arXiv:2101.07147v1 [gr-qc] 18 Jan 2021 e,freape 2 7.W su rmteKr erci h Boyer- the in Kerr the from issue We [7]. - c [2] another or example, one for co in under metric see, corresponding problem the the write on to Depending convenient is [1]. it Kerr by gravit given describing was equations mass Einstein spinning vacuum the of solution exact An Introduction 1 nteKr erci ycrnu eeec frame reference synchronous a in metric Kerr the On ewrs eea eaiiy ersaeie ycrnu fra synchronous ; Kerr relativity; general keywords: 83C57 83C15; classes: MSC 04.70.Bw 04.20.Jb; Nos.: PACS h iglrte fwihaeehutdb h hsclKer physical the by exhausted ring). singularity the are of which analogue of direct singularities the the and frame synchronous a have the value initial still Modifying as interpretation their coordinates. (keeping as way certain trajectories th of particles set for certain conditions initial a and time proper using by h ermti scniee nasnhoosfaeo refer of frame synchronous a in considered is metric Kerr The fSbra rnhRsinAaeyo Sciences of Academy Russian Branch Siberian of -alades [email protected] address: E-mail ukrIsiueo ula Physics Nuclear of Institute Budker ooiis,609,Russia 630090, Novosibirsk, ..Khatsymovsky V.M. Abstract 1 tlredsacs,we distances), large at s tfel oealong move freely at ecodntsi a in coordinates se iglrt (the singularity r eatemetric, Lemaitre me neobtained ence odnt system; oordinate toa edo a of field ational nsideration, Lindquist 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 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 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 . (16) 2 2 3 √rg r r (r + a ) − √rg Z " # The contravariant metric tensor becomes evenp more simplified,

2 1 ρ2 4 sin (2θ) 2 a2 sin(2θ) 1 a√rg rg + 2 △ 2 + a 2 I 2 I 2 2 r1 r(r +a ) ρ √r1 ρ − √r1 r +a j k 2 x1x1  h a sin(2θ) i 1  g = 2 I 2 0 . (17) √r1 ρ ρ  1 a√rg 1   2 2 0 2 2 2   − √r1 r +a (r +a ) sin θ    Note that now we can consider r1 and τ not necessarily arbitrarily large, and ex- pression (15) for g j k will still be accurate, only the interpretation of r1 as the initial x1x1 value for r will not necessarily take place. 3/2 2r1 /(3√rg) is an analogue of the Lemaitre [12] radial coordinate in the Schwarz- schild case (expressing the Lemaitre metric in terms of r1 instead of that coordinate was proposed in [13]). At a = 0, we have the Lemaitre metric,

2 2 r1 2 2 2 3/2 3/2 2 ds = dτ + dr1 + r (r1, τ)dΩ , r = r1 √rgτ. (18) − r(r1, τ) − 3

In Fig. 1, a section along r1, τ passing through the singularity r = 0, θ = π/2 is shown. There are two regions, r =0, 0 θ <π/2 and r > 0, θ = π/2 (the inside and ≤ outside of the singularity ring in the equatorial plane), on either side of the singularity line r = 0, θ = π/2, compared to the Lemaitre metric, for which there is only r> 0 on one side. From (16), we can find for the differentials

2 ρ 2 √r1dr1 = √rgdτ + dr + a I sin(2θ)dθ. (19) r(r2 + a2)

If we exclude r1 in favor of r, we shouldp obtain an analogue of the Painlev´e-Gullstrand metric [14, 15] for the Schwarzschild geometry, at least in the way of obtaining. This 7 turns out to be just the Doran metric [3], r r ρ2 r r ds2 = 1+ g dτ 2 + dr2 + ρ2dθ2 +2 g dτdr − ρ2 r2 + a2 r2 + a2   r r r r r +2a g sin2 θdrdϕ +2a g sin2 θdτdϕ r2 + a2 1 ρ2 1 r r r + r2 + a2 + a2 g sin2 θ sin2 θdϕ2. (20) ρ2 1   Vice versa, substituting r = r(τ,r1, θ), we can obtain a synchronous metric from the

Doran one. In the present consideration, an asymptotic (at τ, r1 large) connection between these new coordinates and the coordinates bound to the set of freely moving particles is shown.

4 Conclusion

Thus, we have made a binding of coordinates to the set of timelike which represent the motion of freely falling particles with (E, L, Q) = (1, 0, 0). This gives a synchronous frame. The following two points seem to be interesting. First, we cannot set the Carter’s constant Q to be zero from the beginning, but we should carefully tend to zero, starting with small positive values. Second, the metric components (11) in the obtained 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 the initial radial coordinate r1 are chosen large and at the same time corresponding to the points of interest in the original coordinates (Boyer-Lindquist). Moreover, we can take the asymptotic (at large τ, r1) form (16) of the transformation (13) from r to r1 and

3 2 (τ τ0)√rg −

r = 0 θ = π/2

r = 0 θ = const

r = 0 r = const θ = 0 θ = π/2

r3/2 r3/2 1 − 1(0) 3I a3/2 0 − 0

Figure 1: A section along r1, τ passing through the singularity r = 0, θ = π/2.

∞ 4 1/2 2 I0 = 0 (1 + y )− dy = [Γ(1/4)] /(4√π). R 8 get the metric in a slightly modified synchronous frame of reference (15). This metric is singular only in the true Kerr singularity (the singularity ring) and is the direct analogue of the Lemaitre metric in the Schwarzschild case, only the interpretation of r1 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 the Russian Federation.

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