# Spinning test particle motion around a traversable wormhole

Carlos A. Benavides-Gallego<EMAIL_ADDRESS>Shanghai Astronomical
Observatory, 80 Nandan Road, Shanghai 200030, P. R. China  Wen-Biao Han
<EMAIL_ADDRESS>Shanghai Astronomical Observatory, 80 Nandan Road, Shanghai
200030, P. R. China  Daniele Malafarina<EMAIL_ADDRESS>Department of Physics, Nazarbayev University, 53 Kabanbay Batyr avenue, 010000
Nur-Sultan, Kazakhstan  Bobomurat Ahmedov<EMAIL_ADDRESS>Ulugh Beg
Astronomical Institute, Astronomy St. 33, Tashkent 100052, Uzbekistan
National University of Uzbekistan, Tashkent 100174, Uzbekistan  Tashkent
Institute of Irrigation and Agricultural Mechanization Engineers, Kori
Niyoziy, 39, Tashkent 100000, Uzbekistan  Ahmadjon Abdujabbarov
<EMAIL_ADDRESS>Shanghai Astronomical Observatory, 80 Nandan Road, Shanghai
200030, P. R. China  Ulugh Beg Astronomical Institute, Astronomy St. 33,
Tashkent 100052, Uzbekistan  National University of Uzbekistan, Tashkent
100174, Uzbekistan  Tashkent Institute of Irrigation and Agricultural
Mechanization Engineers, Kori Niyoziy, 39, Tashkent 100000, Uzbekistan
Institute of Nuclear Physics, Tashkent 100214, Uzbekistan

###### Abstract

The motion of spinning test particles around a traversable wormhole is
investigated using the Mathisson-Papapetrous-Dixon equations, which couple the
Riemann tensor with the antisymmetric tensor $S^{\alpha\beta}$, related to the
spin of the particle. Hence, we study the effective potential, circular
orbits, and innermost stable circular orbit (ISCO) of spinning particles. We
found that the spin affects significantly the location of the ISCO, in
contrast with the motion of non-spinning particles, where the ISCO is the same
in both the upper and lower universes. On the other hand, since the dynamical
four-momentum and kinematical four-velocity of the spinning particle are not
always parallel, we also consider a superluminal bound on the particle’s
motion. In the case of circular orbits at the ISCO, we found that the motion
of particles with an adimensional spin parameter lower (greater) than $s=-1.5$
($1.5$) is forbidden. The spin interaction becomes important for Kerr black
hole orbiting super massive wormholes (SMWH).

##  I Introduction

In the last years of his life, Einstein tried to find a theory to unify
electromagnetism with general relativity (GR). In contrast to some scientists,
Einstein was concerned by the idea of considering material particles as
singularities of the fields:“a singularity brings so much arbitrariness in the
theory that it nullifies its laws”Einstein:1935tc . In this sense, it was a
fundamental principle for Einstein to remove singularities in a field theory.
Hence, in 1935, Einstein and Rosen tried to answer the following question:

> “Is an atomistic theory of matter and electricity conceivable which, while
> excluding singularities in the field, makes use of no other field variables
> than those of the gravitational field ($g_{\mu\nu}$) and those of the
> electromagnetic field in the sense of Maxwell (vector
> potentials)?”Einstein:1935tc .

One example of such singularities is the curvature singularity that appears in
the well-known Schwarzschild solution Schwarzschild:1916uq ;
Schwarzschild:1916ae . When solving the field equations for a vacuum, static
and spherically symmetric space-time (with and without an electrostatic
field), it is clear that singularities at the center of symmetry must emerge.
Nevertheless, by reinterpreting the Schwarzschild solution, Einstein and Rosen
were able to find a way to avoid the singularity problem. Their idea consisted
of treating the physical space as two concurrent sheets, where (neutral and
charged) particles are seen as the portions of the space-time that connect the
two sheets. In simple words, Einstein and Rosen thought of particles as
“bridges”111It is necessary to point out that the Einstein–Rosen bridge was
originally discovered by Flamm (1916) after Schwarzschild published his
solution G.W.Gibbons2015 . connecting two regions of the space-time.

With the paradigm of particles as bridges, Einstein and Rosen tried to explain
the atomistic character of matter222For example, according to this idea, the
problem of the nature of electrons and protons, which are described as point-
like particles in electrodynamics, can be addressed by considering it as a two
bridge problem. without introducing new variables other than the metric and
the vector potential. It is worth noticing that for some time the idea of the
existence of “bridges”, or wormholes333The term “wormhole” was coined by
Wheeler and Misner in 1957 Misner:1957mt . as they are called today, was
considered more appealing than that of black holes. However, in 1962, Wheeler
and Fuller showed that the Einstein-Rosen bridge (also known as Schwarzschild
wormhole) is unstable and will pinch off in a finite time. Hence, although the
Schwarzschild space-time is static, a proper analysis shows that the
Schwarzschild geometry is changing with time Fuller:1962zza . As a
consequence, Schwarzschild wormholes are not traversable.

The possibility of traversable wormholes in general relativity appeared for
the first time in the works of Ellis HGEllis:1973  and Bronnikov
Bronnikov:1973fh  (1973). In his paper, Ellis obtained a solution of
Einstein’s field equations for a vacuum space-time, modified by the inclusion
of a scalar field coupled to the Ricci tensor. The solution is known as the
Ellis drain-hole. Thus, by analyzing the topology and geodesics, Ellis was
able to show that it is geodesically complete, horizonless, singularity-free,
and fully traversable in both directions. The drain-hole solution requires two
parameters to be described: $m$, which fixes the strength of its gravitational
field, and $n$, which determines the curvature of its spatial cross-sections.
When $m=0$, the drain-hole gravitational field vanishes, forming a non-
gravitating, purely geometric, traversable wormhole HGEllis:1973 .

Traversable wormholes motivated numerous studies during the ’80s and the ’90s
Morris:1988tu ; Morris:1988cz ; Visser:1989kh ; Visser:1989kg ; Visser:1989vq
; AzregAinou:1989wr ; Visser:1989am ; Poisson:1989zz ; Visser:1990wj ;
Frolov:1990si ; Visser:1990wi . These works provided a deeper understanding of
the physics behind each particular solution. In Ref. Morris:1988cz , for
example, Morris and Thorne considered the following question: What properties
a classical wormhole should have to be traversable? Usually, as Visser pointed
out in Ref. Visser:1989kh , the process to analyze classical wormholes starts
by considering a Lagrangian, solving the Einstein field equations, and then
looking for different geometries among these solutions. However, the authors
of Ref. Morris:1988cz  used a different approach. There, Morris and Thorne
assumed the wormhole geometry to derive, via Einstein field equations, the
stress-energy tensor and then investigate the physics. In this way, they were
able to show that “exotic” matter should be present at the throat of a
traversable wormhole.

On the other hand, the idea of wormholes also has inspired the possibility of
time travel Morris:1988cz ; Morris:1988tu . Although this possibility has been
considered speculatively (because the existence of wormholes is constrained to
existence of “exotic” matter, which violates the weak energy condition), we
cannot entirely dismiss it. In fact, the discovery that black holes can
evaporate Hawking:1974rv  has suggested that quantum fields can violate the
energy conditions and has led some to speculate on the validity of energy
conditions Martin-Moruno:2013 . One example related to wormholes is the
quantum creation of particles in Morris:1988cz .

After the work of Einstein and Rosen (1935) Einstein:1935tc , the interest in
wormhole solutions remained dormant for almost twenty years. Then, with the
works of Wheeler (1955) and Misner (1957) Wheeler:1955zz ; Misner:1957mt
wormholes started to be considered again by the scientific community as viable
astrophysical objects. Recently, a lot of theoretical researchers have studied
different aspects of traversable wormholes within Einstein’s gravity
Echeverria:1991nk ; Deser:1992ts ; Deser:1993jr ; Hochberg:1997wp ;
Abdujabbarov:2009ad ; Abdujabbarov:2016efm  as well as in alternative theories
of gravity Moffat:1991xp ; Carlini:1992jda ; Bhawal:1992sz ; Letelier:1993cj ;
Vollick:1998qf . Furthermore, some authors have considered the effects that
wormholes would produce from the observational point of view Perlick:2003vg ;
Perlick:2004tq ; Nandi:2006ds ; Kardashev:2006nj ; Bambi:2013jda ;
Tripathi:2019trz  with the aim of putting constraints on their possible
detection. In Bambi:2013jda , for example, it is studied the K$\alpha$ iron
line of several wormhole solutions based on the idea that supermassive black
hole candidates at the center of galaxies might be wormholes formed in the
early universe. By calculating the K$\alpha$ iron line produced by accretion
disks in the space-time of these solutions, one could compare it with that
produced by a Kerr black hole. It was found that the K$\alpha$ iron line
produced around non-rotating or slow-rotating wormholes may mimic the one
obtained in vicinity of Kerr black holes (with mid or high spins). Moreover,
the results are still marginally compatible with current observations. Hence,
the possibility that the supermassive black hole candidates in galactic nuclei
could in fact be these objects is still not ruled out. On the other hand, in
the case of wormholes with spin parameter $a_{*}>0.02$, the iron line is
indeed different from the one produced in the space-time of a Kerr black hole
and therefore, their existence may already be excluded via current
observations Bambi:2013jda .

One aspect that is important to consider when dealing with accretion disks is
the spin of the particles in the accretion disk’s gas. In fact nowadays, the
spin interaction of relativistic systems has become an important subject of
study Kidder:1992fr ; Apostolatos:1994mx ; Suzuki:1996gm ; Suzuki:1997by ;
Saijo:1998mn ; Semerak:1999qc ; Kyrian:2007zz ; Plyatsko:2013xza ;
Hackmann:2014tga ; Nucamendi:2019qsn ; Conde:2019juj ; Toshmatov:2019bda ;
Boonserm:2019nqq ; Han:2016cdh ; Han:2016djt ; Toshmatov:2020wky . For this
reason, it is important to properly understand the dynamics of test particles
in curved space-time that includes classical spin. In 1937, Mathisson studied
the problem of extended bodies in GR. In his work, he demonstrated an existing
interaction between the Riemann curvature tensor and the spin of the moving
particle in the equations of motion Mathisson:1937zz . Papapetrou also
considered the same problem in Refs. Papapetrou:1951pa ; Corinaldesi:1951pb
developing a similar approach as Mathisson. It is worth remembering that
later, Tulczyjew improved on the methods of Mathisson tulczyjew1959motion ;
BWTulzcyjew1962  while Moller and others made improvements in the definition
of center-of-mass in Refs. moller1949definition ; beiglbock1967center ;
dixon1964covariant ; Dixon:1970zza ; Dixon:1970zz ; ehlers1977dynamics .
Today, the equations that describe the motion of extended bodies with spin and
mass are known as the Mathisson-Papapetrous-Dixon equations (MPD).

In this work, we investigate the motion of spinning particles around a
traversable wormhole. We organize our work as follows: In Sec. II, we review
properties of traversable wormholes discussed in Ref. Morris:1988cz . Next, in
Sec. III, we discuss the motion of spinning particles in a spherically
symmetric space-time. In this section, we use the MPD equations to obtain the
effective potential and the superluminal condition. Then, in Sec. IV and Sec.
V, we apply the results of Sec. III to the Morris-Thorne wormhole. We compute
the effective potential, circular orbits, and the innermost stable circular
orbit (ISCO). We also use the superluminal bound to find a constraint for the
allowed spin of the particle. Finally, in Sec. VI, we summarize our work and
discuss the implications of the results. Throughout the manuscript, we use
geometrized units setting $G=c=M=1$.

##  II Morris-Thorne wormholes

In this section, we review properties of wormholes discussed by Morris and
Thorne in Ref. Morris:1988cz , where the authors assumed the wormhole’s space-
time geometry, and then, via the field equations, computed the corresponding
energy-momentum tensor. Therefore, the discussion starts by assuming a static
and spherically symmetric space-time for which the line element has the
following form

$ds^{2}=-e^{2\Phi(r)}dt^{2}+\frac{dr^{2}}{\left(1-\frac{b(r)}{r}\right)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (1)

where $\Phi(r)$ and $b(r)$ are arbitrary functions of the radial coordinate
$r$ known as the “redshift function” and the “shape function”, respectively.
The fact that Eq. (1) can represent a wormhole with a throat is easily
explained by embedding the line element in a three-dimensional space at a
fixed time slice $t$. In addition, given the spherical symmetry of the line
element in Eq. (1), we can limit the attention to the equatorial plane
$\theta=\pi/2$ and consider the restricted line element

$ds^{2}=\left(1-\frac{b(r)}{r}\right)^{-1}dr^{2}+r^{2}d\varphi^{2}.$ (2)

Figure 1: Representation of the wormhole embedded in a three dimensional
space-time. For the plot we use the line element
$ds^{2}=-dt^{2}+dl^{2}+(b^{2}_{0}+l^{2})(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})$, which is a special case of Eq. (1), where the coordinate $l$
is related to $r$ by $r^{2}=b^{2}_{0}+l^{2}$. Here the “redshift function”
vanishes ($\Phi=0$) and the “shape function” is given by $b(r)=b^{2}_{0}/r$.
In the plot we chose $b_{0}=10M$.

One can interpret Eq. (2) as a “picture” of the space-time observed from the
equatorial plane at time $t$. The line element in Eq. (2) can be embedded in
the three-dimensional space in cylindrical coordinates $\\{r,z,\varphi\\}$
with $z=z(r)$ as

$ds^{2}=\left[1+\left(\frac{dz}{dr}\right)^{2}\right]dr^{2}+r^{2}d\varphi^{2},$
(3)

if we compare Eqs. (2) and (3) to obtain $dz/dr$, which is given by

$\frac{dz}{dr}=\pm\left[\frac{r}{b(r)}-1\right]^{-\frac{1}{2}}.$ (4)

As an example, we consider one of the wormhole solutions obtained in Ref.
Morris:1988cz  with $b(r)=b_{0}^{2}/r$ (see Fig. 1 for details). Hence, after
integration, the coordinate $z$ as a function of radial coordinate $r$ takes
the following form

$z(r)=\pm
b_{0}\ln\left[\frac{r}{b_{0}}+\sqrt{\left(\frac{r}{b_{0}}\right)^{2}-1}\right].$
(5)

Note that $dz/dr$ in Eq (4) (the slope) diverges at $r=b(r)=b_{0}$. Moreover,
one can observe that Eq. (5) has two branches: one with positive sign and the
other with negative sign. In Fig. 2 we show the plot of $z$ vs. $r$. From the
figure, it is possible to see the two branches of Eq. (5), which correspond to
the upper ($+$) and lower ($-$) universes. Here the value $b_{0}$ is the
wormhole’s throat radius, where $dz/dr$ diverges. For a complete three-
dimensional view of the wormhole, it is necessary to rotate the function
$z(r)$ around the $z$-axis. The result is shown in Fig. 1.

Figure 2: Plot of Eq. (5) with $b_{0}=10M$ (the radius of the throat). One can
see the upper (continous line) and lower (dotted line) universes.

Now, knowing that the line element in Eq. (2) does represent a spherically
symmetric wormhole, the next step is to make sure it is a solution of the
field equations. In this sense, it is necessary to compute the energy-momentum
tensor via Einstein’s field equations. Here, the reason for considering an
energy-momentum tensor different from zero has to do with the fact that while
non-traversable wormholes, like the Einstein-Rosen bridge, are vacuum
solutions of the field equations for traversable wormholes one must have an
energy-momentum tensor different from that of a vacuum located at the throat
in order to allow the matching of the upper and lower universes.

The analysis presented in Ref. Morris:1988cz  is performed using a “proper
reference frame”, this means that physical “observations” are performed by a
local observer, who remains at rest with respect to the coordinate system
$\\{t,r,\theta,\varphi\\}$. The “proper reference frame” is constructed using
the tetrad formalism with the basis vectors Chandrasekhar:1998

$\begin{array}[]{ccc}\textbf{e}_{(a)}=e^{\;\;\;\;\mu}_{(a)}\partial_{\mu}&\text{and}&\textbf{e}^{(a)}=e^{(a)}_{\;\;\;\;\mu}dx^{\mu}.\end{array}$
(6)

Here we use the same notation as in Ref. Chandrasekhar:1998 , where tetrad
indices are enclosed in parentheses to distinguish from the tensor indices. In
the case of the line element described by Eq. (1), the orthonormal basis of a
“proper reference frame” is given by $e^{\;\;\;\;\mu}_{(a)}$. In terms of the
metric, the tetrad has the form

$\displaystyle e^{\;\;\;\;\mu}_{(0)}$ $\displaystyle=(e^{-\Phi},0,0,0)$ (7)
$\displaystyle e^{\;\;\;\;\mu}_{(1)}$
$\displaystyle=\left(0,\left[1-\frac{b(r)}{r}\right]^{1/2},0,0\right)$
$\displaystyle e^{\;\;\;\;\mu}_{(2)}$ $\displaystyle=(0,0,r^{-1},0)$
$\displaystyle e^{\;\;\;\;\mu}_{(3)}$
$\displaystyle=(0,0,0,(r\sin\theta)^{-1}).$

The purpose of using a “proper reference frame” (or, equivalently, a “local
Lorentz frame”) in the analysis of the physical quantities is to diagonalize
the energy momentum tensor $T_{(a)(b)}$ and then be able to relate each
(diagonal) component to a physical quantity of clear meaning, e. i. the total
density of mass-energy $\rho(r)$, the tension per unit area $\tau(r)$, and the
pressure $p(r)$. Hence, from the physical point of view, the observer in a
“local Lorentz frame” interprets $\rho(r)$, $\tau(r)$, and $p(r)$ as

$\begin{array}[]{ccc}T_{(0)(0)}=\rho,&T_{(1)(1)}=-\tau(r),&T_{(2)(2)}=T_{(3)(3)}=p(r).\end{array}$
(8)

The field equations of course are given by

$G_{(a)(b)}=8\pi T_{(a)(b)},$ (9)

where $G_{(a)(b)}$ is the Einstein tensor.

Using Eq. (8) and the field equations (9) Morris and Thorne obtained the
following relations between the physical quantities $\rho(r)$, $\tau(r)$,
$p(r)$ and the metric quantities $\Phi(r)$ and $b(r)$ Morris:1988cz

$\displaystyle\rho$ $\displaystyle=\frac{b^{\prime}(r)}{8\pi r^{2}}$ (10)
$\displaystyle\tau$ $\displaystyle=\frac{b(r)/r-2(r-b(r))\Phi^{\prime}}{8\pi
r^{2}}$ $\displaystyle p$
$\displaystyle=\frac{r}{2}\left[(\rho-\tau)\Phi^{\prime}-\tau^{\prime}\right]-\tau,$

where ${f}^{\prime}$ denotes the partial derivative of $f$ with respect to
$r$. Therefore, solving the field equations in the case of a wormhole
corresponds to solving the set of equations (10). Notice that these are three
equations in five unknown quantities. Accordingly, Morris and Thorne proposed
to use this freedom to control the functions $\Phi(r)$ and $b(r)$ (related to
the wormhole’s geometry) in such a way that the solution generates a
traversable wormhole.

According to Ref. Morris:1988cz , there are several properties that must be
satisfied for the geometry to describe a traversable wormhole. As already
mentioned, the assumption that the space-time be static and spherically
symmetric greatly simplifies the calculations (for stationary wormholes see
for example stationary-wormhole , while for static but non spherically
symmetric wormholes see axial-wormhole1 ; axial-wormhole2 ; axial-wormhole3 ).
Secondly, any wormhole solution should contain a throat that connects two
asymptotically flat regions of space-time. This property is clearly shown in
Figs. 1 and 2 for a particular form of the shape function $b(r)$. As said, the
existence of a throat is related to the divergence of $dz/dr$ at
$r=b(r)=b_{0}$, while to demonstrate that the solution is asymptotically flat,
noticing that the radial coordinate is ill-define at the throat, Morris and
Thorne considered the proper radial distance Morris:1988cz

$l(r)=\pm\int^{r}_{b_{0}}\left[1-\frac{b(\tilde{r})}{\tilde{r}}\right]^{-\frac{1}{2}}d\tilde{r},$
(11)

which is finite everywhere if $1-b(r)/r\geq 0$. Using the proprer radial
distance, Morris and Thorne showed that $dz/dl\rightarrow 0$ as
$l\rightarrow\pm\infty$, e.i. the space-time is asymptotically flat.

Finally, the solution must not contain a horizon. Wormholes allow to causally
connect two different portions of the space-time by the throat. Therefore, the
presence of a horizon would prevent the two universes to be causally
connected. This condition is satisfied by demanding $\Phi(r)$ to be finite
everywhere.

These properties, along with the field equations (10), are the “basic wormhole
criteria”. Nevertheless, the authors have also discussed the possibility of
tuning the wormhole’s parameters to make it possible for humans to travel
through them; this requires the crossing-time to be finite for any observer
and the tidal force felt by the astronaut to be small. On the other hand, one
of the most important results is the necessity of “exotic matter” to generate
the wormhole’s space-time curvature. This conclusion comes from the fact that
in the throat

$\tau_{0}>\rho_{0}.$ (12)

Physically, this means that the tension $\tau_{0}$ in the throat must be so
large as to exceed the total density mass-energy $\rho_{0}$, which implies
that an observer passing through the throat, with a radial velocity close to
the speed of light, would perceives a negative $\tilde{T}_{(0)(0)}$
Morris:1988cz

$\tilde{T}_{(0)(0)}=\gamma^{2}[\rho_{0}-\tau_{0}]+\tau_{0}.$ (13)

Physicists have established a series of energy conditions to ensure the
physical viability of $T_{\mu\nu}$, such as for example avoiding negative
energy densities. However, energy conditions are considered to be valid for
classical matter fields but may be violated at quantum level Visser:1989am .
Therefore, while Eq. (12) violates the weak energy condition, from a quantum
perspective, there exist some situations in which such violation may be
physically valid, such as, for example, the quantum mechanical creation of
particles Martin-Moruno:2013sfa . In this sense one can not entirely rule out
the possibility of the existence of the exotic material required for the
throat of a traversable wormhole to hold Morris:1988cz .

Finally, with the help of Eqs. (10), Morris and Thorne established a method to
obtain traversable wormhole solutions. The method is as follows. First, given
$\Phi(r)$ and $b(r)$ satisfying the conditions for a traversable wormhole, one
can use the first relation in Eq. (10) to obtain $\rho(r)$. Then, using the
second relation in Eq. (10) one can obtain $\tau(r)$. Finally, with $\rho(r)$
and $\tau(r)$, one can find $p(r)$. Following this method, Morris and Thorne
obtained three solutions: the zero-tidal-force wormhole, a solution with a
finite radial cutoff of the stress-energy, and a solution with exotic matter
limited to the throat’s vicinity. In this paper, we will focus our attention
in the zero-tidal-force solution with

$\Phi(r)=-\frac{b_{0}}{r}\;\;\text{and}\;\;b(r)=\frac{b^{2}_{0}}{r}.$ (14)

##  III Equations of motion for a spinning particle

The equations of motion of spinning particles are given by the MPD equations
that can be expressed as

$\displaystyle\frac{Dp^{\alpha}}{d\lambda}$
$\displaystyle=-\frac{1}{2}R^{\alpha}_{\;\;\beta\delta\sigma}u^{\beta}S^{\delta\sigma},$
(15) $\displaystyle\frac{DS^{\alpha\beta}}{d\lambda}$
$\displaystyle=p^{\alpha}u^{\beta}-p^{\beta}u^{\alpha},$

where $D/d\lambda\equiv u^{\alpha}\nabla_{\alpha}$ is the projection of the
covariant derivative along the particle’s trajectory,
$u^{\mu}=dx^{\mu}/d\lambda$ is the 4-velocity of the test particle,
$p^{\alpha}$ is the canonical 4-momentum, $R^{\alpha}_{\;\;\beta\delta\sigma}$
is the Riemann curvature tensor, and $\lambda$ is an affine parameter. The
second rank tensor $S^{\alpha\beta}$ is antisymmetric444Note that for any
antisymmetric tensor, the diagonal components vanishe identically, i.e.
$S^{\alpha\beta}=0$ if $\alpha=\beta$., $S^{\alpha\beta}=-S^{\beta\alpha}$. In
general relativity, the geodesic equation is given by

$\partial_{\beta}u^{\alpha}u^{\beta}+\Gamma^{\alpha}_{\;\;\;\sigma\beta}u^{\sigma}u^{\beta}=0,$
(16)

which can be expressed in terms of the 4-momentum of the particle and the
projection of the covariant derivative along the particle’s trajectory as

$\frac{Dp^{\alpha}}{d\lambda}=0.$ (17)

Therefore, if one compares Eqs. (15) and (17), one concludes that spinning
particles do not follow a geodesic due to the interaction between the Riemann
curvature tensor and the antisymmetric tensor $S^{\alpha\beta}$.

To solve the MPD, it is necessary to fix the center of mass of the spinning
particle by including the condition tulczyjew1959motion ; Saijo:1998mn

$S^{\alpha\beta}p_{\alpha}=0,$ (18)

which is the so-called Tulczyjew Spin Supplementary Condition (SSC)
Saijo:1998mn . From Eq. (18), it turns out that the canonical momentum and the
spin of the particle provide two independent conserved quantities given by the
relations

$\displaystyle p^{\alpha}p_{\alpha}$ $\displaystyle=-m^{2},$ (19)
$\displaystyle s^{2}$
$\displaystyle=\frac{1}{2}S^{\alpha\beta}S_{\alpha\beta}.$

However, although the canonical momentum of the spinning particle is
conserved, it is important to point out that the squared velocity does not
necessarily satisfy the condition

$u_{\alpha}u^{\alpha}=-1,$ (20)

because the 4-vectors $p^{\alpha}$ and $u^{\alpha}$ are not always parallel.
Therefore, it is necessary to impose an additional condition known as the
superluminal bound to ensure that the particle’s 4-velocity is always smaller
than the speed of light.

In addition to the conserved quantities resulting from the Tulczyjew-SSC
condition, there exist also the conserved quantities associated to the space-
time symmetries given by the Killing vectors $\xi^{\mu}$, which can be
expressed as

$p^{\alpha}\xi_{\alpha}-\frac{1}{2}S^{\alpha\beta}\nabla_{\beta}\xi_{\alpha}=p^{\alpha}\xi_{\alpha}-\frac{1}{2}S^{\alpha\beta}\partial_{\beta}\xi_{\alpha}=\text{constant}$
(21)

where we have used the fact that the term
$S^{\alpha\beta}\Gamma^{\gamma}_{\;\;\beta\alpha}$ in the convariant
derivative vanishes because $S^{\alpha\beta}$ is antisymmetric while
$\Gamma^{\gamma}_{\;\;\beta\alpha}$ is symmetric.

###  III.1 Effective potential

As mentioned before, the line element of a spherically symmetric wormhole is
given by Eq. (1). Here, we assume $b(r)=b^{2}_{0}/r$, where $b_{0}$ is the
wormhole’s throat. In general, $b_{0}$ is interpreted as the mass of the
wormhole in the Newtonian limit (see for example Ref. Li:2014coa ). Also we
make a common choice for the redshift function Li:2014coa ; Harko:2008vy
taking Eq. (14). We base this choice on the behavior of circular orbits around
the wormhole, which are unstable when $\partial\Phi/\partial r<0$ and stable
when $\partial\Phi/\partial r>0$ (see Ref. Harko:2008vy  for details).

Using the proper distance $l$, related to $r$ by the relation
$r^{2}=b^{2}_{0}+l^{2}$, as a new radial coordinate, the line element Eq. (1)
takes the form:

$ds^{2}=-e^{2\Phi(l)}dt^{2}+dl^{2}+(b^{2}_{0}+l^{2})(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (22)

with

$\Phi(l)=-\frac{b_{0}}{\sqrt{b^{2}_{0}+l^{2}}}.$ (23)

With this choice the throat of the wormhole is located at $l=0$ and we are
then able to consider the lower universe ($l<0$) and the upper universe
($l>0$) in a unified manner. Since the space-time is static and spherically
symmetric, the line element allows two Killing vector fields given by

$\begin{array}[]{ccc}\xi^{\alpha}=\delta^{\alpha}_{t},&&\xi^{\alpha}=\delta^{\alpha}_{\varphi},\end{array}$
(24)

which correspond to time translation and rotations, and have associated two
conserved quantities, i.e. the energy $E$ and the angular momentum $L$. Using
Eq. (24), the conserved quantities can be expressed as Semerak:1999qc ;
Conde:2019juj ; Toshmatov:2019bda

$\displaystyle-E=$ $\displaystyle
p_{t}-\frac{1}{2}g_{t\alpha,\beta}S^{\alpha\beta}=p_{t}-\frac{1}{2}g_{tt,l}S^{tl},$
(25) $\displaystyle L=$ $\displaystyle
p_{\varphi}+\frac{1}{2}g_{\varphi\alpha,\beta}S^{\beta\alpha}=p_{\varphi}+\frac{1}{2}g_{\varphi\varphi,l}S^{l\varphi}.$

Given the spherical symmetry of the geometry we can restrict the attention to
a plane of constant $\theta$, such as the equatorial plane $\theta=\pi/2$.
Then $p^{\theta}=0$ and the metric functions on the equatorial plane depend
only on the radial coordinate $l$. Therefore, since $S^{\theta\alpha}=0$, the
number of independent components of $S^{\alpha\beta}$ reduces to three. Using
Eq. (18) one obtains the following relations:

$\displaystyle S^{t\varphi}$
$\displaystyle=\frac{p_{l}}{p_{\varphi}}S^{lt}=-\frac{p_{l}}{p_{\varphi}}S^{tl},$
(26) $\displaystyle S^{l\varphi}$
$\displaystyle=-\frac{p_{t}}{p_{l}}S^{t\varphi}=\frac{p_{t}}{p_{\varphi}}S^{tl}.$

Now, from the normalization condition and conservation of angular momentum in
Eqs. (19) we have

$\displaystyle p^{2}_{l}$
$\displaystyle=-g_{ll}\left[g^{tt}p^{2}_{t}+g^{\varphi\varphi}p^{2}_{\varphi}+m^{2}\right],$
(27) $\displaystyle 2S^{2}$
$\displaystyle=S^{\alpha\beta}S_{\alpha\beta}=2S^{lt}\left(S_{tl}-\frac{p_{l}}{p_{\varphi}}S_{t\varphi}+\frac{p_{t}}{p_{\varphi}}S_{l\varphi}\right),$

and

$\displaystyle S_{tl}$
$\displaystyle=g_{t\rho}g_{l\epsilon}S^{\rho\epsilon}=g_{tt}g_{ll}S^{tl},$
(28) $\displaystyle S_{t\varphi}$
$\displaystyle=g_{t\gamma}g_{\varphi\sigma}S^{\gamma\sigma}=-g_{\varphi\varphi}g_{tt}\frac{p_{l}}{p_{\varphi}}S^{tl},$
$\displaystyle S_{l\varphi}$
$\displaystyle=g_{l\lambda}g_{\mu\varphi}S^{\lambda\mu}=g_{ll}g_{\varphi\varphi}\frac{p_{t}}{p_{\varphi}}S^{tl}.$

From which we get

$S^{tl}=\pm\frac{p_{\varphi}s}{\sqrt{-g_{tt}g_{ll}g_{\varphi\varphi}}}=\frac{p_{\varphi}e^{-\Phi(l)}}{\sqrt{b^{2}_{0}+l^{2}}}s,$
(29)

where $s=S/m$ can be positive or negative and represents the direction of the
spin with respect to the direction of $p_{\varphi}$. Now, from the
conservation of energy and angular momentum Eq. (25) we have

$\displaystyle-E$ $\displaystyle=p_{t}-s\mathcal{A}p_{\varphi},$ (30)
$\displaystyle L$ $\displaystyle=p_{\varphi}+s\mathcal{B}p_{t},$

with555Where now prime ′ denotes partial derivatives with respect to radial
coordinate $l$.

$\displaystyle\mathcal{A}$
$\displaystyle=\frac{1}{2}\frac{g_{tt,l}}{\sqrt{-g_{tt}g_{ll}g_{\varphi\varphi}}}=-\frac{e^{\Phi(l)}\Phi^{\prime}(l)}{\sqrt{b^{2}_{0}+l^{2}}},$
(31) $\displaystyle\mathcal{B}$
$\displaystyle=\frac{1}{2}\frac{g_{\varphi\varphi,l}}{\sqrt{-g_{tt}g_{ll}g_{\varphi\varphi}}}=\frac{le^{-\Phi(l)}}{\sqrt{b^{2}_{0}+l^{2}}}.$

After solving, we obtain

$\displaystyle p_{t}$
$\displaystyle=\frac{-E+s\mathcal{A}L}{1+s^{2}\mathcal{A}\mathcal{B}},$ (32)
$\displaystyle p_{\phi}$
$\displaystyle=\frac{L+s\mathcal{B}E}{1+s^{2}\mathcal{AB}},$

and

$\mathcal{AB}=-\frac{l\Phi^{\prime}(l)}{b^{2}_{0}+l^{2}}.$ (33)

Now, after inserting Eq. (32) into Eq. (27), we finally obtain

$(p_{l})^{2}=\frac{1}{\alpha}(\beta E^{2}+2\delta LE+\sigma L^{2}-\gamma),$
(34)

which is quadratic equation for $E$ with

$\displaystyle\alpha$
$\displaystyle=g^{ll}\left(1-\frac{g^{\prime}_{tt}g^{\prime}_{\varphi\varphi}s^{2}}{4g_{tt}g_{ll}g_{\varphi\varphi}}\right)^{2},$
(35) $\displaystyle\beta$
$\displaystyle=-g^{tt}+\frac{g^{\varphi\varphi}(g^{\prime}_{\varphi\varphi})^{2}s^{2}}{4g_{tt}g_{ll}g_{\varphi\varphi}},$
$\displaystyle\delta$
$\displaystyle=\frac{(g^{tt}g^{\prime}_{tt}-g^{\varphi\varphi}g^{\prime}_{\varphi\varphi})s}{2\sqrt{-g_{tt}g_{ll}g_{\varphi\varphi}}},$
$\displaystyle\sigma$
$\displaystyle=-g^{\varphi\varphi}+\frac{g^{tt}(g^{\prime}_{tt})^{2}s^{2}}{4g_{tt}g_{ll}g_{\varphi\varphi}},$
$\displaystyle\gamma$
$\displaystyle=m^{2}\left(1-\frac{g^{\prime}_{tt}g^{\prime}_{\varphi\varphi}s^{2}}{4g_{tt}g_{ll}g_{\varphi\varphi}}\right)^{2}.$

Equation (34) can be expressed as

$(p_{l})^{2}=\frac{\beta}{\alpha}(E-V_{+})(E-V_{-}),$ (36)

where $V_{\pm}$ is a solution of $(p_{r})^{2}=0$ given by

$V_{\pm}=-\frac{\delta
L}{\beta}\pm\sqrt{\frac{\delta^{2}L^{2}}{\beta^{2}}+\frac{\gamma-\sigma
L^{2}}{\beta}}.$ (37)

According to Eq. (36), the energy of the particle must satisfy the conditions

$\begin{array}[]{ccc}E\in(-\infty,V_{-}]&\text{or}&E\in[V_{+},\infty),\end{array}$
(38)

in order to have $(p^{l})^{2}\geq 0$. In the following, we shall focus on the
case in which test particles have positive energy and therefore explore the
effective potential given by $V_{\text{eff}}=V_{+}$.

###  III.2 Superluminal bound

As mentioned, the dynamical four-momentum $p^{\alpha}$ and the kinematical
four-velocity $u^{\alpha}$ of a spinning particle are not always parallel. As
a consequence, although $p_{\alpha}p^{\alpha}=-m^{2}$ is satisfied, the
normalization $u_{\alpha}u^{\alpha}=-1$ does not hold. As the spinning
particle moves closer to the center of symmetry, $u^{\alpha}$ increases, and
eventually, for certain values of the spin $s$ and radius $l$, some components
of the four-velocity may even diverge. Before this happens, the motion of the
particle crosses the boundary between time-like and space-like trajectories
thus becoming superluminal. It is well-known that space-like (or superluminal)
motion does not have any physical meaning, and the transition to
$u_{\alpha}u^{\alpha}>0$ is not allowed for real particles. In this sense, one
must impose a further constrain known as the superluminal bound, defined by
the relation $u_{\alpha}u^{\alpha}=0$. Hence, for the particle to move always
in the time-like region, it is necessary to impose the following condition
Conde:2019juj ; Toshmatov:2019bda

$\frac{u_{\alpha}u^{\alpha}}{(u^{t})^{2}}=g_{tt}+g_{ll}(u^{l})^{2}+g_{\phi\varphi}(u^{\varphi})^{2}\leq
0,$ (39)

with the equality holding at the superluminal bound. To compute $u^{l}$ and
$u^{\varphi}$, we use a method developed in Ref. Hojman:2012me  (see App. A
for the full derivation). This method is based on application of the MPD
equations (15). From the second MPD equation, by using Tulczyjew-SSC, applying
$D/d\lambda$ and solving it for $DS^{tl}/d\lambda$, $DS^{t\varphi}/d\lambda$
and $DS^{l\varphi}/d\lambda$ one can obtain the following system for the non-
zero components of $S^{\alpha\beta}$:

$\displaystyle\frac{DS^{tl}}{d\lambda}$ $\displaystyle=p^{t}u^{l}-u^{t}p^{l},$
(40) $\displaystyle\frac{DS^{t\varphi}}{d\lambda}$
$\displaystyle=p^{t}u^{\varphi}-u^{t}p^{\varphi},$
$\displaystyle\frac{DS^{l\varphi}}{d\lambda}$
$\displaystyle=p^{l}u^{\varphi}-u^{l}p^{\varphi}.$

Following to the gauge choices and invariant relations in Ref. Hojman:2012me ,
we set $\lambda=t$ and can express the above system of equations in terms of
$S^{\varphi l}$ only. This is a consequence of the MPD equations that imply

$\displaystyle u^{l}$
$\displaystyle=\frac{\hat{\mathcal{C}}}{\hat{\mathcal{B}}}\frac{p_{l}}{p_{t}},$
(41) $\displaystyle u^{\varphi}$
$\displaystyle=\frac{\hat{\mathcal{A}}}{\hat{\mathcal{B}}}\frac{p_{\varphi}}{p_{t}},$

with

$\displaystyle\hat{\mathcal{A}}$
$\displaystyle=g^{\varphi\varphi}+R_{tllt}\left(\frac{S^{\varphi
l}}{p_{t}}\right)^{2},$ (42) $\displaystyle\hat{\mathcal{B}}$
$\displaystyle=g^{tt}+R_{\varphi ll\varphi}\left(\frac{S^{\varphi
l}}{p_{t}}\right)^{2},$ $\displaystyle\hat{\mathcal{C}}$
$\displaystyle=g^{ll}+R_{\varphi tt\varphi}\left(\frac{S^{\varphi
l}}{p_{t}}\right)^{2}.$

Now, inserting Eq. (41) into the superluminal bound condition (39) we have

$g_{tt}(\hat{\mathcal{B}})^{2}(p_{t})^{2}+g_{ll}(\hat{\mathcal{C}})^{2}(p_{l})^{2}+g_{\varphi\varphi}(\tilde{A})^{2}(p_{\varphi})^{2}\leq
0,$ (43)

and from the conservation of the four-momentum $p_{\alpha}p^{\alpha}=-m^{2}$,
the superluminal bound condition reduces to

$\mathcal{F}=\left(\frac{p_{t}}{m}\right)^{2}\mathcal{X}+\left(\frac{p_{\varphi}}{m}\right)^{2}\mathcal{Y}-\mathcal{Z}\leq
0,$ (44)

with

$\displaystyle\mathcal{X}$
$\displaystyle=g_{tt}(\hat{\mathcal{B}})^{2}-\frac{(\hat{\mathcal{C}})^{2}}{g_{tt}},$
(45) $\displaystyle\mathcal{Y}$
$\displaystyle=g_{\varphi\varphi}(\hat{\mathcal{A}})^{2}-\frac{(\hat{\mathcal{C})^{2}}}{g_{\varphi\varphi}},$
$\displaystyle\mathcal{Z}$ $\displaystyle=g_{ll}(\hat{\mathcal{C}})^{2}.$

###  III.3 Other bounds on the spin parameter

When dealing with realistic atrophysical scenarios, such as compact stars and
stellar mass black holes orbiting supermassive black holes, we must notice
that the approximations considered here have limits. The mass and size of the
spinning test particle must be negligible with respect to the central object’s
mass and must not affect the geometry. Furthermore, it is crucial to remark
that MPD equations only take into account effects generated by the mass
monopole and spin dipole, the well-known pole-dipole approximation. In this
sense, these equations do not take into account the mass quadrupole effect
related to the tidal coupling. Hence, one can not use them to model more
extended objects Semerak:1999qc ; Hartl:2002ig . Additionally, since the MPD
equations are conservative, the effects of gravitational radiation are
ignored.

In this sense, following Refs. Semerak:1999qc ; Hartl:2002ig , it is important
to point out that we measure the spin parameter $s$ in terms of $mb_{0}$, not
$m^{2}$. Therefore, the system considered in this work is a spinning compact
body of mass $m$ orbiting a super massive wormhole of mass $b_{0}$. In the
case of compact objects, such as black holes, neutron stars, or white dwarfs,
physically realistic values of the spin must satisfy $s<<1$. If the compact
object is a maximally spinning black hole of mass $m$, with spin angular
momentum $m^{2}$ (and $m<<b_{0}$), for example, the spin parameter $s$ is
given by the condition Hartl:2002ig

$s=\frac{S}{mb_{0}}\leq\frac{m^{2}}{mb_{0}}=\frac{m}{b_{0}}<<1.$ (46)

On the other hand, if the compact object is a neutron star, one can take a
maximum value (approximately) of $0.6m^{2}$ Cook:1993qr . From which one gets
$s\lesssim 0.6m/b_{0}$.

When the compact object is a white draft, the bound on the spin is more
difficult to compute. First, one needs to consider the maximum value of $s$
before the star begins to break up. This value is given by
$s_{max}=I\Omega_{max}$, where $\Omega_{max}$ is the maximum angular velocity
and $I$ is the moment of inertia. If we we write $I=\alpha mR^{2}$ and
$\Omega_{max}=\beta\sqrt{m/R^{3}}$ for some constants $\alpha=0.2044$ and
$\beta\lesssim 1$ ($\beta=0.5366$), we have that Hartl:2002ig

$s_{max}=0.110\sqrt{m^{3}R}.$ (47)

Nevertheless, the limit in Eq. (47), depends on the mass-radius ratio between
the objects in question. Therefore, an analytical expression may be necessary
to compute the mass-radius ratio. One example is given by the relation (for
non rotating white dwarfs) Hartl:2002ig

$\frac{R}{R_{\odot}}=0.1125\left(\frac{m}{m_{max}}\right)^{-\frac{1}{3}}f^{\frac{1}{2}}(m),$
(48)

with

$f(m)=1-\left(\frac{m}{m_{max}}\right)^{\frac{4}{3}},$ (49)

and $m_{max}=1.454M_{\odot}$. From which, following the results of Ref.
Lai:1993pa , it is possible to obtain the limit

$s\leq s_{max}=\frac{S_{max}}{mb_{0}}=9\times 10^{-6}<<1.$ (50)

##  IV Spinning particles around the Morris-Thorne wormhole

In this section, we use the results from Sec. III to compute the effective
potential $V_{\text{eff}}$ for a spinning test particle moving around a
wormhole. As we have shown, the effective potential depends on $\alpha$,
$\beta$, $\delta$, $\sigma$, $\gamma$, which are given by Eqs. (35). Note that
these functions depend on the redshift function $\Phi$. As mentioned before,
we focus on particles with positive energy taking $V_{\text{eff}}=V_{+}$ in
Eq. (37).

In general it is useful to rescale all the quantities in such a way that they
are adimensional. Hence, we define

$\begin{array}[]{ccc}l\rightarrow\frac{l}{b_{0}},&s\rightarrow\frac{s}{b_{0}}=\frac{S}{mb_{0}},&\mathcal{L}\rightarrow\frac{\mathcal{L}}{b_{0}}=\frac{L}{mb_{0}}.\end{array}$
(51)

Since the throat of the wormhole can be interpreted as the gravitational mass
of the source, we can rescale the radial distances per unit mass or
equivalently set $b_{0}=M=1$. Therefore, from Eqs. (35), and (37) we obtain

$\begin{array}[]{ccc}\includegraphics[scale={0.3}]{figure3a.png}&\includegraphics[scale={0.3}]{figure3b.png}&\includegraphics[scale={0.3}]{figure3c.png}\\\
\end{array}$

Figure 3: The effective potential $\mathcal{V}_{\text{eff}}$ as a function of
the radial coordinate $l$ with the throat located at $l=0.$ Left panel: Plots
of $\mathcal{V}_{\text{eff}}$ in “plus”(red) and “minus” (black)
configurations. Central panel: two “plus” configurations. Right panel: two
“minus” configurations. The dots show the location of the ISCO in the lower
(black) and upper (gray) universes. In the plots we assume $b_{0}=M=1$.

$\displaystyle\mathcal{V}_{\text{eff}}(l,s,\mathcal{L})$
$\displaystyle=-\frac{\delta\mathcal{L}}{\beta}+\sqrt{\left(\frac{\delta\mathcal{L}}{\beta}\right)^{2}+\frac{\gamma-\mathcal{L}^{2}\sigma}{\beta^{2}}},$
(52)

where we have also defined the adimensional effective potential per unit mass
of test particle by setting $\mathcal{V}_{\text{eff}}\rightarrow
V_{\text{eff}}/m$. With the above adjustments we get

$\displaystyle\beta$
$\displaystyle=\frac{e^{\frac{2}{\sqrt{1+l^{2}}}}[1+l(l-s)][1+l(l+s)]}{(1+l^{2})^{2}},$
(53) $\displaystyle\delta$ $\displaystyle\rightarrow
b_{0}\delta=\frac{ls(1-\sqrt{1+l^{2}})}{(1+l^{2})^{\frac{3}{2}}\sqrt{e^{-\frac{2}{\sqrt{1+l^{2}}}}(1+l^{2})}},$
$\displaystyle\sigma$ $\displaystyle\rightarrow
b^{2}_{0}\sigma=\frac{l^{2}s^{2}}{(1+l^{2})^{4}}-\frac{1}{1+l^{2}},$
$\displaystyle\gamma$
$\displaystyle\rightarrow\frac{\gamma}{m^{2}}=\left(1-\frac{l^{2}s^{2}}{(1+l^{2})^{\frac{5}{2}}}\right)^{2}.$

From Eqs. (52) and (53), it is possible to see the symmetries in
$\mathcal{V}_{\text{eff}}(l)$ depending on the signs of $s$, and
$\mathcal{L}$. We may call $\mathcal{V}^{P}_{\text{eff}}$ the “plus”
configuration with positive signs for $s$ and $\mathcal{L}$. Then we have

$\displaystyle\mathcal{V}^{P}_{\text{eff}}(l,s,\mathcal{L})$
$\displaystyle=\mathcal{V}^{P}_{\text{eff}}(l,-s,-\mathcal{L}).$ (54)

Also we may call $\mathcal{V}^{M}_{\text{eff}}$ the “minus” configuration with
the opposite sign in the first term of Eq. (52). This is obtained when either
$\mathcal{L}$ or $\delta$ or $\beta$ changes sign. Then we have

$\displaystyle\mathcal{V}^{M}_{\text{eff}}(l,s,-\mathcal{L})$
$\displaystyle=\mathcal{V}^{M}_{\text{eff}}(l,-s,\mathcal{L}).$ (55)

$l$ | $s$ | $\mathcal{L}$ | Configuration | $l$ | $s$ | $\mathcal{L}$ | Configuration
---|---|---|---|---|---|---|---
+ | + | + | Plus | - | - | - | Plus
+ | + | - | Minus | - | - | + | Minus
+ | - | + | Minus | - | + | - | Minus
+ | - | - | Plus | - | + | + | Plus
Table 1: Symmetries and configurations for $\mathcal{V}_{\text{eff}}$
depending on the signs of $l$, $s$, and $\mathcal{L}$.

In Table. 1 we show in detail the symmetries for $\mathcal{V}_{\text{eff}}$ in
the upper ($l>0$) and lower ($l<0$) universes. Moreover, the behavior of each
configuration is shown in Fig. 3. In the left panel, we have plot together
$\mathcal{V}^{M}_{\text{eff}}(l,-0.1,1.997394)$ (red) and
$\mathcal{V}^{P}_{\text{eff}}(l,0.1,1.997394)$ (black), as functions of $l$.
Note that the change in the configuration from “plus” to “minus” also changes
the location of the innermost stable circular orbit (ISCO), shown in the
figure with black and gray dots. In the central panel, we plot the “plus”
configuration given by
$\mathcal{V}^{P}_{\text{eff}}(l,0.1,1.997394)=\mathcal{V}^{P}_{\text{eff}}(l,-0.1,-1.997394)$.
Finally, in the right panel, we plot the “minus” configuration given by
$\mathcal{V}^{M}_{\text{eff}}(l,-0.1,1.997394)=\mathcal{V}^{M}_{\text{eff}}(l,0.1,-1.997394)$.

Figure 4: The effective potential for a non-spinning test particle
($\tilde{s}=0$) is symmetric about the throat, hence the ISCO for non-spinning
particles is at the same distance from the throat in the upper and lower
universes. In the plots we assume $b_{0}=M=1$.

$\begin{array}[]{ccc}\includegraphics[scale={0.3}]{figure5a.png}&\includegraphics[scale={0.3}]{figure5b.png}&\includegraphics[scale={0.3}]{figure5c.png}\\\
\end{array}$

Figure 5: Plots of $\mathcal{V}_{\text{eff}}$ as a function of $l$ in
different situations. Left panel: for a non-rotating particle with different
values of $\mathcal{L}$. Central panel: constant angular momentum
$\mathcal{L}=2$ and different values of $s$. Right panel: constant spin
$s=0.2$ with different values of $\mathcal{L}$. In the plots we consider
$b_{0}=M=1$. Furthermore, in the the left and right panels, we vary
$\mathcal{L}$ from $0.7$ to $2.3$ by a step of $0.1$.

One interesting feature one sees from Fig. 3 is that the profile of the
effective potential is not symmetric about the throat $l=0$ due to the spin of
the particle, in contrast to the motion of non-spinning particles ($s=0$) as
shown in Fig. 4. The direction of the spin of the test particle changes the
shape of the effective potential removing its symmetry between upper and lower
universe. In the “plus” configuration, for example, the value of
$\mathcal{V}^{P}_{\text{eff}}$ in the upper universe is larger than that in
the lower universe. On the other hand, in the “minus” configuration, the value
of the effective potential in the lower universe is larger than that in the
upper universe, as can be seen from Fig. 3. For spinning particles the
symmetry of $\mathcal{V}_{\text{eff}}$ depends on which universe (i.e. sign of
$l$) and direction of the spin (i.e. sign of $s$), so that particles with
positive spin in the upper universe have the same motion as particles with
negative spin in the lower universe.

$\begin{array}[]{ccc}\includegraphics[scale={0.3}]{figure6a.png}&\includegraphics[scale={0.3}]{figure6b.png}&\includegraphics[scale={0.3}]{figure6c.png}\\\
\includegraphics[scale={0.3}]{figure6d.png}&\includegraphics[scale={0.3}]{figure6e.png}&\includegraphics[scale={0.3}]{figure6f.png}\end{array}$

Figure 6: Contour plots for the first and second derivatives of
$\mathcal{V}_{\text{eff}}(l,\mathcal{L})$ and location of the ISCO shown using
dots with black a red colors. In the first row we have $s=0.1$, in the second
row $s=1.3$. Left panel: Contour plot of $d\mathcal{V}_{\text{eff}}/dl={\rm
const.}$. The ISCO is located on the curve $d\mathcal{V}_{\text{eff}}/dl=0$.
Middle panel: Contour plot of $d^{2}\mathcal{V}_{\text{eff}}/dl^{2}={\rm
const.}$. The ISCO is located on the curve
$d^{2}\mathcal{V}_{\text{eff}}/dl^{2}=0$. Right panel: Superposition of
$d\mathcal{V}_{\text{eff}}/dl$ and $d^{2}\mathcal{V}_{\text{eff}}/dl^{2}$. The
ISCO (when the first and second derivatives vanish) is given by the point of
intersection of the two curves
$d\mathcal{V}_{\text{eff}}/dl=d^{2}\mathcal{V}_{\text{eff}}/dl^{2}=0$. In the
plots we assume $b_{0}=M=1$.

In Fig. 5, the behavior of $\mathcal{V}_{\text{eff}}$ as function of $l$ in
different situations is plotted. In the left panel, we show the effective
potential for a non-rotating particle with different values of the angular
momentum $\mathcal{L}$, varying from $0.7$ to $2.3$ and increasing with a step
of $0.1$. In the figure, it is possible to see the symmetry about the throat
in the shape of $\mathcal{V}_{\text{eff}}$, which has the same behaviour in
both the lower and upper universes. As the angular momentum increases, the
effective potential increases. For large values of $\mathcal{L}$, such as
$\mathcal{L}=2.1$, $2.2$ and $2.3$, the behavior of $\mathcal{V}_{\text{eff}}$
shows two maxima (unstable circular orbits) and three minima, one of which is
located at the throat. The behavior in the upper (lower) universe outside the
throat resembles the usual behavior in the Schwarzschild geometry. Also, when
the angular momentum $\mathcal{L}$ decreases, the two peaks vanish, and the
shape of $\mathcal{V}_{\text{eff}}$ changes in such a way that only one
minimum value appears at the throat. One can see similar behavior in the right
panel of Fig. 5, where the effective potential of a spinning particle
($s=0.2$) for different values of $\mathcal{L}$ is plotted. However here we
see that the presence of the spin removes the symmetry of
$\mathcal{V}_{\text{eff}}$ about the throat and the effective potential does
not have the same behavior in both universes. This is clearly highlighted in
the central panel of Fig. 5 where $\mathcal{V}_{\text{eff}}$ is plotted for
various values of $s$, while keeping the angular momentum ($\mathcal{L}=2$)
constant. As above mentioned, the profile of $\mathcal{V}_{\text{eff}}$ is
symmetric only when the spin of the particle is zero (red color curve in the
central panel of Fig. 5). Finally, it is important to point out that
$\mathcal{V}_{\text{eff}}$ tends to $1$ when $l\rightarrow\pm\infty$.

##  V Innermost stable circular orbits

Now, we focus our attention on circular orbits of spinning particles in the
space-time of a wormhole given by Eq. (22). Circular motion occurs when the
radius is constant and the radial acceleration of the particle vanishes.
Mathematically, this means that the radial velocity of the test particle
vanishes $dl/d\lambda=0$, which implies that
$\mathcal{E}=\mathcal{V}_{\text{eff}}(l)$ (see Eq. (36)), where we have
defined the energy of the test particle per unit mass as $\mathcal{E}=E/m$.
The radial acceleration of the particle also vanishes, i.e.
$d^{2}l/d\lambda^{2}=0$, from which $d\mathcal{V}_{\text{eff}}/dl=0$.
Nevertheless, this condition does not guarantee that circular orbits are
stable. The stability of a circular orbit can be evaluated from the second
derivative of the effective potential with respect to the radial coordinate,
which must be positive, namely

$\frac{d^{2}\mathcal{V}_{\text{eff}}}{dl^{2}}\geq 0.$ (56)

When $d^{2}\mathcal{V}_{\text{eff}}/dl^{2}=0$, one can obtain the marginally
stable circular orbit, corresponding to the smallest allowed value for stable
circular orbits, also known as the innermost stable circular orbit or ISCO.

Using the conditions $\mathcal{E}=\mathcal{V}_{\text{eff}}$ and
$d\mathcal{V}_{\text{eff}}/dl=0$ we obtain the values of $\mathcal{E}$ and
$\mathcal{L}$ in terms of the circular orbit radius $l$ and then from
$d^{2}\mathcal{V}_{\text{eff}}/dl^{2}=0$ the value of the radius of the ISCO
for a spinning test particle. Since the process involves the solution of a
non-linear system of equations for $l$ and $\mathcal{L}$, we must in general
solve it numerically. Also, to ensure that motion of the spinning particle is
physically valid, we check that the superluminal bound is satisfied using Eq.
(44).

In Fig. 6, we show the contour plots for the first and second derivatives of
the effective potential for two different values of the spin $s=0.1$ and
$1.3$. When we consider $s=0.1$ and the “plus” configuration (both
$\mathcal{L}$ and $s$ positives), the contour plot shows two possible values
for the ISCO: one located in the upper universe
($l_{\text{ISCO}}=1.946354727$) and the other is in the lower universe
($l_{\text{ISCO}}=-1.470809497$). It is important to point out that each value
of the ISCO corresponds to a different value in the angular momentum. Hence,
in the upper universe, a particle with spin $s=0.1$ moves in a stable circular
orbit at the ISCO radius $l=1.946354727$ with angular momentum
$\mathcal{L}=1.979967293191$. In the lower universe, on the other hand, a
particle with spin $s=0.1$ moves in a stable circular orbit at the ISCO radius
$l=1.470809497$ with angular momentum $\mathcal{L}=1.997394095017$. Note that
the ISCO in the lower universe has a higher angular momentum, and it is closer
to the wormhole’s throat. When we change the “minus” configuration, the roles
are inverted. See the black and red dots in the figure. In the “minus”
configuration ($s$ positive and $\mathcal{L}$ negative), the ISCO represented
by the red dot is at the lower universe.

$\begin{array}[]{ccc}\includegraphics[scale={0.3}]{figure7a.png}&\includegraphics[scale={0.3}]{figure7b.png}&\includegraphics[scale={0.3}]{figure7c.png}\end{array}$

Figure 7: First and second derivatives of the effective potential as a function of $l$. In the left panel with $s=0.1$ and $\mathcal{L}=1.997394095017$, in the middle panel with $s=0.1$ and $\mathcal{L}=1.979967293191$ and in the right panel, we consider the case in which $s=-1.5$ and $\mathcal{L}=0.868763508655$. In the plots we consider $b_{0}=M=1$, we also include, in the small frame, a plot of the effective potentials for comparison. The ISCO is given by $\mathcal{V}^{\prime}_{\text{eff}}=\mathcal{V}^{\prime\prime}_{\text{eff}}=0$ and we can see that for $s=0.1$ there are two ISCO, one for $l<0$ and one for $l>0$, while for $s=-1.5$ there is only one ISCO for $l<0$.  Table 2:  Values for the radius $l_{\text{ISCO}}$, the angular momentum $\mathcal{L}_{\text{ISCO}}$ and the energy $\mathcal{E}_{\text{ISCO}}$ of the innermost stable circular orbit for different values of the spin $s$. $s$ | $l_{\text{ISCO}}$ | $\mathcal{L}_{\text{ISCO}}$ | $\mathcal{E}_{\text{ISCO}}$ | $s$ | $l_{\text{ISCO}}$ | $\mathcal{L}_{\text{ISCO}}$ | $\mathcal{E}_{\text{ISCO}}$
---|---|---|---|---|---|---|---
$\mp$ 1.6 |  $\mp$ 1.920360608 | 0.722911422497 | 0.8612999409591453 |  $\mp$ 0.7 |  $\pm$ 0.429511438 | 1.402514112333 | 0.6351506341677197
$\mp$ 1.5333See the right panel of Fig. 7. |  $\mp$ 2.141902280 | 0.868763508655 | 0.8742579899441784 |  $\mp$ 0.6 |  $\mp$ 2.534233144 | 1.713278206766 | 0.8872844627862982
$\mp$ 1.4 |  $\mp$ 2.322036598 | 0.993562472504 | 0.8814014855529899 |  $\mp$ 0.6 |  $\pm$ 0.526940168 | 1.554036117274 | 0.682349979652707
$\mp$ 1.3222See the second row of Fig. 6. |  $\mp$ 2.452395852 | 1.105236494336 | 0.8855957905171136 | $\mp 0.5$ |  $\mp$ 2.464002952 | 1.780911539067 | 0.8851281413233677
$\mp$ 1.2 |  $\mp$ 2.541002804 | 1.208099326345 | 0.8881263685022468 |  $\mp$ 0.5 |  $\pm$ 0.634414281 | 1.682361990518 | 0.7239081520286046
$\mp$ 1.1 |  $\mp$ 2.596660369 | 1.304401541915 | 0.889587124437354 |  $\mp$ 0.4 |  $\mp$ 2.373390705 | 1.842804901024 | 0.8822631357857148
$\mp$ 0.9 |  $\mp$ 2.631997715 | 1.481545447701 | 0.8903365076796163 |  $\mp$ 0.4 |  $\pm$ 0.763940166 | 1.795084655659 | 0.7618079197491819
$\mp$ 0.9 |  $\pm$ 0.230465057 | 0.928262126118 | 0.50216126965439 |  $\mp$ 0.2 |  $\mp$ 2.119571659 | 1.944481810011 | 0.8735759021557974
$\mp$ 0.8 |  $\mp$ 2.618174005 | 1.563289723767 | 0.8898509886736642 |  $\mp$ 0.2 |  $\pm$ 1.183708000 | 1.962437611319 | 0.0.824239661146142
$\mp$ 0.8 |  $\pm$ 0.334514462 | 1.209867268321 | 0.5781567283131726 |  $\mp$ 0.1111See the first row of Fig. 6, and the left and central panels in Fig. 7. |  $\mp$ 1.946354727 | 1.979967293191 | 0.8669581544832881
$\mp$ 0.7 |  $\mp$ 2.585463638 | 1.640598254787 | 0.8888394718973344 |  $\mp$ 0.1111See the first row of Fig. 6, and the left and central panels in Fig. 7. |  $\pm$ 1.470809497 | 1.997394095017 | 0.8443614729992029
$\mp$ 0.7 |  $\pm$ 0.429511438 | 1.402514112333 | 0.6351506341677197 | 0.0 |  $\mp$ 1.732050808 | 1.999999999999 | 0.8577638849606256

For values of the spin in the range $-1<s<1$, we have two values for the ISCO
at each universe. Nevertheless, for $s<-1$ or $s>1$, we have only one ISCO;
located only in one of the universes: upper or lower. We can see this behavior
from the contour plots in the second row of Fig. 6, where we impose $s=1.3$.

$\begin{array}[]{ccc}\includegraphics[scale={0.32}]{figure8a.png}&\includegraphics[scale={0.32}]{figure8b.png}&\\\
\includegraphics[scale={0.32}]{figure8c.png}&\includegraphics[scale={0.32}]{figure8d.png}&\end{array}$

Figure 8: Plots of ISCO radius $l_{\text{ISCO}}$, energy
$\mathcal{E}_{\text{ISCO}}$, angular momentum $\mathcal{L}_{\text{ISCO}}$, and
$\mathcal{F}$ (see Eq. (44)) as a function of $s$. The curves in red color
corresponds to the values of the ISCO closer to the wormhole’s throat (see
table 2). The region in gray color represents the zone where the particle’s
motion is space-like. In the plots we assume $b_{0}=M=1$.

The above statement is clearly illustrated in Fig. 7. Nevertheless, taking
into account the discussion of Sec.III.3, the interval $|s|>1$ does not
represent realistic values of the spin. In the case of an extended body like
Jupiter, for example, the ratio $S/m^{2}$ could be $\mathcal{O}(10^{3})$, then
$s\sim 1$. Therefore, although Jupiter may survive a traversable wormhole
(because there is no tidal), the MPD equations are inappropriate to model its
motion. Only those values of $s<<1$ have physical meaning.

In the figure, we plot the first and second derivatives of the effective
potential when $s=0.1$ (left and central panels) and $s=-1.5$ (right panel),
with the corresponding values of the angular momentum being
$\mathcal{L}_{\text{ISCO}}=1.979967293191$, $1.997394095017$ and
$0.868763508655$, respectively. In the left panel, we see how the derivatives
intersect at $l_{\text{ISCO}}=-1.4708095$ (lower universe) when
$\mathcal{L}=1.997394$. Moreover, from the small frame, we see that the values
of the corresponding effective potential (Dot-dashed line) are greater than
the values of $\mathcal{V}_{\text{eff}}$ when the angular momentum is
$\mathcal{L}=1.979967293191$ (Gray-line). In the central panel of Fig. 7, we
see the intersection of the first and second derivatives at
$l_{\text{ISCO}}=+1.946354727$ (upper universe). In the small frame we plot
the effective potential when $\mathcal{L}=1.979967293191$ (Dot-Dashed line)
and $\mathcal{L}=1.997394095017$ (gray line) for reference. In the right
panel, we consider the case when $s=-1.5$ and $\mathcal{L}=0.868763508655$ and
in this case, the first and second derivatives intersect at $l=-2.141902280$
(lower universe). Note that this time we considered the “minus” configuration,
but analogue results hold for the “plus” configuration. Other examples are
listed in table 2.

Using the values of table 2, one can also check the superluminal bound by
means of Eq. (44). Hence, for a wormhole described by Eqs. (22) and (23), we
have

$\displaystyle\frac{p^{t}}{m}$
$\displaystyle=-\frac{1}{1-\frac{l^{2}s^{2}}{\left(l^{2}+1\right)^{5/2}}}\left(\frac{l\sqrt{e^{-\frac{2}{\sqrt{l^{2}+1}}}\left(l^{2}+1\right)}s\mathcal{L}}{\left(l^{2}+1\right)^{5/2}}+\mathcal{E}\right)$
(57) $\displaystyle\frac{p^{\varphi}}{m}$
$\displaystyle=\frac{1}{1-\frac{l^{2}s^{2}}{\left(l^{2}+1\right)^{5/2}}}\left(\frac{ls\mathcal{E}}{\sqrt{e^{-\frac{2}{\sqrt{l^{2}+1}}}\left(l^{2}+1\right)}}+\mathcal{L}\right),$

and

$\displaystyle\mathcal{X}$
$\displaystyle=e^{\frac{2}{\sqrt{1+l^{2}}}}\left\\{\left(\frac{l^{2}s^{2}}{(1+l^{2})^{\frac{5}{2}}}-1\right)^{2}-\frac{[(1+l^{2})^{2}-s^{2}]^{2}}{(1+l^{2})^{4}}\right\\},$
(58) $\displaystyle\mathcal{Y}$
$\displaystyle=\Bigg{\\{}\left[(1+l^{2})^{\frac{7}{2}}-\left(1-l^{2}(1+2l^{2})+l^{2}\sqrt{1+\l^{2}}\right)s^{2}\right]^{2}-(1+l^{2})^{2}\left[(1+l^{2})^{\frac{5}{2}}-l^{2}s^{2}\right]\Bigg{\\}}\frac{1}{(1+l^{2})^{8}},$
$\displaystyle\mathcal{Z}$
$\displaystyle=1-\frac{l^{2}s^{2}}{(1+l^{2})^{\frac{5}{2}}}.$

In Fig. 8, we show the behavior of $l_{ISCO}$, $\mathcal{E}_{ISCO}$,
$\mathcal{L}_{ISCO}$ and $\mathcal{F}_{ISCO}$ as a function of $s$. In the
first row left panel, for example, we plot $l_{ISCO}$ as a function of $s$. As
mentioned before, note there is only one value for $l_{ISCO}$ (in the upper or
lower universes) when $|s|\geq 1$ and two values of $l_{ISCO}$ when $-1<s<1$
(one in the upper universe and the other in the lower universe). For this
reason, we have two different colors representing these intervals: black (when
$l_{ISCO}$ is far from the throat) and red (for those values of $l_{ISCO}$
closer to the throat) colors. We have a similar situation for
$\mathcal{E}_{ISCO}$, $\mathcal{L}_{ISCO}$ and $\mathcal{F}_{ISCO}$. In the
lower universe, the figure shows that $l_{ISCO}$ decreases when the spin of
the particle $s\leq-1$. Then, in the interval $-1<s<1$, $l_{ISCO}$ increases
and gets closer to the throat of the wormhole. In the upper universe, on the
other hand, the situation is inverted. Note that we have chosen
$\mathcal{L}_{ISCO}$ always positive so that the “plus” and “minus”
configurations are given only by the sign of $s$.

In the first row right panel of Fig. 8, we show the behavior of
$\mathcal{E}_{ISCO}$ as a function of $s$. As expected, when $s=0$, the value
of $\mathcal{E}_{ISCO}$ is the same in both universes due to the symmetry in
the effective potential. On the other hand, when $-1<s\leq 0$ (“minus”
configuration), note that particles in the lower universe move with a higher
value of energy than those moving in the upper universe, but it increases as
$s$ increases reaching the same value at $s=0$. The situation is inverted when
we change the configuration to “puls” ($0\leq s<1$).

In the second row left panel of Fig. 8, we show the behavior of
$\mathcal{L}_{ISCO}$ as a function of $s$. From the figure, when $-1<s\leq 0$,
it is possible to see that particles in the lower universe move with higher
angular momentum than those in the upper universe. However, at some value of
$s$ (around $-0.25$), particles in the upper universe begin to move with
higher angular momentum with a maximum value
$\mathcal{L}_{ISCO}=2.0022010837825808$, see the small frame added in the
figure. Note that this maximum value occurs far from the throat when
$l_{ISCO}=1.627928721886677$. Once again, the situation is inverted when we
consider the “plus” configuration.

Finally, in the second row right panel of Fig. 8, we show the value of
$\mathcal{F}_{ISCO}$ (see Eq. (44)). From the figure, it is clear that the
condition $\mathcal{F}<0$ is always satisfied when the particle’s spin belongs
to the interval $-1<s<1$. The last statement is true for particles moving in
both universes. However, when $|s|>1$, the value of $\mathcal{F}$ increases
and reaches the zero value at $s=\mp 1.5$, which is the well-known
superluminal bound. Therefore, when the spin is in the interval $-1.5<s<1.5$,
the motion of a spinning particle at the ISCO is time-like and has physical
meaning. Nevertheless, when $|s|>1.5$, the motion of a spinning particle at
the ISCO is space-like and meaningless from the physical point of view; see
the gray region in all the figures.

##  VI Conclusions

In this work, we have studied the motion of spinning test particles around a
traversable wormhole using the MPD equations. These equations relate the
Riemann curvature tensor to the second rank and skew-symmetric tensor
$S^{\alpha\beta}$, which is related to the particle spin $s$. Hence, following
Refs. Conde:2019juj ; Toshmatov:2019bda , we were able to compute the
effective potential for a spinning test particle moving in the background of a
traversable wormhole, described by the Morris-Thorne solution given in Refs.
Li:2014coa ; Harko:2008vy .

Our analysis shows how the effective potential depends on the angular
momentum, the throat of the wormhole $b_{0}$, the proper radial distance $l$
and the spin of the particle $s$. In particular, we discussed the symmetries
of the system and the location of the ISCO in upper and lower universes. These
symmetries depend on the signs of the angular momentum and the spin of the
particle. When $-1<s<1$, one interesting feature we found is the existence of
two radii allowed for the ISCO on both sides of the wormhole’s throat
depending on whether the particle’s spin is co-rotating or counter-rotating
with respect to the particle’s angular momentum. On the other hand, when
$|s|>1$, only one ISCO exists for co-rotation in the upper universe and
counter-rotation in the lower universe. This would produce some peculiar
features for accretion disks around wormholes that will be investigated in
future works.

Finally, we studied the superluminal bound for spinning particles at the ISCO
and showed that they are allowed to move in a time-like trajectory if
$-1.5<s<1.5$. In this region, the value of $\mathcal{F}$ is negative,
satisfying Eq. (44). Nevertheless, for values of $|s|>1.5$, the motion of the
spinning test particles is space-like, and therefore meaningless from the
physical point of view. On the other hand, in the case of compact objects
orbiting a traversable wormhole, our analysis indicates that they move in a
time-like trajectory because the spin parameter of these objects is $s<<1$. In
this particular case, our analysis shows that compact objects orbiting a
traversable wormhole in the lower universe will move with higher values of
energy $\mathcal{E}_{ISCO}$ than those moving in the upper universe. We can
see similar behavior in the case of the angular momentum $\mathcal{L}_{ISCO}$:
compact objects in the lower universe move with higher angular momentum than
those in the upper universe. However, at some value of the spin
$s\approx-1.5$, compact objects in the upper universe begin to move with
higher angular momentum.

When considering astrophysical sources such as stellar mass black hole
candidates or supermassive black hole candidates, then the motion of test
particles orbiting the central object may allow us to determine the features
that distinguish a central black hole from a wormhole Bambi:2021qfo ;
Jusufi:2021lei ; Tripathi:2019trz . Although, considering the discussion of
Sec.III.3, it is important to remember that the MPD equations are obtained
under the assumption that the mass and size of the spinning test particle must
be negligible with respect to the central object’s mass and must not affect
the background geometry, the presence of the spin may still produce observable
effects on the motion of test particles. Such test particles may be in the
form of gas particles in the accretion disk but also in the form of large
objects such as asteroids and planets orbiting stellar mass candidates and
rapidly rotating black holes and neutron stars orbiting supermassive
candidates. In this respect then, the spin of the test particle may become an
important element to consider when describing the motion of such objects and
the observations that would allow us to conclude if these objects are black
holes or wormholes.

###### Acknowledgements.

The work of C.A.B.G and A.A. is supported by a postdoc fund through PIFI of
the Chinese Academy of Sciences. WH is supported by NSFC No. 11773059. A.A.
and B.A. acknowledge the support of the Uzbekistan Ministry for Innovative
Development.

##  Appendix A Derivation of $u^{l}$ and $u^{\varphi}$

Let us start with the first relation in Eq. (40). From the Tulczyjew-SSC
condition $S^{\alpha\beta}p_{\alpha}=0$, we obtain

$S^{tl}p_{t}+S^{\varphi l}p_{\varphi}=0.$ (59)

Applying the operator $D/d\lambda$, we get

$\displaystyle\frac{D}{d\lambda}(S^{tl}p_{l})+\frac{D}{d\lambda}(S^{\varphi
l}p_{\varphi})$
$\displaystyle=p_{t}\frac{DS^{tl}}{d\lambda}+S^{tl}\frac{Dp_{t}}{d\lambda}$
(60) $\displaystyle+p_{\varphi}\frac{DS^{\varphi l}}{d\lambda}+S^{\varphi
l}\frac{Dp_{\varphi}}{d\lambda}=0,$

from which, after recalling that $S^{tl}=-(p_{\varphi}/p_{t})S^{\varphi l}$,
solving for $DS^{tl}/d\lambda$, and inserting into the MPD equation, we obtain

$\frac{DS^{tl}}{d\lambda}=\frac{p_{\varphi}S^{\varphi
l}}{(p_{t})^{2}}\frac{Dp_{t}}{d\lambda}-\frac{p_{\varphi}}{p_{t}}\frac{DS^{\varphi
l}}{d\lambda}-\frac{S^{\varphi
l}}{p_{t}}\frac{Dp_{\varphi}}{d\lambda}=p^{t}u^{l}-u^{t}p^{l}.$ (61)

In a similar way, we proceed with $DS^{t\varphi}/d\lambda$. The Tulczyjew-SSC
condition takes the form

$S^{\alpha\varphi}p_{\alpha}=S^{l\varphi}p_{l}+S^{t\varphi}p_{t}=0.$ (62)

Then, applying the operator $D/d\lambda$, we obtain

$\displaystyle\frac{D}{d\lambda}(S^{l\varphi}p_{l})+\frac{D}{d\lambda}(S^{t\varphi}p_{t})$
$\displaystyle=p_{l}\frac{DS^{l\varphi}}{d\lambda}+S^{l\varphi}\frac{Dp_{l}}{d\lambda}$
(63)
$\displaystyle+p_{t}\frac{DS^{t\varphi}}{d\lambda}+S^{t\varphi}\frac{Dp_{t}}{d\lambda}=0.$

Recalling that $S^{t\varphi}=(p_{l}/p_{t})S^{\varphi l}$, taking into account
the $S^{l\varphi}=-S^{\varphi l}$, solving for $DS^{t\varphi}/d\lambda$, and
inserting into the MPD equation, we obtain

$\frac{DS^{t\varphi}}{d\lambda}=\frac{p_{l}}{p_{t}}\frac{DS^{\varphi
l}}{d\lambda}+\frac{S^{\varphi
l}}{p_{t}}\frac{Dp_{l}}{d\lambda}-\frac{p_{l}S^{\varphi
l}}{(p_{t})^{2}}\frac{Dp_{t}}{d\lambda}=p^{t}u^{\varphi}-u^{t}p^{\varphi}.$
(64)

Now, from the MPD equation for $S^{\varphi l}$, the last two of Eqs. (40)
reduce to

$\displaystyle\frac{S^{\varphi
l}}{p_{t}}\left(p_{\varphi}\frac{Dp_{t}}{d\lambda}-p_{t}\frac{Dp_{\varphi}}{d\lambda}\right)$
$\displaystyle=(p_{\varphi}p^{\varphi}+p_{t}p^{t})u^{l}$ (65)
$\displaystyle-p^{l}p_{\varphi}u^{\varphi}-p^{l}p_{t}u^{t},$
$\displaystyle\frac{S^{\varphi
l}}{p_{t}}\left(p_{t}\frac{Dp_{l}}{d\lambda}-p_{l}\frac{Dp_{t}}{d\lambda}\right)$
$\displaystyle=(p_{l}p^{l}+p_{t}p^{t})u^{\varphi}$
$\displaystyle-p^{\varphi}p_{l}u^{l}-p^{\varphi}p_{t}u^{t}.$

Now we use the second MPD equation to obtain $Dp_{t}/d\lambda$,
$Dp_{\varphi}/d\lambda$ and $Dp_{l}/d\lambda$. Hence from

$\frac{Dp_{\alpha}}{d\lambda}=-\frac{1}{2}R_{\alpha\beta\delta\sigma}u^{\beta}S^{\delta\sigma}$
(66)

we obtain

$\frac{Dp_{t}}{d\lambda}=-\frac{1}{2}\left(2R_{tt\delta\sigma}u^{t}S^{\delta\sigma}+2R_{tl\delta\sigma}u^{l}S^{\delta\sigma}+2R_{t\varphi\delta\sigma}u^{\varphi}S^{\delta\sigma}\right).$
(67)

The factor 2 in the Riemann tensor components comes due to the antisymmetry of
both $S^{\alpha\beta}$ and $R_{\alpha\beta\delta\sigma}$. Therefore, we have
to count twice in the sum because
$R_{ab\delta\sigma}S^{\delta\sigma}=R_{ab\sigma\delta}S^{\sigma\delta}$.
Consequently, we get

$\frac{Dp_{t}}{d\lambda}=-R_{tltl}u^{l}S^{tl}-R_{t\varphi
t\varphi}u^{\varphi}S^{t\varphi}.$ (68)

Recalling that $S^{tl}=-\frac{p_{\varphi}}{p_{t}}S^{\varphi l}$ and
$S^{t\varphi}=\frac{p_{l}}{p_{t}}S^{\varphi l}$, the last equation reduces to

$\frac{Dp_{t}}{d\lambda}=\frac{S^{\phi
l}}{p_{t}}\left(p_{\varphi}R_{tltl}u^{l}-p_{l}R_{t\varphi
t\varphi}u^{\varphi}\right).$ (69)

Following a similar procedure, we obtain the following relations for
$Dp_{l}/d\lambda$ and $Dp_{\varphi}/d\lambda$

$\displaystyle\frac{Dp_{l}}{d\lambda}$ $\displaystyle=\frac{S^{\varphi
l}}{p_{t}}(p_{\varphi}R_{lttl}u^{t}+p_{t}R_{l\varphi l\varphi}u^{\varphi}),$
(70) $\displaystyle\frac{Dp_{\varphi}}{d\lambda}$
$\displaystyle=\frac{S^{\varphi l}}{p_{t}}(p_{t}R_{\varphi
ll\varphi}u^{l}-p_{l}R_{\varphi tt\varphi}u^{t}).$

Then, inserting them into the system of equations (65), we obtain

$\displaystyle
u^{l}\left[(p_{\varphi})^{2}\hat{\mathcal{A}}+(p_{t})^{2}\hat{\mathcal{B}}\right]$
$\displaystyle=u^{\varphi}\hat{\mathcal{C}}p_{l}p_{\varphi}+u^{t}\hat{\mathcal{C}}p_{l}p_{t},$
(71) $\displaystyle
u^{\varphi}\left[(p_{l})^{2}\hat{\mathcal{C}}+(p_{t})^{2}\hat{\mathcal{B}}\right]$
$\displaystyle=u^{l}\hat{\mathcal{A}}p_{l}p_{\varphi}+u^{t}\hat{\mathcal{A}}p_{\varphi}p_{t}.$

Finally, solving the system, we obtain

$\displaystyle u^{l}$
$\displaystyle=\frac{\hat{\mathcal{C}}}{\hat{\mathcal{B}}}\frac{p_{l}}{p_{t}},$
(72) $\displaystyle u^{\varphi}$
$\displaystyle=\frac{\hat{\mathcal{A}}}{\hat{\mathcal{B}}}\frac{p_{\varphi}}{p_{t}},$

with

$\displaystyle\hat{\mathcal{A}}$
$\displaystyle=g^{\varphi\varphi}+R_{tllt}\left(\frac{S^{\varphi
l}}{p_{t}}\right)^{2},$ (73) $\displaystyle\hat{\mathcal{B}}$
$\displaystyle=g^{tt}+R_{\varphi ll\varphi}\left(\frac{S^{\varphi
l}}{p_{t}}\right)^{2},$ $\displaystyle\hat{\mathcal{C}}$
$\displaystyle=g^{ll}+R_{\varphi tt\varphi}\left(\frac{S^{\varphi
l}}{p_{t}}\right)^{2}.$

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