Nonlinearly charged black holes in the scalar-tensor modified gravity theory

Starting from the action of the scalar-tensor modified gravity theory coupled to the power law nonlinear electrodynamics and making use of the suitable transformation relations, the corresponding action of the Einstein-power-Maxwell-dilaton gravity theory has been obtained. Thermodynamical properties of the new charged dilatonic black holes have been investigated in the presence of a power Maxwell field as the nonlinear electrodynamics. Making use of a special form of a scalar field, it has been found that the scalar potential can be written as the linear combination of three Liouville-type potentials. Three new classes of spherically symmetric charged dilatonic black hole solutions have been obtained as the exact solutions to the field equations of the Einstein-power-Maxwell-dilaton theory. The conserved and thermodynamical quantities, related to the new black hole solutions, have been calculated from geometrical and thermodynamical approaches separately. Through comparison of the results obtained from these two alternative approaches, the validity of the first law of black hole thermodynamics has been confirmed for either of three new black hole solutions. Thermal stability or phase transition of the new black hole solutions has been analyzed, making use of the canonical ensemble method, and by calculating the black hole heat capacity at the fixed black hole charge. The points of type one and type two phase transitions as well as the ranges at which the black holes are locally stable have been determined precisely. The nonlinearly charged string black hole solutions have been obtained from their Einstein’s counterparts. The thermodynamical properties as well as the thermal stability of the new charged string black holes have been investigated by utilizing the canonical ensemble method.

Figure 1

$\mathrm{\Psi }(r)$ versus $r$ for $\nu =-3/4,L=1,\mathrm{\Lambda }=-3,Q=1$, and $M=1$ [Eq. (3.14)]. Left: $p=0.53$, $b=1$, and $\alpha =0.52$, 0.538, 0.56, 0.58 for black, red, green, and blue curves, respectively. Middle: $\alpha =0.6$, $b=1$, and $p=0.514$, 0.5167, 0.52, 0.523 for black, red, green, and blue curves, respectively. Right: $\alpha =0.6$, $p=0.53$, and $b=1.004$, 1.009, 1.0135, 1.018 for black, red, green, and blue curves, respectively.

Figure 2

$\mathrm{\Psi }(r)$ versus $r$ for $\nu =-1/2,\mathrm{\Lambda }=-3,M=3,p=(2-\alpha {)}^{-1}$ [Eq. (3.14)]. Left: $Q=1$, $b=1$, and $\alpha =0.54$, 0.532, 0.523, 0.515 for black, red, green, and blue curves, respectively. Middle: $\alpha =0.54$, $Q=1$, and $b=1$, 1.008, 1.015, 1.02 for black, red, green, and blue curves, respectively. Right: $\alpha =0.54$, $b=1$, and $Q=1.022$, 1.0145, 1.008, 1 for black, red, green, and blue curves, respectively.

Figure 3

$\mathrm{\Psi }(r)$ versus $r$ for $\mathrm{\Lambda }=-3,M=2.56,b=1$, and $Q=2$ [Eq. (3.14)]. Left: $\alpha =1$, $p=0.8$, and $\nu =-0.2,-0.176,-0.15,-0.12$ for black, red, green, and blue curves, respectively. Middle: $\nu =-0.02,p=0.8$, and $\alpha =0.5$, 0.57, 0.65, 0.75 for black, red, green, and blue curves, respectively. Right: $\nu =-0.02,\alpha =1$, and $p=0.76$, 0.75, 0.73, 0.715 for black, red, green, and blue curves, respectively.

Figure 4

$T$ and ${({\partial }^{2}M/\partial S^2)}_Q$ versus $r_{+}$ for $\nu =-3/4,\mathrm{\Lambda }=-3,Q=1,b=1$ [Eqs. (4.1) and (6.2)]. Left: $p=0.8$ ($T:\alpha =0.7$, 0.8 for brown and red curves, respectively). [$0.5{({\partial }^{2}M/\partial S^2)}_Q$: $\alpha =0.7$, 0.8 for black and blue curves, respectively]. Right: $\alpha =0.7$ ($T:p=0.75$, 0.8 for brown and red curves, respectively). [$0.5{({\partial }^{2}M/\partial S^2)}_Q$: $p=0.75$, 0.8 for black and blue curves, respectively].

Figure 5

$0.1T$ [Eq. (4.1)] and ${({\partial }^{2}M/\partial S^2)}_Q$ [Eq. (6.3)] versus $r_{+}$ for $\nu =-1/2,\mathrm{\Lambda }=-3,Q=5,b=0.8$, and $p=(2-\alpha {)}^{-1}$. $0.1T:\alpha =0.4$, 0.6 for brown and red curves, respectively. ${({\partial }^{2}M/\partial S^2)}_Q$: $\alpha =0.4$, 0.6 for black and blue curves, respectively.

Figure 6

$0.5T$ [Eq. (4.1)] and ${({\partial }^{2}M/\partial S^2)}_Q$ [Eq. (6.5)] versus $r_{+}$ for $\mathrm{\Lambda }=-3,Q=1,b=1$. Left: $p=0.8$ $\alpha =0.6$ ($0.5T:\nu =-0.65,-0.7$ for brown and red curves, respectively). (${({\partial }^{2}M/\partial S^2)}_Q$: $\nu =-0.65,-0.7$ for black and blue curves, respectively). Middle: $p=0.8$, $\nu -0.65$ ($0.5T:\alpha =0.6$, 0.5 for brown and red curves, respectively). [${({\partial }^{2}M/\partial S^2)}_Q:\alpha =0.6$, 0.5 for black and blue curves, respectively]. Right: $\alpha =0.6,\nu =-0.65$ ($0.5T:p=0.775$, 0.85 for brown and red curves, respectively). [${({\partial }^{2}M/\partial S^2)}_Q:p=0.775$, 0.85 for black and blue curves, respectively].

Figure 7

$B(r)$ versus $r$ for $\nu =-3/4,L=1,\mathrm{\Lambda }=-3,Q=1,M=1$, and $c=\frac{\alpha p}{1-p}$. Left: $p=0.53$, $b=1$, and $\alpha =0.52$, 0.54, 0.56, 0.58 for black, red, brown, and blue curves, respectively. Middle: $\alpha =0.6$, $b=1$, and $p=0.514$, 0.5167, 0.52, 0.523 for black, red, brown, and blue curves, respectively. Right: $\alpha =0.6$, $p=0.53$, and $b=1.004$, 1.009, 1.0135, 1.018 for black, red, brown, and blue curves, respectively.

Figure 8

$B(r)$ versus $r$ for $\nu =-1/2,\mathrm{\Lambda }=-3,M=3,p=(2-\alpha {)}^{-1}$, and $c=\frac{\alpha }{1-\alpha }$. Left: $Q=1$, $b=1$, and $\alpha =0.54$, 0.532, 0.5235, 0.515 for black, red, brown, and blue curves, respectively. Middle: $\alpha =0.54$, $Q=1$, and $b=1$, 1.008, 1.015, 1.022 for black, red, brown, and blue curves, respectively. Right: $\alpha =0.54$, $b=1$, and $Q=1.022$, 1.0145, 1.0075, 1 for black, red, brown, and blue curves, respectively.

Figure 9

$B(r)$ versus $r$ for $\mathrm{\Lambda }=-3,M=2.56,b=1,Q=2$, and $c=\frac{\alpha p}{1-p}$. Left: $\alpha =1$, $p=0.8$, and $\nu =-0.2,-0.176,-0.145,-0.1$ for black, red, brown, and blue curves, respectively. Middle: $\nu =-0.02,p=0.8$, and $\alpha =0.45$, 0.56, 0.68, 0.78 for black, red, brown, and blue curves, respectively. Right: $\nu =-0.02,\alpha =1$, and $p=0.77$, 0.75, 0.7303, 0.715 for black, red, brown, and blue curves, respectively.