Kerr Geodesics in horizon-penetrating Kerr coordinates: description in terms of Weierstrass functions
Zuzanna Bakun Angelika Łukanty Anastasiia Untilova Adam Cieślik Patryk Mach



Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland


Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland


Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland



Institute of Theoretical Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland


Institute of Theoretical Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland




§ ABSTRACT 
    We revisit the theory of timelike and null geodesics in the (extended) Kerr spacetime. This work is a sequel to a recent paper by Cieślik, Hackmann, and Mach, who applied the so-called Biermann–Weierstrass formula to integrate Kerr geodesic equations expressed in Boyer–Lindquist coordinates. We show that a formulation based on the Biermann–Weierstrass theorem can also be applied in horizon-penetrating Kerr coordinates, resulting in solutions that are smooth across Kerr horizons. Horizon-penetrating Kerr coordinates allow for an explicit continuation of timelike and null geodesics between appropriate regions of the maximal analytic extension of the Kerr spacetime. A part of this work is devoted to a graphic visualisation of such geodesics.


Kerr Geodesics in horizon-penetrating Kerr coordinates: description in terms of Weierstrass functions
    Patryk Mach
    September 9, 2024
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§ INTRODUCTION


The Kerr metric is usually written in Boyer–Lindquist coordinates <cit.>. They lead to a simple form of the metric tensor with a single off-diagonal term—an advantage allowed by the fact that the Kerr metric is circular (or orthogonally transitive)—but suffer a coordinate singularity at the black hole horizon. A system of coordinates on the Kerr spacetime regular at the horizon has already been proposed in the original paper of Kerr in 1963 <cit.>. Here we will work with its more popular variant, obtained by performing an additional transformation t = u - r (in the original notation of Kerr <cit.>) and a change in the convention regarding the black hole spin (a → -a). These coordinates are often referred to simply as Kerr or Kerr–Schild coordinates <cit.>. Since the latter term is usually reserved for a popular Cartesian-type coordinate system on the Kerr spacetime, we prefer to use the term “horizon-penetrating Kerr coordinates.”

Separability of the geodesic motion in the Kerr spacetime was discovered in 1968 by Carter <cit.> (see also <cit.>), who worked in the original coordinate system introduced by Kerr (save for the change a → - a). Nowadays, the majority of works on Kerr geodesics use Boyer–Lindquist coordinates (a sample of papers published after 2000 includes <cit.>). While algebraic simplicity is an obvious advantage, using Boyer–Lindquist coordinates for Kerr geodesics can be, in some cases, misleading, as it leads to a non-physical near-horizon behavior—generic geodesics tend to wind up around the horizon. Boyer–Lindquist coordinates were also used in a recent paper <cit.>, which introduced a uniform description of all generic timelike and null Kerr geodesics, based on Weierstrass functions. The advantage of this work was three-fold: Solutions were given in a form depending explicitly on the constants of motion and initial positions. No a priori knowledge of radial turning points was needed. Finally, geodesics with no radial turning points were described by explicitly real formulas. 

In this paper we extend the analysis of <cit.> to horizon-penetrating coordinates, which allow us to remove the singular behavior of geodesics at the black hole horizon, depending on the direction of motion. Quite surprisingly, little of the original simplicity of the analysis presented in <cit.> and based on Boyer–Lindquist coordinates is lost. Radial and polar equations and solutions remain unchanged. Additional terms appear in the azimuthal and time equations, which can readily be integrated. All solutions can still be written in terms of standard Weierstrass functions, and a single set of formulas remains applicable to all generic geodesics. As in <cit.>, solutions are fully specified by the values of constants of motion—the energy, the angular momentum, the Carter constant—and the initial position.

We show explicit examples of Kerr geodesics, attempting to visualize the behavior of those geodesics that cross the horizons. Such a visualization is especially tricky for geodesics which continue to negative values of the radius, allowed in the maximal analytic extension of the Kerr spacetime.

We prove that a future-directed timelike geodesic originating outside the event horizon and plunging into the black hole can, in horizon-penetrating Kerr coordinates, be continued smoothly through the event horizon. We also show that if this coordinate system admits a smooth transition of a given geodesic trajectory through one of horizons in a given radial direction, such a smooth transition is not allowed in the opposite direction.
 



§ GEODESIC EQUATIONS IN THE KERR SPACETIME




 §.§ Metric conventions


In this work, we employ standard geometric units in which the speed of light c and the gravitational constant G are set to 1. The metric signature is (-, +, +, +).

In Boyer–Lindquist coordinates x^μ = (t,r,θ,φ) the Kerr metric can be written as

    g=-(1-2Mr/ρ^2) dt^2 -4Marsin^2θ/ρ^2 dtdφ+ ρ^2/Δ dr^2+ρ^2 dθ^2+(r^2+a^2+2Ma^2r sin^2θ/ρ^2) sin^2θ dφ^2,

where


    Δ    :=     r^2 - 2Mr + a^2, 
    ρ^2     :=     r^2 + a^2 cos^2θ.


We will only consider the case with a^2 < M^2, due to its physical relevance. Kerr horizons are located at the zeros of Δ, i.e., at r_± = M ±√(M^2 - a^2). We will refer to horizons at r = r_- and r = r_+ as the Cauchy and the event horizon, respectively.

The coordinate singularity at the horizons can be removed by a coordinate transformation to horizon-penetrating Kerr coordinates (t^',r,θ,φ^') defined in terms of one-forms

    dt^' = dt+2Mr/Δdr,     
        dφ'=dφ+a/Δdr,

or more explicitly, by setting

    t'=t+2M∫rdr/r^2-2Mr+a^2,     φ'=φ+a∫dr/r^2-2Mr+a^2.

In horizon-penetrating Kerr coordinates the Kerr metric reads

    g     =     -(1-2Mr/ρ^2)dt^'^2-4Mr/ρ^2asin^2θ dt^' dφ^' + 4Mr/ρ^2 dt^' dr - 2 a (1 + 2Mr/ρ^2)sin^2θ dr dφ^'
           + (1+2Mr/ρ^2) dr^2 + ρ^2 dθ^2 + [ (r^2+a^2)^2-a^2 Δsin^2θ]sin^2θ/ρ^2dφ^'^2.


In this paper we will restrict ourselves to a region of the maximally extended Kerr spacetime covered by horizon-penetrating Kerr coordinates with the ranges t^'∈ℝ, r ∈ℝ, θ∈ [0,π], φ∈ [0,2π). It spans across three causal diamonds or Boyer–Lindquist blocks, numbered as I, II, and III, according to a convention used in <cit.>. They are defined by the following ranges of the radius. Block I: r > r_+; Block II: r_- < r < r_+; Block III: r < r_-. We depict this region on a Penrose conformal diagram corresponding to the symmetry axis of the Kerr spacetime in Fig. <ref>. To show the properties of the time foliation defined by the horizon-penetrating Kerr coordinates, we also plot, in Fig. <ref>, hypersurfaces of constant time t^'. They remain regular across the event horizon r = r_+ joining Blocks I and II and across the Cauchy horizon r = r_-, joining Blocks II and III. Further details of the construction of the Penrose diagram shown in Fig. <ref> are given in Appendix <ref>.


 



 §.§ Geodesic equations


Geodesic equations can be written in the Hamiltonian form as

    dx^μ/dτ̃=∂ H/∂ p_μ,   dp_ν/dτ̃=-∂ H/∂ x^ν,

where p^μ=dx^μ/dτ̃, H(x^α,p_β)=1/2g^μν(x^α)p_μ p_ν=-1/2m^2, and m is the particle rest mass. The four-velocity u^μ = d x^μ/dτ is normalised as g_μνu^μ u^ν=-δ_1, where

    δ_1 =  1    for timelike geodesics, 
    0    for null geodesics.

The above conventions imply that the affine parameter τ̃ and the proper time τ are related by τ̃= τ/m.

Let the metric g be given by Eq. (<ref>) or (<ref>). A standard reasoning shows that H, E := - p_t and l_z := p_φ are constants of motion. The fourth constant, referred to as the Carter constant 𝒦, can be derived by a separation of variables in the corresponding Hamilton–Jacobi equation <cit.>. This separation can be performed both in Boyer–Lindquist and in horizon-penetrating Kerr coordinates. Moreover, in both coordinate systems constants H, E, l_z, and 𝒦 have the same numerical values. In particular, momentum components p_t and p_φ transform as p_t^' = p_t = -E and p_φ^' = p_φ = l_z.

In Boyer–Lindquist coordinates geodesic equations can be written as



    ρ^2dr/dτ̃    =    ϵ_r√(R(r)),
    ρ^2dθ/dτ̃    =    ϵ_θ√(Θ(θ)),
    ρ^2dφ/dτ̃    =    a[(r^2+a^2)E-al_z]/Δ+1/sin^2θ(l_z-aEsin^2θ),
    ρ^2dt/dτ̃    =    (r^2+a^2)[(r^2+a^2)E-al_z]/Δ+a(l_z-aEsin^2θ),


where we denoted


    R(r)     :=     [(r^2+a^2)E-al_z]^2-Δ(m^2r^2+𝒦), 
    Θ(θ)     :=    𝒦-m^2a^2cos^2θ- ( l_z/sinθ-asinθ E )^2.


We will refer to R(r) and Θ(θ) as the radial and polar effective potentials, respectively. The signs ϵ_r = ± 1 and ϵ_θ = ± 1 indicate the direction of motion with respect to the radial and polar coordinates.

There is a useful parametrization of Kerr geodesics, introduced by Mino in <cit.>, which allows to partially decouple Eqs. (<ref>). The so-called Mino time s̃ is defined by

    ρ^2dx^μ/dτ̃=dx^μ/ds̃

or

    τ̃= ∫_0^s̃ρ^2ds.

Using s̃ as the geodesic parameter, we obtain



    dr/ds̃    =    ϵ_r√(R(r)), 
    dθ/ds̃    =    ϵ_θ√(Θ(θ)), 
    dφ/ds̃    =    a[(r^2+a^2)E-al_z]/Δ+1/sin^2θ(l_z-aEsin^2θ), 
    dt/ds̃    =    (r^2+a^2)[(r^2+a^2)E-al_z]/Δ+a(l_z-aEsin^2θ).




Geodesic equations in the horizon-penetrating Kerr coordinates can be obtained simply by the following vector transformation


    dt'/ds̃    =    ∂ t'/∂ tdt/d s̃+∂ t'/∂ rdr/d s̃=dt/ds̃+2Mr/Δdr/ds̃, 
    dφ'/ds̃    =    ∂φ'/∂φd φ/d s̃+∂φ '/∂ rdr/d s̃=dφ/ds̃+a/Δdr/ds̃.


This gives



    dr/ds̃    =    ϵ_r√(R(r)),
    dθ/ds̃    =    ϵ_θ√(Θ(θ)),
    dφ'/ds̃    =    a[(r^2+a^2)E-al_z]/Δ+1/sin^2θ(l_z-aEsin^2θ)+a/Δϵ_r√(R(r)),
    dt'/ds̃    =    (r^2+a^2)[(r^2+a^2)E-al_z]/Δ+a(l_z-aEsin^2θ)+2Mr/Δϵ_r√(R(r)).



In the remainder of this paper we will use the following dimensionless variables:

    a=Mα,      t'=MT',   r=Mξ,   E=mε,  𝒦=M^2m^2κ,    l_z=Mmλ_z,   s̃=1/Mms.

Dimensionless radii corresponding to Kerr horizons will be denoted by ξ_± = 1 ±√(1 - α^2). For null geodesics, for which m = 0, the parameter m in the above equations can be replaced with any mass parameter m̃ > 0.

In dimensionless variables (<ref>), geodesic equations (<ref>) have the form


    dξ/ds    =    ϵ_r√(R̃), 
    dθ/ds    =    ϵ_θ√(Θ̃), 
    dφ'/ds    =    α[(ξ^2+α^2)ε-αλ_z]/ξ^2-2ξ+α^2+1/sin^2θ(λ_z-αεsin^2θ)+α/ξ^2-2ξ+α^2ϵ_r√(R̃), 
    dT'/ds    =    (ξ^2+α^2)[(ξ^2+α^2)ε-αλ_z]/ξ^2-2ξ+α^2+α(λ_z-αεsin^2θ)+2ξ/ξ^2-2ξ+α^2ϵ_r√(R̃),


where


    R̃    :=     [(ξ^2+α^2)ε-αλ_z]^2-(ξ^2-2ξ+α^2)(δ_1ξ^2+κ),
    
    Θ̃    :=    κ-δ_1α^2cos^2θ-1/sin^2θ(λ_z-αεsin^2θ)^2.


Note that the coordinate transformation from Boyer–Lindquist to horizon-penetrating Kerr coordinates affects only the equations for φ^' and T^', while the radial and polar equations remain unchanged. Equations (<ref>) can also be obtained directly by working in the horizon-penetrating Kerr coordinates and by separating the variables in the Hamilton–Jacobi equation. We emphasise a connection with the Boyer–Lindquist form (<ref>), in order to make use of solutions to Eqs. (<ref>) derived in <cit.>. In the next section, we review the solutions of the radial and polar equations obtained in <cit.> and focus on equations for φ^' and T^'.



§ SOLUTIONS OF GEODESIC EQUATIONS




 §.§ Biermann–Weierstrass formula


The form of solutions used in this work predominantly rely on the following result due to Biermann and Weierstrass <cit.>. Proofs of this theorem can be found in <cit.>.

Let f be a quartic polynomial

    f(x) = a_0 x^4 + 4 a_1 x^3 +6 a_2 x^2 + 4a_3 x + a_4,

and let g_2 and g_3 denote Weierstrass invariants of f:



    g_2     =      a_0 a_4 - 4a_1 a_3 + 3 a_2^2, 
    
            g_3     =     a_0 a_2 a_4 + 2a_1 a_2 a_3 -a_2^3 -a_0 a_3^2 - a_1^2 a_4.


Denote

    z(x) = ∫^x_x_0dx^'/√(f(x^')),

where x_0 can be any constant. Then x can be expressed as

    x = x_0 + - √(f(x_0))℘'(z)
      + 1/2 f'(x_0) [ ℘(z) - 1/24f”(x_0) ]
      + 1/24 f(x_0) f”'(x_0)  /2 [ ℘(z)
      - 1/24 f”(x_0) ]^2 - 1/48 f(x_0) f^(4)(x_0) ,

where ℘(z) =℘(z;g_2,g_3) is the Weierstrass function with invariants
(<ref>). In addition


    ℘(x)     =    √(f(x)f(x_0))+f(x_0)/2(x-x_0)^2+f'(x_0)/4(x-x_0)+f”(x_0)/24, 
    ℘' (x)     =     -[f(x)/(x-x_0)^3-f'(x_0)/4(x-x_0)]√(f(x_0)) - [f(x)/(x-x_0)^3+f'(x_0)/4(x-x_0)]√(f(x)).
 




 §.§ Radial motion


The dimensionless radial potential R̃ can be written as

    R̃(ξ) = a_0 ξ^4 + 4 a_1 ξ^3 + 6 a_2 ξ^2 + 4a_3 ξ + a_4,

where



    a_0     =    ε^2 - δ_1, 
    
            a_1     =    1/2δ_1, 
    
            a_2     =     -1/6(δ_1 α^2 + κ - 2α^2 ε^2 + 2 αελ_z) , 
    
            a_3     =    1/2κ, 
    
            a_4     =    α^4 ε^2 - α^2 κ - 2 α^3 ελ_z + α^2 λ_z^2 = -α^2 [κ - (αε - λ_z)^2].


Weierstrass invariants associated with the coefficients (<ref>) will be denoted by



    g_R̃,2    =     a_0 a_4 - 4 a_1 a_3 + 3 a_2^2, 
    
        g_R̃,3    =     a_0 a_2 a_4 + 2 a_1 a_2 a_3 - a_2^3 - a_0 a_3^2 - a_1^2 a_4.


A direct application of the Biermann–Weierstrass theorem to Eq. (<ref>) yields the formula for ξ=ξ(s),

    ξ(s)=ξ_0+-ϵ_r,0√(R̃(ξ_0))℘'_R̃(s)+1/2R̃'ξ_0)[℘_R̃(s)-1/24R̃”(ξ_0)]+1/24R̃(ξ_0)R̃”'(ξ_0)]/2[℘_R̃(s)-1/24R̃”(ξ_0)]^2-1/48R̃(ξ_0) R̃^(4)(ξ_0),

where ℘_R̃(s)=℘_R̃(s;g_R̃,2g_R̃,3) is the Weierstrass function with invariants (<ref>), and ξ_0 = ξ(0) is an arbitrarily selected initial radius corresponding to s = 0. In Equation (<ref>) the sign ϵ_r,0 = ± 1 denotes a value of ϵ_r corresponding to the initial location ξ_0. In other words, ϵ_r,0 is a part of initial data (initial parameter), while the sign ϵ_r can change along a given geodesic.



 §.§ Polar motion


Equation (<ref>) can be transformed to the Biermann–Weierstrass form by a substitution μ=cosθ, which provides a one to one mapping for 0 ≤θ≤π. Defining g(μ) := sin^2 θΘ̃(θ), we get

    dμ(s)/ds = - ϵ_θ√(g(μ)).

The function g(μ) is a polynomial with respect to μ given by

    g(μ) = b_0 μ^4 + 6 b_2 μ^2 + b_4,

where the coefficients b_0, b_2, and b_4 can be expressed as


    b_0     =     - α^2 ( ε^2 - δ_1 ) = - α^2 a_0, 
    
        b_2     =    1/6( - α^2 δ_1 + 2 α^2 ε^2 - κ - 2 αελ_z ) = a_2, 
    
        b_4     =     - α^2 ε^2 + κ + 2 αελ_z - λ_z^2 = κ - (αε - λ_z)^2 = - a_4/α^2,


and a_0, a_2, and a_4 are given by Eqs. (<ref>). Weierstrass invariants associated with coefficients b_0, b_2, and b_4 can be written as


    g_g,2    =     b_0 b_4 + 3 b_2^2, 
    
    g_g,3    =     b_0 b_2 b_4 - b_2^3.



Again, using the Biermann–Weierstrass formula, one can write the expression for μ=μ(s) in the form

    μ(s)=μ_0+ϵ_θ ,0√(g(μ_0))℘'_g(s)+1/2g'(μ_0)[℘_g(s)-1/24g”(μ_0)]+1/24g(μ_0)g”'(μ_0)]/2[℘_g(s)-1/24g”(μ_0)]^2-1/48g(μ_0) g^(4)(μ_0),

where ℘_g(s)=℘_g(s;g_g,2g_ g,3) and μ_0 = μ(0) = cosθ_0 =cosθ(0) represents an initial value corresponding to s=0. As in Eq. (<ref>), the sign ϵ_θ ,0 = ± 1 is a parameter equal to ϵ_θ at θ = θ_0, and it remains constant along the entire geodesic. A more detailed discussion of solutions (<ref>) and (<ref>) can be found in <cit.>.



 §.§ Azimuthal motion


Equation (<ref>) consists of a component related to the azimuthal motion in Boyer–Lindquist coordinates and an additional term, dependent on ξ(s), arising from transformation (<ref>). We will first treat the two components separately and then discuss their sum, which may remain regular across the horizons. An integration with respect to the Mino time s yields

    φ^'(s)-φ^'(0)     =    ∫_0^s{α [(ξ^2(s̅)+α^2)ε-αλ_z]/ξ^2(s̅)-2ξ(s̅)+α^2+1/sin^2θ(s̅)[ λ_z-αεsin^2θ(s̅)] } ds̅+∫_0^sαϵ_r √(R̃(s̅))/ξ^2(s̅)-2ξ(s̅)+α^2 ds̅
        =    α∫_0^s2 ξ(s̅) ε - αλ_z/ξ^2(s̅)-2ξ(s̅)+α^2 ds̅ + λ_z ∫_0^s d s̅/sin^2 θ(s̅) + α∫_0^sϵ_r √(R̃(s̅))/ξ^2(s̅)-2ξ(s̅)+α^2 ds̅
        =     J_BL(s) + J_ξ(s),

where J_BL(s) := J̃_BL(s) + J_θ(s) and

    J̃_BL(s)     :=    α∫_0^s2 ξ(s̅) ε - αλ_z/ξ^2(s̅)-2ξ(s̅)+α^2 ds̅, 
    
        J_θ(s)     :=    λ_z ∫_0^s d s̅/sin^2 θ(s̅), 
    
        J_ξ(s)     :=    α∫_0^sϵ_r  √(R̃ (s̅))/ξ^2(s̅)-2ξ(s̅)+α^2 ds̅.


Integrals J̃_BL and J_θ were expressed in terms of Weierstrass functions in <cit.>. To obtain a solution for Eq. (<ref>), we write

    J_ξ(s)= α∫_0^s d ξ(s̅)/d s̅/ξ^2(s̅)-2ξ(s̅)+α^2 ds̅.

For a segment of a geodesic along which ξ = ξ(s) is monotonic, one change the integration variable to ξ and write

    J_ξ(s)= α∫_ξ_0^ξ(s)dξ̅/ξ̅^2-2 ξ̅+α^2=α[ arctan(ξ(s)-1/√(α^2 -1))/√(α^2 -1)- arctan(ξ_0 - 1/√(α^2 -1))/√(α^2 -1)],

which, together with Eq. (<ref>), provides the solution. Note that due to the symmetry of Eq. (<ref>), expression (<ref>) remains valid also for trajectories passing through radial turning points.

The integrals J_BL and J_ξ are, generically, divergent at the horizons, i.e., for s such that ξ(s) = ξ_±, but the sum J_BL + J_ξ can remain regular. A direct calculation shows that J_BL(s) and J_ξ (s) can be expressed in the form

    J̃_BL(s)+J_ξ (s) = α∫_0^s{δ_1 ξ ^2(s̅)+κ/[ξ^2(s̅)+α^2]ε -αλ_z - d ξ(s̅)/d s̅ - ε}ds̅.

The integrand in Eq. (<ref>) can be divergent, if

    [ξ^2(s)+α^2] ε -αλ_z - d ξ(s)/d s = 0

or, equivalently,

    [ξ^2(s)+α^2]ε -αλ_z = ϵ_r √(R̃).

By computing the square of the above equation we see that this can only happen, if

    [ξ^2(s) - 2 ξ(s) + α^2][δ_1 ξ^2(s) + κ] = 0,

that is at ξ(s) = ξ_±. On the other hand, the expression [ξ^2(s)+α^2]ε -αλ_z - d ξ(s)/d s can be clearly non-zero at the horizon, depending on the sign ϵ_r. In our examples discussed in Sec. <ref> this happens for ϵ_r = -1, i.e., for incoming geodesics. In general, if the sign of [ξ^2(s) + α^2] ε - αλ_z can be controlled, one can exclude the possibility that J̃_BL diverges at the horizon. For instance, for  [ξ^2(s) + α^2] ε - αλ_z > 0 and ϵ_r = -1, the denominator in Eq. (<ref>) remains strictly positive. We discuss this problem in more detail in Sec. <ref>.



 §.§ Time coordinate


Similarly to the equation for φ^', the right-hand side of Eq. (<ref>) also consist of a Boyer–Lindquist term and a term associated with the transformation to horizon-penetrating Kerr coordinates. Integrating Eq. (<ref>) one gets

    T^'(s)-T^'(0)     =    ∫_0^s{[ξ^2(s̅)+α^2][(ξ^2(s̅)+α^2)ε-αλ_z]/ξ^2(s̅)-2ξ(s̅)+α^2 + α [λ_z-αεsin^2θ(s̅)] }ds̅
           +∫_0^s 2ξ(s̅) ϵ_r √(R̃(s̅))/ξ^2(s̅)-2ξ(s̅)+α^2ds̅
        =     N_BL(s) + N_ξ(s).

Here

    N_BL(s)     :=    ∫_0^s{[ξ^2(s̅)+α^2][(ξ^2(s̅)+α^2)ε-αλ_z]/ξ^2(s̅)-2ξ(s̅)+α^2+α(λ_z-αεsin^2θ(s̅)) } ds̅
        =    Ñ_BL(s) - α^2 ε N_θ(s),

where

    Ñ_BL(s)     :=    ∫_0^s [ ξ^2(s̅) + α^2 ]^2 ε - 2 αλ_z ξ(s̅)/ξ^2(s̅) - 2 ξ(s̅) + α^2 d s̅, 
    
        N_θ(s)     :=    ∫_0^s sin^2 θ(s̅) d s̅,

and 

    N_ξ(s) := 2∫_0^sξ(s̅) ϵ_r √(R̃(s̅))/ξ^2(s̅)-2ξ(s̅)+α^2ds̅.

The integrals Ñ_BL(s) and N_θ(s) were computed in <cit.>.

As before, we start by evaluating the integral in Eq. (<ref>), which we write in the form

    N_ξ(s)= 2∫_0^sξ(s̅)/ξ^2(s̅)-2ξ(s̅)+α^2d ξ(s̅)/d s̅ ds̅.

Changing the integration variable form s to ξ=ξ(s) we obtain

    N_ξ (s) = 2 ∫_ξ_0^ξ(s)ξ̅/ξ̅^2-2ξ̅+α^2dξ̅ = log[ ξ^2(s) - 2 ξ(s) +α^2/ξ_0^2-2 ξ_0+α^2] + 2 [arctan(ξ(s) - 1/√(α^2 - 1))/√(α^2 -1)- arctan(ξ_0 - 1/√(α^2 -1))/√(α^2 -1)],

which again together with Eq. (<ref>) provides the solution.

A regularization of the sum Ñ_BL(s) + N_ξ(s) can be done in many ways. One of them is, however, particularly convenient. Note that

    (ξ^2 + α^2)^2 ε - 2 αλ_z ξ/ξ^2 - 2 ξ + α^2 = (ξ^2 + α^2) ε + 2 ξ[ (ξ^2 + α^2) ε - αλ_z ]/ξ^2 - 2 ξ + α^2.

Therefore,

    Ñ_BL(s) + N_ξ(s)     =    ∫_0^s {2 ξ(s̅) [ (ξ^2(s̅) + α^2) ε - αλ_z + dξ(s̅) / ds̅]/ξ^2(s̅) - 2 ξ(s̅) + α^2 + [ξ^2(s̅) + α^2] ε} d s̅
        =    ∫_0^s {2 ξ(s̅) [ δ_1 ξ^2(s̅) + κ]/[ξ^2(s̅) + α^2] ε - αλ_z - d ξ(s̅) / d s̅ + [ξ^2(s̅) + α^2] ε} d s̅.

The advantage of this choice is that now a potentially problematic denominator in Eq. (<ref>) has the same form as in Eq. (<ref>).



 §.§ Regularity at horizons



We see from preceding subsections that the regularity of the expressions for φ^' and T^' depends on the signs of

    A := (ξ^2 + α^2) ε - αλ_z

and ϵ_r. It can be shown that for timelike future-directed geodesics at ξ > ξ_+ (in Boyer–Lindquist Block I), one has A > 0. Thus, by continuity, for ϵ_r = -1 (incoming geodesics) and R̃ > 0, we have A - d ξ(s)/ds = A - ϵ_r √(R̃) > 0 up to the horizon at ξ_+. Hence, for incoming future-directed timelike geodesics both φ^'
and T^' remain regular at the event horizon joining Blocks I and II.

A proof that for a timelike future-directed geodesic at ξ > ξ_+, one must have A > 0 can be found in <cit.>, but we repeat it here for completeness. Note first that the vector

    N = ( 1 + 2 M r/ρ^2) ∂_t^' - 2 M r/ρ^2∂_r

is normal to hypersurfaces of constant time t^'. Lowering the indices in N one gets N_μ = (-1,0,0,0). Since g(N,N) = - (1 + 2 M r/ρ^2), it is also timelike, as long as r > 0. This vector defines a time orientation. Consider a vector

    X := (r^2 + a^2) ∂_t^' + a ∂_φ^'.

O'Neill refers to X as one of “the canonical Kerr vector fields” (<cit.>, p. 60).
It satisfies g(X,X) = - ρ^2 Δ, and thus it is timelike for Δ > 0, i.e., for r < r_- or r > r_+ (in Blocks I and III). It is also future-directed, since g(N,X)= - dt^'(X) = -(r^2 + a^2) < 0. On the other hand,

    p(X) = (r^2 + a^2) p_t^' + a p_φ^' = - [ (r^2 + a^2) E - a l_z ] = - M^2 m [ (ξ^2 + α^2) ε - αλ_z ],

where p denotes the momentum covector. A future-directed momentum p has to satisfy p(X) < 0, and hence A = (ξ^2 + α^2) ε - αλ_z > 0. We emphasise that negative values of ε are still allowed within the ergosphere.

In Block II, where Δ < 0, i.e., for r_- < r < r_+, Rioseco and Sarbach <cit.> propose to use the vector

    Y := 2 M r ∂_t^' + Δ∂_r + a ∂_φ^'.

It satisfies g(Y,Y) = ρ^2 Δ and, consequently, it is timelike for r_- < r < r_+. Since g(N,Y) = -dt^'(Y) = - 2 M r, it is also future-directed in r_- < r < r_+. On the other hand,

    p(Y) = 2 M r p_t^' + Δ p_r + a p_φ^' = - 2 M r E + Δ p_r + a l_z.

Since Δ p_r = ϵ_r √(R) + 2 M r E - a l_z, we get p(Y) = ϵ_r √(R). Thus in Block II, the momentum p can be future-directed only for ϵ_r = -1.


Turning to the “regularization” applied in Eqs. (<ref>) and (<ref>) note that it is based on the observation that

    A^2 - R̃ = (ξ^2 - 2 ξ + α^2)(δ_1 ξ^2 + κ),

and hence A^2 - R̃ has two real zeros, precisely at ξ = ξ_±. On the other hand

    A^2 - R̃ = ( A - ϵ_r √(R̃)) ( A + ϵ_r √(R̃)) = ( A - d ξ/ds) (A + d ξ/ds).

Thus, if A - ϵ_r √(R̃) remains non-zero at a given radius ξ = ξ_+ or ξ = ξ_-, then A + ϵ_r √(R̃) must be zero for the same radius, and vice versa.  In other words, horizon-penetrating Kerr coordinates do not allow for a description in which a given trajectory can cross smoothly horizons at ξ = ξ_+ or ξ = ξ_- in both radial directions ϵ_r = ± 1 at the same time. This is, of course, consistent with the behavior shown in Fig. <ref>. Suppose that a given trajectory passes from Block I to Block II, and then to Block III, where it encounters a radial turning point and continues further with ϵ_r = +1. Such a trajectory will hit the Cauchy horizon at ξ = ξ_- with ϵ_r = +1, where both T^' and φ^' would diverge. We show this behavior in particular examples in Sec. <ref>. It would also be tempting to illustrate this situation directly in the Penrose diagram in Fig. <ref>, which is however plotted for points at the axis, and assuming that d θ = d φ = 0. An illustration of this kind should be possible in terms of projection diagrams defined in <cit.>. Unfortunately, from the computational point of view, such diagrams are much more difficult to draw exactly.





§ EXAMPLES





In this section we discuss a collection of sample solutions obtained with the help of formulas derived in preceding sections.

We perform our computations using Wolfram Mathematica <cit.>. The formulas for ξ(s) and μ(s), or equivalently θ(s), can be encoded directly. The formulas for the azimuthal angle φ^'(s) and the time T^'(s) require a regularization, if a geodesic crosses one of the horizons. In practice, we substitute in Eq. (<ref>) expressions for ξ(s) and θ(s) given by Eqs. (<ref>) and (<ref>), and evaluate the resulting integral numerically. In principle, the derivative d ξ(s)/ds in Eq. (<ref>) could be expressed as d ξ(s)/ds = ϵ_r √(R̃), by virtue of Eq. (<ref>). Using this form turns out to be problematic, as it requires knowledge of the sign ϵ_r, which changes at the radial turning points. To circumvent this difficulty, we use in Eq. (<ref>) the derivative d ξ(s)/ds obtained by a direct differentiation of Eq. (<ref>). Although the integral (<ref>) can be evaluated analytically, similarly to the calculation presented in <cit.>, we find computing it numerically to be quite effective. In a sense, we try to combine in our implementation the best of the two approaches—elegant formulas for the radial and polar coordinates and relatively straightforward numerical integrals providing φ^' and the time coordinate T^'.

Figures <ref> to <ref> depict our solutions obtained for various parameters, collected in Table <ref>. They show the orbits in ξ–θ and ξ–φ^' planes and in the three-dimensional space. We use Cartesian coordinates (x,y,z) defined as 



    x     =    ξcosφ^'sinθ, 
    
            y     =    ξsinφ^'sinθ, 
    
            z     =    ξcosθ.


In Table <ref> we provide also real zeros of R̃(ξ), corresponding to radial turning points, as well as zeros of Θ̃(θ). The latter define the angular ranges available for the motion along a given geodesic. We illustrate these ranges by drawing appropriate cones in three-dimensional plots or lines in ξ–θ plane plots. Kerr horizons at ξ = ξ_± are depicted as spheres or circles. Finally, as a double-check of our results, we plot solutions obtained by solving numerically the Kerr geodesic equations. These numerical solutions are depicted with dotted lines. For simplicity, we omit the prime in φ^' in labels of all figures in this paper.

Figure <ref> shows an unbound timelike trajectory, plunging into the black hole. The trajectory crosses smoothly both horizons at ξ = ξ_± and encounters a radial turning point located at 0 < ξ < ξ_-. The solution for φ^' can be continued smoothly up to the Cauchy horizon at ξ = ξ_-, where it diverges.

Figure <ref> shows a standard timelike bound geodesic, corresponding to a “Keplerian” motion. In Fig. <ref>, we depict an unbound timelike orbit, which does not plunge into the black hole.

A very interesting case is shown in Fig. <ref>, depicting a timelike unbound trajectory crossing smoothly both horizons at ξ = ξ_±. This trajectory does not encounter a radial turning point and consequently it continues, in the extended Kerr spacetime, to negative radii ξ (Boyer–Lindquist Block III, see also <cit.>, p. 163). In the terminology of <cit.> such orbits are referred to as “transits”. We illustrate the transition from ξ > 0 to ξ < 0 by plotting a segment of the geodesic corresponding to ξ > 0 in orange, and a segment corresponding to ξ < 0 in purple. Note that Cartesian coordinates defined by Eq. (<ref>) allow for a change from positive to negative values of the radius, which corresponds simply to the reflection x → -x, y → - y, z → -z, or equivalently, to a change of angular coordinates. To avoid confusion, apart from using two colors in the graphs, we also provide a plot of the radius ξ versus the Mino time s. Since no horizons occur for ξ < 0, the trajectory can be continued smoothly to ξ→ - ∞. Note that the apparent reflection of Cartesian coordinates associated with the transition from ξ > 0 to ξ < 0 creates (in the plots) a false impression of a reflection in the polar angle θ, as well as an impression that the trajectory leaves the bounds of the allowed polar motion, given by zeros of Θ̃(θ). In reality, θ remains safely within the allowed range along the entire geodesic trajectory.

Figure <ref> shows an example of an unbound null geodesic, scattered by the black hole. In Fig. <ref>, we also deal with a null geodesic, but it is much more interesting. This trajectory crosses smoothly both horizons and continues to negative values of ξ, where it encounters a radial turning point. It then re-enters the region with ξ > 0 and can be continued up to the Cauchy horizon at ξ = ξ_-. In Fig. <ref> a segment corresponding to negative radii is again marked in purple, while a segment with positive radii is plotted in orange.

Figure <ref> illustrates a behavior similar to the one depicted in Fig. <ref>, but obtained for a null geodesic. The trajectory plunges into the black hole and continues to negative radii. Figure <ref> shows another timelike geodesic with a behavior similar to the one illustrated in Fig. <ref>.

Finally, in Fig. <ref> we plot a collection of more or less random timelike geodesics plunging into the black hole. In this case we only show segments of geodesics located outside the black hole horizon (although they can be continued into the black hole). With this plot we aim to show that, when visualized in horizon-penetrating Kerr coordinates, the geodesics plunging into the black hole do not make an impression of swirling around the horizon—a behavior present in the Boyer–Lindquist coordinates. Our Fig. <ref> should be contrasted, e.g., with Fig. 6 in <cit.>. 










 


























§ SUMMARY


We provided a description of timelike and null Kerr geodesics in horizon-penetrating Kerr coordinates, extending a recent analysis of <cit.>. From the technical point of view, the change from Boyer–Lindquist coordinates used in <cit.> to the horizon-penetrating Kerr coordinate system only slightly affects our formalism. The radial and polar equations remain unchanged, which allows us to use relatively compact formulas (<ref>) and (<ref>) for the radial and polar coordinates. Additional terms appear in equations for the azimuthal and time coordinates, but they can be integrated, once the solution to the radial equation is known. Our formulas (<ref>) and (<ref>) provide, together with Eq. (<ref>), all necessary expressions.

The horizon-penetrating Kerr coordinate system allows for a continuation of geodesics across horizons within the region of the extended Kerr spacetime in which it is well-defined, i.e., within Boyer–Lindquist Blocks I, II, and III (Fig. <ref>). At the level of geodesic equations, regularity at the horizons depends explicitly on the radial direction of motion. A reasoning given in Sec. <ref> and our examples of Sec. <ref> provide the following generic picture. A future-directed timelike or null geodesic originating outside the black hole (in Block I) and moving inward (ϵ_r = -1) can encounter a radial turning point at r > r_+ or it can pass smoothly through the event (r = r_+) and the Cauchy (r = r_-) horizons, transiting through Block II to Block III. In Block III a trajectory may continue smoothly to r → - ∞ (the so-called transit orbit), or it may get reflected at a radial turning point. In the latter case, the Kerr coordinate system only allows for a continuation up to the Cauchy horizon at r = r_-, where both t^' and φ^' diverge. We leave aside an obvious case of a geodesic hitting the ring singularity at ρ^2 = 0. It is known (see a proof in <cit.>, p. 288) that a timelike or null trajectory can only hit the ring singularity, if it is located entirely within the equatorial plane.


A. C. and P. M. acknowledge a support of the Polish National Science Centre Grant No. 2017/26/A/ST2/00530.






§ PENROSE DIAGRAM AND HYPERSURFACES OF CONSTANT KERR TIME



The diagram shown in Fig. <ref> is computed by a nearly standard Kruskal procedure, but a few details should be given in order to explain the plots of hypersurfaces of constant time t^'.

The conformal diagram at the symmetry axis is constucted for the two dimensional metric

    ^(2)g = - . Δ/ρ^2|_cos^2 θ = 1 dt^2 + . ρ^2/Δ|_cos^2 θ = 1 dr^2 = - F(r) dt^2 + 1/F(r) dr^2,

where

    F(r) := r^2 - 2 M r + a^2/r^2 + a^2.

The usual Kruskal construction starts by defining a new coordinate

    r_∗ := ∫^r dr^'/F(r^') = ∫^r r^'^2 + a^2/r^'^2 - 2 M r^' + a^2 dr^'.

The above integral can be evaluated analytically. In terms of dimensionless quantities, we get,

    ξ_∗ = ξ + 1/√(1 - α^2)log| (ξ - ξ_+)^ξ_+/(ξ - ξ_-)^ξ_-|,

where r_∗ = M ξ_∗. Expression (<ref>) is well defined on ξ∈ℝ, except at ξ = ξ_±, where it diverges.

We will discuss some of the details, working for simplicity in Block I. A standard construction assumes the following coordinate transformations:

    u := t - r_∗,    v := t + r_∗

and

    û := - exp (- c u),   v̂ := exp (c v),

where c is a constant. We choose c = F^'(r_+)/2. This gives

    ^(2)g = -F dt^2 + 1/F dr^2 = -F/c^2 exp (2 c r_∗) d û d v̂.

Variables û and v̂ can be compactified by setting

    U := arctan(û),    V := arctan(v̂).

Finally, one defines Cartesian type coordinates T and X by

    U = T - X,    V = T + X,

or, equivalently,

    T := U + V/2,    X := V - U/2.

so that

    - d û d v̂ = 1/cos^2(T - X) cos^2(T + X)(-dT^2 + dX^2).

This yields the metric in the form

    ^(2)g = -F dt^2 + 1/F dr^2 = F/c^2 exp (2 c r_∗) cos^2(T - X) cos^2(T + X)(-dT^2 + dX^2).

Combining these transformations, we get

    - tanh( c r_∗) = cos (2 X)/cos (2 T),   tanh (c t) = sin (2 T)/sin (2 X).

Equations (<ref>) allow for drawing the lines of constant r_∗ and t. We use these formulas to plot Blocks I and III, assuming the following ranges of X and T: In Block I: X ∈ (0,π/2), T ∈ (-π/4,π/4). In Block III: X ∈ (-π/2,0), T ∈ (π/4,3π/4).

In Block II, formulas (<ref>) have to be defined in a slightly different way. We set

    - tanh (c r_∗) = cos (2 T)/cos (2 X),   tanh (c t) = sin (2 X)/sin (2 T),

while the ranges of X and T are X ∈ (-π/4,π/4), T ∈ (0,π/2).

Plotting the lines of constant t^' is more involved. From Eq. (<ref>), we have t^' = t - r + r_∗. We plot the lines t^' = const by inverting, numerically, the relation r_∗ = r_∗(r), defined by Eq. (<ref>).




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