Equilibrium configuration of perfect fluid orbiting around black holes in some classes of alternative gravity theories

Before proceeding to alternative theories we shall first consider a solution in GR itself. This will help the reader to understand various physical quantities of interest in the other theories better. The line element for the RN black hole has the following expression

$$\begin{eqnarray}&&{\rm d}{{s}^{2}}=-\left( 1-\frac{2M}{r}+\frac{{{Q}^{2}}}{{{r}^{2}}} \right){\rm d}{{t}^{2}}+{{\left( 1-\frac{2M}{r}+\frac{{{Q}^{2}}}{{{r}^{2}}} \right)}^{-1}}{\rm d}{{r}^{2}}+{{r}^{2}}{\rm d}{{\Omega }^{2}}\end{eqnarray} \tag{ 28 }$$

where Q is the charge of the black hole and M is its mass. The potential $W(r,\theta )$, introduced by equations (9) and (16) for the above metric takes the following form

$$\begin{eqnarray}&&W(r,\theta )=\frac{1}{2}{\rm ln} \left[ \frac{{{r}^{2}}{{{\rm sin} }^{2}}\theta \left( {{r}^{2}}-2Mr+{{Q}^{2}} \right)}{{{r}^{4}}{{{\rm sin} }^{2}}\theta -\left( {{r}^{2}}-2Mr+{{Q}^{2}} \right){{\ell }^{2}}} \right].\end{eqnarray} \tag{ 29 }$$

Hence from the reality criteria of the potential we arrive at the following constraints

$$\begin{eqnarray}&&\left( 1-\frac{2M}{r}+\frac{{{Q}^{2}}}{{{r}^{2}}} \right)\geqslant 0\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&{{\ell }^{2}}\leqslant \ell _{ph}^{2}=\frac{{{r}^{4}}}{{{r}^{2}}-2Mr+{{Q}^{2}}}\end{eqnarray} \tag{ 31 }$$

where ℓ_ph corresponds to angular momentum associated with the photon orbit. Also we have the following result for $({\rm d}\theta /{\rm d}r)$ from equation (17) as

$$\begin{eqnarray}&&\frac{{\rm d}\theta }{{\rm d}r}=\frac{{{r}^{4}}\left( Mr-{{Q}^{2}} \right){{{\rm sin} }^{2}}\theta -{{\ell }^{2}}{{\left( {{r}^{2}}-2Mr+{{Q}^{2}} \right)}^{2}}}{r{{\ell }^{2}}{{\left( {{r}^{2}}-2Mr+{{Q}^{2}} \right)}^{2}}}{\rm tan} \theta .\end{eqnarray} \tag{ 32 }$$

Now there are two horizons of this spacetime: one is the regular event horizon, located at $r={{r}_{{\rm eh}}}$ while the other, known as the Cauchy horizon, is located at $r={{r}_{{\rm ch}}}$. These have the following expressions

$$\begin{eqnarray}&&{{r}_{{\rm eh}}}=M+\sqrt{{{M}^{2}}-{{Q}^{2}}}\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&{{r}_{{\rm ch}}}=M-\sqrt{{{M}^{2}}-{{Q}^{2}}}.\end{eqnarray} \tag{ 34 }$$

Another important radius is the photon circular orbit radius, whose location in this spacetime is given by the following expression:

$$\begin{eqnarray}&&{{r}_{ph}}=\frac{3M\pm \sqrt{9{{M}^{2}}-8{{Q}^{2}}}}{2}.\end{eqnarray} \tag{ 35 }$$

The extrema of the potential $W(r,\theta )$ on the equatorial plane are being determined by an angular momentum density ℓ_K² from equation (22) with the following expression

$$\begin{eqnarray}&&\ell _{K}^{2}=\frac{{{r}^{4}}\left( Mr-{{Q}^{2}} \right)}{{{\left( {{r}^{2}}-2Mr+{{Q}^{2}} \right)}^{2}}}.\end{eqnarray} \tag{ 36 }$$

Then we also have the following expression for derivative of ℓ_K²:

$$\begin{eqnarray}&&\frac{\partial \ell _{K}^{2}}{\partial r}=\frac{\left( {{r}^{2}}-2Mr+{{Q}^{2}} \right)\left( 5M{{r}^{4}}-4{{r}^{3}}{{Q}^{2}} \right)-4{{r}^{4}}\left( r-M \right)\left( rM-{{Q}^{2}} \right)}{{{\left( {{r}^{2}}-2r+{{Q}^{2}} \right)}^{3}}}.\end{eqnarray} \tag{ 37 }$$

The minima of the photon angular momentum can be obtained by plugging equation (35) into equation (31) which leads to:

$$\begin{eqnarray}&&\ell _{ph({\rm min} )}^{2}=\frac{{{\left( 3M+\sqrt{9{{M}^{2}}-8{{Q}^{2}}} \right)}^{4}}}{8\left( 3{{M}^{2}}-2{{Q}^{2}}+M\sqrt{9{{M}^{2}}-8{{Q}^{2}}} \right)}.\end{eqnarray} \tag{ 38 }$$

Thus we have presented all these results for a well known solution in GR to facilitate easy comparison with other gravity theories. This also brings out the key parameters involved in these computations which will be helpful as we go along discussing various alternative gravity theories.

In this section we will restrict ourselves to four dimensional spacetime. The action in this four dimensional spacetime with F(R) gravity in the presence of a matter field can be presented as:

$$\begin{eqnarray}&&S=\frac{1}{16\pi }\int {{d}^{4}}x\sqrt{-g}\left[ F(R)+{{\mathcal{L}}_{{\rm matter}}} \right].\end{eqnarray} \tag{ 39 }$$

In the above expression for action F(R) is an arbitrary function of the Ricci scalar R and ${{\mathcal{L}}_{{\rm matter}}}$ is the Lagrangian for matter fields. Then variation of the above action with respect to the metric leads to the field equation [9, 59]:

$$\begin{eqnarray}&&{{R}_{ab}}{{F}_{R}}-{{\nabla }_{a}}{{\nabla }_{b}}{{F}_{R}}+\left( \square {{F}_{R}}-\frac{1}{2}F(R) \right){{g}_{ab}}=T_{ab}^{{\rm matter}}\end{eqnarray} \tag{ 40 }$$

where R_ab is the Ricci tensor, ${{F}_{R}}\equiv dF(R)/dR$ and $T_{ab}^{{\rm matter}}$ is the standard matter stress-energy tensor, derived from the matter part of the Lagrangian ${{\mathcal{L}}_{{\rm matter}}}$ as given in action (39). Also the trace of the above equation connects the trace of the stress-energy tensor to the scalar curvature. We are interested in determining a spherically symmetric solution to the above field equation. Following the analysis presented in [60] we arrive at a charged solution in this F(R) gravity model with $F(R)=R-\lambda {\rm exp} (-\xi R)$. It should be noted as emphasized in section 1 that these modifications in EH action should pass all tests starting from clustering of galaxies down to Solar System tests. For this special exponential correction factor it has been shown [61] that there is no contradiction with Solar System tests. Also the solutions of this model are indistinguishable from the standard EH solution except for a possible change in Newton's constant [62]. The solution to the above field equation presented in equation (40) is topologically charged and can be presented following equation (11) as

$$\begin{eqnarray}&&f(r)=1-\frac{\Lambda }{3}{{r}^{2}}-\frac{M}{r}+\frac{{{Q}^{2}}}{{{r}^{2}}}\end{eqnarray} \tag{ 41 }$$

where $\Lambda =\lambda \left[ (4+2\xi R)/(8{{e}^{\xi R}}) \right]$. In order for this metric ansatz to satisfy equation (40) we should have some constraint among the parameters appearing in our theory, which are the following [60]:

$$\begin{eqnarray}&&1+\frac{\lambda \xi }{{\rm exp} (\xi R)}=0\end{eqnarray} \tag{ 42 }$$

$$\begin{eqnarray}&&\frac{\lambda }{{\rm exp} (\xi R)}+\frac{R}{2}\left( \frac{\lambda \xi }{{\rm exp} (\xi R)}-1 \right)=0.\end{eqnarray} \tag{ 43 }$$

Although this solution looks similar to the charged black hole, its higher dimensional behavior is different. The charge term in the above expression goes as ${{r}^{-(d-2)}}$ in d dimension, while it goes as ${{r}^{-2(d-3)}}$ in standard charged higher dimensional black hole spacetime. Therefore it is interesting to ask whether this term is originating from the scalar-tensor representation of F(R) gravity. In order to obtain the representation the standard way is to start with conformal transformations on the metric elements. However in this case no new physical interpretation can be obtained except its relation to the scale of the problem [10, 60, 63].

The above result can also be interpreted in a slightly different manner: uncharged solutions in $F(R)=R+f(R)$ theory show exact similarity with Einstein gravity in the presence of a cosmological constant. Thus in those situations f (R) plays the role of cosmological constant. As shown in [60] this charged solution is equivalent to Einstein gravity coupled with a conformally invariant Maxwell field, such that the f(R) term in this situation has the role of electromagnetic field. This acts as an origin of the charge term.

Next we will consider the equilibrium configurations of perfect fluid rotating around this black hole. For notational simplicity we will assume M = 2, Q = 1 and $y=\Lambda /3$. Then the line element for this black hole spacetime reduces to the following form

$$\begin{eqnarray}&&{\rm d}{{s}^{2}}=-\left( 1-y{{r}^{2}}-\frac{2}{r}+\frac{1}{{{r}^{2}}} \right){\rm d}{{t}^{2}}+\frac{{\rm d}{{r}^{2}}}{\left( 1-y{{r}^{2}}-\frac{2}{r}+\frac{1}{{{r}^{2}}} \right)}+{{r}^{2}}{\rm d}{{\Omega }^{2}}.\end{eqnarray} \tag{ 44 }$$

We must stress that static region exists in this spacetime with sub-critical values of cosmological parameter y as

$$\begin{eqnarray}&&y\lt {{y}_{c}}=\frac{1}{16}.\end{eqnarray} \tag{ 45 }$$

Thus equilibrium configurations are possible in this spacetime provided cosmological parameter y satisfies the constraint given by equation (45). The equipotential surfaces are being given by

$$\begin{eqnarray}&&W(r,\theta )={\rm ln} \left[ \frac{r{\rm sin} \theta \sqrt{\left( 1-y{{r}^{2}}-\frac{2}{r}+\frac{1}{{{r}^{2}}} \right)}}{\sqrt{{{r}^{2}}{{{\rm sin} }^{2}}\theta -{{\ell }^{2}}\left( 1-y{{r}^{2}}-\frac{2}{r}+\frac{1}{{{r}^{2}}} \right)}} \right]\end{eqnarray} \tag{ 46 }$$

and we also have,

$$\begin{eqnarray}&&\frac{{\rm d}\theta }{{\rm d}r}=\frac{\left( 2r-2-2y{{r}^{4}} \right){{{\rm sin} }^{2}}\theta -2{{\ell }^{2}}{{\left( 1-y{{r}^{2}}-\frac{2}{r}+\frac{1}{{{r}^{2}}} \right)}^{2}}}{2r{{\ell }^{2}}{{\left( 1-y{{r}^{2}}-\frac{2}{r}+\frac{1}{{{r}^{2}}} \right)}^{2}}}{\rm tan} \theta .\end{eqnarray} \tag{ 47 }$$

Note that for y = 0 it reduces to the RN scenario discussed earlier. The best idea about the nature of an $\ell ={\rm constant}$ surface can be extracted from the behavior of the potential $W(r,\theta )$ in equatorial plane ($\theta =\pi /2$). There we have two reality conditions imposed on the potential,

$$\begin{eqnarray}&&\left( 1-y{{r}^{2}}-\frac{2}{r}+\frac{1}{{{r}^{2}}} \right)\geqslant 0\end{eqnarray} \tag{ 48 }$$

$$\begin{eqnarray}&&{{\ell }^{2}}\leqslant \ell _{ph}^{2}\equiv \frac{{{r}^{2}}}{\left( 1-y{{r}^{2}}-\frac{2}{r}+\frac{1}{{{r}^{2}}} \right)}.\end{eqnarray} \tag{ 49 }$$

The function ℓ_ph represents the angular momentum of the photon's orbit. Further extremizing the potential $W(r,\theta =\pi /2)$ we arrive at the particular expression for angular momentum density

$$\begin{eqnarray}&&{{\ell }^{2}}=\ell _{K}^{2}(r,y)\equiv \frac{\left( 2r-2-2\;y{{r}^{4}} \right)}{2{{\left( 1-y{{r}^{2}}-\frac{2}{r}+\frac{1}{{{r}^{2}}}. \right)}^{2}}}\end{eqnarray} \tag{ 50 }$$

The specific energy of circular geodesics, corresponding to local extrema of the effective potential, can be expressed as

$$\begin{eqnarray}&&{{E}_{c}}(r,y)=\frac{1-y{{r}^{2}}-\frac{2}{r}+\frac{1}{{{r}^{2}}}}{\sqrt{1-\frac{3}{r}+\frac{2}{{{r}^{2}}}}}.\end{eqnarray} \tag{ 51 }$$

Now we turn back to the horizons in this spacetime. For $y\gt 0$, the photon angular momentum $\ell _{ph}^{2}(r,y)$ diverges at the black hole horizon r_h and cosmological horizon r_c determined by the equality in equation (48). This leads to the following expression for horizons,

$$\begin{eqnarray}&&{{r}_{h}}=\frac{1}{2}\left( \frac{1}{\sqrt{y}}-\frac{\sqrt{1-4\sqrt{y}}}{\sqrt{y}} \right)\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}&&{{r}_{c}}=\frac{1}{2}\left( \frac{1}{\sqrt{y}}+\frac{\sqrt{1-4\sqrt{y}}}{\sqrt{y}} \right).\end{eqnarray} \tag{ 53 }$$

Also the photon circular orbit radius can be obtained from the extrema of ℓ_ph². This has the numerical value: ${{r}_{ph}}=3.56155$. For this circular orbit the photon angular momentum density could be given by,

$$\begin{eqnarray}&&\ell _{ph(c)}^{2}\equiv \ell _{ph}^{2}({{r}_{ph}},y)=\frac{12.6846}{0.359611-12.9846y}.\end{eqnarray} \tag{ 54 }$$

Additionally the zero point of ℓ_K² leads to a radius called static radius and in this black hole spacetime is expressed parametrically as,

$$\begin{eqnarray}&&y({{r}_{s}})=\frac{{{r}_{s}}-1}{r_{s}^{4}}.\end{eqnarray} \tag{ 55 }$$

Note that ℓ_K² is not well defined in the region $r\gt {{r}_{s}}$, being negative there. At this static radius black hole attraction is compensated for cosmological repulsion. Then local extrema of ℓ_K² corresponds to the following expression for the cosmological parameter as,

$$\begin{eqnarray}&&{{y}_{ms}}(r)=\frac{{{(r-1)}^{4}}(r-4)}{{{r}^{4}}(12-15r+4{{r}^{2}})}.\end{eqnarray} \tag{ 56 }$$

This determines the marginally stable circular geodesics. The local maxima of this function y_ms give the critical value for the cosmological parameter y admitting stable circular orbits,

$$\begin{eqnarray}&&{{y}_{ms}}=0.000692.\end{eqnarray} \tag{ 57 }$$

For $y\lt {{y}_{ms}}$, there exists an inner or outer marginally stable circular geodesic at ${{r}_{ms(i)}}$ or ${{r}_{ms(o)}}$. There exists another special value of y, which corresponds to the situation where the minimum value of ℓ_ph² equals the maximum of ℓ_K². This value is denoted by y_e and has the following numerical value,

$$\begin{eqnarray}&&{{y}_{e}}\sim 4.8\times {{10}^{-5}}.\end{eqnarray} \tag{ 58 }$$

We can now separate out four different classes depending on the behavior of the functions ℓ_ph² and ℓ_K². These four classes are defined according to the cosmological parameter in the following way,

• (i)  
  $0\lt y\lt {{y}_{e}}$
• (ii)  
  $y={{y}_{e}}$
• (iii)  
  ${{y}_{e}}\lt y\lt {{y}_{ms}}$
• (iv)  
  ${{y}_{ms}}\lt y\lt {{y}_{c}}$.

For all these classes the variation of the two angular momentum quantities is illustrated in figure 1. In all these figures the descending parts of the curve $\ell _{K}^{2}(r,y)$ connect to unstable circular geodesics, along with the growing part (if it exists) determining stable circular geodesics. The extrema of $\ell _{K}^{2}(r,y)$ have an important significance: the minima determines ${{\ell }_{ms(i)}}$ at ${{r}_{ms(i)}}$, the inner marginally stable circular orbit and the maxima determines ${{\ell }_{ms(o)}}$ at ${{r}_{ms(o)}}$, which corresponds to the outer marginally stable circular orbit.

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The equipotential surfaces are also astrophysically quite important. Their properties can be established by studying the properties of the potential $W(r,\theta )$ in the equatorial plane. Note that the potential $W(r,\theta =\pi /2,y)$ has closely related properties with the effective potential of geodesic motion. It is worthwhile to mention that in the limit $r\to {{r}_{h}}$ or $r\to {{r}_{c}}$ $W(r,\theta =-\pi /2,y)\to -\infty$ hence the topological properties of the equipotential surfaces are directly inferred from the behavior of the potential $W(r,\theta =\pi /2,y)$. As pointed out earlier, local extrema of the potential are determined by the condition,

$$\begin{eqnarray}&&{{\ell }^{2}}=\ell _{K}^{2}(r,y).\end{eqnarray} \tag{ 59 }$$

Hence the decreasing part of ℓ_K² determines the maxima of the potential $W(r,\theta =\pi /2,y)$. These correspond to cusps such that at these radii matter moves along an unstable geodesic orbit. While the rising part in ℓ_K² determines the minima of the potential. These correspond to central rings of the equilibrium configurations along which matter moves in stable geodesics.

Next we provide a complete study of the behavior of equipotential surfaces along with the potential $W(r,\theta =\pi /2,y)$. We begin with the situations of astrophysical importance.

• (i)  
  $\;0\lt y\lt {{y}_{e}}$. This is the situation illustrated in figure 1(a). Here we discuss eight different configurations according to the values of $\ell ={\rm constant}$ satisfying $\ell \gt 0$.

  • (a)  
    $\ell \lt {{\ell }_{ms(i)}}$. Open surfaces are the only ones in existence. No disks are possible while surfaces with outer cusps exist (figure 2(a1–2)).
  • (b)  
    $\ell ={{\ell }_{ms(i)}}$. An infinitesimally thin, unstable ring located at ${{r}_{ms(i)}}$ exists. Open surfaces with outer cusps also remain in the picture (figure 2(b1–2)).
  • (c)  
    ${{\ell }_{ms(i)}}\lt \ell \lt {{\ell }_{mb}}$. Closed surfaces come into existence. Many equilibrium configurations even without any cusps are now possible. We also have surfaces with inner and outer cusps (figure 2(c1–2)).
  • (d)  
    $\ell ={{\ell }_{mb}}$. Here also equilibrium configurations without cusps are possible. A surface comes into existence with both inner and outer cusp. Along with inflow into the black hole due to mechanical non-equilibrium, an outflow from the disk also occurs. This comes due to the repulsive nature of the cosmological parameter. This is an astrophysically important scenario (figure 2(d1–2)).
  • (e)  
    ${{\ell }_{mb}}\lt \ell \lt {{\ell }_{ph(c)}}$. Equilibrium configurations are possible but accretion is not possible since we have no closed surface with inner cusp. The surface with an inner cusp is an open equipotential surface, while that with an outer cusp is a closed equipotential surface. This makes outflow from the disk possible (figure 2(e1–2)).
  • (f)  
    $\ell ={{\ell }_{ph(c)}}$. The potential $W(r,\theta =\pi /2,y)$ diverges at the photon circular orbit. Thus the inner cusp completely disappears. Closed equipotential surfaces with outer cusps still exist enabling outflow from the disk (figure 2(f1–2)).
  • (g)  
    $\ell ={{\ell }_{ms(o)}}$. An infinitesimally thin, unstable ring located at ${{r}_{ms(o)}}$ exists with coalescing outer and central cusp (figure 2(g1–2)).
  • (h)  
    $\ell \gt {{\ell }_{ms(o)}}$. Only open equipotential surfaces exist with no cusp (figure 2(h1–2)).
• (ii)  
  $y={{y}_{e}}.$ For this special y value the variation of ℓ_K² and ℓ_ph² has been illustrated in figure 1(b). For this case also we obtain all the hypersurfaces with similar nature. All of them fall into a single class, given by

  • (a)  
    $\ell ={{\ell }_{ph(c)}}={{\ell }_{ms(o)}}$. Here there exists no inner cusp, only an outer cusp mixing with the center exists (figure 2(i1–2)).
• (iii)  
  ${{y}_{e}}\lt y\lt {{y}_{ms}}$ This situation is illustrated by figure 1(c). However in this situation many other choices come into existence. We classify them below:

  • (a)  
    ${{\ell }_{mb}}\lt \ell \lt {{\ell }_{ms(o)}}$. This condition is equivalent to the condition (1e) illustrated above.
  • (b)  
    $\ell ={{\ell }_{ms(o)}}.$ This situation is slightly different. Here there exists an inner cusp of an open equipotential surface, while the outer cusp gets mixed with the center corresponding to a thin ring at the radius ${{r}_{ms(o)}}$ (figure 2(j1–2)).
  • (c)  
    ${{\ell }_{ms(o)}}\lt \ell \lt {{\ell }_{ph(c)}}$. This condition has only open equipotential surfaces; among them, one has an inner cusp (figure 2(k1–2)).
• (iv)  
  ${{y}_{ms}}\lt y\lt {{y}_{c}}$. For this choice of cosmological parameter the variations of ℓ_K² and ℓ_ph² are shown in figure 1(d). Note that in this situation ℓ_K² has only a declining part. Thus we can have only maxima of the potential and existence of open equipotential surfaces. This leads to the fact that in these spacetimes stable circular geodesics do not exist. In this case there are only two physically meaningful intervals.

  • (a)  
    $\ell \lt {{\ell }_{ph(c)}}$. This situation has equipotential surfaces identical to the situation illustrated in (1a).
  • (b)  
    $\ell \geqslant {{\ell }_{ph(c)}}$ This is similar to the situation (2a) discussed earlier.

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We should also mention in this connection that the following numerical values are obtained for angular momentum densities, which are important for the study of fluid orbiting around a black hole. The angular momentum density for an inner marginally stable orbit is ${{\ell }_{ms(i)}}=3.03965$, while that for an outer marginally stable orbit is given by ${{\ell }_{ms(o)}}=4.9$ for the cosmological parameter being $y={{10}^{-6}}$. Two other important angular momentum densities are that of the marginally bound orbit, with ${{\ell }_{mb}}=3.28852$, and photon circular orbit, ${{\ell }_{ph(c)}}=4.27392$. These expressions are of importance for studying the structure of accretion disk.

Equilibrium configurations originating from test particle fluid, which is rotating in a given spacetime, are determined by the equipotential surfaces, where the gravitational and inertial forces are being compensated by the pressure gradient. (For an axially symmetric spacetime the rotation axis of the equilibrium configuration coincides with the axis of symmetry for the spacetime, while for a spherically symmetric case this axis can be any radial line; see for example [48]).

The influence of a non-zero dilaton charge coming from an effective string theory on the character of the equipotential surfaces of marginally stable orbits has been studied for configuration rotating around such black holes. Static uncharged black holes in general relativity are described by the well known and well studied Schwarzschild solution. However even for large mass black holes (compared to Planck mass) the Schwarzschild solution is a good approximation to describe uncharged black holes in string theory except for regions near the singularity. This situation is completely different for the Einstein–Maxwell solution in string inspired theory due to dilaton coupling.

The dilaton couples with F² with the implication that every solution having non zero ${{F}_{\mu \nu }}$ couples with the dilaton. Thus the charged black hole solution in general relativity (the RN solution) appears in string theory with the presence of the dilaton. Thus the effective four dimensional low energy action is given by,

$$\begin{eqnarray}&&S=\int {{d}^{4}}x\sqrt{-g}\left[ -R+{{e}^{-2\Phi }}{{F}^{2}}+2{{(\nabla \Phi )}^{2}} \right]\end{eqnarray} \tag{ 60 }$$

where ${{F}_{\mu \nu }}$ is the Maxwell field and we have set other gauge and antisymmetric tensor fields, e.g. the Kalb–Ramond field ${{H}_{\mu \nu \rho }}$, to zero in order to focus on the dilaton field Φ [27–30, 64]. Extremizing the above action with respect to the $U(1)$ potential ${{A}_{\mu }}$, dilaton field Φ and metric ${{g}_{\mu \nu }}$ lead to the following field equations,

$$\begin{eqnarray}&&{{\nabla }_{\mu }}\left( {{e}^{-2\Phi }}{{F}^{\mu \nu }} \right)=0\end{eqnarray} \tag{ 61 }$$

$$\begin{eqnarray}&&{{\nabla }^{2}}\Phi +\frac{1}{2}{{e}^{-2\Phi }}{{F}^{2}}=0\end{eqnarray} \tag{ 62 }$$

$$\begin{eqnarray}&&{{R}_{\mu \nu }}=2{{\nabla }_{\mu }}\Phi {{\nabla }_{\nu }}\Phi +2{{e}^{-2\Phi }}{{F}_{\mu \lambda }}F_{\nu }^{\lambda }-\frac{1}{2}{{g}_{\mu \nu }}{{e}^{-2\Phi }}{{F}^{2}}.\end{eqnarray} \tag{ 63 }$$

The static spherically symmetric solution to the above field equations yields the line element as [27]:

$$\begin{eqnarray}&&{\rm d}{{s}^{2}}=-(1-\frac{2M}{r}){\rm d}{{t}^{2}}+\frac{1}{(1-\frac{2M}{r})}{\rm d}{{r}^{2}}+r(r-{{e}^{2{{\Phi }_{0}}}}\frac{{{Q}^{2}}}{M}){\rm d}{{\Omega }^{2}}\end{eqnarray} \tag{ 64 }$$

where ${\rm d}{{\Omega }^{2}}={\rm d}{{\theta }^{2}}+{{{\rm sin} }^{2}}\theta {\rm d}{{\phi }^{2}}$. Due to isometry we can confine our motion in the equatorial plane such that ${\rm d}{{\Omega }^{2}}={\rm d}{{\phi }^{2}}$ [i.e. $\theta =\pi /2$]. Here ${{\Phi }_{0}}$ gives the asymptotic value of the dilaton field and Q represents the black hole charge. This solution is almost identical to the Schwarzschild metric with the difference being that areas of the two spheres depend on Q. The surface $r=\frac{{{Q}^{2}}{{e}^{2{{\Phi }_{0}}}}}{M}$ is singular and $r=2M$ is the regular event horizon. The solution of the scalar field Φ as a function of r in terms of its asymptotic value ${{\Phi }_{0}}$ can be obtained from equation (62) leading to the following result [27]:

$$\begin{eqnarray}&&{{e}^{-2\Phi }}={{e}^{-2{{\Phi }_{0}}}}-\frac{{{Q}^{2}}}{Mr}.\end{eqnarray} \tag{ 65 }$$

From the above result note that we have the following limit, $r\to \infty$ leading to $\Phi \to {{\Phi }_{0}}$. Thus the dilaton charge can be defined as,

$$\begin{eqnarray}&&D=\frac{1}{4\pi }\int {{{\rm d}}^{2}}{{\sigma }^{\mu }}{{\nabla }_{\mu }}\Phi \end{eqnarray} \tag{ 66 }$$

where the integral is over the two spheres located at spatial infinity and ${{\sigma }^{\mu }}$ is the normal to the corresponding two spheres at spatial infinity. For the charged black hole we are considering this leads to,

$$\begin{eqnarray}&&D=-\frac{{{Q}^{2}}{{e}^{2{{\Phi }_{0}}}}}{2M}.\end{eqnarray} \tag{ 67 }$$

Note that dilaton charge D depends on the asymptotic value of dilaton field ${{\Phi }_{0}}$, which is determined solely by M and Q and is always negative.

One point that must be mentioned while passing is that this charge originates from the coupling of the electromagnetic field with a scalar field, and the actual charge present in the system is the electromagnetic charge Q. If there were no electromagnetic charge the dilaton charge D defined through equation (67) would have been zero. Hence this dilaton charge is not a scalar charge, it is a coupling between the electromagnetic charge and the scalar field at asymptotic infinity. This charge is also responsible for long range attractive force between black holes [27].

Note that the actual dependence on the dilaton field depends on the Planck scale described by ${{e}^{-\Phi /{{M}_{pl}}}}$. Since we have worked in the unit ${{M}_{pl}}\sim 1$ the term modified to ${{e}^{-\Phi }}$. As $\Phi \to {{\Phi }_{0}}\sim {{M}_{pl}}$, this term becomes significant. For notational convenience we shall define a quantity $q=-D$ and by virtue of the above discussion q is always positive. Hence our solution is parameterized by the variable q and choosing the dimensionless expression such that M = 1 we get the line element as

$$\begin{eqnarray}&&{\rm d}{{s}^{2}}=-\left( 1-\frac{2}{r} \right){\rm d}{{t}^{2}}+{{\left( 1-\frac{2}{r} \right)}^{-1}}{\rm d}{{r}^{2}}+r\left( r-2q \right){\rm d}{{\Omega }^{2}}.\end{eqnarray} \tag{ 68 }$$

Then we have two horizons in our system given by ${{r}_{h1}}=2$ and ${{r}_{h2}}=2q$. We shall take the choice in which ${{r}_{h1}}\gt {{r}_{h2}}$ or equivalently $q\leqslant 1$. Thus equilibrium configurations are possible only for $r\gt 2$. Now the equipotential surfaces are possible only in these spacetimes. Now, the equipotential surfaces are determined by the result,

$$\begin{eqnarray}&&W(r,\theta )=\frac{1}{2}{\rm ln} \frac{\left[ r(r-2)(r-2q){{{\rm sin} }^{2}}\theta \right]}{\left[ {{r}^{2}}(r-2q){{{\rm sin} }^{2}}\theta -(r-2){{\ell }^{2}} \right]}\end{eqnarray} \tag{ 69 }$$

and

$$\begin{eqnarray}&&\frac{{\rm d}\theta }{{\rm d}r}=\frac{{{r}^{2}}{{(r-2q)}^{2}}{{{\rm sin} }^{2}}\theta -{{\ell }^{2}}{{(r-2)}^{2}}(r-q)}{{{\ell }^{2}}r{{(r-2)}^{2}}(r-2q)}{\rm tan} \theta .\end{eqnarray} \tag{ 70 }$$

For q = 0 the above results reduce to the well known Schwarzschild result [37]. The best insight into the nature of the $\ell ={\rm constant}$ fluid configurations is obtained by examining the behavior of $W(r,\theta )$ in the equatorial plane ($\theta =\pi /2$). There are two reality conditions on $W(r,\theta =\pi /2)$:

$$\begin{eqnarray}&&r\gt 2;r\gt 2q\end{eqnarray} \tag{ 71 }$$

and

$$\begin{eqnarray}&&{{r}^{2}}(r-2q)\gt {{r}^{2}}{{\ell }^{2}}.\end{eqnarray} \tag{ 72 }$$

The first condition is just the statement that the fluid should remain outside the black hole horizons and we shall calculate its property in that non singular region of spacetime. While the second condition implies,

$$\begin{eqnarray}&&{{\ell }^{2}}\leqslant \ell _{ph}^{2}(r;q)\equiv \frac{{{r}^{2}}(r-2q)}{{{r}^{2}}}.\end{eqnarray} \tag{ 73 }$$

This function $\ell _{ph}^{2}(r;q)$ can be thought of as an effective potential of the photon geodesic motion, also note that $\ell =\frac{{{U}_{\phi }}}{{{U}_{t}}}$ has a close correspondence with the impact parameter for photon geodesic motion [56]. Further, the condition to obtain local extrema of the potential $W(r,\theta =\pi /2)$ is identical to the condition for vanishing of the pressure gradient ($\partial {{U}_{t}}/\partial r=0,\partial {{U}_{t}}/\partial \theta =0$). Since at the equatorial plane we have $\partial {{U}_{t}}/\partial \theta =0$ independently along with the criteria $\ell ={\rm constant}$, and the following relation:

$$\begin{eqnarray}&&\frac{\partial {{U}_{t}}}{\partial r}=\frac{{{r}^{2}}{{(r-2q)}^{2}}-{{\ell }^{2}}{{(r-2)}^{2}}(r-q)}{{{\left[ {{r}^{2}}(r-2q)-{{\ell }^{2}}(r-2) \right]}^{3/2}}{{\left[ r(r-2)(r-2q) \right]}^{1/2}}}.\end{eqnarray} \tag{ 74 }$$

From which we arrive at the particular expression for angular momentum density,

$$\begin{eqnarray}&&{{\ell }^{2}}=\ell _{K}^{2}(r;q)\equiv \frac{{{r}^{2}}{{(r-2q)}^{2}}}{(r-q){{(r-2)}^{2}}}.\end{eqnarray} \tag{ 75 }$$

The extrema for $W(r,\theta =\pi /2)$ correspond to the spacetime points, where fluid moves along a circular geodesics, since $\ell _{K}^{2}(r;q)$ relates to the distribution of the angular momentum density of circular geodesic orbits. Clearly this leads to,

$$\begin{eqnarray}&&{{W}_{{\rm extr}}}(r,\theta =\pi /2;q)={\rm ln} {{E}_{c}}(r,y)\end{eqnarray} \tag{ 76 }$$

where

$$\begin{eqnarray}&&{{E}_{c}}(r,q)=\frac{(r-2)\sqrt{(r-q)}}{\sqrt{r}\sqrt{(r-q)(r-2)-(r-2q)}}\end{eqnarray} \tag{ 77 }$$

is defined as the specific energy along these circular geodesics. Important properties of the potential $W(r,\theta )$ are determined by its behavior at the equatorial plane, and especially by the properties of the functions $\ell _{ph}^{2}(r;q)$ and $\ell _{K}^{2}(r,q)$. Discussion of these properties enables us to classify the dilaton gravity spacetime according to the properties of equipotential surfaces of test particle fluid. For pure Schwarzschild spacetime (q = 0) the analysis can be found in [35].

From the analytical form of $\ell _{ph}^{2}(r,q)$ it is evident that the quantity diverges at the black hole horizon (r = 2). However it shows no such divergence for the inner horizon given by $r=2q$. The local minimum of the function $\ell _{ph}^{2}(r,q)$ corresponds to the radius of the photon circular orbit and has the following expression,

$$\begin{eqnarray}&&{{r}_{ph}}=\frac{3+q\pm \sqrt{{{(q+3)}^{2}}-16q}}{2}\end{eqnarray} \tag{ 78 }$$

with the impact parameter,

$$\begin{eqnarray}&&\ell _{ph(c)}^{2}=\ell _{ph({\rm min} )}^{2}\equiv {{r}^{2}}(r-q).\end{eqnarray} \tag{ 79 }$$

The function $\ell _{K}^{2}(r,q)$, determining the Keplerian (geodesic) circular orbits, has a zero point at the so-called static radius given by,

$$\begin{eqnarray}&&{{r}_{s}}=2q\end{eqnarray} \tag{ 80 }$$

which coincides with the inner horizon, also note that the function has divergent nature at r = 2 and as well as at r = q. The function $\ell _{K}^{2}(r;q)$ diverges at the black hole horizon i.e. $\ell _{K}^{2}(r\to 2)\to +\infty$. Since,

$$\begin{eqnarray}\begin{array}{rcl} \frac{\partial \ell _{K}^{2}}{\partial r} & = & \frac{4r(r-2q)}{{{(r-2)}^{2}}} \\ {} & - & \frac{{{r}^{2}}{{(r-2q)}^{2}}(3r-2-2q)}{{{(r-q)}^{2}}{{(r-2)}^{3}}} \\ \end{array}\end{eqnarray} \tag{ 81 }$$

the local extrema of $\ell _{K}^{2}(r,q)$ are given by the condition ${{q}_{ms}}=1$ determining the marginally stable circular orbits. For $q\lt {{q}_{ms}}$, there exists an inner (outer) marginally stable circular geodesic at ${{r}_{ms(i)}}$ (${{r}_{ms(o)}}$).

Other special values of q correspond to the situation where the value of the minimum of $\ell _{ph}^{2}(r,q)$ equals the maximum of $\ell _{K}^{2}(r,q)$. We denote this value by q_e and it is a solution of the algebraic equation,

$$\begin{eqnarray}&&q_{e}^{5}-22q_{e}^{4}+80q_{e}^{3}-64q_{e}^{2}-32=0.\end{eqnarray} \tag{ 82 }$$

However this situation is less important as the above equation has no physical solution. Thus we will not have any inner cusp present in the system. Hence this has less attractive features, still interesting, which we will present now.

In this situation the behavior of ℓ_ph² and ℓ_K² is similar to that of the RN scenario, while we have only four available situations. As illustrated in figure 3 we have a minimum for ℓ_K² having value ${{\ell }_{ms}}=3.5739$. We also have a minimum in ℓ_ph² which corresponds to the value ${{\ell }_{ph(c)}}=4.2786$. Thus all together we have four cases in this dilaton gravity model from the behavior of the potential $W(r,\theta =\pi /2,q)$. These cases are given for the following intervals of ℓ:

• (i)  
  $\ell \lt {{\ell }_{ms}}$. We have only open equipotential surfaces in this angular momentum range (see figure 4(a1–2)).
• (ii)  
  $\ell ={{\ell }_{ms}}$. An infinitesimally thin and unstable ring comes into existence at the marginally stable radius (see figure 4(b1–2)).
• (iii)  
  ${{\ell }_{ms}}\lt \ell \leqslant {{\ell }_{ph(c)}}$. Closed equipotential surfaces come into existence, with one such surface having a cusp, allowing inflow to the black hole (see figure 4(c1–2)).
• (iv)  
  $\ell \gt {{\ell }_{ph(c)}}$. Closed equipotential surfaces exist, however cusps do not exist. Thus in the near horizon region equipotential surfaces cannot cross the equatorial plane (see figure 4(d1–2)).

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Thus we have obtained the structure of equipotential surfaces in dilaton gravity. We have established how the inflow occurs in this spacetime along with necessary criteria on the angular momentum density; also the behavior that determines the angular momentum density which admits no minima. Hence this can be thought of as having less structure than the previous one while still having quite interesting features.

In high energy physics, assuming spacetime to have more than four dimensions is a common practice. In these scenarios, the spacetime is assumed to be a four dimensional brane, embedded on a higher dimensional bulk. Though ordinary matter fields are confined to these branes, gravity can interact via the bulk as well. The EH action needs to be modified too and the most natural choice would be to include higher order terms in the action. A famous second order term is the GB term, under whose presence the modified action looks like,

$$\begin{eqnarray}&&S=\int {\rm d}{{x}^{5}}\sqrt{-g}[R+\alpha ({{R}_{abcd}}{{R}^{abcd}}-4{{R}_{ab}}{{R}^{ab}}+{{R}^{2}})+{{F}_{ab}}{{F}^{ab}}]\end{eqnarray} \tag{ 83 }$$

where R, R_ab and R_abcd are the Ricci scalar, Ricci tensor and Riemann tensor respectively. F_ab is the electromagnetic tensor field and α is the GB coupling coefficient with dimension of length squared. In order to get equations of motion we need to vary the action with respect to the metric g_ab and electromagnetic field F_ab respectively. This leads to [65],

$$\begin{eqnarray}\begin{array}{rcl} {{R}_{ab}} & - & \frac{1}{2}{{g}_{ab}}R-\alpha \left[ \frac{1}{2}{{g}_{ab}}\left( {{R}_{pqrs}}{{R}^{pqrs}} \right. \right. \\ {} & - & \left. 4{{R}_{mn}}{{R}^{mn}}+{{R}^{2}} \right)-2R{{R}_{ab}}+4{{R}_{am}}R_{b}^{m} \\ {} & + & \left. 4{{R}^{mn}}{{R}_{ambn}}-2R_{a}^{\,pqr}{{R}_{bpqr}} \right]={{T}_{ab}} \\ \end{array}\end{eqnarray} \tag{ 84 }$$

$$\begin{eqnarray}&&{{\nabla }_{a}}{{F}^{ab}}=0\end{eqnarray} \tag{ 85 }$$

where T_ab is the usual stress tensor for the electromagnetic field. The important thing to notice is that the field equation only contains second order derivatives of the metric, and no higher derivatives are present. This is expected since GB gravity is a subclass of Lovelock gravity, which does not contain higher derivative terms of the Riemann tensor.

Now it is possible to obtain a static spherically symmetric solution to this field equation having the form of equation (11). It turns out that those solutions are asymptotically de Sitter or anti-de Sitter. Thus this situation would be identical to that in section 4.2. This immediately tells us that only a correspondence between the parameters in this theory with those of the charged F(R) scenario discussed in an earlier section is required.

To this end we must mention that the solution obtained above corresponds to a five dimensional spacetime, i.e. the line element takes the form: ${\rm d}{{s}^{2}}=-f(r){\rm d}{{t}^{2}}+{{f}^{-1}}(r){\rm d}{{r}^{2}}+{{r}^{2}}{\rm d}\Omega _{3}^{2}$, where ${\rm d}\Omega _{3}^{2}={\rm d}\theta _{1}^{2}+{{{\rm sin} }^{2}}{{\theta }_{1}}({\rm d}\theta _{2}^{2}+{{{\rm sin} }^{2}}{{\theta }_{2}}{\rm d}\theta _{3}^{2})$. However, as we have been emphasizing, this solution can be interpreted from a brane world point of view, such that the visible brane is characterized by ${{\theta }_{3}}={\rm constant}$ hypersurface. Hence the solution reduces to four dimensions with $(t,r,{{\theta }_{1}},{{\theta }_{2}})$ as a set of coordinates. Thus on the visible brane the spherically symmetric solution can be treated on the same footing as those discussed earlier. Such spherically symmetric solutions are obtained in [65] and have the particular form with reference to equation (11) as

$$\begin{eqnarray}&&f(r)=K+\frac{{{r}^{2}}}{4\alpha }\left[ 1\pm \sqrt{1+\frac{8\alpha \left( m+2\alpha |K| \right)}{{{r}^{4}}}-\frac{8\alpha {{q}^{2}}}{3{{r}^{6}}}} \right]\end{eqnarray} \tag{ 86 }$$

where K determines the scalar curvature of the spacetime, to which we attribute a positive value. Then from Solar System tests and neutrino oscillation experiments [66, 67] it turns out that ${{\alpha }^{-1}}$ has stringent bounds. Also in all astrophysical scenarios the distance from black hole r is quite large, thus for our study we can make a power series expansion of the terms inside the square root and arrive at the following form:

$$\begin{eqnarray}&&f(r)=1+\frac{{{r}^{2}}}{3\alpha }+\frac{m+2\alpha }{{{r}^{2}}}-\frac{{{q}^{2}}}{3{{r}^{4}}}.\end{eqnarray} \tag{ 87 }$$

Then we readily identify this with equation (44) leading to the following mapping, $\alpha =-1/(2y)$. The possibility that α may be negative has been discussed in detail by [68] and [69]. With this identification we can run all our machinery described in previous sections and obtain all the relevant physical quantities.

All the numerical values for cosmological parameters discussed earlier are directly mapped to those of α. We can also conclude that it has identical structure to equipotential surfaces, with the existence of inner and outer cusps. Thus matter outflow can also occur in these topological dS or AdS black holes along with the possibility of inflow to it.

Some numerical estimates can be made for this model following previous discussions. The angular momentum densities for inner and outer marginally stable orbits are 3.04 and 4.9 respectively with $\alpha =5\times {{10}^{5}}$. Also marginally bound and photon circular orbits have similar expressions.

Thus for various alternative gravity theories we have obtained the equipotential surfaces for rotating perfect fluid. For some of them we have observed dynamical accretion disks requiring inflow, outflow and cusps, while in some cases the structure was simpler and did not have such dynamical behavior. Thus we may conclude that black holes in different alternative theories affect accretion disk structure differently.