## arXiv:0809.1191v2 [gr-qc] 28 Jan 2009

### The Magnetosphere of Oscillating Neutron Stars in General Relativity

### Ernazar B. Abdikamalov

^{1,}

^{2⋆}

### , Bobomurat J. Ahmedov

^{2,}

^{3}

### and John C. Miller

^{1,}

^{4}

1SISSA, International School for Advanced Studies, and INFN–Trieste, Via Beirut 2-4, 34014 Trieste, Italy

2Institute of Nuclear Physics, Ulughbek, Tashkent 100214, Uzbekistan

3Ulugh Beg Astronomical Institute, Astronomicheskaya 33, Tashkent 100052, Uzbekistan

4Department of Physics (Astrophysics), University of Oxford, Keble Road, Oxford OX1 3RH, UK

Accepted<date>. Received<date>; in original form<date>

ABSTRACT

Just as a rotating magnetised neutron star has material pulled away from its surface to populate a magnetosphere, a similar process can occur as a result of neutron- star pulsations rather than rotation. This is of interest in connection with the overall study of neutron star oscillation modes but with a particular focus on the situation for magnetars. Following a previous Newtonian analysis of the production of a force-free magnetosphere in this way Timokhin et al. (2000), we present here a corresponding general-relativistic analysis. We give a derivation of the general relativistic Maxwell equations for small-amplitude arbitrary oscillations of a non-rotating neutron star with a generic magnetic field and show that these can be solved analytically under the assumption of low current density in the magnetosphere. We apply our formalism to toroidal oscillations of a neutron star with a dipole magnetic field and find that the low current density approximation is valid for at least half of the oscillation modes, similarly to the Newtonian case. Using an improved formula for the determination of the last closed field line, we calculate the energy losses resulting from toroidal stellar oscillations for all of the modes for which the size of the polar cap is small. We find that general relativistic effects lead to shrinking of the size of the polar cap and an increase in the energy density of the outflowing plasma. These effects act in opposite directions but the net result is that the energy loss from the neutron star is significantly smaller than suggested by the Newtonian treatment.

Key words: stars: magnetic field – stars: neutron – stars: oscillations – pulsars:

general

1 INTRODUCTION

Study of the internal structure of neutron stars (NSs) is of fundamental importance for subatomic physics since these objects provide a laboratory for studying the properties of high-density matter under very extreme conditions. In particular, there is the intriguing possibility of using NS oscillation modes as a probe for constraining models of the equation of state of matter at supranuclear densities. It was suggested long ago that if a NS is oscillating, then traces of this might be revealed in the radiation which it emits (Pacini & Ruderman 1974; Tsygan 1975; Boriakoff 1976; Bisnovatyi-Kogan 1995;

Ding & Cheng 1997; Duncan 1998). Recently, a lot of interest has been focussed on oscillations of magnetized NSs because of the discovery of gamma-ray flare activity in Soft Gamma-Ray Repeaters (SGRs) which are thought to be the very highly magnetised NSs known as magnetars (for recent review on the SGRs see Woods & Thompson 2006; Watts & Strohmayer 2007). The giant flares in these objects are thought to be powered by global reconfigurations of the magnetic field and it has been suggested that the giant flares might trigger starquakes and excite global seismic pulsations of the magnetar crust

⋆ E-mail:abdik@sissa.it

(Thompson & Duncan 1995, 2001; Schwartz et al. 2005; Duncan 1998). Indeed, analyses of the observations of giant flares have revealed that the decaying part of the spectrum exhibits a number of quasi-periodic oscillations (QPOs) with frequencies in the range from a few tens of Hz up to a few hundred Hz (Israel et al. 2005; Strohmayer & Watts 2006; Watts & Strohmayer 2006) and there has been a considerable amount of theoretical effort attempting to identify these with crustal oscillation modes (Glampedakis et al. 2006; Samuelsson & Andersson 2007; Levin 2007; Sotani et al. 2007a,b). While there is substantial evidence that the observed SGR QPOs are caused by neutron star pulsations, there is a great deal of uncertainty about how stellar surface motion gets translated into the observed features of the X-ray radiation (Strohmayer 2008; Strohmayer & Watts 2006; Timokhin et al. 2007). To make progress with this, it is necessary to develop a better understanding of the processes occurring in the magnetospheres of oscillating neutron stars.

Standard pulsars typically have magnetic fields of around 10^{12} G while magnetars may have fields of up to 10^{14}−10^{15}
G near to the surface. Rotation of a magnetized star generates an electric field:

E^{rot}∼ΩR

c B , (1)

where B is the magnetic field strength, c is the speed of light and Ω is the angular velocity of the star with radius R.

Depending on the rotation velocity and the magnetic field strength, the electric field may be as strong as 10^{10}V cm^{−1}and it
has a longitudinal component (parallel toB) which can be able to pull charged particles away from the stellar surface, if the
work function is sufficiently small, and accelerate them up to ultra-relativistic velocities. This result led Goldreich & Julian
(1969) to suggest that a rotating NS with a sufficiently strong magnetic field should be surrounded by a magnetosphere
filled with charge-separated plasma which screens the accelerating electric field and thus hinders further outflow of charged
particles from the stellar surface. Even if the binding energy of the charged particles is sufficiently high to prevent them being
pulled out by the electric field, the NS should nevertheless be surrounded by charged particles produced by plasma generation
processes (Sturrock 1971; Ruderman & Sutherland 1975), which again screen the longitudinal component of the electric field.

These considerations led to the development of a model for pulsar magnetospheres which is frequently called the “standard model” (an in depth discussion and review of this can be found in, e.g., Michel 1991; Beskin et al. 1993; Beskin 2005).

Timokhin, Bisnovatyi-Kogan & Spruit (2000) (referred to as TBS from here on) showed that an oscillating magnetized NS should also have a magnetosphere filled with charge-separated plasma, even if it is not rotating, since the vacuum electric field induced by the oscillations would have a large radial component which can be of the same order as rotationally-induced electric fields. One can show this quantitatively by means of the following simple arguments. To order of magnitude, the radial component of the vacuum electric field generated by the stellar oscillations is given by

E^{osc}∼ ωξ

c B , (2)

whereωis the oscillation frequency andξis the displacement amplitude. Using this together with Eq.(1), it follows immediately that the electric field produced by oscillations will be stronger than the rotationally induced one for sufficiently slowly-rotating neutron stars, having

Ω. ωξ

R . (3)

For stellar oscillations with ξ/R∼0.001 andω ∼1 kHz, the threshold is Ω∼1 Hz. Within this context, TBS developed a formalism extending the basic aspects of the standard pulsar model to the situation for a non-rotating magnetized NS undergoing arbitrary oscillations. This formalism was based on the assumption of low current densities in the magnetosphere, signifying that the influence of currents outside the NS on electromagnetic processes occurring in the magnetosphere is negligibly small compared to that of currents in the stellar interior. This assumption leads to a great simplification of the Maxwell equations, which then can be solved analytically. As an application of the formalism, TBS considered toroidal oscillations of a NS with a dipole magnetic field, and obtained analytic expressions for the electromagnetic field and charge density in the magnetosphere. (Toroidal oscillations are thought to be particularly relevant for magnetar QPO phenomena.) They found that the low current density approximation (LCDA) is valid for at least half of all toroidal oscillation modes and analyzed the energy losses due to plasma outflow caused by these modes for cases where the size of the polar cap (the region on the stellar surface that is crossed by open magnetic field lines) is small, finding that the energy losses are strongly affected by the magnetospheric plasma. For oscillation amplitudes larger than a certain critical value, they found that energy losses due to plasma outflow were larger than those due to the emission of the electromagnetic waves (assuming in that case that the star was surrounded by vacuum). Recently, Timokhin (2007) considered spheroidal oscillations of a NS with a dipole magnetic field, using the TBS formalism, and found that the LCDA again holds for at least half of these modes. Discussion in Timokhin (2007) also provided some useful insights into the role of rotation for the magnetospheric structure of oscillating NSs.

The TBS model was a very important contribution and, to the best of our knowledge, remains the only model for the magnetosphere of oscillating NSs available in the literature. However, it should be pointed out that it does not include several ingredients that a fully consistent and realistic model ought to include. Most importantly, it does not treat the magnetospheric

currents in a fully consistent way: although it gives a consistent solution for around half of the oscillation modes, the remaining solutions turn out to be unphysical and, as TBS pointed out, this is a symptom of the LCDA failing there. Also, rotation and the effects of general relativity can be very relevant; in particular, several authors have stressed that using a Newtonian approach may not give very good results for the structure of NS magnetospheres (see, e.g., Beskin 1990; Muslimov & Tsygan 1992; Mofiz & Ahmedov 2000; Morozova et al. 2008). However, a more realistic model would naturally be more complicated than the TBS one whose relative simplicity can be seen as a positive advantage when using it as the basis for further applications.

The aim of the present paper is to give a general relativistic reworking of the TBS model so as to investigate the effects of the changes with respect to the Newtonian treatment. We derive the general relativistic Maxwell equations for arbitrary small-amplitude oscillations of a non-rotating spherical NS with a generic magnetic field configuration and show that they can be solved analytically within the LCDA as in Newtonian theory. We then apply this solution to the case of toroidal oscillations of a NS with a dipole magnetic field and find that the LCDA is again valid for at least half of all toroidal oscillation modes, as in Newtonian theory. Using an improved formula for the determination of the last closed field line, we calculate the energy losses resulting from these oscillations forallof the modes for which the size of the polar cap is small and discuss the influence of GR effects on the energy losses.

The paper is organized as follows. In Section 2 we introduce some definitions and derive the quasi-stationary Maxwell equations in Schwarzschild spacetime as well as the boundary conditions for the electromagnetic fields at the stellar surface.

In Section 3 we sketch our method for analytically solving the Maxwell equations for arbitrary NS oscillations with a generic magnetic field configuration. In Section 4 we apply our formalism to the case of purely toroidal oscillations of a NS with a dipole magnetic field and also discuss the validity of the LCDA and the role of GR effects. In Section 5 we calculate the energy losses due to plasma outflow caused by the toroidal oscillations. Some detailed technical calculations related to the discussion in the main part of the paper are presented in Appendices A-C.

We use units for which c = 1, a space-like signature (−,+,+,+) and a spherical coordinate system (t, r, θ, φ). Greek indices are taken to run from 0 to 3 while Latin indices run from 1 to 3 and we adopt the standard convention for summation over repeated indices. We indicate four-vectors with bold symbols (e.g.u) and three-vectors with an arrow (e.g.~u).

2 GENERAL FORMALISM

2.1 Quasi-stationary Maxwell equations in Schwarzschild spacetime

The study of electromagnetic processes related to stellar oscillations in the vicinity of NSs should, in principle, use the coupled
system of Einstein-Maxwell equations. However, such an approach would be overly complicated for our study here, as it is for
many other astrophysical problems. Here we simplify the problem by neglecting the contributions of the electromagnetic fields,
the NS rotation and the NS oscillations to the spacetime metric and the structure of the NS^{1}, noting that this is expected
to be a good approximation for small-amplitude oscillations. Indeed, for a star with average mass-energy density ¯ρ, massM
and radiusR, the maximum fractional change in the spacetime metric produced by the magnetic field is typically of the same
order as the ratio between the energy density in the surface magnetic field and average mass-energy density of the NS, i.e.,

B^{2}

8πρc¯^{2} ≃10^{−7}

„ B
10^{15} G

«2„ 1.4M⊙

M

« „ R 10 km

«3

. (4)

The corresponding fractional change in the metric due to rotation is of order 0.1

„ Ω ΩK

«2

= 10^{−7}

„ Ω 1 Hz

«2„ 1 kHz

ΩK

«2

(5) where ΩKis the Keplerian angular velocity at the surface of the NS. Moreover, in the case of magnetars, which we consider in our study, the oscillations are thought to be triggered by the global reconfiguration of the magnetic field. Due to this reason, the corrections due to the oscillations should not exceed the contribution due to the magnetic field itself given by estimate (4). Therefore, we can safely work in the background spacetime of a static spherical star, whose line element in a spherical coordinate system (t, r, θ, φ) is given by

ds^{2}=g00(r)dt^{2}+g11(r)dr^{2}+r^{2}dθ^{2}+r^{2}sin^{2}θdφ^{2} , (6)

while the geometry of the spacetime external to the star (i.e. forr>R) is given by the Schwarzschild solution:

ds^{2}=−N^{2}dt^{2}+N^{−2}dr^{2}+r^{2}dθ^{2}+r^{2}sin^{2}θdφ^{2} , (7)

1 Several authors have, in fact, studied the equilibrium configurations of magnetars by solving the Einstein-Maxwell equations in full general relativity (Bocquet et al. 1995; Bonazzola et al. 1996; Cardall et al. 2001) or by using perturbative techniques (Colaiuda et al.

2008; Haskell et al. 2008).

whereN≡(1−2M/r)^{1/2} andM is the total mass of the star. For the part of the spacetime inside the star, we represent the
metric in terms of functions Λ and Φ as

g00=−e^{2Φ(r)} , g11=e^{2Λ(r)}=

„

1−2m(r) r

«−1

, (8)

wherem(r) = 4πRr

0 r^{′2}ρ(r^{′})dr^{′}is the volume integral of the total energy densityρ(r) over the spatial coordinates. The form of
these functions is given by solution of the standard TOV equations for spherical relativistic stars (see, e.g., Shapiro & Teukolsky
1983) and they are matched continuously to the external Schwarzschild spacetime through the relations

g00(r=R) =N_{R}^{2} , g11(r=R) =N_{R}^{−2} , (9)

whereN_{R} ≡(1−2M/R)^{1/2}. Within the external part of the spacetime, we select a family of static observers with four-velocity
components given by

(u^{α})_{obs}≡N^{−1} 1,0,0,0

!

. (10)

and associated orthonormal frames having tetrad four vectors {eµˆ}= (e_{ˆ}

0,e_{ˆ}_{r},e_{ˆ}

θ,e_{ˆ}

φ) and 1-forms{ω^{ˆ}^{µ}}= (ω^{ˆ}^{0},ω^{r}^{ˆ},ω^{θ}^{ˆ},ω^{φ}^{ˆ}),
which will become useful when determining the “physical” components of the electromagnetic fields. The components of the
vectors are given by equations (6)-(9) of Rezzolla & Ahmedov (2004) (hereafter Paper I).

The general relativistic Maxwell equations have the following form (Landau & Lifshitz 1987)

3F_{[αβ,γ]}=Fαβ,γ+Fγα,β+Fβγ,α= 0, (11)

F^{αβ}_{;β}= 4πJ^{α}, (12)

whereF^{αβ} is the electromagnetic field tensor andJ is the electric-charge 4-current. We consider the region close to the star
(the near zone), at distances from the NS much smaller than the wavelengthλ= 2πc/ω. In the near zone the electromagnetic
fields are quasi-stationary, therefore we neglect the displacement current term in the Maxwell equations. Once expressed in
terms of the physical components of the electric and magnetic fields, equations (11) and (12) become (see Section 2 of Paper
I for details of the derivation)

sinθ∂r

“
r^{2}B^{ˆ}^{r}”

+N^{−1}r∂θ

“

sinθB^{ˆ}^{θ}”

+N^{−1}r∂φB^{φ}^{ˆ}= 0, (13)

(rsinθ)∂B^{ˆ}^{r}

∂t =Nh

∂φE^{θ}^{ˆ}−∂θ

“sinθE^{φ}^{ˆ}”i

, (14)

`N^{−1}rsinθ´∂B^{θ}^{ˆ}

∂t =−∂φE^{ˆ}^{r}+ sinθ∂r

“rN E^{φ}^{ˆ}”

, (15)

`N^{−1}r´∂B^{φ}^{ˆ}

∂t =−∂r

“rN E^{θ}^{ˆ}”

+∂θE^{r}^{ˆ}, (16)

Nsinθ∂r

“r^{2}E^{r}^{ˆ}”
+r∂θ

“sinθE^{θ}^{ˆ}”

+r∂φE^{φ}^{ˆ}= 4πρer^{2}sinθ , (17)

h∂θ

“sinθB^{φ}^{ˆ}”

−∂φB^{θ}^{ˆ}i

= 4πrsinθJ^{ˆ}^{r} , (18)

∂φB^{ˆ}^{r}−sinθ∂r

“rN B^{φ}^{ˆ}”

= 4πrsinθJ^{θ}^{ˆ}, (19)

∂r

“
N rB^{θ}^{ˆ}”

−∂θB^{r}^{ˆ}= 4πrJ^{φ}^{ˆ} , (20)

whereρe is the proper charge density. We further assume that the force-free condition,

E~_{SC}·B~= 0 , (21)

is fulfilled everywhere in the magnetosphere, implying that the magnetosphere of the NS is populated with charged particles
that cancel the longitudinal component of the electric field. The charge densityρ_{SC}responsible for the electric field E~_{SC} (cf.

equation 17) is the characteristic charge density of the force-free magnetosphere; this is appropriate for describing the charge
density in the inner parts of the NS magnetosphere. We will refer toE~_{SC} as the space-charge (SC) electric field, while toρ_{SC}
as the SC charge density.

Finally, we introduce the perturbation of the NS crust in terms of its four-velocity, with the components being given by
w^{α}≡e^{−Φ} 1,dx^{i}

dt

!

=e^{−Φ} 1, e^{−Λ}δv^{ˆ}^{r},δv^{θ}^{ˆ}
r , δv^{φ}^{ˆ}

rsinθ

!

, (22)

whereδv^{i}=dx^{i}/dtis the relative oscillation three-velocity of the conducting stellar surface with respect to the unperturbed
state of the star.

2.2 Boundary conditions at the surface of star

We now begin our study of the internal electromagnetic field induced by the stellar oscillations. We assume here that the material in the crust can be treated as a perfect conductor and the induced electric field then depends on the magnetic field and the pulsational velocity field according to the following relations (see Paper I for details of the derivation):

E_{in}^{r}^{ˆ} =−e^{−Φ}h

δv^{θ}^{ˆ}B^{φ}^{ˆ}−δv^{φ}^{ˆ}B^{θ}^{ˆ}i

, (23)

E_{in}^{θ}^{ˆ} =−e^{−Φ}h

δv^{φ}^{ˆ}B^{ˆ}^{r}−δv^{ˆ}^{r}B^{φ}^{ˆ}i

, (24)

E_{in}^{φ}^{ˆ} =−e^{−Φ}h

δv^{r}^{ˆ}B^{ˆ}^{θ}−δv^{θ}^{ˆ}B^{r}^{ˆ}i

. (25)

Boundary conditions for the magnetic field at the stellar surface (r = R) can be obtained from the requirement of continuity for the radial component, while leaving the tangential components free to be discontinuous because of surface currents:

B_{ex}^{r}^{ˆ} |^{r=R}=B_{in}^{ˆ}^{r} |^{r=R}, (26)

B_{ex}^{θ}^{ˆ} |^{r=R}=B_{in}^{θ}^{ˆ}|^{r=R}+ 4πi^{φ}^{ˆ} , (27)

B_{ex}^{φ}^{ˆ}|r=R=B_{in}^{φ}^{ˆ}|r=R−4πi^{θ}^{ˆ}, (28)

where i^{ˆ}^{i} is the surface current density. Boundary conditions for the electric field at the stellar surface are obtained from
requirement of continuity of the tangential components, leavingE^{r}^{ˆ}to have a discontinuity proportional to the surface charge
density Σs:

E_{ex}^{r}^{ˆ} |^{r=R}=E_{in}^{r}^{ˆ}|^{r=R}+ 4πΣs=−N_{R}^{−1}h

δv^{θ}^{ˆ}B^{φ}^{ˆ}−δv^{φ}^{ˆ}B^{θ}^{ˆ}i

|^{r=R}+ 4πΣs , (29)

E_{ex}^{θ}^{ˆ} |^{r=R}=E_{in}^{θ}^{ˆ}|^{r=R}=−N_{R}^{−1}h

δv^{φ}^{ˆ}B^{r}^{ˆ}−δv^{r}^{ˆ}B^{φ}^{ˆ}i

|^{r=R}, (30)

E_{ex}^{φ}^{ˆ}|^{r=R}=E_{in}^{φ}^{ˆ}|^{r=R}=−N_{R}^{−1}h

δv^{ˆ}^{r}B^{θ}^{ˆ}−δv^{θ}^{ˆ}B^{r}^{ˆ}i

|^{r=R} , (31)

where Σs is the surface charge density.

2.3 The low current density approximation

The low current density approximation was introduced by TBS, and in the present section we present a brief introduction to it for completeness. Close to the NS surface, the current flows along the magnetic field lines, and so in the inner parts of the magnetosphere it can be expressed as

J~=α(r, θ, φ)·B ,~ (32)

whereαis a scalar function. The system of equations (13)–(20), (21) and (32) forms a complete set but is overly complicated for solving in the general case. However, within the LCDA these equations can, as we show below, be solved analytically for arbitrary oscillations of a NS with a generic magnetic field configuration.

The LCDA scheme is based on the assumption that the perturbation of the magnetic field induced by currents flowing in the NS interior is much larger than that due to currents in the magnetosphere, which are neglected to first order in the oscillation parameter ¯ξ≡ξ/R:

4π

c J~≪ ∇ ×B ,~ (33)

and

∇ ×B~^{(1)}= 0, (34)

whereB~^{(1)} is the first order term of the expansion in ¯ξ. This also implies that the current density satisfies the condition

J≪1 r

„ B(0)

ξ R

«

c≈ρ_{SC}(R)c“ c
ωr

”

, (35)

whereρ_{SC}(R) is the SC density near to the surface of the star. Here we have used the relationρ_{SC}(R)≃B^{(0)}η/cR, whereη
is the velocity amplitude of the oscillation andωis its frequency.

In regions of complete charge separation, the maximum current density is given byρ_{SC}c. Since the absolute value ofρ_{SC}
decreases with increasingrand becauser≪c/win the near zone, condition (35) is satisfied in the magnetosphere if there is
complete charge separation there. Since the current in the magnetosphere flows along magnetic field lines, its magnitude does
not change and so condition (35) is also satisfied along magnetic field lines in non-charge-separated regions as long as they
have crossed regions with complete charge separation.

In the following, we solve the Maxwell equations assuming that condition (35) is satisfied throughout the whole near zone.

As discussed above, a regular solution of the system of equations (13)-(20), (21) and (32) should exist for arbitrary oscillations and arbitrary configurations of the NS magnetic field and so, as shown by TBS, if a solution has an unphysical behaviour, this would imply that the LCDA fails for this oscillation and that the accelerating electric field cannot be screened only by a stationary configuration of the charged-separated plasma. In some regions of the magnetosphere, the current density could be as high as

J≃ρ_{SC}c“ c
ωr

” . (36)

For a more detailed discussion of the LCDA and its validity, we refer the reader to Sections 2.3 and 3.2.1 of TBS.

3 THE LCDA SOLUTION

3.1 The electromagnetic field in the magnetosphere

We now begin our solution of the Maxwell equations, assuming that the LCDA condition (35) is satisfied everywhere in the magnetosphere. Within the LCDA, equations (18)-(20) for the magnetic field in the magnetosphere take the form

∂θ

“

sinθB^{φ}^{ˆ}”

−∂φB^{θ}^{ˆ}= 0, (37)

∂φB^{ˆ}^{r}−sinθ∂r

“
rN B^{φ}^{ˆ}”

= 0, (38)

∂r

“N rB^{θ}^{ˆ}”

−∂θB^{r}^{ˆ}= 0. (39)

As demonstrated in Paper I, the components of the magnetic field B^{r}^{ˆ}, B^{θ}^{ˆ} and B^{φ}^{ˆ} can be expressed in terms of a scalar
functionS in the following way:

B^{r}^{ˆ}=− 1

r^{2}sin^{2}θ[sinθ∂θ(sinθ∂θS) +∂φ∂φS] , (40)

B^{θ}^{ˆ}= N

r∂θ∂rS , (41)

B^{φ}^{ˆ}= N

rsinθ∂φ∂rS . (42)

Substituting these expressions into the Maxwell equations (14)–(16), we obtain a system of equations for the electric field components which has the following general solution

E_{SC}^{r}^{ˆ} =−∂r(Ψ_{SC}), (43)

E_{SC}^{θ}^{ˆ} =− 1

N rsinθ∂t∂φS− 1

N r∂θ(Ψ_{SC}), (44)

E_{SC}^{φ}^{ˆ} = 1

N r∂t∂θS− 1

N rsinθ∂φ(Ψ_{SC}) , (45)

where Ψ_{SC}is an arbitrary scalar function. The terms proportional to the gradient of Ψ_{SC}are responsible for the contribution
of the charged particles in the magnetosphere. The vacuum part of the electric field is given by the derivatives of the scalar
functionS. Substituting (43)-(45) into equation (17), we get an expression for the SC charge density in terms of Ψ_{SC}:
ρ_{SC}=− 1

4πr^{2}

» N ∂r`

r^{2}∂rΨSC´
+ 1

N△^{Ω}ΨSC

–

, (46)

where△^{Ω} is the angular part of the Laplacian:

△^{Ω}= 1

sinθ∂θ(sinθ∂θ) + 1

sin^{2}θ∂φφ. (47)

3.2 The equation for Ψ_{SC}

Substituting expressions (40)–(42) and (43)–(45) for the components of the electric and magnetic fields into the force-free
condition (21), we get the following equation for Ψ_{SC}

1

sin^{2}θ[sinθ∂θ(sinθ∂θS) +∂φ∂φS]∂r(Ψ_{SC})− 1

sinθ[∂φ∂tS∂θ∂rS−∂θ∂tS∂φ∂rS]

−∂θ∂rS∂θ(Ψ_{SC})− 1

sin^{2}θ∂φ∂rS∂φ(Ψ_{SC}) = 0. (48)

If the amplitude of the NS oscillations is suitably small ( ¯ξ ≪ 1), the function S can be series expanded in terms of the dimensionless perturbation parameter ¯ξand can be approximated by the sum of the two lowest order terms

S(t, r, θ, φ) =S0(r, θ, φ) +δS(t, r, θ, φ). (49)

Here the first term S0 corresponds to the unperturbed static magnetic field of the NS, while δSis the first order correction
to it. At this level of approximation, equation (48) for Ψ_{SC}takes the form

1

sin^{2}θ[sinθ∂θ(sinθ∂θS0) +∂φ∂φS0]∂r(Ψ_{SC})− 1

sinθ[∂φ∂t(δS)∂θ∂rS0−∂θ∂t(δS)∂φ∂rS0]

−∂θ∂rS0∂θ(Ψ_{SC})− 1

sin^{2}θ∂φ∂rS0∂φ(Ψ_{SC}) = 0. (50)
Next we expandS in terms of the spherical harmonics:

S=

∞

X

ℓ=0 ℓ

X

m=−ℓ

Sℓm(t, r)Yℓm(θ, φ). (51)

where the functionsSℓm are given in terms of Legendre functions of the second kindQℓby (Rezzolla et al. 2001)
Sℓm(t, r) =−r^{2}

M^{2}
d
dr

» r

„ 1−2M

r

« d drQℓ

“1− r M

”–

sℓm(t). (52)

Note that all of the time dependence in (52) is contained in the integration constantssℓm(t) which, as we will see later, are determined by the boundary conditions at the surface of the star. We now series expand the coefficientsSℓm(t, r) andsℓm(t) in terms of ¯ξ

Sℓm(t, r) =S0ℓm(r) +δSℓm(t, r), sℓm(t) =s0ℓm+δsℓm(t), (53)

where all of the time dependence is now confined within the coefficients δSℓm(r, t) and δsℓm(t), while the coefficients S0ℓm

ands0ℓm are responsible for the unperturbed static magnetic field of the star. Using these results, we can also expressS and δS in terms of a series inYℓm(θ, φ) in the following way

S0=

∞

X

ℓ=0 ℓ

X

m=−ℓ

S0ℓm(r)Yℓm(θ, φ), (54)

δS=

∞

X

ℓ=0 ℓ

X

m=−ℓ

δSℓm(t, r)Yℓm(θ, φ). (55)

The variablesrandtin the functionsSℓm(t, r) andSℓm(t, r) can be separated using relation (52):

S0ℓm(r) =−r^{2}
M^{2}

d dr

» r

„ 1−2M

r

« d drQℓ

“1− r M

”–

s0ℓm , (56)

Sℓm(t, r) =−r^{2}
M^{2}

d dr

» r

„ 1−2M

r

« d drQℓ

“1− r M

”–

δsℓm(t). (57)

3.3 The boundary condition forΨ_{SC}

We now derive a boundary condition for Ψ_{SC} at the stellar surface using the behaviour of the electric and magnetic fields in
that region. Following TBS, we assume that near to the stellar surface the interior magnetic field has the same behaviour as

the exterior one:

B^{r}^{ˆ}=− C1

r^{2}sin^{2}θ[sinθ∂θ(sinθ∂θS) +∂φφS] , (58)

B^{θ}^{ˆ}=C1

e^{−Λ}

r ∂θ∂rS , (59)

B^{φ}^{ˆ}=C1

e^{−Λ}

rsinθ∂φ∂rS . (60)

Using the continuity condition for the normal component of the magnetic fieldˆ
B^{r}^{ˆ}˜

= 0 at the stellar surface (Pons & Geppert
2007) together with the condition e^{−Λ}|^{r=R} ≡ NR, one finds that the integration constantC1 is equal to one. The interior
electric field components can then be obtained by substituting (58) – (60) (withC1= 1) into (23) – (25):

Ein^{r}^{ˆ} =−e^{−(Φ+Λ)}
rsinθ

n

δv^{θ}^{ˆ}∂φ∂rS−sinθδv^{φ}^{ˆ}∂θ∂rSo

, (61)

Ein^{θ}^{ˆ} =e^{−(Φ+Λ)}
rsinθ

(

δv^{r}^{ˆ}∂φ∂rS+δv^{φ}^{ˆ}e^{Λ}

rsinθ [sinθ∂θ(sinθ∂θS) +∂φ∂φS]

)

, (62)

E_{in}^{φ}^{ˆ} =−e^{−(Φ+Λ)}
r

(

δv^{r}^{ˆ}∂θ∂rS+ δv^{θ}^{ˆ}e^{Λ}

rsin^{2}θ[sinθ∂θ(sinθ∂θS) +∂φ∂φS]

)

. (63)

The continuity condition for theθcomponent of the electric field across the stellar surface (30) gives a boundary condition
for∂θΨ_{SC}|^{r=R}:

Ψ_{SC,θ}|r=R=−
( δv^{φ}^{ˆ}

Rsin^{2}θ[sinθ∂θ(sinθ∂θS) +∂φ∂φS] +N δv^{r}^{ˆ}

sinθ∂φ∂rS+ 1 sinθ∂t∂φS

)

|r=R , (64)

while the continuity condition forE^{φ}^{ˆ}(31) gives a boundary condition for∂φΨ_{SC}|r=R:
Ψ_{SC,φ}|^{r=R}=

( δv^{θ}^{ˆ}

Rsinθ[sinθ∂θ(sinθ∂θS) +∂φ∂φS] +N δv^{r}^{ˆ}sinθ∂θ∂rS+ sinθ∂t∂θS
)

|^{r=R}. (65)

Integration of equation (64) or equation (65) overθ or φrespectively, gives a boundary condition for Ψ_{SC}. We will use the
result of integrating equation (64) overθ. Assuming that the perturbation depends on timetase^{−iωt}, we obtain the following
condition, correct to first order in ¯ξ,

Ψ_{SC}|r=R=−
Z (

δv^{φ}^{ˆ}

Rsin^{2}θ[sinθ∂θ(sinθ∂θS0) +∂φ∂φS0] +N δv^{ˆ}^{r}

sinθ ∂φ∂rS0+ 1

sinθ∂t∂φ(δS) )

dθ|r=R+e^{iωt}F(φ), (66)
whereF(φ) is a function only ofφwhich we will determine below.

The components of the stellar-oscillation velocity field are continuously differentiable functions ofr, θandφ. The boundary
conditions for the electric field (30)-(31) imply that the tangential components of the electric fieldE~_{SC} must be finite. The
vacuum terms on the right-hand side of (44)-(45) and the terms on both sides of equation (45) are also finite. Consequently,
the term

−∂φ(Ψ_{SC})

sinθ |^{r=R} (67)

should also be finite. Hence we obtain that∂φ(Ψ_{SC})|^{θ=0,π;r=R} = 0 and so the functionF(φ) in the expression for boundary
condition (66) must satisfy the condition (Ψ_{SC})|^{θ=0,π;r=R}=C e^{−iω t}, whereCis a constant. Using gauge invariance, we choose

Ψ_{SC}|^{θ=0;r=R}= 0, (68)

and from this and equation (66), we obtain our expression for the boundary condition for Ψ_{SC} at the stellar surface:

Ψ_{SC}|^{r=R}=−
Z θ

0

( δv^{φ}^{ˆ}

Rsin^{2}θ[sinθ∂θ(sinθ∂θS0) +∂φ∂φS0] +N δv^{r}^{ˆ}

sinθ ∂φ∂rS0+ 1

sinθ∂t∂φ(δS) )

dθ|^{r=R}. (69)

4 TOROIDAL OSCILLATIONS OF A NS WITH A DIPOLE MAGNETIC FIELD

As an important application of this formalism, we now consider small-amplitude toroidal oscillations of a NS with a dipole
magnetic field. For toroidal oscillations in the (ℓ^{′}, m^{′}) mode, a generic conducting fluid element is displaced from its initial

location (r, θ, φ) to a perturbed location (r, θ+ξ^{θ}, φ+ξ^{φ}) with the velocity field (Unno et al. 1989),
δv^{r}^{ˆ}= 0, δv^{θ}^{ˆ}=dξ^{θ}

dt =e^{−iωt}η(r) 1

sinθ∂φY_{ℓ}′m^{′}(θ, φ), δv^{φ}^{ˆ}=dξ^{φ}

dt =−e^{−iωt}η(r)∂θY_{ℓ}′m^{′}(θ, φ), (70)
where ωis the oscillation frequency and η(r) is the transverse velocity amplitude. Note that in the above expressions (70),
the oscillation mode axis is directed along thez-axis. We use a prime to denote the spherical harmonic indices in the case of
the oscillation modes.

4.1 The unperturbed exterior dipole magnetic field

If the static unperturbed magnetic field of the NS is of a dipole type, then the coefficientss0ℓminvolved in specifying it have the following form (see eq. 117 of Paper I)

s0 10=−

√3π

2 µcosχ , s0 11=

r3π

2 µsinχ , (71)

whereµis the magnetic dipole moment of the star, as measured by a distant observer, andχis the inclination angle between the dipole moment andz-axis. Substituting expressions (71) into (56) and then the latter into (54), we get

S0=−3µr^{2}
8M^{3}

»

lnN^{2}+2M
r

„ 1 +M

r

«–

(cosθcosχ+e^{iφ}sinθsinχ) (72)

The corresponding magnetic field components have the form
B_{0}^{r}^{ˆ}=− 3µ

4M^{3}

»

lnN^{2}+2M
r

„ 1 +M

r

«–

(cosχcosθ+ sinχsinθe^{iφ}), (73)

B0^{θ}^{ˆ}= 3µN
4M^{2}r

» r

M lnN^{2}+ 1
N^{2} + 1

–

(cosχsinθ−sinχcosθe^{iφ}), (74)

B_{0}^{φ}^{ˆ}= 3µN
4M^{2}r

» r

M lnN^{2}+ 1
N^{2} + 1

–

(−i sinχe^{iφ}). (75)

At the stellar surface, these expressions for the unperturbed magnetic field components become

B_{R}^{r}^{ˆ} =f_{R}B_{0}(cosχcosθ+ sinχsinθe^{iφ}), BR^{θ}^{ˆ} =h_{R}B_{0}(cosχsinθ−sinχcosθe^{iφ}), B_{R}^{φ}^{ˆ} =−ih_{R}B_{0}(sinχe^{iφ}), (76)
where B0 is defined as B0 = 2µ/R^{3}. In Newtonian theoryB0 would be the value of the magnetic strength at the magnetic
pole but this becomes modified in GR. The GR modifications are contained within the parameters

h_{R}=3R^{2}N_{R}
8M^{2}

»R

M lnN_{R}^{2} + 1
N_{R}^{2} + 1

–

, f_{R}=−3R^{3}
8M^{3}

»

lnN_{R}^{2} +2M
R

„ 1 +M

R

«–

. (77)

For a givenµ, the magnetic field near to the surface of the NS is stronger in GR than in Newtonian theory, as already noted by Ginzburg & Ozernoy (1964).

4.2 The equation for Ψ_{SC}

SubstitutingS0 from (72) into equation (50), we obtain a partial differential equation containing two unknown functions Ψ_{SC}
andδS for arbitrary oscillations of a NS with a dipole magnetic field

−2r^{2}q1(r)“

cosθcosχ+e^{iφ}sinθsinχ”

∂r(Ψ_{SC}) +∂rˆ

r^{2}q1(r)˜“

sinθcosχ−e^{iφ}cosθsinχ”

∂θ(Ψ_{SC}) (78)

−∂rˆ

r^{2}q1(r)˜e^{iφ}sinχ

sinθ ∂φ(Ψ_{SC}) +∂r

ˆr^{2}q1(r)˜
sinθ

h“

sinθcosχ−e^{iφ}cosθsinχ”

∂φ∂t(δS) +ie^{iφ}sinχsinθ∂θ∂t(δS)i

= 0, where we have introduced a new functionq1(r) for simplicity of notation [see Eq. (A2) for the definition ofq1(r)].

From (69), the boundary condition for Ψ_{SC} at the stellar surface is
Ψ_{SC}|^{r=R}=

Z θ 0

B0Rf_{R}δv^{φ}^{ˆ}∂θ

“

cosθcosχ+e^{iφ}sinθsinχ”

− 1

sinθ∂t∂φ(δS) ﬀ

dθ|^{r=R}. (79)

Using the expressions for the velocity field of the toroidal oscillations (70) and for the boundary conditions for the partial derivatives of the SC potential (64)-(65), we find that∂tδSis given by (see Appendix A for details of the derivation)

∂tδS(r, t) =

∞

X

ℓ=0 ℓ

X

m=−ℓ

B0Rf_{R}η˜_{R}
ℓ(ℓ+ 1)

r^{2}qℓ(r)

R^{2}qℓ(R) (80)

× Z

4π

h∂θYℓm

“sinθcosχ−e^{iφ}cosθsinχ”

+ie^{iφ}∂φYℓmsinθsinχiY_{ℓ}^{∗}′m^{′}(θ, φ)
sinθ dΩ.

From here on, for simplicity, we will consider only the case withχ= 0. Although our solution depends on the angle between the magnetic field axis and the oscillation mode axis, focusing on the caseχ= 0 does not actually imply a loss of generality because any mode with its axis not aligned with a given direction can be represented as a sum of modes with axes along this direction. We have developed a MATHEMATICA code for analytically solving equation (50) and hence obtaining analytic expressions for the electric and magnetic fields and for the SC density.

The solution of equation (78) for the caseχ= 0 is given in Appendix B, where we show that the general solution has the following form

Ψ_{SC}=−1
2

m^{′2}

ℓ^{′}(ℓ^{′}+ 1)B0Rf_{R}η˜_{R}
Z r

R

∂r^{′}[r^{′2}q1(r^{′})]

q1(r^{′})

qℓ^{′}(r^{′})
R^{2}qℓ^{′}(R)

Yℓ^{′}m^{′}(θ(r^{′}), φ)

cosθ(r^{′}) dr^{′}+ Φ2

hp−r^{2}q1(r) sinθ, φ, ti

, (81)

wherer^{′}is the integration variable. In order to solve this integral, the functionθ(r^{′}) is expressed in termsr^{′}and a constantϕ2

through the characteristic equation (B6) and, after performing the integration,ϕ2 is removed again using (B6). The unknown
function Φ2 is determined using the boundary condition for Φ2|^{r=R}given by (B13). Once the integral on the right-hand side
of (B13) has been evaluated, we then express all of the trigonometric functions resulting from the integral, in terms of sinθ.

Thisθ is the value atr=R. To obtain an expression for the value of Φ2 at a general radius, we write thisθ (atr=R) in terms of the value ofθat a general point (withr > R) using the characteristic relation (B6), i.e.,

sinθ→p

[r^{2}q1(r)]/[R^{2}q1(R)]×sinθ , (82)

so that cosθ→

»

1− r^{2}q1(r)
R^{2}q1(R)sin^{2}θ

–1/2

sign(cosθ), (83)

where sign(x) is defined such that sign(x) = +1 if x >0, and sign(x) = −1 if x <0. There are then different expressions
for Ψ_{SC} in the two regionsθ ∈[0, π/2] andθ∈[π/2, π]. If these two expressions do not coincide at the equatorial plane for
r > R, then there will be a discontinuity in Ψ_{SC} atθ=π/2, and quantities that depend on∂θΨ_{SC}will become singular there.

As shown by TBS, the function Ψ_{SC} is indeed discontinuous atθ =π/2 for some oscillation modes and, as we discussed in
Section 2.3 above, this unphysical behaviour indicates that the LCDA ceases to be valid for those modes. In these cases, the
accelerating electric field cannot be canceled without presence of strong currents which may become as large as (36) in some
regions of the magnetosphere. The occurrence of such singularities was explained by TBS and the reader is referred to Section
3.2 of their paper for a detailed discussion.

Next we discuss how GR effects contribute to our solution. As discussed above, for a given magnetic moment µ (as measured by a distant observer) the strength of the unperturbed magnetic field near to the surface of the NS is larger in GR than in Newtonian theory. Due to the linearity of the Maxwell equations, a perturbation of a stronger magnetic field should produce a larger electric field for the same oscillation parameters. This in turn should lead to a larger absolute value of the SC density in GR, since the SC density takes the value necessary to cancel the electric field. In the next Section, we will give a more quantitative analysis of the GR contribution in our solution.

We point out that the function Ψ_{SC} does not depend on ℓ >1 perturbations to the stellar magnetic field in the case
of axisymmetric (m^{′} = 0) toroidal modes. This can be seen from the fact that these perturbations are confined within the
δS terms which enter equation (78) for the function Ψ_{SC} only through a derivative with respect toφ; hence vanish for the
axisymmetric modes. Therefore, the only perturbation to the magnetic field is due to the ℓ= 1 term, and the solution for
these modes is much simpler than that for non-axisymmetric (m^{′}6= 0) modes. It is then convenient to discuss separately the
axisymmetric and non-axisymmetric cases.

The solution (81) for case them^{′}= 0 modes atr=Rhas the following form
Ψ_{SC}(r, θ, φ, t)|^{r=R}=−B0Rf_{R}η˜_{R}

Z θ 0

cosϑ ∂ϑYℓ^{′}0(ϑ, φ)dϑ . (84)

Using the properties of the spherical harmonics, we can express Ψ_{SC}(r, θ, φ, t)|^{r=R}for oddℓ^{′} modes in the general form
Ψ_{SC}(r, θ, φ, t)|^{r=R}∼

N

X

n=1

A2nsin^{2n}θ , (85)

while for evenℓ^{′}modes, the general form of Ψ_{SC} is
Ψ_{SC}(r, θ, φ, t)|^{r=R}∼(A+Bcosθ)

N

X

n=1

A2nsin^{2n}θ , (86)

where the coefficientsA,BandA2n do not depend onrandθ. The value ofN equalsℓ^{′}/2 + 1 for evenℓ^{′} and (ℓ^{′}+ 1)/2 for

1 1.2 1.4 1.6 1.8 2
*r / R*

1.2 1.3 1.4

(Ψ SC) GR / (Ψ SC) Newt

### 1 1.2 1.4 1.6 1.8 2

*r / R*

### 1

### 1.5 2 2.5 3 3.5

(ρ SC) GR / (ρ SC) Newt

*l’*=2, *m’*=0
*l’*=1, *m’*=1
*l’*=3, *m’*=0

Figure 1. Left panel: The ratio (Ψ_{SC})_{GR}/(Ψ_{SC})_{Newt} along the polar axis plotted as a function of the distance from the star, for
axisymmetric toroidal modes of a NS with compactnessM/R= 0.2.Right panel:The ratio (ρ_{SC})_{GR}/(ρ_{SC})_{Newt} in the equatorial plane
plotted as a function ofr, for a star withM/R= 0.2, for toroidal oscillation modes (2,0), (1,1) and (3,0).

oddℓ^{′}. As we discussed above, in order to obtain the solution for Ψ_{SC}forr > R, one has use sinθ→p

r^{2}q1(r)/R^{2}q1(R) sinθ
on the right-hand sides of (85) and (86). Thus the GR effects contribute to the solution form^{′}= 0 modes only through terms
f_{R}ˆ

r^{2}q1(r)/R^{2}q1(R)˜n

, wheren>1. Note that the factorf_{R}in this term appears due to the boundary condition at the surface
of the star, namely from the continuity of the tangential components of the electric field, while the factorˆ

r^{2}q1(r)/R^{2}q1(R)˜n

appears due to the presence of charged particles in the magnetosphere. The second factor is equal to 1 at the stellar surface
and approaches its Newtonian value (R/r)^{n} at larger and smallM/R. Sincef_{R} >1, the absolute value of Ψ_{SC} should be
greater in GR than in Newtonian theory. For example, in the case of small θ, the only term which is important is that with
n = 1 and hence we get (Ψ_{SC})_{GR}/(Ψ_{SC})_{Newt} =f_{R}r^{3}q1(r)/R^{3}q1(R). This quantity is shown in Figure 1 (left panel), where
we can see that, near to the stellar surface, the function Ψ_{SC} is larger in GR than in Newtonian theory, while at largerr it
asymptotically approaches its Newtonian value.

Analysis of the GR contribution to the solution in the case of non-axisymmetric modes is more complicated because in this case the solution depends not only on ℓ= 1 perturbations to the magnetic field but also onℓ >1 perturbations, which are contained in the term∂t∂φδS of equation (78), and contribute to the solution due to the integral in (81). Nevertheless, some rough estimates of the GR effects can be made in the following way. Near to the stellar surface the integral in (81) can be approximated as

f_{R}
Z r

R

∂z[z^{2}q1(z)]

q1(z)

qℓ(z)
R^{2}qℓ(R)

Yℓm(θ(z), φ)

cosθ(z) dz≃f_{R} r^{2}qℓ(r)
R^{2}qℓ(R)

Yℓm(θ(r), φ)

cosθ(r) . (87)

Close to the star, (r^{2}qℓ(r))/(R^{2}qℓ(R))≃1 and so the leading GR contribution comes from the factorf_{R} which increases the
absolute value of Ψ_{SC} with respect to the Newtonian case. Further away from the star,r^{2}qℓis approximately proportional to
r^{−ℓ} and so the integral in (87) can be approximated as∼f_{R}(R/r)^{ℓ}^{′}^{+1+m}^{′}^{/2} to leading order in R/r. Therefore, while this
integral makes an important contribution to Ψ_{SC} near to the star, it becomes negligibly small forℓ >1 atr > Ras compared
with Φ2. The GR effects contribute to Φ2 through the termsf_{R}ˆ

r^{2}q1(r)/R^{2}q1(R)˜n

in a similar way to their contribution to
Ψ_{SC} for the axisymmetric modes discussed above. This increase of the function Ψ_{SC} due to the GR effects also lead to an
increase in the absolute values of the SC densityρ_{SC} near to the star, as shown in Figures 1 (right panel) and 2 for some
toroidal oscillation modes.

5 ENERGY LOSSES

It was shown by TBS that the kinetic energy of the stellar oscillations should be lost through being passed to plasma near to the stellar surface which then flows out along the open magnetic field lines. Note that within the framework of the TBS model, electromagnetic fields are considered only in the near zone and so the existence of the plasma outflow cannot be shown explicitly; however, qualitatively, the mechanism for the plasma outflow should be the following. The charged particles that were accelerated to high energies by the longitudinal electric field move along the magnetic field lines in the near zone. If the kinetic energy density of the plasma at the equator becomes comparable to the energy density of the magnetic field at some point, then the field line which crosses the equator at that point becomes open. Plasma flowing along open field lines forms an electromagnetically driven wind which closes at infinity. There is then an electric current flowing along the stellar surface

### 0 1 2 3 -0.04

### -0.02 0 0.02

### ρ SC

*l’*=1, *m’*=0

### 0 1 2 3

### θ [rad]

### -0.02 0 0.02

*l’*=1, *m’*=1

### 0 1 2 3

### 0 0.2

*l’*=3, *m’*=0

Figure 2.The space-charge densityρ_{SC} atr= 1.5R(in units withB0 =η= 1) plotted as a function ofθfor the toroidal oscillation
modes (1,0), (1,1) and (3,0). The solid lines refer to the SC densityρ_{SC} in the relativistic theory, while its Newtonian value is shown
by the dashed lines.

between positive and negative emission regions. Because this current must cross the magnetic field lines at the stellar surface, it exerts a braking torque on the NS oscillations and thus reduces their kinetic energy (see Section 3.2.2 of TBS for more details).

In the following, we carry out a GR calculation of the energy lost by toroidal stellar oscillations due to plasma outflow.

First we calculate the energy losses due to the outflow of a particle along an open field line from a given point on the stellar surface. For this purpose, we start by considering the motion of the charged particle along theθ-direction on the stellar surface (where it crosses the magnetic field lines, hence exerts a braking torque on the stellar oscillations). The equation of motion for a test particle of massmin a generic electromagnetic field has the general form (Landau & Lifshitz 1987)

mDw^{α}

dτ =eF^{αβ}wβ , (88)

whereD/dτ is a comoving derivative,w^{α}is the four-velocity of the particle given by
wα= uα+vα

√1−v^{2} , (89)

u^{α} is the 4-velocity of the static observer (10), andv^{α} is the velocity of the particle relative to the static observer.

Because of time-invariance, there exists a timelike Killing vectorξ^{α}such thatξ^{α}ξα=−N^{2}. The four-velocity of the static
observer can be expressed in terms ofξ^{α} asu^{α}=N^{−1}ξ^{α}, and therefore the energy of the particle is given by

E=−pαξ^{α}=−mwαξ^{α} . (90)

Contracting the equation of motion (88) with the Killing vectorξαgives
ξαmw_{;β}^{α}w^{β}=−eF^{αβ}uα+vα

√1−v^{2}ξβ , (91)

The right-hand side of this can be rewritten as

−eF^{αβ}uα+vα

√1−v^{2}ξβ =eF^{αβ} vβξα

√1−v^{2} =eF^{αβ}N uβvα

√1−v^{2} =eE^{α}^{ˆ}vαˆN

√1−v^{2} , (92)

while the left-hand side can be transformed as

ξαmw_{;β}^{α}w^{β}=m(wαξ^{α});βw^{β}−mξα;βw^{α}w^{β} , (93)

The second term on the right-hand of this equation vanishes due to antisymmetry of the tensorξα;β. Therefore, the projection of the equation of motion onto the Killing vector can be written as

dE

dτ =eN E^{α}^{ˆ}vαˆ

√1−v^{2} . (94)

For particles moving along theθ-direction, this equation takes the form
dE =eE^{α}^{ˆ}RN_{R}^{2}

√1−v^{2}dθ . (95)