Spinning test particle motion around a rotating wormhole

In 1935, Einstein and Rosen explored the possibility of an atomistic theory of matter and electricity without singularities Einstein:1935tc . The main idea behind their work was to consider the physical space as two identical sheets connected by a “bridge.” In this geometrical representation, particles were the very bridges connecting these sheets. Hence, using the metric tensor $g_{\mu\nu}$ of general relativity and the fields $\varphi_{\mu}$ of electromagnetism, they modified the gravitational equations, demonstrating that it is possible to obtain regular spherically symmetric solutions. These solutions are currently known as Einstein-Rosen bridges or “wormholes”.

The representation of particles as “bridges” could have been discovered back in 1916. According to G. W. Gibbons G.W.Gibbons2015 , previous to Einstein-Rosen’s paper, and a few months after the Schwarzschild solutions Schwarzschild:1916uq ; Schwarzschild:1916ae , Ludwig Flamm submitted a manuscript in which he explores some geometrical aspects of both exterior and interior solutions of Schwarzschild space-time L.Flamm1916 . In the former case (the exterior solution), Flamm was able to show that “the planar section is isometric to a surface of revolution, where the meridional curve is a parabola.” However, Flamm never contemplates the possibility of interpreting this result as a “bridge” connecting to regions of space-time. For this reason, we assume that the work of Einstein and Rosen gave birth to the modern study of “wormholes”.

Initially, the idea of “bridges” was considered more attractive than that of black holes until Wheeler and Fuller showed that the Einstein-Rosen bridge¹¹1Also known as the Schwarzschild wormhole. is unstable. Using a proper analysis, they discovered that the bridge would pinch-off in a finite time Fuller:1962zza . Therefore, Schwarzschild wormholes are not traversable. Nevertheless, the possibility of traversable wormholes was considered years latter in several works Ellis:1973yv ; Bronnikov:1973fh ; Morris:1988tu ; Morris:1988cz ; Visser:1989kh ; Visser:1989kg ; Visser:1989vq ; AzregAinou:1989wr ; Visser:1989am ; Poisson:1989zz ; Visser:1990wj ; Frolov:1990si ; Visser:1990wi . In particular, in Ref Morris:1988cz , Morris and Thorne used a different approach which allowed them to propose a set of “basic wormhole criteria.” Thus, by assuming the wormhole geometry, they used the Einstein field equations to compute physical quantities, such as the total energy density, the tension per unit area, and the pressure, in terms of what they call the “redshift” and “shape” functions. Their analysis establishes three properties to describe a wormhole. Firstly, the space-time is assumed to be static and spherically symmetric. Secondly, any wormhole solution should contain a throat that connects two asymptotically flat regions of space-time. This property is deeply related to the “shape function”. Finally, the solution must be horizonless. Since wormholes causally connect two different portions of the space-time by the throat, the presence of a horizon would prevent the two universes from being connected causally. This condition is satisfied by demanding the “redshift function” to be finite everywhere.

On the other hand, the most relevant conclusion in Morris and Thorne’s analysis is the necessity of “exotic matter” to generate a traversable wormhole. According to the authors, an observer passing through the throat with a radial velocity close to the speed of light would perceive negative energy. Therefore, from the classical point of view, traversable wormhole solutions violate the well-known energy conditions, established precisely to avoid negative energy densities. However, from the quantum point of view, there are some situations in which such violations may be physically valid. The quantum mechanical creation of particles, for example 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.

The idea of wormholes has been considered in different scenarios, such as Einstein’s gravity Echeverria:1991nk ; Deser:1992ts ; Deser:1993jr ; Hochberg:1997wp ; Abdujabbarov:2016efm ; Abdujabbarov:2009ad ; Pugliese:2011xn ; Willenborg:2018zsv ; Arganaraz:2019fup ; Stuchlik:2021tcn ; Benavides-Gallego:2021lqn and alternatives theories of gravity Moffat:1991xp ; Carlini:1992jda ; Bhawal:1992sz ; Letelier:1993cj ; Vollick:1998qf . In Ref. Benavides-Gallego:2021lqn , Benavides-Gallego et al. investigated the motion of spinning test particles around traversable wormholes. Using the Mathisson-Papapetrous-Dixon (MPD) equations, the authors computed the effective potential and showed that it is affected by the adimensional spin $s$ of the test particle. When $s=0$, the particle follows the geodesic equation, and its effective potential is symmetric: its behavior is the same in both the lower and upper universes. Therefore, the innermost stable circular orbit (ISCO) is at the same distance from the wormhole’s throat.

On the other hand, when the spin $s\neq 0$, there is a different kind of symmetry which depends on the sing of the particle’s spin $s$ and $\mathcal{L}$, the particle’s dimensionless angular momentum. In that case, there are two possible configurations which define two different symmetries: “plus” and “minus”. In this sense, the effective potential has the same behavior if it has the same configuration. For example, a particle with negative $s$ and negative $\mathcal{L}$ will have the same behavior as a particle with both $s$ and $\mathcal{L}$ positive. In other words, the effective potential of a particle with $s$ parallel to the symmetry axis ($s>0$) and moving counterclockwise ($\mathcal{L}>0$) is the same as that of a particle with $s$ antiparallel to the symmetry axis ($s<0$) and moving clockwise around the wormhole ($\mathcal{L}<0$), see Fig. 3 of Ref. Benavides-Gallego:2021lqn . The authors also found a “mirror behavior” when the “plus” and “minus” configurations are considered at the same time. To explain this behavior, the authors consider two spinning particles moving counterclockwise with the same angular momentum $\mathcal{L}$, one in the lower universe (with $s>0$) and the other in the upper universe (with $s<0$). In this case, the effective potential has the same behavior for each particle, resembling the case of a non-spinning test particle, see the first panel in figure 3 of Ref. Benavides-Gallego:2021lqn .

The existence of two configurations affects the location of the ISCO. As mentioned before, in the case of non-spinning test particles, one finds a single value for $l_{ISCO}$ located at the same distance from the wormhole’s throat in both universes²²2$l$ is the radial coordinate in Ref. Benavides-Gallego:2021lqn . The value $l=0$ represents the wormhole’s throat.. However, the behavior is different when we consider spinning test particles. If $|s|\geq 1$, one finds only one possible value for the ISCO. On the other hand, if $s$ belongs to the interval $-1<s<1$, one encounters two possible values for the ISCO. One of these values is always closer to the wormhole’s throat.

One significant conclusion from Ref. Benavides-Gallego:2021lqn is the constrain obtained for the particle’s spin $s$. It is well-known that the dynamical four-momentum $p^{\alpha}$ and the kinematical four-velocity $u^{\alpha}$ of a spinning test particle are not always parallel. In this sense, although $p_{\alpha}p^{\alpha}=-m^{2}$ holds, the normalization $u_{\alpha}u^{\alpha}=-1$ does not. Therefore, while 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 diverge. This means that for a certain value of $s$, the particle’s trajectory changes from time-like to space-like, becoming, in this way, superluminal. From the physical point of view, the space-like motion does not have any meaning because the transition to $u_{\alpha}u^{\alpha}>0$ is not allowed for real particles. Therefore, one must impose an additional constrain defined by the relation $u_{\alpha}u^{\alpha}=0$, the superluminal bound. In the case of a non-rotating wormhole, spinning test particles will always move in the time-like region as long as $|s|<1.5$ Benavides-Gallego:2021lqn .

In this paper, we consider the motion of spinning test particles around a rotating wormhole, generalizing in this way the results obtained by Benavides-Gallego et al. in Ref. Benavides-Gallego:2021lqn . We organize our work as follows. In Sec. II, we review the properties of rotating wormholes discussed in Ref. Teo:1998dp by E. Teo and its connection with the ideas of traversable wormholes described in Ref. Morris:1988cz . Next, in Sec. III, we discuss the motion of spinning particles in a static and axially symmetric space-time, where we use the MPD equations to obtain the effective potential and the superluminal condition. Then, in Sec. IV, we apply the results of Sec. III to Teo’s wormhole solution. 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. V, we summarize our work and discuss the implications of the results. Throughout the manuscript, we use geometrized units setting $G=c=1$ and $b_{0}=M=1$.

In Ref. Teo:1998dp , Teo follows the same paradigm as Morris-Thorne in Ref. Morris:1988cz , i.e. he first assumes the space-time geometry and then uses the Einstein field equations to deduce the form of the matter required to maintain the wormhole. Hence, the author begins by considering a stationary and axially symmetric space-time.

It is well-known that space-times are said to be stationary if it possesses a time-like Killing vector field $\xi^{\alpha}\equiv(\partial/\partial t)^{\alpha}$, which generates invariant time translations. On the other hand, a space-time is axisymmetric if it possesses a space-like killing vector field $\psi^{\alpha}\equiv(\partial/\partial\varphi)^{\alpha}$, related to invariant rotations with respect to $\varphi$. Therefore, a space-time is stationary and axisymmetric if it possesses both $\xi^{\alpha}$ and $\psi^{\alpha}$ killing vector fields satisfying the following commutation relation Wald:1984rg ,

The commutativity of $\xi^{\alpha}$ and $\psi^{\alpha}$ in Eq. (1) allow us to choose a coordinate system in such a way that the Killing vectors represent the directions in which the space-time has symmetries Wald:1984rg . Hence, one can set $x^{0}=t$, $x^{1}=\varphi$, $x^{2}$, and $x^{3}$ as a coordinate system. The stationary and axisymmetric character of the space-time requires the metric components to be independent of $t$ and $\varphi$. Therefore, the metric takes the form,

From the physical point of view, stationary and axisymmetric space-times have been of considerable interest in the study of black holes and stars since this kind of geometry describes the exterior gravitational field of rotating bodies, see Refs Hartle:1967he ; Hartle:1968si ; Thorne:1971R and references therein.

Thorne discusses the properties of static and axisymmetric space-times in Ref. Thorne:1971R . There, he stars by pointing out that $(t,\varphi,x^{2},x^{3})$ and $(t,\varphi+2\pi,x^{2},x^{3})$ represent the same point. This behavior is due to the fact that $\varphi$ is an angular coordinate about the rotation axis. As a consequence, $\varphi$ belongs to the interval $[0,2\pi)$.

On the other hand, because the space-time is axially symmetric, it must be invariant under a simultaneous inversion of $t$ and $\varphi$, i.e. the space-time does not change if $t\rightarrow-t$ and $\varphi\rightarrow-\varphi$. As a consequence, the metric coefficients $g_{t2}$, $g_{t3}$, $g_{\varphi 2}$ and $g_{\varphi 3}$ must vanish because they change the sign under simultaneous inversion of $\varphi$ and $t$. Therefore, the line element of Eq. (2) simplifies even more and reduces to Papapetrou:1966zz ; Carter:1969zz

with $i,j=2,3$. Here, the presence of the term $g_{01}$ is related to the well-know dragging effect, see Appendix A.

The coordinates in Eq. (3) are uniquely determine up to a coordinate transformation of the form Thorne:1971R

The freedom in such transformations can be used to simply the mathematics in the Einstein’s field equations or adapt the geometry to specific problems. Under transformation of the form given in Eq. (4), the components $g_{00}$, $g_{01}$, and $g_{11}$ are invariant. Hence, $g_{00}=\xi_{\alpha}\xi^{\alpha}$, $g_{01}=\psi_{\alpha}\xi^{\alpha}$, and $g_{11}=\psi_{\alpha}\psi^{\alpha}$ Wald:1984rg ; Thorne:1971R .

Finally, the space-time described in Eq. (3) must be asymptotically flat. This means that $g_{00}\rightarrow 1$, $g_{01}\rightarrow 1/r$ and $g_{11}\rightarrow r^{2}\sin^{2}\theta$ as $r\rightarrow\infty$. The asymptotically-flatness of the line element (3) is important to define the mass, and the angular momentum³³3Here $\theta$ and $r$ are the usual spherical coordinates, but not necessarily the same as $x^{2}$ and $x^{3}$Thorne:1971R .

In this section, we review the theoretical background necessary to investigate the motion of spinning test particles. It is well-known that this problem was considered for the first time by Mathisson in 1937 when he studied the problem of extended bodies in general relativity (GR). According to Mathisson, the motion of spinning test particles does not follow the usual geodesic equation of GR due to the coupling between the Riemann curvature tensor and the spin of the moving particle Mathisson:1937zz . Papapetrou considers the same problem in Refs. Papapetrou:1951pa ; Corinaldesi:1951pb , where he established a similar approach. Later, Tulczyjew improved on the methods of Mathisson tulczyjew1959motion ; BWTulzcyjew1962 while Moller and others made improvements in the definition of center-of-mass 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 (MPD) equations. Recently, some authors modified the MPD equations, see Refs. Deriglazov:2017jub ; Deriglazov:2018vwa .

Mathematically, the MPD equations are given by

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 antisymmetric, $S^{\alpha\beta}=-S^{\beta\alpha}$.

Expressed in that form, Eq. (17) shows us the coupling between the Riemann tensor $R^{\alpha}_{\;\;\beta\delta\sigma}$ and the spin of the moving particle. To see this coupling clearly, let’s consider the well-known geodesic equation of GR

In terms of the particle 4-momentum and the projection of the covariant derivative along the particle’s trajectory, Eq. (18) reduces to

Therefore, comparing Eqs. (17) and (19), one can see how the interaction between the Riemann curvature tensor and the antisymmetric tensor $S^{\alpha\beta}$ does affect the motion of spinning test particles in curved space-times.

A crucial aspect in the MPD equations is related to the center of mass of the spinning test particle. In this sense, to solve the system in Eq. (17), one needs to fix its center of mass. This is done by including the condition tulczyjew1959motion ; Saijo:1998mn

This condition is known as the Tulczyjew Spin Supplementary Condition (SSC) Saijo:1998mn . From Eq. (20), the canonical momentum and the spin of the particle provide two independent conserved quantities given by the relations

Nevertheless, in contrast to the spinning test particle’s canonical momentum conservation, it is important to point out that the squared velocity does not necessarily satisfy the condition

because the 4-vectors $p^{\alpha}$ and $u^{\alpha}$ are not always parallel. In this sense, to ensure that the particle’s 4-velocity is always smaller than the speed of light, one needs to impose an additional condition: the superluminal bound, see Sec. III.2.

Additionally, to the conserved quantities resulting from the Tulczyjew-SSC condition, one also has the conserved quantities associated to the space-time symmetries. As remarked in Sec.II, the line element in Eq. (5) has associated two Killing vector fields. One generating invariant time translations ($\xi^{\alpha}$) and the other related to invariant rotations with respect to $\varphi$ ($\psi^{\alpha}$). Hence, the conserved quantities associated to them can be obtained from the following relation,

where $\kappa^{\alpha}$ is the Killing vector field. In the last expression, we used the fact that the term $S^{\alpha\beta}\Gamma^{\gamma}_{\;\;\beta\alpha}$ in the covariant derivative $S^{\alpha\beta}\nabla_{\beta}\kappa_{\alpha}$ vanishes because $S^{\alpha\beta}$ is antisymmetric while $\Gamma^{\gamma}_{\;\;\beta\alpha}$ is symmetric.