1. Introduction
Within the realm of braneworld gravity models, various constructs have been documented in the literature. The Randall-Sundrum II (RS-II) [1] and Dvali–Gabadadze–Porrati (DGP) [2] models are among these, and will be the topics of presentation in this paper. The latter boasts several intriguing attributes concerning astronomical observations. Presently, our universe's expansion is a well-established phenomenon, attributed to the positive value of the cosmological constant within Einstein's field equations. In the DGP brane model, a positive cosmological constant naturally emerges, interlinked with a cross-scale within the theory that governs the transition from five to four dimensions. This facet elucidates the observed self-accelerating nature of the Universe within the DGP brane model. This explanation underscores the sustained interest in this area, as evident in recent works [3–8]. Notably, investigations have also yielded charged black hole solutions within the DGP model [9–11].
In the context of the Randall-Sundrum II (RS-II) model [1], the effective Einstein equations on the brane were formulated by Shiromizu et al [12] using the Gauss–Codazzi approach. This equation has been extensively investigated, leading to the proposition of corresponding braneworld black hole solutions. For instance, the static black hole solution within this effective equation was presented in [13], where the metric function exhibits the form of the Reissner–Nordstrom solution while introducing a ‘charge' term in the metric. This charge is interpreted as the ‘tidal charge,' attributed to the geometrical effect of the bulk. In scenarios where the brane is not vacuum, containing localized Maxwell fields, this tidal charge lacks coupling with the $U\left(1\right)$ gauge fields. Consequently, no electromagnetic interaction occurs between a charged probe and the neutral braneworld black hole featuring a tidal charge, as reported in [13]. The RS-II model's non-vacuum brane, housing localized Maxwell fields, was examined in [14], yielding a static charged black hole akin to the Reissner–Nordstrom solution of the Einstein–Maxwell theory. The limit of the vanishing electric charge in the solution by Chamblin et al [14] reduces to that in [13], rendering these black holes asymptotically flat. The work by Aliev et al [15] achieved a solution for rotating and charged black holes within the RS-II model, employing the Kerr–Schild ansatz to solve the Hamilton constraint derived from the effective Einstein equation on the brane. Investigating the (A)dS black holes on the brane, [16] took into account the effective Einstein equation with the presence of the four-dimensional cosmological constant. Furthermore, the study of topological black holes on the brane was undertaken in [17], unveiling a class of solutions describing static charged Anti-de Sitter (AdS) black holes residing on the brane.
In the domain of general relativity, the integration of nonlinear electrodynamics (NED) has garnered notable interest in recent years. In ongoing research, gravitational configurations within an electrovacuum setting are scrutinized, encompassing nonlinear constitutive relations and matter equations that are invariant under transformations. This theory, referred to as ‘ModMax,' extends duality invariance and is elucidated through the Hamiltonian formalism [18]. In this study, we examine configurations of gravitational fields within an electrovacuum setting. These configurations involve a nonlinear constitutive relation and matter equations that remain unchanged under Hodge duality rotations and conformal transformations. This extended theory, which preserves duality invariance through conformal extensions, was recently introduced in [19]. An alternative derivation of the theory's Lagrangian is available in [18]. This formalism offers insights, such as highlighting the unique nature of this nonlinear extension in standard electrodynamics. Among the noteworthy features, the question of birefringence arises when considering extensions to Maxwell's electrodynamics [20].
In this paper, we explore the braneworld scenarios of RS-II and DGP, incorporating localized modified Maxwell (ModMax) electrodynamics on the 3-brane. Essentially, the matter energy-momentum tensor considered on the brane is described by the ModMax Lagrangian. We employ solution-finding methodologies outlined in [15, 21] for the RS-II case and [22] for the DGP scenario. Notably, we obtain significant outcomes; the metric and vector solutions take on a straightforward generalization of those found with the consideration of standard Maxwell electrodynamics. This finding closely aligns with the results in [23], wherein the static charged black hole within the Einstein–ModMax theory adopts the Reissner–Nordstrom metric form and an associated gauge field solution featuring a screening factor on the electric charge.
The organization of this paper is as follows. The next section provides short reviews on the equations of motion in ModMax, RS-II, and DGP theories. Section 3 contains the novel solutions of the braneworld black holes with localized ModMax electrodynamics on the 3-brane. Discussions on the particle motions in these new spacetime solutions are given in section 4 . Finally we present a conclusion. Throughout the paper, we use the convention of natural units c = G4 = ℏ = 1.
2. Review of some effective equations
2.1. Einstein–ModMax Lagrangian
ModMax electrodynamics is described by the Lagrangian [19]
$\begin{eqnarray}{{ \mathcal L }}_{\mathrm{MM}}=-X\cosh \left(\nu \right)+\sqrt{{X}^{2}+{Y}^{2}}\sinh \left(\nu \right),\end{eqnarray}$
where $X=\tfrac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }$ and $Y=\tfrac{1}{4}{F}_{\mu \nu }{\tilde{F}}^{\mu \nu }$. As in the ordinary Maxwell theory, the field-strength tensor Fμν is given by Fμν = ∂μAν − ∂νAμ, whereas its dual is ${\tilde{F}}_{\mu \nu }=\tfrac{1}{2}{\varepsilon }_{\mu \nu \alpha \beta }{F}^{\alpha \beta }$. Obviously, the Lagrangian above depends on the nonlinear parameter ν. As indicated in [19], one should consider ν ≥ 0 to have the causality fulfilled. For the null nonlinear parameter, the ModMax theory reduces to the original Maxwell electrodynamics. The field equation in ModMax theory is given by $\begin{eqnarray}{{\rm{\nabla }}}_{\mu }{P}^{\mu \nu }=0,\end{eqnarray}$
where Pμν is the Plebanski dual variable, $\begin{eqnarray}\begin{array}{rcl}{P}_{\mu \nu } & = & \left(\cosh \left(\nu \right)-\displaystyle \frac{X}{\sqrt{{X}^{2}+{Y}^{2}}}\sinh \left(\nu \right)\right){F}_{\mu \nu }\\ & & -\displaystyle \frac{Y\sinh \left(\nu \right)}{\sqrt{{X}^{2}+{Y}^{2}}}{\tilde{F}}_{\mu \nu }.\end{array}\end{eqnarray}$
Now let us consider the generalization of Einstein–Maxwell theory to the Einstein–ModMax scenario, where the action can be written as2.5 ) is the well-known Einstein–Hilbert action. The Einstein–ModMax equations can be obtained after varying equation (2.4 ) with respect to the metric tensor gμν, namely
$\begin{eqnarray}S={S}_{\mathrm{EH}}+{S}_{\mathrm{MM}}\end{eqnarray}$
where $\begin{eqnarray}{S}_{\mathrm{EH}}=\displaystyle \frac{1}{16\pi }\int {{\rm{d}}}^{4}x\sqrt{-g}R,\end{eqnarray}$
and $\begin{eqnarray}{S}_{\mathrm{MM}}=\displaystyle \frac{1}{4\pi }\int {{\rm{d}}}^{4}x\sqrt{-g}{{ \mathcal L }}_{\mathrm{MM}}.\end{eqnarray}$
In the action above, R denotes the Ricci scalar and equation ( $\begin{eqnarray}{R}_{\mu \nu }-\displaystyle \frac{1}{2}{g}_{\mu \nu }R=8\pi {T}_{\mu \nu },\end{eqnarray}$
with the energy-momentum tensor $\begin{eqnarray}{T}_{\mu }^{\nu }=\displaystyle \frac{1}{4\pi }\left({\delta }_{\mu }^{\nu }{{ \mathcal L }}_{\mathrm{MM}}-{F}_{\mu \beta }{P}^{\beta \nu }\right).\end{eqnarray}$
In [23], the authors derived a family of dyonic static charged black hole solutions that solve equations (2.2 ) and (2.7 ). Essentially, the authors extended the dyonic Reissner–Nordstrom spacetime from Einstein–Maxwell theory to the framework of Einstein–ModMax. It turns out that the resulting solution closely resembles the conventional dyonic Reissner–Nordstrom fields within Einstein–Maxwell theory. The notable distinction is the presence of an additional term e−ν, interpreted as a screening factor for the black hole charge. Effectively this factor shields the actual charge of the black hole.
If one only considers the electric charge and mass parameters, the metric solution takes the form2.9 ), in solving the Einstein–ModMax equations of motion is2.9 ), the horizons of a static charged ModMax black hole are given by2.9 ), have been worked out in [24].
$\begin{eqnarray}\begin{array}{l}{\rm{d}}{s}^{2}=-\left(1-\displaystyle \frac{2M}{r}+\displaystyle \frac{{Q}^{2}{{\rm{e}}}^{-\nu }}{{r}^{2}}\right){\rm{d}}{t}^{2}\\ \quad +{\left(1-\displaystyle \frac{2M}{r}+\displaystyle \frac{{Q}^{2}{{\rm{e}}}^{-\nu }}{{r}^{2}}\right)}^{-1}{\rm{d}}{r}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}_{2}^{2},\end{array}\end{eqnarray}$
where ${\rm{d}}{{\rm{\Omega }}}_{2}^{2}$ denotes a unit 2-sphere. Clearly, when the nonlinear parameter ν is set to zero, the metric above simplifies to the familiar Reissner–Nordstrom solution characterized by mass M and electric charge Q. The accompanying one-form for the vector solution corresponding to the metric, equation ( $\begin{eqnarray}{\boldsymbol{A}}={A}_{\mu }{\rm{d}}{x}^{\mu }=\displaystyle \frac{Q{{\rm{e}}}^{-\nu }}{r}{\rm{d}}t.\end{eqnarray}$
This vector also reduces to the one for Reissner–Nordstrom in the ν = 0 consideration. From the metric function in equation ( $\begin{eqnarray}{r}_{\pm }=M\pm \sqrt{{M}^{2}-{Q}^{2}{{\rm{e}}}^{-\nu }}.\end{eqnarray}$
The extremal state is denoted by the overlapping of inner and outer horizons, namely M = Qe−ν/2. Stability and pair production of scalars in the charged Einstein–ModMax black hole spacetime described by the line element, equation (2.2. RS-II brane
The derivation of the effective Einstein equation on the brane involves employing the Gauss–Codazzi method to project the bulk equation onto the brane, while enforcing ${{\mathbb{Z}}}_{2}$ symmetry to attain the desired equation. In the work by Shiromizu et al [12], specific Gaussian normal coordinates were chosen to further this process. However, a more general coordinate setting has been utilized to derive the effective Einstein equation on the brane from the five-dimensional gravitational equations in the bulk [25]. The form of the effective Einstein equations on the brane, incorporating a cosmological constant, can be derived as demonstrated in [12, 15].
$\begin{eqnarray}{R}_{\mu \nu }-\displaystyle \frac{1}{2}{g}_{\mu \nu }R=8\pi {T}_{\mu \nu }+{\kappa }_{5}^{4}\left({S}_{\mu \nu }-\displaystyle \frac{1}{2}{g}_{\mu \nu }{S}_{\alpha }^{\alpha }\right)-{E}_{\mu \nu }.\end{eqnarray}$
Here, Gμν represents the four-dimensional Einstein tensor, Λ4 stands for the brane's four-dimensional cosmological constant, and gμν corresponds to the associated metric tensor. In the subsequent section, we will proceed to articulate the expression for the five-dimensional cosmological constant in relation to the AdS5 curvature radius, denoted as ℓ, and formulated as follows $\begin{eqnarray}{{\rm{\Lambda }}}_{5}=-\displaystyle \frac{6}{{{\ell }}^{2}},\end{eqnarray}$
where the interconnection between the five- and four-dimensional Newton constants is established as G5 = ℓG4. Here, we adopt the convention G4 = 1, leading to direct implications such as G5 = ℓ and ${\kappa }_{5}^{4}=64{\pi }^{2}{{\ell }}^{2}$. Concretely, the AdS radius ℓ is expressed as ${\ell }=6{\left(\lambda {\kappa }_{5}^{2}\right)}^{-1}$, with λ representing the tension of the brane.In the scope of this study, we explore the scenario where the brane is endowed with localized ModMax fields. These fields contribute to the electric charge of a black hole and facilitate electromagnetic interactions between the black hole and charged entities. Consequently, the energy-momentum tensor associated with these phenomena on the brane is that given in equation (2.8 ) and the ‘square' of Tμν in equation (2.12 ) is given by
$\begin{eqnarray}{S}_{\mu \nu }=-\displaystyle \frac{1}{4}\left({T}_{\mu \alpha }{T}_{\nu }^{\alpha }-\displaystyle \frac{1}{2}{g}_{\mu \nu }{T}_{\alpha \beta }{T}^{\alpha \beta }\right).\end{eqnarray}$
In the next section, solutions to the effective Einstein equations on the 3-brane in the RS-II scenario are given.2.3. DGP brane
A complete review on the effective Einstein equations on the DGP brane can be found in [2, 10, 11]. Since in this work we aim to investigate the static and charged black hole in the DGP brane with localized ModMax fields, the Hamiltonian condition can be written as
$\begin{eqnarray}\begin{array}{l}{R}_{\mu \nu }{R}^{\mu \nu }+{\lambda }^{2}R-\displaystyle \frac{{R}^{2}}{3}+{\kappa }^{4}{T}_{\mu \nu }{T}^{\mu \nu }\\ \,-2{\kappa }^{2}{R}_{\mu \nu }{T}^{\mu \nu }=0,\end{array}\end{eqnarray}$
where Tμν is the energy-momentum tensor on the brane. The equation of motion within the brane scenario can be derived through the utilization of Israel's junction condition [10, 11], by incorporating it into the Einstein equations within the bulk, specifically for z ≠ 0. This equation is expressed as $\begin{eqnarray*}{R}_{\mu \nu }-\displaystyle \frac{1}{2}{g}_{\mu \nu }R+{E}_{\mu \nu }=-\displaystyle \frac{{\kappa }^{4}}{{\lambda }^{2}}\left({T}_{\mu }^{\alpha }{T}_{\alpha \nu }-\displaystyle \frac{1}{2}{g}_{\mu \nu }{T}_{\alpha \beta }{T}^{\alpha \beta }\right)\end{eqnarray*}$
$\begin{eqnarray*}-\,\displaystyle \frac{1}{{\lambda }^{2}}\left({R}_{\mu }^{\alpha }{R}_{\alpha \nu }-\displaystyle \frac{2}{3}{{RR}}_{\mu \nu }+\displaystyle \frac{1}{4}{g}_{\mu \nu }{R}^{2}-\displaystyle \frac{1}{2}{g}_{\mu \nu }{R}_{\alpha \beta }{R}^{\alpha \beta }\right)\end{eqnarray*}$
$\begin{eqnarray}+\,\displaystyle \frac{{\kappa }^{2}}{{\lambda }^{2}}\left({R}_{\mu }^{\alpha }{T}_{\alpha \nu }+{T}_{\mu }^{\alpha }{R}_{\alpha \nu }-\displaystyle \frac{2}{3}{{RT}}_{\mu \nu }-{g}_{\mu \nu }{R}_{\alpha \beta }{T}^{\alpha \beta }\right).\end{eqnarray}$
Within the presented equation, Eμν corresponds to the traceless component referred to as the ‘electric part' of the five-dimensional Weyl tensor. Notably, we have designated $\lambda =2{\kappa }^{2}/{\tilde{\kappa }}^{2}$ and assumed κ2 = 8π.3. Braneworld black holes with ModMax on the brane
Now let us solve the the effective Einstein equations reviewed in the previous section by following the approach presented in [15, 21] for the RS-II case, and [9–11, 22] for the DGP brane. To get the static and charged black hole solution on the brane with ModMax electrodynamics, let us consider a general Kerr–Schild ansatz with the coordinate $(u,r,x=\cos \theta ,\phi )$ 2.12 ), which yields the Hamilton constraint, gives us a differential equation for the H(r) function in the metric ansatz above3.5 ), yields the trace of the energy-momentum tensor on the brane to be vanished.
$\begin{eqnarray}{\rm{d}}{s}^{2}={\rm{d}}{s}_{0}^{2}+H\left(r\right){\rm{d}}{u}^{2}\end{eqnarray}$
where $\begin{eqnarray}{\rm{d}}{s}_{0}^{2}=-{\rm{d}}{u}^{2}+{r}^{2}{{\rm{\Delta }}}_{x}{\rm{d}}{\phi }^{2}+\displaystyle \frac{{r}^{2}}{{{\rm{\Delta }}}_{x}}{\rm{d}}{x}^{2}-2{\rm{d}}u{\rm{d}}r,\end{eqnarray}$
where Δx = 1 − x2. Note that this Kerr–Schild ansatz will be employed in both RS-II and DGP cases below. The trace of equation ( $\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{2}H\left(r\right)}{{\rm{d}}{r}^{2}}+\displaystyle \frac{4}{r}\displaystyle \frac{{\rm{d}}H\left(r\right)}{{\rm{d}}r}+\displaystyle \frac{{l}^{2}{Q}^{4}{{\rm{e}}}^{-2\nu }}{{r}^{8}}+\displaystyle \frac{2}{{r}^{2}}H\left(r\right)=0.\end{eqnarray}$
The solution for H(r), which reduces to the known four-dimensional static charged black hole in Einstein–ModMax theory reads $\begin{eqnarray}H(r)=\displaystyle \frac{2M}{r}-\displaystyle \frac{\beta +{Q}^{2}{{\rm{e}}}^{-\nu }}{{r}^{2}}-\displaystyle \frac{{l}^{2}{Q}^{4}}{20{r}^{6}},\end{eqnarray}$
and the corresponding gauge vector as a one-form is $\begin{eqnarray}{\boldsymbol{A}}={A}_{\mu }{\rm{d}}{x}^{\mu }=\displaystyle \frac{Q{{\rm{e}}}^{-\nu }}{r}{\rm{d}}u.\end{eqnarray}$
The ν = 0 case of this solution is just the charged and static braneworld black hole solution presented in [15] and [21]. Note that the antisymmetric nature of the field-strength tensor with the vector field, equation (Note that the tidal charge β appears in the metric function, as expected from the RS-II braneworld black hole [15]. This charge is interpreted as the effect from the extra dimension that influences the motion of test bodies on the brane in contrast to scenarios where such a charge is absent. However, as the tidal charge is unrelated to the electromagnetic interaction, there is no e−ν factor coupling to it to diminish its effective magnitude. Interestingly, the last term in equation (3.4 ), containing a coupling between the AdS bulk radius and the black hole charge, does not contain correction from the nonlinear parameter ν as well. Thus, it can be deduced that the nonlinear parameter ν serves as a local property of the brane, with no coupling to bulk or extra-dimensional parameters.
Conversely, in the DGP case, the spacetime solution can be derived in two scenarios, where the relevant quantities are intuitively chosen to satisfy the Hamiltonian condition, equation (2.15 ). This methodology has been explored in [10, 11]. The first one is known as the flat case, obeying3.1 ), where the Ricci scalar gives an equation to the metric function3.5 ).
$\begin{eqnarray*}R=0,\,{R}_{\alpha \beta }{R}^{\alpha \beta }=\displaystyle \frac{4{Q}^{4}{{\rm{e}}}^{-2\nu }}{{r}^{8}}\,\,,\end{eqnarray*}$
$\begin{eqnarray}{R}_{\mu \nu }{T}^{\mu \nu }=\displaystyle \frac{{Q}^{4}{{\rm{e}}}^{-2\nu }}{2\pi {r}^{8}},\,{T}_{\mu \nu }{T}^{\mu \nu }=\displaystyle \frac{{Q}^{4}{{\rm{e}}}^{-2\nu }}{16{\pi }^{2}{r}^{8}}.\end{eqnarray}$
Note the null of the Ricci scalar in this consideration. The second one is the non-flat case, satisfying $\begin{eqnarray*}R=12{\lambda }^{2},\,\,{R}_{\mu \nu }{T}^{\mu \nu }=\displaystyle \frac{{Q}^{4}{{\rm{e}}}^{-2\nu }}{2\pi {r}^{8}}\,,\end{eqnarray*}$
$\begin{eqnarray}{R}_{\alpha \beta }{R}^{\alpha \beta }=36{\lambda }^{4}+\displaystyle \frac{4{Q}^{4}{{\rm{e}}}^{-2\nu }}{{r}^{8}},\,{T}_{\mu \nu }{T}^{\mu \nu }=\displaystyle \frac{{Q}^{4}{{\rm{e}}}^{-2\nu }}{16{\pi }^{2}{r}^{8}},\end{eqnarray}$
which corresponds to the non-vanishing constant Ricci scalar. The solution to the second case can be constructed by using the ansatz, equation ( $\begin{eqnarray*}\displaystyle \frac{{{\rm{d}}}^{2}H\left(r\right)}{{\rm{d}}{r}^{2}}+\displaystyle \frac{4}{r}\displaystyle \frac{{\rm{d}}H\left(r\right)}{{\rm{d}}r}+\displaystyle \frac{2}{{r}^{2}}H\left(r\right)=12\lambda .\end{eqnarray*}$
The solution can found as $\begin{eqnarray}H(r)=\displaystyle \frac{2M}{r}-\displaystyle \frac{{Q}^{2}{{\rm{e}}}^{-\nu }}{{r}^{2}}+{r}^{2}{\lambda }^{2}.\end{eqnarray}$
The non-flat case of the function above is the limit λ → 0, which appears as the static charged black hole in Einstein–ModMax theory [23]. Similar to the gauge field solution in the RS-II with localized ModMax interaction on the 3-brane, the gauge vector associated with the tensor metric when solving the effective Einstein equations in the DGP scenario is simply equation (The Boyer–Lindquist type of the above metric can be obtained by employing the coordinate transformation3.9 ), maps the gauge field, equation (3.5 ), to take the form
$\begin{eqnarray}{\rm{d}}u={\rm{d}}t-\displaystyle \frac{{r}^{2}}{{\rm{\Delta }}}{\rm{d}}r\end{eqnarray}$
where $\begin{eqnarray}{{\rm{\Delta }}}_{\mathrm{RS}-\mathrm{II}}={r}^{2}-2{Mr}+\beta +{Q}^{2}{{\rm{e}}}^{-\nu }+\displaystyle \frac{{l}^{2}{Q}^{4}{{\rm{e}}}^{-2\nu }}{20{r}^{4}},\end{eqnarray}$
for the RS-II case, and $\begin{eqnarray}{{\rm{\Delta }}}_{\mathrm{DGP}}={r}^{2}-2{Mr}+{Q}^{2}{{\rm{e}}}^{-\nu }-{\lambda }^{2}{r}^{4},\end{eqnarray}$
for the DGP. In a general form, the resulting spacetime metrics describing static charge braneworld black holes can be expressed as $\begin{eqnarray}{\rm{d}}{s}_{{\rm{k}}}^{2}=-\displaystyle \frac{{{\rm{\Delta }}}_{{\rm{k}}}}{{r}^{2}}{\rm{d}}{t}^{2}+\displaystyle \frac{{r}^{2}}{{{\rm{\Delta }}}_{{\rm{k}}}}{\rm{d}}{r}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}_{2},\end{eqnarray}$
where the subscript k is used to represent the brane case under consideration. In the equations above, dΩ2 is the element of the unit 2-sphere. Furthermore, the transformation, equation ( $\begin{eqnarray}{A}_{\mu }{\rm{d}}{x}^{\mu }=\displaystyle \frac{Q{{\rm{e}}}^{-\nu }}{r}{\rm{d}}t,\end{eqnarray}$
where we have performed a gauge transformation to get rid of the r-component of the gauge vector.Before discussing the motions of charged particles, let us analyze the gtt component in each metric mentioned above to understand how the nonlinear parameter affects the positions of the black hole horizons. It is understood that the horizons in the spacetime are given by the roots of gtt. Unfortunately, for 3-brane with ModMax electrodynamics discussed in this paper, finding the roots of gtt is solving the polynomial with the power of six for the RS-II brane, and with the power of four for the DGP brane. Such problems are indeed difficult to solve. However, some qualitative analysis can be given for such equations. In the RS-II brane, the contributions of extra dimension can be acknowledged from the appearance of the tidal charge β and AdS5 bulk curvature radius l in the ΔRS−II function. These parameters modify the horizon radius compared to the generic ModMax black hole [23]. Meanwhile, the function to determine the black hole horizon in the DGP brane resembles that for (anti)-de Sitter spacetime. In such a case, in addition to outer and inner black hole horizons, a cosmological horizon exists that corresponds to the largest root of the ΔDGP = 0 equation.
In the subsequent numerical examples, the presence of the nonlinear parameter ν results in a reduction in the distance between the inner and outer horizons of the black hole. This holds true for both RS-II and DGP braneworlds, as well as the broader Einstein–ModMax considerations [23]. In the context of the DGP brane discussion, it is important to be mindful of the existence of additional horizons since, in general, the function, equation (3.11 ), has four roots. However, for our specific discussion, we focus exclusively on the inner and outer black hole horizons. Illustrations for some properties of gtt in these brane cases are given in figures 1–5.
4. Charged particle motions
In this section, our focus is on investigating the trajectories of a charged test object within the spacetimes derived in the preceding section. We are particularly interested in understanding the implications of the nonlinear and tidal charge parameters. From an intuitive standpoint, given that the nonlinear parameter is always non-negative, it diminishes the effective charge and alters the Coulomb-like interaction between the black hole and the test object. Conversely, the tidal charge β and the inverse of the crossover scale λ, conceived as an additional dimensional influence, have the potential to modify the effective value of the black hole's charge. We will elucidate these aspects in the subsequent discussion.
The associated Lagrangian governing a massive test particle with charge is given by [26]4.1 ), namely the conserved energy
$\begin{eqnarray}{ \mathcal L }=\displaystyle \frac{1}{2}{g}_{\mu \nu }\displaystyle \frac{{{\rm{d}}{x}}^{\mu }}{{\rm{d}}\tau }\displaystyle \frac{{{\rm{d}}{x}}^{\nu }}{{\rm{d}}\tau }+{{qA}}_{\mu }\displaystyle \frac{{{\rm{d}}{x}}^{\mu }}{{\rm{d}}\tau },\end{eqnarray}$
where we have considered the unit mass for the test object, τ is the proper time, and q is the corresponding charge-to-mass ratio. Obviously the test particle Lagrangian above can apply if we neglect the back-reaction and radiative effects in the process. Moreover, this Lagrangian yields the familiar equation of motion for an electrically charged test object in a curved background, namely [27] $\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{2}{x}^{\mu }}{{\rm{d}}{\tau }^{2}}+{{\rm{\Gamma }}}_{\alpha \beta }^{\mu }\displaystyle \frac{{{\rm{d}}x}^{\alpha }}{{\rm{d}}\tau }\displaystyle \frac{{{\rm{d}}x}^{\beta }}{{\rm{d}}\tau }={{qF}}^{\mu \nu }\displaystyle \frac{{{\rm{d}}x}_{\nu }}{{\rm{d}}\tau }.\end{eqnarray}$
For the spacetime under consideration in this paper, one can deduce the conserved quantities associated with an entity governed by the Lagrangian, equation ( $\begin{eqnarray}E=-\displaystyle \frac{\partial { \mathcal L }}{{\rm{d}}\dot{t}}={{qA}}_{t}-{g}_{{tt}}\dot{t},\end{eqnarray}$
and angular momentum $\begin{eqnarray}L=\displaystyle \frac{\partial { \mathcal L }}{{\rm{d}}\dot{\phi }}={g}_{\phi \phi }\dot{\phi }.\end{eqnarray}$
Subsequently, the final two equations yield $\begin{eqnarray}\dot{t}=\displaystyle \frac{{g}_{\phi \phi }\left(E-{{qA}}_{t}\right)}{{\rm{\Delta }}},\end{eqnarray}$
$\begin{eqnarray}\dot{\phi }=-\displaystyle \frac{{g}_{{tt}}L}{{\rm{\Delta }}}.\end{eqnarray}$
Within the spacetime metric above, we adopt a metric primarily of the positive type. Consequently, the trajectories of particle motion can be formulated as follows
$\begin{eqnarray}{g}_{{tt}}{\dot{t}}^{2}+{g}_{{rr}}{\dot{r}}^{2}+{g}_{\phi \phi }{\dot{\phi }}^{2}=-\delta ,\end{eqnarray}$
where δ = 1 is associated with a time-like particle, whereas δ = 0 denotes a null object. Particularly, on the equatorial plane where the relationship ${\rm{\Delta }}={g}_{t\phi }^{2}-{g}_{{tt}}{g}_{\phi \phi }$ holds, the time-like effective potential ${V}_{{\rm{e}}{\rm{f}}{\rm{f}}}\left(r\right)={E}^{2}-{\dot{r}}^{2}$ can be expressed as $\begin{eqnarray}{V}_{{\rm{eff}}}={E}^{2}+\displaystyle \frac{{\rm{\Delta }}-{g}_{{tt}}{L}^{2}-{g}_{\phi \phi }{\left(E-{{qA}}_{t}\right)}^{2}}{{g}_{{rr}}{\rm{\Delta }}}.\end{eqnarray}$
For an object engaged in circular motion with $\dot{r}=0$, the constants of motion E and L can be derived by simultaneously solving the equations Veff = 0 and $V{{\prime} }_{\mathrm{eff}}=0$.
Figure 1. Plots of −gtt of a charged black hole in Einstein–ModMax theory presented in [23]. We consider Q = 0.5M together with the vanishing of β and l. The solid, dashed, and dashed–dot curves represent the cases of ν = 0, ν = 0.2, and ν = 0.4, respectively. |
Figure 2. Plots of −gtt in figure 1 for a range of radius around the outer horizon. |
Figure 3. Plots of −gtt of the static RS-II braneworld black hole with ModMax electrodynamics on the 3-brane. We consider Q = 0.5M and l = 0. The solid, dashed, and dashed–dot curves represent the cases of ν = 0, ν = 0.2, and ν = 0.4, respectively. |
Figure 4. Plots of −gtt of the static DGP braneworld black hole with ModMax electrodynamics on the 3-brane. We consider Q = 0.5M and λM = 0.1. The solid, dashed, and dashed–dot curves represent the cases of ν = 0, ν = 0.2, and ν = 0.4, respectively. |
Figure 5. Plots of −gtt in figure 4 for a range of radius around the outer horizon. |
However, in this context, our aim is to illustrate the connection between the particle motion and parameters ν, β, and λ by showcasing their respective effective potentials through plots. When discussing the RS-II, we choose to neglect the influence of the AdS5 curvature radius l based on our findings that its impact on the numerical plots depicted below would be trivial due to its small value. Various numerical plots for Veff in the RS-II case are presented in figures 6 and 7, while the DGP cases are illustrated in figures 8, 9, 10, and 11. On the whole, these examples highlight that a higher value of ν corresponds to an elevated Veff for the case negative q, whereas the opposite results for positive q. This relationship can be associated with the reduction of an effective charge of the black hole, resulting in a weaker interaction between the black hole and the charged test object. Conversely, in the RS-II case, the presence of the tidal charge parameter β modifies the motions of a test object solely due to the spacetime curvature, as it does not contribute to the electromagnetic interaction between the black hole and the test object. In the DGP, the inverse of the crossover scale λ can be viewed effectively as the square root of the cosmological constant Λ. The effective potentials depicted in figures 8, 9, 10, and 11 mirror those in the ModMax–Einstein–de Sitter scenario [23].
Figure 6. Plots of Veff in the RS-II case for some particular values of nonlinear parameters. The solid, dashed, and dashed–dot curves represent the cases of ν = 0, ν = 0.2, and ν = 0.4, respectively. In obtaining these plots, we have used the numerical values L = 5M, Q = 0.7M, E = 0.9, and β = 0.5M2. The black curves relate to q = −2, whereas the blue ones correspond to q = 2. |
Figure 7. The value of Veff in the RS-II case at r = 2M for some particular values of the β parameter. The solid, dashed, and dashed–dot curves represent the cases of β = 0, β = 0.1M2, and β = −0.1M2, respectively. In obtaining these plots, we have used the numerical values L =2M, q = − 2, Q = 0.5M, and E = 0.9. |
Figure 8. Plots of Veff in the DGP case for λM = 0.5 and some values of ν's. In obtaining the plot, we have used the numerical values L = 5M, Q = 0.7M, and E = 0.9. The solid, dashed, and dashed–dot curves represent the cases of ν = 0, ν = 0.2, and ν = 0.4, respectively. The black curves relate to q = −2, whereas the blue ones correspond to q = 2. |
Figure 9. The value of Veff in the DGP case at r = 2M for some particular values of the λ parameter. The solid, dashed, and dashed–dot curves represent the cases of λ = 0, λM = 0.1, and λM = 0.2, respectively. In obtaining these plots, we have used the numerical values L =2M, q = −2, Q = 0.5M, and E = 0.9. |
Figure 10. The value of Veff in the DGP case at r = 3M for some particular values of the λ parameter. The solid, dashed, and dashed–dot curves represent the cases of λ = 0, λM = 0.1, and λM = 0.2, respectively. In obtaining these plots, we have used the numerical values L =2M, q = − 2, Q = 0.5M, and E = 0.9. |
Figure 11. The value of Veff in the DGP case at r = 5M for some particular values of the λ parameter. The solid, dashed, and dashed–dot curves represent the cases of λ = 0, λM = 0.1, and λM = 0.2, respectively. In obtaining these plots, we have used the numerical values L = 2M, q = − 2, Q = 0.5M, and E = 0.9. |
5. Conclusion
In this paper, we consider the ModMax electrodynamics that is localized in the RS-II and DGP branes. We obtain the corresponding 3-brane spacetime solution, obeying the effective Einstein equations in each case. The motions of charge objects in the spacetime are discussed, finding the significance of the nonlinear parameter ν. As one would expect, similar to the ModMax black hole studied in [23], it suppresses the effective charge of the black hole due to its definite positive value. By setting ν to be vanished, one recovers the charged black hole with the tidal parameter discussed in [15] in the RS-II case, and the charged black hole in the DGP brane worked out in [22] for the DGP case. For future work, we would like to investigate the compatibility of Euler–Heisenberg electrodynamics on the 3-brane (both RS-II and DGP cases). Incorporating Euler–Heisenberg electrodynamics in Einstein gravity can yield a black hole spacetime solution, whose metric is more complicated that that analyzed in this paper [28]. It is challenging to verify its compatibility with the braneworld scenario and investigate some of its physical aspects.