1. Introduction
Einstein's general relativity (GR) remained a well-defined theory in which black hole (BH) solutions to the field equations were predicted as simple mathematical models. This viewpoint, however, has changed as the existence of BHs in the Universe has been confirmed by several modern observations in recent years, e.g. gravitational waves (GWs) [1, 2] and shadows of BHs by the Event Horizon Telescope (EHT) [3, 4], both of which come to play the most important role in testing the nature of spacetime geometry in the close surroundings of BHs. GR fails to explain the inevitable occurrence of singularities inside BHs, where it forecasts its own demise and loses its applicability. Hence, curvature singularity has long been regarded as a fundamental difficulty of GR as it cannot be explained by the theory itself. As a result of their efforts to find a new solution without singularity, numerous researchers proposed BH solutions in GR, usually known as regular BHs [5–8]. Since quantum theory gravity has not been well established, regular BHs can be used in BH physics and to explain geodesically complete spacetimes. In addition, GR is not sufficient as a potential explanation for dark matter, not directly detected, as well as the accelerated expansion of the Universe. In this regard, GR remains an incomplete theory. To that, is widely believed that promising alternative theories are required to approach a fundamental understanding in relation to these objects.
A well-established theory was proposed as an alternative to GR, which refers to scalar tensor vector gravity (STVG, i.e. modified gravity (MOG)) resolving issues in relation to dark matter as well as dark energy (see details in [8]). In STVG theory, a description of the expansion of the Universe and the weak interaction of massive particles, referred to as scalar and vector fields, was effectively described by solving its field equations to give solutions resulting in the MOG theory, which can help to explain gravitational systems with their predictable nature. Afterwards, STVG theory opened up an active research field devoted to the study of its remarkable nature and has attracted lots of attention in recent years. It should be emphasized that STVG solutions were widely applied to dark matter problems [9] as well as recent astrophysical observations [10–12]. There has since been extensive analysis that has considered various situations in the context of the MOG field theory; we give some representative references here [13–25].
From the astrophysical point of view, it is particularly important to gain a deeper understanding about the nature of the existing fields in the BH surroundings and about their impacts on the geodesics of massive and massless particles. Hence, such fields may alter the geodesics of massive and massless particles, i.e. observable properties such as the shadow, lensing, etc. In a realistic astrophysical scenario, it is increasingly important to consider the impact arising from dark energy in the surrounding environment of BHs as well as on a large scale. Relying on the observations, it is widely believed that the expansion of the Universe is accelerating as that of the vacuum energy, usually referred to as the cosmological constant Λ in GR. However, the quintessence scalar field has been well defined and regarded as a promising alternative to the vacuum energy to examine the behaviour of dark energy [26–28]. First, Kiselev came up with a new solution including the quintessence scalar field, together with the state equation with p = wq ρ [29]. Here, it should be noted that the equation of state parameter wq exhibits the quintessence field in the following range (−1;−1/3) as well as (−1;-2/3) [30]. Following Kiselev [29], the quintessence and dark matter fields have since been considered in a variety of contexts [31–39].
It is to be emphasized that the BH image was a fundamental question in past decades. However, much progress has been made in observational studies of BHs with the help of recent modern astrophysical observations. To that end, the recent triumphal discovery in relation to the detection of the first shadow image of a supermassive BH referred to as the M87* galaxy [3, 4] has confirmed not only the observational progress but also their existence in our universe. After that, the BH in astrophysics has now been taking center stage and its image has been tested not only by GR but also by modified theories of gravity. It is well defined that gravitational lensing causes the light to be strongly bent from its original path, giving rise to the BH shadow as a result of reflection. It happens because the light cannot escape the BH's pull due to the strong deflection, thereby resulting in a dark disc referred to as the BH shadow. Hence, the BH shadow and the actual deflection angle are potentially crucial to help observers in testing the nature of the geometry around the BH horizon. Note that the BH shadow as a disc was considered by Synge [40] and Luminet [41] focusing on studying the light deflection in the close vicinity of the Schwarzschild BH. Over the past years, BH shadows have been widely investigated and modeled theoretically: see, for example, [42–59]. In recent years, BH shadows have been considered in the context of MOG theory [60–62], addressing the shadows of non-rotating and rotating BHs. Note that a faraway observer cannot distinguish a BH geometry from other compact objects such as scalar boson and Proca stars being alternatives to BHs. For clear distinction, the shadow can be used as a viable test to explore their geometry. To this end, BH shadows have been investigated very successfully, determining their geometry and gaining insights into scalar boson and Proca stars through analytical fitting models using numerical solutions [63, 64]. Shadows and photon rings of regular black holes and geonic horizonless compact objects has been studied in [55]. Also, observational properties of relativistic fluid spheres with thin accretion disks have been considered via extensive analysis [65]. Recent EHT observations are very important in studying BH shadows and helping to have restrictions on some theoretical models [66].
Another fundamental phenomenon is the gravitational lensing that occurs because of the light deviation under the strong gravitational field of a massive central object, i.e. the light ray is bent from its original path due to the deflection. It is well established that the deflection can widely describe the gravitational lensing in GR. Hence, we further focus on studying the deflection angle around a BH, which not only provides the primary source of information pertaining to gravity in the strong field regime and but also shows some departures of the geometry of compact objects. It is worth noting that GR was successfully tested first by the gravitational lensing effect to provide some evidence regarding BH geometry [67]. Additionally, it was suggested that gravitational lensing can be used to constrain the variation of the state equation provided that it varies with time [68]. With this in view, gravitational lensing has since been widely used as a key element to provide convincing comparisons and explanation for arbitrary theories of gravity. In this respect, a large amount of work has been implemented extensively to study gravitational lensing in recent years [69–79]. Furthermore, the gravitational lensing effects were considered powerful tools for testing the Weyl and bumblebee theories of gravity as well as the Lorentz invariance violation in bumblebee gravity [80, 81]. It should be also noted that there exist investigations that added valuable contributions to understand deeply BH properties and gravitational lensing in the context of modified gravity and quintessence scalar fields: see, for example, [82–84]. Later on, it was also extended to modified theories of gravity [85–87]. It must also be noted that Futamase and Yoshida proposed a method to potentially measure the time variability of vacuum energy using strong gravitational lensing events [88]. The authors used the example of the Einstein cross lens HST 14 176 + 5226 so as to demonstrate that there would be a chance to determine the time dependence of the vacuum energy and the density parameter with an accuracy of approximately 0.1 by accurately measuring the velocity dispersion with an accuracy of ±5 km/s in the case of keeping the lens model fixed. This investigation suggests that strong gravitational lensing events could offer insights into quintessence and density parameters [88]. In this current paper, we focus on the weak gravitational lensing effects and do not explore strong gravitational lensing at this time, but we acknowledge its potential for future investigations nevertheless. Gravitational lensing events have also been approached using different methods, e.g. one was proposed by Gibbons and Werner [89, 90]. We further consider this method and give details on studying the weak deflection angle around the MOG BH with a quintessence field.
The STVG MOG theory successfully explains Solar System observations [8], galaxy cluster dynamics [91] and predictions for the cosmic microwave background [92]. In addition, recent studies reveal that BH solutions in the MOG gravity model can successfully recreate GR's exact solutions [60, 61]. This physical solution for a MOG BH inspired us to expand our research to examine the properties of BHs in the presence of a scalar field, i.e. a quintessence field, as well as the properties of compact objects in such an extension of MOG theory. Furthermore, the findings could lead to direct astrophysical studies of BH-type objects in modified gravity theories.
In the present work, we consider a MOG BH surrounded by quintessence field, as described by the line element in the next section. This geometry was proposed as a well-defined alternative to GR. We further aim to investigate the combined impact of the MOG and quintessence fields on horizons evolution, shadow, and the weak deflection angle for a static and spherically symmetric MOG BH in the presence of quintessence field.
This paper is organized as follows. In section 2 , we briefly describe the metric of a static BH in the STVG theory with a quintessence field. Furthermore, we obtain numerical results and plots for the horizons. In section 3 , we analyse the BH shadow and we study the impact of the model parameters on the photon sphere and shadow radius. In section 4 , we study the weak deflection angle around the BH. Conclusions and final remarks are given in section 5 . We use a system of units in which GN = c = 1 throughout the manuscript.
2. Regular MOG BH surrounded by a quintessence field
2.1. A brief review of the spacetime
In this section, we describe the metric of a static regular BH in the STVG theory (the MOG gravity) with a quintessence scalar field. The total action in the theory of S-T-V MOG is in the form of
$\begin{eqnarray}S={S}_{{GR}}+{S}_{M}+{S}_{\phi }+{S}_{S},\end{eqnarray}$
where SGR is the Einstein–Hilbert action, SM is the action of all possible matter sources, Sφ is the action of the vector field, and SS is the action of three scalar fields: $\begin{eqnarray}{S}_{{GR}}=\displaystyle \frac{1}{16\pi }\int {{\rm{d}}}^{4}x\sqrt{-g}\displaystyle \frac{1}{G}R,\end{eqnarray}$
$\begin{eqnarray}{S}_{\phi }=-\int {{\rm{d}}}^{4}x\sqrt{-g}\left(\displaystyle \frac{1}{4}{B}^{\mu \nu }{B}_{\mu \nu }-\displaystyle \frac{1}{2}{\mu }^{2}{\phi }^{\mu }{\phi }_{\mu }-{J}^{\mu }{\phi }_{\mu }\right),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}{S}_{S}=\int {{\rm{d}}}^{4}x\sqrt{-g}\displaystyle \frac{1}{{G}^{3}}\left(\displaystyle \frac{1}{2}{g}^{\mu \nu }{{\rm{\nabla }}}_{\mu }G{{\rm{\nabla }}}_{\nu }G-V\left(G\right)\right)\\ \,+\int {{\rm{d}}}^{4}x\sqrt{-g}\displaystyle \frac{1}{{\mu }^{2}G}\left(\displaystyle \frac{1}{2}{g}^{\mu \nu }{{\rm{\nabla }}}_{\mu }\mu {{\rm{\nabla }}}_{\nu }\mu -V\left(\mu \right)\right).\end{array}\end{eqnarray}$
Here, gμν is the spacetime metric, g is the determinant of the metric, R is the Ricci scalar, φμ is a proca-type massive vector field such that Bμν = ∂μφν − ∂νφμ, $G\left(x\right)$ and $\mu \left(x\right)$ are scalar fields and V(G) and V(μ) are the corresponding potentials.Given that G is considered a constant independent of the spacetime coordinates, using a vacuum solution will simplify the action to8 ) as a regular MOG BH with a quintessence field.
$\begin{eqnarray}S=\displaystyle \frac{1}{16\pi G}\int {{\rm{d}}}^{4}x\sqrt{-g}\left(R-\displaystyle \frac{1}{4}{B}^{\mu \nu }{B}_{\mu \nu }\right),\end{eqnarray}$
where α is a dimensionless parameter, $G={G}_{N}\left(1+\alpha \right)$ and GN is the Newtonian constant. From varying this action with respect to gμν, the following field equations for the matter-free MOG BH metric spacetimes are obtained [8]: $\begin{eqnarray}{R}_{\mu \nu }-\displaystyle \frac{1}{2}{g}_{\mu \nu }R=8\pi {G}_{N}\left(1+\alpha \right){T}_{\mu \nu }.\end{eqnarray}$
A metric of the MOG BH surrounded by quintessence can be obtained by decomposing the energy momentum tensor in the Einstein field equation into two distinct components [29]: $\begin{eqnarray}{T}_{\mu \nu }={T}_{\mu \nu }^{q}+{T}_{\mu \nu }^{\phi },\end{eqnarray}$
where ${T}_{\mu \nu }^{q}$ is for the quintessence part. Finally, we write a static and spherically symmetric BH solution in the presence of a quintessence field in the STVG theory as [29, 60, 93]: $\begin{eqnarray}\begin{array}{l}{\rm{d}}{s}^{2}=-f\left(r\right){\rm{d}}{t}^{2}+{f}^{-1}(r){\rm{d}}{r}^{2}\\ \quad +\,{r}^{2}\left({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}f\left(r\right)=1-\displaystyle \frac{2\left(1+\alpha \right){{Mr}}^{2}}{{\left({r}^{2}+\alpha \left(1+\alpha \right){M}^{2}\right)}^{3/2}}\\ \quad +\,\displaystyle \frac{\alpha \left(1+\alpha \right){M}^{2}{r}^{2}}{{\left({r}^{2}+\alpha \left(1+\alpha \right){M}^{2}\right)}^{2}}-\displaystyle \frac{c\left(1+\alpha \right)}{{r}^{3{w}_{q}+1}},\end{array}\end{eqnarray}$
in which, M is the mass of the gravitating object, c is a normalization factor referred to as the quintessence field parameter that depicts the quintessence field's intensity and wq is the state parameter of the quintessence. The allowed values for wq are in the range −1 < wq < − 1/3, as mentioned previously [29, 30]. We will refer to the BH metric described by equation (Next, to examine the behaviour of the MOG BH with a quintessence solution, i.e. we explore the curvature scalars. The Ricci scalar for the given BH geometry is defined by
$\begin{eqnarray}\begin{array}{rcl}R & = & \displaystyle \frac{6{M}^{3}\alpha {\left(1+\alpha \right)}^{2}(-{r}^{4}+2{M}^{3}{\alpha }^{2}\left(1+\alpha \right)(2M\left(1+\alpha \right)-\sqrt{{r}^{2}+\alpha \left(1+\alpha \right){M}^{2}})}{{\left({r}^{2}+\alpha \left(1+\alpha \right){M}^{2}\right)}^{9/2}}\\ & & +\displaystyle \frac{6{M}^{4}{r}^{2}{\alpha }^{2}{\left(1+\alpha \right)}^{2}\left(3M\left(1+\alpha \right)+2\sqrt{{r}^{2}+\alpha \left(1+\alpha \right){M}^{2}}\right)}{{\left({r}^{2}+\alpha \left(1+\alpha \right){M}^{2}\right)}^{9/2}}+\displaystyle \frac{3{{cw}}_{q}\left(3{w}_{q}-1\right)\left(1+\alpha \right)}{{r}^{3\left({w}_{q}+1\right)}}.\end{array}\end{eqnarray}$
From the above equation, the Ricci scalar exhibits a physical singularity at r = 0 in the limit of α = 0 and c = 0. Figures 1 and 2 provide a comprehensive analysis of the properties of MOG BHQ (black hole with quintessence) spacetime in terms of scalar invariants. Clearly, both scalar invariants follow the same behaviours and are positive-definite. Figures 1 and 2 show that the MOG and quintessence parameters influence the variation of the Ricci scalar and the Kretshmann effectively with increasing c and α, thus resulting in increasing both scalars slowly. Furthermore, we observe that for very large values of r, all of the scalars approach to zero.
Figure 1. Variation of Ricci and Kretschmann scalars versus r for several values of c. Here, M = 1, α = 0.2 and wq = − 2/3. |
Figure 2. Variation of Ricci and Kretschmann scalars versus r for several values of α. Here, M = 1, c = 0.02 and wq = − 2/3. |
It is well defined that f(r) = 0 can solve to give the BH horizon, which is given by11 )) numerically. The numerical results of these three horizons are tabulated in table 1, in which we consider the role of the quintessence state parameter wq and the MOG parameter α on BH horizons: see also figure 3. Table 1 shows that the cosmological horizon is particularly sensitive to the effect of wq, as wq drops, the cosmological horizon becomes much larger. It is to be emphasized that the cosmological horizon turns out to be non-vanishing whether the source exists or not. This happens because the quintessence scalar field can be regarded as a promising alternative to the vacuum energy for examining dark energy behaviour. The cosmological horizon does, therefore, exist at larger distances far away from the BH. We also illustrate the impact of MOG parameter α and quintessence parameter c on BH horizons in figure 4. As can be observed from figure 4, both the Cauchy and event horizons increase, while the cosmological horizon decreases as a consequence of the increase in the value of MOG parameter α. A rise in the quintessence parameter c also leads to a decrease in the Cauchy radius and in the cosmological horizon, while to an increase in the event horizon. It is worth noting that both parameters α and c have a significant impact on the cosmological horizon.
$\begin{eqnarray}\begin{array}{l}\left(1-c\left(1+\alpha \right)r\right){\left({r}^{2}+\alpha \left(1+\alpha \right){M}^{2}\right)}^{2}\\ \quad +\,\alpha \left(1+\alpha \right){M}^{2}{r}^{2}\\ \quad -\,2\left(1+\alpha \right){{Mr}}^{2}\sqrt{{r}^{2}+\alpha \left(1+\alpha \right){M}^{2}}\,=\,0.\end{array}\end{eqnarray}$
The above equation has theee roots that can represent the Cauchy radius r−, event horizon r+ and cosmological horizon r∞. However, we further explore the horizon equation (equation (
Figure 3. Metric function ( |
Figure 4. Plot of the three horizons versus c with different values of α. Here, wq = − 2/3. |
Table 1. Numerical results for the Cauchy radius, r−, event horizon r+ and cosmological horizon r∞ of the MOG BH surrounded by a quintessence field. |
| wq = − 2/3 | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| α = 0 | α = 0.1 | α = 0.2 | α = 0.3 | α = 0.4 | |||||||||
| c | 0 | 0.01 | 0.02 | 0.03 | 0.01 | 0.02 | 0.03 | 0.01 | 0.02 | 0.03 | 0.01 | 0.02 | 0.03 |
| r− | x | 0.1626 | 0.1624 | 0.1622 | 0.314 | 0.313 | 0.312 | 0.480 | 0.477 | 0.474 | 0.667 | 0.659 | 0.652 |
| r+ | 2 | 2.12 | 2.18 | 2.25 | 2.18 | 2.26 | 2.35 | 2.23 | 2.34 | 2.46 | 2.27 | 2.40 | 2.57 |
| r∞ | x | 88.65 | 43.13 | 27.92 | 80.86 | 39.11 | 25.13 | 74.23 | 35.67 | 22.73 | 68.51 | 32.67 | 20.61 |
| wq = − 4/9 | |||||||||||||
| r− | x | 0.1621 | 0.1614 | 0.1608 | 0.3132 | 0.3111 | 0.3089 | 0.4787 | 0.4737 | 0.4688 | 0.6651 | 0.6549 | 0.6451 |
| r+ | 2 | 2.0998 | 2.1339 | 2.1693 | 2.1588 | 2.2020 | 2.2471 | 2.2000 | 2.2548 | 2.3121 | 2.2195 | 2.2893 | 2.3621 |
| r∞ | x | 751 308 | 93 907 | 27 819 | 578 697 | 72 330 | 21 426 | 455158 | 56 888 | 16 850 | 364 423 | 45 545 | 13 489 |
3. Shadows of the MOG BH with a quintessence field
In this section, we study the shadow formation. To do this, we first consider the Hamilton–Jacobi equation for the system of a test particle around the MOG BH with a quintessence field, which is written as follows:18 ) can be rewritten as24 ) and (25 ), which is nothing but the solution of25 ) yields the following result:27 ) is extremely difficult to solve analytically. As a result, we present numerical analysis and plots demonstrating the effects of the quintessence parameter c and the MOG parameter α on the photon (circular) radius of mass particles and the radius of the BH shadow. Table 2 displays the results of the photon sphere rps and the BH shadow Rsh. For a constant value of c, our results show that rps and Rsh increase as α increases. When α is the same, increasing the parameter c increases rps and Rsh. In figure 5, we show how the parameters α and c influence the rps of massive particles surrounding a MOG BHQ. It is clear that rps increases as α and c increase. Similarly, as shown in figure 6, the radius of the BH shadow increases as α and c increase.
$\begin{eqnarray}\displaystyle \frac{\partial { \mathcal S }}{\partial \sigma }=-\displaystyle \frac{1}{2}{g}^{\mu \nu }\displaystyle \frac{\partial { \mathcal S }}{\partial {x}^{\mu }}\displaystyle \frac{\partial { \mathcal S }}{\partial {x}^{\nu }},\end{eqnarray}$
where ${ \mathcal S }$ is the Jacobi action. Consider the following Jacobi action separable solution: $\begin{eqnarray}{ \mathcal S }\,=\,-{Et}+{\ell }\phi +{{ \mathcal S }}_{r}\left(r\right)+{{ \mathcal S }}_{\theta }\left(\theta \right),\end{eqnarray}$
where the two constants E (the energy) and ℓ (angular momentum) are given by $\begin{eqnarray}\begin{array}{l}E=\displaystyle \frac{{\rm{d}}L}{{\rm{d}}\dot{t}}\,=\,-\left(1-\displaystyle \frac{2\left(1+\alpha \right){{Mr}}^{2}}{{\left({r}^{2}+\alpha \left(1+\alpha \right){M}^{2}\right)}^{3/2}}\right.\\ \quad \left.+\displaystyle \frac{\alpha \left(1+\alpha \right){M}^{2}{r}^{2}}{{\left({r}^{2}+\alpha \left(1+\alpha \right){M}^{2}\right)}^{2}}-\displaystyle \frac{c\left(1+\alpha \right)}{{r}^{3{w}_{q}+1}}\right)\,\dot{t}\end{array}\end{eqnarray}$
$\begin{eqnarray}{\ell }=\displaystyle \frac{{\rm{d}}L}{{\rm{d}}\dot{\phi }}\,=\,{r}^{2}{\sin }^{2}\theta \dot{\phi }.\end{eqnarray}$
We can obtain the equations of motion, namely, $\begin{eqnarray}\displaystyle \frac{{\rm{d}}t}{{\rm{d}}\sigma }=\displaystyle \frac{E}{f},\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}\phi }{{\rm{d}}\sigma }=-\displaystyle \frac{{\ell }}{{r}^{2}{\sin }^{2}\theta },\end{eqnarray}$
$\begin{eqnarray}{r}^{2}\displaystyle \frac{{\rm{d}}r}{{\rm{d}}\sigma }=\pm \sqrt{{ \mathcal R }\left(r\right)},\end{eqnarray}$
$\begin{eqnarray}{r}^{2}\displaystyle \frac{{\rm{d}}\theta }{{\rm{d}}\sigma }=\pm \sqrt{{\rm{\Theta }}\left(\theta \right)},\end{eqnarray}$
where $\begin{eqnarray}{ \mathcal R }\left(r\right)={r}^{4}{E}^{2}-\left({ \mathcal K }+{{\ell }}^{2}\right){r}^{2}f,\end{eqnarray}$
$\begin{eqnarray}{\rm{\Theta }}\left(\theta \right)={ \mathcal K }-{{\ell }}^{2}\cot \theta ,\end{eqnarray}$
and ${ \mathcal K }$ is the Carter separation constant. Let us define the two quantities $\eta =\tfrac{{ \mathcal K }}{{E}^{2}}$ and ζ = $\tfrac{{\ell }}{E}$ (which stand for the impact parameters). Shadow casts can generally be obtained by using unstable null circular orbits. Therefore, equation ( $\begin{eqnarray}{\left(\displaystyle \frac{{\rm{d}}r}{{\rm{d}}\sigma }\right)}^{2}+{V}_{{\rm{eff}}}=0,\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{l}{V}_{{\rm{eff}}}\left(r\right)={E}^{2}\left[\displaystyle \frac{\eta +{\zeta }^{2}}{{r}^{2}}\left(1-\displaystyle \frac{2\left(1+\alpha \right){{Mr}}^{2}}{{\left({r}^{2}+\alpha \left(1+\alpha \right){M}^{2}\right)}^{3/2}}\right.\right.\\ \quad \left.\left.+\displaystyle \frac{\alpha \left(1+\alpha \right){M}^{2}{r}^{2}}{{\left({r}^{2}+\alpha \left(1+\alpha \right){M}^{2}\right)}^{2}}-\displaystyle \frac{c\left(1+\,\alpha \right)}{{r}^{3{w}_{q}+1}}\right)-1\right].\end{array}\end{eqnarray}$
The following are the conditions for stable photons in circular orbits corresponding to the maximum effective potential: $\begin{eqnarray}{V}_{{\rm{eff}}}\left(r\right)\left|{}_{r={r}_{{ps}}}\right.={V}_{{\rm{eff}}}^{{\prime} }\left(r\right)\left|{}_{r={r}_{{ps}}}\right.=0\end{eqnarray}$
or $\begin{eqnarray}{ \mathcal R }\left(r\right)\left|{}_{r={r}_{{ps}}}\right.={{ \mathcal R }}_{{\rm{eff}}}^{{\prime} }\left(r\right)\left|{}_{r={r}_{{ps}}}\right.=0,\end{eqnarray}$
where rps is the radius of the unstable photon sphere orbit. We can obtain rps using equations ( $\begin{eqnarray}{r}_{{ps}}f^{\prime} ({r}_{{ps}})-2f({r}_{{ps}})=0,\end{eqnarray}$
or $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{3}{2}{{cr}}^{-3{w}_{q}-1}\left(1+{w}_{q}\right)\left(1+\alpha \right){\left({r}^{2}+\alpha \left(1+\alpha \right){M}^{2}\right)}^{3}\\ \quad -\,\left({r}^{6}+3{M}^{4}{r}^{2}{\alpha }^{2}{\left(1+\alpha \right)}^{2}+{M}^{6}{\alpha }^{3}{\left(1+\alpha \right)}^{3}\right)\\ \quad +\,{{Mr}}^{4}\left(1+\alpha \right)\left(5M\alpha -3\sqrt{{r}^{2}+\alpha \left(1+\alpha \right){M}^{2}}\right)=0.\end{array}\end{eqnarray}$
Equation ( $\begin{eqnarray}\eta +{\zeta }^{2}=\displaystyle \frac{4{r}_{{ps}}^{2}}{2f({r}_{{ps}})+{r}_{{ps}}f^{\prime} ({r}_{{ps}})}.\end{eqnarray}$
Equation (
Figure 5. Dependence of the photon sphere radius on the α and c parameters. |
Figure 6. Variation of the shadow radius with the parameters α and c. |
Table 2. Numerical results for the rps and Rsh of the MOG BHQ. Here, wq = − 2/3. |
| c = 0.01 | c = 0.02 | c = 0.03 | c = 0.04 | |||||
|---|---|---|---|---|---|---|---|---|
| α | rps/M | Rsh/M | rps/M | Rsh/M | rps/M | Rsh/M | rps/M | Rsh/M |
| 0 | 3.0464 | 5.4450 | 3.0958 | 5.7285 | 3.1487 | 6.0556 | 3.2055 | 6.4387 |
| 0.1 | 3.2030 | 5.8546 | 3.2688 | 6.2287 | 3.3401 | 6.6748 | 3.4180 | 7.2189 |
| 0.2 | 3.3467 | 6.2612 | 3.4328 | 6.7470 | 3.5276 | 7.3485 | 3.6330 | 8.1203 |
| 0.3 | 3.4770 | 6.6658 | 3.5882 | 7.2885 | 3.7128 | 8.0947 | 3.8544 | 9.1964 |
| 0.4 | 3.5928 | 7.0690 | 3.7354 | 7.8594 | 3.8979 | 8.9377 | 4.0872 | 10.5358 |
| 0.5 | 3.6924 | 7.4709 | 3.8742 | 8.4670 | 4.0854 | 9.9131 | 4.3383 | 12.2997 |
| 0.6 | 3.7728 | 7.8710 | 4.0044 | 9.1202 | 4.2786 | 11.0746 | 4.6174 | 14.8262 |
| 0.7 | 3.8286 | 8.2679 | 4.1254 | 9.8304 | 4.4820 | 12.5098 | 4.9393 | 19.0008 |
| 0.8 | 3.8494 | 8.6577 | 4.2358 | 10.612 | 4.7020 | 14.3736 | 5.3269 | 28.4836 |
Here, it should be noted that from the behaviour of the spacetime metric there exists the cosmological horizon, which can be assumed to be located far away from the BH at a larger distance. Hence, we suppose that observer, r0, can be located at the cosmological horizon r∞. In the following, we will investigate how the size of the MOG BHQ shadow radius, Rs, varies with α and c. We further use the celestial coordinates X and Y [94] to locate the shadow for a more accurate representation. For that, these coordinates are defined by29 ) and (30 ) follow
$\begin{eqnarray}X=\mathop{\mathrm{lim}}\limits_{{r}_{0}\to {r}_{\infty }}\left(-{r}_{0}\sin {\theta }_{0}{\left.\displaystyle \frac{{\rm{d}}\phi }{{\rm{d}}r}\right|}_{{r}_{0},{\theta }_{0}}\right),\end{eqnarray}$
$\begin{eqnarray}Y=\mathop{\mathrm{lim}}\limits_{{r}_{0}\to {r}_{\infty }}\left({r}_{0}{\left.\displaystyle \frac{{\rm{d}}\theta }{{\rm{d}}r}\right|}_{{r}_{0},{\theta }_{0}}\right),\end{eqnarray}$
where (r0, θ0) are the position coordinates of the observer. Assuming the observer is on the equatorial hyperplane, equations ( $\begin{eqnarray}{X}^{2}+{Y}^{2}=\eta +{\zeta }^{2}={R}_{{\rm{sh}}}^{2}.\end{eqnarray}$
The size of the shadow can be analysed to determine the parameters, including the parameters of the MOG BHQ. The variation of the size of the BH shadow is depicted in figure 7.
Figure 7. The shadow of the MOG BH surrounded by a quintessence field for different values of c (left panel) and α (right panel). Note that we have set wq = − 2/3. |
Now we turn to the application of theoretical results to the recent EHT observations, which can help us to have restrictions on the BH parameters. For our purpose, we suppose that both Sgr A⋆ and M87⋆ can be regarded as static and spherically symmetric BHs in our model. As a matter of fact, they may not support our approach considered here. However, they satisfy well our assumption. We aim to constrain the lower limits of both parameters α and c, relying on the EHT observational data in connection with the shadows, together the angular diameter of the BH shadow θ, the distance D and the mass of the supermassive BHs Sgr A⋆ and M87⋆, e.g. ${\theta }_{{\rm{M}}{87}^{\star }}\,=\,42\pm 3\mu {as}$, D = 16.8 ± 0.8 Mpc between Earth and M87⋆ and ${M}_{{\rm{M}}{87}^{\star }}\,=\,6.5\pm 0.7\times {10}^{9}{M}_{\odot }$ for M87⋆ and ${\theta }_{{{\rm{SgrA}}}^{\star }}\,=\,48.7\pm 7\mu $, D = 8277 ± 9 ± 33pc and ${M}_{{{\rm{SgrA}}}^{\star }}\,=\,4.297\pm 0.013\times {10}^{6}{M}_{\odot }$ for Sgr A⋆ (VLBI: very long baseline interferometery), respectively [3, 4]. Taking these observational data into consideration, we determine the shadow diameter per unit mass of the BH as stated by the following expression:
$\begin{eqnarray}{d}_{{\rm{sh}}}=\displaystyle \frac{D\theta }{M}.\end{eqnarray}$
Following dsh = 2Rsh, we further obtain the BH shadow's diameter. As a result, we define it as follows: ${d}_{{\rm{sh}}}^{{\rm{M}}{87}^{\star }}\,=(11\pm 1.5)M$ for M87⋆ and ${d}_{{\rm{sh}}}^{{{\rm{Sgr}}}^{\star }}=(9.5\pm 1.4)M$ for Sgr A⋆. Based on the observational EHT data, we explore constraints on both parameters α and c numerically for Sgr A⋆ and M87⋆. By applying the observational data, we then show the lower values of α and c in figure 8. As can be observed from figure 8, one can expect lower values of both parameters from the observational data of Sgr A⋆ compared to that of M87⋆. However, it is worth noting that the observational data from the shadow of the two objects considered here are only facts to obtain the best-fit constraints on the parameters α and c.
Figure 8. Constraint values of α and c for M87⋆ and Sgr A⋆. Here, we have set M = 1 and wq = − 2/3. |
4. Weak deflection angle around the MOG BH surrounded by a quintessence field
Here, we turn to the study of the weak deflection angle around the MOG BH surrounded by a quintessential field. To this end, we approach this issue from the perspective of the well-accepted Gauss–Bonnet theorem (GBT) developed by Gibbons and Werner as a new thought experiment, helping one in evaluating the weak deflection angle of spherically symmetric BH spacetimes (see details in [89, 90]). Afterwards, this thought experiment was also extended to various situations, e.g. axisymmetric spacetimes [95] and non-asymptotically flat spacetimes [96, 97], as well as BH spacetimes with a plasma medium [98]. There has since been extensive analysis performed using the GBT method in recent years (see, for example, [52, 99–107]).
In the following, we also consider the weak deflection angle of BH within the context of the GBT method. For this, we shall further restrict the motion to null geodesics for the photon, and we shall then consider the optical metric for the MOG BH surrounded by a quintessence field with the corresponding line element, which can be defined accordingly by
$\begin{eqnarray}{\rm{d}}{\sigma }^{2}={g}_{{kl}}^{\mathrm{opt}}{\rm{d}}{x}^{k}{\rm{d}}{x}^{l}=\displaystyle \frac{1}{f(r)}\left(\displaystyle \frac{{\rm{d}}{r}^{2}}{f(r)}+{r}^{2}{\rm{d}}{\phi }^{2}\right),\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{l}f\left(r\right)=1-\displaystyle \frac{2\left(1+\alpha \right){{Mr}}^{2}}{{\left({r}^{2}+\alpha \left(1+\alpha \right){M}^{2}\right)}^{3/2}}\\ \quad +\displaystyle \frac{\alpha \left(1+\alpha \right){M}^{2}{r}^{2}}{{\left({r}^{2}+\alpha \left(1+\alpha \right){M}^{2}\right)}^{2}}-\displaystyle \frac{c\left(1+\alpha \right)}{{r}^{3{w}_{q}+1}}.\end{array}\end{eqnarray*}$
The above-mentioned optical metric helps us in determining the Gaussian curvature K with the terms of the radial coordinate, i.e. it is given with linear order of M as follows: $\begin{eqnarray}\begin{array}{l}K{\rm{d}}S=\displaystyle \frac{1+\alpha }{4{r}^{2}{\left(1-(\alpha +1){cr}\right)}^{5/2}}\\ \quad \times \,\left[(1+\alpha ){c}^{2}{r}^{3}\left((1+\alpha ){cr}-1\right)\right.\\ \quad \left.-\,\left(5(1+\alpha ){cr}\left(3(1+\alpha \right){cr}-4)+8\right)M\right],\end{array}\end{eqnarray}$
where we have set ωq = − 2/3. In relation to the contribution stemming from the MOG field, which is given to the Gaussian curvature, one can write the following form for the geodesic curvature [89, 98]: $\begin{eqnarray}{\left.\displaystyle \frac{{\rm{d}}\sigma }{{\rm{d}}\phi }\right|}_{{C}_{R}}\,=\,{\left(\displaystyle \frac{{r}^{2}}{f(R)}\right)}^{1/2}.\end{eqnarray}$
The above expression leads to the following form in the limiting case: $\begin{eqnarray}{\left.\mathop{\mathrm{lim}}\limits_{R\to \infty }{\kappa }_{g}\displaystyle \frac{{\rm{d}}\sigma }{{\rm{d}}\phi }\right|}_{{C}_{R}}\approx 1.\end{eqnarray}$
Keeping the above expressions in mind together with the limiting case R → ∞ and parametrization $r=b/\sin \phi $, we can further determine the deflection angle on the basis of the GBT method, which is rewritten as [89]: $\begin{eqnarray}\begin{array}{rcl}{\alpha }_{d} & = & {\left.{\displaystyle \int }_{0}^{\pi +{\alpha }_{d}}\left[{\kappa }_{g}\displaystyle \frac{{\rm{d}}\sigma }{{\rm{d}}\phi }\right]\right|}_{{C}_{R}}{\rm{d}}\phi -\pi \\ & = & \,-\mathop{\mathrm{lim}}\limits_{R\to \infty }{\displaystyle \int }_{0}^{\pi }{\displaystyle \int }_{\tfrac{b}{\sin \phi }}^{\infty }K\,{\rm{d}}S,\end{array}\end{eqnarray}$
where we have defined b as the impact parameter. As a result, the deflection angle around the MOG BH with the quintessence can be determined by the following approximate form: $\begin{eqnarray}{\alpha }_{d}\approx \displaystyle \frac{(1+\alpha )M\left[4+5\pi (1+\alpha )c\,b\mathrm{log}(2b)\right]}{b}.\end{eqnarray}$
We will examine the impact of MOG and quintessence fields on the deflection angle in the weak form as stated by the GBT method considered here. To be more informative and to understand more deeply, we show the dependence of the deflection angle αd on the impact parameter b for various combinations of MOG and quintessence field parameters. As can be observed from figure 9, the deflection angle decreases as the impact parameter increases. It is, however, obvious that the parameters α and c have a physical impact that can shift the deflection angle upward toward to its larger values. This is consistent with the physical meaning of both parameters as attractive gravitational charges, thereby manifesting the model for MOG and quintessence fields and their profile on the weak deflection angle.
Figure 9. The deflection angle around the MOG BH surrounded by quintessence for various combinations of MOG parameter α for fixed c = 0.01 (left) and of quintessence parameter c for fixed α = 0.1 (right panel). Note that we have set M = 1 and ωq = − 2/3. |
5. Conclusion
In this paper, we investigated the effect of MOG and the quintessence scalar fields on the optical properties of static and spherically symmetric regular BHs. In pursuit of this goal, we focused on studying the horizon evolution, BH shadows and the constraints on the BH parameters through the EHT observational data, as well as the weak gravitational lensing.
The first property of the spacetime of MOG BH surrounded by quintessence is the locations of three horizons. We obtained numerical solutions for the locations of the horizons represented in table 1 and figure 4. It was found that that the quintessence state parameter wq significantly affects the cosmological horizon, reducing its radius significantly. We also showed that the MOG parameter α increases the Cauchy and event horizons, while it decreases the cosmological horizon. It was also shown that the decrease in the Cauchy radius and event horizon as well as a rapid fall in the cosmological horizon occur due to the increase in the quintessence parameter c.
We further focused on studying the BH shadow and explored the photon sphere and the shadow radius numerically (table 2). We demonstrated that MOG α and quintessence field c parameters have a significant impact on the BH shadow and photon sphere (figures 5–6). Based on our findings, we demonstrated that the combined effects of MOG and the quintessence field parameters can raise the values of the BH shadow and photon sphere radii in comparison to the Schwarzschild case. Additionally, taking into account results regarding to the BH shadow, we obtained constraints on the parameters α and c inferred from the EHT observational data for Sgr A⋆ and M87⋆. As a result, we showed that both parameters can take lower values as a consequence of the observational data of Sgr A⋆ in contrast to that of M87⋆. This is in good agreement with the fact that M87⋆ is more massive and larger than Sgr A⋆.
Furthermore, the authors of [57–59] delved into the shadow of the BH for diferent BH backgrounds. It was discovered that all BH parameters have a considerable impact on the shadow and other physical observables. In our research, we investigated the MOG BH with a quintessence field, and just as their findings revealed changes in shadow, we discovered that the MOG and quintessence parameters have a substantial impact on the BH shadow and photon sphere.
We also studied the impact of MOG and quintessence fields together on the deflection angle and we showed that the deflection angle decreases as a consequence of the increase in the value of the impact parameter. Both the parameters α and c have a similar effect, thereby resulting in increasing the deflection angle. However, the combined effect of both α and c on the deflection angle becomes more powerful, as can be observed in figure 9. This behaviour is in good agreement with the physical meaning of both α and c as attractive gravitational charges, thus strengthening the gravity and manifesting the model more effective on the weak deflection angle.
We would like to emphasize that our approach brought out the combined effects of MOG and quintessence scalar fields on the optical properties of static and spherically symmetric regular BH with detailed analysis. We have also provided an interpretation of the MOG and quintessence parameters as attractive gravitational charges that can physically manifest to strengthen the BH gravity, thus making the results comprehensive and accurate, while exhibiting striking differences from previous analyses. The theoretical studies and results can help to open up new avenues for observational testing of these kinds of objects, as well as distinguishing between different geometries of compact objects. Research into applying a similar investigation to rotating BHs might be conducted in the future.