Topological classes of black holes in de-Sitter spacetime

Abstract

In this paper, we investigate the topological number of de-Sitter black hole solutions with different charges (q) and rotational (a) parameters. By using generalized free energy and Duan’s $\phi$-mapping topological current theory, we find that the topological numbers of black holes can still be classified into three types. In addition, we interestingly found the topological classes for de-Sitter (dS) spacetime with distinct horizons, i.e., black hole event horizon and cosmological horizon, will be different. Moreover, we also investigate topological classifications of dS black hole solutions in higher dimensions with or without the Gauss-Bonnet term.

1 Introduction

The black hole is one of the most fascinating objects in the universe and has received increased attention both in theoretical and observational physics. The classical properties of black holes can be well described by general relativity (GR). However, the singularity theorem [1] indicates that GR has intrinsic limitations. Particularly, when one considers the quantum effect, people found that black hole is not black at all [2] and surprisingly radiate like a black body and have rich thermodynamical properties [3,4,5]. Under this framework, many issues such as the information paradox arise and are discussed [6].

Very recently, based on Duan’s topological current $\phi$-mapping theory [7], Wei et al. [8, 9] innovatively proposed that black holes can be viewed as topological thermodynamic defects, and successfully classify different black hole solutions with their global topological charges. The black hole solutions can be divided into three different topological classes according to their different topological numbers. This method offers us a new perspective on black hole solutions and their thermodynamic property.

Along this line, many works continue to discuss the topological classes of different types of black holes. To go beyond the static solutions, in [10], stationary black holes are considered. The topological number of Kerr and Kerr-Newman black holes are calculated respectively. Moreover, they also calculate the singly-rotating black hole in higher dimensions as well as in the Anti-de-Sitter case [10, 11]. Topological charge and phase transition of AdS black hole are also explored in [12]. In addition to research on topology classes in general relativity, some researchers are also extending these classifications into modified gravity theories such as Gauss-Bonnet gravity [13,14,15] or Lovelock gravity [16]. The topological numbers are quite different in these cases, which in turn provides us with a new perspective to consider the difference between GR and modified gravity.

Since the study of the topological classifications of black holes are still in their infancy and the topological number of black holes de-Sitter(dS) spacetime remains a virgin territory, it deserves to be explored deeply. Our current universe is in a state of accelerated expansion. The simplest explanation of such accelerated expansion is a positive cosmological constant [17]. More importantly, in dS spacetime there are two kinds of horizons, both emitting Hawking radiation [18, 19], different from the thermodynamics of flat or Anti de-Sitter(AdS) spacetime. Thus investigating dS spacetime has theoretical and practical significance [21]. Extending these approaches to the dS case may provide us with some new insight into these universal phenomena. With these strong motivations in hand, in this paper, we follow this newly-hewn path to study the topological classes of black holes in dS spacetime.

At the same time, many works about modified gravity, such as adding higher curvature terms in GR or considering higher dimensions, have been done to reconcile GR with other quantum theories. As a result, de-Sitter spacetime also exists in many modified gravity theories. In this article, we will concentrate on the thermodynamics of a dS black hole including its event horizon and cosmological horizon which can be seen as two different thermodynamic systems [19] and then calculate the topological charge when the angular momentum and electric charge are taken different values. To get more information, we subsequently explore dS black holes in different backgrounds, e.g. the Gauss-Bonnet (GB) gravity and the higher dimensional case.

This paper is organized as follows: In Sect. 2, we briefly review some useful results of thermodynamics of asymptotically de-Sitter spacetimes. Then we use the generalized free energy to establish a parameter space, and find the zero points with their topological charge. In Sect. 3, we follow the same step and get the topological number of high dimensional dS black holes with or without the Gauss-Bonnet term. Finally, we summarize our results in Sect. 4 and made a comparison with Anti-de-Sitter and flat cases.

2 Topological classes of four dimensional dS black hole solutions

2.1 Thermodynamics of four dimensional dS black hole

In this section, we first give a brief review of thermodynamics of dS black hole. We use the generalized off-shell free energy [22] so that we could classify different black hole solutions. This means black holes with the same energy (and electric charge or angular momentum, if any) can be in different temperatures. The Kerr-Newman de-Sitter metric can be written in the Boyer-Lindquist type coordinates as follows [21, 23]

$$\begin{aligned} d s^2= & {} -\frac{\Delta _r}{R^2}\left( d t-\frac{a}{\Xi } \sin ^2 \Theta d \Phi \right) ^2+R^2\left( \frac{d r^2}{\Delta _r}+\frac{d \Theta ^2}{\Delta _\theta }\right) \nonumber \\{} & {} \quad +\frac{\Delta _\Theta \sin ^2 \Theta }{R^2}\left( a d t-\frac{r^2+a^2}{\Xi } d \Phi \right) ^2, \end{aligned}$$

(2.1)

where

$$\begin{aligned}{} & {} R^2=r^2+a^2 \cos ^2 \Theta , \quad \Xi =1+\frac{a^2}{L^2}, \end{aligned}$$

(2.2)

$$\begin{aligned}{} & {} \Delta _r=\left( r^2+a^2\right) \left( 1-\frac{r^2}{L^2}\right) -2 m r+q^2, \end{aligned}$$

(2.3)

$$\begin{aligned}{} & {} \Delta _\theta =1+\frac{a^2}{L^2} \cos ^2 \Theta , \quad \frac{1}{L^2}=\frac{\Lambda }{3}. \end{aligned}$$

(2.4)

Here m, a and q respectively represent the mass, rotating parameter and electric charge of the dS black hole. L is the radius of curvature of spacetime and satisfy $L^{2}=3/\Lambda$ with $\Lambda$ being the positive cosmological constant. Throughout the paper, we use natural units, namely $\hbar =c=G=1$ as well as $k_{B}=1$. The metric (2.1) solves the Einstein-Maxwell equations with electromagnetic potential given by

$$\begin{aligned} A_{t}=\frac{q r}{R^{2}}, \quad A_{\phi }=\frac{q r}{R^{2} \Xi } a \sin ^{2} \Theta . \end{aligned}$$

(2.5)

The metric is singular where $\Delta _{r}$ vanishes. The algebraic equation $\Delta _{r}=0$ has four roots which are three positive solutions and one negative solution in the condition that the relation

$$\begin{aligned}{} & {} {\left[ \left( L^{2}-a^{2}\right) ^{2}-12 L^{2}\left( a^{2}+q^{2}\right) \right] ^{3}}\nonumber \\{} & {} \quad > \Bigg [\left( L^{2}-a^{2}\right) ^{3}+36 L^{2}\left( L^{2}-a^{2}\right) \nonumber \\{} & {} \qquad \times \left( a^{2}+q^{2}\right) -54 m^{2} L^{4}\Bigg ]^{2} \end{aligned}$$

(2.6)

The largest positive solution is the cosmological horizon $r_{c}$, the smallest positive solution is the inner black hole horizon, and the other positive solution is the black hole event horizon $r_{h}$. The negative solution has no physical meaning. In this paper, we assume that Eq.(2.6) is satisfied, and only focus on the event horizon and cosmological horizon. It has been known for a long time that if using the Euclidean solution to investigate thermodynamics of dS black hole solution, one finds that the imaginary time periods required to avoid conical singularity at two horizons do not match. So we cannot remove the singularities of dS black hole solutions at the same time. An observer located in this spacetime would receive the radiation from black hole event horizon as well as cosmological horizon. The way to handle this problem is to consider the two horizons as different thermal states and discuss them separately [18, 19]. When one observes either one of two horizons as a thermodynamic system, then the other should be viewed as a boundary where the parameters are fixed and there will be no field equations to satisfy. Once one type of spacetime manifold is set, its Euclidean action as well as thermodynamical quantities can be obtained. For Kerr de Sitter spacetime, mass and angular momentum are given by [19] as

$$\begin{aligned}{} & {} M_{h}=\frac{m_{k}}{\Xi ^{2}}, \quad J_{h}=\frac{m_{k} a_{k}}{\Xi ^{2}},\end{aligned}$$

(2.7)

$$\begin{aligned}{} & {} M_{c}=-\frac{m_{k}}{\Xi ^{2}}, \quad J_{c}=-\frac{m_{k} a_{k}}{\Xi ^{2}} \end{aligned}$$

(2.8)

where subscript h means that it is the physical quantity associated with the black hole event horizon and c for the cosmological horizon. k means it is the quantity of Kerr spacetime. An interesting observation is that if one replaces $l^{2}$ with $-l^{2}$, the conserved charges for dS black hole event horizon agree with the results of anti-de Sitter black hole, which are first obtained in [20]. Also, conserved charges for the cosmological horizon are different from those for black hole horizon only its signs. These two facts make the authors of [23] deduce that in KN-dS spacetime, the conserved charges for black hole horizon and cosmological horizon take the form of

$$\begin{aligned} M_{h}= & {} \frac{m}{\Xi ^{2}}, \quad J_{h}=\frac{m a}{\Xi ^{2}}, \quad Q_{h}=\frac{q}{\Xi },\end{aligned}$$

(2.9)

$$\begin{aligned} M_{c}= & {} -\frac{m}{\Xi ^{2}}, \quad J_{c}=-\frac{m a}{\Xi ^{2}}, \quad \nonumber \\ Q_{c}= & {} -\frac{q}{\Xi }, \end{aligned}$$

(2.10)

and cosmological constant should also be taken differently as

$$\begin{aligned} \quad \Lambda _{h}=\Lambda , \quad \Lambda _{c}=-\Lambda . \end{aligned}$$

(2.11)

The rationality of taking these forms is as follow. In [20], $\Lambda$ or l has been seen as a variable so that the modified Bekenstein-Smarr mass formula

$$\begin{aligned} M^{2}= & {} \frac{S}{4 \pi }+\frac{\pi }{4 S}(4 J^{2}+Q^{4})+\frac{J^{2}}{l^{2}}+\frac{Q^{2}}{2}\nonumber \\{} & {} \quad +\frac{S}{2 \pi l^{2}}\left( Q^{2}+\frac{S}{\pi }+\frac{S^{2}}{2 \pi ^{2} l^{2}}\right) \end{aligned}$$

(2.12)

still holds in AdS black hole thermodynamics (S in the above equation represents the entropy of AdS black hole). By the same account, in the dS case we also have to consider $\Lambda _{h}$ and $\Lambda _{c}$ as dynamical variables of the thermodynamic systems. And [23] concludes that only when the total electrical charges are taken the forms of (2.9) and (2.10), the generalized Smarr formula for KN-dS black hole event horizon and cosmological horizon can be realized. By virtue of analytical continuation of Lorentzian metric by $t \rightarrow -i\tau$, $a\rightarrow ia$ and identifying the Euclidean metric, the entropy of the two thermodynamic systems can be obtained respectively as [23]

$$\begin{aligned} S_{h}=\frac{\pi (r_{h}^{2}+a^2)}{\Xi }, \quad S_{c}=\frac{\pi (r_{c}^{2}+a^2)}{\Xi }. \end{aligned}$$

(2.13)

For KN-dS black hole horizon, $M_{h}$ plays the role of energy U. So the generalized Helmholtz free energy, defined as $F=U-S/\tau$ [22], can be written as

$$\begin{aligned} F_{h}= & {} M_{h}-\frac{S_{h}}{\tau _{h}}=\frac{a^2 L^2-a^2 r_{h}^2+L^2 q^2+L^2 r_{h}^2-r_{h}^4}{2 L^2 r_{h} \left( \frac{a^2}{L^2}+1\right) ^2}\nonumber \\{} & {} \quad -\frac{\pi \left( a^2+r_{h}^2\right) }{\tau _{h} \left( \frac{a^2}{L^2}+1\right) }. \end{aligned}$$

(2.14)

Following the same step we get the generalized free energy of cosmological horizon as

$$\begin{aligned} F_{c}= & {} M_{c}-\frac{S_{c}}{\tau _{c}}=-\frac{a^2 L^2-a^2 r_{c}^{2}+L^2 q^2+L^2 r_{c}^2-r_{c}^4}{2 L^2 r_{c} \left( \frac{a^2}{L^2}+1\right) ^2}\nonumber \\{} & {} \quad -\frac{\pi \left( a^2+r_{c}^2\right) }{\tau _{c} \left( \frac{a^2}{L^2}+1\right) }. \end{aligned}$$

(2.15)

They are off-shell except at $\tau =1/T$ which means the systems are in the maximal mixed state (T represents the Hawking temperature of the horizons).

2.2 Topological classes of dS black hole event horizon

When the generalized free energy is in hand, we could apply the method in [8] to establish a parameter space and find the zero point of the vector field in it. Profoundly, the zero points are exactly corresponding to the on-shell black hole solution. We can calculate the topological number of them by virtue of Duan’$\phi$-mapping topological current theory [7]. The number can be seen as a characteristic value of the on-shell black hole solution. Following the spirit of [8], we define the vector as

$$\begin{aligned} \phi =(\phi ^{r_{h}},\phi ^{\theta })=\left( \frac{\partial F}{\partial r_{h}},-\cot \theta \csc \theta \right) \end{aligned}$$

(2.16)

with $\theta \in [0,\pi ]$ for convenience. The vector field is on $\theta -r_{h}$ space, and we can see that $\phi ^{\theta }$ is divergent when $\theta =0,\pi$, making the direction of vectors point vertically outward at this boundary. The zero point, corresponding to $\tau =1/T$ [9], can only be obtained when $\theta =\pi /2$. Now we introduce the topological current as

$$\begin{aligned} j^\mu =\frac{1}{2 \pi } \epsilon ^{\mu \nu \rho } \epsilon _{a b} \partial _\nu n^a \partial _\rho n^b, \quad \mu , \nu , \rho =0,1,2 \end{aligned}$$

(2.17)

where $\partial _\nu =\left( \partial / \partial x^\nu \right)$ and $x^\nu =\left( \tau , r_h, \theta \right)$. The unit vector is defined as $n^a=\left( \phi ^a /\Vert \phi \Vert \right) (a=1,2)$ by its Euclidean metric. The conservation law of the current, $\partial _\mu j^\mu =0$ is obvious according to the definition of $j^\mu$, and $\tau$ here serves as a time parameter of the topological defect. By using the Jacobi tensor $\epsilon ^{a b} J^\mu (\phi / x)=$ $\epsilon ^{\mu \nu \rho } \partial _\nu \phi ^a \partial _\rho \phi ^b$ and the two-dimensional Laplacian Green function $\Delta _{\phi ^a} \ln \Vert \phi \Vert =2 \pi \delta ^2(\phi )$, the topological current can be written as

$$\begin{aligned} j^\mu =\delta ^2(\phi ) J^\mu \left( \frac{\phi }{x}\right) . \end{aligned}$$

(2.18)

where $j^\mu$ is nonzero only at $\phi ^a\left( x^i\right) =0$, and we denote its i-th solution as $\textbf{x}=\textbf{z}_i$. The topological current density then reads [24]

$$\begin{aligned} j^0=\sum _{i=1}^N \beta _i \eta _i \delta ^2\left( \textbf{x}-\textbf{z}_i\right) , \end{aligned}$$

(2.19)

where $\beta _i$ is the Hopf index, which counts the number of the loops that $\phi ^a$ makes in the vector $\phi$ space when $x^\mu$ goes around the zero point $z_i$. Thus Hopf index is always positive. $\eta _i$ is the Brouwer degree and satisfy $\eta _i=$ ${\text {sign}}\left( J^0(\phi / x)_{z_i}\right) =\pm 1$. Given a parameter region $\Sigma$, the corresponding topological number can be obtained as

$$\begin{aligned} W=\int _{\Sigma } j^0 d^2 x=\sum _{i=1}^N \beta _i \eta _i=\sum _{i=1}^N w_i, \end{aligned}$$

(2.20)

where $w_i$ is the winding number for the i-th zero point of $\phi$ contained in $\Sigma$ and its value does not depend on the shape of the region where we perform the calculation. Usually, distinct zero points of the vector field are isolated, making Jacobian $J_0(\phi / x) \ne 0$. If Jacobian $J_0(\phi / x) = 0$, it means that the defect bifurcates [25]. Equation (2.20) shows that in any given region, the global topological number is the sum of the winding number of each zero point which reflects the local property of the topological defect. Based on the approach introduced above, we would investigate dS black holes with different horizons.

Fig. 1

Solution curves in $\tau _{h}-r_{h}$ plane with different values of $\Lambda$, where k is an arbitrary positive constant. It can be seen that for dS black hole the curves are always monotonically increasing which is the same as $\Lambda =0$ case. But for AdS black hole, the curve go down to zero so there are two zero points for a given $\tau$

Next, we consider the dS black hole as a thermodynamical object in a “box”, the boundary of which is at the cosmological horizon $r_{c}$. We first concentrate on $a=0$ and $q=0$ cases whose generalized free energy is reduced to

$$\begin{aligned} F_{h}=M_{h}-\frac{S_{h}}{\tau _{h}}=\frac{L^2 r_{h}-r_{h}^3}{2 L^2 }-\frac{\pi r_{h}^2}{\tau _{h} }. \end{aligned}$$

(2.21)

We define the vector field as

$$\begin{aligned} \phi= & {} \left( \phi ^{r_{h}},\phi ^{\theta }\right) \nonumber \\= & {} \left( -\frac{3 r_h^2}{2 L^2}-\frac{2 \pi r_h}{\tau _{h} }+\frac{1}{2},-\cot \theta \csc \theta \right) . \end{aligned}$$

(2.22)

By solving the equation $\phi =0$, we acquire the relation

$$\begin{aligned} \tau _{h}=\frac{4 \pi L^2 r_h}{L^2-3 r_h^2}, \quad \theta =\frac{\pi }{2}. \end{aligned}$$

(2.23)

We take different values of $\Lambda$ and depict the on-shell solution curve on $\tau _{h}-r_{h}$ plane as shown in Fig. 1. When $\tau _{h}\approx 5.16 k$, different from the Schwarzchild-AdS case, there would be one zero point in $\theta -r_{h}$ plane at $(r_{h}/k,\theta )=(0.3,\pi /2)$, which is illustrated in Fig. 2. The loop surrounding the zero point sets the boundary of a given region so we can use Eq. (2.20) to acquire the winding number of point P. In this case, it can be easily calculated as $w=-1$. Its sign is the same as the heat capacity of Schwarzchild-dS black hole [18]. Therefore, as same as the result in [8], the stability of the black hole solution can be read through its winding number. Since there is only one defect in the parameter space, it gives rise to the global topological number $W=-1$ for the non-rotating and uncharged Schwarzchild-dS black hole solution.

Fig. 2

Unit vector field in parameter space with $a=0$, $q=0$ and $\Lambda =3/k^{2}$ Schwarzchild-dS solution as $\tau _{h}\approx 5.16 k$. P marked with black dot at $(r_{h}/k,\theta )=(0.3,\pi /2)$ is the zero point of the vector field. The black contour is a closed loop enclosing the zero point and we can performing the calculation of Eq. (2.20) inside it. The shape and the size of the loop will not affect the winding number

Now we would further explore the rotating cases ($a\ne 0, q=0$). The generalized free energy reads

$$\begin{aligned} F_{h}=\frac{-a^2 r_h^2+a^2 L^2+L^2 r_h^2-r_h^4}{2 L^2 \left( \frac{a^2}{L^2}+1\right) ^2 r_h}-\frac{\pi \left( a^2+r_h^2\right) }{\tau _{h} \left( \frac{a^2}{L^2}+1\right) }. \end{aligned}$$

(2.24)

As a result, we define the vector field as

$$\begin{aligned} \phi =\left( \phi ^{r_{h}},\phi ^{\theta }\right) =\left( -\frac{L^2 \left( \tau _{h} \left( a^2-L^2\right) r_h^2+4 \pi \left( a^2+L^2\right) r_h^3+a^2 L^2 \tau _{h} +3 \tau _{h} r_h^4\right) }{2 \tau _{h} \left( a^2+L^2\right) ^2 r_h^2},-\cot \theta \csc \theta \right) . \end{aligned}$$

(2.25)

Solving the equation $\phi =0$ give us

$$\begin{aligned} \tau _{h}=-\frac{4 \pi \left( a^2+L^2\right) r_h^3}{\left( a^2-L^2\right) r_h^2+a^2 L^2+3 r_h^4}, \quad \theta =\frac{\pi }{2}. \end{aligned}$$

(2.26)

Fig. 3

Solution curves in $\tau -r_{h}$ plane with different values of $\Lambda$ and a. There is always one turning point in the curve as long as $a\ne 0$

Fig. 4

Vector field in $\theta -r_{h}$ plane with $a\ne 0$. We take $\tau =6.25k$. The points marked in black dot are zero points of the field. They are respectively at $P_{1}=(r_{h}/k,\theta )=(0.12,\pi /2)$ and $P_{2}=(r_{h}/k,\theta )=(0.3,\pi /2)$

Upon the value of $\Lambda$ and a are determined, we could acquire a curve in $\tau _{h}-r_{h}$ plane. For instance, as shown in Fig. 3, when taking $\Lambda =3/k^{2}$ and $a=k/10$, the blue curve goes down rapidly to a peak at $(r_{h},\tau _{h})\approx (0.17k,3.87k)$ and climbs as $r_{h}$ gets larger. Consequently, the vector field possesses two zero points for large $\tau _{h}$, in contrast to one zero point for Schwarzschild-dS black hole case. As $\tau _{h}=6.25k$, the two intersection points are respectively at $r_{h}\approx 0.12k$ and $r_{h}\approx 0.3k$. We illustrate the vector field and the zero points in Fig. 4. When $\tau =\tau _{cri}\approx 3.87k$, the intersection points coincide, and for smaller $\tau _{h}$, annihilate. It is easy to check the critical point satisfy $d^{2}\tau _{h}/dr_{h}^{2}>0$, which belongs to generation point. For $\tau >\tau _{cri}$, we find that the winding number of the two zero points are $w_{1}=1$ and $w_{2}=-1$ respectively. Thus the global topological number for Kerr-dS solution is $W=w_{1}+w_{2}=0$, different from non-rotating case.

When taking electric charge into account, the generalized off-shell free energy of dS black hole is given by Eq. (2.14). Following the same step, we get the on-shell solution curve in $\tau _{h}-r_{h}$ plane, which satisfies

$$\begin{aligned} \tau _{h}=-\frac{4 \pi \left( a^2+L^2\right) r_h^3}{\left( a^2-L^2\right) r_h^2+L^2 \left( a^2+q^2\right) +3 r_h^4}. \end{aligned}$$

(2.27)

Fig. 5

Solution curves in $\tau _{h}-r_{h}$ plane with different values of $\Lambda$, a and q. We found that non-trivial q does not bring the essential difference to the trend of the curve compared with $a\ne 0$ but $q=0$ case

Interestingly, we find that the electric charge does not change the rough trend of the curve. As shown in Fig. 5, there is invariably one generation point in $\tau _{h}-r_{h}$ plane, and $\tau _{cri}$ merely gets larger as the value of q taken larger. There are two zero points as well. We calculate the winding number of them for a given $\tau _{h}$ and find $w_{1}=1$, $w_{2}=-1$. See Fig. 6 as an example. The global topological number of Kerr-Newman-dS black hole solution is $W=0$, which is the same as Kerr-dS black hole. We also calculate the non-rotating charged cases whose global topological number is $W=w_{1}+w_{2}=0$. Hence from the perspective of topological charge, these three kinds of black holes are just the same. Moreover, compared with the negative cosmological constant case, for instance, the charged Reissner-Nordstrom Anti-de-Sitter (RN-AdS) black hole in four dimensions [8], RN-dS black hole solution evidently has fewer zero points and the topological number are distinct. We plot their on-shell solution curves in Fig. 7. We could conclude the sign of cosmological constant may have an influence on the topological number and quantity of defects of black holes of the same type.

Fig. 6

Vector field in $\theta -r_{h}$ plane with $a\ne 0$ and $q\ne 0$. We take $\tau _{h}\approx 55.8k$. The points marked in the black dot are zero points of the field. They are respectively at $P_{1}=(r_{h}/k,\theta )\approx (0.27,\pi /2)$ and $P_{2}=(r_{h}/k,\theta )\approx (0.45,\pi /2)$

Fig. 7

On-shell charged solution curves for different cosmological constant. The trend of the curves are the same when $r_{h}$ is small, but look different as $r_{h}$ gets larger

2.3 Topological classes of dS solutions with cosmological horizon

In the following, we shall discuss thermodynamics and the corresponding topological charge of cosmological horizon. Now the black hole event horizon $r_{c}$ serves as a boundary, which plays the analogous role of coordinate origin of empty de Sitter space [23]. The form of generalized free energy of the cosmological horizon resembles the black hole event horizon counterpart. For $a=0$, $q=0$ case, it reads

$$\begin{aligned} F_{c}=-\frac{L^2 r_c^2-r_c^4}{2 L^2 r_c}-\frac{\pi r_c^2}{\tau _c}. \end{aligned}$$

(2.28)

The vector field can be defined as

$$\begin{aligned} \phi =\left( \phi ^{r_{c}},\phi ^{\theta }\right) =\left( \frac{3 r_c^2}{2 L^2}-\frac{2 \pi r_c}{\tau _c}-\frac{1}{2},-\cot \theta \csc \theta \right) .\nonumber \\ \end{aligned}$$

(2.29)

By solving the equation $\phi =0$, we acquire the relation

$$\begin{aligned} \tau _{c}=-\frac{4 \pi L^2 r_c}{L^2-3 r_c^2}. \end{aligned}$$

(2.30)

Fig. 8

Solution curves in $\tau _{c}-r_{c}$ plane with different values of $\Lambda$. It can be seen that the curves are always monotonically decreasing

We could see that the only difference of the $r-\tau$ relation with the black hole event horizon is the sign of Eq. (2.30). But it makes the winding number distinct as we will show. We take various values of $\Lambda$ and depict the on-shell solution curve on $\tau _{c}-r_{c}$ plane as shown in Fig. 8. When $\tau _{c}\approx 8\pi k/11$, there would be one zero point in $\theta -r_{c}$ plane at $(r_{c}/k,\theta )=(2,\pi /2)$, which is illustrated in Fig. 9. Under Eq. (2.20), we get the winding number of the single zero point $w=1$ which is different from the non-rotating uncharged black hole event horizon case. Given the heat capacity of the cosmological horizon is negative, the winding number can no longer characterize the stability of the solution.

Fig. 9

Unit vector field in parameter space with $a=0$, $q=0$ and $\Lambda =3/k^{2}$ solution as $\tau _{c}\approx 8\pi k/11$. P marked with black dot at $(r_{c}/k,\theta )=(2,\pi /2)$ is the zero point of the vector field. We perform the calculation of Eq. (2.20) inside the black contour and obtain the winding number $w=1$

Fig. 10

Solution curves in $\tau _{c}-r_{c}$ plane with different values of a and q, with $\Lambda =3/k^2$

Fig. 11

Unit vector field in parameter space with $a=k/5$, $q=k/10$ and $\Lambda =3/k^{2}$ solution as $\tau _{c}\approx 10k$. $P_{1}$ and $P_{2}$ are marked with black dot at $(r_{c}/k,\theta )\approx (0.21,\pi /2)$ and $(r_{c}/k,\theta )\approx (0.81,\pi /2)$. We find that $w_{1}=-1$ and $w_{2}=1$. It makes the global topological number $W=0$

For $a\ne 0$, $q\ne 0$, or rotating and charged case, we find that there are invariably two zero points in the vector field. Although the global topological charge is $W=0$ again, the winding number of defects with a smaller radius, $w_{1}=-1$, or with a larger radius, $w_{2}=1$ is respectively different with black hole event horizon case. We plot the solution curve in Fig. 10 and calculate the topological charge when taking $\tau _{c}=10k$, $a=k/5$, $q=k/10$ as well as $\Lambda =3/k^{2}$ as an example, as shown in Fig. 11.

3 Topological classes of higher dimensional dS black hole solutions with Gauss-Bonnet term

To gain more knowledge of topological classes of dS black holes, we further explore high dimensional solutions and add higher derivative curvature terms in the form of Gauss-Bonnet gravity. For simplicity, we only focus on uncharged static black hole solutions. The most general d-dimensional spherically symmetrical metric solution reads [26, 27]

$$\begin{aligned} d s^{2}=-f(r) d t^{2}+\frac{d r^{2}}{f(r)}+r^{2} d \Omega _{d-2}^{2}, \end{aligned}$$

(3.1)

where

$$\begin{aligned}{} & {} f(r)=1+\frac{r^{2}}{2 \alpha }\mp \frac{r^{2-d / 2} \sqrt{r^{d}+4 \alpha \left( r \omega _{d-3}+r^{d} / L^{2}\right) }}{2 \alpha },\nonumber \\ \end{aligned}$$

(3.2)

$$\begin{aligned}{} & {} \omega _{d-3}=\frac{16 \pi M}{(d-2) \Omega _{d-2}}, \end{aligned}$$

(3.3)

$\alpha$ represents Gauss-Bonnet coefficient and $L^2=(d-1) (d-2)/2 \Lambda$. M is an integration constant which is the total mass of the solution [26]. According to [26, 28], the branch with $''+''$ in (3.2) is unstable while $''-''$ branch is stable and the graviton is free of ghost. Thus in the following, we merely concentrate on the solution branch with $''-''$ sign which is of much physical interest.

As $f(r)=0$, there are two positive real roots which can be seen as two horizons of the spacetime. Just as the case in KN-dS solution, the large root represents the cosmological horizon, $r_{c}$ and the small one event horizon, $r_{h}$. Additionally, the two horizons emitting radiation are also at distinct temperature. So we have to think of it as two separated thermodynamic systems. [26] gives the specific expressions of the total energy of two system as

$$\begin{aligned} U_{h}= & {} M = \frac{(d-2) \Omega _{d-2} r_{h}^{d-3}}{16 \pi }\left( 1+\frac{\alpha }{r_{h}^{2}}-\frac{r_{h}^{2}}{L^{2}}\right) \end{aligned}$$

(3.4)

$$\begin{aligned} U_{c}= & {} -M = -\frac{(d-2) \Omega _{d-2} r_{c}^{d-3}}{16 \pi }\left( 1+\frac{\alpha }{r_{c}^{2}}-\frac{r_{c}^{2}}{L^{2}}\right) \end{aligned}$$

(3.5)

and the entropy of the two systems are

$$\begin{aligned} S_{h}= & {} \frac{\Omega _{d-2} r_{h}^{d-2}}{4}\left( 1+\frac{2(d-2) \alpha }{(d-4) r_{h}^{2}}\right) , \end{aligned}$$

(3.6)

$$\begin{aligned} S_{c}= & {} \frac{\Omega _{d-2} r_{c}^{d-2}}{4}\left( 1+\frac{2(d-2) \alpha }{(d-4) r_{c}^{2}}\right) . \end{aligned}$$

(3.7)

So we get the generalized free energy, $F=U-S/\tau$, as

$$\begin{aligned} F_{h}= & {} \frac{(d-2) \Omega _{d-2} r_{h}^{d-3}}{16 \pi }\left( 1+\frac{\alpha }{r_{h}^{2}}-\frac{r_{h}^{2}}{L^{2}}\right) \nonumber \\{} & {} -\frac{\Omega _{d-2} r_{h}^{d-2}}{4\tau _{h}}\left( 1+\frac{2(d-2) \alpha }{(d-4) r_{h}^{2}}\right) , \end{aligned}$$

(3.8)

$$\begin{aligned} F_{c}= & {} -\frac{(d-2) \Omega _{d-2} r_{c}^{d-3}}{16 \pi }\left( 1+\frac{\alpha }{r_{c}^{2}}-\frac{r_{c}^{2}}{L^{2}}\right) \nonumber \\{} & {} -\frac{\Omega _{d-2} r_{c}^{d-2}}{4\tau _{c}}\left( 1+\frac{2(d-2) \alpha }{(d-4) r_{c}^{2}}\right) . \end{aligned}$$

(3.9)

We first calculate the topological classes of black hole event horizon. Following the same step, the vector field can be defined as Eq. (2.16). So we have

$$\begin{aligned} \phi ^{r_{h}}=-\frac{r_{h}^{d-6} \Omega _{d-2} \left( \tau _h \left( -\alpha \left( d^2-7 d+10\right) -\left( d^2-5 d+6\right) r_{h}^2+2 \Lambda r_{h}^4\right) +4 \pi (d-2) r_{h} \left( 2 \alpha +r_{h}^2\right) \right) }{8 \pi \tau _h}. \end{aligned}$$

(3.10)

Fig. 12

On-shell solution curve of uncharged black hole for different $\alpha$ in AdS or dS spacetime of dimension $d=5$

Fig. 13

On-shell solution curve of uncharged black hole for different $\alpha$ in AdS or dS spacetime of dimension $d=6$

Solving the equation $\phi ^{r_{h}}=0$, we get the on-shell solution curves on $\tau _{h}-r_{h}$ plane for the given value of $\alpha$ and dimensions, as illustrated in Figs. 12 and 13. We could then calculate the topological charge for the on-shell solution. For instance, when $\alpha =0$ and $d=5$, we find that there is always one zero point whose winding number is $w=-1$. Whereas considering higher curvature term, the vector field of uncharged dS black hole would possess two zero points, whose winding numbers are $w_{1}=1$ and $w_{2}=-1$ respectively. Thus the total topological number is 0, in contrast to 1 in the AdS case. We depict the vector field of five-dimensional dS black hole in Fig. 14 as $\tau _{h}\approx 21.18$.

Fig. 14

Vector field of five dimensional uncharged dS black hole when taking $\tau _{h}\approx 21.18$ and $\Lambda =1/10$. The two zero points are at $(r_{h},\theta )\approx (0.8,\pi /2)$ and $(r_{h},\theta )\approx (1.91,\pi /2)$, marked in black dots. The winding number are $w_{1}=1$ and $w_{2}=-\,1$

Fig. 15

On-shell solution curve of uncharged black hole in dS spacetime for different dimensions and $\alpha$

Table 1 The number of zero points and global topological number for different black hole solutions

Then we will take a view of cosmological horizons. By solving the equation

$$\begin{aligned} \phi ^{r_{c}}=\frac{\partial F_{c}}{\partial r_{c}}=0, \end{aligned}$$

(3.11)

we also acquire the solution curve on $\tau _{c}-r_{c}$ plane for the given value of $\alpha$ and dimensions, as illustrated in Fig. 15. Using the same analytical approach, we surprisingly conclude that the dimension does not alter the quantity of zero points and the global topological number no matter whether a higher curvature term is added. In contrast, for the uncharged black hole horizon, the number of zero points changes when introducing the Gauss-Bonnet term. In addition, the dimension plays an important role in varying the topological number of Gauss-Bonnet dS black holes. But for cosmological horizon, there is always 1 zero point whose winding number is 1. In this case, dimension or higher curvature term may be incapable to bring intrinsical change to the thermodynamic system of cosmological horizon from a topological perspective.

4 Conclusion

In this paper, we use the generalized free energy of dS black hole solution with different horizons to define a vector field in a parameter space $\theta -r_{h}$ or $\theta -r_{c}$. We find the zero point of the field and obtain the winding number by applying Duan’s $\phi$-mapping topological current theory.

It is discovered that dS black hole solution with smaller horizon $r_{h}$ has one zero point for $a=q=0$ and two for rotating and charged cases. The global topological charges are − 1 and 0 respectively. Whereas for solution with the cosmological horizon $r_{c}$, the topological number of the non-rotating uncharged case is 1, inverse of the event horizon case. But as $a\ne 0$ or $q\ne 0$, the number obtained is the same, 0. As well, we find the conjecture that the winding number can characterize the stability of the on-shell solution may not be applied to the cosmological horizon. We also discuss the black hole horizon in higher dimension case ($d=5$ or $d=6$) with the Gauss-Bonnet term and conclude that uncharged static dS black holes are of the same category as static asymptotic flat black hole solution. To have an overview of the results for different black hole solutions in general relativity or higher dimensional modified gravity, we list them in Table 1.

The previous works [8, 10] indicate that all black hole solutions in the pure Einstein-Maxwell gravity theory should be classified into three different topological classes for four and higher spacetime dimensions. This observation is further enhanced in this paper. Moreover, there are many issues that deserve further investigation. Generalize our results to stationary cases and compare with the existing result will be interesting. Moreover, another interesting object is to investigate the topological number of the black hole solutions in the supergravity with or without positive/negative cosmological constant. We leave these interesting topics for future studies.