aa r X i v : . [ g r- q c ] S e p Rotating De Sitter Space
G. F. Chapline a P. Marecki b a Lawrence Livermore National Laboratory, Livermore, CA 94550 b Institut fuer Theoretische Physik, D-04009 Leipzig, Germany
Abstract
An exact solution of the vacuum Einstein equations with a cosmological constantis exhibited which can perhaps be used to describe the interior of compact rotatingobjects. The physical part of this solution has the topology of a torus, which mayshed light on the origin of highly collimated jets from compact objects.
Key words: de Sitter, rotating space-times, black holes
PACS:
In this paper we exhibit an exact solution to the Einstein field equations thatmay help resolve two outstanding puzzles in theoretical astrophysics. The firstpuzzle is to describe the nature of space-time inside a rotating object that issufficiently compact that it lies entirely inside a surface where classical generalrelativity predicts that an event horizon would form. The conventional view isthat such an object is a “black hole”. However, both non-rotating and rotatingblack holes have features such as singularities and “reversal of space and time”that may be unphysical. In addition, the Kerr solution for a rotating black hole[1] shares in common with other rotating solutions of Einstein’s equations thepathological feature that there are closed time-like curves. It has been pointedout [2,3] that in the case of non-rotating compact objects the objectionablefeatures of the non-rotating black hole interior space-time would be removed ifthe interior Schwarzchild space-time were replaced with de Sitter’s “interior”cosmological solution [4]. This space is non-singular and removes the “reversal
Preprint submitted to Elsevier 24 October 2018 f space and time” that plagues the interior Schwarzschild solution. Further-more the de Sitter interior solution can be made to exactly match the exteriorSchwarzschild solution at the event horizon if the vacuum energy is chosen sothat the total mass-energy of the interior de Sitter solution matches the blackhole mass. According to this new picture of non-rotating compact objectsspace-time would not be analytically smooth at the event horizon, so classicalgeneral relativity would fail there. However, it has long been recognized thatquantum effects become important near an event horizon, and therefore it isquite plausible that classical general relativity fails in the vicinity of an eventhorizon. References [2] and [3] offer two different scenarios as to what actuallyhappens at the event horizon. However, for the purposes of this paper it notnecessary to understand in detail what happens at the event horizon; insteadwe will focus on question as to whether there is a candidate space-time thatcould serve as a non-singular model for the bulk interior space-time insiderotating compact objects.In accordance with the expectation that the interior space-time of a collapsedobject should be obtained by continuous “squeezing” of a condensate vacuumstate [5], we expect that this space-time should have a large vacuum energy;i.e. this interior space-time should locally resemble classical flat space-timewith a cosmological constant; i.e. it should locally look like a region of deSitter space-time. One nagging question concerning the proposals of references[2] and [3], though, is what should replace de Sitters interior solution in thecase of a rotating compact object. To our knowledge a rotating version ofde Sitter space-time has never been explicitly discussed in the literature. Inthe following we address this deficiency by exhibiting a mathematically exactsolution of Einstein’s field equations that is in fact a rotating generalizationof de Sitter’s interior solution. Not only does this solution provide a plausiblepicture for the nature of space-time in the interior of rotating compact objects,but as a bonus this solution provides a new insight into the nature of thehighly collimated jets that have been observed to be emanating from compactastrophysical objects. 2
The metric
Our proposed interior metric is (we use units such that 8 πG/c = 1) ds = (cid:20) − Λ3 ( r − a cos θ ) (cid:21) dt + 2 a (cid:20) − Λ3 r cos θ (cid:21) dtdϕ − r + a cos θr − Λ3 r + a dr − r + a cos θ − Λ3 a cos θ cot θ dθ (1) − ( r sin θ − a cos θ ) dϕ where a is the angular momentum per unit mass. This metric is a limiting caseof a class of metrics discovered by Carter [6] and independently by Plebanski[7]. In the limit a → r = 0 for any value of θ . The apparent singu-larities in the g rr and g θθ components of the metric tensor can be removed bya change of variables [7], and represent event horizons where g g ϕϕ − g ϕ = 0.The singularity in g rr is associated with a spherical event horizon located at r H = 32Λ + " + 3 a Λ / (2)In the limit a → r H becomes the de Sitter horizon q / Λ. In addition to thespherical event horizon (2) there is a conical event horizon located attan θ H = −
12 + s Λ3 a + 14 (3)In the case of slow rotation Λ a ≪ g ϕϕ is positive for some values of r . In our case this means that r sin θ − a cos θ < ̺ < a , where ̺ is the horizontal distance from theaxis of rotation. Thus for slow rotation closed time-like curves will appear veryclose to the rotation axis. This is very reminiscent of the situation with space-time spinning strings [8]. Actually for all values of the rotation parameter a the conical horizon (3) lies inside the region where closed time-like curves3ppear, and the critical angle where the inequality in (4) becomes an equalityis precisely the horizon angle θ H where r = r H .The event horizon for the Kerr solution will match the event horizon (2) if themass parameter for the Kerr solution is m = Λ6 r H (5)Curiously this is the same condition that was used in ref. [2] to match deSitter’s interior solution to the exterior Schwarzschild solution in the caseof a non-rotating compact object. If we impose the condition (5) the eventhorizon for our “interior” solution will occur at precisely the same radius asthe horizon for the Kerr solution. In this case it might be reasonable to supposethat the “exterior” space-time outside the horizon (2) is just the usual exteriorKerr solution. Near to the spherical event horizon (2) the angular part of ourrotating metric (1) has the form ds = − a sin θ − Λ3 a cos θr H + a cos θ dt − r H a dϕ ! − r H + a cos θ − Λ3 a cos θ cot θ dθ . (6)For comparison the angular part of the Kerr metric when expressed in Boyer-Lindquist coordinates [9] and evaluated on the event horizon is ds = − a sin θr H + a cos θ dt − r H a dϕ ! − ( r H + a cos θ ) dθ . (7)It can be seen that except near to θ = 0 the angular part of our metric near tothe spherical event horizon is not too different to the angular part of the Kerrmetric at the event horizon for all values of a such that Λ a <
1. Significantdifference do appear near to θ = 0, which as we discuss below is due to theappearance of new physics near to the axis of rotation.Inside the spherical event horizon (2) the behavior of our metric is completelydifferent from that of the interior Kerr metric. For example, there are nospace-time singularities. In the case of the Kerr solution g < r + a cos θ = 2 mr . Although our g is negative at the event horizon(and close to the Kerr g ), it is actually positive for all values of r inside r = (3 / Λ) / + a cos θ , which for small Λ a would be almost everywhere inthe interior. In addition in contrast with the Kerr solution the radial metriccoefficient g rr is negative for all values of r inside the spherical event horizon.Thus the problematic reversal of the roles of time and radial distance in theinterior Kerr solution is alleviated. The Kerr solution has the property that4nside the ergosphere particles cannot be at rest but must rotate about theaxis. At the event horizon the frame in which particles could be at rest rotateswith the “frame dragging” angular velocity dϕdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r H = ar H + a . (8)For the metric (1) g < a/r H , so particles in our space-time will also rotate as theyapproach the event horizon from the inside. Indeed our interior metric containsa reflection of the usual exterior Kerr ergosphere, with an inside boundary at r = (3 / Λ) / + a cos θ . Thus in our picture of rotating compact objects themetric just inside the event horizon is a reflection of the metric just outside,at least away from θ = 0. Reflection symmetry between the inner and outermetrics at an event horizon is just the matching condition for metrics sug-gested in ref. [2], and is a consequence of replacing the smooth geometry at anevent horizon that is predicted by classical general relativity with a quantumcritical layer. One might guess that the metric (1) could also be derived from Demianski’swell known generalization of the Kerr solution to include a cosmological con-stant [10]. Indeed taking the m = 0 limit of Demianski’s metric yields ds = h − λ ( r + a sin θ ) i dt + 2 a sin θ · λ ( r + a ) dtdϕ − r + a cos θ ( r + a )(1 − λr ) dr − r + a cos θ λa cos θ dθ (9) − ( a + r ) sin θ h λa i dϕ , where λ = Λ3 . As was the case for metric (1) the variables θ and ϕ representthe polar angles on a sphere. At first sight (1) and (9) appear to be different.However, both metrics (1) and (9) can be obtained as special cases of thePlebanski metric [7], which has the general form ds = Q p + q ( dt − p dσ ) − P p + q ( dt + q dσ ) − p + q P dp − p + q Q dq (10)5hen the mass, NUT charge, and electric and magnetic charges are all zerothen the functions P ( p ) and Q ( q ) have the simple forms P = b − ǫp − λp and Q = b + ǫq − λq . In this case the Weyl tensor vanishes and the Plebanskimetrics are conformally flat. For both metrics (1) and (9) p = a cos θ , q = r , σ = ϕ/a , and b = a . However, for metric (1) ǫ = 1 and τ = t , while for themetric (9) ǫ = 1 − λa and τ = t + a σ .Evidently the essential difference between the two geometries lies in the valueof ǫ . However it is known that Plebanski metrics with different values of arerelated by a certain scaling transformation. This scaling transformation hasthe form p ′ = p/α, q ′ = q/α, σ ′ = α σ, τ ′ = ατ, b ′ = b/α , ǫ ′ = ǫ/α , (11)where α is the scaling parameter. The value of Λ is unchanged. Because b = a for both metrics we may replace a by √ b so that both metrics depend onlyon the parameters b that appear in the original Plebanski metric (10). Thatthe metrics (1) and (9) are locally isometric can now be seen as follows: westart with the m = 0 Demianski metric (9) and rescale it using the scalingtransformation (11) and α = 1 − λa . This leads to the metric (1) with ǫ =1 and a = b/ (1 − Λ b/ m = 0 Demianski’s geometry islocally isometric to the geometry of (1). The question to whether the m = 0Demianski geometry is isomorphic to our rotating solution is more complicatedbecause the value of b affects the ranges of p and ϕ . In particular since ϕ = 0 isidentified with ϕ = 2 π , then in the scaled metric σ = 0 is identified with σ =2 π/ √ b . In addition the range of p is restricted because p ∈ [ √ b, + √ b ]. Thuswhereas the metric (1) is defined for all values of θ and ϕ the m = 0 Demianskigeometry corresponds to only a part of the sphere (cf. Fig. 1). Amusingly thelatitudes covered by the m = 0 Demianski are just those outside the conicalhorizon eq. (3). In accordance with our a priori expectations regarding thenature of the vacuum state inside compact objects, both metrics are locallyisometric to de Sitter space-time. As noted above the metric (1) is plagued by time-like closed curves near to theaxis of rotation. Closed time-like curves are extremely pathological from thepoint of view of quantum mechanics. The pathological nature of the space-time corresponding to the metric (1) near to the axis of rotation can also beseen from the signature of the metric. Physical space-times should have thesignature + − − − . In the case of the Plebanski metrics this is only possibleif P > Q > P > Q <
0. If P <
0, as is the case inside he conical6 ig. 1. The region of the spacetime (1) isometric (covered) by the Demianski’sspacetime (9). horizon, the signature is + + + − , so the space-time is not physical. Bothof these considerations suggest that space-time undergoes some sort of phasetransition near to the axis of rotation.Recently it has been suggested [11] that the way to resolve the difficulties withclassical rotating space-times that have closed time-like curves is to supposethat the rotation is actually carried by space-time “spinning strings” [8], in amanner analogous to the way rotation of superfluid helium inside a rotatingcontainer is carried by quantized vortices. The spinning strings resolve thequestion of the consistency of rotating space-times with quantum mechanicsbecause the vorticity of space-timewould be concentrated into the cores ofthe spinning strings where the condensate density would be very low and theEinstein equations are modified by the appearence of torsion. As shown in ref.[11] averaging over the vorticity of many perfectly aligned spinning stringsleads to a Godel-like space-time. In a similar way it is reasonable to guessthat the correct physical picture for the space-time inside the region whereclosed time-like curves appear in our solution for a rotating compact objectis a Godel-like space-time. Indeed the Som-Raychaudhuri metric exhibitedin ref. [11] may be a good approximation for the metric in this region. Thismetric would be applicable inside the critical radius where the local speed7f frame rotation is equal to the speed of light. The equation of motion forparticles in a Godel-like space-time is well known [12]. In general this flow ofparticles will be collimated since the particles are confined to lie inside thecylinder where the velocity of frame rotation is less than the speed of light. Inour situation the radius of this cylinder will equal the angular momentum perunit mass parameter a used in eq. (1); i.e. where the closed time-like curvesfirst appear in our solution as the axis of rotation is approached. On the otherhand particles in a Godel-like space-time are free to move parallel to the axis,so that for slow rotation of our compact object the flow of particles along theaxis of rotation will be highly collimated. In summary, the metric (1) provides an interior solution for rotating compactobjects that avoids many of the unphysical features of the interior Kerr so-lution. It does not avoid the appearance of closed time-like curves, but theresults of ref. [11] suggest that near to the axis of rotation a Godel-like phaseof space-time appears where the vacuum energy is much smaller than in thebulk condensate of the rotating object and solid body-like rotation of thespace-time appears.Finally we should note that our metric (1) may also serve as a model for thelarge scale structure of our universe, where there are hints from observationsof the large scale anisotropy of the cosmic microwave background that theuniverse might be rotating [13].The authors are very grateful for numerous discussions with Pawel Mazur.
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