Magnetic brane solutions of Lovelock gravity with nonlinear electrodynamics

In this paper, we consider logarithmic and exponential forms of nonlinear electrodynamics as a source and obtain magnetic brane solutions of the Lovelock gravity. Although these solutions have no curvature singularity and no horizon, they have a conic singularity with a deficit angle. We investigate the effects of nonlinear electrodynamics and the Lovelock gravity on the value of the deficit angle and find that various terms of Lovelock gravity do not affect the deficit angle. Next, we generalize our solutions to spinning cases with maximum rotating parameters in arbitrary dimensions and calculate the conserved quantities of the solutions. Finally, we consider nonlinear electrodynamics as a correction of the Maxwell theory and investigate the properties of the solutions.

Figure 1

LNEF branch of EN gravity: $f_{\mathrm{EN}}(\rho )$ vs $\rho$ for $l=3$, $q=1$, and $d=4$. Left panel: $m=0.5$, $\beta =2$ (continuous line), $\beta =2.5$ (doted line), and $\beta =5$ (dashed line). Right panel: $\beta =5$, $m=0.5$ (continuous line), $m=1$ (doted line), and $m=2$ (dashed line).

Figure 2

LNEF branch of GB gravity: $f_{\mathrm{GB}}(\rho )$ vs $\rho$ for $l=2$, $q=0.1$, $m=0.005$, and $d=7$. Left panel: $\beta =10$, ${\alpha }_2=0.01$ (continuous line), ${\alpha }_2=0.03$ (doted line), and ${\alpha }_2=0.06$ (dashed line). Right panel: ${\alpha }_2=0.01$, $\beta =2.6$ (continuous line), $\beta =3.2$ (doted line), and $\beta =10$ (dashed line).

Figure 3

LNEF branch of TOL gravity: $f_{\mathrm{TOL}}(\rho )$ vs $\rho$ for $l=2$, $q=0.1$, $m=0.005$, and $d=7$. Left panel: $\beta =10$, ${\alpha }_2=0.01$ (continuous line), ${\alpha }_2=0.03$ (doted line), and ${\alpha }_2=0.06$ (dashed line). Right panel: ${\alpha }_2=0.01$, $\beta =2.6$ (continuous line), $\beta =3.2$ (doted line), and $\beta =10$ (dashed line).

Figure 4

$\delta \varphi /\pi$ vs $\beta$ for $d=4$, $l=1$ and $r_{+}=2$. Left panel (ENEF): $q=1$ (continuous line), $q=2$ (doted line), and $q=3$ (dashed line). Right panel (LNEF): $q=1$ (continuous line), $q=2$ (doted line), and $q=3$ (dashed line).

Figure 5

$\delta \varphi /\pi$ vs $\beta$ for $d=4$, $l=1$, and $q=1$. Left panel (ENEF): $r_{+}=1$ (continuous line), $r_{+}=1.2$ (doted line), and $r_{+}=1.4$ (dashed line). Right panel (LNEF): $r_{+}=1$ (continuous line), $r_{+}=1.2$ (doted line), and $r_{+}=1.4$ (dashed line).

Figure 6

$\delta \varphi /\pi$ vs $r_{+}$ for $d=4$, $l=2$ and $q=1$. Left panel (ENEF): $\beta =2$ (continuous line), $\beta =3$ (doted line), and $\beta =5$ (dashed line). Right panel (LNEF): $\beta =1.1$ (continuous line), $\beta =1.5$ (doted line), and $\beta =5$ (dashed line).

Figure 7

$\delta \varphi /\pi$ vs $r_{+}$ for $\beta =4$, $l=1$, and $q=1$. Left panel (ENEF): $d=5$ (continuous line), $d=7$ (doted line), and $d=11$ (dashed line). Right panel (LNEF): $d=5$ (continuous line), $d=7$ (doted line), and $d=11$ (dashed line).

Figure 8

$\delta \varphi /\pi$ vs $\beta$ for $r_{+}=2$, $l=1$ and $q=1$. Left panel (ENEF): $d=5$ (continuous line), $d=7$ (doted line), and $d=11$ (dashed line). Right panel (LNEF): $d=5$ (continuous line), $d=7$ (doted line), and $d=11$ (dashed line).

Figure 9

$\delta \varphi /\pi$ vs $\eta$ for $l=1$ and $d=4$. Left panel: $r_{+}=2$, $q=2$ (continuous line), $q=2.5$ (doted line), and $q=3$ (dashed line). Right panel: $q=3$, $r_{+}=2$ (continuous line), $r_{+}=2.1$ (doted line), and $r_{+}=2.2$ (dashed line).

Figure 10

$\delta \varphi /\pi$ vs $r_{+}$ for $l=1$. Left panel: $q=3$, $d=4$, $\eta =0.02$ (continuous line), $\eta =0.05$ (doted line), and $\eta =0.08$ (dashed line). Right panel: $q=1$, $\eta =0.05$, $d=5$ (continuous line), $d=8$ (doted line), and $d=11$ (dashed line).

Figure 11

$\delta \varphi /\pi$ vs $r_{+}$ for $l=1$, $q=4$ and $d=4$. Left panel: Maxwell case (continuous line), ENEF case for $\beta =5$ (doted line), and LNEF case for $\beta =5$ (dashed line), respectively. Right panel: Maxwell case (bold line), MC case for $\eta =0.02$ (doted line), $\eta =0.05$ (dashed line), and $\eta =0.08$ (continuous line).