AAntipodal identification in the Schwarzschild spacetime
Miguel SocolovskyInstituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, Cd.Universitaria, 04510, Ciudad de M´exico, M´[email protected]
Abstract
Through a M¨obius transformation, we study aspects like topology, ligth cones, horizons,curvature singularity, lines of constant Schwarzschild coordinates r and t , null geodesics, andtransformed metric, of the spacetime ( SKS/ (cid:48) that results from: i) the antipode identifica-tion in the Schwarzschild-Kruskal-Szekeres ( SKS ) spacetime, and ii) the suppression of theconsequent conical singularity. In particular, one obtains a non simply-connected topology:(
SKS/ (cid:48) ∼ = R ∗ × S and, as expected, bending light cones. Keywords: antipodal identification; Schwarzschild spacetime
In 1965, Rindler suggested the possibility of the antipode identification of coordinates (
V, U ) ≡ ( − V, − U )in the maximal analytic extension of the Schwarzschild metric, the Schwarzschild-Kruskal-Szekeres ( SKS )spacetime. Later, Sanchez and Whiting (1987) and more recently ’t Hooft (2018) used it (or a variation ofit) for the study of quantum field theory in black holes. The identification allows to obtain
SKS/
2, closerthan
SKS to the physical spacetime associated with the total collapse of a spherically symmetric star: boththe mirror image of “our” asymptotically flat region, and the white hole region together with its associatedpast singularity s p , dissappear. The conical singularity appearing as a consequence of this identificationmust be excluded from the spacetime, leading to the space ( SKS/ (cid:48) . Inserting its ( V, U ) part in the upperpart of a complex half plane, a M¨obius-type complex transformation φ exhibits ( SKS/ (cid:48) with the topology( S \ ( { N, S } ∪ φ ( s f ) ∪ int ( φ ( s f )))) × S ∼ = R ∗ × S , where N and S are the north and south poles of theunit 2-sphere S and s f is the future singularity. Precisely due to the suppression of the conical singularity,its boundary, image of the past horizon -which dissappears in a physical collapse- can not be associatedto a closed causal (null) curve. Also, the same suppression produces a non simply-connected space, since π (( SKS/ (cid:48) ) ∼ = Z . The maximal analytic extension of the Schwarzschild metric (with coordinates ( t, r, θ, ϕ ), t ∈ ( −∞ , + ∞ ), r > θ ∈ (0 , π ), ϕ ∈ [0 , π )), is that of Kruskal-Szekeres (1960) with coordinates ( V, U, θ, ϕ ) ( V ∈ ( −∞ , + ∞ ): temporal, U ∈ ( −∞ , + ∞ ): spatial, θ, ϕ as in Schwarzschild, angular coordinates). The V /U part is given by the following diagram: a r X i v : . [ phy s i c s . g e n - ph ] S e p igure 1: SKS solution
In Fig. 1: U : “our” universe; ¯ U : anti-universe; B : black hole; W : white hole; dashed red lines: futureand past singularities s f and s p , respectively given by the hyperboles V = ±√ U , which are asymptoticto the future and past horizons h + , h − ; h + , h − and s f , s p consist of points where r = 2 M and r = 0respectively; the remaining hyperbole filling the V /U plane correspond to fixed values of r , and the linesthrough the origin correspond to fixed values of t ; each point has associated with it a 2-sphere S = S r of radius r , S ( r, θ, ϕ ); the light cones are at ± ◦ everywhere: particle trajectories in their interior aretimelike and on their boundaries are null (light rays) ; s f , s p and the shaded regions do not belong tothe SKS spacetime, which turns out to be non compact, 1-connected, globally hyperbolic (it has a globalCauchy surface) and geodesically incomplete; M is the gravitating mass; since V, U ∈ ( −∞ , + ∞ ), then,topologically, SKS ∼ = R × S . (1)It can be shown that the “lines” h ± (together with the corresponding S M = S ( r = 2 M, θ, ϕ )) are nullhypersurfaces; for the homotopy groups, π k ( SKS ) ∼ = π k ( R × S ) ∼ = π k ( S ) (2) k =1, Z for k =2,3, Z for k =4, etc.); U and ¯ U are asymptotically flat at r → + ∞ (Minkowskispacetime).The SKS metric is ds = 32 M r e r/ M ( dV − dU ) − r ( dθ + sin θdϕ ) , (3)where the relations between the ( r, t ) and ( V, U ) is given by(1 − r M ) e r/ M = V − U (4)which gives r = r ( V, U ) through the Lambert W function defined via µe µ = ν ⇒ µ = W ( ν ), (Lambert,1758), and t M = T h − (( VU ) sg ( r/ M − × ) , r (cid:54) = 2 M, (5)with t = ±∞ at the horizons V = ± U . ds has the symmetry P T with P the spatial inversion U → − U and T the time inversion V → − V , sothat P T : (
V, U ) → ( − V, − U ). Since ( P T ) = Id , the associated symmetry group is ( Id, P T ) ∼ = Z . Thispermits the antipode (through the origin ( V, U ) = (0 , − V, − U ) ≡ ( V, U ) and allows toremain with only the “half” spacetime in the Fig. 1, which we call
SKS/
2. This is illustrated in Fig. 2. ~ ~ Figure 2:
SKS/ If S ( r, θ, ϕ ) were identified with S ( r, π − θ, ϕ + π ), then at each r one should have the projective space S / Z ∼ = R / R ∗ = R P which would destroy spatial orientability. Requiring this condition, we restrict theantipode identification only to the V, U coordinates.¯ U and W dissapear, remaining U , B , s f h + and h − . Along h − , points denoted by ∼ are identified. Thepossibility of this identification was suggested by Rindler (Rindler, 1965), and also previously mentioned y Szekeres (Szekeres, 1960). The diagram in Fig. 2 is more related than that in Fig. 1 to the diagramcorresponding to the final collapse of a spherical symmetric star and the formation of a real black hole,where ¯ U , W , s p and h − do not exist. (This is the reason why the diagram in Fig. 1 is said to represent anideal eternal black hole.) SKS/
SKS Z turns out to be a manifold with boundary ∂ ( SKS/
2) = h − ∼ = [0 , + ∞ ) × S M = Cy , (6)an infinite 3-dimensional hypercylinder, and a conical singularity at ( V, U ) = (0 , SKS/ ∼ = ( R × S ) ∪ Cy . (7)The conical singularity must be taken off from the spacetime, resulting( SKS/ (cid:48) ∼ = ( R × S ) ∪ ( Cy ) ∗ (8)with h (cid:48)− ∼ = h (cid:48) + ∼ = ( Cy ) ∗ = R ∗ × S M . (See Fig. 3.) ~ ~ Figure 3: (
SKS/ (cid:48) The dropping of the conical singularity seems to be done by hand; however, it is done to mantaindifferentiability at all points of the spacetime.
We can ask ourselves for another picture of the topology of (
SKS/ (cid:48) . With this aim, we consider the halfplane “above” h (cid:48)− as the complex half-plane C / { z = x + iy, x ∈ R , x (cid:54) = 0 , y ∈ (0 , + ∞ ) } (9) ith the x -axis identified with h (cid:48)− ( z = x + i
0) (later we make the identification x ∼ − x ), and the half y -axis identified with h (cid:48) + ( z = iy ), and consider its image into the complex plane C = { w = ξ + iη, ξ, η ∈ R } (10)through the M¨obius-type mapping φ : ( SKS/ (cid:48) → C , z (cid:55)→ φ ( z ) = w := z − iz + i = x + i ( y − x + i ( y + 1) = ξ ( x, y ) + iη ( x, y ) , (11)and S r (cid:55)→ S r , with ξ ( x, y ) = x + y − x + ( y + 1) , η ( x, y ) = − xx + ( y + 1) . (12)The relation between the ( V, U ) coordinates and the ( y, x ) coordinates is V = y − x , U = y + x . (13)(The previous φ is a particular case of z (cid:55)→ φ ( z ) = e iγ ( z − z z − ¯ z ) , γ ∈ R , Im ( z ) > γ = 0 and z = i . φ maps the half plane Imz = y > | w | <
1, and the boundary of that halfplane ( h (cid:48)− ) onto the boundary of that disk.)The inverse of φ is given by z = z ( w ) = φ − ( w ) = − i w + 1 w − x ( ξ, η ) + iy ( ξ, η ) , (14)with x ( ξ, η ) = − η ( ξ − + η , y ( ξ, η ) = − ( ξ −
1) + η ( ξ − + η . (15)The Cauchy-Riemann ( C − R ) equations for φ and φ − are ∂ξ∂x = ∂η∂y , ∂ξ∂y = − ∂η∂x (16)and ∂y∂η = ∂x∂ξ , ∂y∂ξ = − ∂x∂η , (17)respectively. The differentiability of φ and φ − guarantees that the coordinate transformation ( x, y ) → ( η, ξ )is a genuine transformation in the context of general relativity. φ is analytic: φ (cid:48) ( z ) = dwdz = 2 i ( z + i ) (18)and 1-1. Then, ( SKS/ (cid:48) is onto its image and therefore homeomorphic and diffeomorphic to it, whichturns out to be φ (( SKS/ (cid:48) ) = S × ( S \ ( { N, S } ∪ A )) , A = φ ( s f ) ∪ int ( φ ( s f )) , (19)where S is the unit 2-sphere, the identification ∼ is done, S = φ (0) = ( ξ = − , iη = 0), and N = ( ξ =1 , iη = 0) = lim φ ( x + i
0) as x → ±∞ . (See Fig. 4.) In turn, one has the homeomorphism φ (( SKS/ (cid:48) ) ∼ = R ∗ × S (20)since S \ ( { S }∪ A ) ∼ = R \{ S } ∼ = R ∗ , with fundamental group π ( R ∗ × S ) ∼ = π ( R ∗ ) ∼ = Z . Homotopically,then, ( SKS/ (cid:48) ∼ = φ (( SKS/ (cid:48) ) (cid:39) S × S . (21) ~ (cid:31) (cid:30) Figure 4: Image of (
SKS/ (cid:48) under φ ; p (cid:48) → q (cid:48) : null radial geodesic; M = 1 Since φ is analytic and φ (cid:48) ( z ) (cid:54) = 0, φ is also conformal, and so preserves the angles between tangents tointersecting curves. In particular this will be applied to the transformation of the light cones in ( SKS/ (cid:48) .The arrows in the image of h (cid:48)− , φ ( h (cid:48)− ) = ∂ ( φ (( SKS/ (cid:48) )) = ( S \ { N, S }∼ ) × S M , (22)corresponds to the path from x = −∞ to x = + ∞ ( x (cid:54) = 0) along h (cid:48)− , and that in the image of h (cid:48) + corresponds to the path from y = 0 + to y = + ∞ along h (cid:48) + . The image of s f is given by the red dashedcurve, obtained from φ ( z ) with y = − x , x ∈ ( −∞ , Q one has Q = 0 . i .
4. The factthat
S / ∈ φ ( h (cid:48) + ) implies that there is no (though infinite) closed causal (null) curve ( h (cid:48)− in Fig. 3 or φ ( h (cid:48)− )in Fig. 4). Anyway, h (cid:48)− dissappears in a physical collapse. r and t constant lines r lines i) B region:For 0 < r < M , r = 2 αM with α ∈ (0 , V − U = (1 − α ) e α and using (13), y = − (1 − α ) e α x ,which replaced in (12) gives w B ( x ; α ) = ( x + (1 − α ) e α − x ) + i ( − x ) x + (1 − α ) e α − x (1 − α ) e α + x , x ∈ ( −∞ , . (23)ii) U region:For 2 M < r , r = 2 βM with β ∈ (1 , + ∞ ). From (4), y = ( β − e β x , which replaced in (12) gives w U ( x ; β ) = ( x + (1 − β ) e β − x ) + i ( − x ) x + (1 − β ) e β − x (1 − β ) e β + x , x ∈ (0 , + ∞ ) . (24)For β → + ∞ , y → + ∞ , w U ( x ; + ∞ ) = 1 + i N . In Fig. 4 we plot some of these r = const. lines. t lines From (5), (13), and (15) we obtain
T h ( t M ) = ξ − η ∓ ηξ − η ± η (25)with upper and lower signs respectively corresponding to the r > M ( U ) and r < M ( B ) regions. If wecall τ := T h ( t M ) ∈ ( − , +1) , (26)it is easy to obtain η X ( ξ ; τ ) for X = U and X = B : η U ( ξ ; τ ) = ( 1 + τ − τ ) − (cid:114) ( 1 + τ − τ ) + (1 − ξ ) , η B ( ξ ; τ ) = − ( 1 + τ − τ ) + (cid:114) ( 1 + τ − τ ) + (1 − ξ ) . (27)At the horizons φ ( h (cid:48)− ) and φ ( h (cid:48) + ), t = −∞ and t = + ∞ respectively. For t = 0 we have η U ( ξ ; 0) =1 − (cid:112) − ξ , η B ( ξ ; 0) = − (cid:112) − ξ . These lines are plotted in Fig. 4. In the ( ξ, η )-plane (remember that topologically C ∼ = R ), the metric (3) becomes that of φ (( SKS/ (cid:48) ), andis given by ds = − M r e r/ M (( ∂x∂ξ )( ∂x∂η )( dη − dξ ) − ( ∂x∂η − ∂x∂ξ ) dηdξ ) − r ( dθ − sin θdϕ ) (28)where the C − R equations (16) and (17) were used, and ∂x∂ξ = 4( ξ − η (( ξ − + η ) , ∂x∂η = − ξ − − η (( ξ − + η ) . (29) = r ( ξ, η ) through the Lambert W function and (13) and (15). (28) tells us that in the ( ξ, η ) coordinatesthe light cones bend.Given that φ ( z ) is analytic and, by (18), φ (cid:48) ( z ) (cid:54) = 0, φ is conformal and therefore preserves anglesbetween intersecting curves. Given two such curves in the ( V, U ) plane (e.g. the lines corresponding to theboundary of the light cones), then their common rotation angle in the ( ξ, η ) plane is given by the argumentof φ (cid:48) ( z ) = | φ (cid:48) ( z ) | e iδ φ (cid:48) ( z ) : δ φ (cid:48) ( z ) = tg − ( x + ( y + 1) x ( y + 1) ) = tg − ( 12 ( 1 − ξη − ( 1 − ξη ) − )) . (30)Some of these bended light cones are shown in Fig. 4. We analize here the image by φ of a typical radial null geodesic p → q in Fig. 1. In SKS , p → q isdescribed by the equation V = − U + 2 with ( V ( p ) , U ( p )) = (0 , s f at the point q where − U + 2 = + √ U which implies U = 3 / V = 5 /
4; so ( V ( q ) , U ( q )) = (5 / , / ξ ( V, U ) = 2( V + U ) − V + U + V + U ) + 1 , η ( V, U ) = 2( V − U )2( V + U + V + U ) + 1 ; (31)so the image of p → q in the ( ξ, η ) plane is ξ ( U, − U + 2) = 4( U − U ) − U − U ) + 13 , η ( U, − U + 2) = − U − U − U ) + 13 . (32)In particular, for p (cid:48) = φ ( p ), and q (cid:48) = φ ( q ), we obtain( ξ ( p (cid:48) ) , η ( p (cid:48) )) = (7 / , − / (cid:39) (0 . , − . , ( ξ ( q (cid:48) ) , η ( q (cid:48) )) = (13 / , / (cid:39) (0 . , . . (33)It is then easily verified that q (cid:48) ∈ φ ( s f ) i.e. φ ( p → q ) = p (cid:48) → q (cid:48) dies at φ ( s f ), as it must be. (See Fig. 4.)A similar analysis can be done with any other null geodesic in the image φ (( SKS/ (cid:48) ). The antipodal identification (
V, U ) ≡ ( − V, − U ) in the Schwarzschild-Kruskal-Szekeres ( SKS ) metric canbe done without the introduction of additional singularities, since the requirement of differentiability makesit necessary to eliminate from the spacetime the emerging conical singularity. At the same time, thissuppression guarantees the non existence of closed (though infinite) causal (null) curves. The M¨obiustransformation makes easier to study the topology of the resulting spacetime (
SKS/ (cid:48) which, as expected,and in contradistinction with SKS , becomes non simply connected: φ (( SKS/ (cid:48) ) ∼ = R ∗ × S (cid:39) S × S ,where (cid:39) denotes homotopy type. The picture, however, of light cones, r and t constant lines, metric, andnull geodesics, becomes much more involved than before the transformation. Acknowledgments
The author thanks Leonardo J. M´endez for numerical calculations, and Oscar Brauer for drawing the figures. eferences Churchill, R.V. “Complex Variables and Applications”, 4th. edition (1984), pp. 194-195.Kruskal, M.D. “Maximal Extension of Schwarzschild Metric”, Phys. Rev. (1960) 1743-5.Lambert, J.H. “Observationes variae in mathesin purae”, Acta Helveticae physico-mathematico-anatomico-botanico-medica, Band III (1758) 128-68. (Lambert W function, Wikipedia, 1-15.)Rindler,W. “Elliptic Kruskal-Schwarzschild Space”, Phys. Rev. Lett. (1965) 1001-2.Sanchez, N. and Whiting, B.F. “Quantum Field Theory and the Antipodal Identification of BlackHoles”, Nucl. Phys. B (1987) 605-23.Szekeres, Gy. “On the Singularities of a Riemannian Manifold”, Publicationes Mathematicae Debrecent (1960) 285; reprinted: Gen. Rel. Grav. (2002) 2001-16.’t Hooft, G. “Virtual Black Holes and the Space-Time Structure”, Found. Phys. (2018) 1134-49.(2018) 1134-49.