On the Kerr metric in a synchronous reference frame
aa r X i v : . [ g r- q c ] J a n On the Kerr metric in a synchronous reference frame
V.M. Khatsymovsky
Budker Institute of Nuclear Physicsof Siberian Branch Russian Academy of SciencesNovosibirsk, 630090, RussiaE-mail address: [email protected]
Abstract
The Kerr metric is considered in a synchronous frame of reference obtainedby using proper time and initial conditions for particles that freely move alonga certain set of trajectories as coordinates. Modifying these coordinates in acertain way (keeping their interpretation as initial values at large distances), westill have a synchronous frame and the direct analogue of the Lemaitre metric,the singularities of which are exhausted by the physical Kerr singularity (thesingularity ring).
PACS Nos.: 04.20.Jb; 04.70.BwMSC classes: 83C15; 83C57keywords: general relativity; Kerr spacetime; synchronous frame
An exact solution of the vacuum Einstein equations describing gravitational field of aspinning mass was given by Kerr [1]. Depending on the problem under consideration,it is convenient to write the corresponding metric in one or another coordinate system;see, for example, [2] - [7]. We issue from the Kerr metric in the Boyer-Lindquist1coordinates [2],d s = (cid:18) − r g rρ (cid:19) d t + ρ △ d r + ρ d θ + (cid:18) r + a + a r g rρ sin θ (cid:19) sin θ d ϕ − a r g rρ sin θ d ϕ d t, (1)where ρ = r + a cos θ, △ = r − r g r + a . The contravariant metric tensor is (cid:13)(cid:13) g λµ (cid:13)(cid:13) = − Σ /ρ △ − ar g r/ρ △ △ /ρ /ρ − ar g r/ρ △ △ − a sin θ ) /ρ △ sin θ , (2)where Σ = (cid:0) r + a (cid:1) − a △ sin θ. One of the important coordinate systems is the synchronous reference system. Byfixing four components of the metric tensor, g λ = ( − , , , N, N ) = (1 , ) in the Arnowitt-Deser-Misner formalism [8], a way to transfer the physics of the phenomenon to thetrue canonical coordinate (spatial metric), leaving ( N, N ) = const, which may beinteresting in quantum theory.In what follows, we will transform the metric (1) into a synchronous frame of refer-ence, in which the coordinates are the proper time and initial conditions for a certainset of trajectories of freely moving particles, using a technique based on the Hamilton-Jacobi equation for a particle. The resulting metric has singularities in addition tothe true ring Kerr singularity. Then we modify the definition of the new coordinatesto some ”asymptotic” form, so that their interpretation as initial values for the origi-nal coordinates (Boyer-Lindquist) will not necessarily be correct (it is correct at largedistances), but the metric is simplified and only has the true ring Kerr singularity. A way of passing to a synchronous frame is to use the Hamilton-Jacobi equation for aparticle with action τ (as mentioned, for example, in the textbook [9]), g λµ ∂τ∂x λ ∂τ∂x µ + 1 = 0 . (3)For that, its solution is required, τ = f ( ξ , x , t ) + A ( ξ ) , (4)which depends on four constants ξ , A ( ξ ) as parameters, of which A ( ξ ) is considered asan arbitrary function of ξ . The equations of motion are f ,ξ j ( ξ , x , t ) + A ,ξ j ( ξ ) = 0 . (5)We consider the set of trajectories corresponding to a given fixed ξ . We take τ as thenew time coordinate and set A ( ξ ) = 0 for the given ξ (this is tantamount to redefining τ by shifting). If at τ = 0 the trajectory passing through x , t has coordinates x , t ( x ),then x , τ are the new coordinates of the point x , t . We have τ = f ( ξ , x , t ) ,f ( ξ , x , t ) = 0 ⇒ t = t ( x ) ,f ,ξ j ( ξ , x , t ( x )) = f ,ξ j ( ξ , x , t ) . (6)The contravariant metric tensor in the new coordinates x , τ has the components g ττ = g λµ ∂f ( ξ , x , t ) ∂x λ ∂f ( ξ , x , t ) ∂x µ = − ,g x j τ = ∂x j ∂x λ ∂τ∂x µ g λµ = ∂x j ∂f ,ξ k ( ξ , x , t ( x )) ∂f ,ξ k ( ξ , x , t ( x )) ∂x λ ∂τ∂x µ g λµ = ∂x j ∂f ,ξ k ( ξ , x , t ( x )) ∂f ,ξ k ( ξ , x , t ) ∂x λ ∂f ( ξ , x , t ) ∂x µ g λµ = 12 ∂x j ∂f ,ξ k ( ξ , x , t ( x )) ∂∂ξ k (cid:20) ∂f ( ξ , x , t ) ∂x λ ∂f ( ξ , x , t ) ∂x µ g λµ ( x , t ) (cid:21) = 0 ,g x j x k = ∂x j ∂x λ ∂x k ∂x µ g λµ . (7)The Hamilton–Jacobi equation is completely separable in the Kerr geometry [10](see also the review [11]), and the completely separated solution is f ( ξ , x , t ) = − Et + Lϕ + Z r √ R △ d r + Z θ √ Θd θ,R = (cid:2)(cid:0) r + a (cid:1) E − aL (cid:3) − △ (cid:2) Q + ( L − aE ) + r (cid:3) , Θ = Q + (cid:20)(cid:0) E − (cid:1) a − L sin θ (cid:21) cos θ, (8)where △ = r − r g r + a . The constants of motion are energy E , angular momentum L and a new constant Q .We choose E = 1 and L = 0, as for freely falling particles that start with zero velocityat infinity, with which the Lemaitre frame [12] can be associated in the Schwarzschildcase. Then the values Q < Q = q .Equations (6) relating x , τ and x , t at ξ = ( E, L, q ) = (1 , , q ) have the form τ = − t + Z r √ R △ d r + qθ,f ,E : Z r r + q √ R d r − qθ + Z θ a cos θq d θ = − t + Z r ( r + a ) − a △△√ R d r + Z θ a cos θq d θ,f ,L : − Z r ar g r △√ R d r + ϕ = − Z r ar g r △√ R d r + ϕ,f ,q : − Z r q √ R d r + θ = − Z r q √ R d r + θ. (9)At q = 0, the set of trajectories with ( E, L, q ) = (1 , , q ) does not reach sufficientlysmall r ( R < r ). Therefore, we pass to the limit q →
0. Equation (9)obtained from f ,q gives θ − θ = O ( q ) →
0, which allows finding ( θ − θ ) /q in theequation from f ,E . The system (9) gives a relation between the differentials of thecoordinates, and we find the nontrivial 3 × g x j x k , (cid:13)(cid:13)(cid:13) g x j x k (cid:13)(cid:13)(cid:13) = (10) e (cid:16) ρ △ R + h ρ (cid:17) e hρ e η h ρ △ R (cid:16) − ηη (cid:17) + h ρ i e hρ ρ η hρ e η h ρ △ R (cid:16) − ηη (cid:17) + h ρ i η hρ η (cid:20) ρ △ R (cid:16) − ηη (cid:17) + h ρ (cid:21) + ρ − r g rρ △ sin θ , where e = √ R ρ , η = ar g r ρ △ , R = r g r ( r + a ) , h = a sin(2 θ ) Z r r d r √ R , and the subscript 1 at functions means the replacement r → r . The covariant non-trivial (spatial-spatial) components are as follows, g r r = ρ r ( r + a ) 1 ρ (cid:18) r − a r g r ρ △ r sin θ + a r g r ρ △ Σ sin θ (cid:19) ,g r θ = − a ρ p r ( r + a ) r sin(2 θ ) ρ ˜ I (cid:18) − a r g r ρ △ sin θ (cid:19) ,g r ϕ = a √ r g ρ p r ( r + a ) sin θρ (cid:18) r − r ρ △ Σ (cid:19) ,g θ θ = ρ + a r sin (2 θ ) ρ ˜ I ,g θ ϕ = − a √ r g r sin(2 θ ) sin θρ ˜ I,g ϕ ϕ = Σ ρ sin θ, (11)where ˜ I = Z r r d r p r ( r + a ) . The determinant is det k g x j x k k = ρ r ( r + a ) r ( r + a ) sin θ. (12)The dependence of r on r and τ for a given θ (= θ ) is determined from the relation τ √ r g = Z r r r + a cos θ p r ( r + a ) d r. (13)Note that such a geodesic was obtained in [7] in the Doran coordinates [3] τ , r , θ . The metric obtained is singular if r , ρ or △ are equal to zero. However, thesesingularities are transient, since they are excluded if the moment in time is somewhatgreater than τ = 0. Namely, if τ > τ , τ √ r g = Z r + r + a p r ( r + a ) d r = Z r + r r + a r d r, (14)where r + is the larger of the roots of △ = 0 (horizon radius), then equation (13) forany θ has a solution r ≥ r > r + , and we are left only with the physical ringsingularity at ρ = 0.Moreover, we note that we can take both r and τ arbitrarily large, while keeping2 r / / (3 √ r g ) − τ finite and hence other coordinates in the physical region of interest.Then the (covariant) metric tensor is simplified, (cid:13)(cid:13)(cid:13) g x j x k (cid:13)(cid:13)(cid:13) = r rρ −√ r a r sin(2 θ ) ρ I √ r a √ r g r sin θρ −√ r a r sin(2 θ ) ρ I ρ + a r sin (2 θ ) ρ I − a √ r g r sin(2 θ ) sin θρ I √ r a √ r g r sin θρ − a √ r g r sin(2 θ ) sin θρ I (cid:16) r + a + a r g rρ sin θ (cid:17) sin θ , where I = Z ∞ r d r p r ( r + a ) , (15)and r is regarded as a function of τ , r , θ via τ = 23 r / − r / √ r g + Z ∞ r " r + a cos θ p r ( r + a ) − √ r d r √ r g . (16)The contravariant metric tensor becomes even more simplified, (cid:13)(cid:13)(cid:13) g x j x k (cid:13)(cid:13)(cid:13) = r h r g + ρ △ r ( r + a ) + a (2 θ ) ρ I i a √ r sin(2 θ ) ρ I − √ r a √ r g r + a a √ r sin(2 θ ) ρ I ρ − √ r a √ r g r + a r + a ) sin θ . (17)Note that now we can consider r and τ not necessarily arbitrarily large, and ex-pression (15) for g x j x k will still be accurate, only the interpretation of r as the initialvalue for r will not necessarily take place.2 r / / (3 √ r g ) is an analogue of the Lemaitre [12] radial coordinate in the Schwarz-schild case (expressing the Lemaitre metric in terms of r instead of that coordinatewas proposed in [13]). At a = 0, we have the Lemaitre metric,d s = − d τ + r r ( r , τ ) d r + r ( r , τ )dΩ , r / = r / − √ r g τ. (18)In Fig. 1, a section along r , τ passing through the singularity r = 0, θ = π/ r = 0, 0 ≤ θ < π/ r > θ = π/ r = 0, θ = π/
2, compared to the Lemaitre metric, for which there is only r > √ r d r = √ r g d τ + ρ p r ( r + a ) d r + a I sin(2 θ )d θ. (19)If we exclude r in favor of r , we should obtain an analogue of the Painlev´e-Gullstrandmetric [14, 15] for the Schwarzschild geometry, at least in the way of obtaining. Thisturns out to be just the Doran metric [3],d s = (cid:18) − r g rρ (cid:19) d τ + ρ r + a d r + ρ d θ + 2 r r g rr + a d τ d r +2 a r r g rr + a sin θ d r d ϕ + 2 a r g rρ sin θ d τ d ϕ + (cid:18) r + a + a r g rρ sin θ (cid:19) sin θ d ϕ . (20)Vice versa, substituting r = r ( τ, r , θ ), we can obtain a synchronous metric from theDoran one. In the present consideration, an asymptotic (at τ , r large) connectionbetween these new coordinates and the coordinates bound to the set of freely movingparticles is shown. Thus, we have made a binding of coordinates to the set of timelike geodesics whichrepresent the motion of freely falling particles with (
E, L, Q ) = (1 , , Q to be zero from the beginning, but we should carefully tendto zero, starting with small positive values. Second, the metric components (11) in theobtained frame of reference has singularities additional to the true Kerr singularity.However, these singularities are absent if the coordinates of the proper time τ and theinitial radial coordinate r are chosen large and at the same time corresponding to thepoints of interest in the original coordinates (Boyer-Lindquist). Moreover, we can takethe asymptotic (at large τ , r ) form (16) of the transformation (13) from r to r and r / − r / − I a / r = 0 θ = 0 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) r = 0 θ = const (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) r = 0 θ = π/ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) r = const θ = π/ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ( τ − τ ) √ r g Figure 1: A section along r , τ passing through the singularity r = 0, θ = π/ I = R ∞ (1 + y ) − / d y = [Γ(1 / / (4 √ π ).get the metric in a slightly modified synchronous frame of reference (15). This metricis singular only in the true Kerr singularity (the singularity ring) and is the directanalogue of the Lemaitre metric in the Schwarzschild case, only the interpretation of r as the initial value for r will not necessarily take place. Acknowledgments
The present work was supported by the Ministry of Education and Science of theRussian Federation.
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