s− s2)√

2

),

y = −√

2 log[√

2|k| cosh

(k(s−s2)√

2

)],

(8)

where px now is a constant of motion, and k, s1, and s2are constants of integration. Moreover, the Hamiltonianconstraint imposes that p2x = k2. The asymptotic com-ponents of the velocity are x = px and y −−−−−→

s→±∞∓|k| =

∓|px|. In the x − y plane, all solutions look like a ballbouncing off an exponential slope and rolling inertiallyto infinity at a 45 angle.

From Fig. 1 we can see that the volume v ∝ AB2 ofour fiducial region is convex (as a function of s), goingto zero as s→ ±∞, and reaching a unique maximum inbetween.

We can calculate the Ricci tensor on the solution (8),and only one component turns out to be nonzero:

Rµν =(k2 − p2x

)δ0µδ

0ν , (9)

and if we impose the Hamiltonian constraint p2x = k2, thespacetime we get is Ricci-flat. To highlight the location ofthe singularity, we can calculate the Kretschmann scalar:

RµνρσRµνρσ =

3ν4(e√2k(s2−s) + 1

)6k4e−

√2k(2s1+4s2)

, (10)

and see that it diverges when sign(k)s→ −∞.

s = const. r = const.r =

2M ,

r = 2M ,

r = 2M

r = ∞

r = ∞

r = 0 , k s = -∞

k s = +∞

ρ = +∞

ρ = -∞

k s = +∞

FIG. 2. The shaded region corresponds to the patch ofSchwarzschild spacetime that is covered by the s, ρ coordi-nates as it appears in the Penrose–Carter diagram. The bor-ders of this coordinate patch are represented by the two reddots, the singularity and the horizon.

OBTAINING THE SCHWARZSCHILD METRIC

The Schwarzschild solution can be obtained by setting2

k(s1 − s2) =√

2 log ν , (11)

then one can see that A and B, expressed in terms of thesolution x, y of (8) through the relations (5), satisfy theequation

A2 = 2MB − 1 , (12)

where, as it turns out, 2M =√2|k|ν2 . We can re-

cover the full Schwarzschild metric by making a timereparametrization s → r that transforms B[s(r)] = r(which is legitimate because, as is easy to check, on-shellB is definite, and B is therefore monotonic), which gives

s = s2 − 1√2k

log(2Mr − 1

), (13)

and transforms the lapse into

N [s(r)]∂s∂r =(√

2|k|ν2r − 1

)−1/2=(2Mr − 1

)−1/2. (14)

Notice that all solutions (8) represent a Schwarzschildspacetime. Those whose integration constants fail to sat-isfy (11) are just associated to a rescaled metric:

N = α(2Mr − 1

)−1/2, A = α

(2Mr − 1

)1/2, B = αr ,

(15)

2 In this Section we assume that px = k, the other case px = −kcan be straightforwardly worked out analogously.

3

where α = ν e− k(s1−s2)−x0√

2 , and therefore a redefinition ofunits can reabsorb this.

The time redefinition (13) gives us another way toidentify the values of s corresponding to the singularity(which is at r → 0). The reparametrization monotoni-cally maps r ∈ (0, 2M) into sign(k)s ∈ (−∞,∞). Thesingularity r → 0+ coincides with sign(k)s → −∞, i.e.when x→ −∞. The other limit x→ +∞ coincides withthe horizon r → (2M)−.

SHAPE SPACE AND ORIENTATION

One linear combination of the x and y variables corre-sponds to the scale degree of freedom, while the otheris conformally invariant and determines the shape ofour spatial hypersurface (in particular, it determinesthe ratio between the radial extension of our coordinatepatch and its areal radius). To disentangle scale andshape, consider the determinant of the spatial metric,

det g = A2B4 = ν−6e√2(x+2y), which is a pure scale de-

gree of freedom. Therefore x + 2y determines the scale,while the orthogonal direction in the (x, y) plane deter-mines the shape. The following linear canonical transfor-mation separates between scale z and shape w:

x =1√3

(2w − z) , y =1√3

(2z − w) ,

px =pz + 2pw√

3, py =

2pz + pw√3

,

(16)

so that now det g = ν−6e√6z depends on z alone. In the

new variables, the Hamiltonian constraint takes a simpleform:

H = 12

(p2w − p2z

)− e√

23 (2z−w) , (17)

notice that, as usual in a constant-mean-extrinsic-curvature foliation, the scale degree of freedom gives anegative contribution to the kinetic term [9].

Notice now that the coordinate change from the (w, z)variables to the original (A,B) variables,

A =e

1√6(z−2w)

ν, B =

e1√6(w+z)

ν, (18)

is not surjective: it only maps R2 to the first quad-rant (A > 0, B > 0) of R2. Normally this would notbe a problem, because the metric (1) depends only onthe square of A and B, and the configuration space ofKantowski–Sachs metrics is more appropriately definedas the quotient of the (A,B) plane by reflections of A anB. However, there is a bit of information that is erasedby this quotienting procedure, which we might want tokeep track of instead. This is the orientation of our spa-tial manifold, which is encoded, for example, in the triadformulation of the metric [10]

gij = δabeaiebj , (19)

-20 -10 10 20w

-15

-10

-5

5

10

z

FIG. 3. The same solutions of Fig. 1, this time plotted inthe w− z plane. The singularity is reached asymptotically asw → −∞, while the horizon is at w → +∞.

the associated volume form e1 ∧ e2 ∧ e3 defines an ori-entation on our manifold. In this formulation, underthe Kantowski–Sachs ansatz the frame field componentsare linear in A and B, and the volume form readse1 ∧ e2 ∧ e3 = AB2. Therefore, the sign of A determinesthe orientation of our spatial hypersurface.

The variable z parametrizes the scale degree of free-dom, while w determines the shape of our spatial mani-fold, and it makes sense to include the information re-garding the orientation into the “shape space” of ourmodel [2, 9]. We can then extend the shape space, bydefining two coordinate patches, w+ ∈ R and w− ∈ R,which are mapped to the two possible signs of A:

A =

e

1√6(z−2w+)

ν, if A > 0 ,

− e1√6(z−2w−)

ν, if A < 0 .

(20)

The above map sends two copies of R onto the two halvesof the real line.

The two possible signs of A correspond to the choicebetween left- and right-handed triads compatible withthe metric, e±. Taking the Schwarzschild solution asour guide we expect that as we approach the singular-ity |A| → ∞. The singularity is potentially a point oftransition between e+ and e−, hence a point at whichthe orientation of our space may change. By extendingour description to the coordinate patches w± we allowfor our dynamics to distinguish between orientations.

At this point we could propose a continuation theoremalong the lines of what was done in Bianchi IX [2–4], how-ever such a theorem would be, in the present case, trivial.This is because the theorem of [2–4] depends on the pres-ence of more than one shape degree of freedom, and itbecomes trivial in the case of a one-dimensional shape

4

space. In fact, at the core of the continuation result, isthe fact that one can decouple the scale degree of free-dom (which is singular at the singularity) from the shapeones, and express the dynamics as a differential systemin which the change of one shape degree of freedom isexpressed in terms of the change in the others. This isthe fundamental idea behind the “Shape Dynamics” for-mulation of General Relativity [9], and the papers [2–4]show how this intrinsic dynamics of pure shapes is regularat the singularity and can be continued deterministicallythrough it. However, when we have only one shape de-gree of freedom, its change cannot be expressed in termsof other shape degrees of freedom. The intrinsic shapedynamics reduces to the prediction of an unparametrizedcurve on a one-dimensional manifold (a circle), and thereis only one such curve. The fact that this curve continuesthrough the singularity (which is located at a particularpoint on the circle) is a trivial statement.

For this reason, we are compelled to add some moreshape degrees of freedom, in order to have a shape spaceof dimension at least two, where the fact that the intrinsicshape dynamics continues uniquely through the singular-ity is a nontrivial statement. The simplest way to do thisis to add a homogeneous scalar field, which contributeswith one shape degree of freedom. Notice that in [2, 4]too we were forced to add (at least) one scalar field, butfor a different reason. In fact, in these papers we wereinterested in the Bianchi IX cosmological model, whichalready comes equipped with a two-dimensional shapespace. However, unless a stiff matter source is added, thismodel has an essential singularity at the big bang, whichmakes continuation impossible. The the simplest formof stiff matter is a scalar field without mass or potential,the addition of which causes the system to transition toa state that is known as “quiescence”, after which thedynamics ceases to be chaotic and admits a determinis-tic continuation through the singularity. In the presentcase, we add the scalar field just because we need addi-tional scale degrees of freedom and that is the simplestoption. The dynamics of the Kantowski–Sachs model canbe continued through the singularity independently of thepresence of scalar fields or stiff matter sources, becauseit is not chaotic like Bianchi IX.

HOMOGENEOUS SCALAR FIELD

To include a homogeneous scalar field to the Einstein–Hilbert Lagrangian (3) we need to add the following term:

Lϕ = −∫d3x√−g[

1

2gµν∂µϕ∂νϕ+ V (ϕ)

]=

= 4πλAB2

[1

2N−1(ϕ)2 −N V (ϕ)

].

(21)

Notice that the homogeneous ansatz for the scalar fieldcorresponds, in the limit ks→∞ (r → (2M)−), to a field

that is constant on the horizon. This could be taken asthe s-wave contribution in an expansion in spherical har-monics around a Schwarzschild background. We can nowshow how the Hamiltonian (6) generalizes in presence of aminimally-coupled homogeneous scalar field ϕ (with the

convenient choice of lapse N = 1ν e

x+2y√2 ):

H = 12

(p2x − p2y + 1

κπ2ϕ

)+ U(x, y, ϕ) , (22)

where πϕ is the momentum canonically conjugate to ϕ,

and U(x, y, ϕ) = −e√2y + κ

ν2 e√2(x+2y)V (ϕ) is the sum of

the geometric and the scalar field potentials.The first thing to notice is that the potential term

breaks the conservation of the momentum px, and is ca-pable of making the variable x non-monotonic, poten-tially preventing it from reaching the singularity x →−∞. However, under certain not particularly restrictiveconditions on the form of V (ϕ),3 one can see that there

will be large classes of solutions in which e√2(x+2y)V (ϕ)

asymptotes to zero, and x and y asymptote to thestraight-line motion that ends in the singularity at x →−∞. This argument traces closely the more in-depth dis-cussion developed in [4] with regards to scalar field (andinflationary) potentials.

We are interested in solutions that reach the singular-ity, and, by what we have just observed, these are suchthat the scalar field asymptotes to free dynamics (thepotential V (ϕ) becomes negligible near the singularity),and the solutions are identical to Eqs. (3), with the ad-dition of ϕ = pu (s − s3), pu = const.. What changes isthe form of the Hamiltonian constraint:

p2x +1

κπ2ϕ − k2 = 0 , (23)

this implies that the asymptotic motion in the x−y planeis not at 45, but at a steeper angle. Eq. (23) has anotherconsequence: the Ricci tensor vanishes only when πϕ = 0.It is only in absence of the scalar field that spacetime isRicci-flat, and isometric to the Schwarzschild metric.

SHAPE SPACE WITH ORIENTATION,AND ITS COMPACTIFICATION

First, it is convenient to change the scalar field vari-able ϕ to a dimensionless one, by means of the followingcanonical transformation:

u =√κϕ , pu = πϕ/

√κ . (24)

Then, we can repeat the transformation (16) in order toseparate scale and shape degrees of freedom. In the new

3 Essentially, V (ϕ) can go to infinity as ϕ→ ±∞, but it has to doso slower than exp(|ϕ|1+ε) at least in one direction [4].

5

u-

β

|tanβ| det e < 0

det e > 0

w+

w-

u+

α

FIG. 4. Shape space with orientation: each hemisphere repre-sents an orientation, and each point on the sphere representsdifferent values of the shape degrees of freedom (w, u). Thepoles coincide with the value u = w = 0, while the equatorcorresponds to the border of the (w, u) plane at infinity. Asolution curve is shown on the top plane, together with itsprojection on the northern hemisphere.

variables, the Hamiltonian constraint takes this form:

H = 12

(p2w + p2u − p2z

)+ U(w, u, z) . (25)

The map (20) still applies in presence of a scalar field,however now the two fixed-orientation shape spaces aretwo-dimensional planes, coordinatized by (w−, u−) ∈ R2

and (w+, u+) ∈ R2. This extends also to any numberof additional fields: the shape space consists of two N -dimensional hyperplanes, one for each orientation.

We can now discuss one of the crucial steps allowingus to establish a continuation result: as we did in [2–4], we impose a particular topology on shape-space-with-orientation, which joins the borders of its two fixed-orientation connected components, making the overallspace connected. This is done by compactifying shapespace through the gnomonic projection: each of the twofixed-orientation planes is mapped onto one of the hemi-spheres of a 2-sphere, with the origins mapped to the twopoles, and the asymptotic borders mapped to the equator(see Fig. ). The gnomonic projection maps the coordi-nates (w±, u±) into the spherical coordinates β ∈ [0, π]and α ∈ [0, 2π) as follows:

| tanβ|(cosα, sinα) =

(w+, u+) , if β < π/2 ,

(w−, u−) , if β > π/2 .(26)

In terms of these variables, and their conjugate momentapα and pβ , the Hamiltonian (25) takes the following form:

H = 12 cot2 β p2α + 1

2 cos4 β p2β − 12p

2z + U(α, β, z) . (27)

CONTINUATION THROUGH THESINGULARITY

In complete analogy with the cosmological models dis-cussed in [2–4], under the conditions described in Sec. forthe scalar field potential, the Hamiltonian on the shapesphere (27) generates a dynamics that, near the singu-larity, asymptotes to that of a free point particle on theshape sphere (i.e. a particle moving along great circles).The angle β grows monotonically in this regime, and itcan therefore be used as the independent variable, ex-pressing the equations of motion in terms of derivativesof all the other variables with respect to β. In this formu-lation, all equations of motion except those for pβ and zare regular. However, the following change of variables:

J = pβ cos2 β , v = z +tanβ pz

J, (28)

gives a system of differential equations that are smooth

at the singularity β →(π2

)±[4]:

dv

dβ= − pz p

2α

sin2 βJ3,

dα

dβ=

pα

sin2 βJ,

dJ

dβ=

cosβ p2αsin3 βJ

− cos2 β∂U

∂β,

dpαdβ

=∂U

∂α,

dpzdβ

=∂U

∂v,

(29)

where

U(α, β, v, J) = −e√

83 (v− tan β pz

J )e−√

23 | tan β| cosα

+κ

ν2e√6(v− tan β pz

J )V(| tan β| sinα√

κ

).

(30)

Note that although J is frequently in the denominatorof these equations, J 6= 0 on solutions. Following fromequation 28, the only possibilities that allow for J = 0would be if the momentum pβ were to vanish, or at thesingularity where cosβ can vanish. The former case isexcluded dynamically as β is increasing towards the sin-gularity. At the singularity pβ →∞ such that J remainsfinite and non-zero.

Just as in our previous results [2–4], Eqs. (29) satisfythe assumptions of the existence and uniqueness theo-rem (the Picard-Lindelof theorem) for solutions of ordi-nary differential equations, and therefore, to each solu-tion reaching the singularity from one hemisphere we canassociate one and only one solution reaching the samepoint on the equator from the other hemisphere.

The Schwarzschild solution is a special case of theabove system, in which there is no matter potential(V = 0) and no scalar field momentum. In such a case itcan be verified that α = pα = 0 is a solution to the equa-tions of motion, which is represented by a great circlethrough the poles on the shape sphere. At the equatorthe solution continues along the great circle and crosses

6

s = const. r = const.

r = 2M

,

r = 2M ,

r = 2M

r = ∞

r = ∞

r = 0 , k s = -∞

k s = +∞

ρ = +∞

ρ = -∞

k s = +∞

FIG. 5. The continuation of the Schwarzschild solution. Atthe singularity, the shape system remains well defined, andconnects two Schwarzschild interiors described by right andleft-handed triads.

from one hemisphere to the other. On each hemisphere ofshape space, the solution describes a black hole interiorwith either a left- or right-handed triad. The Picard-Lindelof theorem shows then that there is a unique con-tinuation of the Schwarzschild interior beyond the singu-larity - it is an orientation-flipped interior of an otherwiseidentical black hole.

DISCUSSION

Our generalized dynamical system allows to continuesingular solutions through the Schwarzschild singularityuniquely. As can be deduced by looking at the shapesphere in Fig. , a great circle that crosses the equatorwon’t be invariant under reflections with respect to theequator’s plane (unless we’re in the special case of a ver-tical, “meridian” circle). Then the solution continues toone that is objectively different: it is not simply the time-reversed repetition of the initial solution. After crossingthe singularity, the shape degrees of freedom w and u willhave a different evolution and will go through differentpairs of values.

A legitimate question, at this point, is: what is thestructure of the spacetime that corresponds to these con-tinued solutions? The first thing we might investigateis its causal structure, which is entirely codified in theevolution of the shape variable w. We know the causalstructure associated to any half of each solution that is

confined to one hemisphere: it is that of the region ofSchwarzschild’s spacetime that is inside the horizon: theshaded region in Fig. 2. A full solution can then be as-sociated to two such causal patches, and it is temptingto glue them at the singularity in the manner of Fig. 5:one has two regions with opposite spatial orientations,looking like a black hole interior glued to a white hole in-terior. Extending these spacetimes beyond the horizons,one finds two asymptotically flat regions of opposite ori-entations, one in the causal past and one in the future.

This picture, however, is tentative and does not nec-essarily reflect actual physics. A Penrose diagram makessense as an effective description of the causal relations be-tween test particles propagating in a background space-time, in a regime in which the backreaction of the par-ticles on the geometry can be neglected. This is a rea-sonable assumption around most points in the Penrosediagram 5, but not in the vicinity of the singularity. Wecannot say, at the moment, what a test particle would ex-perience upon crossing the singularity: that would needa dedicated analysis. Until that is done, we cannot besure that timelike worldlines would behave smoothly atthe singularity in the Penrose diagram 5, and thereforethe physical meaning of that diagram remains unclear.

This paper has shown how spacelike singularities atthe center of black holes do not represent the end of thedeterminism of the solution. Together with [2–4], thishints that the resolution of spacelike singularities maybe a generic feature of the relational approach. However,this is far from the end of the problem of singularities.The Hawking-Penrose theorems still hold, and as yet itis not known how to extend geodesics beyond the singu-larity itself. Recent work [11], see also [12, 13] has shownthat despite these problems, given some extensions ofspacetime beyond a singularity certain matter degrees offreedom can be deterministically evolved beyond thesepoints. A tantalizing prospect is that relational descrip-tions may resolve the issues of singularities entirely clas-sically. The ramifications for quantum gravity searches,many of which have their sights set on resolution of sin-gularities, would be profound.

Another issue that should be investigated beforeproposing causal structures for our singularity-crossingsolutions (and, in particular, before extending thesestructures outside of the horizons, is the fact that theSchwarzschild spacetime represents an eternal black hole,while realistic black holes are created through the col-lapse of matter. This is better discussed within a mattercollapse model that creates the black hole metric in itswake (e.g. a thin-shell [14, 15] or a Lemaitre–Tolman–Bondi model). Then, the study of the behaviour of thecollapsing matter upon crossing the singularity shouldreveal the nature of the region beyond the singularity.A compelling possibility is that the singularity turns thecollapse of the matter into an expansion, and the ex-panding matter leaves behind a pocket of spacetime with

7

a white-hole metric.

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