1 Introduction

General relativity (GR) is the standard theory of gravity, which is proposed by Einstein in 1915 and has been successfully tested using different experiments and observations. Particularly, if solar system tests [1] of GR can be considered as a test in weak field regime, then recent observation of gravitational waves [2] and shadow of M87 [3, 4] can be considered as a test of GR in strong field regime. However, GR has few limitations in sense of describing the nature of the singularity during the gravitational collapse, cosmic acceleration, rotation curve of galaxies, incompatibility of GR with quantum field theory, etc. In order to resolve these problems, one needs to either break the symmetries of GR or introduce modifications or alternative theories of gravity. Indeed, there are few experimental and theoretical studies of Lorentz symmetry violations [5,6,7].

Among the different modifications and extensions of GR built on symmetry breaks, the Einstein-Æther theory became attractive for theoretical studies. Particularly, Einstein-Æther theory has been constructed in a such a way that Lorentz symmetry breaks down in microscale within the four-dimensional spacetime. In the action describing the spacetime together with the standard Hilbert term, there is extra additional term constructed using a unit timelike vector field. This unit vector assigns all timelike vectors of arbitrary direction and is called Æther field. The Lorentz symmetry is violated due to this Lagrangian containing Æther field and constructed using derivatives of Æther field with respect to four parameters. For given combinations of these parameters, one may obtain exact solutions of field equations describing the gravitating objects with additional parameters in the Einstein-Æther gravity. Particularly, in Ref. [8] the authors have obtained two exact solutions describing the static and electrically charged black holes within Einstein-Æther gravity using two parameters of Æther field. Numerical solutions of the field equations in the Einstein-Æther gravity have been obtained in [9].

The shadow of black hole described by the Einstein-Æther theory has been studied in [10], and authors have obtained constraints on the Æther parameter using data from Event Horizon Telescope on image of M87 [11]. Another constraint on parameters of Æther field has been obtained by analysing the data from gravitational wave observation [12]. Innermost stable circular orbits of test particles, periodic orbits and quasiperiodic oscillations have been extensively studied in [13]. In [13], authors have also explored astrophysical applications of their toy model using few known microquasars. In [14], authors have studied magnetized particle motion around black hole immersed in external magnetic field within Einstein-Æther theory.

Light bending in the vicinity of the compact gravitating object is one of the main consequences of the metric theories of gravity. The effect of gravitational lensing is responsible for the deflection of the photon from straight line. In fact, the first pioneering verification of Einstein’s general relativity during the observation of solar eclipse in 1919 [15] was related to the gravitational lensing by Sun. Gravitational lensing is extensively discussed from theoretical point of view in the recent literature (see, e.g. [16,17,18,19,20,21,22,23,24]). The numerous works devoted to study the photon motion around compact objects, and effect of gravitational lensing can be found, for example, in Refs.  [25,26,27,28,29,30,31,32,33,34] including the study of gravitational lensing in the presence of plasma in Refs. [35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51]. It is worth noting that shadow of black hole is formed by photon motion around black holes [52]. The detailed study of shadow of various black holes can be found, for example, in Refs. [53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73].

The treatment of particles’ motion in environment of compact gravitating objects is powerful tool to test the spacetime properties in the strong gravitational field regime (see [74,75,76,77]). The particles orbits and their motion are very sensitive to the parameters of the gravitational field [78,79,80,81,82,83,84]. Thus study of test (mass) particles leads to study the parameters of field and consequently to study the theory of gravity in strong field regime. And also study of the energy extracted from a black hole is quite extended for various spacetime metrics by several authors in Refs. [85,86,87,88,89,90,91].

In this paper, we plan to study the test particle dynamics and gravitational lensing in the vicinity of black hole in Einstein-Æther gravity. Particularly, we plan to investigate the photon motion and gravitational lensing in the presence of plasma around black hole with nonzero Æther parameter. The work is arranged as follows: In Sect. 2, we study the properties of Einstein-Æther gravity and corresponding black hole solutions. Particle motion around BH in the Einstein-Æther gravity is investigated in Sect. 3. In Sect. 4, gravitational weak lensing is studied for various distributions of plasma around Einstein-Æther BH. Image source magnification in the presence of uniform and non-uniform plasma is explored in Sect. 5. We have also discussed the energy extraction process from the Einstein-Æther BH in Sect. 6. Section 7 is devoted to analysis of the obtained results. We use the geometrical system of units where \(G=1=c\). Latin (Greek) indices run from 1(0) to 3.

2 Black hole in Einstein-Æther gravity

The action of Einstein-Æther theory contains the Einstein–Hilbert action with an additional term corresponding to a dynamical, unit time-like Æther field [92,93,94], which cannot vanish anywhere, and breaks local Lorentz symmetry. The complete action has the following form:

$$\begin{aligned} S_{\ae }=\frac{1}{16\pi G_{\ae }}\int d^4x\sqrt{-g}\left( R+{{{\mathcal {L}}}}_{\ae }\right) , \end{aligned}$$
(1)

where \(g=|g_{\mu \nu }|\) is the determinant of the metric tensor of spacetime described by the Einstein-Æther gravity. The Lagrangian of Æther field in the action (1) has the following form:

$$\begin{aligned} {{{\mathcal {L}}}}_{\ae }=-M^{\alpha \beta }_{~~~\mu \nu }(D_{\alpha }u^{\mu })(D_{\beta }u^{\nu })+\lambda (g_{\mu \nu }u^{\mu }u^{\nu }+1) , \end{aligned}$$
(2)

here \(D_{\alpha }\) is the covariant derivative with respect to \(x^\alpha \), \(\lambda \) is the Lagrangian multiplier which is responsible for the timelike Æther four-velocity \(u^{\alpha }\), and \(M^{\alpha \beta }_{~~~\mu \nu }\) is defined as

$$\begin{aligned} M^{\alpha \beta }_{~~~\mu \nu }=c_1g_{\mu \nu }g^{\alpha \beta }+c_2 \delta ^{\alpha }_{\mu }\delta ^{\beta }_{\nu }+c_3\delta ^{\alpha }_{\nu }\delta ^{\beta }_{\mu }-c_4 u^{\alpha }u^{\beta } g_{\mu \nu } , \end{aligned}$$
(3)

where \(c_i\) (\(i=1,2,3,4\)) are dimensionless coupling constants introduced in this theory. Æther gravitational constant has the following form:

$$\begin{aligned} G_{\ae }=\frac{G_N}{1-\frac{1}{2}c_{14}} , \end{aligned}$$
(4)

where \(G_N\) is the Newtonian gravitational constant.

The exact solution of field equations in the Einstein-Æther theory (1) describing the spacetime of spherically symmetric black hole has the following form in the spherical polar coordinates [95]:

$$\begin{aligned} ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2(d\theta ^2+\sin ^2\theta d\phi ^2) , \end{aligned}$$
(5)

with

$$\begin{aligned} f(r)=1-\frac{2 M}{r}\left( 1+\frac{2 c_{13}-c_{14}}{4(1-c_{13})}\frac{M}{r}\right) , \end{aligned}$$
(6)

and \(c_{13}=c_1+c_3\) and \(c_{14}=c_1+c_4\) are the coupling constants of the Einstein-Æther theory.

Now we study horizon structure of BH in Einstein-Æther gravity using \(f(r)=0\) and easily get event horizon of BH which is represented in Fig. 1. We explored different values of the parameters \(c_{13}\) and \(c_{14}\) in Einstein-Æther theory and observed that as the parameter \(c_{13}\) increases, the event horizon is increased and, in opposite interestingly, as the parameter \(c_{14}\) increases, the event horizon is decreased.

Fig. 1
figure 1

Dependence of the event horizon radius \(r_h\) on parameters \(c_{13}\) and \(c_{14}\) for BH in Einstein-Æther gravity. We set \(M=1\)

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3 Particle dynamics around black hole in Einstein-Æther gravity

In this section, we investigate the motion of massive and massless particles around a black hole in Einstein-Æther gravity.

3.1 Massive particles motion around black hole in Einstein-Æther gravity

First we study massive particle motion around the regular black hole in Einstein-Æther gravity. In order to find the trajectory of the test particle, one needs to consider Lagrangian for the test particle of mass m of the following form

$$\begin{aligned} L'=\frac{1}{2}g_{\mu \nu }u^{\mu }u^{\nu } , \quad u^{\mu }=\frac{dx^{\mu }}{d\tau }, \end{aligned}$$
(7)

where \(\tau \) is an affine parameter, \(x^{\mu }\) and \(u^{\mu }\) are the coordinates and four-velocity of the test particle, respectively. The conservative quantities of the motion such as the energy \({{{\mathcal {E}}}}\) and the angular momentum \({{{\mathcal {L}}}}\) of the test particle can be written in the following form:

$$\begin{aligned} {{{\mathcal {E}}}}= & {} \frac{\partial L'}{\partial u^t}=-f(r)\frac{dt}{d\tau } , \nonumber \\ {{{\mathcal {L}}}}= & {} \frac{\partial L'}{\partial u^{\phi }}=r^2\sin ^2\theta \frac{d\phi }{d\tau } . \end{aligned}$$
(8)

By inserting expressions in (8) into the normalization condition \(g_{\mu \nu }u^{\mu }u^{\nu }=-\epsilon \), one can easily find the equations of motion of test particle in the following form:

$$\begin{aligned} \frac{\mathrm{d}r}{\mathrm{d}\tau }= & {} \sqrt{{{{\mathcal {E}}}}^2-f(r)\left( \epsilon +\frac{{{{\mathcal {K}}}}}{r^2}\right) }, \end{aligned}$$
(9)
$$\begin{aligned} \frac{\mathrm{d}\theta }{\mathrm{d}\tau }= & {} \frac{1}{r^2}\sqrt{{{{\mathcal {K}}}}-\frac{{{{\mathcal {L}}}}^2}{\sin ^2\theta }}, \end{aligned}$$
(10)
$$\begin{aligned} \frac{\mathrm{d}\phi }{\mathrm{d}\tau }= & {} \frac{{{{\mathcal {L}}}}^2}{r^2\sin ^2\theta }, \end{aligned}$$
(11)
$$\begin{aligned} \frac{\mathrm{d}t}{\mathrm{d}\tau }= & {} \frac{{{{\mathcal {E}}}}}{f(r)}, \end{aligned}$$
(12)

where \({{{\mathcal {K}}}}\) is the Carter constant and the parameter \(\epsilon \) is defined as:

$$\begin{aligned} \epsilon = {\left\{ \begin{array}{ll} 1,\quad \mathrm{for\ \ timelike \ \ geodesics} \\ 0,\quad \mathrm{for\ \ null\ \ geodesics} \\ -1,\quad \mathrm{for \ \ spacelike\ \ geodesics}. \end{array}\right. } \end{aligned}$$
(13)

For simplicity, one can consider the motion of the particles in the equatorial plane in which \(\theta =\frac{\pi }{2}\) and \(\frac{d\theta }{d\tau }=0\). In this special case, Carter constant takes form \({{{\mathcal {K}}}}={{{\mathcal {L}}}}^2\) and the equation for the radial motion takes the following form:

$$\begin{aligned} \left( \frac{dr}{d\tau }\right) ^2={{{\mathcal {E}}}}^2-V_{eff}(r)={{{\mathcal {E}}}}^2-f(r)\left( 1+\frac{{{{\mathcal {L}}}}^2}{r^2}\right) , \end{aligned}$$
(14)

with

$$\begin{aligned} V_{eff}=f(r)(1+\frac{ {{{\mathcal {L}}}}^2}{r^2}), \end{aligned}$$
(15)

being the effective potential of the radial motion of test particles. The radial dependence of the effective potential for the massive test particle around BH in Einstein-Æther theory for different values of \(c_{13}\) and \(c_{14}\) parameters is represented in Fig. 2. It can be seen from the graphs that under the influence of the \(c_{13}\) parameter the stable circular orbits move towards the central compact object and the effect of the \(c_{14}\) parameter has an opposite behaviour, i.e. the stable circular orbits move outwards the central compact object.

In order to derive the circular motion of the neutral particle around BH, one can use the following conditions \(\dot{r}=0\) and \(\ddot{r}=0\). These conditions allow one to obtain expressions for the energy \({{\mathcal {E}}}\) and the angular momentum \({{\mathcal {L}}}\) of the test particle in the following form

Fig. 2
figure 2

Dependence of \(V_{eff}\) on radial coordinate r of massive particle for the different values of Einstein-Æther BH parameters \(c_{13}\) and \(c_{14}\)

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$$\begin{aligned} {{{\mathcal {L}}}}^2=\frac{M r^2 (M (c_{14}-2 c_{13})+2 (c_{13}-1) r)}{2 M^2 (2 c_{13}-c_{14})-6 (c_{13}-1) M r+2 (c_{13}-1) r^2}, \end{aligned}$$
(16)

and

Fig. 3
figure 3

Radial dependence of \({{{\mathcal {L}}}}\) and \({{\mathcal {E}}}\) for the different values of Æther parameters

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$$\begin{aligned} {{{\mathcal {E}}}}^2=\frac{\left( -2 c_{13} (M-r)^2+c_{14} M^2+2 r (r-2 M)\right) ^2}{4 (c_{13}-1) r^2 \left( M^2 (2 c_{13}-c_{14})-3 (c_{13}-1) M r+(c_{13}-1) r^2\right) } . \end{aligned}$$
(17)

In order to get more information about conservative quantities, one can make plot for them, and it is shown in Fig. 3. The graphs indicate that under the influence of the \(c_{13}\) parameter the radial coordinates are shifted towards the centre, and conversely, the effect of the \(c_{14}\) parameter shifts the radial coordinates outwards direction.

Now we may consider the inner stable circular orbit \(r_{ISCO}\). In order to find ISCO radius, one needs to use the following conditions:

$$\begin{aligned} {\left\{ \begin{array}{ll} V'_\mathrm{eff}=0\\ V''_\mathrm{eff}=0. \end{array}\right. } \end{aligned}$$
(18)

Due to the complex form of the effective potential, we cannot get exact analytical expression of \(r_\mathrm{ISCO}\). However, one may represent the radius of ISCO using the graphs, and it is represented for various parameters \(c_{13}\) and \(c_{14}\) in Fig. 4. From this, one can get information about inner stable circular orbits dependence on parameter \(c_{13}\) and \(c_{14}\) in Einstein-Æther gravity. Particularly, ISCO radius increases with the increase in parameter \(c_{13}\). ISCO radius decreases with the increase in parameter \(c_{14}\)  in Fig. 4.

Fig. 4
figure 4

Relation of \(r_{_{\mathrm{ISCO}}}/M\) on parameters \(c_{13}\) and \(c_{14}\) in Einstein-Æther gravity

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3.2 Massless particle (photon) motion

In this subsection, we explore massless particle (photon) motion in spacetime of BH in Einstein-Æther gravity. Using Lagrangian (7), equation of photon motion around compact object can be obtained by setting \(\epsilon =0\) (13) and at the equatorial plane can be expressed in the following form:

$$\begin{aligned} \dot{r}^2= & {} {{{\mathcal {E}}}}^2-f(r)\frac{{{{\mathcal {L}}}}^2}{r^2}, \end{aligned}$$
(19)
$$\begin{aligned} \dot{\phi }= & {} \frac{{{{\mathcal {L}}}}}{r^2}, \end{aligned}$$
(20)
$$\begin{aligned} \dot{t}= & {} \frac{{{{\mathcal {E}}}}}{f(r)}. \end{aligned}$$
(21)

Using Eq. (19), one can easily get expression of effective potential \(V_{eff}\) for the radial motion of photon as

$$\begin{aligned} V_{eff}=f(r)\frac{ {{{\mathcal {L}}}}^2}{r^2} . \end{aligned}$$
(22)
Fig. 5
figure 5

Dependence of \(V_\mathrm{eff}\) on radial coordinate r of photon for the different values of Einstein-Æther BH parameters \(c_{13}\) and \(c_{14}\)

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The radial dependence of effective potential for radial motion of photons is shown in Fig. 5. From this figure, one can easily see that photon orbits shift towards the central BH under the effect of the \(c_{13}\) parameter and vice versa of the \(c_{14}\) parameter.

Radius of photon’s circular orbit \(r_{ph}\) around BH in Einstein-Æther gravity can be found as solution of the following equation:

$$\begin{aligned} V'_{eff}=0 , \end{aligned}$$

which leads to the following solution

$$\begin{aligned} r_{ph}= & {} \frac{-\sqrt{c_{13}^2 M^2-10 c_{13} M^2+4 c_{13} c_{14} M^2-4 c_{14} M^2+9 M^2}}{2 \left( c_{13}-1\right) }\nonumber \\&+ \frac{3 c_{13} M-3 M}{2 \left( c_{13}-1\right) } \end{aligned}$$
(23)

The dependence of the radius of photon orbits \(r_{ph}\) on Æther parameters \(c_{13}\) and \(c_{14}\) of Einstein-Æther gravity is shown in Fig. 6. One can see that the radius of the photon orbit is increased with the increase in the parameter \(c_{13}\), while the effect of the parameter \(c_{14}\) on radius of photon orbits is opposite.

Fig. 6
figure 6

Dependence of radius of photon \(r_{ph}\) on parameters \(c_{13}\) and \(c_{14}\) of Einstein-Æther gravity

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4 Weak gravitational lensing in Einstein-Æther gravity

Fig. 7
figure 7

The dependence of the deflection angle \({\hat{\alpha }}_{\text {uni}}\) on the impact parameter b for the different values of parameters \(c_{13}\) and \(c_{14}\) (upper panel) and plasma medium (lower panel)

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Fig. 8
figure 8

The dependence of the deflection angle \({\hat{\alpha }}_{\text {uni}}\) on plasma parameters for the fixed value of impact parameter \(b = 5M\)

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Fig. 9
figure 9

The dependence of the deflection angle \({\hat{\alpha }}_{\text {sis}}\) on the impact parameter for the different values of parameters \(c_{13}\) and \(c_{14}\) (upper panel) and plasma parameters (lower panel)

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In this section, we test optical properties of BH in Einstein-Æther gravity using gravitational weak lensing effect. For a weak-field approximation, one can use the following notation of metric tensor as [35, 40]

$$\begin{aligned} g_{\alpha \beta }=\eta _{\alpha \beta }+h_{\alpha \beta } , \end{aligned}$$
(24)

where \(\eta _{\alpha \beta }\) and \(h_{\alpha \beta }\) refer to the expressions for Minkowski spacetime and perturbation gravity field describing Einstein-Æther theory, respectively. One has to require the following properties for \(\eta _{\alpha \beta }\) and \(h_{\alpha \beta }\)

$$\begin{aligned}&\eta _{\alpha \beta }=diag(-1,1,1,1) , \nonumber \\&h_{\alpha \beta } \ll 1, \quad h_{\alpha \beta } \rightarrow 0 \quad under\quad x^{\alpha }\rightarrow \infty ,\nonumber \\&g^{\alpha \beta }=\eta ^{\alpha \beta }-h^{\alpha \beta }, \quad h^{\alpha \beta }=h_{\alpha \beta }. \end{aligned}$$
(25)

Using the fundamental equation, we can get the expression for the angle of deflection around a compact object in Einstein-Æther gravity as follows [35]:

$$\begin{aligned} {\hat{\alpha }}_{\text {b}}=\frac{1}{2}\int _{-\infty }^{\infty }\frac{b}{r}\left( \frac{\mathrm{d}h_{33}}{\mathrm{d}r}+\frac{1}{1-\omega ^2_e/ \omega }\frac{\mathrm{d}h_{00}}{\mathrm{d}r}-\frac{K_e}{\omega ^2-\omega ^2_e}\frac{\mathrm{d}N}{\mathrm{d}r} \right) \mathrm{d}z , \end{aligned}$$
(26)

where \(\omega \) and \(\omega _{e}\) are known as quantities representing the photon and plasma frequencies, respectively. One may rewrite metric element (5) in the form

$$\begin{aligned} \mathrm{d}s^2= & {} \mathrm{d}s^2_0+\frac{2 M}{r}\left( 1+\frac{(2 c_{13}-c_{14} )M}{4 (1-c_{13}) r}\right) \mathrm{d}t^2\nonumber \\&+\frac{2 M}{r}\left( 1+\frac{(2 c_{13}-c_{14} )M}{4 (1-c_{13}) r}\right) \mathrm{d}r^2 , \end{aligned}$$
(27)

where \(ds^2_0=-dt^2+dr^2+r^2(d\theta ^2+\sin ^2\theta d\phi ^2)\).

Fig. 10
figure 10

The dependence of the deflection angle \({\hat{\alpha }}_{\text {sis}}\) on plasma for the fixed value of impact parameter \(b = 5M\)

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Fig. 11
figure 11

The dependence of the deflection angle \({\hat{\alpha }}_{\text {b}}\) on the impact parameter (upper panel) and the plasma parameters (lower panel)

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Fig. 12
figure 12

The dependence of the total magnification \(\mu _{tot}\) on the plasma parameter for the different values of parameter \(c_{13}\) and \(c_{14}\) corresponding to the fixed value of impact parameter \(b = 5M\)

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Fig. 13
figure 13

Image magnification in the presence of uniform plasma. The fixed parameters used are \(M=1\), \(b=3M\), \(c_{13}\)=-0.2 and \(c_{14}\)=0.4

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Fig. 14
figure 14

The total magnification of the images as a function of non-uniform plasma. Here impact parameter is set as \(b = 5M\)

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Now one may easily find components \(h_{\alpha \beta }\) of metric element in Cartesian coordinates in the form

$$\begin{aligned} h_{00}= & {} \frac{2 M}{r}\left( 1+\frac{(2 c_{13}-c_{14} )M}{4 (1-c_{13}) r}\right) \end{aligned}$$
(28)
$$\begin{aligned} h_{ik}= & {} \left( \frac{2 M}{r}\left( 1+\frac{(2 c_{13}-c_{14}) M}{4 (1-c_{13}) r}\right) \right) n_i n_k \end{aligned}$$
(29)
$$\begin{aligned} h_{33}= & {} \frac{2 M}{r}\left( 1+\frac{(2 c_{13}-c_{14}) M}{4 (1-c_{13}) r}\right) \cos ^2\chi , \end{aligned}$$
(30)

where \(\cos ^2\chi =z^2/(b^2+z^2)\) and \(r^2=b^2+z^2\). The derivatives of \(h_{00}\) and \(h_{33}\) by radial coordinate are defined as:

$$\begin{aligned} \frac{\mathrm{d}h_{00}}{\mathrm{d}r}=-\frac{c_{14} M^2}{(c_{13}-1) r^3}+\frac{2c_{13} M^2}{(c_{13}-1) r^3}-\frac{2c_{13} M}{(c_{13}-1) r^2}+\frac{2 M}{(c_{13}-1) r^2} , \end{aligned}$$
(31)
$$\begin{aligned} \frac{\mathrm{d}h_{33}}{\mathrm{d}r}=-\frac{2c_{14} M^2 z^2}{(c_{13}-1) r^5}+\frac{4 c_{13} M^2 z^2}{(c_{13}-1) r^5}-\frac{6c_{13} M z^2}{(c_{13}-1) r^4}+\frac{6 M z^2}{(c{13}-1) r^4} . \end{aligned}$$
(32)

We can write the following expression for the deflection angle [49]

$$\begin{aligned} \hat{\alpha _b}=\hat{\alpha _1}+\hat{\alpha _2}+\hat{\alpha _3} , \end{aligned}$$
(33)

with

$$\begin{aligned} \hat{\alpha _1}= & {} \frac{1}{2}\int _{-\infty }^{\infty } \frac{b}{r}\frac{\mathrm{d}h_{33}}{\mathrm{d}r}\mathrm{d}z ,\nonumber \\ \hat{\alpha _2}= & {} \frac{1}{2}\int _{-\infty }^{\infty } \frac{b}{r}\frac{1}{1-\omega ^2_e/ \omega }\frac{\mathrm{d}h_{00}}{\mathrm{d}r}\mathrm{d}z\ ,\nonumber \\ \hat{\alpha _3}= & {} \frac{1}{2}\int _{-\infty }^{\infty } \frac{b}{r}\left( -\frac{K_e}{\omega ^2-\omega ^2_e}\frac{\mathrm{d}N}{\mathrm{d}r} \right) \mathrm{d}z . \end{aligned}$$
(34)

Now we plan to check and evaluate the deflection angle for different plasma density distributions.

4.1 Uniform plasma

In this subsection, gravitational deflection angle around BH in Einstein-Æther gravity in the presence of a uniform plasma can be written as sum [49]

$$\begin{aligned} {\hat{\alpha }}_\mathrm{uni}={\hat{\alpha }}_{uni1}+{\hat{\alpha }}_{uni2}+{\hat{\alpha }}_{uni3}. \end{aligned}$$
(35)

From Eqs. (34), (30) and (33), we may easily get the expression of deflection angle around BH in Einstein-Æther gravity in a uniform plasma medium as

$$\begin{aligned} {\hat{\alpha }}_{uni}= & {} \frac{2 M}{b}+\frac{\pi c_{14} M^2}{8 b^2 (c_{13}-1)}-\frac{\pi c_{13} M^2}{4 b^2 (c_{13}-1)}+\nonumber \\&+ \left( \frac{2 M}{b}+\frac{\pi c_{14} M^2}{4 b^2 (c_{13}-1)}-\frac{\pi c_{13} M^2}{2 b^2 (c_{13}-1)}\right) \frac{\omega ^2}{\omega ^2-\omega ^2_e}. \end{aligned}$$
(36)

Using the above equation, we plot the dependence of the angle of deflection on the impact parameter b for different values of parameters \(c_{13}\), \(c_{14}\), \(\omega ^2_{e}/\omega ^2\) in spacetime of BH in Einstein-Æther gravity, and it is represented in Fig. 7. Also we have explored the dependence of deflection angle on plasma parameters for selected values of Einstein-Æther parameters \(c_{13}\) and \(c_{14}\) and they are presented in Fig. 8

4.2 Non-uniform plasma

In this part of our current analysis, we consider that non-singular isothermal sphere (SIS) is the most favourable model to comprehend peculiar features of gravitational weak lensed photons around BH. Generally, SIS is spherical gas cloud with a singularity located at its centre where the density tends to infinity. The density distribution of a SIS is given by [35]:

$$\begin{aligned} \rho (r)=\frac{\sigma ^2_{\nu }}{2\pi r^2} , \end{aligned}$$
(37)

where \(\sigma ^2_{\nu }\) refers to a one-dimensional velocity dispersion. The plasma concentration admits the following analytic expression [35]

$$\begin{aligned} N(r)=\frac{\rho (r)}{k m_p} , \end{aligned}$$
(38)

where \(m_p\) is proton mass and k is a dimensionless constant coefficient generally associated with the dark matter universe. The plasma frequency is:

$$\begin{aligned} \omega ^2_e=K_e N(r)=\frac{K_e \sigma ^2_{\nu }}{2\pi k m_p r^2} . \end{aligned}$$
(39)

Now we explore non-uniform plasma (SIS) effect on deflection angle in spacetime of Einstein-Æther black hole. We may write the expression of deflection angle around BH in Einstein-Æther gravity as [49]

$$\begin{aligned} {\hat{\alpha }}_\mathrm{SIS}={\hat{\alpha }}_{SIS1}+{\hat{\alpha }}_{SIS2}+{\hat{\alpha }}_{SIS3} . \end{aligned}$$
(40)

Combining Eqs. (30), (34), (40), deflection angle can be written in the following form:

$$\begin{aligned} {\hat{\alpha }}_{SIS}= & {} \frac{3 \pi c_{14} M^2}{8 b^2 (c_{13}-1)}-\frac{3 \pi c_{13} M^2}{4 b^2 (c_{13}-1)}+\frac{4 M}{b}+ \nonumber \\&+\frac{3 c_{14} k M^4}{4 b^4 (c_{13}-1)}-\frac{3 c_{13} k M^3}{2 b^4 (c_{13}-1)}-\frac{4 k M^2}{\pi b}+\frac{16 k M^3}{3 \pi b^3}\ . \end{aligned}$$
(41)

These calculations bring up a supplementary plasma constant \(\omega ^2_c\), which has the following analytic expression [40]

$$\begin{aligned} \omega ^2_c=\frac{K_e \sigma ^2_{\nu }}{2\pi k m_p R^2_S} . \end{aligned}$$
(42)

From Eq. (41), we do graphs for the dependence of the angle of deflection on the impact parameter b for different values of parameters \(c_{13}\), \(c_{14}\), \(\omega ^2_{c}/\omega ^2\) for non-uniform plasma medium in Einstein-Æther spacetime, and it is shown in Fig. 9. Then, we demonstrate the dependence of the angle of deflection of a light ray around a BH in the presence of a non-uniform plasma in Figs. 9 and 10.

In addition, we compared the different effects of plasma on the Einstein-Æther BH deflection angle with gravity, as shown in Fig. 11.

5 Magnification of gravitationally lensed image

Now we explore the brightness of the image in the presence of a plasma through the deflection angle of light rays around BH in the Einstein-Æther gravity. Using the lens equation, we can write the combination of light angles around BH in Einstein-Æther gravity (\({\hat{\alpha }}\), \(\theta \) and \(\beta \)) [37, 40] as

$$\begin{aligned} \theta D_{\mathrm {s}}=\beta D_{\mathrm {s}}+{\hat{\alpha }}D_{\mathrm {ds}} , \end{aligned}$$
(43)

where \(D_{\mathrm {s}}\), \(D_{\mathrm {d}}\) and \(D_{\mathrm {ds}}\) are the distances from the source to the observer, from the lens to the observer and from the source to the lens, respectively, and \(\theta \) and \(\beta \) refer to the angular position of the image and the source. Now, from the above equation, we can rewrite equation for \(\beta \) as follows:

$$\begin{aligned} \beta =\theta -\frac{D_{\mathrm {ds}}}{D_{\mathrm {s}}}\frac{\xi (\theta )}{D_{\mathrm {d}}}\frac{1}{\theta }\ , \end{aligned}$$
(44)

where \(\xi (\theta )=|{\hat{\alpha }}_b|b\) and \(b=D_{\mathrm {d}}\theta \) have been used in [40]. If the shape of the image looks like a ring, it is defined as Einstein’s ring and radius of Einstein’s ring is \(R_s=D_{\mathrm {d}}\theta _E\). The angular \(\theta _E\) due to spacetime geometry between the images of the source in a vacuum [20] can be written as

$$\begin{aligned} \theta _E=\sqrt{2R_s\frac{D_{ds}}{D_dD_s}} . \end{aligned}$$
(45)

Now, we explore the expression of the magnification of brightness [20]

$$\begin{aligned} \mu _{\Sigma }=\frac{I_{\mathrm {tot}}}{I_*}=\underset{k}{\sum }\bigg |\bigg (\frac{\theta _k}{\beta }\bigg )\bigg (\frac{\mathrm{d}\theta _k}{\mathrm{d}\beta }\bigg )\bigg |, \quad k=1,2, \ldots , j , \end{aligned}$$
(46)

where \(I_*\) and \(I_{\mathrm {tot}}\) are the unlensed brightness of the source and the total brightness of all images, respectively. The expressions of the magnification of the source are written in the following form [20]:

$$\begin{aligned}&\mu ^{\mathrm {pl}}_\mathrm {+}=\frac{1}{4}\bigg (\frac{x}{\sqrt{x^2+4}}+\frac{\sqrt{x^2+4}}{x}+2\bigg ), \end{aligned}$$
(47)
$$\begin{aligned}&\mu ^{\mathrm {pl}}_\mathrm {-}=\frac{1}{4}\bigg (\frac{x}{\sqrt{x^2+4}}+\frac{\sqrt{x^2+4}}{x}-2\bigg )\ , \end{aligned}$$
(48)

where \(x={\beta }/{\theta _0}\) is a dimensionless quantity [40], and \(\mu _{+}^{\mathrm {pl}}\) and \(\mu _{-}^{\mathrm {pl}}\) are the images. Using Eqs. (47) and (48), one can get formula for the total magnification in the following form:

$$\begin{aligned} \mu ^{\mathrm {pl}}_{\mathrm {tot}}=\mu ^{\mathrm {pl}}_{+}+\mu ^{\mathrm {pl}}_{-}=\frac{x^2+2}{x\sqrt{x^2+4}}\ . \end{aligned}$$
(49)

Now we investigate the magnification in the presence of plasma in BH environment with different distributions: (i) uniform and (ii) non-uniform cases.

5.1 Uniform Plasma

In this subsection, we explore the effect of uniform plasma on magnification of image in Einstein-Æther theory and we may write expressions for total magnification \(\mu ^{pl}_{tot}\) and angle \(\theta ^\mathrm{pl}_\mathrm{uni}\) as

$$\begin{aligned} \mu ^\mathrm{pl}_\mathrm{tot}=\mu ^{pl}_{+}+\mu ^{pl}_{-}=\dfrac{x^2_\mathrm{uni}+2}{x_\mathrm{uni}\sqrt{x^2_\mathrm{uni}+4}}\ , \end{aligned}$$
(50)

and

$$\begin{aligned} \theta ^\mathrm{pl}_\mathrm{uni}= & {} \theta _0\sqrt{\frac{1}{2}\left( ( A+1)\frac{1}{1-\frac{\omega ^2_0}{\omega ^2}}+\frac{A}{2}+1\right) } , \end{aligned}$$
(51)

with dimensionless for simplification

$$\begin{aligned} A=\frac{\pi \left( c_{14}-2 c_{13}\right) M}{8 b \left( c_{13}-1\right) } , \end{aligned}$$
(52)

where \(x_\mathrm{uni}\), \((\mu ^\mathrm{pl}_+)_\mathrm{uni}\) and \((\mu ^{pl}_-)_{uni}\) are defined as

$$\begin{aligned} x_\mathrm{uni}= & {} \frac{\beta }{(\theta ^{pl}_E)_\mathrm{uni}}=\nonumber \\= & {} \frac{x_0}{\sqrt{\frac{1}{2}\left( ( A+1)\frac{1}{1-\frac{\omega ^2_0}{\omega ^2}}+\frac{A}{2}+1\right) }} \end{aligned}$$
(53)
$$\begin{aligned} (\mu ^\mathrm{pl}_+)_\mathrm{uni}= & {} \frac{1}{4}\left( \dfrac{x_\mathrm{uni}}{\sqrt{x^2_\mathrm{\mathrm uni}+4}}+\dfrac{\sqrt{x^2_\mathrm{uni}+4}}{x_\mathrm{uni}}+2\right) \end{aligned}$$
(54)
$$\begin{aligned} (\mu ^{pl}_-)_{uni}= & {} \frac{1}{4}\left( \dfrac{x_\mathrm{uni}}{\sqrt{x^2_\mathrm{uni}+4}}+\dfrac{\sqrt{x^2_\mathrm{uni}+4}}{x_\mathrm{uni}}-2\right) \end{aligned}$$
(55)

The total magnification of the image in the presence of a plasma \(\mu ^{\mathrm {pl}}_{\mathrm {tot}}\) in Einstein-Æther gravity is represented in Fig. 12, and the total magnification is decreased with \({\omega ^2_{e}}/{\omega ^2}\) for the different values of parameters \(c_{13}\) and \(c_{14}\) in Einstein-Æther spacetime. Plot for dependence of the total magnification on \(x_{0}\) in the presence of uniform plasma for fixed values of the parameters \(c_{13}\) and \(c_{14}\) is demonstrated in Fig. 13.

5.2 Non-uniform plasma

Here, we study the effect of non-uniform plasma (as SIS medium) on magnification, which can be written as:

$$\begin{aligned} (\mu ^\mathrm{pl}_\mathrm{tot})_\mathrm{SIS}=(\mu ^\mathrm{pl}_{+})_\mathrm{SIS}+(\mu ^\mathrm{pl}_{-})_\mathrm{SIS}=\dfrac{x^2_\mathrm{SIS}+2}{x_\mathrm{SIS}\sqrt{x^2_\mathrm{SIS}+4}}\ , \end{aligned}$$
(56)

with

$$\begin{aligned}&(\mu ^\mathrm{pl}_+)_\mathrm{SIS}=\frac{1}{4}\left( \dfrac{x_\mathrm{SIS}}{\sqrt{x^2_\mathrm{SIS}+4}}+\dfrac{\sqrt{x^2_\mathrm{SIS}+4}}{x_\mathrm{SIS}}+2\right) \ , \end{aligned}$$
(57)
$$\begin{aligned}&(\mu ^\mathrm{pl}_-)_\mathrm{SIS}=\frac{1}{4}\left( \dfrac{x_\mathrm{SIS}}{\sqrt{x^2_\mathrm{SIS}+4}}+\dfrac{\sqrt{x^2_\mathrm{SIS}+4}}{x_\mathrm{SIS}}-2\right) \ , \end{aligned}$$
(58)

and

$$\begin{aligned} x_\mathrm{SIS}= & {} \frac{\beta }{(\theta ^\mathrm{pl}_E)_\mathrm{SIS}}\nonumber \\= & {} \frac{x_0}{\sqrt{\frac{3\omega ^2_c M^2( c_{14} M-2c_{13}) }{16 b^3 \omega ^2 \left( c_{13}-1\right) }+\frac{4 \omega ^2_c M^2}{3 \pi \omega ^2 b^2}-\frac{M\omega ^2_c}{\pi \omega ^2 }+\frac{3A}{4}+1}} . \end{aligned}$$
(59)
Fig. 15
figure 15

Image magnification in the presence of SIS. The fixed parameters used are \(M=1\), \(b=3M\), \(c_{13}\)=-0.2 and \(c_{14}\)=0.4

Full size image
Fig. 16
figure 16

Image magnifications in the presence of uniform plasma and SIS. The fixed parameters used are \(M=1\), \(b=3M\), \(\frac{\omega ^2_0}{\omega ^2}=0.5\), \(\frac{\omega ^2_c}{\omega ^2}=0.5\), \(c_{13}\)=-0.2 and \(c_{14}\)=0.4

Full size image

Using Eq. (56), one can find the dependence of total magnification on plasma parameter and with the increase in plasma parameter magnification is decreased, which is represented in Fig. 14. Also, the plot of dependence of total magnification on \(x_{0}\) in the presence of plasma for the fixed values of the parameters \(c_{13}\) and \(c_{14}\) is presented in Fig. 15. Finally, we compare in Fig. 16 these two cases, which are for uniform and non-uniform plasma distributions.

6 Extracted energy by collisions

Now we study the particle acceleration process of two charged particles colliding near the horizon of the black hole in the Einstein-Æther gravity. We consider two particles that have rest masses \(m_1\) and \(m_2\) at a distance far away from BH, and we can write equations of motion as

$$\begin{aligned} p^{\alpha }= & {} mu^{\alpha } , \end{aligned}$$
(60)
$$\begin{aligned} u^t= & {} \frac{{{\mathcal {E}}}}{f(r)} , \end{aligned}$$
(61)
$$\begin{aligned} u^{\phi }= & {} \frac{{{\mathcal {L}}}}{r^2} , \end{aligned}$$
(62)
$$\begin{aligned} u^r= & {} \sqrt{{{{\mathcal {E}}}}^2-f(r)\left( 1+\frac{{{{\mathcal {L}}}}^2}{r^2}\right) }\ . \end{aligned}$$
(63)

The extracted energy \(E_{C.M.}\) by collision between two particles is defined by [74]

$$\begin{aligned} \frac{E_{C.M.}}{2 m_1 m_2}=\dfrac{m^2_1+m^2_2}{2 m_1 m_2}-g_{\alpha \beta }u^{\alpha }_1 u^{\beta }_2 . \end{aligned}$$
(64)

By using Eqs. (61), (62) and (63), we can rewrite Eq. (64) for the energy extracted as

$$\begin{aligned} \frac{E_{C.M.}}{2 m_1 m_2}= & {} 1+\dfrac{(m_1-m_2)^2}{2 m_1 m_2}+\frac{{{{\mathcal {E}}}}_1{{{\mathcal {E}}}}_2}{f(r)}-\frac{{{{\mathcal {L}}}}_1{{{\mathcal {L}}}}_2}{r^2}\nonumber \\&-\frac{1}{f^2(r)}\sqrt{{{{\mathcal {E}}}}^2_1-f(r)\left( 1+\frac{{{{\mathcal {L}}}}^2_1}{r^2}\right) }\sqrt{{{{\mathcal {E}}}}^2_2-f(r)\left( 1+\frac{{{{\mathcal {L}}}}^2_2}{r^2}\right) }\ , \end{aligned}$$
(65)

where f(r) is the known lapse function of the gravity.

Fig. 17
figure 17

The relationship of \(E_{C.M}\) to radial coordinate r/M is described for different \({{{\mathcal {L}}}}_1\) and \({{{\mathcal {L}}}}_2\) values (upper panel) and for different Æther parameters(lower panel)

Full size image

The radial dependence of \(E_{C.M.}\) energy of BH can be seen in Fig. 17. From Fig. 17 (upper panel), the minimum energy shows that two particles have opposite angular momentum. This graph is drawn for \(c_{13}\) = \(c_{14}\) = 0.2 case. Figure 17 (lower panel) illustrates the radial dependence of centre-of-mass energy of colliding for two non-spinning particles for the different values of the Æther parameters. In this plot considered head-on collision with the specific angular momentum \({{{\mathcal {L}}}}_1=-{{{\mathcal {L}}}}_2=2M\). We can see that the value of energy increases with increasing parameter \(c_{13}\) and decreasing parameter \(c_{14}\).

7 Conclusions

In this work, we have studied the optical and energetic properties of BH in the Einstein-Æther gravity for mass and massless particles, and from this work, we can draw the following conclusions:

  • We have studied the structure of the horizon of BH in the Einstein-Æther gravity. It can be seen from the obtained results that with the increase of the parameter \(c_{13}\) radius of the event horizon is increased, and for the parameter \(c_{14}\) it is vice versa as demonstrated in Fig. 1.

  • For particle motion around compact object, we have discussed effective potential. We also studied the dependence of effective potential of massive and massless particles on radial coordinate for different values of the Einstein-Æther parameters \(c_{13}\) and \(c_{14}\).

  • ISCO is investigated for the massive particles around BH in the Einstein-Æther gravity including parameters \(c_{13}\) and \(c_{14}\). It is demonstrated that with the increase in parameter \(c_{13}\) ISCO radius is slightly increased, but influence of parameter \(c_{14}\) is opposite as demonstrated in Fig. 4.

  • Radius of photon sphere is explored around BH in Einstein-Æther gravity. Orbits of photons are obtained using the general method based on study of the effective potential and demonstrated with more details in Fig. 6.

  • We also studied optical properties a black hole in Einstein-Æther gravity through light rays, that is a gravitational weak lensing. First, we focus on deflection angle around BH in Einstein-Æther gravity in the presence of plasma (uniform and non-uniform cases). In this part of our work, we have discussed and found that the influence of the uniform plasma on the deflection angle is greater than that in the non-uniform plasma case for fixed parameters of BH in Einstein-Æther gravity.

  • Magnification of image is also explored using deflection angle of light rays, and it is shown with more details in Figs. 13 and 15.

  • Finally, we investigated centre mass energy in vicinity BH with non-vanishing Einstein-Æther parameters \(c_{13}\) and \(c_{14}\). We explored it for different values of the angular momentum of two particles and plotted the dependence of the energy of the centre of mass on the radial coordinate.