Generalized Hayward spacetimes: Geometry, matter, and scalar quasinormal modes

Bardeen’s 1968 idea of a regular black hole spacetime was revived by Hayward in 2006 through the construction of a new example of such a geometry. Later it was realized by Neves and Saa, that a wider, two-parameter class exists, with Bardeen and Hayward spacetimes as special cases. In this article, we revisit and generalize the Hayward spacetime by applying the Damour-Solodukhin (DS) prescription. Recalling the DS suggestion of a deformed Schwarzschild spacetime where $g_{tt}=-(1-\frac{2M_1}{r})$, $g_{rr}={(1-\frac{2M_2}{r})}^{-1}$ and $M_1\ne M_2$, we propose a similar deformation of the Hayward geometry. The $g_{tt}$ and $g_{rr}$ in the original Hayward line element remain functionally the same, albeit with changes introduced via differently valued metric parameters, following the DS idea. This results in a plethora of spacetime geometries, known as well as new, and including singular black holes, wormholes or regular black holes. We first study the geometric features and matter content of each of such spacetimes in some detail. Subsequently, we find the scalar quasinormal modes corresponding to scalar wave propagation in these geometries. We investigate how the real and imaginary parts of the quasinormal modes depend on the values and ranges of the metric parameters used to classify the geometries. Finally, we argue how our results on this family of spacetimes suggest their utility as black hole mimickers.

Figure 1

Parameter space ($\sigma$, $\kappa$) showing different spacetimes for specific, different ranges of the parameters.

Figure 2

Plot of $z(\zeta )$ showing embedding of the $t=\mathrm{const}$, $\theta =\pi /2$ slice of the Schwarzschild wormhole.

Figure 3

Plot of $z(\zeta )$ showing embedding of the $t=\mathrm{const}$, $\theta =\pi /2$ slice of the Hayward wormhole for different $\kappa$ values.

Figure 4

Embedding diagram for the Case IV spacetime with $\kappa >\frac{4}{3\sqrt{3}}$ where the function can be smoothly extended to $r=0$.

Figure 5

Plot of $z(\zeta )$ showing embedding of the $t=\mathrm{const}$, $\theta =\pi /2$ slice of the Hayward regular black hole.

Figure 6

Variation of $V_{\mathrm{eff}}$ for different parameter values of $(\sigma ,\kappa )$ and angular momentum number $m$ (Cases I–IV). We observe that the double-barrier nature of the potential depends on the value of $\sigma$. Same $\kappa$ wormholes will have the same throat radius but the tortoise coordinate will be different because of its dependence on $\sigma$. (a) $\kappa =0,\sigma =1$ (Case I) (b) $\kappa =0,\sigma =0$ (Case II) (c) $\kappa =0,\sigma =0.1$ (Case III) (d) $\kappa =0,\sigma =0.7$ (Case III) (e) $\kappa =0,\sigma =0.9$ (Case III) (f) $\kappa =0,\sigma =0.99$ (Case III) (g) $\kappa =0.1,\sigma =0$ (Case IV) (h) $\kappa =0.7,\sigma =0$ (Case IV).

Figure 7

Plots of the effective potential $V_{\mathrm{eff}}$ for Cases V and VI. Once again the double-barrier potential is observed for values of $\sigma$ close to 1. (a) $\kappa =0.1,\sigma =1$ (Case V) (b) $\kappa =0.7,\sigma =1$ (Case V) (c) $\kappa =0.1,\sigma =01.1$ (Case VI) (d) $\kappa =0.1,\sigma =0.9$ (Case VI) (e) $\kappa =0.7,\sigma =0.1$ (Case VI) (f) $\kappa =0.7,\sigma =0.75$ (Case VI).

Figure 8

Typical Schwarzschild black hole ringdown profile as observed at $r_{*}=30$ with initial Gaussian profile $\psi =e^{-\frac{(u-10{)}^2}{100}}$ and grid spacing $h=0.5$.

Figure 9

QNM variation for Case II wormhole with angular momentum $m$. The $\mathrm{Im}(\omega )$ decreases with increasing $m$ as observed in most wormhole geometries. (a) Variation of $\mathrm{Re}(\omega )$ (in $M^{-1}$) with $m$. (b) Variation of $\mathrm{Im}(\omega )$ (in $M^{-1}$) with $m$.

Figure 10

TD profile for a Damour-Solodukhin wormhole with $\sigma$ values close to 1 observed at $r_{*}=30$ with initial Gaussian profile $\psi =e^{-\frac{(u-10{)}^2}{100}}$, grid spacing $h=0.5$ and $m=1$. (a) $\sigma =0.9$ (b) $\sigma =0.9999$.

Figure 11

Analytical fitting function mentioned in the inset boxes in the figures indicate the dependence of the $\mathrm{Re}(\omega )$ and $\mathrm{Im}(\omega )$ (in $M^{-1}$) on $\sigma$ for Case III wormholes and is shown by the red curve for $m=1$. (a) Fitting $\mathrm{Re}(\omega )$ (b) Fitting $\mathrm{Im}(\omega )$.

Figure 12

Variation of $\mathrm{Re}(\omega )$ (in $M^{-1}$) with $m$ for different $\kappa$ Hayward wormholes.

Figure 13

Variation of the imaginary component of the QNM frequency (in units of $M^{-1}$) for different $\kappa$ Hayward wormholes with $m$.

Figure 14

Analytical fitting function mentioned in the inset boxes in the plot indicates the dependence of the $\mathrm{Re}(\omega )$ (in $M^{-1}$ units) on $\kappa$ for Hayward wormholes and is shown by the red curve with $m=1$. It matches well with the QNM values in blue.

Figure 15

Analytical fitting functions mentioned in the inset boxes in the plots indicate the dependence of the $\mathrm{Im}(\omega )$ (in $M^{-1}$ units) on $\kappa$ for Hayward wormholes and is shown by the red curve with $m=4$ and $m=6$. (a) Fitting $\mathrm{Im}(\omega )$ for $m=4$ (b) Fitting $\mathrm{Im}(\omega )$ for $m=6$.

Figure 16

Analytical fitting functions mentioned in the inset boxes in the plots indicate the dependence of the $\mathrm{Re}(\omega )$ and $\mathrm{Im}(\omega )$ (in $M^{-1}$) on $\kappa$, for Hayward regular black hole and is shown by the red curve with $m=1$. (a) Fitting $\mathrm{Re}(\omega )$ (b) Fitting $\mathrm{Im}(\omega )$.

Figure 17

Time domain profile showing distinct echoes as observed at $r_{*}=30$ for a $\sigma =0.9999$, $\kappa =0.5$ Case VI wormhole. The grid spacing $h=0.5$ and a Gaussian initial profile are used.

Figure 18

Analytical fitting functions mentioned in the inset boxes in the figures indicate the dependence of the $\mathrm{Re}(\omega )$ and $\mathrm{Im}(\omega )$ (in $M^{-1}$) on $\kappa$ for Case VI wormholes and is shown by the red curve with $m=1$ and fixed $\sigma$. (a) Fitting $\mathrm{Re}(\omega )$ for $\sigma =0.1$ (b) Fitting $\mathrm{Im}(\omega )$ for $\sigma =0.1$ (c) Fitting $\mathrm{Re}(\omega )$ for $\sigma =0.5$ (d) Fitting $\mathrm{Im}(\omega )$ for $\sigma =0.5$.

Figure 19

The analytical fitting functions mentioned in the figures indicate the dependence of the $\mathrm{Re}(\omega )$ and $\mathrm{Im}(\omega )$ (in $M^{-1}$) on $\kappa$ for Case VI wormholes and is shown by the red curve with $m=1$ and fixed $\sigma$. (a) Fitting $\mathrm{Re}(\omega )$ for $\sigma =0.88$ (b) Fitting $\mathrm{Im}(\omega )$ for $\sigma =0.88$ (c) Fitting $\mathrm{Re}(\omega )$ for $\sigma =0.9$ (d) Fitting $\mathrm{Im}(\omega )$ for $\sigma =0.9$.

Figure 20

Analytical fitting functions mentioned in the inset boxes in the figures indicate the dependence of the $\mathrm{Re}(\omega )$ and $\mathrm{Im}(\omega )$ (in $M^{-1}$) on $\sigma$ for Case VI wormholes and is shown by the red curve with $m=1$ and fixed $\kappa$. (a) Fitting $\mathrm{Re}(\omega )$ for $\kappa =0.1$ (b) Fitting $\mathrm{Im}(\omega )$ for $\kappa =0.1$ (c) Fitting $\mathrm{Re}(\omega )$ for $\kappa =0.7$ (d) Fitting $\mathrm{Im}(\omega )$ for $\kappa =0.7$.

Figure 21

Time domain profiles for Case II wormhole (red) and a Case III Damour-Solodukhin wormhole (blue) with $\sigma =0.01$. The signal is observed at $r_{*}=5$, grid size $h=0.5$ and initial Gaussian signal of the form $e^{\frac{-(u-10{)}^2}{100}}$. With $\sigma$ values being close to 0, the two wormhole geometries are difficult to distinguish as is evident from their TD profiles and QNM values.

Figure 22

TD using exact form of tortoise coordinate.

Figure 23

TD using tortoise coordinate obtained by numerical interpolation.