Spherical Symmetric Solutions for the Motion of Relativistic Membranes in the Schwarzschild-Anti de Sitter Space-Time

Abstract

This paper concerns motion of relativistic membranes in the Schwarzschild-anti de Sitter space-time. We derive a nonlinear equation for relativistic membranes moving in the Schwarzschild-anti de Sitter space-time, discuss spherical symmetric solutions for the motion equations, and obtain some interesting physical results.

1. Introduction

This paper concerns the motion of relativistic strings in the Schwarzschild-anti de Sitter space-time. The metric reads 𝑑𝑠²=−1−2𝑚𝑟−𝜆3𝑟²𝑑𝑡²+1−2𝑚𝑟−𝜆3𝑟²⁻¹×𝑑𝑟²+𝑟²𝑑𝜃²+sin²𝜃𝑑𝜑², (1) which is a spherically symmetric solution of the Einstein field equations. The Schwarzschild-anti de Sitter space-time is a fundamental physical space-time; it plays an important role in general relativity, the theory of black holes, and modern cosmology.

As is well known, the theory of minimal surfaces/submanifolds has a long history originating from the papers of Lagrange in 1760 and the famous Plateau problem. And this theory plays a significant role in general relativity, the theory of black holes, particle physics, and so on. A good deal of attention has been paid to the theory of minimal surfaces in the Euclidean space ℝ^𝑛 and Riemannian manifolds in recent years. On this topic, we refer to two classical books [1, 2]. For the theory of extremal surfaces/submanifolds in the Minkowski space-time, many important results have been derived ([3–6]). Particularly, the theory of extremal surfaces/submanifolds is very important in elementary particle physics. It is a relativistic string model that deals with a one-dimensional relativistic object, whose world surface is an extremal surface in the Minkowski space-time [7]. For the relativistic string theory, we refer to an excellent book by Barbashov and Nesterenko [8].

The authors in [9] simplified the description of relativistic membrane in the Minkowski space-time by the light-cone gauge. By variables transformations, a relationship was established between the dynamics of relativistic membrane and two-dimensional fluid dynamics. Moreover, in [10] they obtained a vector-valued equation of first order for relativistic membrane by introducing the orthonormal (1+3)-gauge, and then they deduced a second-order equation for minimal graph 𝑧=𝑧(𝑡,𝑥₁,𝑥₂) in the Minkowski space-time ℝ¹⁺³ by the hodograph technique. Furthermore, using reduction of membrane equation in light-cone gauge, they obtained a second-order partial differential equation for the velocity potential. In the paper [11], a lot of simplifications of the equations of motion for the relativistic membrane were exhibited such as the orthonormal light-cone gauge, minimal graph method, and level set method. According to these reformulations, Hoppe found some classical solutions for the equations governing the motion of relativistic membrane in the Minkowski space-time.

Here we want to mention a result in [12]: the authors investigated the basic equations for the motion of relativistic membranes in the Schwarzschild space-time and got a nonlinear wave equation, and then they studied a spherical symmetric solution for the motion of relativistic membranes, giving many new physical results. By variational and geometrical methods, Huang and Kong in [13] obtained a new kind of equation describing the motion of relativistic membranes in the Minkowski space-time ℝ^1+𝑛(𝑛⩾3), gave some interesting properties, and showed that all plane-wave solutions of these equations were light-like extremal submanifolds and vice versa except for a type of special solution.

Kong and Zhang studied the motion of relativistic closed strings in the Minkowski space ℝ^1+𝑛 in [14]; particularly, the authors obtained a general solution formula for this system of nonlinear equations. Based on the solution formula, they showed that the motion of closed strings was always time-periodic and extended the solution formula to finite relativistic strings. Moreover, in [15], Kong et al. investigated the dynamics of relativistic strings in the Minkowski space-time ℝ^1+𝑛  (𝑛≥2). They first obtained a system with 𝑛 nonlinear wave equations of Born-Infeld type describing the motion of the string, and then they showed that this system enjoyed some interesting geometric properties; in the end, they gave a sufficient and necessary condition for the global existence of extremal surfaces without a space-like point in ℝ^1+𝑛. Furthermore, they made a lot of numerical analyses demonstrating that various topological singularities developed in finite time in the motion of the string.

This paper mainly focuses on the equations and spherical symmetric solutions for the motion of relativistic membranes in the Schwarzschild-anti de Sitter space-time. Concretely, we derive an interesting nonlinear wave equation for relativistic membranes and study systematically the spherical symmetric solutions for the motion of membranes.

The paper is organized as follows. In Section 2, we recall the basic equations for the motion of relativistic membranes in the Schwarzschild-anti de Sitter space-time. Section 3 is devoted to a systematical study on the spherical symmetric solutions of the equations for the motion of relativistic membranes; at the same time, some new and interesting physical phenomena are discovered and illustrated. Some discussions are given in Section 4.

2. Basic Equation

A four-dimensional Lorentzian manifold (𝑀,𝑔_𝜇𝜈) is called the Schwarzschild-anti de Sitter space-time if the metric of 𝑀 can be written as (2) with 𝜆<0 ([16]). Consider 𝑑𝑠²=𝑔_𝜇𝜈𝑑𝑥^𝜇𝑑𝑥^𝜈=−1−2𝑚𝑟−𝜆3𝑟²𝑑𝑡²+1−2𝑚𝑟−𝜆3𝑟²⁻¹×𝑑𝑟²+𝑟²𝑑𝜃²+sin²𝜃𝑑𝜑², (2) where 𝑚 is a positive constant representing the mass of the universe and 𝜆 is the cosmological constant. Assume 𝑋=(𝑡,𝑟,𝜃,𝜑) is a position vector of a point in the Schwarzschild-anti de Sitter space-time. Moreover, let 𝑟₊ be the largest root of the equation 1−2𝑚/𝑟−(𝜆/3)𝑟²=0. Obviously, it holds that 0<𝑟₊<2𝑚. Since we are only interested in the motion of membrane in the region 𝑟>𝑟₊, we may suppose that 𝑟₀>𝑟₊.

Consider the motion of a relativistic membrane in the Schwarzschild-anti de Sitter space-time 𝑡,𝜃,𝜑⟶𝑡,𝑟𝑡,𝜃,𝜑,𝜃,𝜑. (3) In the coordinates (𝑡,𝜃,𝜑), the induced metric of the submanifold ℳ is 𝑑𝑠²=𝑑𝑡,𝑑𝜃,𝑑𝜑𝒢𝑑𝑡,𝑑𝜃,𝑑𝜑^𝑇, (4) where 𝑔𝒢=⁰⁰𝑔₀₁𝑔₀₂𝑔₁₀𝑔₁₁𝑔₁₂𝑔₂₀𝑔₂₁𝑔₂₂, (5) in which 𝑔₀₀=−1−2𝑚𝑟−𝜆3𝑟²+1−2𝑚𝑟−𝜆3𝑟²⁻¹𝑟²_𝑡,𝑔₀₁=𝑔₁₀=1−2𝑚𝑟−𝜆3𝑟²⁻¹𝑟_𝑡𝑟_𝜃,𝑔₀₂=𝑔₂₀=1−2𝑚𝑟−𝜆3𝑟²⁻¹𝑟_𝑡𝑟_𝜑,𝑔₁₁=𝑟²+1−2𝑚𝑟−𝜆3𝑟²⁻¹𝑟²_𝜃,𝑔₁₂=𝑔₂₁=1−2𝑚𝑟−𝜆3𝑟²⁻¹𝑟_𝜃𝑟_𝜑,𝑔₂₂=𝑟²sin²𝜃+1−2𝑚𝑟−𝜆3𝑟²⁻¹𝑟²_𝜑. (6)

We assume that the submanifold ℳ is 𝐶² and time-like; that is, Δ≜det𝒢=−1−2𝑚𝑟−𝜆3𝑟²𝑟⁴sin²𝜃−𝑟²𝑟²_𝜑−𝑟²sin²𝜃𝑟²_𝜃+1−2𝑚𝑟−𝜆3𝑟²⁻¹×𝑟⁴sin²𝜃𝑟²_𝑡<0. (7)

The area element of ℳ is 𝑑𝒜=√−Δ𝑑𝑡𝑑𝜃𝑑𝜑. (8) And the submanifold ℳ is called extremal if 𝑟=𝑟(𝑡,𝜃,𝜑) is a critical point of the area functional ℐ=√−Δ𝑑𝑡𝑑𝜃𝑑𝜑. (9)

By calculations, we obtain the corresponding Euler-Lagrange equation as follows: 1−2𝑚𝑟−𝜆3𝑟²⁻¹−1Δ1−2𝑚𝑟−𝜆3𝑟²⁻²𝑟⁴sin²𝜃𝑟²_𝑡×𝑟⁴sin²𝜃𝑟_𝑡𝑡−𝑟²+1Δ𝑟⁴𝑟²_𝜑𝑟_𝜑𝜑−11+Δ𝑟²sin²𝜃𝑟²_𝜃𝑟²sin²𝜃𝑟_𝜃𝜃+2Δ×1−2𝑚𝑟−𝜆3𝑟²⁻¹𝑟⁶sin²𝜃𝑟_𝑡𝑟_𝜑𝑟_𝑡𝜑−2Δ𝑟⁴sin²𝜃𝑟_𝜃𝑟_𝜑𝑟_𝜃𝜑+2Δ1−2𝑚𝑟−𝜆3𝑟²⁻¹𝑟⁶sin⁴𝜃𝑟_𝑡𝑟_𝜃𝑟_𝑡𝜃+𝑟³sin²𝜃2−3𝑚𝑟−𝜆𝑟²−𝑟𝑟²_𝜑−𝑟𝑟²_𝜃sin²𝜃−𝑟²𝑟_𝜃sin2𝜃+𝑟²_𝑡𝑟³sin²𝜃1−2𝑚𝑟−𝜆3𝑟²⁻²×2−5𝑚𝑟−𝜆3𝑟²+𝑟²𝑟_𝜑×2Δ−2𝑟𝑟_𝜑𝑟²_𝜃sin²𝜃+−4+6𝑚𝑟+2𝜆𝑟²×𝑟³sin²𝜃𝑟_𝜑+1−2𝑚𝑟−𝜆3𝑟²⁻²×𝑟²_𝑡𝑟³sin²𝜃𝑟_𝜑4−10𝑚𝑟−2𝜆3𝑟²−2𝑟𝑟³_𝜑+𝑟²𝑟_𝜃sin²𝜃×−2Δ1−2𝑚𝑟−𝜆3𝑟²𝑟⁴sin2𝜃−2𝑟𝑟_𝜃𝑟²_𝜑−2𝑟𝑟³_𝜃sin²𝜃−𝑟²𝑟²_𝜃+sin2𝜃1−2𝑚𝑟−𝜆3𝑟²⁻¹𝑟²_𝑡𝑟⁴sin2𝜃+𝑟³𝑟_𝜃sin²𝜃𝑟²_𝑡1−2𝑚𝑟−𝜆3𝑟²⁻²×4−10𝑚𝑟−2𝜆3𝑟²+𝑟³𝑟_𝜃sin²𝜃−4+6𝑚𝑟+2𝜆𝑟²−𝑟⁴𝑟_𝑡sin²𝜃2Δ1−2𝑚𝑟−𝜆3𝑟²⁻¹×−2𝑟𝑟_𝑡𝑟²_𝜃sin²𝜃+𝑟³𝑟_𝑡sin²𝜃×−4+6𝑚𝑟+2𝜆𝑟²+𝑟³𝑟³_𝑡sin²𝜃×1−2𝑚𝑟−𝜆3𝑟²⁻²×4−10𝑚𝑟−2𝜆3𝑟²−2𝑟𝑟_𝑡𝑟²_𝜑=0. (10)

3. Spherical Symmetric Solutions

This section is concentrated on spherical symmetric solutions for the motion of relativistic membranes. Now we study the spherical symmetric solutions 𝑟=𝑟(𝑡) for the motion of relativistic membranes. Therefore, in the present situation, (10) can be simplified as the following ordinary differential equation: 𝑟_𝑡𝑡=2𝑟−𝑚−(5/3)𝜆𝑟³𝑟𝑟−2𝑚−(𝜆/3)𝑟³𝑟²_𝑡−𝑟−2𝑚−(𝜆/3)𝑟³2𝑟−3𝑚−𝜆𝑟³𝑟³. (11)

Consider the initial value problem for (11) with the initial data 𝑟(0)=𝑟₀,𝑟_𝑡(0)=𝑟₁, (12) where 𝑟₀ and 𝑟₁ are two constants meaning the initial position and initial velocity of the membrane, respectively. Since we are only interested in the motion of the membrane in the region 𝑟>𝑟₊, without loss of generality, we may assume that 𝑟₀>𝑟₊.

Let 𝑧=𝑟_𝑡. (13)

Equation (11) becomes the following form: 𝑑𝑧=()𝑑𝑟2𝑟−𝑚−5/3𝜆𝑟³𝑟𝑟−2𝑚−(𝜆/3)𝑟³𝑧−𝑟−2𝑚−(𝜆/3)𝑟³2𝑟−3𝑚−𝜆𝑟³𝑟³1𝑧. (14)

Using (14), we derive 𝑑𝑧𝑑𝑟𝑟𝑟−2𝑚−²(𝜆/3)𝑟³³=𝑑1𝑑𝑟𝑟𝑟−2𝑚−(𝜆/3)𝑟³³𝑧²+2𝑧𝑟𝑟−2𝑚−(𝜆/3)𝑟³³𝑑𝑧2𝑑𝑟=−2𝑟−3𝑚−𝜆𝑟³𝑟⁴𝑟−2𝑚−(𝜆/3)𝑟³²=𝑑1𝑑𝑟𝑟³𝑟−2𝑚−(𝜆/3)𝑟³. (15) Integrating (15), we have 𝑧²=𝑟−2𝑚−(𝜆/3)𝑟³²𝑟²×1+𝐹𝑟₀,𝑟₁𝑟³𝜆𝑟−2𝑚−3𝑟³, (16) where 𝐹𝑟₀,𝑟₁=𝑟²₁𝑟₀𝑟₀−2𝑚−(𝜆/3)𝑟³₀³−1𝑟³₀𝑟₀−2𝑚−(𝜆/3)𝑟³₀. (17) Noting (16), we obtain from (11) that 𝑟_𝑡𝑡=𝑟−2𝑚−(𝜆/3)𝑟³𝑟³22𝑚−3𝜆𝑟³+𝐹𝑟₀,𝑟₁𝑟³𝜆𝑟−2𝑚−3𝑟³×52𝑟−𝑚−3𝜆𝑟³. (18)

Define two functions 𝑓(𝑟) and 𝑔(𝑟) as follows: 𝑓(𝑟)=𝑟³𝜆𝑟−2𝑚−3𝑟³,𝑔(𝑟)=𝑟³𝜆𝑟−2𝑚−3𝑟³52𝑟−𝑚−3𝜆𝑟³. (19) Obviously 𝑓(𝑟) and 𝑔(𝑟) are strictly increasing positive functions when 𝑟>𝑟₊. Hence, using the notations 𝑓(𝑟) and 𝑔(𝑟), (16) and (18) are written as 𝑧²=𝑟−2𝑚−(𝜆/3)𝑟³²𝑟²1+𝐹𝑟₀,𝑟₁𝑓(𝑟,𝑟_𝑡𝑡)=𝑟−2𝑚−(𝜆/3)𝑟³𝑟³22𝑚−3𝜆𝑟³+𝐹𝑟₀,𝑟₁𝑔(𝑟), (20)(21) respectively.

In order to solve the Cauchy problem (11) and (12), that is, (21) and (12), we distinguish several cases as follows.

Case 1. Consider 𝑟₁⩾𝑟₀−2𝑚−(𝜆/3)𝑟³₀𝑟₀. (22)
In this case, noting 𝑟₀>𝑟₊, it holds that 𝐹𝑟₀,𝑟₁⩾0,𝑟_𝑡(0)=𝑟₁>0,𝑟_𝑡𝑡(0)>0. (23) According to the properties of 𝑓(𝑟) and 𝑔(𝑟), it follows from (20) that 𝑧=𝑟_𝑡>0, (24) which implies that 𝑟 is a strictly increasing function of 𝑡. Therefore, it follows that 𝑟(𝑡)>𝑟(0)=𝑟₀>𝑟₊ for 𝑡>0. Thus, we obtain from (21) that 𝑟_𝑡𝑡>0. (25)
Case 1(a) 𝑟₁=(𝑟₀−2𝑚−(𝜆/3)𝑟³₀)/𝑟₀.
In the present situation, we have 𝐹(𝑟₀,𝑟₁)=0, and 𝑟_𝑡=𝑟−2𝑚−(𝜆/3)𝑟³𝑟,𝑟_𝑡𝑡=𝑟−2𝑚−(𝜆/3)𝑟³𝑟³22𝑚−3𝜆𝑟³. (26) Clearly, because 𝑟_𝑡>0 and 𝑟_𝑡𝑡>0, 𝑟(𝑡) tends to the infinity. We next prove that 𝑟(𝑡) goes to the positive infinity in finite time. In fact, it follows from (26) that 𝑡=^𝑟_𝑟0𝑟𝑟−2𝑚−(𝜆/3)𝑟³𝑑𝑟. (27) Let 𝑇_max=^∞_𝑟0𝑟𝑟−2𝑚−(𝜆/3)𝑟³𝑑𝑟. (28) It is easy to see that 𝑡≤𝑇_max. Now we show that 𝑇_max<∞. For a certain large 𝑟̃, 𝑇_max=_𝑟𝑟₀̃𝑟𝑟−2𝑚−(𝜆/3)𝑟³𝑑𝑟+^∞𝑟̃𝑟𝑟−2𝑚−(𝜆/3)𝑟³𝑟̃𝑑𝑟<_𝑟0𝑟𝑟−2𝑚−(𝜆/3)𝑟³𝑑𝑟+^∞𝑟̃𝑟−(𝜆/3)𝑟³=𝑟̃𝑑𝑟_𝑟0𝑟𝑟−2𝑚−(𝜆/3)𝑟³3𝑑𝑟−𝜆𝑟̃<∞, (29) which implies 𝑟(𝑡)⟶+∞,when𝑡↗𝑇_max. (30) Equation (30) shows that the solution of the initial value problem (11) and (12) must blow up in finite time and the life span is [0,𝑇_max), where 𝑇_max is defined by (28). Different from the case that initial velocity is equal to critical velocity in [12], where the solution 𝑟=𝑟(𝑡) of the Cauchy problem exists globally on the time 𝑡∈[0,∞), in this paper the solution of the initial value problem (11) and (12) blows up in finite time when initial velocity is equal to critical velocity. See Figure 1 for the graph of the solution 𝑟=𝑟(𝑡) in the present situation.
Case 1(b) 𝑟₁>(𝑟₀−2𝑚−(𝜆/3)𝑟³₀)/𝑟₀.
In this case, we have 𝐹(𝑟₀,𝑟₁)>0. Since 𝑟_𝑡>0 and 𝑟_𝑡𝑡>0, 𝑟(𝑡) goes to the infinity. In what follows, we prove that 𝑟(𝑡) tends to the positive infinity in finite time.
Similar to the above case, it follows from (16) that 𝑡=^𝑟_𝑟0𝑟𝑟−2𝑚−(𝜆/3)𝑟³×11+𝐹𝑟₀,𝑟₁𝑟³𝑟−2𝑚−(𝜆/3)𝑟³𝑑𝑟. (31) Denote 𝑇_max=^∞_𝑟0𝑟𝑟−2𝑚−(𝜆/3)𝑟³×11+𝐹𝑟₀,𝑟₁𝑟³𝑟−2𝑚−(𝜆/3)𝑟³𝑑𝑟. (32) Obviously, we have 𝑡⩽𝑇_max.
We next show that 𝑇_max<∞. For a certain large 𝑟, 𝑇_max=^𝑟_𝑟0𝑟𝑟−2𝑚−(𝜆/3)𝑟³×11+𝐹𝑟₀,𝑟₁𝑟³𝑟−2𝑚−(𝜆/3)𝑟³+𝑑𝑟_∞𝑟𝑟𝑟−2𝑚−(𝜆/3)𝑟³×11+𝐹𝑟₀,𝑟₁𝑟³𝑟−2𝑚−(𝜆/3)𝑟³<𝑑𝑟_𝑟𝑟0𝑟𝑟−2𝑚−(𝜆/3)𝑟³×11+𝐹𝑟₀,𝑟₁𝑟³𝑟−2𝑚−(𝜆/3)𝑟³𝑑𝑟+^∞_𝑟𝑟−(𝜆/3)𝑟⁶−(𝜆/3)𝐹𝑟₀,𝑟₁=𝑑𝑟_𝑟𝑟0𝑟𝑟−2𝑚−(𝜆/3)𝑟³×11+𝐹𝑟₀,𝑟₁𝑟³𝑟−2𝑚−(𝜆/3)𝑟³+1𝑑𝑟−(4/3)𝜆−(𝜆/3)𝐹𝑟₀,𝑟₁𝑟⁴<∞. (33)
This proves that 𝑟(𝑡)⟶+∞,when𝑡↗𝑇_max. (34) Equation (34) implies that the solution of the Cauchy problem (11)-(12) must blow up in finite time and the life span is the interval [0,𝑇_max), where 𝑇_max is defined by (32). Compared to the case that initial velocity is larger than critical velocity in [12], in the present paper 𝑇_max is controlled by the constant 𝜆. See Figure 1 for the graph of the solution 𝑟=𝑟(𝑡).

Figure 1 

The graph of the solution 𝑟=𝑟(𝑡) for Cases 1(a) and 1(b).

Case 2. Consider 𝑟₀−2𝑚−(𝜆/3)𝑟³₀𝑟₀2𝑟₀−3𝑚−𝜆𝑟³₀2𝑟₀−𝑚−(5/3)𝜆𝑟³₀<𝑟₁<𝑟₀−2𝑚−(𝜆/3)𝑟³₀𝑟₀. (35)
In this situation, we obtain from (17) and (21) that −2𝑚−(2/3)𝜆𝑟³₀𝑔𝑟₀<𝐹𝑟₀,𝑟₁𝑟<0,_𝑡(0)=𝑟₁>0,𝑟_𝑡𝑡(0)>0. (36) By the properties of 𝑓(𝑟) and 𝑔(𝑟), there exists a point 𝑟^∗∈(𝑟₊,+∞) such that 𝑓𝑟^∗1=−𝐹𝑟₀,𝑟₁. (37) Thus, it follows from (20) that 𝑧=𝑟_𝑡=0,when𝑟=𝑟^∗. (38)
On the one hand, noting (20) and the second inequality in (36), 𝑟(𝑡) will increase until a time 𝑡^∗>0. At the time 𝑡^∗, it holds that 𝑟_𝑡𝑡<0,when𝑟=𝑟^∗. (39) By (36), we have 𝑟_𝑡𝑡>0,when𝑟=𝑟₀. So there exists a point 𝑟_⋆∈(𝑟₀,𝑟^∗), such that 𝑟_𝑡𝑡=0,when𝑟=𝑟_⋆. (40)
On the other hand, when 𝑟→𝑟₊+0, 𝑟_𝑡⟶0,𝑟_𝑡𝑡⟶0. (41)
For 𝑟→𝑟₊+𝛿 (𝛿 is sufficiently small and positive), both 𝑟−2𝑚−(𝜆/3)𝑟³ and 𝑔(𝑟) are small enough and positive; thus 𝑟_𝑡𝑡 is sufficiently small and 𝑟_𝑡𝑡>0. Combined with (39), it holds that there exists a point 𝑟_∗∈(𝑟₊+𝛿,𝑟^∗) such that 𝑟_𝑡𝑡=0,when𝑟=𝑟_∗. (42)
Now we will discuss the developing with time of the membrane, that is, the properties enjoyed by the solution of the Cauchy problem (11) and (12). At the time 𝑡^∗, it holds that 𝑟=𝑟^∗,𝑟_𝑡=0,𝑟_𝑡𝑡<0. (43) Therefore, 𝑟_𝑡(𝑡)<0 for 𝑡>𝑡^∗. That is, when 𝑡>𝑡^∗, 𝑟(𝑡) is a strictly decreasing function; then there exists a time 𝑡_∗ such that 𝑟𝑡_∗=𝑟_∗,𝑟_𝑡𝑡_∗<0,𝑟_𝑡𝑡𝑡_∗=0. (44) Hence, by (20) and (21), we obtain that for 𝑡>𝑡_∗𝑟(𝑡)<𝑟_∗,𝑟_𝑡(𝑡)<0,𝑟_𝑡𝑡(𝑡)>0. (45) Furthermore, from the fact that (42) holds when 𝑟→𝑟₊+0, we see that the solution of the Cauchy problem (11) and (12) exists globally on the time 𝑡∈[0,∞), and the solution satisfies the following decay properties: 𝑟(𝑡)⟶𝑟₊,𝑟_𝑡(𝑡)⟶0,𝑟_𝑡𝑡(𝑡)⟶0, (46) when 𝑡→∞. Figure 2 is the graph of the solution for Case 2.

Figure 2 

The graph of the solution 𝑟=𝑟(𝑡) for Cases 2 and 3.

Case 3. Consider 0<𝑟₁⩽𝑟₀−2𝑚−(𝜆/3)𝑟³₀𝑟₀×2𝑟₀−3𝑚−𝜆𝑟³₀2𝑟₀−𝑚−(5/3)𝜆𝑟³₀. (47)
In this situation, it follows from (17) and (21) that −1𝑓𝑟₀<𝐹𝑟₀,𝑟₁()⩽−2𝑚−2/3𝜆𝑟³₀𝑔𝑟₀,𝑟_𝑡(0)=𝑟₁>0,𝑟_𝑡𝑡(0)⩽0. (48) Similar to Case 2, we can exactly analyze the properties enjoyed by the solution of the Cauchy problem (11) and (12). The graph of the solution is shown in Figure 2. Figure 2 shows that, in this case, the solution of the Cauchy problem (11) and (12) exists globally on the time 𝑡∈[0,∞), and the solution satisfies the decay condition (46).

Case 4. Consider −𝑟₀−2𝑚−(𝜆/3)𝑟³₀𝑟₀×2𝑟₀−3𝑚−𝜆𝑟³₀2𝑟₀−𝑚−(5/3)𝜆𝑟³₀<𝑟₁⩽0. (49)
In this case, it holds that −1𝑓𝑟₀≤𝐹𝑟₀,𝑟₁()<−2𝑚−2/3𝜆𝑟³₀𝑔𝑟₀,𝑟_𝑡(0)=𝑟₁⩽0,𝑟_𝑡𝑡(0)<0. (50) Similar to the discussion in Cases 2 and 3, we can exactly analyze the properties enjoyed by the solution of the Cauchy problem (11) and (12). See Case 4 in Figure 3 for the graph of the solution. In this case, the solution of the Cauchy problem (11) and (12) also exists globally on the time 𝑡∈[0,∞), and the solution 𝑟=𝑟(𝑡) satisfies the decay condition (46).

Figure 3 

The graph of the solution 𝑟=𝑟(𝑡) for Cases 4 and 5.

Case 5. Consider 𝑟₁𝑟⩽−⁰−2𝑚−(𝜆/3)𝑟³₀𝑟₀×2𝑟₀−3𝑚−𝜆𝑟³₀2𝑟₀−𝑚−(5/3)𝜆𝑟³₀. (51)
In this case, we have 𝐹𝑟₀,𝑟₁()⩾−2𝑚−2/3𝜆𝑟³₀𝑔𝑟₀,𝑟_𝑡(0)=𝑟₁<0,𝑟_𝑡𝑡(0)⩾0. (52) In the present situation we only consider the case that 𝐹(𝑟₀,𝑟₁)⩾0; then it follows from (21) that 𝑟_𝑡𝑡>0, (53) provided that 𝑟>𝑟₊.
Therefore, noting (53) and using (42), we know that the solution of the Cauchy problem (11) and (12) exists globally on the time 𝑡∈[0,∞), and the solution 𝑟=𝑟(𝑡) satisfies the decay conditions (46). The graph and detailed information of the solution in the present situation are shown in Figure 3.

4. Discussions

The Schwarzschild-anti de Sitter space-time is a fundamental physical space-time and plays an important role in general relativity and the theory of black holes. The notion of extremal surfaces/submanifolds is used to formulate the Wilson criterion of quark confinement in gauge models of strong interactions. The theory of extremal surfaces/submanifolds is important in the elementary particle physics. The world sheets of relativistic membranes moving in physical space-times are nothing but extremal surfaces/submanifolds in these space-times. Some beautiful and deep results have been obtained, but unfortunately only a few results have been known for relativistic membranes moving in the Schwarzschild-anti de Sitter space-time.

In the present paper, we study the equations and spherical symmetric solutions for relativistic membranes moving in the Schwarzschild-anti de Sitter space-time. First, a nonlinear wave equation for the motion of relativistic membranes in the Schwarzschild-anti de Sitter space-time is derived, and then spherical symmetric solutions for the motion of relativistic membranes are studied systematically. We can summarize the main results as follows. In this paper we consider a spherical membrane centered at the black hole 𝑟=0 and assume that 𝑟₀ and 𝑟₁ represent the initial position and initial velocity of the membrane, respectively. For a given initial position 𝑟₀>𝑟₊, we find a critical initial velocity 𝑣_𝑐=(𝑟₀−2𝑚−(𝜆/3)𝑟³₀)/𝑟₀ such that the motion with a different initial velocity has essentially different behavior. Exactly, (i) when 𝑟₁>𝑣_𝑐, the solution for motion of the spherical membrane must blow up in finite time and the life span of the solution is [0,𝑇_max), where 𝑇_max is defined by (32) and 𝑇_max<∞. (ii) When 𝑟₁=𝑣_𝑐, the solution also blows up in finite time and the life span of the solution is [0,𝑇_max), where 𝑇_max is given by (28) and 𝑇_max<∞. (iii) When 𝑟₁<𝑣_𝑐, the solution exists globally on the time 𝑡∈[0,∞), and the solution satisfies the following decay conditions: 𝑟(𝑡)⟶𝑟₊,𝑟_𝑡(𝑡)⟶0,𝑟_𝑡𝑡(𝑡)⟶0, (54) when 𝑡→∞.

In geometry, our results show that, for a given spherical membrane centered at the black hole 𝑟=0, (i) if the initial velocity is larger than the critical velocity 𝑣_𝑐, then singularities in the motion will develop in finite time, precisely, at the time 𝑇_max; that is, the spherical membrane disappears at the time 𝑇_max in this case. (ii) If the initial velocity is equal to the critical velocity 𝑣_𝑐, similar to the above case, singularities will develop in finite time, and the spherical membrane disappears at the time 𝑇_max in this situation. (iii) If the initial velocity is less than the critical velocity 𝑣_𝑐, then motion of the spherical membrane will not stop at any finite time, and the spherical membrane converges to the event horizon 𝑟=𝑟₊ when the time tends to infinity. From the graphs we see that motion of the spherical membrane may be different due to the different initial velocity. In this case, the moving velocity of the membrane goes to zero when the membrane tends to the event horizon.

Acknowledgments

This work was supported in part by the NNSF of China (Grant no. 11271323) and the Qiu-Shi Professor Fellowship from Zhejiang University, China.