1 Introduction: conformal Weyl gravity and null geodesics

Conformal Weyl gravity (CWG) was born of an attempt to unify gravity and electromagnetism based on the principle of local invariance of a manifold, described by the metric \(g_{\mu \nu }(x)\), under the change

$$\begin{aligned} g_{\mu \nu }(x)\rightarrow \varOmega ^2(x)\,g_{\mu \nu }(x), \end{aligned}$$
(1)

where \(\varOmega (x)\) is a smooth, strictly positive function [1,2,3,4]. The CWG theory can be obtained from the conformally invariant action

$$\begin{aligned} I_W=2\,\alpha _w \, \int \mathrm{d}^4 x\,\sqrt{-g}\, \left[ R_{\mu \nu }\,R^{\mu \nu }-\frac{1}{3} \left( R^{\mu }_{\mu }\right) ^2\right] , \end{aligned}$$
(2)

where \(\alpha _w\) is a dimensionless parameter chosen to be positive if Eq. (2) is a positive definite Euclidean action. The vacuum field equations associated with this action are solved by the static, spherically symmetric line element given by [5,6,7,8,9]

$$\begin{aligned} \mathrm{d}\tilde{s}^{2}=-B(\tilde{r})\,\mathrm{d}\tilde{t}^{2}+\frac{\mathrm{d}\tilde{r}^{2}}{B(\tilde{r})}+ \tilde{r}^{2}(\mathrm{d}\tilde{\theta }^{2}+\sin ^{2}\tilde{\theta }\, \mathrm{d}\tilde{\phi }^{2}), \end{aligned}$$
(3)

where the coordinates are defined in the range \(-\infty< \tilde{t} < \infty \), \(\tilde{r}\ge 0\), \(0\le \tilde{\theta }\le \pi \), \(0\le \tilde{\phi }\le 2\pi \), and the lapse function \(B(\tilde{r})\) is given by

$$\begin{aligned} B(\tilde{r})=1-\frac{\tilde{\beta }\,(2-3\tilde{\beta }\,\tilde{\gamma })}{\tilde{r}}-3\tilde{\beta }\,\tilde{\gamma }+\tilde{\gamma } \tilde{r} - \tilde{k} \tilde{r}^{2}. \end{aligned}$$
(4)

Here \(\tilde{\beta }\), \(\tilde{k}\) and \(\tilde{\gamma }\) are positive constants associated with the central mass, cosmological constant and the measurements of the departure of the Weyl theory from the Einstein–de Sitter, respectively. Clearly, taking the limit \(\tilde{\gamma }=0=\tilde{k}\) recovers the Schwarzschild case, so that we can identify \(\tilde{\beta } = M\).

A study of the basis and properties, together with applications of the motion of massive and massless particles in this geometry can be found, for example, in [10,11,12,13,14,15,16,17,18,19], and can be obtained using the standard Lagrange procedure [20,21,22,23,24,25,26,27], which allows a Lagrangian \(\mathscr {L}\) to be associated with the metric and then the equation of motion to be obtained from the Lagrange’s equations,

$$\begin{aligned} \dot{\varPi }_{q} - \frac{\partial \mathscr {L}}{\partial q} = 0, \end{aligned}$$
(5)

where \(\varPi _{q} = \partial \mathscr {L}/\partial \dot{q}\) are the conjugate momenta to the coordinate q, and the dot denotes a derivative with respect to the affine parameter \(\tau \) along the geodesic. Thus, in Sect. 2, and following the procedure performed by Klemm [28], we perform analytical continuations to obtain a non-trivial topology associated with toroidal topological black holes coming from CWG. In particular, we focus on the toroidal AdS black hole. Other studies associated with topological black holes can be found, for example, in Refs. [29,30,31,32,33,34,35,36]. Then we obtain the conserved quantities together with the equations of motion for massless particles on these manifolds. In Sect. 3 the radial motion is analyzed for photons going to spatial infinity or to the singularity, while Sect. 4 is devoted to obtaining analytically the trajectory for photons with non-zero angular momentum, for which we employ an analysis in terms of Jacobi elliptic functions. In Sect. 5 we apply the methods outlined by Sakurai, Tartaglia, Rizzi and Ruggiero, among others, to obtain an analogy to the Aharanov–Bohm effect to describe the Sagnac effect for this space-time. Finally, in Sect. 6 we conclude and summarize our results.

2 Toroidal topology

In order to obtain a toroidal topological black hole, we perform the following analytical continuation of the metric Eq. (3):

$$\begin{aligned}&\tilde{t} \rightarrow \sqrt{\alpha }\,t, \quad \tilde{r}\rightarrow \frac{r}{\sqrt{\alpha }},\quad \tilde{\phi }\rightarrow \phi ,\\&\tilde{\theta }\rightarrow \sqrt{\alpha }\,\theta , \quad \tilde{\beta } \rightarrow \frac{\beta }{\sqrt{\alpha }}, \quad \tilde{\gamma } \rightarrow \frac{\gamma }{\sqrt{\alpha }},\quad \tilde{k} \rightarrow k. \end{aligned}$$

Then, by taking the limit \(\alpha \rightarrow 0\), the line element becomes

$$\begin{aligned} \mathrm{d}s^2=-B(r)\,\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{B(r)}+r^{2}(\mathrm{d}\theta ^{2}+\theta ^{2}\,\mathrm{d}\phi ^{2}), \end{aligned}$$
(6)

with the lapse function

$$\begin{aligned} B(r)=\frac{3\,\beta ^2\,\gamma }{r}-3\,\beta \gamma +\gamma \,r-k\,r^2. \end{aligned}$$
(7)

In this case, it is possible to prove that the metric induced on the space-like surface of constant t and r corresponds to a compact orientable surface with genus \(g=1\), i.e., a torus, so the topology of this four-dimensional manifold becomes \(\mathbb {R}^2\times S^1\times S^1\) [28]. Therefore, performing \(\gamma =-2\eta /L\), \(\beta =\sqrt{L/3}\), \(k=-1/\ell ^2\), and then evaluating at the limit \(L\rightarrow \infty \), the lapse function becomes

$$\begin{aligned} B(r)=-\frac{2\eta }{r}+\frac{r^2}{\ell ^2}, \end{aligned}$$
(8)

which for \(\eta >0\) describes the AdS uncharged static toroidal black hole [37,38,39,40,41,42,43] with the event horizon placed at

$$\begin{aligned} r_+=(2\,\eta \, \ell ^2)^{1/3}. \end{aligned}$$
(9)

The vanished Lagrangian associated with the photons that move on this manifold can be expressed as

$$\begin{aligned} \mathscr {L}=-\frac{1}{2} B(r) \dot{t}^2+\frac{1}{2}\frac{\dot{r}^2}{B(r)}+\frac{1}{2}r^2 (\dot{\theta }^2+\theta ^2\,\dot{\phi }^2)=0. \end{aligned}$$
(10)

Conversely, the toroidal metric given by Eqs. (6) and (8) admits the following Killing vector field:

  • The time-like Killing vector \(\chi =\partial _t\) is related to the stationarity of the metric. The conserved quantity is given by

    $$\begin{aligned} g_{\mu \nu }\,\chi ^{\mu }\,u^{\nu }=-B(r)\,\dot{t}=-\sqrt{E} \end{aligned}$$
    (11)

    where E is a constant of motion that cannot be associated with the total energy of the test particle because this metric is not asymptotically flat.

  • The most general space-like Killing vector is given by

    $$\begin{aligned} \mathbf {\chi }= & {} \left( A\,\cos \phi +B\,\sin \phi \right) \,\partial _{\theta } \nonumber \\&+ \left[ C-A\frac{\,\sin \phi }{\theta }+B\,\frac{\cos \phi }{\theta }\right] \,\partial _{\phi }, \end{aligned}$$
    (12)

    where A, B and C are arbitrary constants. Is easy to see that it is a linear combination of the three Killing vectors

    $$\begin{aligned} \chi _1= & {} \partial _{\phi },\\ \chi _2= & {} \cos \phi \, \partial _{\theta }-\frac{\sin \phi }{\theta }\,\partial _{\phi },\\ \chi _3= & {} \sin \phi \, \partial _{\theta }+\frac{\cos \phi }{\theta } \partial _{\phi } \end{aligned}$$

    which are the angular momentum operators for this space-time. The conserved quantities are given by

    $$\begin{aligned} g_{\alpha \beta }\,\chi _{1}^{\alpha }\,u^{\beta }= & {} r^{2}\,\theta ^2\,\dot{\phi } =L_1, \end{aligned}$$
    (13)
    $$\begin{aligned} g_{\alpha \beta }\,\chi _{2}^{\alpha }\,u^{\beta }= & {} r^{2}\,(\cos \phi \,\dot{\theta }-\theta \,\sin \phi \,\dot{\phi }) =L_2, \end{aligned}$$
    (14)
    $$\begin{aligned} g_{\alpha \beta }\,\chi _{3}^{\alpha }\,u^{\beta }= & {} r^{2}\,(\sin \phi \,\dot{\theta }+\theta \, \cos \phi \,\dot{\phi }) =L_3, \end{aligned}$$
    (15)

    where \(L_1\), \(L_2\) and \(L_3\) are constants associated with the angular momentum of the particles.

It is interesting to point out that Eqs. (14) and (15) imply that

$$\begin{aligned} J^2\equiv L_2^2+L_3^2=r^4 (\dot{\theta }^2+\theta ^2\,\dot{\phi }^2), \end{aligned}$$
(16)

such that, if we focus our attention on the invariant plane \(\theta =\theta _0\), so \(\dot{\theta }=0\), we get \(L_1=\theta _0 \,J\). Thus, using Eqs. (11) and (16) in Eq. (10) we obtain the radial equation of motion corresponding to the one-dimensional problem

$$\begin{aligned} \dot{r}^{2}=E-\mathscr {V}(r), \end{aligned}$$
(17)

where \(\mathscr {V}(r)\) is the effective potential defined as

$$\begin{aligned} \mathscr {V}(r)=J^2\,\frac{B(r)}{r^2}\equiv J^2 \,V(r). \end{aligned}$$
(18)

Here \(V(r)=B(r)/r^2\) is the effective potential per unit of \(J^2\).

On the other hand, without lack of generality we choose \(\theta _0=1\) and combining Eqs. (16) and (17), we obtain the angular motion equation

$$\begin{aligned} \left( \frac{1}{r^2}\frac{dr}{d\phi }\right) ^{2}= \frac{1}{b^2}-V(r)=\frac{2\,\eta }{r^3}-\left( \frac{1}{\ell ^2}-\frac{1}{b^2}\right) , \end{aligned}$$
(19)

where \(b=J/\sqrt{E}\) is the impact parameter.

In Fig. 1 we have plotted the effective potential (per unit of \(J^2\)) as a function of the radial coordinate. In the next sections the motion of massless particles will be analyzed analytically by integration of the equations of motion.

Fig. 1
figure 1

Graphs of the topological toroidal effective potential as a function of the radial coordinate. The critical impact parameter \(b_c\) corresponds to the value of b at \(r\rightarrow \infty \)

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3 Radial motion

Radial motion corresponds to a trajectory with null angular momentum \(J=0\), so photons fall toward the singularity or to the spatial infinity. From Eq. (18) we can see that photons in radial motion have a null effective potential \(\mathscr {V}(r)=0\), so that Eqs. (11) and (17) become

$$\begin{aligned} \frac{dr}{d\tau }=\pm \sqrt{E}, \end{aligned}$$
(20)

and

$$\begin{aligned} \frac{\text {d}r}{\text {d}t}=\pm B(r)=\pm \frac{1}{\ell ^2}\left( \frac{r^3-r_+^3}{r}\right) , \end{aligned}$$
(21)

where the sign − (\(+\)) corresponds to photons falling to the event horizon (spatial infinity). Assuming that \(t=\tau =0\) at \(r=r_i\), then a straightforward integration of Eq. (20) leads to

$$\begin{aligned} \tau (r)=\pm \frac{r-r_i}{\sqrt{E}}, \end{aligned}$$
(22)

while an integration of Eq. (21) becomes

$$\begin{aligned} t(r)= & {} \pm \frac{\ell ^2}{\sqrt{3}\,r_+}\left\{ \arctan \left( \frac{2r+r_+}{\sqrt{3}\,r_+}\right) -\arctan \left( \frac{2r_i+r_+}{\sqrt{3}\,r_+}\right) \right. \nonumber \\&\quad \left. +\log \left[ \frac{r_i^3-r_+^3}{r^3-r_+^3}\left( \frac{r-r_+}{r_i-r_+}\right) ^3 \right] \right\} . \end{aligned}$$
(23)

Obviously, in the proper system photons cross the event horizon in a finite time \(\tau (r_+)\equiv \tau _+=(r_i-r_+)/\sqrt{E}\) and, eventually, arrive at the singularity in a finite time \(\tau (0)\equiv \tau _0=r_i/\sqrt{E}\). Also, they eternally approach the spatial infinity i.e., \(\tau (\infty )\rightarrow \infty \). On the other hand, an observer at \(r_i\) sees that photons take an infinite coordinate time even to arrive at \(r_+\), while it takes a finite coordinate time even to arrive at the spatial infinity, given by

$$\begin{aligned} t_{\infty }=\frac{\ell ^2}{\sqrt{3}\,r_+}\left\{ \frac{\pi }{2}-\arctan \left( \frac{2r_i+r_+}{\sqrt{3}\,r_+}\right) +\log \left[ \frac{r_i^3-r_+^3}{(r_i-r_+)^3} \right] \right\} . \end{aligned}$$
(24)

The existence of this time is due to the presence of the cosmological term on the toroidal topology and depends on the position of the observer \(r_i\). A similar feature was reported before by Villanueva and Vásquez, but in the context of Lifshitz black holes [44]. The behavior of both proper time and coordinate time is shown in Fig. 2.

Fig. 2
figure 2

Temporal null geodesics for radial motion described by Eqs. (22) and (23). This shows some equivalence with its spherical counterpart previously studied by Cruz et al. [21] for photons moving to \(r_+\). The novel result appears for photons moving to infinity in which, as is measured by an observer in \(r_i\), it takes a finite coordinate time \(t_{\infty }\) to reach infinity. This feature was reported before by Villanueva and Vásquez but in the context of Lifshitz space-times [44]

Full size image

4 Angular motion

This section is devoted to studying the angular motion of the test particles (\(J\ne 0\)), which depends on the value of the impact parameter b. From Fig. 1 we can see that there are two distinct regions separated by the critical impact parameter \(b_c=\ell \). Thus, if \(b>b_c\) the motion will be confined with a turning point, the apoastron distance \(r_t\), placed at

$$\begin{aligned} r_t(b)=\frac{r_+}{\left[ 1-\left( \frac{b_c}{b}\right) ^2\right] ^{1/3}}, \end{aligned}$$
(25)

whereas if \(b<b_c\) the turning point does not exist, and the motion is unbound with a negative characteristic distance (without physical meaning), the magnitude of which is

$$\begin{aligned} r_D(b)=\frac{r_+}{\left[ \left( \frac{b_c}{b}\right) ^2-1\right] ^{1/3}}. \end{aligned}$$
(26)

Both distances, \(r_t\) and \(r_D\), play an important role in the determination of trajectories because they depend strongly on the impact parameter b (see Figs. 3 and 5).

Fig. 3
figure 3

Plot for the turning point or apoastron, \(r_t\) as a function of the impact parameter b. The validity of this function is in \(b_c<b<\infty \)

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4.1 Confined motion

Returning to the general equation (19), we first consider the case when the impact parameter lies between \(b_c<b<\infty \), so using the variable \(u=1/r\) with \(u_t=1/r_t\), we can write

$$\begin{aligned} \frac{\mathrm{d}u}{\mathrm{d}\phi }= & {} \pm \sqrt{2\,\eta }\,\sqrt{u^3-u_t^3}\nonumber \\= & {} \pm \sqrt{2\,\eta }\,\sqrt{(u-u_t)\left[ \left( u+\frac{u_t}{2}\right) ^2+\frac{3\, u_t^2}{4}\right] }. \end{aligned}$$
(27)

Using the substitution

$$\begin{aligned} u=u_t\,\left( \sqrt{3}\,\cot ^2 \frac{\xi }{2}+1\right) , \end{aligned}$$
(28)

where the range of u is \(u_t\le u<\infty \), and the corresponding range of \(\xi \) is \(\xi _t=\pi \le \xi <2\pi \), Eq. (27) can be reduced to the elliptic form

$$\begin{aligned}&\pm&\alpha _1\,\int \limits _{\phi _t}^{\phi }\mathrm{d}\phi '=\int \limits _{\pi }^{\xi }\frac{\mathrm{d}\varphi }{\sqrt{1-k_1^2\,\sin ^2 \varphi }}\nonumber \\= & {} \int \limits _{0}^{\xi }\frac{\mathrm{d}\varphi }{\sqrt{1-k_1^2\,\sin ^2 \varphi }}-\int \limits ^{\pi }_{0}\frac{\mathrm{d}\varphi }{\sqrt{1-k_1^2\,\sin ^2 \varphi }}, \end{aligned}$$
(29)

where

$$\begin{aligned} \alpha _1= \sqrt{2\sqrt{3}\,\eta \,u_t},\qquad k_1=\sqrt{\frac{2-\sqrt{3}}{4}}. \end{aligned}$$
(30)

So, we may write the solution for \(\phi \) as

$$\begin{aligned} \pm \alpha _1\, \phi =2K(k_1)-F(\xi , k_1), \end{aligned}$$
(31)

where K(k) and \(F(\psi , k)\) are the complete and incomplete elliptic integrals of the first kind, respectively, and we have assumed that \(\phi _t=0\). Therefore, by using some formulas and identities of the Jacobian elliptic functions, we can write the equation of the trajectory as

$$\begin{aligned} r(\phi , b)=\frac{r_t(b)}{1+\sqrt{3}\,\text {tn}^2\left( \frac{\alpha _1(b)\,\phi }{2}, k_1\right) \,\text {dn}^2\left( \frac{\alpha _1(b)\,\phi }{2}, k_1\right) }, \end{aligned}$$
(32)

where tn(x, k) and dn(x, k) are Jacobi’s elliptic functions (see Appendix 1 and Refs. [45,46,47]). Obviously, this trajectory depends on the impact parameter and is shown in Fig. 4 for photons falling from \(r_t\).

Fig. 4
figure 4

Polar plot for a confined trajectory of photons from a distance \(r_t\)

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4.2 Unbounded motion

We now consider the case when the impact parameter lies between \(0<b<b_c\), which means that the real root is a negative one \(r_n\equiv -r_D\), where \(r_D\) is given by Eq. (26), so there is no turning point. Obviously, this negative root lacks physical interpretation, but it is important for determining the shape of the trajectory. Therefore, setting \(u=1/r\) again with \(u_D=1/r_D\), the equation of motion (19) can be written as

$$\begin{aligned} \frac{\mathrm{d}u}{\mathrm{d}\phi }= & {} \pm \sqrt{2\,\eta }\,\sqrt{u^3+u_D^3}\nonumber \\= & {} \pm \sqrt{2\,\eta }\,\sqrt{(u+u_D)\left[ \left( u-\frac{u_D}{2}\right) ^2+\frac{3\, u_D^2}{4}\right] }. \end{aligned}$$
(33)

In order to integrate Eq. (33), we now make the substitution

$$\begin{aligned} u=u_D\left( \sqrt{3}\,\cot ^2 \frac{\xi }{2}-1\right) , \end{aligned}$$
(34)

where the range is

$$\begin{aligned}&u\rightarrow 0\quad (\mathrm{infinity}), \quad \xi _{\infty }=\arccos \left( -\frac{2-\sqrt{3}}{2}\right) , \end{aligned}$$
(35)
$$\begin{aligned}&u\rightarrow \infty \quad (\mathrm{singularity}), \quad \xi _{s}=\pi , \end{aligned}$$
(36)

so the quadrature becomes

$$\begin{aligned} \pm \alpha _2\,\int \limits _{\phi _{\infty }}^{\phi }\mathrm{d}\phi '= & {} \int \limits _{\xi _{\infty }}^{\xi }\frac{\mathrm{d}\varphi }{\sqrt{1-k_2^2\,\sin ^2 \varphi }}\nonumber \\= & {} \int \limits _{0}^{\xi }\frac{\mathrm{d}\varphi }{\sqrt{1-k_2^2\,\sin ^2 \varphi }}-\int \limits ^{\xi _{\infty }}_{0}\frac{\mathrm{d}\varphi }{\sqrt{1-k_2^2\,\sin ^2 \varphi }},\nonumber \\ \end{aligned}$$
(37)

with

$$\begin{aligned} \alpha _2= \sqrt{2\sqrt{3}\,\eta \,u_D},\qquad k_2=\sqrt{\frac{2+\sqrt{3}}{4}}. \end{aligned}$$
(38)

Note from Eqs. (30) and (38) that the module of one trajectory corresponds to the complementary module of the other, \(k_1=\sqrt{1-k_2^2}=k'_2\) and \(k_2=\sqrt{1-k_1^2}=k'_1\). Therefore, assuming that \(\phi _{\infty }=0\) we may write

$$\begin{aligned} \pm \alpha _2\,\phi =F(\xi , k_2)-F(\xi _{\infty }, k_2), \end{aligned}$$
(39)

which implies that the trajectory now is described by the polar equation

$$\begin{aligned} r(\phi , b)=\frac{r_D(b)}{\sqrt{3}\,\mathrm{cs}^2(\varTheta , k_2) \,\mathrm{nd}^2(\varTheta , k_2)-1}, \end{aligned}$$
(40)

where cs(x, k) and nd(x, k) are Jacobi’s elliptic functions (see Appendix 1), and the phase \(\varTheta \) is given by

$$\begin{aligned} \varTheta =\frac{F(\xi _{\infty }, k_2)-\alpha _2(b)\,\phi }{2}. \end{aligned}$$
(41)

In Fig. 6 we have plotted the unbounded trajectory (40) for photons coming from infinity.

Fig. 5
figure 5

Plot for the modulus of the real negative root \(r_D\) as a function of the impact parameter associated with the unbounded motion of photons, i.e. \(0<b<b_c\)

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Fig. 6
figure 6

Polar plot for the unbounded trajectory, Eq. (40), followed by photons whose impact parameter lies between \(0<b<b_c\)

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5 The Sagnac effect

In this section we describe the Sagnac effect [48] by applying the formalism developed by Sakurai [49], Tartaglia [50], Rizzi and Ruggiero [51,52,53,54], among others, to the exterior space-time of the topological toroidal black hole. This approach was used to calculate this effect in the exterior space-time to an uncharged spherical symmetric black hole in conformal Weyl gravity [55]. For the non-rotating toroidal metric given by Eqs. (6) and (8) written in the usual Schwarzschild-like coordinates (\(ct', r', \theta ', \phi '\)):

$$\begin{aligned} \mathrm{d}s^2= & {} -\left( -\frac{2\eta }{r'}+\frac{r'^2}{\ell ^2}\right) \,c^2\mathrm{d}t'^{2}+\frac{\mathrm{d}r'^{2}}{\left( -\frac{2\eta }{r'}+\frac{r'^2}{\ell ^2}\right) }\nonumber \\&\quad +r'^{2}(\mathrm{d}\theta '^{2}+\theta '^{2}\,\mathrm{d}\phi '^{2}), \end{aligned}$$
(42)

so the transformation to the azimuthal frame of the rotating platform

$$\begin{aligned} ct=ct',\quad r=r',\quad \theta =\theta ',\quad \phi =\phi '-\varOmega t', \end{aligned}$$
(43)

where \(\varOmega \) is the constant angular velocity of the physical frame, yields the metric (after setting \(r=R\) and \(\theta =1\))

$$\begin{aligned} \mathrm{d}s^2= & {} -\left( -\frac{2\eta }{R}+\frac{R^2}{\ell ^2}-\frac{\varOmega ^2 R^2}{c^2}\right) \,\mathrm{d}(ct)^{2}+R^{2}\mathrm{d}\phi ^{2} \nonumber \\&\quad +2\frac{\varOmega }{c}\,R^2\,\mathrm{d}\phi \,\mathrm{d}(ct). \end{aligned}$$
(44)

Therefore, the non-zero components of the unit vector \(\gamma ^{\alpha }\) along the trajectory \(r=R\) are given by

$$\begin{aligned} \gamma ^t= & {} \frac{1}{\sqrt{-g_{tt}}}=\gamma _J, \end{aligned}$$
(45)
$$\begin{aligned} \gamma _t= & {} -\sqrt{-g_{tt}}=-\gamma _J^{-1}, \end{aligned}$$
(46)
$$\begin{aligned} \gamma _{\phi }= & {} g_{\phi t}\,\gamma ^t=\frac{\varOmega }{c}\,R^2\,\gamma _J, \end{aligned}$$
(47)

where

$$\begin{aligned} \gamma _J=\left[ R^2\left( \frac{1}{\ell ^2}-\frac{\varOmega ^2}{c^2}\right) -\frac{2\eta }{R}\right] ^{-1/2}. \end{aligned}$$
(48)

In terms of this component, the gravito-electric and gravito-magnetic potentials are given by

$$\begin{aligned} \phi ^G= & {} -c^2\,\gamma ^t=-c^2\,\gamma _J, \end{aligned}$$
(49)
$$\begin{aligned} A_{\phi }^G= & {} c^2 \frac{\gamma _{\phi }}{\gamma _t}=- c\,\varOmega \, R^2\, \gamma _J^2. \end{aligned}$$
(50)

As was shown in [51], it is possible to express the phase shift and time delay between light beams detected by a co-moving observer on the interferometer in terms of the gravito-magnetic induction field, \(\mathbf {B}^G=\mathbf {\nabla }\times \mathbf {A}^G\), by means of the expressions

$$\begin{aligned} \varDelta \varPhi =\frac{2\, \epsilon \, \gamma _t}{\hbar \,c^3}\int \limits _{S}\mathbf {B}^G \cdot \mathrm{d}\mathbf {S}=\frac{2\, \epsilon \, \gamma _t}{\hbar \,c^3}\int \limits _{\zeta (S)}\mathbf {A}^G \cdot \mathrm{d}\mathbf {r} \end{aligned}$$
(51)

and

$$\begin{aligned} \varDelta \tau =\frac{2\gamma _t}{c^3}\int \limits _{S}\mathbf {B}^G \cdot \mathrm{d}\mathbf {S}=\frac{2 \gamma _t}{c^3}\int \limits _{\zeta (S)}\mathbf {A}^G \cdot \mathrm{d}\mathbf {r}, \end{aligned}$$
(52)

where \(\epsilon \) is the relative energy of the photon as measured in the interferometer. Therefore, the phase shift turns out to be

$$\begin{aligned} \varDelta \varPhi =\frac{4\pi \epsilon }{\hbar c}\frac{\tilde{\varOmega }R}{\sqrt{1-\tilde{\varOmega }^2-\frac{r_+^3}{R^3}}}, \end{aligned}$$
(53)

while the time delay turns out to be

$$\begin{aligned} \varDelta \tau =\frac{4\pi }{c}\frac{\tilde{\varOmega }R}{\sqrt{1-\tilde{\varOmega }^2-\frac{r_+^3}{R^3}}}, \end{aligned}$$
(54)

where \(\tilde{\varOmega }=\varOmega /\varOmega _{\ell }\) is the dimensionless angular velocity, and \(\varOmega _{\ell }\equiv c/\ell \). Thus, from this last equation, we can see that if \(\varOmega =0\), i.e., performing a measure of the propagation time in a non-rotating frame, no Sagnac effect arises. Also, there is an upper limit for the angular velocity \(\varOmega _R\) related to the radius of the orbit R (see Fig. 7) given by

$$\begin{aligned} \varOmega <\varOmega _R \equiv \varOmega _{\ell }\,\sqrt{1-\left( \frac{r_+}{R}\right) ^3}. \end{aligned}$$
(55)
Fig. 7
figure 7

Plot for the maximum angular velocity \(\varOmega _R\) as a function of the radius of the orbit R for an interferometer in the Sagnac effect

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6 Summary

In this paper we have studied the null structure of the geodesics for a toroidal topological space-time which surrounds a black hole in the conformal Weyl gravity. First, we obtained an explicit behavior of radial photons to conclude that, while no changes in the motion to the singularity with respect to the Schwarzschild anti-de Sitter counterpart is found, there is a non-trivial coordinate time \(t_{\infty }\) for the description of the motion to the spatial infinity (see Eq. (24)). A similar result was obtained by Villanueva and Vásquez but in the context of the Lifshitz space-time [44]. Next, following the standard Lagrangian procedure, we have obtained analytically the trajectories of the confined and unbounded angular motion for photons in terms of Jacobi elliptic functions, Eqs. (32) and (40), and we have shown our results in Figs. 4 and 6, respectively. Obviously, these trajectories depend on the impact parameter b and, due to the topology, always fall to the singularity, which is a characteristic of AdS space-times. Finally, the Sagnac effect has been studied for this topological space-time. Our result is consistent with those obtained previously in other geometries in the sense that no Sagnac effect arises for a non-rotating frame. In addition, we have found a strong condition for its existence, which depends on the theory’s parameters \(\{\eta , \ell \}\) (in \(r_+\) and \(\varOmega _{\ell }\)) as well as on the radius of the circular orbit R. This condition is given in Eq. (55); cf.

$$\begin{aligned} \varOmega <\varOmega _R \equiv \varOmega _{\ell }\,\sqrt{1-\left( \frac{r_+}{R}\right) ^3}, \end{aligned}$$

for which the upper limit for the angular velocity \(\varOmega _R\) was plotted in Fig. 7 as a function of the radius R.

Finally, our study provides a simple physical visualization of the null trajectories and their main characteristics, and complements other studies carried out in the standard and/or trivial topology in conformal Weyl gravity [11,12,13,14,15], as well as in physically similar situations such as time-like geodesics on toroidal space-times [56] and motion of particles in toroidal magnetic fields on a Schwarzschild background [57], among others [58, 59].