Newtonian analogue of Schwarzschild–de Sitter spacetime: Influence on the local kinematics in galaxies

The late time accelerated expansion of the Universe demands that even in local galactic scales it is desirable to study astrophysical phenomena, particularly relativistic accretion related phenomena in massive galaxies or in galaxy mergers and the dynamics of the kiloparsecs-scale structure and beyond, in the local galaxies in Schwarzschild–de Sitter (SDS) background, rather than in Schwarzschild or Newtonian paradigm. Owing to the complex and nonlinear character of the underlying magnetohydrodynamical equations in general relativistic (GR) regime, it is quite useful to have a Newtonian analogous potential containing all the important GR features that allows us to treat the problem in Newtonian framework for study of accretion and its related processes. From the principle of conserved Hamiltonian of the test particle motion, here, a three dimensional Newtonian analogous potential has been obtained in spherical geometry corresponding to SDS/Schwarzschild–anti–de Sitter spacetime, that reproduces almost all of the GR features in its entirety with remarkable accuracy. The derived potential contains an explicit velocity dependent term of the test particle that renders an approximate relativistic modification of Newtonian-like potential. The complete orbital dynamics around SDS geometry and the epicyclic frequency corresponding to SDS metric have been extensively studied in the Newtonian framework using the derived potential. Applying the derived analogous potential it is found that the current accepted value of $\mathrm{\Lambda }\sim {10}^{-56}{\mathrm{cm}}^{-2}$ moderately influences both sonic radius as well as Bondi accretion rate, especially for spherical accretion with smaller values of adiabatic constant and temperature, which might have interesting consequences on the stability of accretion disk in active galactic nuclei/radio galaxies.

Figure 1

Variation of potential $V_{\mathrm{DS}}$ [Eq. (16)] with radial distance $r$. The solid and long-dashed lines in Fig. 1 correspond to $V_{\mathrm{DS}}$ in low energy limit $E_{\mathrm{DS}}\sim 0$ for SDS geometry with $\mathrm{\Lambda }{r}_s^2=1\times 10^{-27}$ with ${\lambda }_{\mathrm{DS}}=(0,9.5)$, respectively. The short-dashed curve represents Schwarzschild spacetime ($\mathrm{\Lambda }=0$). Figure 1 is for semirelativistic energy $E_{\mathrm{DS}}\sim 0.5$, otherwise resemble 1. Figures 1 and 1 correspond to effective potential $V_{\mathrm{DS}}^{\text{Eff}}$, with other parameters same to those in Figs 1 and 1, respectively. Figure 1 corresponds to GR effective potential for SDS geometry with the same physical parameters as those in Figs. 1, but independent of conserved energy. $E_{\mathrm{DS}}$ and ${\lambda }_{\mathrm{DS}}$ are expressed respectively in units of $c^2$ and $GM/c$.

Figure 2

Variation of the difference between $V_{\mathrm{DS}}$ ($\mathrm{\Lambda }{r}_s^2=1\times 10^{-27}$) and $V_{\mathrm{SW}}$ with $r$. Solid and short-dashed curves in Fig. 2 show the difference between $V_{\mathrm{DS}}$ and $V_{\mathrm{SW}}$, corresponding to $E_{\mathrm{DS}}\sim 0$ and $E_{\mathrm{DS}}\sim 0.5$, respectively. Solid curve in Fig. 2 shows the variation of the relative deviation (in percentage) ${\xi }_{횤}$ with $r$, for both $E_{\mathrm{DS}}\sim 0$ and $E\mathrm{DS}\sim 0.5$ (which coincide). The two extreme vertical dashed lines, denoted as $x_{\min }$ and $x_{\max }$ (in units of $r_s$, correspond to the value of ${\xi }_{횤}\sim 0.01\%$ and $\sim 200\%$, respectively). The vertical dashed line in the middle represents $r$ at which ${\xi }_{횤}\sim 10\%$. $E_{\mathrm{DS}}$ is expressed in units of $c^2$. Note that ${\lambda }_{\mathrm{DS}}$ has no effect on the nature of potentials when $r>100r_s$.

Figure 3

Variation of potential $V_{\mathrm{DS}}$ and the PNP in Eq. (22) with $r$ corresponding to circular motion trajectory for $\mathrm{\Lambda }{r}_s^2=1\times 10^{-27}$. The solid and short-dashed lines correspond to $V_{\mathrm{DS}}$ in Eq. (11) and PNP in Eq. (22), respectively. Figure 3 covers the entire distance, Figs. 3focus only on the outer radii, near to the static radius and the cosmological horizon, respectively, whereas Fig. 3 focuses on the inner radii near the BH event horizon.

Figure 4

Variation of angular momentum with $r$ for circular orbit trajectory. The short-dashed curve in Fig. 4 corresponds to $V_{\mathrm{DS}}$ with $\mathrm{\Lambda }{r}_s^2=1\times 10^{-27}$ that coincides with the corresponding GR results. The solid curves in Figs. 4 and 4 are due to $V_{\mathrm{SW}}$. The long-dashed line in Fig. 4 corresponds to $V_{\mathrm{DS}}$ which coincides with the angular momentum profile corresponding to the PNP given in Eq. (22).

Figure 5

Radial variation of the relative deviation (in percentage) in angular momentum. The two extreme vertical dashed lines, represented by $x_{\min }$ and $x_{\max }$, correspond to value of ${\xi }_{횤}\sim 0.01\%$ and $\sim 200\%$, respectively. The curve is truncated at $x_{\max }$ which also corresponds to the static radius. The vertical dashed line in the middle represents $r$ at which ${\xi }_{횤}\sim 10\%$.

Figure 6

Variation of Hamiltonian with $r$ for test particle in circular orbit corresponding to $\mathrm{\Lambda }{r}_s^2=1\times 10^{-27}$. The solid and long-dashed lines in Fig. 6 are for the Hamiltonian corresponding to $V_{\mathrm{SW}}$ and $V_{\mathrm{DS}}$, respectively. The profile of $E_{\mathrm{DS}}^C$ corresponding to $V_{\mathrm{DS}}$ coincides with that in general relativity. The solid, long-dashed and short-dashed lines in Figs. 6 are for the Hamiltonian corresponding to $V_{\mathrm{SW}}$, $V_{\mathrm{DS}}$ and the PNP in Eq. (22), respectively. Figures 6 focus on the outer radii; near to the static radius and the cosmological horizon, respectively. Figure 6 resembles Figs. 2 and 5, however, showing the relative deviation in Hamiltonian (in percentage), as a function of radial distance $r$.

Figure 7

The variation of the angular frequency and the epicyclic frequency with $r$ for the test particle in circular orbit. Solid, long-dashed and short-dashed curves in Fig. 7 depict angular frequency profiles corresponding to $V_{\mathrm{DS}}$ with $\mathrm{\Lambda }{r}_s^2=1\times 10^{-27}$, the GR counterpart, and the PNP in Eq. (22) respectively. Note that ${\omega }_{\mathrm{DS}}^C$ coincides with the angular frequency profile corresponding to $V_{\mathrm{SW}}$, both in the inner as well as at the outer radii. Figure 7 depicts the variation of epicyclic frequency with $r$, corresponding to $V_{\mathrm{DS}}$.

Figure 8

Variation of radial velocity of the test particle falling radially towards the central BH. Solid, long-dashed and short-dashed curves in Fig. 8 are for $v_r$ corresponding to $V_{\mathrm{DS}}$ with $\mathrm{\Lambda }{r}_s^2=1\times 10^{-27}$, $V_{\mathrm{SW}}$ and PNP in Eq. (22), respectively, for test particle with $E_{\mathrm{DS}}\sim 0$. Figure 8 is similar to that in 8, however, for $E_{\mathrm{DS}}\sim 0.5$. In Figs. 8 which resemble Figs. 8, the detailed nature of the corresponding profiles at the outer radii is depicted. In Figs. 8 we show the relative deviation (in percentage) in $v_r$, corresponding to $E_{\mathrm{DS}}\sim 0$ and $E_{\mathrm{DS}}\sim 0.5$, respectively. The vertical lines in the figures are same to those in Fig. 2. $E_{\mathrm{DS}}$ is expressed in units of $c^2$.

Figure 9

Variation of $dr/d\mathrm{\Omega }$ in radial direction $r$ corresponding to $\mathrm{\Lambda }{r}_s^2=1\times 10^{-27}$. The solid, long-dashed and short-dashed curves in all the figures correspond to $V_{\mathrm{DS}}$, PNP in Eq. (22), and $V_{\mathrm{SW}}$. Figures 9 and 9 are for $E_{\mathrm{DS}}\sim 0$ corresponding to angular momentum ${\lambda }_{\mathrm{DS}}=(1.1,9.5)$, respectively. Figures 9 and 9 resemble Figs. 9, however for $E_{\mathrm{DS}}\sim 0.5$. $E_{\mathrm{DS}}$ and ${\lambda }_{\mathrm{DS}}$ are expressed in units of $c^2$ and $GM/c$, respectively.

Figure 10

The profile of elliptical-like trajectory of particle orbit in equatorial plane due to $V_{\mathrm{DS}}$ with $\mathrm{\Lambda }{r}_s^2=1\times 10^{-27}$. In Fig. 10, the solid line is due to $V_{\mathrm{SW}}$ and the dotted line for $V_{\mathrm{DS}}$. The particle trajectory coincides with that in general relativity. Figure 10 shows the profile of perihelion advancement $\mathrm{\Psi }$ as a function of eccentricity corresponding to $V_{\mathrm{DS}}$ for $r_a=8\times 10^8{r}_s$. The solid and long-dashed curves correspond to $V_{\mathrm{SW}}$ and $V_{\mathrm{DS}}$ respectively.

Figure 11

Variation of the location of outermost stable circular and outermost bound circular orbit with repulsive $\mathrm{\Lambda }$, obtained with $V_{\mathrm{DS}}$. The solid and dashed lines in Fig. 11 show the variation of $r_{\max }^{\mathrm{SC}}$ and $r_{\max }^{\mathrm{BC}}$ with $\mathrm{\Lambda }$, respectively. Figures 11 show the variation of Hamiltonian, angular momentum and angular frequency along $r_{\max }^{\mathrm{SC}}$ with corresponding values of $\mathrm{\Lambda }$, obtained with $V_{\mathrm{DS}}$. The y axis in Fig. 11 is in logarithmic scale.

Figure 12

Variation of sonic radius and Bondi accretion rate with temperature $T_{\infty }$ obtained with $V_{\mathrm{DS}}$, for $\mathrm{\Lambda }\sim {10}^{-56}{\mathrm{cm}}^{-2}$. Figure 12 shows the variation of the ratio of sonic radius in the SDS background to the sonic radius with $\mathrm{\Lambda }=0$, in temperature. Figure 12 is similar to 12, but for the ratio of Bondi accretion rate in SDS background to Bondi accretion rate with $\mathrm{\Lambda }=0$. Solid, long-dashed and dotted curves in the Figs. 12 are for adiabatic constant $\gamma =(4/3,1.5,1.66)$, respectively. BH mass is chosen to be $\sim 10^9{M}_{\odot }$.