Comparing spin supplementary conditions for particle motion around traversable wormholes

The dynamics of extended bodies is a crucial problem in any theory of gravity Almonacid:2013xnu ; Almonacid:2017gbz . In the particular case of general relativity (GR), the problem has been investigated since Einstein’s theory was published, and two approaches have been developed throughout history MPD03 . In the first approach, based on Einstein’s philosophy, bodies were considered as a set of elementary particles Einstein:1938yz ; Einstein:1940mt ; Fock1964 . In the second approach, on the other hand, bodies were treated using multipole moments and assuming them small so that they do not affect the spacetime background. This approach corresponds to the work of Mathisson, Papapetrou, Tulczyjew, and Dixon MPD03 ; MPD01 ; MPD01new ; MPD02 ; Tulczyjew1959 .

In contrast to particles without internal structure, extended bodies do not follow a geodesic¹¹1 Initially, in the case of point particles, this was a postulate of the theory, but later, Einstein showed it was a consequence of his field equations EinsteinI1927 ; EinsteinII1927 .. Therefore, obtaining the equations of motion is more challenging mainly because GR describes physical bodies using a four-dimensional region of spacetime, which is not independent of the relationship between gravity and geometry. In this sense, it is troublesome to establish the laws of motion of the body since its trajectory (world-line) is affected by itself, the gravitational field, and the geometry, which depends on the momentum-energy distribution Almonacid:2013xnu .

In particular, if one considers extended bodies as a set of elementary particles (Einstein’s point of view), it is clear that this approach generates problems when facing the question of the motion of celestial bodies because it would be necessary treating them as an assembly of elementary particles. According to Dixon, this requires “a general relativistic version of statistical mechanics, a formidable task” Dixon2015 . Therefore, from the astronomical point of view, a more suitable approach is to assume sufficiently small individual bodies so that one can treat them as particles Robertson1937 . That, however, required answering two important questions Dixon2015 . First, what point to choose to describe the position of the body? Second, how to describe its structure? Mathisson was the first to address these questions in 1937. To solve them, he started by selecting an arbitrary world-line within the body to characterize its motion and position and considering the energy-momentum tensor as an infinite set of multipole moments. In this way, the covariant conservation of the energy-momentum tensor becomes a set of equations representing the evolution of the multipole moments. All these considerations are contained in what Mathisson named “the gravitational skeleton of a body” Dixon2015 . Hence, chronologically speaking, he was the first to introduce the concepts of multipole moments and multipole particles in GR MPD01 .

In 1951, Papapetrou also considered an extended test body as described by an energy-momentum tensor, $T^{\mu\nu}$, and he defined in a non-covariant way its multipole moments MPD02 . Then, using the conservation equation $\nabla_{\beta}T^{\alpha\beta}=0$, he obtained the equations for a line inside the world-tube of the body under the assumption that all the moments higher than the dipole moment can be neglected. Furthermore, similarly to Mathisson, Papapetrou imposed the supplementary condition $V_{\alpha}S^{\alpha\beta}=0$ to make the equations fully determinate; however, in contrast to Mathisson, where $V_{\alpha}$ was the four-velocity, $u_{\alpha}$, in Papapetrou’s development, $V_{\alpha}$ corresponds to a vector field determined by the metric and not related to the body under consideration. Back then, the advantage of considering such a vector lay in avoiding the nonphysical helical motions allowed by Mathisson’s equations. Today, thanks to the work of Costa et al., we know that helical motions are perfectly valid and physically equivalent to the dynamics of a spinning body; the only difference is the choice of the representative point of the particle, a gauge choice Costa:2011zn .

In 1959, Tulczyjew simplified Mathisson’s theory by describing the particle not by a singularity in a linearized disturbance but by a singular energy-momentum tensor Dixon2015 . He obtained the same equations as Mathisson without imposing the supplementary condition $u_{\alpha}S^{\beta\alpha}=0$, and showing that the momentum vector of the particle, $p^{\alpha}$, and its four-velocity, $u^{\alpha}$, are not parallel Tulczyjew1959 . In a succeeding work BTulczyjew1962 , Tulczyjew considered Papapetrou’s approach in a more covariant form by introducing the world-line of the center of mass, related to the condition $p_{\beta}S^{\alpha\beta}=0$ Dixon2015 ; BTulczyjew1962 . Finally, in 1964, Dixon proposed a new treatment to the problem of extended bodies in GR, in which he included the effect of the electromagnetic field and a covariant definition of the center of mass. This allowed him to obtain the equations of motion in the pole-dipole approximation Dixon:1970zza ; Dixon:1970zz . According to Dixon, these equations are applicable to macroscopic bodies such as a planet orbiting the Sun but not to bodies of atomic scales since GR breaks down when quantum phenomena become important.

Nowadays, the motion of extended bodies in the pole-dipole approximation is described by the Mathisson-Papapetrou-Dixon (MPD) equations MPD01 ; MPD01new ; MPD02 ; MPD03 ,

	$\displaystyle\frac{Dp^{\mu}}{d\lambda}=$	$\displaystyle-\frac{1}{2}R^{\mu}_{\nu\rho\sigma}u^{\nu}S^{\rho\sigma}$		(1)
	$\displaystyle\frac{DS^{\mu\nu}}{d\lambda}=$	$\displaystyle p^{\mu}u^{\nu}-u^{\mu}p^{\nu},$		(2)

where the spinning test particle is characterized by a velocity vector, $u^{\mu}$, and momentum, $p^{\mu}$, in a curved spacetime background with the Riemann tensor defined as

$$R^{\mu}_{\nu\kappa\lambda}=\Gamma^{\mu}_{\kappa\alpha}\Gamma^{\alpha}_{\lambda\nu}-\Gamma^{\mu}_{\lambda\alpha}\Gamma^{\alpha}_{\kappa\nu}-\partial_{\lambda}\Gamma^{\mu}_{\kappa\nu}+\partial_{\kappa}\Gamma^{\mu}_{\lambda\nu}.$$

(3)

In Eqs. (1) and (2), $\frac{D}{d\lambda}\equiv u^{\mu}\nabla_{\mu}$ is the absolute derivative and $\lambda$ is an affine parameter. The antisymmetric tensor, $S^{\mu\nu}=-S^{\nu\mu}$, is related to the particle’s spin.

As mentioned above, note that in contrast to non-spinning test particles, where the absolute derivative of the four-momentum $p^{\mu}$ vanishes (the geodesic equation), spinning test particles follow an equation of motion coupled to $S^{\mu\nu}$ and $R^{\mu}_{\nu\kappa\lambda}$. From the physical point of view, this means that spinning test particles do not follow a geodesic. As a consequence, $u^{\mu}$ and $p^{\mu}$ are not parallel, and one needs to introduce two concepts of mass, the dynamical and the kinematical rest masses, defined by

	$\displaystyle\mu^{2}=$	$\displaystyle-p_{\alpha}p^{\alpha},$		(4)
	$\displaystyle m=$	$\displaystyle-p_{\alpha}u^{\alpha}.$		(5)

The non-parallel behavior between the velocity and the momentum is obtained by contracting the second MPD Eq.  (2) with the velocity and taking into account the normalization condition $u_{\alpha}u^{\alpha}=-1$. Hence, one obtains the following relation

$$p^{\alpha}=mu^{\alpha}+p^{\alpha}_{\text{hidden}},$$

(6)

where the term $p^{\alpha}_{\text{hidden}}=u_{\beta}\frac{DS^{\alpha\beta}}{d\lambda}$ is called the hidden momentum, which depends on the behavior of the spin tensor along the trajectory.

The MPD system of Eqs.  (1) and (2), together with the normalization condition for the velocity, is a set of eleven equations relating fourteen unknown variables: $[p^{\alpha},u^{\alpha},S^{\alpha\beta}]$. To “close” this system, it is necessary to introduce a constraint equation, usually known as the spin supplementary condition (SSC); one can find several SSCs in the literature represented by the general form

$$V_{\mu}S^{\mu\nu}=0,$$

(7)

where $V^{\mu}$ is a future-oriented reference vector satisfying the normalization condition $V_{\alpha}V^{\alpha}=-1$. As mentioned above, while developing the spinning test particle dynamics, Mathisson, Papapetrou, Pirani, and Tulczyjew used different SSCs MPD03 ; Tulczyjew1959 ; Pirani1956 ; Pirani2009 . Other examples are the the Ohashi-Kyrian-Semeràk (OKS) OKS01 ; OKS02 ; OKS03 , Corinaldesi-Papapetrou Corinaldesi:1951pb and the Newton-Wigner Newton:1949cq SSCs. From the physical point of view, choosing a particular reference vector $V^{\mu}$ corresponds to fixing the centroid of the body.

The MPD equations have been used widely in the literature, see Wald:1972sz ; Barker:1979dy ; Tod:1976ud ; Carmeli:1976mq ; Hojman:1976kn ; Semerak:1999qc ; Kyrian:2007zz ; Shibata:1993uk ; Mino:1995fm ; Costa:2017kdr and references therein. For example, The effects of the spin-curvature interaction were discussed by Wald (1972) and Barker $\&$ O’Connell (1979), Refs. Wald:1972sz and Barker:1979dy , respectively. The motion of spinning test particles in the field of a black hole was investigated by K. P. Tod et al. in Ref. Tod:1976ud , while the dynamics in Vaidya’s radiating metric and the Kerr-Newman spacetime were considered in Refs. Carmeli:1976mq and Hojman:1976kn , respectively. Latter, Semeràk, and K. Kyrian and Semeràk numerically solved the MPD equations to investigate the trajectories of spinning particles in the Kerr black hole using the TD SSC Semerak:1999qc ; Kyrian:2007zz . There, when the pole-dipole approximation is considered, the author found that no significant spin effects are expected if one considers astrophysical scenarios. Nevertheless, during the inspiral of a spinning particle onto a rotating compact body, important effects may occur that would modify the gravitational waves generated by the system. The gravitational wave generated by a spinning test particle falling or orbiting a black hole was investigated by Masaru Shibata and Yasushi Mino et al. in Refs. Shibata:1993uk and Mino:1995fm , respectively.

Recently, the MPD equations have been used to investigate the motion of spinning test particles in different spacetimes Plyatsko:2013xza ; Hackmann:2014tga ; Jefremov:2015gza ; Harms:2016ctx ; Zhang:2017nhl ; Toshmatov:2019bda ; Conde:2019juj ; Larranaga:2020ycg ; Timogiannis:2021ung ; Benavides-Gallego:2021lqn ; Abdulxamidov:2022ofi ; Timogiannis:2022bks ; Ladino:2022aja ; Li:2022sjb . The Kerr spacetime was considered in Refs.Plyatsko:2013xza ; Hackmann:2014tga ; Jefremov:2015gza ; Harms:2016ctx , in the last reference the authors also investigated the motion of spinning test particle in the Schwarzschild case. The properties of the innermost circular orbits (ISCO) for spinning test particles were investigated using the Kerr-Newman background in Ref. Zhang:2017nhl . The $\gamma$-metric, the Maxwell dilaton black hole, the charged Hayward black hole background, and the rotating black hole surrounded by the perfect fluid dark matter were studied in Refs. Toshmatov:2019bda , Conde:2019juj , Larranaga:2020ycg and Li:2022sjb , respectively. The dynamics of spinning test particles in a quantum-improved rotating black Hole (RBH) spacetime was examined by M. Ladino and E. Larranaga in Ref. Ladino:2022aja . In the case of wormholes, the motion of spinning test particles was investigated in Refs. Benavides-Gallego:2021lqn and Abdulxamidov:2022ofi using the Morris-Thorne non-rotating traversable wormhole Morris1988 and the rotating traversable wormhole obtained by Teo Teo:1998dp , respectively.

Finally, in Refs. Timogiannis:2021ung and Timogiannis:2022bks , the authors examine whether the equatorial circular orbits around a massive black hole are affected when the particle’s centroid (associated with the SSC) changes to another centroid (different SSC). To do so, the authors established an analytical algorithm to obtain the orbital frequency of a spinning body moving around an arbitrary stationary, axisymmetric spacetime, focusing on three SCCs: the Tulzcyjew-Dixon (TD), the Mathisson-Pirani (MP), and the Ohashi-Kyrian-Semeràk SSCs (OKS). In the case of the Schwarzschild black hole, they investigated the discrepancies in the orbital frequency employing a power series expansion of the spin for each SSC, imposing corrections to improve the convergence between the SSC. They found that the shifting from one circular equatorial orbit to another, the coincidence between the SSCs only holds up to the third order in the orbital frequency.

In the Kerr spacetime, on the other hand, the authors considered the convergence of the orbital frequency for prograde and retrograde equatorial circular orbits. Following a similar approach used for the Schwarzschild case, they expanded the orbital frequencies in powers of the particle’s spin for the same SSCs, i.e., the TD, MP, and OKS SSCs. The authors also introduced a novel method to compute the ISCO radius for any SCC. Similar to the Schwarszchild case, there is a convergence in the power series of the frequencies of the SSCs. However, there is a limit to this convergence because, in the spinning body approximation, one only considers the first two multipoles (pole-dipole) of the body and ignores the higher ones.

In this work, we follow Refs. Timogiannis:2021ung ; Timogiannis:2022bks to compare the orbital frequency of equatorial circular orbits in the background of a traversable non-rotating wormhole. We consider three of the most well-known SSCs: the TD MPD03 ; Tulczyjew1959 , MP Pirani1956 ; Pirani2009 , and the OKS OKS01 ; OKS02 ; OKS03 supplementary conditions. This work is organized as follows. In Sec. II, we roughly describe the Morris-Thorne traversable wormhole. In Sec. III, we obtain the orbital frequencies for equatorial circular orbits using the analytical algorithm proposed in Ref. Timogiannis:2021ung and obtain the analytical expressions for the Morris-Thorne wormhole using different centroids. Then, in Sec. IV, we discuss the orbital parameters; i.e., the orbital frequency and the ISCO radius, using different SSCs. In Sec. V, we use the centroid corrections to explain the differences between the SSCs. Here we consider the corrections to the position of the centroid and corrections to the spin. Finally, in Sec. VI, we review and conclude our work.

In the manuscript, we use dimensionless units, the Riemann curvature tensor is defined as Eq. (3), and the metric has the signature $(-,+,+,+)$.

The Morris-Thorne traversable wormhole is a spherically symmetric spacetime given by the line element Morris1988 ; Visser2002

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

(8)

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. (8) represents a wormhole with a throat connecting two different regions of the spacetime can be easily depicted by embedding the line element in a three-dimensional space at a fixed time slice $t$, see Fig. 1.

The metric of Eq.  (8) was obtained assuming first the wormhole’s spacetime as spherically symmetric, then, via the field equations, computing the corresponding energy-momentum tensor. In contrast to black holes, wormholes do not have an event horizon or singularities; this means the redshift function $\Phi(r)$ is everywhere finite, and the spacetime has a throat connecting two asymptotically flat regions of the same universe.

On the other hand, a wormhole is traversable if the tidal gravitational forces experienced by any traveler are bearable small Morris1988 . Moreover, the time needed to cross the wormhole must be finite and reasonably small. From the physical point of view, this means that the proper time measured by a traveler and the observers outside the wormhole must be finite and small. Hence, in the case of a zero-tidal-force solution, the redshift and the shape functions have the form Bambi:2013jda

$$\Phi(r)=-\frac{b_{0}}{r}\quad\text{ and }\quad b(r)=\left(\frac{b_{0}}{r}\right)^{\gamma},$$

(9)

where $b_{0}$ is the throat of the wormhole, usually associated with the wormhole’s mass. We use these functions to compare the different SSCs, focusing on the values $\gamma=1$ and $\gamma=2$.

Fig. 2 illustrates the behavior of the shape function $b(r)$ as a function of $r$ for various values of the wormhole throat $b_{0}$, and both $\gamma=1$ and $\gamma=2$ cases of the solution. Here we provide a preliminary comparison of the properties of the two wormhole solutions. The figure shows that as the wormhole throat $b_{0}$ increases, the shape function, $b(r)$, also increases in both solutions. Furthermore, the value of $b(r)$ for the $\gamma=2$ solution is asymptotically smaller and closer to $r=0$ than that of the $\gamma=1$ solution. Note that the shape function $b(r)$ has the same value for all cases at the wormhole throat point $b_{0}$.

The properties mentioned above are essential for traversable wormholes, what Morris and Thorne refer to as basic wormhole criteria. These criteria are deeply related to the form of the energy-momentum tensor, which depends on the matter and fields that generate wormholes. Even though the energy-momentum tensor is not physically reasonable since exotic mater (negative energy density) is required to create the wormhole’s spacetime curvature at its throat, it is possible to tune the wormhole’s parameters to make its constitution material compatible with the form of matter allowed by the laws of physics Morris1988 .

Without loss of generality, we can consider a spherically symmetric and static spacetime, described by the following line element:

$$ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\varphi\varphi}d\varphi^{2}.$$

(10)

For circular equatorial orbits, we fix the spatial coordinates as $r=\text{costant}$, $\theta=\frac{\pi}{2}$, and $\varphi=\Omega t$, where $\Omega=u^{\varphi}/u^{t}$ is the orbital frequency of the spinning test body. To maintain circularity in all the SSCs, the radial and polar four-velocity and four-momentum components must vanish; i.e., $u^{r}=u^{\theta}=0$ and $p^{r}=p^{\theta}=0$, respectively. Moreover, the normalization condition of the four-velocity implies

$$u^{t}=\frac{1}{\sqrt{-g_{tt}-g_{\varphi\varphi}\Omega^{2}}}.$$

(11)

Considering a spinning test body moving in the equatorial plane, with a spin $S$ aligned (or anti-aligned) with the total angular momentum $J_{z}$ (perpendicular to the equatorial plane), it follows that the spin four-vector $S_{\mu}$ takes the form:

$$S_{\mu}:=-\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}V^{\nu}S^{\rho\sigma},$$

(12)

and

$$S^{\rho\sigma}=-\epsilon^{\rho\sigma\nu\kappa}S_{\nu}V_{\kappa}.$$

(13)

Therefore, we can write $S^{\mu}\equiv S^{\theta}\delta_{\theta}^{\mu}$, and then the magnitude of the spin would be $S=\pm\sqrt{g_{\theta\theta}}S^{\theta}$. Hence, introducing the SSC in the general form of Eq.  (7) and considering the definition of the spin angular momentum as

$$S^{2}=\frac{1}{2}S^{\mu v}S_{\mu\nu},$$

(14)

the only non-vanishing components of the spin tensor are

$\displaystyle\begin{cases}&S^{tr}=-S^{rt}=-S\sqrt{-\frac{g_{\theta\theta}}{g}}V_{\varphi},\\ &S^{r\varphi}=-S^{\varphi r}=-S\sqrt{-\frac{g_{\theta\theta}}{g}}V_{t},\end{cases}$

(15)

with $g=g_{tt}g_{rr}g_{\theta\theta}g_{\varphi\varphi}$ the determinant of the metric. To relate the non-vanishing components of the spin tensor with the conserved quantities, we use the Killing equation

$$C_{\xi}=\xi^{\mu}P_{\mu}-\frac{1}{2}S^{\mu\nu}\nabla_{\nu}\xi_{\mu};$$

(16)

where $\xi^{\mu}$ is a Killing vector field associated with the conserved quantity.

In the case of a static and spherically symmetric spacetime, represented by the line element of Eq. (10), two Killing vectors exist. These killing vectors, given by $\xi_{(t)}^{\mu}=\delta_{t}^{\mu}$ and $\xi_{(\varphi)}^{\mu}=\delta_{\varphi}^{\mu}$, are related to the conservation of both the energy $C_{(t)}=-E$ and the $z$ component of the total angular momentum of the spinning test particle $C_{(\varphi)}=J_{z}$, respectively. Using the Eqs. (15), these conserved quantities can be expressed as Timogiannis:2021ung

$\displaystyle\begin{cases}&E=-p_{t}-\partial_{r}g_{tt}\frac{S}{2}\sqrt{-\frac{g_{\theta\theta}}{g}}V_{\varphi},\\ &J_{z}=p_{\varphi}-\partial_{r}g_{\varphi\varphi}\frac{S}{2}\sqrt{-\frac{g_{\theta\theta}}{g}}V_{t}.\end{cases}$

(17)

Hence, with the help of the above considerations and results, the MPD equations for equatorial circular orbits reduce to Timogiannis:2021ung

	$\displaystyle\Gamma^{r}_{\rho\sigma}u^{\rho}p^{\sigma}=$	$\displaystyle-\frac{1}{2}R^{r}_{\nu\rho\sigma}u^{\nu}S^{\rho\sigma}$		(18)
	$\displaystyle\Gamma^{t}_{\rho r}u^{\rho}S^{r\varphi}+\Gamma^{\varphi}_{\rho r}u^{\rho}S^{tr}=$	$\displaystyle p^{t}u^{\varphi}-p^{\varphi}u^{t}.$		(19)

In the following subsections, we apply the MPD Eqs. (18) and (19) to different SSCs.

The differences in the orbital frequency obtained using the three spin supplementary conditions, shown in Tables 1 and 2, are usually explained by noting that each SSC defines a particular centroid Timogiannis:2022bks ; Lukes2017 and therefore, all the moments are evaluated correspondingly. We will assume that the choice of the SSC corresponds to a centroid’s correction in the form $z^{\mu}\rightarrow\tilde{z}^{\mu}=z^{\mu}+\delta z^{\mu}$. In order to compare the results obtained using the three SSCs presented in this paper, we will consider the TD-SSC as a reference because the corresponding centroid is uniquely determined. Hence, the quantities under the TD-SSC will be denoted by a tilde, $\tilde{}$.

Although the momentum components do not depend on the choice of the centroid, the spin tensor does change according to

$$S^{\mu\nu}\rightarrow\tilde{S}^{\mu\nu}=S^{\mu\nu}+p^{\mu}\delta z^{\nu}-p^{\nu}\delta z^{\mu}.$$

(54)

Hence, following the discussion in the appendix B of Timogiannis:2021ung , we will impose the constraint $V_{\alpha}\delta^{\alpha}=0$, which is equivalent to the relation

$$\tilde{p}_{\alpha}\delta z^{\alpha}=0$$

(55)

together with the condition that the centroid cannot have non-radial shifts between the TD- and the MP-/OKS-SSCs. This implies that the change in the centroid will be

$$\delta z^{\alpha}=\delta r=\frac{\tilde{p}_{\mu}S^{\mu r}}{\tilde{\mu}^{2}},$$

(56)

where $\tilde{\mu}^{2}=-\tilde{g}_{\alpha\beta}p^{\alpha}p^{\beta}$. Furthermore, in the previous section, we analyzed the relative differences of the ISCO frequency parameters $\Delta x_{ISCO}$ for the MP- and OKS-SSCs with respect to the TD-SSC, as done in Lukes2017 . Meanwhile, in this section, to compare the different centroid corrections, we will not calculate the relative difference, $\Delta x_{ISCO}$, with respect to the TD-SSC but in terms of each SSC correction. In this sense, to relate the ISCO frequency parameters $x_{ISCO}$ given by the MP-SSC centroid correction applied to the TD-SSC (denoted as TD-MP) with $x_{ISCO}$ given by the MP-SSC without correction, we introduce the relative difference: