Spherically symmetric static vacuum solutions in hybrid metric-Palatini gravity

We consider vacuum static spherically symmetric solutions in the hybrid metric-Palatini gravity theory, which is a combination of the metric and Palatini $f(R)$ formalisms unifying local constraints at the Solar System level and the late-time cosmic acceleration. We adopt the scalar-tensor representation of the hybrid metric-Palatini theory, in which the scalar-tensor definition of the potential can be represented as a Clairaut differential equation. Due to their mathematical complexity, it is difficult to find exact solutions of the vacuum field equations, and therefore we adopt a numerical approach in studying the behavior of the metric functions and of the scalar field. After reformulating the field equations in a dimensionless form, and by introducing a suitable independent radial coordinate, the field equations are solved numerically. We detect the formation of a black hole from the presence of the Killing horizon for the timelike Killing vector in the metric tensor components. Several models, corresponding to different functional forms of the scalar field potential, are considered. The thermodynamic properties of these black hole solutions (horizon temperature, specific heat, entropy, and evaporation time due to Hawking luminosity) are also investigated in detail.

Figure 1

Variation of the metric tensor components $e^{\nu }$ in the vacuum outside a spherically symmetric compact object in HMPG with a vanishing scalar field potential. Left figure: The initial value of scalar field, ${\varphi }_0$, is fixed at ${\varphi }_0=1$ while the initial value of its derivative is taken to be: $u_0=4\times 10^{-9}$ (solid curve), $u_0=8\times 10^{-9}$ (dotted curve), $u_0=1.6\times 10^{-8}$ (short dashed curve), $u_0=3.2\times 10^{-8}$ (dashed curve), $u_0=6.4\times 10^{-8}$ (long dashed curve). Right figure: The initial value of the derivative of the scalar field, $u_0$, is fixed at $u_0=5.12\times 10^{-7}$ while the initial value of the scalar field is taken to be: ${\varphi }_0=0.5$ (solid curve), ${\varphi }_0=1$ (dotted curve), ${\varphi }_0=2$ (short dashed curve), ${\varphi }_0=4$ (dashed curve), ${\varphi }_0=8$ (long dashed curve).

Figure 2

Variation of the metric tensor components $e^{-\lambda }$ in the vacuum outside a spherically symmetric compact object in HMPG with a vanishing scalar field potential. Left: The initial value of scalar field, ${\varphi }_0$, is fixed at ${\varphi }_0=1$ while the initial value of its derivative is taken to be: $u_0=4\times 10^{-9}$ (solid curve), $u_0=8\times 10^{-9}$ (dotted curve), $u_0=1.6\times 10^{-8}$ (short dashed curve), $u_0=3.2\times 10^{-8}$ (dashed curve), $u_0=6.4\times 10^{-8}$ (long dashed curve). Right: The initial value of the derivative of the scalar field, $u_0$, is fixed at $u_0=5.12\times 10^{-7}$ while the initial value of the scalar field is taken to be: ${\varphi }_0=0.5$ (solid curve), ${\varphi }_0=1$ (dotted curve), ${\varphi }_0=2$ (short dashed curve), ${\varphi }_0=4$ (dashed curve), ${\varphi }_0=8$ (long dashed curve).

Figure 3

Variation of the effective mass function $M_{\text{eff}}(\xi )$ in the vacuum outside a spherically symmetric compact object in HMPG with a vanishing scalar field potential. Left: The initial value of scalar field, ${\varphi }_0$, is fixed at ${\varphi }_0=1$ while the initial value of its derivative is taken to be: $u_0=4\times 10^{-9}$ (solid curve), $u_0=8\times 10^{-9}$ (dotted curve), $u_0=1.6\times 10^{-8}$ (short dashed curve), $u_0=3.2\times 10^{-8}$ (dashed curve), $u_0=6.4\times 10^{-8}$ (long dashed curve). Right: The initial value of the derivative of the scalar field, $u_0$, is fixed at $u_0=5.12\times 10^{-7}$ while the initial value of the scalar field is taken to be: ${\varphi }_0=0.5$ (solid curve), ${\varphi }_0=1$ (dotted curve), ${\varphi }_0=2$ (short dashed curve), ${\varphi }_0=4$ (dashed curve), ${\varphi }_0=8$ (long dashed curve).

Figure 4

Comparison of the fitting function ${\xi }_S({\varphi }_0,u_0)=1-148411\times u_0/{\varphi }_0-2.737\times 10^{11}\times u_0^2$ for the position of the event horizon of the HMPG theory black holes and the numerical data for the vanishing potential case, for ${\varphi }_0\in [0.15,5.7665]$ and $u_0\in [3\times 10^{-11},5.9\times 10^{-7}]$.

Figure 5

Variation of the metric tensor components $e^{\nu }$ in the vacuum outside a spherically symmetric compact object in HMPG with a Higgs-type potential $V=\alpha {\varphi }^2+\beta {\varphi }^4$ of the scalar field. The initial value of the scalar field is fixed at ${\varphi }_0=1$, and its derivative is fixed at $u_0=10^{-8}$, respectively. Left figure: the parameter $\alpha$ is fixed at $\alpha =10^{-10}$ while the parameter $\beta$ is taken to be: $\beta =2\times 10^{-10}$ (solid curve), $\beta =3\times 10^{-10}$ (dotted curve), $\beta =4\times 10^{-10}$ (short dashed curve), $\beta =5\times 10^{-10}$ (dashed curve), $\beta =6\times 10^{-10}$ (long dashed curve). Right figure: the parameter $\beta$ is fixed at $\beta =10^{-10}$ while the parameter $\alpha$ is taken to be: $\alpha =2\times 10^{-10}$ (solid curve), $\alpha =3\times 10^{-10}$ (dotted curve), $\alpha =4\times 10^{-10}$ (short dashed curve), $\alpha =5\times 10^{-10}$ (dashed curve), $\alpha =6\times 10^{-10}$ (long dashed curve).

Figure 6

Variation of the metric tensor components $e^{\lambda }$ in the vacuum outside a spherically symmetric compact object in HMPG with a Higgs-type potential $V=\alpha {\varphi }^2+\beta {\varphi }^4$ of the scalar field. The initial value of the scalar field is fixed at ${\varphi }_0=1$, and its derivative is fixed at $u_0=10^{-8}$, respectively. Left figure: the parameter $\alpha$ is fixed at $\alpha =10^{-10}$ while the parameter $\beta$ is taken to be: $\beta =2\times 10^{-10}$ (solid curve), $\beta =3\times 10^{-10}$ (dotted curve), $\beta =4\times 10^{-10}$ (short dashed curve), $\beta =5\times 10^{-10}$ (dashed curve), $\beta =6\times 10^{-10}$ (long dashed curve). Right figure: the parameter $\beta$ is fixed at $\beta =10^{-10}$ while the parameter $\alpha$ is taken to be: $\alpha =2\times 10^{-10}$ (solid curve), $\alpha =3\times 10^{-10}$ (dotted curve), $\alpha =4\times 10^{-10}$ (short dashed curve), $\alpha =5\times 10^{-10}$ (dashed curve), $\alpha =6\times 10^{-10}$ (long dashed curve).

Figure 7

Variation of the effective mass function $M_{\text{eff}}(\xi )$ in the vacuum outside a spherically symmetric compact object in HMPG with a Higgs-type potential $V=\alpha {\varphi }^2+\beta {\varphi }^4$ of the scalar field. The initial value of the scalar field is fixed at ${\varphi }_0=1$, and its derivative is fixed at $u_0=10^{-8}$, respectively. Left figure: the parameter $\alpha$ is fixed at $\alpha =10^{-10}$ while the parameter $\beta$ is taken to be: $\beta =2\times 10^{-10}$ (solid curve), $\beta =3\times 10^{-10}$ (dotted curve), $\beta =4\times 10^{-10}$ (short dashed curve), $\beta =5\times 10^{-10}$ (dashed curve), $\beta =6\times 10^{-10}$ (long dashed curve). Right figure: the parameter $\beta$ is fixed at $\beta =10^{-10}$ while the parameter $\alpha$ is taken to be: $\alpha =2\times 10^{-10}$ (solid curve), $\alpha =3\times 10^{-10}$ (dotted curve), $\alpha =4\times 10^{-10}$ (short dashed curve), $\alpha =5\times 10^{-10}$ (dashed curve), $\alpha =6\times 10^{-10}$ (long dashed curve).

Figure 8

Variation of the black hole temperature $T_{\mathrm{BH}}({\xi }_S)/T_H$ as a function of the horizon radius ${\xi }_S$ of a HMPG black hole in the absence of a scalar field potential, $V=0$, for $u_0\in [2.2\times 10^{-10},1.1264\times 10^{-7}]$, and for different values of ${\varphi }_0$: ${\varphi }_0=0.15$ (solid curve), ${\varphi }_0=0.225$ (dotted curve), ${\varphi }_0=0.3375$ (short dashed curve), ${\varphi }_0=0.50625$ (dashed curve), respectively.

Figure 9

Variation of the specific heat $C_{\mathrm{BH}}({\xi }_S)/C_H$ as a function of the horizon radius ${\xi }_S$ of a HMPG black hole in the absence of a scalar field potential, $V=0$, for $u_0\in [2.2\times 10^{-10},1.1264\times 10^{-7}]$, and for different values of ${\varphi }_0$: ${\varphi }_0=0.15$ (solid curve), ${\varphi }_0=0.225$ (dotted curve), ${\varphi }_0=0.3375$ (short dashed curve), ${\varphi }_0=0.50625$ (dashed curve), respectively.

Figure 10

Variation of the entropy $S_{\mathrm{BH}}({\xi }_S)/C_H$ of a HMPG black hole as a function of the horizon radius ${\xi }_S$ in the absence of a scalar field potential, $V=0$, for $u_0\in [2.2\times 10^{-10},1.1264\times 10^{-7}]$, and for different values of ${\varphi }_0$: ${\varphi }_0=0.15$ (solid curve), ${\varphi }_0=0.225$ (dotted curve), ${\varphi }_0=0.3375$ (short dashed curve), ${\varphi }_0=0.50625$ (dashed curve), respectively.

Figure 11

Variation of the evaporation time ${\tau }_{\mathrm{BH}}({\xi }_S)/{\tau }_H$ as a function of the horizon radius ${\xi }_S$ of a HMPG black hole in the absence of a scalar field potential, $V=0$, for $u_0\in [2.2\times 10^{-10},1.1264\times 10^{-7}]$, and for different values of ${\varphi }_0$: ${\varphi }_0=0.15$ (solid curve), ${\varphi }_0=0.225$ (dotted curve), ${\varphi }_0=0.3375$ (short dashed curve), ${\varphi }_0=0.50625$ (dashed curve), respectively.