Cloud of strings in third order Lovelock gravity

Lovelock theory is a natural extension of the Einstein theory of general relativity to higher dimensions in which the first and second orders correspond, respectively, to general relativity and Einstein–Gauss–Bonnet gravity. We present exact black hole solutions of $D\ge 4$-dimensional spacetime for first-, second-, and third-order Lovelock gravities in a string cloud background. Further, we compute the mass, temperature, and entropy of black hole solutions for the higher-dimensional general relativity and Einstein–Gauss–Bonnet theories and also perform thermodynamic stability of black holes. It turns out that the presence of the Gauss–Bonnet term and/or background string cloud completely changes the black hole thermodynamics. Interestingly, the entropy of a black hole is unaffected due to a background string cloud. We rediscover several known spherically symmetric black hole solutions in the appropriate limits.

Figure 1

Plot of metric function $f(r)$ vs $r$ in various dimensions with $M=1$ for the Einstein–Gauss–Bonnet case. (Left): ${\alpha }_2=0.3$ and $a=0.2,0.4,0.6$, and 0.8 (top to bottom). (Right): ${\alpha }_2=0.8$ and $a=0.2,0.4,0.6$, and 0.8 (top to bottom).

Figure 2

The temperature vs horizon radius $r_{+}$ for the Einstein–Gauss–Bonnet case, with $M=1$, in various dimensions $D=5,6,7$, and 8 (top to bottom). (Left): ${\alpha }_2=0.3$ and $a=0.2,0.4,0.6$, and 0.8. (Right): ${\alpha }_2=0.8$ and $a=0.2,0.4,0.6$, and 0.8.

Figure 3

The specific heat vs horizon radius $r_{+}$ for the Einstein–Gauss–Bonnet case, with $M=1$, in various dimensions $D=5,6,7$, and 8 (top to bottom). (Left): ${\alpha }_2=0.3$ and $a=0.2,0.4,0.6$, and 0.8. (Right): ${\alpha }_2=0.8$ and $a=0.2,0.4,0.6$, and 0.8.

Figure 4

Plot of metric function $f(r)$ vs $r$ for the general Lovelock gravity (${\alpha }_2\ne {\alpha }_3\ne 0$) with ${\alpha }_3=1$ and $M=1$ in various dimensions. (Left): ${\alpha }_2=0.3$ and $a=0.2$ and 0.6 (top to bottom). (Right): ${\alpha }_2=0.8$ and $a=0.2$ and 0.6 (top to bottom).

Figure 5

Plot of metric function $f(r)$ vs $r$ for the general Lovelock gravity (${\alpha }_2\ne {\alpha }_3\ne 0$) with ${\alpha }_2=1$ and $M=1$ in various dimensions. (Left): ${\alpha }_3=0.3$ and $a=0.2$ and 0.6 (top to bottom). (Right): ${\alpha }_3=0.8$ and $a=0.2$ and 0.6 (top to bottom).

Figure 6

Plot of metric function $f(r)$ vs $r$ for the ELL case with ${\alpha }_3=1$ and $M=1$ in various dimensions. (Left): ${\alpha }_2=0.3$ and $a=0.2$ and 0.6 (top to bottom). (Right): ${\alpha }_2=0.8$ and $a=0.2$ and 0.6 (top to bottom).

Figure 7

Plot of metric function $f(r)$ vs $r$ for the ELL case with ${\alpha }_2=1$ and $M=1$ in various dimensions. (Left): ${\alpha }_3=0.3$ and $a=0.2$ and 0.6 (top to bottom). (Right): ${\alpha }_3=0.8$ and $a=0.2$ and 0.6 (top to bottom).