A Second order solution coefficients

Effect of an arbitrary spin orientation on the quadrupolar structure of an extended body in a Schwarzschild spacetime


The influence of an arbitrary spin orientation on the quadrupolar structure of an extended body moving in a Schwarzschild spacetime is investigated. The body dynamics is described by the Mathisson-Papapetrou-Dixon model, without any restriction on the motion or simplifying assumption on the associated spin vector and quadrupole tensor, generalizing previous works. The equations of motion are solved analytically in the limit of small values of the characteristic length scales associated with the spin and quadrupole variables with respect to the characteristic length of the background curvature. The solution provides all corrections to the circular geodesic on the equatorial plane taken as the reference trajectory due to both dipolar and quadrupolar structure of the body as well as the conditions which the nonvanishing components of the quadrupole tensor must fulfill in order that the problem be self-consistent.


I Introduction

An extended test body moving in a given gravitational background is commonly described according to the Mathisson-Papapetrou-Dixon (MPD) model (1); (2); (3); (4); (5); (6); (7); (8); (9). The MPD equations of motion involve a reference world line, properly set up to represent the center-of-mass line of the body, and a number of vector and tensor fields defined along it through a multipole moment expansion, similarly to the standard nonrelativistic theory. The model is fully determined and self-consistent at the dipolar order, i.e., for an extended body endowed with spin only, providing a set of evolution equations for both the linear and angular momentum of the body. For the quadrupolar as well as higher multipolar orders, instead, there are no evolution equations for the quadrupole and higher multipole tensors, and their evolution is fixed entirely by the body’s internal dynamics. The contribution of higher multipoles appears in the form of additional force and torque terms. Therefore, one has to supply the structure of the body as external information, e.g., specifying the equation of state of its matter-energy content. This fact represents a peculiarity of the model itself, which allows for many different approaches.

According to Dixon’s construction, the quadrupole tensor shares the same symmetries of the Riemann tensor and is completely specified by two symmetric and trace-free spatial tensors, i.e., the mass quadrupole (electric-type) and the current quadrupole (magnetic-type) tensors. The most natural and simplifying choice consists in considering the body as “quasi-rigid,” i.e., all unspecified quantities describing its shape are taken constant in the body-fixed frame, i.e., adapted to the 4-momentum of the body itself (10). Alternatively, one can assume the quadrupole tensor be directly related to the Riemann tensor, having the same symmetry properties. For instance, in Ref. (11) the electric and magnetic parts of the quadrupole tensor have been taken proportional to the electric and magnetic parts of the Riemann tensor, respectively, to study quadrupole deformation effects induced by the tidal field of a black hole on the motion of a spinning body. One can also require that the structure of the body be completely determined by its spin, with a quadratic-in-spin quadrupole tensor, which is of purely electric type, being proportional to the trace-free part of the square of the spin tensor (12); (13); (14). The interest in such an approach is the possibility to include spin-induced quadrupole corrections in the post-Newtonian dynamics of a two-body system (see, e.g., Refs. (15); (16); (17); (18); (19); (20); (21)).

Beside the choice of the quadrupole tensor, further simplifications to the problem come from the symmetries of the considered gravitational field as well as the particular kind of motion one is interested in. In the literature, the motion of extended bodies around a compact object is usually assumed to be confined to the equatorial plane of reflection symmetric spacetimes (like Schwarzschild and Kerr backgrounds) taken as the orbital plane, the spin vector being constant in magnitude and necessarily directed orthogonally to it. This is a highly symmetric situation which imposes a strong restriction on the structure of the body, reducing the number of nonvanishing components of the quadrupole tensor. The latter has in general 20 independent components, but only 10 of them are actually relevant, namely those obeying the symmetries of the MPD equations, as shown in Ref. (22). In fact, the quadrupole tensor enters the MPD equations only through certain contractions with the Riemann tensor and its covariant derivative. This is also expected from the standard post-Newtonian formulation of motion of many-body systems. When the motion is restricted to the equatorial plane, the number of nonvanishing effective components of the quadrupole tensor then reduces from 10 to 5, 3 belonging to the mass quadrupole moment, and 2 to the current quadrupole moment (22).

In the present paper we will study the dynamics of an extended body in a Schwarzschild spacetime in the framework of the MPD model, without any simplifying assumption on the body dynamics and structure, allowing it to move off the equatorial plane with an arbitrary orientation of the spin vector, generalizing previous results (23). Following the MPD prescriptions, we only require that the body does not perturb significantly the background field, so that backreaction effects can be neglected and the body structure only produces very small deviations from geodesic motion. This condition is indeed implicit in the MPD model, and allows to treat the equations of motion perturbatively, in the sense that the natural length scales associated with the body, i.e., the “bare” mass as well as the spin and quadrupole characteristic lengths, are taken to be small enough if compared with the length scale associated with the background curvature. The resulting simplified set of differential equations can be integrated analytically. Initial conditions are chosen so that the world line of the extended body has the same starting point and is initially tangent to a timelike circular geodesic on the equatorial plane, taken as a reference world line. It is known that the presence of a spin component in the equatorial plane induces an oscillation of the body path in and out of the equatorial plane to first order in spin (i.e., taking into account only corrections which are linear in spin) (24); (25). The orthogonal component is instead responsible for an oscillating behavior of the radial distance about the reference radius, while the azimuthal motion undergoes similar oscillations plus an additional secular drift. We will show that second order corrections to an initially circular geodesic motion (i.e., spin squared and mass quadrupole terms) introduce secular effects in both radial and polar motions enhancing deviations.

Furthermore, we will discuss the consequences of that general situation on the quadrupolar structure of the body. The spin evolution equations, indeed, provide some compatibility conditions involving spin vector components, first order corrections to the orbit and components of the quadrupole tensor, which turn out to be varying with time. As a result, the mass quadrupole moment associated with the body changes during the evolution, causing its shape to change too, passing from nearly spherical to highly deformed configurations. A variable mass quadrupole moment is usually generated in a binary system because of the tides produced by the higher mass. In such a situation the net gravitational radiation associated with the motion of the smaller mass is due to its orbit, the time varying tides and the interference between them (26); (27). Furthermore, changes in the gravitational quadrupole moment of the companion star are expected to account for most of the observed variations in the orbital parameters of binary pulsar systems (28); (29). Noteworthy, recent radio timing observations of the eclipsing millisecond binary pulsar PSR J2051-0827 have provided evidences for variations of the quadrupole moment in its companion (30); (31); (32). Such variations together with the spin precession of the companion star have been shown to be responsible for the changes of the orbital period, inclination angle and projected semimajor axis of the binary system. Although the underlying mechanism causing a varying quadrupole moment is most likely of non-gravitational nature (e.g., driven by the magnetic activity in close binaries (30); (33)), the purely gravitational effect discussed here may play a role.

We will follow notations and conventions of Ref. (34). Units are chosen so that and the metric signature is . Greek indices run from to , whereas Latin indices from to .

Ii MPD equations in the quadrupole approximation

Consider an extended body endowed with structure up to the quadrupole moment. The MPD equations are


where (with ) is the total 4-momentum of the body with mass , is a (antisymmetric) spin tensor, is the quadrupole tensor, and is the timelike unit tangent vector of the “center of mass line” (with parametric equations ) used to make the multipole reduction, parametrized by the proper time .

In order the model to be mathematically self-consistent the following additional conditions should be imposed (3); (4)


Consequently, the spin tensor can be fully represented by a spatial vector (with respect to ),


where is the spatial (with respect to ) unit volume 3-form with the unit volume 4-form and () the Levi-Civita alternating symbol. It is also useful to introduce the signed magnitude of the spin vector


which is in general not constant along the trajectory of the extended body.

The quadrupole tensor by its definition has the same algebraic symmetries as the Riemann tensor, but enters the MPD equations only through certain combinations, which reduce the number of effective components from 20 to 10 (23); (22); (35). Therefore, in complete analogy with the splitting of the Riemann tensor with respect to a given timelike congruence, it can be written in the form


where and are symmetric and trace-free (STF) spatial tensors as “measured” by an observer comoving with the body, representing the mass quadrupole moment and the flow (or current) quadrupole moment, respectively (see, e.g., Ref. (10)).

In stationary and axisymmetric spacetimes endowed with Killing symmetries the total energy and the angular momentum are conserved quantities along the motion associated with the timelike Killing vector and the azimuthal Killing vector , respectively. They are given by




are the Papapetrou fields associated with the Killing vectors. Note that and as defined above are conserved quantities to all multipole orders in spite of the higher multipolar structure of the body (10).

ii.1 Perturbative approach

Consider a pair of world lines emanating from a common spacetime point, one a geodesic with 4-velocity , the other the world line of an extended body deviating from the reference one because of the combined effects of both the spin-curvature and quadrupole-curvature couplings, with 4-velocity . Introduce a smallness indicator to distinguish between the order of multipolar approximation, so that and . Solutions to the MPD equations can then be found in the general form


The mass of the body is a conserved quantity to first order and the 4-momentum vector is parallel to the 4-velocity , so that one can assume


where denotes the “bare” mass. The second order correction to the mass of the body turns out to be


whereas the unit vectors and are related by


providing four algebraic relations between their components. Substituting the expansions above into the MPD equations (1) and (2) then leads to two different sets of evolution equations for the first order and second order quantities, respectively, neglecting terms of higher order.

It is worth noting that and are unit tangent vectors to different timelike world lines, which are parametrized by different proper times: hence, one should use as the proper time parameter along and as the proper time parameter along . However, from their definitions,


and recalling the normalization condition , one obtains that and can be identified to the second order of approximation, i.e.,


Therefore, although the two world lines are parametrized by different proper times, the latter are synchronized so that can be used unambiguously for that single proper time parametrization of both world lines.

We are interested here in solutions to the MPD equations which describe deviations from geodesic motion due to both the spin-curvature force and the quadrupolar force. Hence, we will choose initial conditions so that the world line of the extended body has the same starting point as the reference geodesic, i.e.,


The two world lines in general have not a common unit tangent vector at ; as increases, then, they deviate from each other. We will require that the 4-velocity be initially tangent to the geodesic 4-velocity , which implies in addition


In the next section we will specialize our analysis to the Schwarzschild background.

Iii Dynamics of extended bodies in a Schwarzschild spacetime

Consider the Schwarzschild spacetime in standard coordinates , with line element written in standard form as


where denotes the lapse function


An orthonormal frame adapted to the static observers, i.e., those following the coordinate time lines with -velocity , is given by


with dual , , and .

Let the reference world line be a circular geodesic in the equatorial plane at radius . The associated -velocity is


where the signs refer to co-rotating and counter-rotating motion with respect to increasing values of the azimuthal coordinate, respectively. Here and (with associated Lorentz factor ) denote the Keplerian angular velocity and linear velocity, respectively, and is a normalization factor defined by


The circular geodesic is thus described by the parametric equations


It is useful to introduce the unit vector along the azimuthal direction in the local rest space of the circular geodesic, orthogonal to in the - plane, i.e.,


An orthonormal frame adapted to is thus given by


with aligned with the (positive) -axis of a naturally defined Cartesian frame.

Finally, the circular geodesic conserved energy and angular momentum are given by



iii.1 First order solution

To first order the set of MPD equations (1) and (2) reduces to


The spin vector must be orthogonal to due to the supplementary conditions (3), so that


and turns out to be parallel transported along the reference circular geodesic due to the spin evolution equations. That leads to a simple rotation of the spin components in the - plane within the local rest space of the circular geodesics. The corresponding solution can then be written as


where a polar representation for the spin vector has been conveniently introduced such that , and , with


The quantities and are constant due to the conservation of the spin magnitude.

The solution for the orbit is then given by (25)




and the dimensionless spin quantities


have been introduced. Here


are respectively the well known epicyclic frequency governing the radial perturbations of circular geodesics and the orbital frequency governing the geodesic oscillations out of the equatorial plane. The latter frequency together with the spin-precession frequency due to the spin oscillation driving term governs the polar angle oscillations about the equatorial plane. Their ratio


will enter most of the relations below, also implying the allowed range for radial distance .

The solution for (which at is aligned with the circular geodesic at ) is then given by , with

When decomposed with respect to the frame (III) adapted to the static observers, the latter writes as , leading to the following relations between frame and coordinate components


Finally, the first order corrections to the circular geodesic conserved energy and angular momentum (25) are given by


respectively, as from Eq. (II).

iii.2 Second order solution

The evolution of the spin vector is completely determined by the first order equations. Therefore, the spin evolution equations (2) to second order simply provide three algebraic relations between the components of and (in agreement with Eq. (12)) plus three compatibility conditions involving spin vector components, first order corrections to the orbit and components of the quadrupole tensor. In fact,


and . A further condition comes from the evolution equation for the second order correction (11) to the mass of the body. Contracting Eq. (1) with leads to


whereas contracting with yields


being orthogonal to , implying that


which involves certain components of the quadrupole tensor and their first derivatives with respect to proper time.

It is convenient to introduce the dimensionless frame components of the spin vector as well as the following combinations of the quadrupole tensor components




are dimensionless quadrupole quantities obtained by suitably rescaling the frame components of the electric and magnetic parts of the quadrupole tensor with respect to the frame (24) adapted to the circular geodesics.

The spin evolution equations (2) then give

together with the following compatibility conditions

The second order correction to the mass of the body is given by Eq. (11), which reads


The condition (41) implies


whose integration yields


Therefore, the solution for turns out to be


Finally, the equations of motion (1) give the evolution equations for the second order corrections to the orbit , i.e.,




from the normalization condition.

The equation for can be integrated straightforwardly. The equations for and are instead coupled. However, taking the derivative of the equation for with respect to and using the equation for leads to an equation for , which can be easily integrated with initial conditions