Perturbation Method for Classical Spinning Particle Motion: II. Vaidya Space-Time

# Perturbation Method for Classical Spinning Particle Motion: II. Vaidya Space-Time

Dinesh Singh Department of Physics, University of Regina
July 12, 2019
###### Abstract

This paper describes an application of the Mathisson-Papapetrou-Dixon (MPD) equations in analytic perturbation form to the case of circular motion around a radially accreting or radiating black hole described by the Vaidya metric. Based on the formalism presented earlier, this paper explores the effects of mass accretion or loss of the central body on the overall dynamics of the orbiting spinning particle. This includes changes to its squared mass and spin magnitude due to the classical analog of radiative corrections from spin-curvature coupling. Various quantitative consequences are explored when considering orbital motion near the black hole’s event horizon. An analysis on the orbital stability properties due to spin-curvature interactions is examined briefly, with conclusions in general agreement with previous work performed for the case of circular motion around a Kerr black hole.

###### pacs:
04.20.Cv, 04.25.-g, 04.70.Bw

## I Introduction

The Mathisson-Papapetrou-Dixon (MPD) equations Mathisson (); Papapetrou (); Dixon1 (); Dixon2 () represent a well-known description of classical spinning particle motion in the presence of a curved space-time background. They comprise the “pole-dipole approximation” for the dynamics of extended bodies with spin angular momentum in the vicinity of black holes, neutron stars, or other sources of space-time curvature where a strong gravitational field is generated. This includes sources which themselves are time-varying for the duration of a spinning particle’s motion along its worldline. As a consequence, the spin-curvature coupling term in the MPD equations, which generates an external force and torque to act on the spinning particle, also becomes time-varying, leading to potentially very interesting dynamical effects experienced by the particle.

One particularly interesting space-time background with an explicit time dependence is known as the Vaidya metric, which describes space-time curvature due to a spherically symmetric compact source that either radially accretes surrounding radiation, or radiates away its central mass. Given that most astrophysical sources have at least some orbital or spin angular momentum during their formation, it is unlikely to find candidate sources in the night sky that carry the properties exactly described by the Vaidya metric. However, because of its relative simplicity compared to the Kerr metric to describe rotating black holes, while also having a time-dependent central mass, the Vaidya metric nonetheless provides an ideal testing ground for understanding subtle properties of the MPD equations for an orbiting spinning particle that is sensitive to a time-varying gravitational field. A recent paper Singh1 () presents an extensive numerical investigation of the MPD equations in a Vaidya background, modelling a point dipole in circular orbit around a much heavier non-rotating black hole described by a monotonically increasing central mass function in terms of known functions. This paper also shows, using only the quadrupole moment formula, that the dynamical background due to a growing central mass can influence the shape and frequency of gravitational waveforms generated by the spinning particle for a sufficiently large mass accretion rate, with potentially useful implications for low-frequency gravitational wave astronomy via the space-based LISA observatory LISA ().

Although a numerical treatment of the MPD equations in a Vaidya background is undoubtedly a useful exercise, an analytical exploration of the same problem is definitely beneficial in many respects. For example, knowing the explicit time-dependence of the mass function within an analytical expression of the MPD equations allows for the study of conditions where instabilities in the dynamical system most likely will occur. It can also potentially give useful insight for knowing when a mass increase or loss will lead to macroscopic changes in the particle’s orbit for a predetermined mass accretion or loss rate. Furthermore, the general results obtained from such a study can provide clues for how a spinning particle may respond due to a more realistic time-dependent source than one described by the Vaidya metric, such as a pulsating star, particularly on determining the most dominant contribution to its response.

A recent development on the study of the MPD equations involves a linear perturbative approach first introduced by Chicone, Mashhoon, and Punsly (CMP) Chicone (), with an application by Mashhoon and Singh Mashhoon2 () for determining a first-order perturbation of a circular orbit around a Kerr black hole due to spin-curvature coupling. This first approach was more recently generalized by Singh Singh0 () to accommodate for higher-order contributions in powers of , where is the Møller radius Mashhoon2 (); Moller () in terms of the particle’s spin magnitude and mass , and is the radial distance from the background mass source to the particle’s location. This generalization can be applied to formally infinite order in the perturbation expansion parameter and makes no reference to any particular space-time metric or symmetries therein. A detailed application of the generalized CMP approximation to the MPD equations was just presented for the case of circular motion around a Kerr black hole. For future reference, this recent paper is now identified as “Paper I” Singh2 (). It would be very interesting to perform the same investigation as found in Paper I, but this time applied to the Vaidya metric.

The purpose of this paper is to apply the generalized CMP approximation of the MPD equations to describe circular motion around a static compact object described by the Vaidya metric, and incorporate both mass accretion from null radiation and outgoing radiation within the formalism. This paper begins with a brief review of the MPD equations and the generalized CMP approximation Mashhoon2 (); Singh0 (); Singh2 (), found in Sec. II. An introduction to the Vaidya metric Singh1 () and its application to the generalized CMP approximation is then presented in Sec. III. Following this, Sec. IV describes the main results for the case of circular motion around the central body to second order in the perturbation expansion parameter, including the “radiative corrections” of the squared mass and spin magnitudes predicted within the underlying formalism Singh0 (); Singh2 (). Afterwards, a discussion of the obtained results is given in Sec. V, followed by a brief conclusion. Consistent with Paper I, the Riemann and Ricci tensors follow the conventions of MTW MTW () with signature , and assuming geometric units of .

## Ii Mathisson-Papapetrou-Dixon (MPD) Equations and the Generalized CMP Approximation

### ii.1 MPD Equations

Given the dynamical degrees of freedom and for the spinning particle’s linear four-momentum and spin tensor, respectively, the MPD equations are

 DPμ\rm dτ = −12RμναβuνSαβ, DSαβ\rm dτ = Pαuβ−Pβuα, (1b)

where is the Riemann curvature tensor and is the four-velocity with affine parametrization . While can be chosen to satisfy to describe proper time, it is not necessary to impose this particular constraint if desired. The combined force equation (1) and torque equation (1b) infer that the particle’s four-momentum precesses around the centre-of-mass worldline, giving rise to non-trivial motion away from time-like geodesic motion.

The MPD equations presented in (1) are underdetermined, and require supplementary equations to specify the system. A commonly accepted constraint is to impose orthogonality between the particle’s linear and spin angular momenta, following Dixon’s approach Dixon1 (); Dixon2 (), such that

 SαβPβ = 0 (2)

As well, the mass and spin parameters and are identified by the constraint equations

 m2 = −PμPμ, s2 = 12SμνSμν, (3b)

which become constants of the motion Chicone () when (2) is implemented within the MPD equations. It is also well-known that the four-velocity can be expressed in terms of and within the MPD formalism Tod (), leading to

 uμ = −P⋅um2⎡⎣Pμ+12SμνRνγαβPγSαβm2+14RαβρσSαβSρσ⎤⎦, (4)

where specification of determines the parametrization constraint for . Clearly, (4) shows that the spin-curvature coupling creates a displacement of the particle’s four-velocity away from geodesic motion.

### ii.2 Generalized CMP Approximation

While a more detailed account of the generalized CMP approximation can be found in Paper I, it is useful to briefly summarize the main points of this approach to the MPD equations. This is a perturbation approach based on the assumption that

 Pμ(ε) ≡ ∞∑j=0εjPμ(j), Sμν(ε) ≡ ε∞∑j=0εjSμν(j) = ∞∑j=1εjSμν(j−1), (5b)

where and are the respective jth-order contributions of the linear momentum and spin angular momentum in , an expansion parameter associated with . In addition, the four-velocity is described as

 uμ(ε) ≡ ∞∑j=0εjuμ(j). (6)

The zeroth-order expressions in then correspond to a spinless particle in geodesic motion, while higher-order contributions are identified with spin-curvature coupling.

The main idea to the generalized CMP approximation is to substitute (5) and (6) into both the MPD equations (1) and the exact expression for according to (4), expand these equations with respect to , and solve for each order of the perturbation expansion iteratively. It follows that the jth-order expressions of the MPD equations are

 DPμ(j)\rm dτ = −12Rμναβj−1∑k=0uν(j−1−k)Sαβ(k), DSαβ(j−1)\rm dτ = 2j−1∑k=0P[α(j−1−k)uβ](k), (7b)

where implies that

 DPμ(0)\rm dτ = 0, (8)

while , corresponding to the CMP approximation Chicone (); Singh0 (); Singh2 (), is

 DPμ(1)\rm dτ = −12Rμναβuν(0)Sαβ(0), DSαβ(0)\rm dτ = 0. (9b)

In addition to (1), the supplementary spin condition equation (2) and constraint equations (3) for the squared mass and spin magnitudes need to be incorporated within this formalism. For the spin condition, it is straightforward to show that

 P(0)μSμν(j) = −j∑k=1P(k)μSμν(j−k),j≥1 (10)

for the (j+1)th-order contribution, where

 P(0)μSμν(0) = 0 (11)

for the first-order perturbation in . As for the squared mass and spin magnitude constraint equations, it is possible to identify a bare mass and bare spin according to

 m20 ≡ −P(0)μPμ(0), (12a) s20 ≡ 12S(0)μνSμν(0), (12b)

such that

 m2(ε) = m20(1+∞∑j=1εj¯m2j), (13a) s2(ε) = ε2s20(1+∞∑j=1εj¯s2j), (13b)

where

 ¯m2j = −1m20j∑k=0P(j−k)μPμ(k), (14a) ¯s2j = 1s20j∑k=0S(j−k)μνSμν(k), (14b)

are dimensionless jth-order “radiative corrections” to and , respectively, due to spin-curvature coupling. Each expression of (14) satisfies

 D¯s2j\rm dτ = D¯m2j\rm dτ = 0. (15)

Solving for the four-velocity (6) requires specifying the parametrization constraint within (4). Following Paper I, the particularly useful choice of

 P⋅u ≡ −m(ε), (16)

 uμ(ε) = ∞∑j=0εjuμ(j) = Pμ(0)m0+ε[1m0(Pμ(1)−12¯m21Pμ(0))] (17) +ε2{1m0[Pμ(2)−12¯m21Pμ(1)−12(¯m22−34¯m41)Pμ(0)]+12m30Sμν(0)RνγαβPγ(0)Sαβ(0)} +ε3{1m0[Pμ(3)−12¯m21Pμ(2)−12(¯m22−34¯m41)Pμ(1)−12(¯m23−32¯m21¯m22+58¯m61)Pμ(0)] +12m30Rνγαβ[1∑n=0Sμν(1−n)n∑k=0Pγ(n−k)Sαβ(k)−32¯m21Sμν(0)Pγ(0)Sαβ(0)]}+O(ε4),

where

 uμ(ε)uμ(ε) = −1+O(ε4), (18)

implying that is indeed the four-velocity with unit normal to third-order in . It is not necessarily true, however, that applied to all orders of corresponds to (16). Extending (18) to fourth-order in and higher requires a more general approach, where

 P⋅u ≡ ∞∑j=0(P⋅u)(j)εj (19)

and the choice for each is determined from constraint equations for each order of as required.

### ii.3 Summary of the Linear Momentum and Spin Angular Momentum Expansion Components

The approach adopted to solve for the linear momentum and spin tensor expansion components in the generalized CMP approximation is to use the tetrad formalism and work in Fermi normal co-ordinates. Full details for obtaining these expressions are shown in Paper I, but it is worthwhile to give a brief outline of the procedure. Suppose that an orthonormal tetrad frame satisfying

 η^α^β = gμνλμ^αλν^β (20)

and parallel transport describes a projection of space-time curvature described by general space-time co-ordinates onto a locally flat tangent space, denoted by . The Fermi co-ordinates are described by in the local neighbourhood about the spinning particle’s centre-of-mass worldline, while general space-time co-ordinates are denoted by . As usual, . Furthermore, the Riemann curvature tensor in the Fermi frame is then given by

 FR^α^β^γ^δ = Rμνρσλμ^αλν^βλρ^γλσ^δ, (21)

and that for ,

 Pμ(j) = λμ^αP^α(j), (22a) Sμν(j) = λμ^αλν^βS^α^β(j). (22b)

Following the approach taken in Paper I and elsewhere Singh0 (); Singh2 (), it is shown from (22) for that

 Pμ(0) = λμ^αP^α(0) = m0λμ^0, (23a) Sμν(0) = λμ^ıλν^ȷS^ı^ȷ(0), (23b)

satisfying the first-order spin condition (11), where and is a constant-valued spatial antisymmetric tensor determined from initial conditions.

For , the linear momentum is straightforwardly determined to be

 Pμ(1) = −12λμ^k∫(FR^k^0^ı^ȷS^ı^ȷ(0))\rm dτ, (24)

while

 Sμν(1) = λμ^αλν^βS^α^β(1) (25) = [14¯s21λμ^ıλν^ȷ−2m0λ[μ^0λν]^ȷP(1)^ı]S^ı^ȷ(0),

subject to

 DSμν(1)\rm dτ = 0 (26)

Contracting (24) into shows that the first-order mass shift contribution is identically

 ¯m21 = 0 (27)

However, the expression for first-order spin shift is still formally undetermined based on (25) alone, and while it is tempting to set in analogy with (27), this is not justified given that only needs to be covariantly constant according to (15), and not necessarily zero. To obtain an expression for requires the direct solving of (26), the details of which are given in Paper I and are presented in Appendix A of this paper.

Solving for the expressions for both and is straightforward, such that

 Pμ(2) = −12λμ^α∫(1m0FR^α^β^k^lP^β(1)S^k^l(0)+FR^α^0^γ^βS^γ^β(1))\rm dτ (28) ≈ −12λμ^α∫(1m0FR^α^β^k^lP^β(1)+14⟨¯s21⟩FR^α^0^k^l−2m0FR^α^0^0^lP(1)^k)S^k^l(0)\rm dτ

for the linear momentum, where

 ⟨¯s2j⟩ = 1T∫T0¯s2j(τ)\rm dτ (29)

is the time-averaged jth-order correction to the squared spin magnitude. The corresponding expression for the spin tensor is given by

 Sμν(2) = 1m0λ[μ^0λν]^ı∫S^ı^ȷ(0)FR^ȷ^0^k^lS^k^l(0)\rm dτ, (30)

the solution to

 DSμν(2)\rm dτ = 1m30P[μ(0)Sν]σ(0)RσγαβPγ(0)Sαβ(0), (31)

after substituting from (17).

### ii.4 Perturbations of the Møller Radius

As noted in Paper I, the Møller radius is closely identified with the strength of spin-curvature coupling experienced by the spinning particle. Previous studies of chaotic dynamics in the Kerr background Suzuki1 (); Suzuki2 (); Hartl1 (); Hartl2 () indicate the possibility that perturbations of may reveal the conditions where a transition from stable to chaotic motion can appear for a spinning particle in a general space-time background. The perturbation expression for the Møller radius is then formally given by

 s(ε)m(ε) = εs0m0{1+ε[12(¯s21−¯m21)] (32) +ε2[12(¯s22−¯m22)−14¯s21¯m21−18(¯s41−3¯m41)] +O(ε3)},

where the contributions due to are retained for completeness’ sake. It is of particular interest to see how (32) behaves for the Vaidya metric, which can then be compared directly with the results obtained in Paper I.

## Iii Generalized CMP Approximation in Vaidya Space-Time

With the formalism of the generalized CMP approximation presented, it is possible to now develop the framework for applications to motion in a Vaidya space-time background. The most immediate challenge is to derive the orthonormal tetrad frame for application of the formalism just outlined. It is very surprising to note that, while the Vaidya metric is much simpler in form compared to the Kerr metric used in Paper I, the relevant computations are technically much more involved, leading to much greater complexity than first anticipated. This is because the Vaidya metric is effectively time-dependent, since the mass function is no longer static, but either grows or shrinks monotonically along null rays. Ultimately, this property must be incorporated within the structure of the orthonormal tetrad.

The Vaidya metric in co-ordinates is described in general form as Singh1 (); Vaidya (); Carmeli ()

 \rm ds2 = −(1−2M(ξ)r)\rm dξ2+2α\rm dξ\rm dr (33) +r2(\rm dθ2+sin2θ\rm dϕ2),

where is a generalized null co-ordinate denoting time development and is a dimensionless parameter chosen such that

 ξ = ν,α=1 (34)

for ingoing radiation Singh1 () along the advanced null co-ordinate , while

 ξ = μ,α=−1 (35)

corresponding to outgoing radiation Carmeli () along the retarded null co-ordinate . For the Vaidya metric, the central mass function is a monotonically increasing or decreasing function of for a given choice of to satisfy the weak energy condition, but is otherwise an arbitrary function. The mass function can also be defined as

 M(ξ) = M0+ΔM(ξ), (36)

where is the static mass for a Schwarzschild black hole and . It will prove useful to express (33) in terms of co-ordinates, where is described by the tortoise co-ordinate condition Singh1 ()

 ξ = t+α[r+2M0ln(r2M0−1)]. (37)

This leads to the Vaidya metric expressed as

 \rm ds2 = −[(1−2M0r)−2ΔMr]\rm dt2 (38) +[4α(1−2M0r)−1ΔMr]\rm dt\rm dr +[(1−2M0r)−1+2(1−2M0r)−2ΔMr]\rm dr2 +r2(\rm dθ2+sin2θ\rm d% ϕ2),

which reduces to the Schwarzschild metric as .

While it is mathematically acceptable to leave unspecified, it creates computational obstacles for an exact treatment of the problem. Therefore, a simplifying assumption adopted is to let , which is well-justified on physical grounds, since the Eddington luminosity limit Heyl () imposes an upper bound mass accretion rate of

 \rm d(ΔM)\rm dt = 3×10−22(1−γ)γ(M0M⊙)≪1, (39)

where is one solar mass and is the energy release efficiency of the outgoing photon flux. This allows for a derivation of the Vaidya tetrad frame in terms of a linear perturbation about , the Schwarzchild tetrad frame for circular motion, which is presented below.

Consider the Schwarzschild orthonormal tetrad frame Mashhoon2 () for circular motion with fixed radius , such that

 λμ^0(Sch) = (EA2,0,0,Lr2sinθ) = uμ(Sch), (40a) λμ^1(Sch) = (−LrAsin(ΩKτ),Acos(ΩKτ),0, (40b) −ErAsinθsin(ΩKτ)), λμ^2(Sch) = (0,0,1r,0), (40c) λμ^3(Sch) = (LrAcos(ΩKτ),Asin(ΩKτ),0, (40d) ErAsinθcos(ΩKτ)),

where

 ΩK = √M0r3 (41)

is the Keplerian frequency of the orbit,

 N = √1−3M0r,A = √1−2M0r, (42)

and the energy and orbital angular momentum for the orbit are

 E = A2N,L = r2ΩKN. (43)

The boundary conditions are determined such that at . Furthermore, given that the Vaidya metric is spherically symmetric, the Cartesian axis centred on the black hole is oriented such that orbital motion is confined to the plane defined by with respect to an assigned -axis.

The next step is to derive the Vaidya orthonormal tetrad in the form

 λμ^α ≈ λμ^α(Sch)+Δλμ^α, (44)

where is the linear perturbation proportional to . While an exact treatment within this perturbation approach is given in Appendices B and C, the outcome is considerably more complicated than for the exact orthonormal tetrad in the Kerr background Mashhoon2 (). Therefore, another simplifying assumption is introduced, in the form of a series expansion with respect to inverse powers of , since any deviations away from the Schwarzschild contribution will only be potentially identifiable when the spinning particle approaches the nearest (photon) orbit of , corresponding to . In addition, an expression for needs to be chosen in terms of that is consistent with both the properties of the metric (38) and the mass accretion rate upper bound (39). This leads to the choice of

 ΔM(τ) ≈ αA∣∣∣\rm d(ΔM)\rm dξ∣∣∣τ, (45)

where the prefactor of in (45) accounts for the direction of radiation flow, and . With these further assumptions incorporated, it can be shown that the Vaidya orthonormal tetrad frame components for a particle in orbit near the event horizon at are

 λ0^0 ≈ 1N+α2N7∣∣∣\rm d(ΔM)\rm dξ∣∣∣(rΩK)C(r,ΩKτ), (46a) λ1^0 ≈ −2αN3∣∣∣\rm d(ΔM)\rm dξ∣∣∣sin2(ΩKτ), (46b) λ2^0 = 0, (46c) λ3^0 = ΩKλ0^0, (46d)

for ,

 λ0^1 ≈ −rΩK[1NA+α2N7∣∣∣\rm d(ΔM)\rm dξ∣∣∣C(r,ΩKτ)] (47a) ×sin(ΩKτ), λ1^1 ≈ Acos(ΩKτ)+2αN3∣∣∣\rm d(ΔM)\rm dξ∣∣∣sin3(ΩKτ), (47b) λ2^1 = 0, (47c) λ3^1 = −[ArN+αΩK2N7∣∣∣\rm d(ΔM)\rm dξ∣∣∣(rΩK)C(r,ΩKτ)] (47d) ×sin(ΩKτ),

for ,

 λ0^2 = 0,λ1^2 = 0, (48a) λ2^2 = 1r,λ3^2 = 0, (48b)

for , and

 λ0^3 ≈ rΩK[1NA+α2N7∣∣∣\rm d(ΔM)\rm dξ∣∣∣C(r,ΩKτ)] (49a) ×cos(ΩKτ), λ1^3 ≈ Asin(ΩKτ) (49b) −2αN3∣∣∣\rm d(ΔM)\rm dξ∣∣∣sin2(ΩKτ)cos(ΩKτ), λ2^3 = 0, (49c) λ3^3 = (49d) ×cos(ΩKτ),

for , where

 C(r,ΩKτ) ≡ 2sin(2ΩKτ) +NrΩK[(1−2rΩK)sin(2ΩKτ)−2ΩKτ].

Given (46)–(49), it is now possible to obtain the Riemann tensor components in the Fermi frame. While the exact expressions for are found in Appendix D, for the special case of and considered in this paper, the dominant nonzero components are

 FR^0^1^0^1 ≈ −Ω2KN2[2A2+r2Ω2K+3αN6∣∣∣\rm d(ΔM)\rm dξ∣∣∣(r3Ω3K)C(r,ΩKτ)]cos2(ΩKτ) = −FR^2^3^2^3, (51a) FR^0^1^0^3 ≈ −Ω2KN2[2A2+r2Ω2K+3αN6∣∣∣\rm d(ΔM)\rm dξ∣∣∣(r3Ω3K)C(r,ΩKτ)]sin(ΩKτ)cos(ΩKτ) (51b) = −FR^1^2^2^3, FR^0^1^1^3 ≈ 3Ω2KN2[A(rΩK)+αN6∣∣∣\rm d(ΔM)\rm dξ∣∣∣(r3Ω3K)C(r,ΩKτ)]cos(ΩKτ) = −FR^0^2^2^3, (51c) FR^0^2^0^2 ≈ Ω2KN2[1+3αN6∣∣∣\rm d(ΔM)\rm dξ∣∣∣(r3Ω3K)C(r,ΩKτ)] = −FR^1^3^1^3, (51d) FR^0^2^1^2 ≈ −3Ω2KN2[A(rΩK)+αN6∣∣∣\rm d(ΔM)\rm dξ∣∣∣(r3Ω3K)C(r,ΩKτ)]sin(ΩKτ) = −FR^0^3^1^3, FR^0^3^0^3 ≈ −Ω2KN2[2A2+r2Ω2K+3αN6∣∣∣\rm d(ΔM)\rm dξ∣∣∣(r3Ω3K)C(r,ΩKτ)]sin2(ΩKτ) (51f) = −