Critical Tidal Currents in General Relativity

Abstract

Relativistic tidal equations are formulated with respect to the rest frame of a central gravitational source and their solutions are studied. The existence of certain relativistic critical tidal currents are thereby elucidated. Specifically, observers that are spatially at rest in the exterior Kerr spacetime are considered in detail; in effect, these fiducial observers define the rest frame of the Kerr source. The general tidal equations for the free motion of test particles are worked out with respect to the Kerr background. The analytic solutions of these equations are investigated and the existence of a tidal acceleration mechanism is emphasized.

1. Introduction

In general relativity (GR), a physically reasonable and geometrically natural method to interpret the local motion of test particles relative to an observer in a gravitational field involves the introduction of a Fermi normal coordinate system along the world line of the reference observer. Imagine a congruence of timelike geodesics. The geodesics are neighboring, but their rates of separation may not be negligibly small compared to the speed of light c. We establish a Fermi coordinate system,^{Xμ^}=(T,X,Y,Z), along the path of a reference geodesic in the congruence and consider the free motion of neighboring particles relative to the fiducial observer that follows the reference geodesic and occupies the spatial origin of the Fermi coordinate system. Limiting our attention to the two-dimensional(T,Z) plane, the equation of relative motion takes the form [1]

^d2Zd^T2+k(T)(1−2^Z˙2)Z+O(^Z2)=0,

where the fiducial observer is fixed atZ=0,Z˙:=dZ/dTandk(T)=_RTZTZis the Gaussian curvature of the(T,Z)plane. Here, we use units such thatc=G=1, unless specified otherwise; moreover, hatted indices refer to the Fermi coordinate system. The timelike nature of geodesic motion requires that

1−^Z˙2+k(T)^Z2+O(^Z3)>0;

in particular, at the position of the fiducial observer (Z=0), we must have|Z˙|≤1 . Neglecting higher-order terms, Equations (1) and (2) are invariant underZ↦−Z . When the speed of separation of neighboring geodesics is negligible, we can ignore the velocity term in Equation (1) and recover the Jacobi equation in the Fermi frame. On the other hand, neglectingO(^Z2) terms, the generalized Jacobi Equation (1) has exact solutions describing uniform motion in the Fermi system withZ˙=±1/2, namely,

_Z±(T)=_Z0±12(T−_T0),

where_Z0and_T0are constants. That is, regardless of the magnitude ofk(T), there is a critical tidal current with speed_Vcrt=c/2≈0.71c such that the corresponding relative motion in the Fermi system is approximately uniform. In GR, a tidal current is relative motion induced by spacetime curvature. The peculiar feature of solution (3) of Equation (1) is that these critical tidal currents are independent of the spacetime curvaturek(T). This particular relativistic feature of tides in general relativity is the focus of the present paper.

If the speed of relative motion is below the critical speed, then the nature of relative motion in Equation (1) is similar to the familiar Jacobi equation. However, for relativistic motion above the critical speed, there is deceleration fork(T)<0toward the critical speed and acceleration fork(T)>0 . Various aspects of this circumstance and its implications for astrophysical jets have been explored in previous work [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. For recent work on general relativistic tidal effects in other contexts, see, for example, Refs. [16,17,18,19,20,21,22] and the references cited therein.

The purpose of the present paper is to illustrate further the nature of critical tidal currents in general relativity. In contrast to previous work regarding certain tidal acceleration phenomena involving energetic particles in astrophysics [1,5], we assume in this paper that the reference observer is spatially static, so that the observer remains at rest in the background gravitational field. Such observers in effect define the rest frame of the background gravitational source. Indeed, we extend here more recent work regarding jets along the axis of a Kerr field [23]. In the cases considered in the present paper, nongravitational forces are needed to keep the observer fixed in space; therefore, the reference observer is generally accelerated. We further assume that the spatially static observer carries an orthonormal tetrad frame that is Fermi-Walker transported along its world line. Using these locally nonrotating basis vectors, we establish a Fermi coordinate system as described in detail in Appendix A. To study the motion of free test particles relative to the fiducial observer, we express the timelike geodesic equation in these Fermi coordinates. In this way, we can describe in an invariant way the local motion of test particles relative to the rest frame of the gravitational source. The resulting generalized Jacobi equation and the timelike condition are given by Equations (A14) and (A15) of Appendix A, respectively. In this case, critical tidal currents are described in Section 2. The analytic solutions of the tidal equations are explored in the exterior Kerr and Schwarzschild spacetimes in Section 3 and Section 4 respectively. Section 5 contains a discussion of our results. Detailed calculations are relegated to Appendix B and Appendix C.

2. Critical Tidal Currents

The tidal equations given in Appendix A are rather complicated. To illustrate a significant feature of these equations involving critical tidal currents, let us imagine a Fermi system of coordinates^{Xμ^}=(T,X,Y,Z) and a circumstance where motion purely along a certain Fermi coordinate direction becomes possible and the corresponding component of acceleration vanishes. Let us call this direction again Z and note that Equations (A14) and (A15) reduce in this case to Equations (1) and (2), respectively, where curvature k is now constant because the fiducial observer is at rest in space and the gravitational field is assumed to be stationary as in the Kerr spacetime (see Appendix B and Appendix C). Ignoring higher-order terms for the moment, we can write the autonomous tidal equation as

Z˙dZ˙dZ+k(1−2^Z˙2)Z=0,

which can be simply integrated. The result is

^Z˙2=12−12−_Z˙int2^e2kZ2

with the initial conditions that atT=0,Z=0andZ˙=_Z˙int . Integrating Equation (5), we find

_∫0Z ^{1+2_Z˙int2−1e2kζ2−1/2}dζ=±_VcrtT,

where only positive square roots are taken into account throughout. Here, as before,_Vcrt=c/2≈0.71cis the critical speed of tidal currents; that is, for_Z˙int2=_Vcrt2=1/2 , Equation (6) impliesZ=±_VcrtT. Clearly, this uniform motion is rather approximate; in fact, higher-order tidal terms introduce significant modifications near the boundary of the Fermi coordinate patch. Furthermore, the Fermi coordinate speed must satisfy the timelike condition, namely,1−^Z˙2+k^Z2>0 in accordance with Equation (2). For insight regarding tidal motions away from critical currents, it proves useful to write Equation (5) as

^Z˙2+12−_Z˙int2^e2kZ2=12

and interpret this relation as an effective energy equation for a classical particle with total energy_Vcrt2=1/2and symmetric effective potential energy_Veff, where

_Veff=(_Vcrt2−_Z˙int2)^e2kZ2.

The effective kinetic energy is always positive; therefore, tidal motion is confined to the region where1≥2_Veff. Moreover, turning points occur where1=2_Veff. Fork>0, the absolute magnitude of the effective potential diverges forZ→±∞. If_Z˙int2<_Vcrt2, the effective potential is then positive and the resulting motion is periodic and confined within a finite spatial region with0<^Z˙2≤_Z˙int2; moreover, the requirement of timelike motion, namely,1−^Z˙2+k^Z2>0is satisfied. However, for_Vcrt2<_Z˙int2<1, the effective potential is negative and the motion accelerates until the timelike motion asymptotically approaches a null ray, in which case1−^Z˙2+k^Z2=0. This speed of light singularity occurs atZ=_ZN, where

1+2k_ZN2=2_Z˙int2−1^e2k_ZN2.

The unique solution of this equation can be expressed by means of the Lambert W function. This function in the real domain has two branches_W0(x)and_W−1(x). The principal branch,_W0, is such that_W0(x)≥−1, while_W−1(x)≤−1. We find

1+2k_ZN2=−_W−1(γ),γ:=1e1−2_Z˙int2,−1e<γ<0.

In connection with tidal acceleration to the speed of light, we note that as the motion approaches the boundary of the Fermi coordinate patch, the higher-order terms that we have thus far neglected come into play and moderate the approach to the speed of light. This issue has been discussed in detail in Refs. [7,11]. The mitigation of the speed of light singularity has been explicitly demonstrated using exact Fermi coordinates in special cases; however, only approximate analytic treatments are possible in the Schwarzschild and Kerr spacetimes [11,15].

Let us next assume thatk<0 . In this case, the effective potential (8) exponentially goes to zero asZ→±∞. For_Z˙int2<_Vcrt2, the motion accelerates toward^Z˙2=_Vcrt2; similarly, for_Z˙int2>_Vcrt2, the motion decelerates toward^Z˙2=_Vcrt2. This means that fork<0, the critical solutions of the tidal equation,Z=±_VcrtT , are attractors. That is, tidal motions with speeds above or below the critical speed eventually tend to approach the critical speed. Further discussion of this circumstance can be found in Ref. [1].

We now turn to a detailed discussion of the analytic solutions of Equations (A14) and (A15) in the Kerr and Schwarzschild spacetimes. Greek indices run from 0 to 3, while Latin indices run from 1 to 3. The signature of the spacetime metric is+2.

3. Exterior Kerr Spacetime: Observer at Rest on the Axis of Rotation

We are interested in the family of test observers that are spatially at rest on the axis of rotational symmetry of the exterior Kerr spacetime. The Kerr metric is given in Boyer-Lindquist coordinates(t,r,θ,ϕ) by [24]

−d^s2=−1−2MrΣd^t2−4MarΣ^sin2θdtdϕ+ΣΔd^r2+Σd^θ2+^r2+^a2+2M^a2rΣ^sin2θ^sin2θd^ϕ2.

Here, M and a are the mass and specific angular momentum of the Kerr source, respectively. Moreover,Σ=^r2+^{a2 cos2}θandΔ=^r2−2Mr+^a2. The spacetime is asymptotically flat and reference observers at rest in space exist from the asymptotic region all the way down to the exterior of the stationary limit surface (Σ=2Mr).

A fiducial observer that is at rest on the axis of symmetry carries a nonrotating orthonormal tetrad frame^λμ _α^such that

_λ0^=^r2+a2Δ1/2_∂t,_λ1^=^Δr2+a21/2_∂r

and(_λ2^,_λ3^) are unit vectors in the plane orthogonal to the rotation axis. The rotational symmetry of the Kerr field about its axis leads to enormous simplification of the general analysis presented in Appendix B and Appendix C. Specifically, we have^λμ _0^=^u¯μ=d^x¯μ/dτ, which is the 4-velocity of the fiducial observer andτis its proper time. Moreover, the observer’s accelerationD^λμ _0^/dτ=^Aμ=^{Ai^λμ} _i^is given by

(_A1^,_A2^,_A3^)=(A,0,0),A=M(^r2−^a2)^{(r2+a2)3/2(r2−2Mr+a2)1/2}.

We establish a Fermi normal coordinate system along the world line of the fiducial observer (see Appendix A). The Fermi coordinates^{Xμ^}=(T,X,Y,Z)in this case are such that X denotes the radial coordinate along the Kerr symmetry axis, while(Y,Z) are the two orthogonal transverse coordinates. In the tidal Equation (A14), the relevant acceleration parameters are then given by Equation (13), while the gravitoelectric and the gravitomagnetic curvature components are given byE= diag(−2E,E,E)andH= diag(−2H,H,H), where

E:=E(θ=0)=Mr(^r2−3^a2)^(r2+a2)3,H:=H(θ=0)=−Ma(3^r2−^a2)^(r2+a2)3;

see Appendix C.

After these preliminary considerations, we now turn to the tidal equations of motion (A14) to linear order in(X,Y,Z), namely,

^d2Xd^T2+(A−2EX)(1−2^X˙2)+^A2X(1+2^X˙2)+23E[X(^Y˙2−^Z˙2)−2X˙(2YY˙+ZZ˙)]+2H(1−^X˙2)(YZ˙−ZY˙)=0,

^d2Yd^T2+EY(1−2^Y˙2)−2[A−(^A2+73E)X]X˙Y˙−23E[Y(^X˙2−2^Z˙2)+5ZY˙Z˙]+2H[2XZ˙+ZX˙−X˙Y˙(YZ˙−ZY˙)]=0,

^d2Zd^T2+EZ(1−2^Z˙2)−2[A−(^A2+73E)X]X˙Z˙−23E[Z(^X˙2−2^Y˙2)+5YY˙Z˙]−2H[2XY˙+YX˙+X˙Z˙(YZ˙−ZY˙)]=0.

To these equations we must add the condition that the tidal motion is timelike, namely,

^(1+AX)2−(^X˙2+^Y˙2+^Z˙2)−E(2^X2−^Y2−^Z2)+13E[2^{(YZ˙−ZY˙)2}−^{(XZ˙−ZX˙)2}−^{(XY˙−YX˙)2}]+4HX(YZ˙−ZY˙)>0.

The azimuthal symmetry of Kerr geometry is reflected in these approximate equations. That is, keeping(X,X˙)unchanged at a given time T, one can show that the(Y,Z)system of equations is invariant under a rotation by a constant angle about the axis of symmetry. Moreover, inspection of the(Y,Z)system reveals thatY(T)=0andZ(T)=0are not separately possible solutions so long asH≠0 . We note from Equation (14) thatH=0if eithera=0, so that the source is spherically symmetric and described by the Schwarzschild spacetime, orr=a/3. In the latter case, the observer must be fixed on the rotation axis atr=a/3, which is possible in the exterior Kerr spacetime if2a>3M.

3.1. Motion Along the Rotation Axis of Kerr Spacetime

IfY(T)=Z(T)=0, then our system reduces to

^d2Xd^T2+(A−2EX)(1−2^X˙2)+^A2X(1+2^X˙2)=0

and

^(1+AX)2−^X˙2−2E^X2>0,

which describe tidal motion along the Kerr rotation axis. In the absence of acceleration (A=0 ), Equation (19) reduces to Equation (1) for the case of constantk=−2E<0 , where the critical currents turn out to be attractors as discussed in Section 2. However, the presence of acceleration drastically changes the dynamical behavior of the system. To see how this comes about in some detail, let us define the Fermi velocity of a free test particle V,

V:=dXdT=X˙

and write Equation (19) in the form

d^V2dX+4[(2E+^A2)X−A]^V2+2[A−(2E−^A2)X]=0.

It is evident that our(X,V)system has a rest point given by(_Xrp,0), where

_Xrp=A2E−^A2.

As expected, forA=0we recover(X,V)=(0,0) , which is the rest point associated with Equation (4).

It is straightforward to integrate Equation (22) by means of an integrating factorI,

I(X)=exp[2(2E+^A2)^X2−4AX],dIdX=4[(2E+^A2)X−A]I.

With the initial condition that atT=0a free test particle atX=0has Fermi coordinate velocityV=_V0,|_V0|<1 , Equation (22) can be integrated once and we find the “energy” equation

I(X)^V2+_Weff(X)=_V02,

where_Weff(X)is the effective potential energy in this case given by

_Weff(X)=2_∫0X[A−(2E−^A2)ξ]I(ξ)dξ.

The path of the free particleX(T) can be simply obtained from integrating Equation (25). To clarify the nature of this motion, we note thatI(X)>0 ; hence, Equation (25) implies that the motion is confined to the region

_Weff(X)≤_V02.

It follows from the definition of the effective potential (26) that this function vanishes atX=0and has an extremum atX=_Xrp. To go further, we need to specify the physical parameters of the configuration under discussion here.

3.2.r≫Mandr≫a

In most astrophysical situations of interest, we deal with motions of energetic particles relative to an environment that constitutes the rest frame of the gravitational source. Employing standard techniques, motions are usually observable that take place at a distance r from the source such thatr≫Mandr≫a. We therefore focus our attention on such situations and note that

A≈M^r21+Mr−3^{a2 r2}+⋯,E≈M^r31−6^{a2 r2}+⋯,

where we have neglected higher-order terms in the small quantitiesM/rand^a2/^r2. Similarly, we can write

_Xrp≈r21+32Mr+3^{a2 r2}+⋯.

It follows from these results and inspection of the expression for the effective potential energy function_Weff(X)that we have a simple potential barrier here that starts from zero atX=0and reaches its maximum value of_νcrt2at_Xrp, where_νcrt>0and

_νcrt2:=_Weff(_Xrp)=2_∫0Xrp [A−(2E−^A2)ξ]I(ξ)dξ.

The energy Equation (25) implies that at the rest point(X,V)=(_Xrp,0),_V02=_νcrt2.

Introducingξ=(r/2)μ in the integrand of Equation (30) and using only the dominant terms in A and E, we find

_νcrt2≈Mr_∫01(1−μ)^{eMr(μ2−2μ)}dμ.

Therefore, to lowest order in the small quantities, we have

_νcrt2≈M2r.

With these results for the effective potential, we now return to the analysis of the energy Equation (25).

3.3. Tidal Acceleration

Imagine a free test particle at the position of the fiducial observer (X=0) with initial velocity_V0,1>_V0>0, such that_V02<_νcrt2 ; that is, in Equation (25),_V02is less than the maximum height of the potential barrier. In this case, the particle reaches a certain maximum distance_Xmax,0<_Xmax<_Xrp, along the radial direction, has a turning point at_V02=_Weff(_Xmax)and falls back toward the gravitational source. On the other hand, for_V02>_νcrt2, the particle tidally accelerates toward the speed of light. Of course, near the boundary of the Fermi coordinate patch, higher-order tidal terms intervene and mitigate the speed of light singularity, which would occur when^V2=^(1+AX)2−2E^X2 in accordance with Equation (20). Nevertheless, considerable tidal acceleration is expected to take place in practice. The implications of this result for the tidal acceleration of astrophysical jets have been extensively studied in Ref. [23], where various possibilities have been thoroughly investigated. Indeed, Equations (19) and (20) are invariant under the transformation(X,A,E)↦(−X,−A,E) in conformity with the double jet structure of astrophysical outflows. The present mechanism for the tidal acceleration of outflows is independent of the presence or absence of an event horizon. As emphasized in Ref. [23], the comparison of these theoretical results with observational data must include electromagnetic forces as well [25].

It is instructive to restate our main physical results here in Newtonian terms. The reference observer is fixed on the rotation axis at the Boyer-Lindquist radial coordinate r sufficiently far from the Kerr source such thatr≫Mandr≫a. The Newtonian escape velocity at the position of reference observer is

_Vesc:=^2GMr1/2.

It follows from Equation (32) that_νcrtis approximately equal to one-half of the Newtonian escape velocity; that is,

_νcrt≈12_Vesc.

Thus if a free test particle is launched outward from the position of the reference observer with a velocity less than about one-half of the Newtonian escape velocity, then the test particle cannot escape the gravitational field of the Kerr source; in fact, as expected, it reaches a maximum height and then falls back toward the source. However, if the initial velocity is more than about one-half of the Newtonian escape velocity, then the particle tidally accelerates away from the Kerr source.

Let us briefly digress here and mention that for purely radial motion in the exterior Schwarzschild spacetime, the escape velocity for a free test particle starting at Schwarzschild radial coordinate r is indeed given by Equation (33), where the escape speed of the particle is measured by the fiducial observer that is fixed at r. On the other hand, our treatment in this section employs the invariantly defined quasi-inertial Fermi normal coordinate system.

We have thus far considered motion along the axis of symmetry of a Kerr source. Fora=0, the Kerr metric reduces to the spherically symmetric Schwarzschild metric, in which case our treatment applies to motion along any radial direction. The corresponding expressions for A and E simplify fora=0 in Equations (13) and (14), respectively, but our main results remain unchanged. Indeed, sufficiently far from any astronomical source, we expect that its gravitational field is dominated by its mass M and the accelerated outflow can occur above the threshold (34) along any radial direction away from mass M.

In our discussion of the solutions of the tidal equations, azimuthal symmetry of the exterior Kerr spacetime has made it possible to do analytic work along the axis of rotational symmetry. This treatment can be extended to any radial direction in the exterior Schwarzschild spacetime due to its spherical symmetry. This symmetry, furthermore, makes it possible to go beyond the radial direction X in the next section and consider the solutions of the tidal equations in the(X,Y)plane with no loss in generality.

4. Exterior Schwarzschild Spacetime

In this section, we assumea=0, so that the reference observer is at rest in the spherically symmetric exterior of Schwarzschild spacetime. Following the discussion of the previous section, we can now setZ(T)=0and consider motion in the(X,Y)plane. The equations of motion then reduce to

^d2Xd^T2+(A−2EX)(1−2^X˙2)+^A2X(1+2^X˙2)+23EY˙(XY˙−4YX˙)=0

and

^d2Yd^T2+EY(1−23^X˙2−2^Y˙2)−2[A−(^A2+73E)X]X˙Y˙=0,

while the timelike condition reduces to

^(1+AX)2−(^X˙2+^Y˙2)−E(2^X2−^Y2)−13E^{(XY˙−YX˙)2}>0.

Inspection of Equations (35) and (36) reveals that an analytic solution of this system is possible and is given by a rest point(X,X˙)=(_XS,0)along the radial direction and a critical current along the transverse direction. That is,

_XS=3A5E−3^A2=3^{[r3(r−2M)]1/2}5r−13M

forr≠13M/5and

Y=_Y0±_Vcrt(T−_T0),

where_Y0and_T0are constants. Forr≫M, we have

_XS=35r1+85Mr+⋯,

and the timelike condition (37) is satisfied in this case. This exact solution of the(X,Y)system means that the Fermi coordinate along the radial direction X remains constant while a critical tidal current Y flows in the transverse direction.

It has not been possible to find other exact solutions of the(X,Y)system. To proceed further, we perturb the(X,Y) system to linear order about the exact solution given in Equations (38) and (39) and then solve the resulting linear perturbation equations via expansions in power series. Let us define a new temporal variableη,

η:=_Y0±_Vcrt(T−_T0)

and substitute

X(η)=_XS+ϵP(η),Y(η)=η+ϵQ(η),

in Equations (35) and (36). Hereϵ,0<ϵ≪1is a constant perturbation parameter. To first order inϵ, we find

^d2Pd^η2−83EηdPdη−2A_XSP+43E_XSdQdη=0,

^d2Qd^η2−4EηdQdη+4(2E_XS−A)dPdη=0.

Substituting Equation (43) and its derivative in Equation (44), we find the third-order homogeneous ordinary differential equation

^d3Pd^η3−_w1η^d2Pd^η2+(_w2^η2−_w3)dPdη+_w4ηP=0,

where_w1,_w2,_w3and_w4are constant coefficients that are given by

_w1=203E,_w2=323^E2,_w3=2E3r−7Mr−2M+8M(r+M)^(5r−13M)2

and

_w4=83^E25r−13Mr−2M,E=M^r3.

The solutions of Equation (45) form a linear manifold. Indeed, the general solution of this homogeneous linear differential equation can be expressed as a linear sum with constant coefficients of three independent solutions [26]. To find explicit solutions of Equation (45), we resort to infinite series in powers ofη. As in the method of Frobenius, we assume

P(η)=^ηρ(_a0+_a1η+_a2^η2+_a3^η3+⋯),

where_an,n=0,1,2,⋯, are constants and_a0≠0 by definition. Substitution of this series in Equation (45) results in an infinite series in increasing powers ofηstarting with^ηρ−3. As in the standard procedure, we recover the relations

ρ(ρ−1)(ρ−2)_a0=0,

ρ(^ρ2−1)_a1=0,

ρ(ρ+1)(ρ+2)_a2=ρ[(ρ−1)_w1+_w3]_a0,

(ρ+1)(ρ+2)(ρ+3)_a3=(ρ+1)(ρ_w1+_w3)_a1

and the recurrence relation

(n+ρ+2)(n+ρ+3)(n+ρ+4)_an+4=(n+ρ+2)[(n+ρ+1)_w1+_w3]_an+2−[(n+ρ)_w2+_w4]_an

forn=0,1,2,⋯.

It follows from Equation (49) thatρ=0,1or 2, since_a0≠0. In each case, we can find power series solutions forP(η)that can be expressed as superpositions of solutions_S0,_S1and_S2defined by

_S0(η):=1−_w44!^η4−(3_w1+_w3)_w46!^η6+⋯,

_S1(η):=η+13!_w3^η3+15!_w5^η5+⋯

and

12_S2(η):=12!^η2+14!(_w1+_w3)^η4+16!_w6^η6+⋯,

where_w5and_w6are constants defined by

_w5:=2_w1w3−2_w2+_w32−2_w4

and

_w6:=(3_w1+_w3)(_w1+_w3)−3(2_w2+_w4).

Forρ=0, it turns out thatP(η)can be written as a linear superposition of the three independent solutions_S0,_S1and_S2with constant coefficients_a0,_a1and_a2, respectively. On the other hand, forρ=1,P(η)is a linear superposition of_S1and_S2with constant coefficients_a0and_a1, respectively, while forρ=2,P(η)is simply proportional to_S2with proportionality constant equal to_a0. Given a solutionP(η) of Equation (45), we can use Equation (43) to find a corresponding solution forQ(η). To illustrate the general case, we assume

P(η)=_a0S0(η)+_a1S1(η)+_a2S2(η),

corresponding toρ=0 . From Equation (43), we can work out the infinite series solution corresponding toQ(η), namely,

Q(η)=_Q0+_{a0 q1w4}8Eη+23E^η3+⋯−_{a1 q1}12!_q2^η2+14!_q4^η4+⋯−_{a2 q1}η+13!_q3^η3+⋯,

where_Q0is an integration constant and

_q1=2E^{[r3(r−2M)]1/2}5r−13M,_q2=12_w3−E3r−7Mr−2M,

_q3=_w1+_w3−23E13r−29Mr−2M,_q4=12_w5−13_w3E17r−37Mr−2M.

Using expressions (59) and (60) forP(η)andQ(η) , respectively, in Equation (42), we have in the neighborhood of the critical solution an approximate power series solution of the tidal equations in the(X,Y)plane that depends on four independent parameters, namely,ϵ_a0,ϵ_a1,ϵ_a2andϵ_Q0. We note that a general solution in terms of the temporal parameterηis expected to depend on four independent parameters corresponding to the initial position and velocity in the(X,Y)plane. The issue of the convergence of such power series solutions as well as their physical interpretations requires further investigation that is beyond the scope of this work.

5. Discussion

In GR, the generalized Jacobi equation is the exact analogue of the Lorentz force law of electrodynamics. Just as the electric and magnetic fields can be measured via the motion of charged particles using the Lorentz force law, the components of the Riemann curvature tensor can be measured, in principle, using the generalized Jacobi equation [20]. This equation contains a critical speed_Vcrt=c/2≈0.71c and the associated critical currents that are explored in the present paper. In particular, we have studied the generalized Jacobi equation for the motion of free test particles with respect to the rest frame of a central Kerr source. Among other things, we have elucidated a general relativistic tidal acceleration mechanism that is relevant for the theory of astrophysical jets [23]. Moreover, the approach adopted in this work can be employed to determine the gravitational influence of the quiescent massive black hole at the Galactic Center on the dynamics of the energetic particles involved in the production of giant bipolar radio bubbles near the Galactic Center [27].

Funding

This research received no external funding.

Acknowledgments

I wish to thank C. Chicone and D. Bini for their past collaborations on the subject of the present paper.

Conflicts of Interest

The author declares no conflict of interest.

Appendix A. Jacobi-Type Equation in Fermi Coordinates

The purpose of this appendix is to consider a gravitational field and the associated curved spacetime manifold within the GR framework. An arbitrary test accelerated observer in this spacetime carries along its path a nonrotating orthonormal tetrad frame. We establish a Fermi normal coordinate system along the world line of this reference observer and investigate the motion of free test particles in the Fermi system. We derive and study the resulting Jacobi-type equation.

Appendix A.1. Fermi Coordinates

We imagine an accelerated test observer with proper timeτfollowing a timelike path^x¯μ(τ)in spacetime. The reference observer carries an orthonormal Fermi-Walker transported tetrad frame^λμ _α^(τ)along its world line. Here,^λμ _0^(τ)=d^x¯μ(τ)/dτ=^u¯μ(τ)is the unit 4-velocity vector of the observer,D^u¯μ(τ)/dτ=^Aμis its acceleration and^λμ _i^(τ),i=1,2,3, constitute the local spatial frame of the observer. At any given eventτalong this path, we consider all spacelike geodesics that originate from this event and are orthogonal to the observer's world line. These form a local spacelike hypersurface. Let^xμbe an event on this hypersurface such that there is a unique spacelike geodesic of proper lengthσthat connects^xμto the observer's world line at^x¯μ(τ). If^ξμ(τ)is the unit spacelike vector tangent to this geodesic at eventτalong the world line, then^ξμ(τ)_u¯μ(τ)=0. We assign Fermi normal coordinates^{Xμ^}=(T,^{Xi^})to event^xμ, where

^{X0^}=T:=τ,^{Xi^}:=σ^ξμ(τ)_λμ ^{i^}(τ).

The fiducial observer has^{Xi^}=0and is thus permanently fixed at the spatial origin of the Fermi coordinate system. Let us note that fori=1,2,3,^ξμ(τ)_λμ ^{i^}(τ)constitute the direction cosines at proper timeτ ; therefore, Fermi coordinates are the natural extension of the inertial Cartesian coordinates to the curved spacetime of GR [28]. Fermi coordinates are admissible in a finite cylindrical domain about the world line of the fiducial observer with|X|≪L(T), whereL(T)is a certain infimum of the observer's acceleration length and the radius of curvature of spacetime. The spacetime metric in Fermi coordinates is given by-d^s2=_gμ^ν^d^{Xμ^}d^{Xν^} , where [29]

_g0^0^=-1-2_Ai^^{Xi^}-(_Ai^Aj^+_R0^i^0^j^)^{Xi^Xj^}+^O(|X|3),

_g0^i^=-23_R0^j^i^k^^{Xj^Xk^}+^O(|X|3),

_gi^j^=_δi^j^-13_Ri^k^j^l^^{Xk^Xl^}+^O(|X|3).

Here, we have defined

_Ai^(T):=_Aμ^λμ _i^

and

_{Rα^β^γ^δ^}(T):=_Rμνρσ^λμ _α^^λν _β^^λρ _γ^^λσ _δ^

to be the components of the acceleration and the Riemann curvature tensor as measured by the reference observer, respectively.

The Christoffel symbols in Fermi coordinates can be computed from the metric; indeed, the nonzero components to linear order in|X| can be obtained from [29]

_Γ0^0^0^=d_Ai^dT^{Xi^},_Γ0^i^0^=_Ai^+(_R0^i^0^j^-_Ai^Aj^)^{Xj^},

_Γi^j^0^=23_R0^(i^j^)k^^{Xk^},_Γ0^0^i^=_Ai^+(_R0^i^0^j^+_Ai^Aj^)^{Xj^},

_Γ0^j^i^=-_R0^k^i^j^^{Xk^},_Γj^k^i^=-23_Ri^(j^k^)l^^{Xl^}.

Appendix A.2. Jacobi-Type Equation

The geodesic equation of motion in Fermi coordinates takes the form

d^{Uμ^}ds+_Γα^β^μ^^{Uα^Uβ^}=0,

where the 4-velocity of the free test particle in the Fermi system can be written as

^{Uμ^}=d^{Xμ^}ds=Γ(1,V),V=dXdT

and the Lorentz factorΓ=dT/dscan be determined via^{Uμ^}_Uμ^=-1. We find

1^Γ2=-_g0^0^-2_g0^i^^{Vi^}-_gi^j^^{Vi^Vj^}>0.

Here, the Fermi velocityVis a coordinate velocity; however, at the location of the reference observerX=0, we must have|V|<1.

It is now straightforward to derive the reduced geodesic equation [1]

^{d2 Xi^}d^T2+_Γα^β^i^-_Γα^β^0^ ^{Vi^}d^{Xα^}dTd^{Xβ^}dT=0,

or, more explicitly,

^{d2 Xi^}d^T2+_Ai^+(_R0^i^0^l^+_Ai^Al^)^{Xl^}-d_Al^dT^{Xl^ Vi^}-2_R0^l^i^j^ ^{Vj^ Xl^}-2[_Aj^+(_R0^j^0^l^-_{Aj^ Al^})^{Xl^}]^{Vi^ Vj^}-23_Ri^j^k^l^+_R0^j^k^l^ ^{Vi^Xl^ Vj^ Vk^}+^O(|X|2)=0.

The LorentzΓfactor now takes the form

1^Γ2=^{(1+_Ai^Xi^)2}-_δi^j^^{Vi^Vj^}+_R0^i^0^j^^{Xi^Xj^}+43_R0^j^i^k^ ^{Vi^ Xj^Xk^}+13_Ri^k^j^l^^{Vi^Vj^Xk^Xl^}+^O(|X|3)>0.

It is clear from this treatment that simple generalizations are possible; for instance, we can consider accelerated motion of test particles in the Fermi coordinate system as well.

Appendix B. Observers at Rest in Exterior Kerr Spacetime

We are interested in the family of test observers that are spatially at rest in the exterior Kerr spacetime. In our considerations, these fiducial observers represent the rest frame of the Kerr source, which is specified by a mass M that rotates uniformly with angular momentum J. In Boyer-Lindquist coordinates(t,r,θ,ϕ) , the Kerr metric can be expressed as [24]

-d^s2=_gαβd^xαd^xβ=-d^t2+ΣΔd^r2+Σd^θ2+(^r2+^a2)^sin2θd^ϕ2+2MrΣ^{(dt-asin2θdϕ)2},

wherea=J/(Mc)is the specific angular momentum parameter and

Σ=^r2+^{a2 cos2}θ,Δ=^r2-2Mr+^a2.

We emphasize that the tidal currents under scrutiny in this work are independent of the existence of an event horizon. Fora≤M, for instance, the source is a Kerr black hole, but this circumstance has no essential impact on the existence of the critical tidal currents in this case.

Let^u¯μ=d^x¯μ/dτbe the unit timelike 4-velocity vector of a member of the family of observers at rest, whereτis the proper time along the world line. We therefore have

u¯=^ΣΣ-2Mr1/2_∂t,τ=^Σ-2MrΣ1/2t,

where we have assumed thatτ=0att=0. Only positive square roots are considered throughout. The reference observers exist outside the stationary limit surface of Kerr spacetime given byΣ-2Mr=0. Next, we need to set up a nonrotating tetrad frame^λμ _α^(τ)along the world line of our reference observer such that^λμ _0^(τ)=^u¯μ(τ). We do this in several steps. First, we set up an orthonormal tetrad frame^eμ _α^at each event in the exterior Kerr spaetime such that

_e0^=1-_gtt∂t,_e1^=1_grr∂r,_e2^=1_gθθ∂θ,_e3^=1^{_gϕϕ-_{gtϕ2 gtt}1/2}-_{gtϕ gtt∂t}+_∂ϕ,

where the tetrad axes are primarily along the Boyer-Lindquist coordinate directions. From

-_{gtϕ gtt}=-2MarΣ-2Mr^sin2θ,^{_gϕϕ-_{gtϕ2 gtt}1/2}=^{ΣΔΣ-2Mr1/2}sinθ,-g=Σsinθ,

we find

_e0^=^ΣΣ-2Mr1/2_∂t,_e1^=^ΔΣ1/2_∂r,_e2^=^1Σ1/2_∂θ,_e3^=-2Marsinθ^{ΔΣ(Σ-2Mr)1/2}_∂t+^{Σ-2MrΔΣ1/2}1sinθ_∂ϕ.

This tetrad frame can be adapted with_e0^=u¯to the reference observers that form a congruence of accelerated, nonexpanding and locally rotating world lines. The lack of expansion of the congruence is due to the alignment of its 4-velocity vector field with the timelike Killing direction of the exterior Kerr spacetime.

The 4-acceleration of the fiducial observers is given by

^Aμ=D^eμ _0^dτ=_Γαβμ^eα _0^^eβ _0^,

where_Γαβμare the Kerr connection coefficients. Thus,

A=MΔ(^r2-^{a2 cos2}θ)^Σ3/2(Δ-^{a2 sin2}θ)_e1^-2Mr^a2sinθcosθ^Σ3/2(Δ-^{a2 sin2}θ)_e2^,

whereΔ-^{a2 sin2}θ=Σ-2Mr. This acceleration counters the attraction of gravity; that is, it is due to forces that are not gravitational in origin and are necessary to keep the reference observer from falling into the source. To set up a nonrotating spatial frame along the world line of each fiducial observer, let^Σμbe a vector that is Fermi-Walker transported along^eμ _0^; then,

d^Σμdτ+_Γαβμ^eα _0^^Σβ=(A·Σ)^eμ _0^-(_e0^·Σ)^Aμ.

We need a vector that is orthogonal to the world line, namely,_e0^·Σ=0; furthermore, such a vector can be expressed in terms of the natural spatial frame^eμ _a^. That is,^Σμ=^{sa^eμ} _a^ , which means via Equation (A24) that in the exterior Kerr spacetime

d^{s1^}dτ=Ma(^r2-^{a2 cos2}θ)sinθ^Σ3/2(Σ-2Mr)^{s3^},

d^{s2^}dτ=-2MarΔcosθ^Σ3/2(Σ-2Mr)^{s3^},

d^{s3^}dτ=Ma^Σ3/2(Σ-2Mr)[-(^r2-^{a2 cos2}θ)sinθ^{s1^}+2rΔcosθ^{s2^}].

These equations can be written in the form

d^{si^}dτ=^{ϵi^j^k^}_Ωj^sk^,

where

^Ωμ=^{Ωa^eμ} _a^=β(cosα^eμ _1^+sinα^eμ _2^)

is the proper precession vector. Here, the angleαand the proper precession frequencyβ>0are defined by

sinα=(^r2-^{a2 cos2}θ)sinθD,cosα=2rΔcosθD

and

β=Ma^Σ3/2(Σ-2Mr)D,

where

D=^{4r2Δcos2θ+(r2-a2 cos2θ)2sin2θ1/2}.

It is interesting to note thatαhas the same range as the polar angleθ, so thatα:0→πwhenθ:0→π.

Let us briefly digress here and use the results given above to correct a few typographical errors that occur in Ref. [23]: In Equation (5) of [23], the temporal component of_e3^ must be multiplied by a, the specific angular momentum of the Kerr source; moreover, in the denominators of Equations (10)-(12) of [23],(Σ-2Mr)must be replaced by^(Σ-2Mr)1/2.

To construct a locally nonrotating spatial frame^λμ _a^along^eμ _0^, we can choose^λμ _1^to be the unit vector along^Ωμ; then,^λμ _2^and^λμ _3^are unit vectors in the plane orthogonal to^λμ _1^and precess with frequencyβabout^λμ _1^. Thus the second step involves the introduction of the orthonormal spatial frame^Eμ _a^,

_E1^=cosα_e1^+sinα_e2^,_E2^=-sinα_e1^+cosα_e2^,_E3^=_e3^.

Then in the final step the Fermi-Walker transported triad^λμ _a^is obtained from^Eμ _a^by a simple rotation about^Eμ _1^with an angle ofβτ,

_λ1^=_E1^,_λ2^=cos(βτ)_E2^+sin(βτ)_E3^,_λ3^=-sin(βτ)_E2^+cos(βτ)_E3^.

Let us combine these rotation matrices in a new matrix S defined by^λμ _i^=_Si ^jeμ _j^, where

S=cosαsinα0-sinαcosψcosαcosψsinψsinαsinψ-cosαsinψcosψ.

Here,ψ=βτ. Moreover, the transpose of the rotation matrix S is given by^ST=^S-1,

^ST=cosα-sinαcosψsinαsinψsinαcosαcosψ-cosαsinψ0sinψcosψ,

so that^eμ _i^=_(^ST)i ^jλμ _j^. It is now possible to express the nongravitational acceleration of the fiducial observer in terms of the Fermi-Walker transported tetrad frame, namely,

A=2Mr(^r2-^{a2 cos2}θ)cosθ^Σ3/2D_λ1^+MΣΔsinθ(Σ-2Mr)D(-cosψ_λ2^+sinψ_λ3^).

This completes the construction of the adapted nonrotating tetrad frame^λμ _α^along the world line of an arbitrary observer that is spatially at rest in the exterior Kerr spacetime; here,^λμ _0^=^eμ _0^=^u¯μ.

To establish a Fermi normal coordinate system along the world line of such a reference observer, it remains to calculate the spacetime curvature components given by Equation (A6).

Appendix C. Measured Components of Curvature

In general, one can take into account the symmetries of the Riemann tensor and express Equation (A6) in the standard manner as a6×6matrix(_RAB), where A and B are indices that belong to the set{01,02,03,23,31,12}. The general form of this matrix is

EH^HTS,

whereEandSare symmetric3×3matrices andHis traceless. Here, the measured gravitoelectric components of the Riemann curvature tensor are represented by the relativistic tidal matrixE. Similarly,HandS represent its gravitomagnetic and spatial components, respectively. In the vacuum region exterior to material sources and free of nongravitational fields, the spacetime is Ricci flat as a consequence of Einstein's gravitational field equations and Equation (A38) becomes

EHH-E,

whereEandHare symmetric and traceless. That is, the Riemann curvature tensor degenerates in the Ricci flat case into the Weyl conformal curvature tensor whose gravitoelectric and gravitomagnetic components are then

_Ea^b^=_Cαβγδ^λα _0^^λβ _a^^λγ _0^^λδ _b^,_Ha^b^=_Cαβγδ*^λα _0^^λβ _a^^λγ _0^^λδ _b^,

where_Cαβγδ*is the unique dual of the Weyl tensor given by

_Cαβγδ*=12^ημν _{αβCμνγδ},

since the right and left duals of the Weyl tensor coincide. Here,_ημνρσis the Levi-Civita tensor and in our convention,_η0^1^2^3^=1, while_η0^a^b^c^=_ϵa^b^c^. Let us note that

_Ha^b^=12^ημν _{αβCμνγδ}^λα _0^^λβ _b^^λγ _0^^λδ _a^=12^ημν _{0^b^Cμν0^a^}=12_C0^a^c^d^^{ϵc^d^} _b^.

We now turn to the explicit computation of curvature components as measured by observers at rest in the exterior Kerr spacetime.

Curvature of Kerr Spacetime as Measured by Observers at Rest

Projected onto the canonical Petrov tetrad of the Kerr field [30], the Weyl tensor takes the form

_E0=E-200010001,_H0=H-200010001,

where,

E+iH=M^(r+iacosθ)3,

so that

E=Mr(^r2-3^{a2 cos2}θ)^Σ3,H=-Ma(3^r2-^{a2 cos2}θ)cosθ^Σ3.

The evident "parallelism" between the gravitoelectric and gravitomagnetic components of curvature in Equation (A43) has to do with the degenerate nature of the Kerr field; in fact, it is of type D in the Petrov classification. We are interested in the curvature of Kerr spacetime as measured by the family of observers at rest. Projected onto the natural frame^eμ _α^with axes that are primarily along the Boyer-Lindquist coordinate directions, we find(^E',^H') . Specifically, the nonvanishing components of the tidal matrix can be obtained from [18]

_E1^1^'=-2EΔ+12^{a2 sin2}θΔ-^{a2 sin2}θ,_E1^2^'=-3asinθH^Δ1/2Δ-^a2sin2θ,_E2^2^'=EΔ+2^{a2 sin2}θΔ-^{a2 sin2}θ,_^E'3^3^=E.

Furthermore, the nonzero elements of the gravitomagnetic part of the Weyl curvature can be obtained from [18]

_H1^1^'=-2HΔ+12^{a2 sin2}θΔ-^{a2 sin2}θ,_H1^2^'=3asinθE^Δ1/2Δ-^a2sin2θ,_H2^2^'=HΔ+2^{a2 sin2}θΔ-^{a2 sin2}θ,_^H'3^3^=H.

We are actually interested in the measured components of curvature along the Fermi-Walker transported tetrad frame^λμ _α^ . In this case, the measured components of the curvature tensor are given by Equation (A39), where(E,H)are related to(^E',^H') via a rotation S given by Equation (A35). Under such a rotation, it is straightforward to show that the gravitoelectric part (i.e., the relativistic tidal matrix) and the gravitomagnetic part of the Weyl tensor undergo a similarity transformation, namely,

E=S^E'S-1,H=S^H'S-1.

To express the measured curvature components explicitly, it proves convenient to definePandQsuch that

P:=12(_E1^1^'-_E2^2^')cos2α+_E1^2^'sin2α,Q:=-12(_E1^1^'-_E2^2^')sin2α+_E1^2^'cos2α.

Then, withψ=βτ, we have

_E1^1^=-12_E3^3^'+P,_E1^2^=Qcosψ,_E1^3^=-Qsinψ,

_E2^2^=-12_E3^3^'+P^cos2ψ+_E3^3^'^sin2ψ,_E2^3^=32_E3^3^'+Psinψcosψ

and

_E2^1^=_E1^2^,_E3^1^=_E1^3^,_E3^2^=_E2^3^,_E3^3^=-_E1^1^-_E2^2^,

since this matrix is symmetric and traceless; moreover, we have exactly the same type of expressions forH. From(E,H), we find_{Rα^β^γ^δ^}(T)employed in the Fermi normal coordinate system.

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