# Three-dimensional general relativistic Poynting-Robertson effect. IV.
Slowly rotating and non-spherical quadrupolar massive source

Vittorio De Falco1<EMAIL_ADDRESS>Maciek Wielgus2,3
<EMAIL_ADDRESS>1 Department of Mathematics and Applications “R.
Caccioppoli”, University of Naples Federico II, Via Cintia, 80126 Naples,
Italy
2 Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA
02138, USA
3 Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA

###### Abstract

We consider a further extension of our previous works in the treatment of the
three-dimensional general relativistic Poynting-Robertson effect, which
describes the motion of a test particle around a compact object as affected by
the radiation field originating from a rigidly rotating and spherical emitting
source, which produces a radiation pressure, opposite to the gravitational
pull, and a radiation drag force, which removes energy and angular momentum
from the test particle. The gravitational source is modeled as a non-spherical
and slowly rotating compact object endowed with a mass quadrupole moment and
an angular momentum and it is formally described by the Hartle-Thorne metric.
We derive the test particle’s equations of motion in the three-dimensional and
two-dimensional cases. We then investigate the properties of the critical
hypersurfces (regions, where a balance between gravitational and radiation
forces is established). Finally, we show how this model can be applied to
treat radiation phenomena occurring in the vicinity of a neutron star.

††preprint: APS/123-QED

##  I Introduction

The motion of matter around compact objects, when it is influenced by an
electromagnetic radiation field (originating for example from the surface of a
neutron star (NS), a boundary layer around a NS, a type-I X-ray burst on the
NS polar caps, a hot corona around a black hole (BH) or an accretion disk
around a NS or a BH) deviates from a geodesic trajectory. In such processes,
the general relativistic Poynting-Robertson (PR) effect plays an important
role in removing energy and angular momentum from the affected body, thus
playing a role of a dissipative force in General Relativity (GR) Poynting
(1903); Robertson (1937).

Recently, a series of programmatic studies on such an effect in GR have been
published. From a theoretical perspective, it is worth to cite: the general
relativistic modeling from the two-dimensional (2D) Bini  _et al._ (2009,
2011a, 2011b, 2015) to the 3D cases in Kerr and other metrics De Falco  _et
al._ (2019); Bakala  _et al._ (2019); Wielgus (2019); De Falco  _et al._
(2020a), its treatment under a Lagrangian formalism, determining for the first
time the analytical form of the Rayleigh potential in GR literature De Falco
_et al._ (2018); De Falco and Battista (2019); De Falco and Battista (2020),
proof that the equatorial ring of the critical hypersurface is a stable
attractor Stahl  _et al._ (2012), and the whole critical hypersurface is a
basin of attraction De Falco and Bakala (2020). There are also several
attempts to apply such effect to describe astrophysical phenomena, like:
analysis of the disk dynamical evolution when it is intercepted by a type-I
X-ray burst Walker and Meszaros (1989); Walker (1992); Lančová  _et al._
(2017); Fragile  _et al._ (2020), modeling the photospheric expansion occuring
during Eddington-luminosity X-ray bursts Wielgus  _et al._ (2015, 2016) and
associated oscillations Wielgus  _et al._ (2012); Bollimpalli  _et al._
(2019), a new method to diagnose the presence of wormholes through the
detection of metric-changes occurring in strong field regimes around black
holes through the PR critical hypersurfaces De Falco  _et al._ (2020b).

The exterior spacetime of rotating NS (or other compact objects such as dark
stars, gravastars, boson stars Cardoso and Pani (2019)) is not unique and
analytically known. However, if we consider such objects in a regime of slow
rigid rotation, modeled as stationary and axially symmetric perfect fluids
through mass $M$, angular momentum $J$, and quadrupole moment $Q$, then the
_Hartle-Thorne metric can be employed to realistically describe them with
accuracy up to the second order in $J$, and first order in $Q$_ Hartle (1967);
Hartle and Thorne (1968). While Schwarzschild, Kerr, and Erez-Rosen metrics
are all exact solutions of Einstein vacuum field equations to respectively
model static, rotating, and static, axially symmetric and non-spherical
compact objects, _the Hartle-Thorne metric, on the other hand, is an
approximate solution of Einstein field equations in the vacuum_.

In this work, we aim to extend our previous works on the 3D modeling of the
general relativistic PR effect De Falco  _et al._ (2019); Bakala  _et al._
(2019); De Falco  _et al._ (2020a) in the Hartle-Thorne metric, using as a
description of the radiation field the model developed in Ref. Bakala  _et
al._ (2019). The paper is organized as follows: in Sec. II we recall the
Hartle-Thorne metric; in Sec. III we derive the test particle’s equations of
motion; in Sec. IV we analyse the critical hypersurfaces, investigating
extensively their properties, and showing also a possibile application to the
NS case; finally in Sec. V we draw our conclusions.

##  II Spacetime geometry

###  II.1 Hartle-Thorne spacetime

Astrophysical objects are not exactly spherical symmetric, the more if they
are rotating. The exterior spacetime of a slowly rotating and slightly
deformed compact object endowed with total mass $M$, angular momentum $J$ and
quadrupole parameter $Q$ 111The quadrupole parameter $Q$ introduced here is
related to the mass quadrupole moment $Q_{HT}$ as defined by Hartle and Thorne
Hartle (1967); Hartle and Thorne (1968) through the formula
$Q=2J^{2}/M-Q_{HT}$. can be accurately described by the Hartle-Thorne metric
Hartle (1967); Hartle and Thorne (1968). Using geometrical units, where
$G=c=1$, and spherical coordinates $(t,r,\theta,\varphi)$, the line element of
this metric is given by Bini  _et al._ (2013)

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f(r)F_{1}(r,\theta)dt^{2}+\frac{F_{2}(r,\theta)}{f(r)}dr^{2}$
$\displaystyle+$ $\displaystyle
r^{2}F_{3}(r,\theta)(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})-\frac{4J}{r}\sin^{2}\theta\ dt\ d\varphi,$

where

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle 1-\frac{2{M}}{r},$
$\displaystyle k_{1}(r)$ $\displaystyle=$
$\displaystyle\frac{J^{2}}{{M}r^{3}}f(-2r)-\frac{5}{8}\frac{Q-J^{2}/{M}}{{M}^{3}}Q_{2}\left(\frac{r}{{M}}-1\right),$
$\displaystyle k_{2}(r)$ $\displaystyle=$ $\displaystyle
k_{1}(r)-\frac{6J^{2}}{r^{4}},$ $\displaystyle k_{3}(r)$ $\displaystyle=$
$\displaystyle
k_{1}(r)+\frac{J^{2}}{r^{4}}-\frac{5}{4}\frac{Q-J^{2}/{M}}{{M}^{2}r\sqrt{f(r)}}Q_{1}\left(\frac{r}{M}-1\right),$
$\displaystyle F_{1}(r,\theta)$ $\displaystyle=$ $\displaystyle
1+2k_{1}(r)P_{2}(\cos\theta)+\frac{2}{f(r)}\frac{J^{2}}{r^{4}}(2\cos^{2}\theta-1),$
$\displaystyle F_{2}(r,\theta)$ $\displaystyle=$ $\displaystyle
1-2k_{2}(r)P_{2}(\cos\theta)-\frac{2}{f(r)}\frac{J^{2}}{r^{4}},$
$\displaystyle F_{3}(r,\theta)$ $\displaystyle=$ $\displaystyle
1-2k_{3}(r)P_{2}(\cos\theta),$ (2)

and $P_{2}(x)$ is the Legendre polynomials of the first kind,
$Q_{1}(x),Q_{2}(x)$ are the associated Legendre polynomials of the second
kind, which all explicitly read as

$\displaystyle P_{2}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(3x^{2}-1),$ $\displaystyle Q_{1}(x)$
$\displaystyle=$
$\displaystyle(x^{2}-1)^{1/2}\left[\frac{3x}{2}\ln\left(\frac{x+1}{x-1}\right)-\frac{3x^{2}-2}{x^{2}-1}\right],$
(3) $\displaystyle Q_{2}(x)$ $\displaystyle=$
$\displaystyle(x^{2}-1)\left[\frac{3}{2}\ln\left(\frac{x+1}{x-1}\right)-\frac{3x^{3}-5x}{(x^{2}-1)^{2}}\right].$

####  II.1.1 Properties of the Hartle-Thorne metric

We report some useful properties of the Hartle-Thorne metric, which will be
useful in the next sections.

  * •

Metric (II.1) reduces to the approximate Kerr metric in the Boyer-Lindquist
coordinates $(t,\ R,\ \Theta,\ \varphi)$ up to second order terms in the
rotation parameter $a$ by considering $J=-M^{2}a$, $Q=J^{2}/M$, and the
following transformation of coordinates Bini  _et al._ (2013)

$\displaystyle r$ $\displaystyle=$ $\displaystyle
R+\frac{a^{2}}{2R}\bigg{[}f(-R)f(2R)$ (4)
$\displaystyle-f(R)f\left(-\frac{3}{2}R\right)\cos^{2}\Theta\bigg{]},$
$\displaystyle\theta$ $\displaystyle=$
$\displaystyle\Theta+\frac{a^{2}}{2R^{2}}f(-R)\sin\Theta\cos\Theta.$

  * •

For $J=0$ metric (II.1) reduces to the linearized Erez-Rosen spacetime
(static, axially symmetric, and non-spherical quadrupolar massive source) with
respect to its quadrupole parameter $Q_{ER}$ Bini  _et al._ (2015).

  * •

Hartle-Thorne spacetime can admit an event horizon $R_{H}$ and ergosphere
$R_{SL}$, whose expressions are obtained by respectively imposing
$g_{t\varphi}^{2}-g_{tt}g_{\varphi\varphi}=0$, and $g_{tt}=0$, and then
solving such equations for $r$ in terms of $\theta$, once $Q,J$ has been
assigned Abramowicz  _et al._ (2003).

  * •

The domain of validity of the Hartle-Thorne approximation around a gravitating
body must always be (see Sec. 2 in Ref. Bini  _et al._ (2013), for further
details)

$r-2M\gg\left(\frac{25J^{2}Q^{2}}{128M^{7}}\right)^{1/3}.$ (5)

###  II.2 Zero angular momentum observers

The Hartle-Thorne spacetime admits, as in the Kerr metric, zero angular
momentum observers (ZAMOs), who are dragged by the rotation of the spacetime
(even though it is not strong) with angular velocity
$\Omega_{\mathrm{ZAMO}}=-g_{\varphi t}/g_{\varphi\varphi}$, while their radial
and latitudinal coordinates remain constant. The four-velocity of ZAMOs,
$\bm{n}$, is Bini  _et al._ (2009, 2011a); De Falco  _et al._ (2019); Bakala
_et al._ (2019),

$\bm{n}=\frac{1}{N}(\bm{\partial_{t}}-N^{\varphi}\bm{\partial_{\varphi}})\,,$
(6)

where $N=(-g^{tt})^{-1/2}$ is the time lapse function, and
$N^{\varphi}=g_{t\varphi}/g_{\varphi\varphi}$ is the spatial shift vector
field. An orthonormal frame adapted to the ZAMOs is Bini  _et al._ (2009,
2011a); De Falco  _et al._ (2019); Bakala  _et al._ (2019)

$\displaystyle\bm{e_{\hat{t}}}=\bm{n},\
\bm{e_{\hat{r}}}=\frac{\bm{\partial_{r}}}{\sqrt{g_{rr}}},\
\bm{e_{\hat{\theta}}}=\frac{\bm{\partial_{\theta}}}{\sqrt{g_{\theta\theta}}},\
\bm{e_{\hat{\varphi}}}=\frac{\bm{\partial_{\varphi}}}{\sqrt{g_{\varphi\varphi}}}.$
(7)

All the indices associated to the ZAMO frame will be labeled by a hat, instead
all the quantities measured in the ZAMO frame will be followed by $(\bm{n})$.

###  II.3 ZAMO kinematical quantities

The properties of the Hartle-Thorne spacetime combines those of the Kerr De
Falco  _et al._ (2019); Bakala  _et al._ (2019) and Erez-Rosen metrics De
Falco  _et al._ (2020a). Therefore, as done in the previous cases it is still
convenient to use the Lie transport (see Bini  _et al._ (1997a, b); De Falco
_et al._ (2018), for further details), where the nonzero ZAMO kinematical
quantities are: acceleration $\bm{a}(\bm{n})=\nabla_{\bm{n}}\bm{n}$, expansion
tensor along the $\hat{\varphi}$-direction
$\bm{\theta_{\hat{\varphi}}}(\bm{n})$, and the signed Lie curvature tensors
$\bm{k}(x^{i},\bm{n})$ relative to the ZAMO $\bm{n}$ four-velocity along the
directions $x^{i}=r,\theta,\varphi$ Bini  _et al._ (1997a, b); De Falco  _et
al._ (2019). They have only nonzero components in the
$\bm{\hat{r}}-\bm{\hat{\theta}}$ ZAMO plane De Falco  _et al._ (2019); Bakala
_et al._ (2019); De Falco  _et al._ (2020a), and can be calculated through the
formulas:

$\displaystyle\bm{a}(\bm{n})$ $\displaystyle=a(\bm{n})^{\hat{r}}\
\bm{e_{\hat{r}}}+a(\bm{n})^{\hat{\theta}}\ \bm{e_{\hat{\theta}}}$ (8)
$\displaystyle=\frac{\partial_{r}N}{N\sqrt{g_{rr}}}\
\bm{\partial_{r}}+\frac{\partial_{\theta}N}{N\sqrt{g_{\theta\theta}}}\
\bm{\partial_{\theta}},$ $\displaystyle\bm{\theta_{\hat{\varphi}}}(\bm{n})$
$\displaystyle=\theta(\bm{n})^{\hat{r}}{}_{\hat{\varphi}}\,\bm{e_{\hat{r}}}+\theta(\bm{n})^{\hat{\theta}}{}_{\hat{\varphi}}\,\bm{e_{\hat{\theta}}}$
$\displaystyle=-\frac{\sqrt{g_{\varphi\varphi}}}{2N}\,\left(\frac{\partial_{r}N^{\varphi}}{\sqrt{g_{rr}}}\,\bm{\partial_{r}}+\frac{\partial_{\theta}N^{\varphi}}{\sqrt{g_{\theta\theta}}}\bm{\partial_{\theta}}\right),$
$\displaystyle\bm{k}(x^{i},\bm{n})$ $\displaystyle=k(x^{i},\bm{n})^{\hat{r}}\
\bm{e_{\hat{r}}}+k(x^{i},\bm{n})^{\hat{\theta}}\ \bm{e_{\hat{\theta}}}$
$\displaystyle=-\frac{\partial_{r}g_{ii}}{2g_{ii}\sqrt{g_{rr}}}\
\bm{\partial_{r}}-\frac{\partial_{\theta}g_{ii}}{2g_{ii}\sqrt{g_{\theta\theta}}}\
\bm{\partial_{\theta}},$

It is important to note that
$\bm{k}(\varphi,\bm{n})=\bm{k}_{\rm(Lie)}(\bm{n})$ De Falco  _et al._ (2019,
2020a). The ZAMO kinematical quantities are expressed in terms of the
derivatives of $\partial_{\alpha}N,\partial_{\alpha}N^{\varphi}$, where
$\alpha=r,\theta$, namely

$\displaystyle\psi_{\alpha}$ $\displaystyle=$ $\displaystyle
g_{t\varphi}^{2}\partial_{\alpha}g_{\varphi\varphi}+g_{\varphi\varphi}(g_{\varphi\varphi}\partial_{\alpha}g_{tt}-2g_{t\varphi}\partial_{\alpha}g_{t\varphi}),$
$\displaystyle\partial_{\alpha}N$ $\displaystyle=$
$\displaystyle-\frac{N^{3}}{2}\frac{\psi_{\alpha}}{(g_{t\varphi}^{2}-g_{tt}g_{\varphi\varphi})^{2}},$
(9) $\displaystyle\partial_{\alpha}N^{\varphi}$ $\displaystyle=$
$\displaystyle\frac{g_{\varphi\varphi}\partial_{\alpha}g_{t\varphi}-g_{t\varphi}\partial_{\alpha}g_{\varphi\varphi}}{g_{\varphi\varphi}^{2}}.$

Therefore, to have theirs explicit expressions we need to calculate the
derivatives of the metric components with respect to the radial $r$ and polar
$\theta$ coordinates. The derivatives with respect to $r$ are

$\displaystyle\partial_{r}g_{tt}=-[\partial_{r}f(r)F_{1}(r,\theta)+f(r)\partial_{r}F_{1}(r,\theta)],$
$\displaystyle\partial_{r}g_{rr}=\frac{f(r)\partial_{r}F_{2}(r,\theta)-\partial_{r}f(r)F_{2}(r,\theta)}{f(r)^{2}},$
$\displaystyle\partial_{r}g_{\theta\theta}=r[2F_{3}+r\partial_{r}F_{3}(r,\theta)],$
(10)
$\displaystyle\partial_{r}g_{\varphi\varphi}=\partial_{r}g_{\theta\theta}\sin^{2}\theta,$
$\displaystyle\partial_{r}g_{t\varphi}=\frac{2J}{r^{2}}\sin^{2}\theta,$

while the derivatives with respect to $\theta$ are

$\displaystyle\partial_{\theta}g_{tt}=-f(r)\partial_{\theta}F_{1}(r,\theta),$
$\displaystyle\partial_{\theta}g_{rr}=\frac{\partial_{\theta}F_{2}(r,\theta)}{f(r)},$
$\displaystyle\partial_{\theta}g_{\theta\theta}=r^{2}\partial_{\theta}F_{3}(r,\theta),$
(11)
$\displaystyle\partial_{\theta}g_{\varphi\varphi}=r^{2}[\sin(2\theta)F_{3}(r,\theta)+\sin^{2}\theta\partial_{\theta}F_{3}(r,\theta)],$
$\displaystyle\partial_{\theta}g_{t\varphi}=-\frac{2J}{r}\sin(2\theta).$

In Table 1 we summarize the explicit expressions of functions’ derivatives
(II.1) and (II.1) in Hartle-Thorne metric.

Table 1:  Explicit expressions of functions’ derivatives (II.1) and (II.1) in
Hartle-Thorne metric. For easing the notations, we define $x=r/M-1$ and
$y=\cos\theta$, where $\partial_{y}P_{2}(y)=3y$ and
$\partial_{r}f(r)=2M/r^{2}$.

---
Metric quantity | Explicit expression
$\partial_{x}Q_{1}$ | $\frac{x(7-6x^{2})+3(x^{2}-1)(x^{2}-1/2)\ln[(x+1)/(x-1)]}{(x^{2}-1)^{3/2}}$
$\partial_{x}Q_{2}$ | $\frac{-8+10x^{2}-6x^{4}+3x(x^{2}-1)^{2}\ln[(x+1)/(x-1)]}{(x^{2}-1)^{2}}$
$\partial_{r}k_{1}(r)$ | $-\frac{J^{2}}{{M}r^{5}}(3r+4M)-\frac{5}{8}\frac{Q-J^{2}/{M}}{{M}^{4}}\partial_{x}Q_{2}(x)$
$\partial_{r}k_{2}(r)$ | $\partial_{r}k_{1}(r)+\frac{24J^{2}}{r^{5}}$
$\partial_{r}k_{3}(r)$ | $\partial_{r}k_{1}(r)-\frac{4J^{2}}{r^{5}}-\frac{5}{4}\frac{Q-J^{2}/{M}}{{M}^{2}}\left[\frac{rf(r)\partial_{x}Q_{1}(x)/M-Q_{1}(x)f(2r)}{r^{2}\sqrt{f(r)^{3}}}\right]$
$\partial_{r}F_{1}(r,\theta)$ | $2\partial_{r}k_{1}(r)P_{2}(y)-\frac{4J^{2}(2\cos^{2}\theta-1)}{r^{6}f(r)^{2}}(2r-3M)$
$\partial_{r}F_{2}(r,\theta)$ | $-2\partial_{r}k_{2}(r)P_{2}(y)+\frac{4J^{2}}{r^{6}f(r)^{2}}(2r-3M)$
$\partial_{r}F_{3}(r,\theta)$ | $-2\partial_{r}k_{3}(r)P_{2}(y)$
$\partial_{\theta}F_{1}(r,\theta)$ | $-\sin(2\theta)\left[3k_{1}(r)+\frac{4J^{2}}{r^{4}f(r)}\right]$
$\partial_{\theta}F_{2}(r,\theta)$ | $3\sin(2\theta)k_{2}(r)$
$\partial_{\theta}F_{3}(r,\theta)$ | $3\sin(2\theta)k_{3}(r)$

##  III Test particle dynamics

###  III.1 Radiation field

In this section we approximate the radiation field by considering it to only
consist locally of photons traveling on trajectories orthogonal to the
rotating emission sphere (corresponding either to the NS surface or a boundary
layer forming around a NS). Such photons are characterized by a four-momentum
component $k^{\theta}=0$. This approach follows previous studies Bini  _et
al._ (2009, 2011a); De Falco  _et al._ (2019); Bakala  _et al._ (2019); De
Falco  _et al._ (2020a), and it simplifies the treatment of the model
significantly. Of course more astrophysically realistic models should take
into account the photon emission from the whole surface including the whole
range of outgoing light ray directions, the angular dependence of the surface
emissivity, and law of emission related to the equation of state of the
emitting surface. Although in the literature there are some attempts along
this direction Abramowicz  _et al._ (1990); Miller and Lamb (1996); Wielgus
(2019), they are based on relativistic models simpler than the one proposed in
this paper. These crucial features are not discussed in the present article,
but they will be part of forthcoming works.

The effective description of the radiation field is thus given by the stress-
energy tensor Bini  _et al._ (2009, 2011a); De Falco  _et al._ (2019); Bakala
_et al._ (2019); De Falco  _et al._ (2020a)

$T^{\alpha\beta}=\Phi^{2}k^{\alpha}k^{\beta},\quad
k^{\alpha}k_{\alpha}=0,\quad k^{\beta}\nabla_{\beta}k^{\alpha}=0,$ (12)

where $\Phi$ is the parameter related to the intensity of the radiation field.
The photon four-momentum $\bm{k}$ can be split in the ZAMO frame as Bini  _et
al._ (2009, 2011a); De Falco  _et al._ (2019); Bakala  _et al._ (2019); De
Falco  _et al._ (2020a)

$\displaystyle\bm{k}=E(\bm{n})[\bm{n}+\bm{\hat{\nu}}(\bm{k},\bm{n})],$ (13)
$\displaystyle\bm{\hat{\nu}}(\bm{k},\bm{n})=\sin\beta\sin\xi\
\bm{e_{\hat{r}}}+\cos\xi\ \bm{e_{\hat{\theta}}}+\sin\xi\cos\beta\
\bm{e_{\hat{\varphi}}},$

where $\bm{\hat{\nu}}(\bm{k},\bm{n})$ is the photon spatial velocity on the
spatial hypersurface orthogonal to $\bm{n}$, and $E(\bm{n})$ is the relative
photon energy in the ZAMO frame Bini  _et al._ (2009, 2011a); De Falco  _et
al._ (2019); Bakala  _et al._ (2019); De Falco  _et al._ (2020a)

$E(\bm{n})=-\bm{k}\cdot\bm{n}=\frac{E}{N}(1+bN^{\varphi}),$ (14)

where $E=-k_{t}>0$ is the conserved photon energy, $\beta$ and $\xi$ are the
two angles in the azimuthal and polar direction, respectively. The case
$\sin\beta>0$ corresponds to outgoing photons (increasing radial distance from
the central source), and $\sin\beta<0$ to incoming photons (decreasing $r$).
The angular momentum along the polar $\bm{\hat{\theta}}$-axis in the local
static observer frame, $L_{\hat{\theta}}(\bm{n})$ is De Falco  _et al._
(2019); Bakala  _et al._ (2019); De Falco  _et al._ (2020a)

$\displaystyle E(\bm{n})\cos\beta\sin\xi$
$\displaystyle=L_{\hat{\theta}}(\bm{n})=\bm{k}(\bm{n})\cdot\bm{e_{\hat{\varphi}}}=\frac{L_{z}}{\sqrt{g_{\varphi\varphi}}},$
(15)

where $L_{z}=k_{\varphi}$ is the conserved photon angular momentum along the
$\bm{\theta}$-axis. From Eqs. (14) and (15), we have

$\cos\beta=\frac{bN}{\sqrt{g_{\varphi\varphi}}(1+bN^{\varphi})},$ (16)

where $b=L_{z}/E$ denotes the azimuthal photon impact parameter associated to
the azimuthal angle $\beta$. Following the same strategy as adopted in
previous studies (see Refs. De Falco  _et al._ (2019); Bakala  _et al._
(2019); De Falco  _et al._ (2020a), for details) we assume that $k^{\theta}=0$
along all the photon trajectories. This implies that the polar angle $\theta$
is conserved along the photon trajectories, namely
$\dot{\theta}=k^{\theta}/g_{\theta\theta}=0$, and so $\theta=const$. Since our
radiation field is emitted radially in the frame of the rigidly rotating
emitting surface, we have that $\xi=\pi/2$ (cf. Eq. (13)) and everything is
expressed only in terms of the parameter $b$ and the angle $\theta$ occupied
by the test particle. This implies that Bakala  _et al._ (2019); De Falco  _et
al._ (2020a)

$b=\left[-\frac{g_{t\varphi}+g_{\varphi\varphi}\Omega_{\star}}{g_{tt}+g_{t\varphi}\Omega_{\star}}\right]_{r=R_{\star}},$
(17)

where $R_{\star}$ and $\Omega_{\star}$ are respectively radius and angular
velocity of the emitting surface.

Therefore, the photon four-momentum $\bm{k}$ is defined in terms of
$b(\theta)$ or equivalently $(\theta,R_{\star},\Omega_{\star})$, whereas the
stress-energy tensor of the radiation field (12) is completely determined by
calculating the quantity $\Phi$. From the conservation equations
$\nabla_{\beta}T^{\alpha\beta}=0$, the absence of photon latitudinal motion
($k^{\theta}=0$), and the axial symmetries of the Hartle-Thorne spacetime, we
have Bini  _et al._ (2009, 2011a); De Falco  _et al._ (2019); Bakala  _et al._
(2019); De Falco  _et al._ (2020a)

$0=\nabla_{\beta}(\Phi^{2}k^{\beta})=\partial_{r}(\sqrt{-g}\Phi^{2}k^{r}).$
(18)

Therefore, we obtain Bini  _et al._ (2011a); De Falco  _et al._ (2019, 2020a)

$\sqrt{-g}\Phi^{2}k^{r}\equiv
NE(\bm{n})\sqrt{g_{\varphi\varphi}g_{\theta\theta}}\sin\beta\Phi^{2}=E\sin\theta\Phi_{0}^{2},$
(19)

where $\Phi_{0}$ is $\Phi$ evaluated at the emitting surface. Then, after some
algebra, we obtain

$\Phi^{2}=\frac{\Phi_{0}^{2}}{(1+bN^{\varphi})r^{2}F_{3}(r,\theta)\sin\beta}.$
(20)

###  III.2 Test particle motion

A test particle moves in the 3D space with four-velocity $\bm{U}$ and spatial
velocity $\bm{\hat{\nu}}(\bm{U},\bm{n})$ with respect to the ZAMO frame, given
respectively by Bini  _et al._ (2009); De Falco  _et al._ (2019)

$\displaystyle\bm{U}$ $\displaystyle=$
$\displaystyle\gamma(\bm{U},\bm{n})[\bm{n}+\bm{\nu}(\bm{U},\bm{n})],$ (21)
$\displaystyle\bm{\hat{\nu}}(\bm{U},\bm{n})$ $\displaystyle=$
$\displaystyle\nu^{\hat{r}}\bm{e_{\hat{r}}}+\nu^{\hat{\theta}}\bm{e_{\hat{\theta}}}+\nu^{\hat{\varphi}}\bm{e_{\hat{\varphi}}}$
$\displaystyle=$ $\displaystyle\nu(\sin\alpha\sin\psi\
\bm{e_{\hat{r}}}+\cos\psi\ \bm{e_{\hat{\theta}}}+\sin\psi\cos\alpha\
\bm{e_{\hat{\varphi}}}),$

where
$\gamma(\bm{U},\bm{n})\equiv\gamma=1/\sqrt{1-||\bm{\nu}(\bm{U},\bm{n})||^{2}}$
is the Lorentz factor,
$\nu^{\hat{\alpha}}(\bm{U},\bm{n})\equiv\nu^{\hat{\alpha}}$ is the spatial
velocity in the ZAMO frame, $\alpha$ and $\psi$ are the azimuthal and polar
angles, respectively, and
$\nu=\sqrt{\nu^{\hat{r}}{}^{2}+\nu^{\hat{\theta}}{}^{2}+\nu^{\hat{\varphi}}{}^{2}}$
is the module of the spatial velocity. The explicit expression of the test
particle velocity components are

$\displaystyle U^{\hat{t}}\equiv\frac{dt}{d\tau}=\frac{\gamma}{N},\quad
U^{\hat{r}}\equiv\frac{dr}{d\tau}=\frac{\gamma\nu^{\hat{r}}}{\sqrt{g_{rr}}},$
(23) $\displaystyle
U^{\hat{\theta}}\equiv\frac{d\theta}{d\tau}=\frac{\gamma\nu^{\hat{\theta}}}{\sqrt{g_{\theta\theta}}},\quad
U^{\hat{\varphi}}\equiv\frac{d\varphi}{d\tau}=\frac{\gamma\nu^{\hat{\varphi}}}{\sqrt{g_{\varphi\varphi}}}-\gamma\frac{N^{\varphi}}{N},$

where $\tau$ is the affine (or proper time) parameter along the test
particle’s world line.

Using the _observer splitting formalism_ , we find that the test particle
acceleration in the Hartle-Thorne spacetime is similar to the that of the
Erez-Rosen metric De Falco  _et al._ (2020a), whose explicit expression is
given by:

$\displaystyle a(\bm{U})^{\hat{r}}$ $\displaystyle=$
$\displaystyle\gamma^{2}\left[a(\bm{n})^{\hat{r}}+2\nu\cos\alpha\sin\psi\theta(\bm{n})^{\hat{r}}{}_{\hat{\varphi}}\right.$
$\displaystyle\left.+\nu^{2}\left(-k(r,\bm{n})^{\hat{\theta}}\sin\alpha\sin\psi\cos\psi\right.\right.$
$\displaystyle\left.\left.+k(\varphi,\bm{n})^{\hat{r}}\sin^{2}\psi\cos^{2}\alpha+k(\theta,\bm{n})^{\hat{r}}\cos^{2}\psi\right)\right]$
$\displaystyle+\gamma\left(\gamma^{2}\sin\alpha\sin\psi\frac{\rm d\nu}{\rm
d\tau}+\nu\cos\alpha\sin\psi\frac{\rm d\alpha}{\rm d\tau}\right.$
$\displaystyle\left.+\nu\cos\psi\sin\alpha\frac{\rm d\psi}{\rm d\tau}\right),$
$\displaystyle a(\bm{U})^{\hat{\theta}}$ $\displaystyle=$
$\displaystyle\gamma^{2}\left[a(\bm{n})^{\hat{\theta}}+2\nu\cos\alpha\sin\psi\theta(\bm{n})^{\hat{\theta}}{}_{\hat{\varphi}}\right.$
$\displaystyle\left.+\nu^{2}\left(k(\varphi,\bm{n})^{\hat{\theta}}\cos^{2}\alpha\sin^{2}\psi\right.\right.$
$\displaystyle\left.\left.+k(r,\bm{n})^{\hat{\theta}}\sin^{2}\alpha\sin^{2}\psi\right.\right.$
$\displaystyle\left.\left.-k(\theta,\bm{n})^{\hat{r}}\sin\alpha\sin\psi\cos\psi\right)\right]$
$\displaystyle+\gamma\left(\gamma^{2}\cos\psi\frac{\rm d\nu}{\rm
d\tau}-\nu\sin\psi\frac{\rm d\psi}{\rm d\tau}\right).$ $\displaystyle
a(\bm{U})^{\hat{\varphi}}$ $\displaystyle=$
$\displaystyle-\gamma^{2}\nu^{2}\left[k(\varphi,\bm{n})^{\hat{\theta}}\sin\psi\cos\alpha\cos\psi\right.$
$\displaystyle\left.+k(\varphi,\bm{n})^{\hat{r}}\sin^{2}\psi\sin\alpha\cos\alpha\right]$
$\displaystyle+\gamma\left(\gamma^{2}\cos\alpha\sin\psi\frac{\rm d\nu}{\rm
d\tau}\right.$ $\displaystyle\left.-\nu\sin\alpha\sin\psi\frac{\rm
d\alpha}{\rm d\tau}+\nu\cos\alpha\cos\psi\frac{\rm d\psi}{\rm d\tau}\right).$

From the orthogonality between $\bm{a}(\bm{U})$ and $\bm{U}$, we can determine
the expression of $a(\bm{U})^{\hat{t}}$ De Falco  _et al._ (2019); Bakala  _et
al._ (2019)

$\displaystyle a(\bm{U})^{\hat{t}}$ $\displaystyle=$
$\displaystyle\nu[a(\bm{U})^{\hat{r}}\sin\alpha\sin\psi+a(\bm{U})^{\hat{\theta}}\cos\psi$
$\displaystyle+a(\bm{U})^{\hat{\varphi}}\cos\alpha\sin\psi]$
$\displaystyle=\gamma^{2}\nu\left[\sin\alpha\sin\psi\left(a(\bm{n})^{\hat{r}}+2\nu\cos\alpha\sin\psi\theta(\bm{n})^{\hat{r}}{}_{\hat{\varphi}}\right)\right.$
$\displaystyle\left.+\cos\psi\left(a(\bm{n})^{\hat{\theta}}+2\nu\cos\alpha\sin\psi\theta(\bm{n})^{\hat{\theta}}{}_{\hat{\varphi}}\right)\right]+\gamma^{3}\nu\frac{\rm
d\nu}{\rm d\tau}.$

We note that such expressions for $J=0$ ($Q=0$) behave similarly to that of
the Erez-Rosen De Falco  _et al._ (2020a) (Kerr De Falco  _et al._ (2019);
Bakala  _et al._ (2019)) metric. Instead, for $J=0,Q=0$, they reduce to that
of the Schwarzschild metric De Falco  _et al._ (2019); Bakala  _et al._
(2019).

###  III.3 Test particle-radiation field interaction

We assume that the radiation-test particle interaction occurs through Thomson
scattering, characterized by a constant momentum-transfer cross section
$\sigma$, independent from direction and frequency of the radiation field. The
radiation force is Bini  _et al._ (2009, 2011a); De Falco  _et al._ (2019);
Bakala  _et al._ (2019); De Falco  _et al._ (2020a)

${\mathcal{F}}_{\rm(rad)}(\bm{U})^{\hat{\alpha}}=-\sigma
P(\bm{U})^{\hat{\alpha}}{}_{\hat{\beta}}\,T^{\hat{\beta}}{}_{\hat{\mu}}\,U^{\hat{\mu}}\,,$
(29)

where
$P(\bm{U})^{\hat{\alpha}}{}_{\hat{\beta}}=\delta^{\hat{\alpha}}_{\hat{\beta}}+U^{\hat{\alpha}}U_{\hat{\beta}}$
projects a vector orthogonally to $\bm{U}$. Decomposing the photon four-
momentum $\bm{k}$ first with respect to the test particle four-velocity,
$\bm{U}$, and then in the local observer frame, $\bm{n}$, we have De Falco
_et al._ (2019)

$\bm{k}=E(\bm{n})[\bm{n}+\bm{\hat{\nu}}(\bm{k},\bm{n})]=E(\bm{U})[\mathbb{U}+\bm{\hat{\mathcal{V}}}(\bm{k},\bm{U})].$
(30)

Exploiting Eq. (30) in Eq. (29), we obtain De Falco  _et al._ (2019); Bakala
_et al._ (2019); De Falco  _et al._ (2020a)

$\displaystyle{\mathcal{F}}_{\rm(rad)}(\bm{U})^{\hat{\alpha}}$
$\displaystyle=-\sigma\Phi^{2}[P(\bm{U})^{\hat{\alpha}}{}_{\hat{\beta}}k^{\hat{\beta}}]\,(k_{\hat{\mu}}U^{\hat{\mu}})$
(31) $\displaystyle=\sigma\,[\Phi
E(\bm{U})]^{2}\,\hat{\mathcal{V}}(\bm{k},\bm{U})^{\hat{\alpha}}\,.$

The equations of motion are
$m\mathbb{a}(\bm{U})=\bm{{\mathcal{F}}_{\rm(rad)}}(\bm{U})$, where $m$ is the
test particle mass. Defined $\tilde{\sigma}=\sigma/m$, we obtain the following
equations De Falco  _et al._ (2019); Bakala  _et al._ (2019); De Falco  _et
al._ (2020a)

$\mathbb{a}(\bm{U})=\tilde{\sigma}\Phi^{2}E(\bm{U})^{2}\,\bm{\hat{\mathcal{V}}}(\bm{k},\bm{U}).$
(32)

Multiplying scalarly Eq. (30) by $\bm{U}$, we find De Falco  _et al._ (2019);
Bakala  _et al._ (2019); De Falco  _et al._ (2020a)

$E(\bm{U})=\gamma E(\bm{n})[1-\nu\sin\psi\cos(\alpha-\beta)].$ (33)

Such splitting permits to determine
$\bm{\hat{\mathcal{V}}}(\bm{k},\bm{U})=\hat{\mathcal{V}}^{t}\bm{n}+\hat{\mathcal{V}}^{r}\bm{e_{\hat{r}}}+\hat{\mathcal{V}}^{\theta}\bm{e_{\hat{\theta}}}+\hat{\mathcal{V}}^{\varphi}\bm{e_{\hat{\varphi}}}$
as De Falco  _et al._ (2019); Bakala  _et al._ (2019); De Falco  _et al._
(2020a)

$\displaystyle\hat{\mathcal{V}}^{\hat{r}}=\frac{\sin\beta}{\gamma[1-\nu\sin\psi\cos(\alpha-\beta)]}-\gamma\nu\sin\psi\sin\alpha,$
(34) $\displaystyle\hat{\mathcal{V}}^{\hat{\theta}}=-\gamma\nu\cos\psi,$ (35)
$\displaystyle\hat{\mathcal{V}}^{\hat{\varphi}}=\frac{\cos\beta}{\gamma[1-\nu\sin\psi\cos(\alpha-\beta)]}-\gamma\nu\sin\psi\cos\alpha,$
(36)
$\displaystyle\hat{\mathcal{V}}^{\hat{t}}=\gamma\nu\left[\frac{\sin\psi\cos(\alpha-\beta)-\nu}{1-\nu\sin\psi\cos(\alpha-\beta)}\right].$
(37)

###  III.4 Equations of motion

The test particle equations of motion are written in terms of magnitude of
spatial velocity $\nu$, polar $\psi$ and azimuthal $\alpha$ angles of the
spatial velocity measured in the local ZAMO frame, radius $r$, polar angle
$\theta$, and independent from the azimuthal angle $\varphi$ due to rotational
symmetry of the PR model De Falco  _et al._ (2019); Bakala  _et al._ (2019);
De Falco  _et al._ (2020a),

$\displaystyle\frac{d\nu}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{1}{\gamma}\bigg{\\{}\sin\alpha\sin\psi\left[a(\bm{n})^{\hat{r}}+2\nu\cos\alpha\sin\psi\theta(\bm{n})^{\hat{r}}{}_{\hat{\varphi}}\right]$
$\displaystyle+\cos\psi\left[a(\bm{n})^{\hat{\theta}}+2\nu\cos\alpha\sin\psi\theta(\bm{n})^{\hat{\theta}}{}_{\hat{\varphi}}\right]\bigg{\\}}$
$\displaystyle+\frac{\tilde{\sigma}[\Phi
E(\bm{U})]^{2}}{\gamma^{3}\nu}\hat{\mathcal{V}}^{\hat{t}},$
$\displaystyle\frac{d\psi}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{\nu}\left\\{-\sin\alpha\cos\psi\left[a(\bm{n})^{\hat{r}}+2\nu\cos\alpha\sin\psi\theta(\bm{n})^{\hat{r}}{}_{\hat{\varphi}}\right]\right.$
$\displaystyle\left.+\sin\psi\left[a(\bm{n})^{\hat{\theta}}+2\nu\cos\alpha\sin\psi\theta(\bm{n})^{\hat{\theta}}{}_{\hat{\varphi}}\right]\right.$
$\displaystyle\left.+\nu^{2}\left[\left(k(\varphi,\bm{n})^{\hat{\theta}}\cos^{2}\alpha+k(r,\bm{n})^{\hat{\theta}}\sin^{2}\alpha\right)\sin\psi\right.\right.$
$\displaystyle\left.\left.-k(\theta,\bm{n})^{\hat{r}}\sin\alpha\cos\psi\right]\right\\}$
$\displaystyle+\frac{\tilde{\sigma}[\Phi
E(\bm{U})]^{2}}{\gamma\nu^{2}\sin\psi}\left[\hat{\mathcal{V}}^{\hat{t}}\cos\psi-\hat{\mathcal{V}}^{\hat{\theta}}\nu\right],$
$\displaystyle\frac{d\alpha}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{\gamma\cos\alpha}{\nu\sin\psi}\left\\{-\left[a(\bm{n})^{\hat{r}}+2\nu\cos\alpha\sin\psi\theta(\bm{n})^{\hat{r}}{}_{\hat{\varphi}}\right]\right.$
$\displaystyle\left.-\nu^{2}\left[\left(k(\varphi,\bm{n})^{\hat{\theta}}-k(r,\bm{n})^{\hat{\theta}}\right)\cos\psi\sin\psi\sin\alpha\right.\right.$
$\displaystyle\left.\left.+k(\varphi,\bm{n})^{\hat{r}}\sin^{2}\psi+k(\theta,\bm{n})^{\hat{r}}\cos^{2}\psi\right]\right\\}$
$\displaystyle+\frac{\tilde{\sigma}[\Phi
E(\bm{U})]^{2}\cos\alpha}{\gamma\nu\sin\psi}\left[\hat{\mathcal{V}}^{\hat{r}}-\hat{\mathcal{V}}^{\hat{\varphi}}\tan\alpha\right],$
$\displaystyle U^{r}$ $\displaystyle\equiv$
$\displaystyle\frac{dr}{d\tau}=\frac{\gamma\nu\sin\alpha\sin\psi}{\sqrt{g_{rr}}},$
(41) $\displaystyle U^{\theta}$ $\displaystyle\equiv$
$\displaystyle\frac{d\theta}{d\tau}=\frac{\gamma\nu\cos\psi}{\sqrt{g_{\theta\theta}}},$
(42) $\displaystyle U^{\varphi}$ $\displaystyle\equiv$
$\displaystyle\frac{d\varphi}{d\tau}=\frac{\gamma\nu\cos\alpha\sin\psi}{\sqrt{g_{\varphi\varphi}}}-\gamma\frac{N^{\varphi}}{N}.$
(43)

Defining $A=\tilde{\sigma}\Phi_{0}^{2}E^{2}$, which is the so-called
luminosity parameter, and can be also written as $A/M=L/L_{\rm EDD}\in[0,1]$,
where $L$ is the luminosity measured by a static observer at infinity, and
$L_{\rm EDD}$ is the Eddington luminosity De Falco  _et al._ (2019); Bakala
_et al._ (2019); De Falco  _et al._ (2020a). Using Eqs. (20) and (33), we
obtain

$\tilde{\sigma}[\Phi
E(\bm{U})]^{2}=\frac{A\gamma^{2}(1+bN^{\varphi})[1-\nu\sin\psi\cos(\alpha-\beta)]^{2}}{N^{2}r^{2}F_{3}(r,\theta)\sin\beta}.$
(44)

##  IV Critical hypersurfaces

The dynamical system governed by Eqs. (III.4) – (43) admits, as the previous
models, a critical hypersurface outside of the emitting surface, where
gravitational attraction and radiation pressure balance. Such region is
analytically determined by the critical radius $r_{\rm crit}$ as function of
$\theta$, i.e., $r_{\rm crit}=r_{\rm crit}(\theta)$, once the parameters
$(J,Q,A,R_{\star},\Omega_{\star})$ are assigned. We consider a test particle
moving along a non-equatorial plane on purely circular orbit (i.e., the
azimuthal and polar angles related to the test particle’ spatial velocity as
measured in the local ZAMO frame are respectively $\alpha=0,\pi$,
$\psi=\pi/2$, and the magnitude of the spatial velocity is
$\nu=\mbox{const}$). Equation (III.4) for $d\nu/d\tau=0$ reduces to Bini  _et
al._ (2009, 2011a); De Falco  _et al._ (2019); Bakala  _et al._ (2019); De
Falco  _et al._ (2020a)

$\tilde{\sigma}[\Phi
E(\bm{U})]^{2}\hat{\mathcal{V}}^{\hat{t}}=0,\quad\Rightarrow\quad\nu=\cos\beta.$
(45)

The velocity of the test particle equates the photon azimuthal velocity. Since
the test particle moves tangentially on the critical hypersurface, we have
$d\alpha/d\tau=0$, and Eq. (III.4) assumes the following form

$\displaystyle
a(\bm{n})^{\hat{r}}+2\nu\theta(\bm{n})^{\hat{r}}{}_{\hat{\varphi}}+\nu^{2}k(\varphi,\bm{n})^{\hat{r}}$
(46) $\displaystyle=\frac{\tilde{\sigma}[\Phi
E(\bm{U})]^{2}}{\gamma^{2}}\hat{\mathcal{V}}^{\hat{r}},$

which is an implicit equation for determining the critical radius $r_{\rm
crit}$ Bini  _et al._ (2009, 2011a); De Falco  _et al._ (2019); Bakala  _et
al._ (2019); De Falco  _et al._ (2020a). The critical hypersurface is axially
symmetric with respect to the polar direction, and can assume either an oblate
or prolate form depending on the interplay between gravitational pull
$a(\bm{n})^{\hat{r}}$, centrifugal forces $k(\varphi,\bm{n})^{\hat{r}}$, frame
dragging effect $2\nu\theta(\bm{n})^{\hat{r}}{}_{\hat{\varphi}}$, and
radiation forces including the PR effect $\frac{\tilde{\sigma}[\Phi
E(\bm{U})]^{2}}{\gamma^{2}}\hat{\mathcal{V}}^{\hat{r}}$.

In Fig. 1, we plot different configurations of critical hypersurfaces by
varying the values of some parameters. Bearing in mind the condition (5),
which tells from which radius the Hartle-Thorne metric is valid, we decided to
display also the _unphysical solutions_ , which are located inside the
emitting source (gray surface) or not respecting the above requirement,
because we would like to highlight how the critical hypersurface
configurations morph in terms of the parameters’ variability. For convenience
we have defined the Hartle-Thorne spin $a\equiv cJ/(GM^{2})\in[0,1]$ and
quadrupole moment $q\equiv-c^{4}Q/(G^{2}M^{3})\in[-1,0]$. It is also important
to note that since the Hartle-Thorne metric is an approximate solution in
terms of $a$ and $q$, it works for $a,-q\ll 1$. We use only negative values of
the quadrupole parameter $q$, otherwise we checked that no critical
hypersurface exists. A physical explanation of the occurrence of such a
phenomenon can be attributed to the combined effect of the centrifugal and
frame-dragging forces, which are responsible for sweeping the test particle
away.

We immediately see that the luminosity parameter $A$ plays a fundamental role
in shaping the critical hypersurface. In particular, high luminosities
$A=L/L_{\rm Edd}\gtrsim 0.7$ are needed to have a critical hypersurface
relatively far from the emitting surface (see upper left panel). Increasing
the spin values, the critical hypersurface becomes more oblate (see upper
right panel), and the same argument holds also for the quadrupole moment (see
lower left panel). The rotation of the emitting surface, $\Omega_{\star}$,
strongly contributes also in shaping the form of the critical hypersurface
(see lower right panel). In conclusion, we can infer that the radiation field
intensity and the different gravitational effects (i.e., curved geometry,
frame dragging, and centrifugal forces) only along the radial direction
contribute to morph the critical hypersurface.

Figure 1: We show in different plots the shapes of the critical hypersurfaces
in terms of varying parameters. In all plots the emitting surface radius is
$R_{\star}=6M$ and is displayed as a gray surface, and the red arrow points in
the positive polar direction. Left upper panel: We fix $a=0.1$, $q=-0.3$,
$\Omega=69\times 10^{-4}M^{-1}$ and change the relative luminosity
$A=0.80,\,0.82,\,0.85,\,0.88,\,0.90$. Right upper panel: We fix $A=0.8$,
$q=-0.3$, $\Omega=0.034M^{-1}$ and change the spin $a=0.1,\,0.2,\,0.3,\,0.5$.
Left lower panel: We fix $A=0.8$, $a=0.1$, $\Omega=0.034M^{-1}$ and change the
quadrupole moment $q=-0.1,\,-0.4$. Right lower panel: We fix $A=0.8$, $a=0.1$,
$q=-0.3$, and change the critical hypersurface angular velocity
$\Omega=0.019M^{-1},0.029M^{-1},0.038M^{-1},0.048M^{-1}$.

In the next sections, we derive the conditions to obtain suspended orbits
(Sec. IV.1), and we apply this model to describe the emission properties of a
NS (Sec. IV.2).

###  IV.1 Suspended orbits

The test particle could move on circular orbits bounded on the critical
hypersurface at constant height $y\neq 0$ (off-equatorial plane), without the
action of the latitudinal drift mechanism (see Refs. De Falco  _et al._
(2019); Bakala  _et al._ (2019), for further details). To obtain such
configurations, the test particle must touch the critical hypersurface with
the following conditions: $\alpha=0,\pi$, $\nu=\cos\beta$, $r=r_{\rm
crit}(\theta)$, and $d\psi/d\tau=0$ (where this last condition is the strong
constraint for not having latitudinal drift towards the equatorial plane).
Vanishing Eq. (III.4), it is possible to determine the value of $\psi$, by
solving this implicit equation Bakala  _et al._ (2019):

$\displaystyle
a(\bm{n})^{\hat{\theta}}+2\nu\sin\psi\theta(\bm{n})^{\hat{\theta}}{}_{\hat{\varphi}}+\nu^{2}k(\varphi,\bm{n})^{\hat{\theta}}$
(47) $\displaystyle+\frac{\tilde{\sigma}[\Phi
E(\bm{U})]^{2}}{\gamma^{2}\nu\sin^{2}\psi}\left[\hat{\mathcal{V}}^{\hat{t}}\cos\psi-\hat{\mathcal{V}}^{\hat{\theta}}\nu\right]=0.$

The value of $\psi$ strongly depends on emitting surface location $R_{\star}$,
angular velocity $\Omega_{\star}$, and compact object quadrupole moment $q$
and spin $a$.

Figure 2: Different configurations of suspended orbits at height $\theta$ on
the critical hypersurface in terms of the angle $\psi$ are shown. We set
$A=0.8$, $\Omega_{\star}=0.01$, and $R_{\star}=3M$. Once we fix $q=-0.4$, and
change the value of the spin $a=0.1,0.2,0.3,0.4$ (see left panel), and then we
fix $a=0.1$ and change the value of the quadrupole moment
$q=-0.1,-0.2,-0.3,-0.4$ (see right panel).

In Fig. 2 we show the angle $\psi$ at which the test particle should touch the
critical hypersurface to reach the fixed height $\theta$, and moving on such
plane on a circular orbit. We note that it is always possible to have
suspended orbits both on and off the equatorial plane. It is interesting to
note that increasing the module of the spin the $\psi$ angle decreases because
the test particle has to contrast stronger forces (see left panel). The same
argument holds also for the module of $q$ (see right panel). These particular
configurations, on which a test particle moves stably, are typical of the
general relativistic PR effect in the 3D space De Falco  _et al._ (2019);
Bakala  _et al._ (2019); De Falco  _et al._ (2020a). Their formation is a
result of the perfect balance among the gravitational contributions (including
also the polar centrifugal force and the frame-dragging effect) and the
radiation forces along the polar direction, see Eq. (47).

We note that off-equatorial suspended orbits are the consequence of our
assumptions on the radiation stress-energy tensor (12), which is constituted
by “a single stream of photons” reaching the test particle at each instant of
time. In other models of the 3D general relativistic PR effect Wielgus (2019),
the radiation source is modeled by a bunch of photons coming from the whole 3D
emitting surface. In this case, the radiation force drives always the test
particle toward the equatorial plane, where it moves stably, since there is a
perfect balance of both gravitational and radiation forces from the two
hemispheres of the emitting surface.

###  IV.2 Radiation effects around a neutron star

The present model of the general relativistic PR effect in the Hartle-Thorne
metric can be used to describe several radiation processes occurring on and
around a NS, like: accretion phenomena, type-I X-ray bursts, photospheric
radius expansion. To see how to apply our developments to a NS, we know that
such an astrophysical object is described by mass $M$, radius $R_{\star}$,
spin frequency $f$ (or angular velocity $\Omega_{\star}$), Hartle-Thorne
angular momentum $a$ and quadrupole moment $q$. Since we have a non-spherical
distribution of the mass, we can also consider that the NS shape is not
anymore spherical, but deformed as an ellipsoid. However, due to the axially
symmetry of the Hartle-Thorne spacetime, it is reasonable to assume that this
ellipsoid is rotationally symmetric and therefore it is defined by the
equatorial $R_{\rm eq}$ and polar $R_{\rm pol}$ radii. Therefore, the NS form
is described by

$\frac{x^{2}+y^{2}}{R_{\rm eq}^{2}}+\frac{z^{2}}{R_{\rm pol}^{2}}=1,\ \
\Leftrightarrow\ \ \begin{cases}x=R_{\rm eq}\sin\theta\cos\varphi,\\\ y=R_{\rm
eq}\sin\theta\sin\varphi,\\\ z=R_{\rm pol}\cos\theta.\end{cases}$ (48)

The polar radius can be also expressed in terms of the ellipticity $e$, namely
$R_{\rm pol}=R_{\rm eq}(1+e)$.

Therefore, the NS is defined by six parameters $\left\\{M,f,R_{\rm
eq},a,q,e\right\\}$. Since we have already several other parameters for
characterizing the radiation processes, we would like to reduce the NS
parameter space. To this end, we follow the approach of Bauböck and
collaborators Bauböck  _et al._ (2013), defining the following set of
parameters:

$\displaystyle f_{0}=\frac{1}{2\pi}\sqrt{\frac{GM}{R_{\rm
eq}^{3}}},\quad\epsilon_{0}=\frac{f}{f_{0}},\quad a=\epsilon_{0}a^{*},$ (49)
$\displaystyle q=\epsilon_{0}^{2}q^{*},\quad
e=\epsilon_{0}^{2}e^{*},\quad\zeta=\frac{GM}{c^{2}R_{\rm eq}},$

where $f_{0}$ is the Keplerian angular velocity of a test particle orbiting at
a radius $R_{\rm eq}$ around a mass $M$, corresponding also to the maximum NS
frequency to which it can be spun up before breakup. We label the parameters
with asterisk in order to highlight that they depend on the particular NS
equation of state considered.

Bauböck and collaborators show that the actual six-parameter space can be
reduced to a three-parameter space spanned by $\left\\{M,R_{\rm
eq},f\right\\}$, which is valid over the astrophysically relevant parameter
range and for a variety of equations of state. Therefore, the remaining three
parameters $\left\\{a,q,e\right\\}$ can be written in terms of
$\left\\{M,R_{\rm eq},f\right\\}$ through the following equations Bauböck  _et
al._ (2013):

$\displaystyle a$ $\displaystyle=$
$\displaystyle\epsilon_{0}(1.1035-2.146\zeta+4.5756\zeta^{2}),$ (50)
$\displaystyle q$ $\displaystyle=$ $\displaystyle
a^{2}\exp\left[-2.014+0.601\log\left(\frac{a}{\epsilon_{0}}\zeta^{-3/2}\right)\right.$
$\displaystyle\left.+1.10\log\left(\frac{a}{\epsilon_{0}}\zeta^{-3/2}\right)^{2}-0.412\log\left(\frac{a}{\epsilon_{0}}\zeta^{-3/2}\right)^{3}\right.$
$\displaystyle\left.+0.0459\log\left(\frac{a}{\epsilon_{0}}\zeta^{-3/2}\right)^{4}\right],$
$\displaystyle e$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{0}^{2}}{32\zeta^{3}}\left\\{2\zeta\left[8\zeta^{2}-32a^{*}\zeta^{7/2}+8a^{*}{}^{2}\zeta^{5}-48a^{*}{}^{2}\zeta^{6}\right.\right.$
$\displaystyle\left.\left.+(a^{*}{}^{2}-q^{*})(45-135\zeta+60\zeta^{2}+30\zeta^{3})+24a^{*}{}^{2}\zeta^{4}\right]\right.$
$\displaystyle\left.+45(a^{*}{}^{2}-q^{*})(1-2\zeta)^{2}\log(1-2\zeta)\right\\}.$

We note that since we assumed an ellipsoid shape, where the NS radius can be
written as

$R_{\rm NS}(\theta)=\sqrt{R_{\rm eq}^{2}\sin^{2}\theta+R_{\rm
pol}^{2}\cos^{2}\theta},$ (53)

we have that the formula to calculate the impact parameter (17) must be
slightly changed in the following form

$b=\left[-\frac{g_{t\varphi}+g_{\varphi\varphi}\Omega_{\star}}{g_{tt}+g_{t\varphi}\Omega_{\star}}\right]_{r=R_{\rm
NS}(\theta)}.$ (54)

We can also relate the angular velocity of the emitting surface
$\Omega_{\star}$ in terms of the frequency $f$ through

$\Omega_{\star}=2\pi f\frac{GM}{c^{3}}.$ (55)

To further reduce our parameter space, we fix the values of the NS mass
$M=1.4M_{\odot}$, and equatorial radius $R_{\rm eq}=6M$, therefore the
remaining free parameter is only $f$. In Fig. 3, we plot different NS critical
hypersurfaces for different values of the luminosity parameter $A$. We have
checked also how the critical hypersurfaces would have altered its shape, if
we had varied $f\in[0,400]$ Hz (physically allowed NS frequency range), but no
significant changes have been found. Therefore, in Fig. 3 we fix $f=400$ Hz.
In such cases, we have the following values of the dependent parameter set:
$R_{\rm pol}=6.23M$, $\Omega_{\star}=17\times 10^{-3}\ M^{-1}$ , $a=0.22$,
$q=0.29$, and $e=0.04$. This example confirms that even if we consider a high
spin frequency, $f=400$ Hz, the above approach results in a rather low spin
parameter, $a=0.22$, within the applicability of the Hartle-Thorne spacetime
model.

Figure 3: Critical hypersurfaces for a NS of mass $M=1.4M_{\odot}$, equatorial
radius $R_{\rm eq}=6M$, and spin frequency $f=400$ Hz, and for different
luminosity parameter values $A=0.82,0.83,0.85,0.88,0.9$ (corresponding
respectively to the orange, blue, green, red, and violet colors of the
displayed surfaces). The gray surface represents the NS and the red arrow the
positive polar direction.

From this plot we also note that for having critical hypersurfaces outside of
the NS surface, we need to have quite high-luminosities, namely $A=L/L_{\rm
Edd}\gtrsim 0.82$. In Fig. 4 we determine the critical luminosity $A_{\rm
crit}$, being the luminosity $A$ at which the critical hypersurfaces touches
the NS surface at height $\theta$. The critical luminosity at the equatorial
plane is $A_{\rm crit}=0.7654$, and at the poles is $A_{\rm crit}=0.836$;
while the average critical luminosity is $A_{\rm crit}=0.800$, and finally the
critical luminosity for $f=0$ (i.e., $a=0$ and $q=0$, the Schwarzschild
metric), remains constant at $A_{\rm crit}=0.817$.

Figure 4: Critical luminosity $A_{\rm crit}$ in terms of the polar angle
$\theta$ for $M=1.4M_{\odot}$, $R_{\rm eq}=6M$, $f=400$ Hz. The red dashed
line represents the average critical luminosity, while the blue continuous
line is the critical luminosity for $f=0$ (i.e., $a=0$ and $q=0$), namely it
is framed in the Schwarzschild spacetime.

##  V Conclusions

We have developed, for the first time in the literature, the full general
relativistic 3D (and 2D by considering $\theta=\psi=\pi/2$) treatment of the
motion of a test particle around a non-spherical, and slowly rotating compact
object, described by the Hartle-Thorne metric, and in the same time affected
by the radiation field, including the general relativistic PR effect, from a
spherical and rigidly rotating emitting surface located outside the compact
object (see Sec. III). The Hartle-Thorne spacetime is an approximate solution
of the Einstein field equations in the vacuum, and it is described in terms of
three parameters: the mass $M$, the angular momentum $J$, and the quadrupole
moment $Q$ (see Sec. II.1).

In order to make our approach more flexible in view of extension of this model
for other metrics more realistic than the Hartle-Thorne description, we have
casted our initial calculations in a modular form (see Sec. II.3). Indeed,
changing the functional form of the functions
$F_{1}(r,\theta),F_{2}(r,\theta),F_{3}(r,\theta)$, and calculating the related
derivatives with respect to $r$ and $\theta$, it is possible to obtain the
ZAMO quantities and then to straightforwardly derive the equations of motion,
the critical hypersurface, and the suspended orbits. In addition in the
Schwarzschild limit (i.e., $q\to 0$ and $a\to 0$), we have that
$F_{1}(r,\theta),F_{2}(r,\theta),F_{3}(r,\theta)\to 1$, and the metric (II.1)
reduces to the Schwarzschild spacetime.

The critical hypersurface equation (46) depends on the luminosity parameter
$A=L/L_{\rm Edd}$, the photon impact parameter $b$ (or equivalently from
radius $R_{\star}$ and angular velocity $\Omega_{\star}$ of the emitting
surface), the spin parameter $a$, and quadrupole moment $q$, see Sec. IV. In
Fig. 1, we have produced different configurations of the critical
hypersurfaces by varying the values of the parameters. We have shown that the
radiation field and the gravitational effects strongly contribute to morph the
critical hypersurfaces. In addition, high luminosities $A\gtrsim 0.7$ permits
to have the PR critical hypersurfaces located outside the emitting surface,
making them physically possible.

We have analysed also the suspended orbits, which are configurations where the
test particle moves on bound off-equatorial circular orbits on the critical
hypersurface at a given $\theta$ height, see Sec. IV.1. In Fig. 2 we have
plotted the angle $\psi$ at which the test particle should be sent in order to
move on a suspended orbit, in terms of the height $\theta$, and once by fixing
the quadrupole moment and changing the angular momentum (see left panel), and
then viceversa (see right panel). Therefore, it is possible to obtain
suspended orbits at all heights for different values of the parameters $a,q$,
due to the perfect equilibrium between gravitational and radiation forces,
without having latitudinal drift motion towards the equatorial plane.

Finally, we have also proposed an application of the PR effect to model NSs,
see Sec. IV.2. In this case, we have found a way to write the parameters
$\left\\{q,a,e,\Omega_{\star}\right\\}$ in terms of the quantities
$\left\\{M,R_{\rm eq},f\right\\}$ through Eqs. (50) – (IV.2), which are valid
for a variety of equations of state. We have considered that the NS is not
anymore a spherical body, but it is an ellipsoid with rotational-azimuthal
symmetry, and determined by the equatorial $R_{\rm eq}$ and polar $R_{\rm
pol}$ radii, where the latter can be expressed also in terms of the
eccentricity $e$. This entailed to slightly change the expression of the
photon impact parameter $b$, see Eq. (54). We can further reduce the parameter
space by setting $M=1.4M_{\odot}$ and $R_{\rm eq}=6M$ (typical NS mass and
radius), so that all the parameters will depend only on the spin frequency
$f$. In Fig. 3 we plotted the critical hypersurfaces, but only by varying the
luminosity parameter $A=L/L_{\rm Edd}$, because there are no significant
change in terms of $f$. Then, we have also analysed at which luminosity
$A=L/L_{\rm Edd}$ the critical hypersurface touches the NS surface at the
height $\theta$, comparing also these configurations with the Schwarzschild
case, see Fig. 4. This can be the initial set up for then developing
astrophysical models involving radiation effects occurring either around or on
the surface of a NS.

As future project, we aim at extending our treatment of the 3D general
relativistic PR effect around a fast rotating and non-spherical quadrupolar
massive source. These astrophysical objects can be modeled by several
sophisticated metrics, which in general due to the complex treatment requires
specific numerical treatments Friedman and Stergioulas (2013).

## Acknowledgements

The authors thank the anonymous referee for the useful remarks and comments
given in the review process. V.D.F. thanks Gruppo Nazionale di Fisica
Matematica of Istituto Nazionale di Alta Matematica for the support. M.W.
acknowledges the support of the Black Hole Initiative at Harvard University,
which is funded by grants from the John Templeton Foundation and the Gordon
and Betty Moore Foundation to Harvard University.

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# Current cross-correlations in a quantum Hall collider at filling factor two

Edvin G. Idrisov  Department of Physics and Materials Science, University of
Luxembourg, Luxembourg  Ivan P. Levkivskyi  Dropbox Ireland, One Park Place,
Hatch Street Upper, Dublin, Ireland  Eugene V. Sukhorukov  Département de
Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland
Thomas L. Schmidt  Department of Physics and Materials Science, University of
Luxembourg, Luxembourg

###### Abstract

We use the non-equilibrium bosonization technique to study the effects of
Coulomb interactions in mesoscopic electron colliders based on quantum Hall
(QH) edge states at filing factor $\nu=2$. The current cross-correlations and
Fano factor, which carry the information about the exclusion statistics, are
calculated. It is shown that both these quantities have a non-analytical
dependence on the source transparency, which scales as $\log(1/T_{s})$ at
small $T_{s}\ll 1$. This is the consequence of electron-electron interactions
in the outgoing non-equilibrium states of the collider.

The progress in experimental techniques at the nanoscale has allowed
experimentalists to design composite systems based on quantum Hall (QH) edge
states [1, 2, 3, 4]. This development has already led to a better
understanding of fundamental phenomena in mesoscopic physics, such as phase
coherence [5, 6, 7, 8, 9, 10, 11], charge and heat quantization [12, 13, 14,
15, 16, 17, 18], as well as equilibration and relaxation [19, 20, 21, 22, 23,
24, 25, 26, 27, 28]. In particular, these novel architectures have started the
field of electron quantum optics, where devices and phenomena known from
optics have been recreated in the electronic domain.

One essential element of these experimental setups is a quantum point contact
(QPC), i.e., a narrow channel between two electrically conducting systems
[29]. In the context of electron quantum optics, such QPCs act as electron
beam splitters and provide a platform for tunneling experiments in the
presence of different kind of injection sources [30, 31]. Particular examples
are electron analogs of the photonic Hong-Ou-Mandel, Hanbury Brown and Twiss,
Mach-Zehnder and Fabry-Perot experiments [32]. The current and shot noise
through a QPC connecting integer and fractional QH edge states have been
measured in several experiments [2]. These studies have made it possible to
distinguish Fermi liquid and non-Fermi liquid phases in current-voltage
characteristics and in the corresponding current noise [33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43]. Apart from that, it is worth mentioning that the
noise provides one of the most straightforward methods to measure the
effective charge of tunneling quasiparticles in the fractional QH regime [44].

Using a combination of QPCs, experimentalists were able recently to build
_anyonic_ colliders at a filling factor $\nu=1/m$ (where $m$ is an odd
integer) [45]. Their measurements of the current cross-correlations at zero
frequency and the generalized Fano factor have provided evidence for anyonic
exchange statistics of charge carriers with an exchange phase $\pi/3$ for
$m=3$. This agrees with the theoretical predictions for the Laughlin state at
$\nu=1/3$ [46]. It is worth mentioning that the measurement of the noise in
such colliders is an alternative method for the investigation of quasiparticle
statistics, compared to the conventional Mach-Zehnder and Fabry-Perot
interferometers, where the measured current or conductance is sensitive to the
exchange phase [47].

Figure 1:  Schematic picture of a mesoscopic electron collider based on QH
edge states at filling factor $\nu=2$. Two independent voltage-biased sources
QPC1 and QPC2 with respective transparencies $T_{1d}$ and $T_{1u}$ generate
non-equilibrium states at the upper and lower edges, which collide at QPC3
with transparency $T$. The zero-frequency current noise $j_{1u}(t)$ and
$j_{1d}(t)$ after QPC3 is measured. The charge density fields are expressed in
terms of non-equilibrium chiral boson fields $\phi_{\alpha s}(x,t)$ where
$\alpha\in\\{1,2\\}$ and $s\in\\{u,d\\}$.

In this paper, we consider a mesoscopic electron collider based on QH edge
states at integer filling factor $\nu=2$, where the incoming non-equilibrium
states are created using two quantum point contacts (QPC1 and QPC2), with
respective transparencies $T_{1s}\ll 1$, where $s\in\\{u,d\\}$ denotes the
upper and lower channels (see Fig. 1 for a schematic illustration). At zero
temperature, the resulting incoming states have “double-step” distribution
functions
$f_{1s}(\epsilon)=\theta(-\epsilon)+T_{1s}\theta(\epsilon)\theta(V-\epsilon)$,
where $V$ is the source voltage. Due to Coulomb interactions the incoming
states split into two modes with different velocities. Subsequently, the non-
equilibrium modes arriving from the upper and lower part of the collider are
mixed by a third QPC (QPC3) with transparency $T$. In order to investigate the
statistical fluctuations after the collision at QPC3, which yield information
about the exclusion statistics, we calculate the zero-frequency cross-
correlations of the current [45]

$S_{1u;1d}(\omega=0)=2\int dt\langle\delta j_{1u}(t)\delta j_{1d}(0)\rangle,$
(1)

where $\delta j_{1s}(t)=j_{1s}(t)-\langle j_{1s}(t)\rangle$ and the average is
taken with respect to non-equilibrium state created by injection currents from
QPC1 and QPC2. This quantity has been measured in recent experiment [45]. For
chiral fermions the current in Eq. (1) is proportional to the charge density,
$j_{1s}(x,t)\propto\psi^{\dagger}_{1s}(x,t)\psi_{1s}(x,t)$. Apart from the
current noise, we are interested in the generalized Fano factor, which
contains information about the exclusion statistics [45]. It is defined as

$P=\frac{S_{1u;1d}(0)}{2eT(1-T)I_{+}},$ (2)

where $I_{+}=e^{2}V(T_{1u}+T_{1d})/2\pi\hbar$ is the total input current. The
theoretical study of current cross-correlations is non-trivial since one has
to deal with a non-equilibrium situation due to the injection QPCs as well as
with strong electron-electron interactions. For this reason, it is convenient
to use the non-equilibrium bosonization approach [48, 49, 50] which we explain
below. Throughout the paper, we focus on the case of zero temperature. We set
$|e|=\hbar=k_{B}=1$ during the calculation and restore the dimensions in main
results for the noise at the end.

Theoretical method. To study quantum Hall edge states at filling factor
$\nu=2$, it is convenient to use their effective field theory [51]. According
to this, the low-energy degrees of freedom of the edge states are collective
fluctuations of the charge densities $\rho_{\alpha s}(x)$, where the
subscripts $\alpha\in\\{1,2\\}$ and $s\in\\{u,d\\}$ denote the channels (see
Fig. 1). The fermionic fields $\psi_{\alpha s}(x,t)\propto\exp[i\phi_{\alpha
s}(x,t)]$ and the charge density fields, $\rho_{\alpha
s}(x,t)=(1/2\pi)\partial_{x}\phi_{\alpha s}(x,t)$, are represented in terms of
chiral boson fields which satisfy the standard equal-time commutation
relations

$[\phi_{\alpha s}(x,t),\phi_{\beta
r}(y,t)]=i\pi\delta_{\alpha\beta}\delta_{sr}\text{sgn}(x-y).$ (3)

The Hamiltonian of the edge states in the presence of Coulomb interactions is
given by

$H=\frac{1}{2}\sum_{s,\alpha\beta}\int\int
dxdyV_{\alpha\beta}(x-y)\rho_{\alpha s}(x)\rho_{\beta s}(y),$ (4)

where the integral kernel $V_{\alpha\beta}(x-y)=2\pi
v_{F}\delta_{\alpha\beta}\delta(x-y)+U_{\alpha\beta}(x-y)$ includes both the
kinetic energy of the free chiral electrons with Fermi velocity $v_{F}$ as
well as the Coulomb interaction between electrons with interaction potential
$U_{\alpha\beta}(x)$. At low energies the characteristic length scales are
much longer than the screening length of the Coulomb interaction. Therefore,
one can further approximate the Coulomb potential as short-ranged, i.e.,
$U_{\alpha\beta}(x-y)=U_{\alpha\beta}\delta(x-y)$, with non-zero diagonal and
off-diagonal interaction strengths $U_{\alpha\beta}$ [32].

The current cross-correlations can be calculated by using the scattering
approach locally at QPC3 [52]. One finds that the cross-correlations of the
outgoing currents after QPC3 at zero frequency are given by a sum of two terms
[46, 45]

$S_{1u;1d}(\omega=0)=\mathcal{P}_{1}(0)+\mathcal{P}_{2}(0).$ (5)

The first term is associated with a transmitted noise and has the simple form,

$\mathcal{P}_{1}(0)=\frac{e^{2}V}{2\pi\hbar}(1-T)T\sum_{s=u,d}T_{1s}(1-T_{1s}).$
(6)

The second term describes the generated noise by QPC3,

$\mathcal{P}_{2}(0)=2(1-T)Ta^{-2}\int dtK_{u}(t)G_{d}(t).$ (7)

where $a$ is the short-distance cut-off from bosonization and

$\displaystyle K_{u}(t)=\langle
e^{-i\phi_{1u}(L,t)}e^{i\phi_{1u}(L,0)}\rangle,$ (8) $\displaystyle
G_{d}(t)=\langle e^{i\phi_{1d}(L,t)}e^{-i\phi_{1d}(L,0)}\rangle.$

All bosonic fields should be evaluated at the position $x=L$ of QPC3 and the
averages in the correlation functions are taken with respect to the non-
equilibrium state created by the source QPC1 and QPC2. Next, we apply the non-
equilibrium bosonization technique [48, 49, 50] to evaluate these correlation
functions.

The Hamiltonian of the edge states in Eq. (4) generates the equation of motion
of the bosonic fields $\phi_{\alpha s}(x,t)$. The latter have to be
supplemented with the appropriate boundary conditions due to source QPCs,
resulting in

$\displaystyle\partial_{t}\phi_{\alpha s}(x,t)$
$\displaystyle=-\frac{1}{2\pi}\sum_{\beta}\int
dyV_{\alpha\beta}(x-y)\partial_{y}\phi_{\beta s}(y,t),$
$\displaystyle\partial_{t}\phi_{\alpha s}(0,t)$ $\displaystyle=-2\pi j^{{\rm
in}}_{\alpha s}(t),$ (9)

where the incoming currents $j^{{\rm in}}_{\alpha s}(t)$ are defined at the
point $x=0^{+}$ right after the source QPC1 and QPC2. These equations of
motion can be solved by diagonalizing the interaction matrix
$V_{\alpha\beta}=2\pi v_{F}\delta_{\alpha\beta}+U_{\alpha\beta}$ via a
Bogoliubov rotation, $V=S(\theta)\Lambda S^{\dagger}(\theta)$, which conserves
the commutation relations of the bosonic fields. The rotation angle is
determined by the strength of the electron-electron interaction and is given
by $\tan(2\theta)=2V_{12}/(V_{11}-V_{22})$. The new collective excitations are
characterized by two velocities, $\Lambda=\text{diag}(u,v)$, where
$u,v=(V_{11}+V_{22})/2\pm\sqrt{(V_{11}-V_{22})^{2}/4+V^{2}_{12}}$ and we used
that $V_{12}=V_{21}$. Imposing the boundary conditions (Current cross-
correlations in a quantum Hall collider at filling factor two) we obtain the
dynamics of bosonic fields in both channels $s\in\\{u,d\\}$,

$\displaystyle\phi_{1s}(x,t)$
$\displaystyle=-\lambda_{1}Q_{1s}(t_{u})-\lambda_{2}Q_{2s}(t_{u})$
$\displaystyle-\lambda^{\prime}_{1}Q_{1s}(t_{v})+\lambda_{2}Q_{2s}(t_{v}),$
(10) $\displaystyle\lambda_{1}$ $\displaystyle=\pi[1-\cos(2\theta)],$
$\displaystyle\lambda_{2}$ $\displaystyle=\pi\sin(2\theta),$
$\displaystyle\lambda^{\prime}_{1}$ $\displaystyle=2\pi-\lambda_{1},$

where we have introduced the injected charges

$\displaystyle Q_{\alpha s}(t)=\int^{t}_{-\infty}dt^{\prime}j^{{\rm
in}}_{\alpha s}(t^{\prime})$ (11)

and the times $t_{u}=t-x/u$ and $t_{v}=t-x/v$. Therefore, the bosonic fields
(Current cross-correlations in a quantum Hall collider at filling factor two)
at observation point $x$ at time $t$ are determined by the charges which
arrive with different delay times $t_{u}$ and $t_{v}$, and these charges are
weighted with coupling constants due to intra- and inter-channel electron-
electron interactions. Now we have all necessary ingredients to calculate the
generated noise from Eq. (7).

Gaussian noise regime. First, we consider the case of Gaussian fluctuations of
the bosonic fields. This means that the correlation function in Eq. (8) can be
written as $\log K_{u}(t)=2\pi\langle j^{{\rm in}}_{1u}\rangle
t+\langle\phi_{1u}(L,t)\phi_{1u}(L,0)\rangle-\langle\phi^{2}_{1u}(L,t)\rangle/2-\langle\phi^{2}_{1u}(L,0)\rangle/2$,
where $\langle j^{{\rm in}}_{1u}\rangle=VT_{1u}/2\pi$ is the current injected
from upper QPC1. Now using Eq. (Current cross-correlations in a quantum Hall
collider at filling factor two) we obtain

$\displaystyle\log
K_{u}(t)=iT_{1u}Vt-\int\frac{d\omega}{\omega^{2}}(1-e^{-i\omega
t})\left(\mathcal{F}_{1}+\mathcal{F}_{2}\right),$ (12)
$\displaystyle\mathcal{F}_{1}=2\pi\left[1-\frac{\lambda_{1}\lambda^{\prime}_{1}}{\pi^{2}}\sin^{2}\left(\frac{\omega
t_{L}}{2}\right)\right]S^{{\rm in}}_{1u}(\omega),$
$\displaystyle\mathcal{F}_{2}=\frac{2\lambda^{2}_{2}}{\pi}\sin^{2}\left(\frac{\omega
t_{L}}{2}\right)S^{{\rm in}}_{2u}(\omega),$

where we have introduced the spectral function of current fluctuations,
$S^{{\rm in}}_{1u}(\omega)=\int dte^{i\omega t}\langle\delta j^{{\rm
in}}_{1u}(t)\delta j^{{\rm in}}_{1u}(0)\rangle$, and the delay time between
wave packets, $t_{L}=L/v-L/u$, where $L$ is the distance between the injection
QPCs and QPC3. A similar expression can be obtained for $G_{d}(t)$ by
replacing the subscript $u$ with $d$.

To obtain the spectral function of the incoming state, we again use the free-
fermion scattering approach [52] because the interactions do not influence the
tunneling at the injection QPCs. The general result for $\alpha\in\\{1,2\\}$
and $s\in\\{u,d\\}$ is then given by the sum of two terms,

$S^{{\rm in}}_{\alpha s}(\omega)=S_{q}(\omega)+(1-T_{\alpha s})T_{\alpha
s}S_{n}(\omega),$ (13)

where $S_{q}(\omega)=\omega\theta(\omega)/2\pi$ is the equilibrium ground
state contribution and $S_{n}(\omega)=\sum_{\pm}S_{q}(\omega\pm
V)-2S_{q}(\omega)$ is the non-equilibrium part which originates from the
voltage-biased injection QPCs. Substituting Eq. (12) into Eq. (7), the second
term in Eq. (5) takes the following final form

$\mathcal{P}_{2}(0)=-\frac{e^{2}V}{2\pi\hbar}2(1-T)T\int\frac{dz}{2\pi}\frac{e^{i(T_{1u}-T_{1d})z}}{(z-i0^{+})^{2}}K_{n}(z),$
(14)

where we have introduced the dimensionless integration variable $z=Vt$ and
used

$\displaystyle\log K_{n}(z)=-2\left[(1-T_{1u})T_{1u}+(1-T_{1d})T_{1d}\right]$
(15)
$\displaystyle\times\int_{0}^{1}\frac{dx}{x^{2}}(1-x)[1-\cos(zx)]\left[1-\frac{\lambda_{1}\lambda^{\prime}_{1}}{\pi^{2}}\sin^{2}\left(\frac{xL}{2L_{ex}}\right)\right],$

where $L_{ex}=uv/[(u-v)V]$ is the characteristic correlation length between
the two modes with velocities $u$ and $v$ and $x=\omega/V$ is the
dimensionless variable of integration.

Figure 2:  The generalized Fano factor $P$ as a function of the dimensionless
length $L/L_{ex}$ for different values of the interaction strength,
parameterized by $\lambda_{1}=\pi[1-\cos(2\theta)]$, in the Gaussian noise
regime. We set $T_{s}=0.15$.

In order to understand and compare the case of Gaussian noise to experimental
data, we consider the generalized Fano factor (2) at $T_{1u}=T_{1d}=:T_{s}$,

$\displaystyle
P=1-T_{s}+\frac{1}{T_{s}}\int\frac{dz}{2\pi}\frac{e^{-4R_{s}T_{s}\Phi(\lambda_{1},L/L_{ex},z)}}{(z-i0^{+})^{2}},$
(16) $\displaystyle\Phi(\lambda_{1},L/L_{ex},z)=$
$\displaystyle\int_{0}^{1}\frac{dx}{x^{2}}(1-x)[1-\cos(zx)]\left[1-\frac{\lambda_{1}\lambda^{\prime}_{1}}{\pi^{2}}\sin^{2}\left(\frac{xL}{2L_{ex}}\right)\right],$

as a function of the dimensionless variable $L/L_{ex}$. Figure 2 shows that
the Fano factor is negative, in contrast to experimental data from Ref. [45],
indicating that the experiment was perhaps not performed in the Gaussian
regime.

Finally, we would like to mention briefly that our results do not apply in the
strictly non-interacting limit. In that case, the two modes have identical
velocities, $u=v$, which leads to a divergence $L_{ex}\to\infty$.

Non-Gaussian noise regime. Next, we consider the fluctuations of the bosonic
fields as non-Gaussian at long distances $L\gg
L_{ex}/\text{min}\\{T_{1u},T_{1d}\\}$. At such distances, the partitioned
charges $Q_{1s}$ at the source QPCs, taken at times $t_{u}$ and $t_{v}$, can
be considered as uncorrelated. Therefore, the correlation functions associated
with the upper and lower parts of the collider in Eq. (8) split into products
of four factors

$\displaystyle
K_{u}(t)=\chi_{1u}(\lambda_{1},t)\chi_{1u}(\lambda^{\prime}_{1},t)\chi_{2u}(\lambda_{2},t)\chi_{2u}(-\lambda_{2},t),$
(17) $\displaystyle
G_{d}(t)=\chi_{1d}(-\lambda_{1},t)\chi_{1d}(-\lambda^{\prime}_{1})\chi_{2d}(-\lambda_{2},t)\chi_{2d}(\lambda_{2},t),$

where each factor represents a generator of full counting statistics (FCS)
[32],

$\chi_{\alpha s}(\lambda,t)=\langle e^{i\lambda Q_{\alpha s}(t)}e^{-i\lambda
Q_{\alpha s}(0)}\rangle.$ (18)

For the non-equilibrium state created at QPC1 and QPC2, the generating
function can be written analytically in the long-time Markovian limit, which
gives the main contribution to integrals in Eq. (7). This is known as the
Levitov-Lee-Lesovik formula [53]. In the Poissonian regime of small
transparencies $T_{1s}\ll 1$, the Levitov-Lee-Lesovik formula for positive
times $t>0$ is given by

$\log\chi_{\alpha
s}(\lambda,t)=\frac{\lambda^{2}}{4\pi^{2}}\log\left(\frac{\gamma}{it+\gamma}\right)-\frac{Vt}{2\pi}T_{\alpha
s}(1-e^{i\lambda}).$ (19)

where $\gamma\propto a/(2\pi v_{F})$ is related to the short-distance cutoff
$a$. For $t<0$ the FCS can be calculated by analytic continuation of Eq. (19),
namely $\chi_{\alpha s}(\lambda,-t)=\chi^{\ast}_{\alpha s}(\lambda,t)$. Since
the channels with subscript $\alpha=2$ are not biased and $T_{2s}=0$, the FCS
generators for these channels are given by the first term of Eq. (19). It is
worth mentioning that this is the ground state (Fermi sea) contribution.
However, the channels with $\alpha=1$ include the second term, which is a non-
Gaussian contribution.

Figure 3:  The generalized Fano factor $P$ as a function of source
transparency $T_{s}$ in the regime of non-Gaussian noise. The solid lines are
numerical results based on Eq. (2) and the dashed lines are the asymptotics in
Eq. (21). The blue and red curves correspond to $\theta=\pi/4$ (strong
interaction) and $\theta=\pi/5$.

Substituting the FCS generator (19) into Eqs. (17) and (7), the second term of
the noise at $T_{1u}=T_{1d}=:T_{s}$ turns out to be equal to

$\mathcal{P}_{2}(0)=-\frac{e^{2}V}{2\pi\hbar}\int\frac{dz}{2\pi}\frac{\exp\left[-2T_{s}f(\lambda_{1})|z|/\pi\right]}{(z-i0^{+})^{2}},$
(20)

where $f(\lambda_{1})=1-\cos(\lambda_{1})$ and $z=Vt$ is the dimensionless
integration variable. Now, substituting Eq. (20) into Eq. (5) and then into
Eq. (2), we get the final formula for the generalized Fano factor to leading
order in the transparency of the injection QPCs

$\displaystyle
P=1-T_{s}+\frac{2f(\lambda_{1})}{\pi^{2}}\left[\mathcal{C}-\pi\tilde{T}_{s}-\log(1/\tilde{T}_{s})\right],$
(21)

where $\mathcal{C}=\log 2+\boldsymbol{\gamma}$, $\boldsymbol{\gamma}\approx
0.5772$ is Euler’s constant, and $\tilde{T}_{s}=T_{s}f(\lambda_{1})$ is the
renormalized transparency of the source QPCs. Note that for $\theta=\pi/4$,
which corresponds to strong electron-electron interactions, one finds
$f(\lambda_{1})=2$. It is worth mentioning that the subleading corrections in
$T_{s}$ show non-analytical behavior as well and are proportional to
$\log(1/\tilde{T}^{2}_{s})$. The result (21) does not depend on length
explicitly because the calculations were done in the limit of large $L$, when
a complete separation of the wave packets with velocities $u$ and $v$ occurs.
The generalized Fano factor as a function of source transparency $T_{s}$ and
for different interaction strengths $\theta$ is plotted in Fig. 3.

To summarize, we have studied the zero-frequency current cross-correlations
and the corresponding generalized Fano factor in a mesoscopic quantum Hall
collider at filling factor $\nu=2$. We have used non-equilibrium bosonization
in order to take into account both the non-equilibrium state created by the
source QPCs as well as the electron-electron interactions. We have shown that
both the noise and the Fano factor consist of two terms. The first term is
associated with the transmitted noise and is proportional to $T_{s}(1-T_{s})$.
This term only contains information about the non-equilibrium conditions after
the injection QPCs. The second term, which mixes the incoming non-equilibrium
states, is related with the generated noise. It shows a non-analytical
dependence on the source transparency and scales as $T_{s}\log(1/T_{s})$ at
small $T_{s}\ll 1$ in case of strong, screened Coulomb interactions
($\theta=\pi/4$). The non-analytical behavior is robust with respect to the
interaction parameter and does not vanish for $\theta\neq\pi/4$. Thus, the
generated noise contains information both about the non-equilibrium state
after the source QPCs and the electron-electron interactions. The same
considerations are valid for the generalized Fano factor. It is known that the
absence of current correlations for free fermions at filling factor $\nu=1$ is
due to perfect spatial exclusion, while the negative correlations for Laughlin
anyons at $\nu=1$ in non-Gaussian noise regime are a signature of a reduced
spatial exclusion [46]. However, for filling factor $\nu=2$ and experimentally
accessible transparencies $T_{s}\geq 0.05$ [45], we have shown that the Fano
factor in the non-Gaussian regime of the injection QPCs is positive, in
contrast to the cases for free fermions and Laughlin quasiparticles. The
positive correlations arise due to the additional Coulomb interactions between
fermions.

We are grateful to G. Fève for sharing experimental data and for fruitful
discussions. EGI and TLS acknowledge financial support from the National
Research Fund Luxembourg under Grant CORE 13579612. EVS acknowledges support
from the Swiss National Science Foundation.

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