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On the Kerr Metric in a Synchronous Reference Frame

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On the Kerr Metric in a Synchronous Reference Frame On the Kerr metric in a synchronous reference frame V.M. Khatsymovsky Budker Institute of Nuclear Physics of Siberian Branch Russian Academy of Sciences Novosibirsk, 630090, Russia E-mail address: [email protected] Abstract The Kerr metric is considered in a synchronous frame of reference obtained by using proper time and initial conditions for particles that freely move along a certain set of trajectories as coordinates. Modifying these coordinates in a certain way (keeping their interpretation as initial values at large distances), we still have a synchronous frame and the direct analogue of the Lemaitre metric, the singularities of which are exhausted by the physical Kerr singularity (the singularity ring). PACS Nos.: 04.20.Jb; 04.70.Bw MSC classes: 83C15; 83C57 arXiv:2101.07147v1 [gr-qc] 18 Jan 2021 keywords: general relativity; Kerr spacetime; synchronous frame 1 Introduction An exact solution of the vacuum Einstein equations describing gravitational field of a spinning 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-Lindquist 1 2 coordinates [2], r r ρ2 r r ds2 = 1+ g dt2 + dr2 + ρ2dθ2 + r2 + a2 + a2 g sin2 θ sin2 θdϕ2 − ρ2 ρ2 r r △ 2a g sin2 θdϕdt, (1) − ρ2 where ρ2 = r2 + a2 cos2 θ, = r2 r r + a2. △ − g The contravariant metric tensor is Σ2/ρ2 0 0 ar r/ρ2 − △ − g △ 0 /ρ2 0 0 gλµ = △ , (2) 2 0 01/ρ 0 2 2 2 2 2 argr/ρ 0 0( a sin θ)/ρ sin θ − △ △ − △ 2 where Σ2 = r2 + a2 a2 sin2 θ. − △ One of the important coordinate systems is the synchronous reference system. By fixing four components of the metric tensor, g =( 1, 0, 0, 0), the synchronous frame 0λ − explicitly leaves us with six physically significant metric functions, the spatial metric. It is also the simplest case of the lapse-shift functions (N, N)=(1, 0) in the Arnowitt- Deser-Misner formalism [8], a way to transfer the physics of the phenomenon to the true canonical coordinate (spatial metric), leaving (N, N) = const, which may be interesting 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 certain set 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 to the true ring Kerr singularity. Then we modify the definition of the new coordinates to 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 large distances), but the metric is simplified and only has the true ring Kerr singularity. 2 Transformation to a synchronous frame A way of passing to a synchronous frame is to use the Hamilton-Jacobi equation for a particle with action τ (as mentioned, for example, in the textbook [9]), ∂τ ∂τ gλµ +1=0. (3) ∂xλ ∂xµ 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 as an arbitrary function of ξ. The equations of motion are f,ξj (ξ, x, t)+ A,ξj (ξ)=0. (5) We consider the set of trajectories corresponding to a given fixed ξ. We take τ as the new 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 x1, t0(x1), then x1, 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 ), ⇒ 0 1 f,ξj (ξ, x1, t0(x1)) = f,ξj (ξ, x, t). (6) The contravariant metric tensor in the new coordinates x1, τ has the components ∂f(ξ, x, t) ∂f(ξ, x, t) gττ = gλµ = 1, ∂xλ ∂xµ − j j j ∂x ∂τ ∂x ∂f (ξ, x , t (x )) ∂τ x1τ 1 λµ 1 ,ξk 1 0 1 λµ g = λ µ g = λ µ g ∂x ∂x ∂f,ξk (ξ, x1, t0(x1)) ∂x ∂x j ∂x1 ∂f,ξk (ξ, x, t) ∂f(ξ, x, t) λµ = λ µ g ∂f,ξk (ξ, x1, t0(x1)) ∂x ∂x 1 ∂xj ∂ ∂f(ξ, x, t) ∂f(ξ, x, t) = 1 gλµ(x, t) =0, 2 ∂f (ξ, x , t (x )) ∂ξ ∂xλ ∂xµ ,ξk 1 0 1 k j k j k ∂x ∂x gx1x1 = 1 1 gλµ. (7) ∂xλ ∂xµ The Hamilton–Jacobi equation is completely separable in the Kerr geometry [10] (see also the review [11]), and the completely separated solution is r √R θ f(ξ, x, t)= Et + Lϕ + dr + √Θdθ, − 2 Z △ Z R = r2 + a2 E aL Q +(L aE)2 + r2 , − −△ − L2 Θ= Q + E2 1 a2 cos2 θ, (8) − − sin2 θ where = r2 r r + a2. △ − g 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 velocity 4 at infinity, with which the Lemaitre frame [12] can be associated in the Schwarzschild case. Then the values Q < 0 are not in the domain of definition. We set Q = q2. Equations (6) relating x1, τ and x, t at ξ =(E, L, q)=(1, 0, q) have the form r √R τ = t + dr + qθ, − Z △ r1 r2 + q2 θ1 a2 cos2 θ f,E : dr qθ1 + dθ √R − q Z Z r (r2 + a2)2 a2 θ a2 cos2 θ = t + − △dr + dθ, − √R q r1Z △ r Z argr argr f,L : dr + ϕ1 = dr + ϕ, − √R − √R Z △ Z △ r1 q r q f,q : dr + θ1 = dr + θ. (9) − √R − √R Z Z At q = 0, the set of trajectories with (E, L, q) = (1, 0, q) does not reach sufficiently 6 small r (R < 0 at these r). Therefore, we pass to the limit q 0. Equation (9) → obtained from f gives θ θ = O(q) 0, which allows finding (θ θ)/q in the ,q 1 − → 1 − equation from f,E. The system (9) gives a relation between the differentials of the coordinates, and we find the nontrivial 3 3 block of the contravariant components of × j k the new metric, gx1x1 , j k gx1x1 = (10) 2 ρ2 h2 h ρ2 η h2 e 1+ △ + 2 e1 2 e1η1 1+ △ 1 + 2 1 R ρ ρ R − η1 ρ h 1 h h i e1 ρ2 ρ2 η1 ρ2 , 2 2 2 2 2 2 ρ η h h 2 ρ η h ρ rgr e1η1 1+ △ 1 + 2 η1 2 η 1+ △ 1 + 2 + 2 − 2 R − η1 ρ ρ 1 R − η1 ρ ρ sin θ △ h i r1 √R1 argr1 2 2 2 dr where e1 = 2 , η1 = 2 , R = rgr(r + a ), h = a sin(2θ) , ρ ρ 1 √R 1 1△ Zr and the subscript 1 at functions means the replacement r r . The covariant non- → 1 5 trivial (spatial-spatial) components are as follows, 4 2 2 2 ρ1 1 a rgr1 2 a rgr1 2 2 gr1r1 = 2 2 2 r 2 2 r sin θ + 4 2 Σ sin θ , r1 (r1 + a ) ρ − ρ1 1 ρ1 1 2 △ 2 △ 2 ρ1 r sin(2θ) ˜ a rgr1 2 gr1θ1 = a I 1 sin θ , 2 2 2 − r (r + a2) ρ − ρ 1 1 1 1△ 2 2 p ρ1 sin θ r1 2 gr1ϕ1 = a rg r Σ , √ 2 2 2 r (r + a2) ρ − ρ 1 1 1 1△ r sin2(2θ) g = ρ2 + ap4 I˜2, θ1θ1 ρ2 2 3 r sin(2θ) sin θ ˜ gθ1ϕ1 = a √rg I, − ρ2 Σ2 g = sin2 θ, (11) ϕ1ϕ1 ρ2 r1 dr where I˜ = . 2 2 r r (r + a ) Z The determinant is p 4 2 2 ρ1r (r + a ) 2 det g j k = sin θ. (12) x1x1 2 2 k k r1 (r1 + a ) The dependence of r on r1 and τ for a given θ (= θ1) is determined from the relation r1 r2 + a2 cos2 θ τ√rg = dr. (13) 2 2 r r (r + a ) Z Note that such a geodesic was obtained inp [7] in the Doran coordinates [3] τ, r, θ. 3 Asymptotic form of the transformation The metric obtained is singular if r , ρ or are equal to zero. However, these 1 1 △1 singularities are transient, since they are excluded if the moment in time is somewhat greater than τ = 0. Namely, if r+ r2 + a2 r+ a2 τ > τ0, τ0√rg = dr = r + dr, (14) 2 2 0 r (r + a ) 0 r Z Z r where r is the larger of the roots ofp = 0 (horizon radius), then equation (13) for + △ any θ has a solution r 0 only for r >r , and we are left only with the physical ring ≥ 1 + singularity at ρ2 = 0. Moreover, we note that we can take both r1 and τ arbitrarily large, while keeping 3/2 2r /(3√rg) τ finite and hence other coordinates in the physical region of interest. 1 − 6 Then the (covariant) metric tensor is simplified, g j k = x1x1 r 2 r sin(2θ) r sin2 θ r 2 r a 2 I r a r 2 1 ρ √ 1 ρ √ 1 √ g ρ − 2 2 2 r sin(2θ) 2 4 r sin (2θ) 2 3 r sin(2θ) sin θ √r1a 2 I ρ + a 2 I a √rg 2 I , − ρ ρ − ρ 2 2 r sin θ 3 r sin(2θ) sin θ 2 2 2 rgr 2 2 √r1a√rg 2 a √rg 2 I r + a + a 2 sin θ sin θ ρ − ρ ρ ∞ dr where I = , (15) 2 2 r r (r + a ) Z and r is regarded as a functionp of τ, r1, θ via 3/2 3/2 2 2 2 2 r r ∞ r + a cos θ dr τ = 1 − + √r .
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