Jacobi Fields and Tidal Effects in Kerr Spacetime

Quasars and active galactic nuclei often exhibit puzzling features involving relativistic outflows. These jets frequently come in pairs and originate from a central source that is believed to be a massive rotating black hole surrounded with an accretion disk. Similar phenomena have been observed in our galaxy and are associated with microquasars, which are binary x-ray sources. Jets are presumably emitted along the rotation axis of a Kerr black hole, a circumstance that appears to be contrary to the strong gravitational attraction of the black hole. Nevertheless, the double-jet feature is consistent with the symmetry of the Kerr gravitational field under reflection about its equatorial plane. Observational data further indicate that in extragalactic jets, jet acceleration away from the black hole results in Lorentz factors that are much greater than unity.

The motion of particles in a jet would be naturally affected by magnetohydrodynamic forces as well as relativistic gravitational effects of general relativity Punsly . The purpose of the present work is to explore further the influence of the gravitational tidal acceleration of the Kerr black hole on the dynamics of astrophysical jets within the framework of general relativity (GR); see Bini:2017uax ; Mashhoon:2020tha and the references cited therein. Unless specified otherwise, in this paper we use gravitational units such that $c=G=1$ throughout; moreover, Greek indices run from 0 to 3, while Latin indices run from 1 to 3, and the signature of the spacetime metric is +2.

The Kerr metric, $-ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$, where $ds$ is the spacetime interval for timelike world lines, can be expressed in Boyer-Lindquist coordinates $x^{\mu}=(\tilde{t},r,\theta,\phi)$ as Chandra ; BON

where $M>0$ and $a>0$ are the mass and specific angular momentum of the Kerr source, $\Sigma=r^{2}+a^{2}\cos^{2}\theta$ and $\Delta=r^{2}-2Mr+a^{2}$. Kerr spacetime is asymptotically flat; in fact, by passing to the limit $r\to\infty$, metric (I) reduces to the Minkowski spacetime metric expressed in spherical polar coordinates. Furthermore, Kerr gravitational field is stationary and axisymmetric with Killing vector fields $\partial_{\tilde{t}}$ and $\partial_{\phi}$. The spacetime is a Kerr black hole for $a\leq M$; otherwise, the Kerr source is a naked singularity. For a Kerr black hole, reference observers can be spatially at rest from the asymptotic region ($r\to\infty$) all the way down to the exterior of the stationary limit surface, where $g_{00}=0$ or $\Sigma=2\,Mr$. On the symmetry axis, $\theta=0$ and the exterior solution is valid for $r$ in the range $M+(M^{2}-a^{2})^{1/2}<r<\infty$.

Astronomers regularly observe energetic outflows and other events relative to the rest frames of the gravitationally collapsed systems that are usually modeled by Kerr gravitational fields. To describe theoretically such phenomena relative to the rest frame of the source, we imagine the existence of reference observers that are spatially at rest in the exterior Kerr spacetime. Such observers in effect define the rest frame of the Kerr source. A detailed description of this family of observers as well as their adapted nonrotating (i.e., Fermi-walker) orthonormal tetrad frames is contained in Bini:2017uax ; Mashhoon:2020tha . In particular, for the present study our fiducial observer is at rest at radial position $r$ along the rotation axis of the exterior Kerr spacetime. The observer has unit 4-velocity vector $\bar{u}^{\mu}=d\bar{x}^{\mu}/d\tau$, where $\tau$ is its proper time. This test observer is accelerated; in fact, nongravitational forces are necessary to keep the observer spatially at rest and prevent it from falling into the black hole. The observer carries an adapted nonrotating orthonormal tetrad frame $e^{\mu}{}_{\hat{\alpha}}$, where $e^{\mu}{}_{\hat{0}}=\bar{u}^{\mu}$,

and $(e_{\hat{2}},e_{\hat{3}})$ 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 Bini:2017uax ; Mashhoon:2020tha . Moreover, the observer’s acceleration $De^{\mu}{}_{\hat{0}}/d\tau=\mathbf{A}^{\mu}=A^{\hat{\alpha}}\,e^{\mu}{}_{\hat{\alpha}}$ is given by

For measurement purposes, the reference observer can in principle set up a quasi-inertial Fermi normal coordinate system based on the local nonrotating tetrad frame along its world line; see TLC ; Synge ; bm77 ; Belinski:2020dmz ; Hoegl:2020hif and the references cited therein. Imagine an arbitrary reference observer with an adapted nonrotating orthonormal tetrad frame $e^{\mu}{}_{\hat{\alpha}}$. The observer follows a timelike world line $\bar{x}^{\mu}(\tau)$ in an arbitrary gravitational field. The observer’s local temporal axis is the timelike unit vector $e^{\mu}{}_{\hat{0}}=d\bar{x}^{\mu}/d\tau=\bar{u}^{\mu}$, while $e^{\mu}{}_{\hat{i}}$, $i=1,2,3$, are orthogonal unit spacelike gyro axes that form the observer’s local spatial frame. Let us consider the class of spacelike geodesics orthogonal to the reference world line at each event $Q(\tau)$ along $\bar{x}^{\mu}(\tau)$. In this way, we have a local spacelike hypersurface at $\tau$. Suppose $P$ is an event with spacetime coordinates $x^{\mu}$ on this local spacelike hypersurface such that there is a unique spacelike geodesic of proper length $\sigma$ that connects $P$ to $Q$. The unit spacelike vector $\xi^{\mu}(\tau)$ is tangent to this spacelike geodesic segment at $Q$. We define the Fermi coordinates of $P$ to be $X^{\hat{\mu}}=(T,X^{\hat{i}})$, $X^{\hat{i}}=(X,Y,Z)$, where

It follows that the Fermi coordinate time $T$ is the proper time of the reference observer that is located at $X=Y=Z=0$; that is, the reference observer is permanently at rest and occupies the spatial origin of Fermi normal coordinates. The reference observer is furthermore locally inertial in the Fermi system by Einstein’s principle of equivalence, but acceleration and curvature terms appear in the metric away from the location of the fiducial observer at the origin of the spatial Fermi coordinates. In the case under consideration, Fermi coordinates $X^{\hat{\mu}}=(T,X,Y,Z)$ for Kerr spacetime are such that $X$ denotes the radial coordinate along the Kerr symmetry axis, while $Y$ and $Z$ are the two transverse coordinates. The metric in Fermi coordinates is given by

where the components of the Fermi metric tensor can be expressed as power series in $X$, $Y$ and $Z$ starting from the corresponding components of the Minkowski metric tensor $\eta_{\hat{\mu}\hat{\nu}}$, namely,

and

The components of the curvature tensor (and its covariant derivatives) as measured by the reference observer appear in the coefficients of these power series. For instance, $R_{\hat{\alpha}\hat{\beta}\hat{\gamma}\hat{\delta}}$ is the projection of the curvature tensor on the tetrad frame of the reference observer, namely, $R_{\mu\nu\rho\sigma}e^{\mu}{}_{\hat{\alpha}}e^{\nu}{}_{\hat{\beta}}e^{\rho}{}_{\hat{\gamma}}e^{\sigma}{}_{\hat{\delta}}$, which, in a Ricci-flat region of spacetime, can in general be expressed via the symmetries of the curvature tensor in the standard manner as a $6\times 6$ matrix of the form

Here, the symmetric and traceless $3\times 3$ matrices $\mathbb{E}$ and $\mathbb{H}$ represent the measured gravitoelectric and gravitomagnetic components of Weyl curvature, respectively. These have been worked out in Mashhoon:2020tha for the general case of fiducial observers at rest in the exterior Kerr spacetime.

Fermi coordinates are properly defined within a coordinate patch whose boundaries are given by the standard GR admissibility conditions. The spacetime coordinates are admissible if the corresponding metric with $(-,+,+,+)$ signature is negative definite. That is, we require the metric tensor in matrix form $(g_{\hat{\mu}\hat{\nu}})$ to be negative definite; equivalently, all of its principal minors must be negative Bini:2012ht ; LL . This means, in particular, that $g_{\hat{0}\hat{0}}<0$ within the Fermi coordinate patch. Therefore, if $g_{\hat{0}\hat{0}}$ vanishes, the corresponding domain where $g_{\hat{0}\hat{0}}=0$ must be part of the boundary of the Fermi coordinate system.

Timelike and null geodesic motions in the exterior Kerr spacetime have been extensively investigated Chandra ; BON . These motions basically refer to the static inertial observers that are at rest at infinity in this asymptotically flat spacetime. However, in connection with motions relative to the Kerr source, we are interested in the timelike motion of free test particles in the Fermi coordinate system established around the world line of our static fiducial observer. The rest mass of such a free test particle is a nonzero constant of its motion; however, we will not explicitly consider the rest mass of the particle further since it is not necessary for the purposes of the present work. The geodesic equation in the Fermi frame can then be obtained from the action principle

where the Lagrangian is given by

The corresponding Euler-Lagrange equation is equivalent to the reduced geodesic equation in Fermi coordinates, namely,

which is a generalization of the Jacobi equation in Fermi coordinates. A detailed description of the Lagrangian and Hamiltonian aspects of such generalized deviation equations is contained in Chicone:2002kb . Furthermore, simple generalizations are possible; for instance, we can consider accelerated motion of test particles in the Fermi coordinate system as well. We must ensure that Eq. (12) refers to timelike motion of the free test particle; to this end, let us define its 4-velocity in the Fermi system

where the Lorentz factor $\Gamma=dT/ds$ can be determined via $U^{\hat{\mu}}\,U_{\hat{\mu}}=-1$. Hence, the timelike condition is given by

Here, it is important to note that $\mathbf{V}$ is a Fermi coordinate velocity; however, we must have $|\mathbf{V}|<1$ at the location of the reference observer ($\mathbf{X}=0$). It is important to compare and contrast these exact general relativistic tidal equations with the corresponding Newtonian tidal equations presented in Appendix A.

The construction of exact Fermi coordinate systems has thus far been possible only for a few simple spatially homogeneous spacetimes Chicone:2005vn ; Klein:2009gz . It was shown explicitly in Chicone:2005vn that it is not possible to express exact Fermi coordinates in Schwarzschild spacetime using only the elementary functions of mathematical physics. In this connection, see also BGJ and the references cited therein. The same argument extends to the exterior Kerr spacetime as well. The natural option is to expand the coordinate transformation from the standard Kerr coordinates to the corresponding Fermi coordinates in infinite series. Unfortunately, obtaining the coefficients of the power series expansion of the corresponding Fermi metric is nontrivial Bini:2005xt ; Klein:2007xj ; in fact, this general computation would seem to require a method that is not at present readily available. In our approximate treatment, we therefore use the expansion of the Kerr metric in Fermi coordinates to second order in the Fermi spatial coordinates $(X,Y,Z)$; hence, the corresponding tidal equations are given to linear order in $(X,Y,Z)$. Higher-order tidal terms have been studied in Marzlin:1994ia ; Chicone:2005da , but not for the Kerr metric.

The Fermi coordinates are introduced here about the world line of a fiducial observer at rest in space in the background stationary Kerr spacetime; therefore, the corresponding Fermi metric must be independent of the Fermi temporal coordinate $T$. It follows that in the case under consideration in this paper, the metric coefficients are of the form $g_{\hat{\mu}\hat{\nu}}(X,Y,Z)$ and $\partial_{T}$ is a timelike Killing vector field. The projection of the 4-velocity vector of a geodesic path on a Killing vector field is a constant of the motion. For a free test particle in the Fermi frame, we thus have

where $\mathcal{E}$ is proportional to the particle’s energy. To see this, let us note that if the free test particle starts out from the location of the fiducial observer with initial speed $V_{0}$, $0<V_{0}<1$, then from Eqs. (14)–(15) and $g_{\hat{\mu}\hat{\nu}}(\mathbf{X}=0)=\eta_{\hat{\mu}\hat{\nu}}$ we have

To monitor the motion of the free test particle within the Fermi frame, we imagine the class of observers that are all spatially at rest within the Fermi patch. The fiducial observer is a member of this class of generally accelerated observers that constitute the natural rest frame of the Kerr source. Let $\Lambda^{\hat{\mu}}{}_{\tilde{\alpha}}(\mathbf{X})$ be the adapted tetrad frame field of these observers; then, $\Lambda^{\hat{\mu}}{}_{\tilde{0}}=(-g_{\hat{0}\hat{0}})^{-1/2}\,\delta^{\hat{\mu}}_{\tilde{0}}$ and the 4-velocity vector of the free test particle as measured by this class of observers is given by

In particular, we can write

Equation (15) then implies

As measured by the class of static observers in the Fermi frame, a free test particle that comes very close to the two-dimensional boundary of the Fermi coordinate system where $g_{\hat{0}\hat{0}}(\mathbf{X})=0$ has a Lorentz factor that approaches infinity and a speed that is very close to the speed of light.

We have placed the fiducial observer on the axis of symmetry in direct connection with the motion of jet particles on or near the Kerr rotation axis, which is the $X$ axis in the Fermi frame. Moreover, the Kerr metric is axially symmetric about its axis of rotation and $\partial_{\phi}$ is a Killing vector field. The position of the reference observer as well as the construction of a Fermi system about its world line respects this axial symmetry of the underlying geometry. Hence, there must be axial symmetry about the $X$ axis in our Fermi system. In particular, introducing spherical polar coordinates via $X=\rho\,\cos\vartheta$, $Y=\rho\,\sin\vartheta\cos\varphi$ and $Z=\rho\,\sin\vartheta\sin\varphi$, the exact Fermi metric tensor (expressed in infinite series) must be independent of $\varphi$. It is straightforward to verify this circumstance explicitly using the approximate Fermi metric tensor employed in this paper. These results are compatible with the existence of the Killing vector field $\partial_{\varphi}$ within the Fermi patch.

Azimuthal symmetry about the $X$ axis indicates that jet particles moving outward along the Kerr rotation axis approach the $g_{\hat{0}\hat{0}}(\mathbf{X})=0$ boundary of the Fermi patch perpendicularly. Consider the surfaces of constant $g_{\hat{0}\hat{0}}(\mathbf{X})$ and their normal vectors

At the position of the fiducial observer $\mathbf{X}=0$ and $g_{\hat{0}\hat{0}}=-1$, and we find from Eqs. (6)–(8) that $N^{\hat{\mu}}=(0,1,0,0)$ coincides with the $X$ axis. Moreover, in the approximation where $O(|\mathbf{X}|^{3})$ terms are dropped in Eqs. (6)–(8), the $X$ axis is again orthogonal to the surfaces of constant $g_{\hat{0}\hat{0}}(\mathbf{X})$. We expect that this should indeed be the case for the exact Fermi system based on the axial symmetry of the configuration under consideration in this paper.

What are the conditions that must be satisfied for a realistic jet scenario? Naturally, to get away from the gravitational attraction of the Kerr source and join the jet current, the free test particle must have an initial speed that is above a certain threshold. This circumstance is in conformity with the notion of escape velocity that is well known in Newtonian gravitation. The rest of this paper is devoted to studying the motion of free test particles using the approximate tidal equations based on the Fermi coordinate system where in Eqs. (6)–(8) third and higher-order terms in the metric tensor are neglected. However, before we move on to the approximate Fermi system, let us mention an important consequence of Eq. (12).

The dynamical system (12) has a rest point given by $\Gamma^{\hat{i}}_{\hat{0}\hat{0}}(X,Y,Z)=0$, which, from the definition of Christoffel symbols and the temporal independence of the Fermi metric under consideration, reduces to $g_{\hat{0}\hat{0},\hat{i}}=0$. Physically, this means the influence of gravity is zero at a rest point. In the Kerr metric, the influence of gravity decreases as one gets far away from the source and vanishes at spatial infinity. The Fermi coordinate patch is local and does not extend to infinity; hence, there should be no rest point in exact Fermi coordinates. The absence of a rest point is also evident in the Newtonian tidal equation. Thus, a reasonable conjecture is that $g_{\hat{0}\hat{0},\hat{i}}$ does not vanish within the Fermi coordinate patch. The approximate first-order tidal equations in Fermi coordinates for the Kerr field do have an unstable rest point, a fact that is compatible with the conjecture. We will prove this elementary result after a more detailed description of these tidal equations and their nondimensionalization. Further discussion of this issue is contained in the following sections.

Keeping terms only up to second order in the infinite series in Eqs. (6)–(8), Eq. (12) takes the form

This is the generalized Jacobi equation Chicone:2002kb , which is the reduced geodesic equation in the Fermi system expanded to first-order in the spatial variables. The corresponding approximate timelike condition can be expressed as

Let us note that in the approximate system (III), the rest point occurs at the solution of the equation

However, as mentioned before, the exact tidal equations do not have rest points.

For the sake of clarity, we briefly digress to explain our approach to approximating tidal force dynamics: We derive consequences of the first-order approximate Eq. (III) as though it were the true equation of motion. Some of the consequences of this equation generally belong to the actual system (e.g., the threshold speed) and some do not (e.g., the rest point). The exact system, given in terms of infinite series in powers of $X$, $Y$ and $Z$, is not explicitly known; therefore, we resort instead to the investigation of the first-order system (III). Limitations of the approximation are emphasized throughout.

Specializing these general equations to the case of the fiducial observer fixed at the Boyer-Lindquist radial coordinate $r$ along the rotation axis of the Kerr source, we note that the relevant acceleration parameter A is given by Eq. (3), while the gravitoelectric and the gravitomagnetic curvature components are given by $\mathbb{E}$ = diag$(-2E,E,E)$ and $\mathbb{H}$ = diag$(-2H,H,H)$, where Bini:2017uax ; Mashhoon:2020tha

The tidal equations of motion to linear order in $(X,Y,Z)$ are then given by

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

A timelike geodesic in the Kerr gravitational field has three integrals of motion. Kerr spacetime has two Killing vector fields, leading to the energy and orbital angular momentum constants of the motion, and a Killing tensor field that leads to Carter’s constant. To take full advantage of these symmetries in the Fermi system, exact Fermi coordinates are required. Unfortunately, approximations derived from truncations of series expansions do not necessarily respect first integrals. Nevertheless, the azimuthal symmetry of Kerr geometry is reflected in our approximate autonomous equations. That is, keeping $(X,\dot{X})$ 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 that $Y(T)=0$ and $Z(T)=0$ are not separately possible solutions so long as $H\neq 0$. We note from Eq. (24) that $H=0$ if either $a=0$, so that the source is spherically symmetric and described by the Schwarzschild spacetime, or $r=a/\sqrt{3}$.

Henceforth, we generally assume for the sake of definiteness that in Kerr spacetime with $a\neq 0$,

which are consistent with observable motions of energetic particles relative to ambient astrophysical environments of interest. Therefore, $E>0$ and $H<0$.

With the metric signature adopted in this paper, the standard admissibility conditions for the Fermi coordinates require that the principal minors of the matrix $(g_{\hat{\mu}\hat{\nu}})$ be negative. For the background exterior Kerr spacetime, the components of the Fermi metric up to second order in $(x,y,z)$ are given by

and

Let us note that the approximate Fermi metric can be written as

where $h_{\hat{\mu}\hat{\nu}}$ is of second order in $(x,y,z)$. Furthermore, the corresponding inverse Fermi metric is given by

where the indices of the second-order perturbation $h_{\hat{\mu}\hat{\nu}}$ are raised and lowered via the Minkowski metric tensor. For instance,

etc.

It follows from a detailed analysis that the admissibility conditions are