On Solutions to the Wave Equation on Nonglobally Hyperbolic Manifold
Abstract
We consider the Cauchy problem for the wave equation on a nonglobally hyperbolic manifold of the special form (Minkowski plane with a handle) containing closed timelike curves (time machines). We prove that the classical solution of the Cauchy problem exists and is unique if and only if the initial data satisfy to some set of additional conditions.
1 Introduction
There is currently a quite developed theory of Cauchy problem for hyperbolic equations on globally hyperbolic manifolds [1]–[4]. Globally hyperbolic manifold is a spacetime oriented with respect to time (i.e., a pair , where is a smooth manifold and is the Lorentz metric) if is diffeomorphic to , where is a Cauchy surface. This definition is equivalent to Leray s definition of global hyperbolicity [3, 5].
Hyperbolic equations on nonglobally hyperbolic spacetimes have been studied considerably less, although numerous examples of such spacetimes are described by wellknown solutions of gravitation field equations; such are the solutions of Gödel, Kerr, Gott and many others [5][18]. These manifolds contain closed timelike curves (time machine) and are nonglobally hyperbolic.
Elementary examples of nonglobally hyperbolic space times are , with the Minkowski metric in which time argument passes a circle, and also antide Sitter space. There are several papers where the hyperbolic equations on nonglobally hyperbolic manifolds were discussed [19][21].
Our purpose here is to study the wave equation on a manifold containing closed timelike curves (CTC). We consider the Minkowski plane with two slits whose edges are glued in a specific manner ( plane with a handle). In paper [22] the Cauchy problem for the wave equation on the Minkowski plane with handle was considered and it was proved that there exists a solution, which is generally discontinuous on the characteristics emerging from the conical points.
In this work we establish the necessary and sufficient conditions on the initial data for existence and uniqueness of the classical (i.e. smooth) solution to the Cauchy problem in the halfplane with exception of slits.
Our motivation is related with studying the possibility of creating “wormholes” and nonglobally hyperbolic regions (mini time machines) in collisions of the highenergy particles [23], also see [24].
Formation of CTC is related with the violation of the null energy condition [25].
Problems of boundary control for wave equation are considered in [26].
We use in this work the following method to obtain the classical solution: we divide upper halfplane into 7 regions , write out the general solutions of wave equation in each of these regions and then we try to satisfy the gluing conditions and initial data, solving a certain system of linear equations. The specific conditions on the initial data (12)–(17) appear in this case. Thus we obtain the classical solution to the wave equation on the plane with handle (theorem 3.1). We also give another method to solving problem using theory of disributions (theorem 5.3). The results obtained by these two methods are equivalent.
2 Setting the problem
We consider two vertical intervals and with length in a halfplane :
(1) 
(2) 
We suppose that
(3) 
We assume that the edges of the intervals are glued as it is shown in Fig.1. The resulting manifold has two conic points – the ends of the intervals.
Every continuous field on this manifold will satisfy certain gluing conditions on the slits and . Conversely, if the field is continuous in domain and satisfies those gluing conditions then it is continuous on the manifold.
Consider the wave equation on that manifold for the function
(4) 
with initial conditions
(5)  
(6) 
where , . Let us set the following gluing conditions:
(7)  
(8)  
(9)  
(10) 
where and the indicated limits exist. We will show below that no extra conditions needed.
Let us define the classical solution:
Definition 2.1. A function is called the classical solution to the problem (4)–(10) if it satisfies conditions (4)–(10), provided the indicated left and righthand side limits exist.
Using characteristic halflines emerging out of the ends of the intervals , we divide the upper halfplane into 7 simply connected domains ,…, (see Fig.2):
3 Theorem of existence of classical solution
Hereafter we will use notations:
and
where can be 1, 2 or empty (in particular, ).
We will prove that the existence of a classical solution is equivalent to fulfilling the conditions
(12) 
(13) 
and conditions of smoothness on characteristics
(14)  
(15)  
(16)  
(17) 
Namely, the following theorem holds.
Theorem 3.1. The classical solution to the problem (4)–(10) exists if and only if the conditions (12)–(17) for hold. Given this, the classical solution is unique and is given by the formula
(18) 
where
(19)  
(20)  
(21)  
(22)  
(23)  
(24)  
(25) 
here ,
(26) 
and
(27) 
Proof. An arbitrary solution to equation (4) in domain is given by
where . We will show that conditions (5)–(10) impose quite strong restrictions to and .
Functions and are calculated directly from via the d’Alembert formulae:
(28) 
From now on we will evaluate through in a manner to make the solution twice continuously differentiable on , including eight characteristics ; here take up such values from that is open halfline. Let us write continuity conditions on , i. e.
(29) 
Analytically halfline is given by . Thus we can write (29) as
Using our notations, we evaluate :
where
Therefore, we have defined function when ; thus it is also defined when ; in addition, equals up to constant.
Similarly, using continuity conditions on we get functions defined when is greater than respectively and equal up to constant.
In a similar way, it is easy to show that functions of are defined when and are equal to up to constants.
Gluing conditions. Now we apply the gluing conditions for functions (7):
i. e.
(30) 
And gluing conditions for derivatives are
(31) 
Let us differentiate (30) on and add it to (31). We obtain
Thus,
Let us note that as this equation holds for
it defines for .
Finally, we have
and
Similarly, using the gluing conditions for and , we have
and
Evaluating constants. We have obtained solution in the form
(32)  
(33)  
(34)  
(35)  
(36)  
(37)  
(38) 
Now we have to find the constants .
Now we will find and , by employing the continuity conditions for solution on the halflines and respectively. We have on , so we can write
as
Recalling , we have
In a similar manner we get
Now we consider the halflines and . Continuity condition on is written as
wherefrom
Similarly, the continuity on is written as
wherefrom, bearing in mind , we get
We have obtained the condition for the functions , :
(39) 
As we will notice, we need two conditions for the continuous solution; the obtained condition will necessarily follow from those two.
So, let us consider the halflines and . We have on . Let us insert it into
We get
Inserting the found constants, we get
Thus we have found the first condition:
(40) 
If we express , through , we will have exactly (12).
Consider . We have on it; computing similarly, we obtain the second condition
(41) 
Easy to see that if we add (40) to (41) we will obtain precisely the condition (39).
We are left to find the last constant . We consider conditions on : let us insert into
We obtain
Recalling (40), we get
One can easily check that the continuity condition on also yields zero .
Differentiability conditions. We will find the conditions for differentiability of the solutions on the halflines . The partial derivatives along halflines exist, as it follows directly from the formulae (19)–(25). Let us write the conditions for continuity of partial derivatives of solution along normals to corresponding halflines.
(42)  
(43)  
(44)  
(45) 
Theorem 3.1 is proved.
3.1 Example
We will discuss an example when all conditions of the theorem are satisfied, and thus, the classical solution exists. We will look for the solution of the rightmode form:
From it follows that we should pick such initial conditions:
Then . We choose as bump function with support in :
4 Discontinuity jumps at slits
In the next section we will study the problem (4)–(10) by means of theory of distributions. We will generalize the method of analysis of the Cauchy problem from [4] to our case of plane with slits. Our method can be of interest in the analysis of generalized solutions of the problem concerned. Here we shall confine ourselves to study some properties of classical solutions of problem (4)–(10) in the “strengthened” setting.
We will use the following notations for the “onesided” limits and discontinuity jumps of functions:
(46) 
For convenience we shall introduce the following class of functions:
Definition 4.1
A function belongs to the class if and there exist the following limits:
where , , .
Let us formulate the main properties of functions from class which comply with the conditions (47)–(50):
Theorem 4.1
Let satisfies the conditions (47)–(50). Let and denote the discontinuity jumps of function and its derivative at the upper slit respectively:
Then one has:

.

, , and for the discontinuity jumps at the lower slit we have
moreover for we have .

Time derivatives satisfy the following gluing conditions:
(51) where .
5 Classical and generalized solutions
In this section we will derive the equation which will be satisfied by every strengthened classical solution of the problem (4)–(10) in sense of distributions . We will use the following notation for the d’Alembert operator: Also let denote the set of distributions from which equal to 0 for .
Theorem 5.1
Let be a strengthened classical solution of the problem (4)–(10). Then the function
satisfies the following equation in the sense of :
(52) 
where
(53) 
The proof is similar to the derivation of the generalized Cauchy problem setting given in [4]. It relies on the fact that , which follows from Theorem 4.1.
Recall the following formula [4]:
(54) 
in sense of . Here denotes the Laplace operator in , function , domain in has partially smooth boundary , , denotes the action of the classical Laplace operator and denotes the discontinuity jump of at the surface . We have obtained the analog of this formula for the d’Alembert operator on the plane with the slits.
Next, by virtue of theorem 4.1
(55) 
Find functions and , equal to 0 for , such that the generalized solution in of the equation
(56) 
(57) 
Note that the conditions (49) and (48) will be satisfied automatically by virtue of conditions (55).
So, the problem of existence and uniqueness of solution has converted to the problem of existence and uniqueness of the discontinuity jumps and satisfying specific conditions. To obtain these conditions we will first find the general solution of equation (56).
5.1 Solution of equation (56)
As is known [4], the solution of the generalized Cauchy problem for equation (56) exists, is unique and is given by a convolution of the fundamental solution with the right hand side defined in (57):
(58) 
Here
is the fundamental solution of operator , where denotes Heaviside step function; for and for .
Let us write out an explicit formula for the convolution (58). For this purpose we use the following formulae:
(59) 
Therefore denoting
(60) 
we obtain the solution of equation (56) in the following form:
(61) 
Here denotes the solution of classical Cauchy problem for wave equation defined by d’Alembert’s fomula:
(62) 
5.2 Gluing conditions
Let us now define the functions and using the gluing conditions. Conditions at the slits (47) (50) take the form
(63) 
(64) 
where .
Note that from (60) follows that