CC BY
59
Вестник Челябинского государственного университета. 2009. № 8 (146). Физика. Вып. 4. С. 62-66.
R. N. Izmailov, N. G. Migranov, K. K. Nandi
rotating wormholes
In the present paper, we obtain, by using a slightly modified form of the Matos-Nunez algorithm, an extended class of asymptotically flat wormhole solutions belonging to Einstein minimally coupled scalar field theory. It is shown that the Ellis I solution of the Einstein minimally coupled theory, when Wick rotated, yields Ellis class III solution. The Wick rotated seed solutions, extended by the algorithm, contain two new parameters a and 5. The solution is discussed.
Key words: Matos-Nunez algorithm, Wick rotation, traversable wormhole, Einstein minimally coupled scalar field theory.
I. Introduction. Recently, there is a revival of interest in the scalar field gravity theories including the Brans-Dicke theory. Brans-Dicke theory has found immense applications in the field of wormhole physics, a field recently re-activated by the seminal work of Morris, Thorne and Yurtsever [1]. Wormholes are topological handles that connect two distant regions of space. These objects are invoked in the investigations of problems ranging from local to cosmological scales, not to mention the possibility of using these objects as a means of interstellar travel [1]. Wormholes require for their construction what is called “exotic matter”. Some classical fields can be conceived to play the role of exotic matter. They are known to occur, for instance, in the R + R2 theories [2], Visser’s thin shell geometries [3] and, of course, in scalar-tensor theories [4] of which Brans-Dicke theory is a prototype. There are several other situations where the energy conditions could be violated [5].
Brans-Dicke theory describes gravitation through a metric tensor (g^v) and a massless scalar field (9). The Brans-Dicke action for the coupling parameter © = -1 can be obtained in the Jordan frame from the vacuum linear string theory in the low energy limit. The action can be conformally rescaled into what is known as the Einstein frame action in which the scalar field couples minimally to gravity. The last is referred to as the Einstein minimally coupled scalar field theory. Several static (mostly nontra-versable) wormhole solutions in Einstein minimally coupled scalar field theory and Brans-Dicke theory have been widely investigated in the literature [6]. However, to our knowledge, exact rotating wormhole solutions are relatively scarce except a recent one in Einstein minimally coupled scalar field theory discussed by Matos and Nunez [7; 8]. In this context, we recall the well known fact that the formal independent solutions of Brans-Dicke theory are not unique. (Of course, the black hole solution is unique
for which the Brans-Dicke or minimal scalar field is trivial in virtue of the so called “no scalar hair” theorem.) Four classes of static Brans-Dicke theory solutions were derived by Brans [9] himself way back in 1962, and the corresponding four classes of Einstein minimally coupled field theory solutions are also known [10]. But recently it has been shown that only two of the four classes of Brans’ solutions are independent [12]; the other two can be derived from them. However, although all the original four classes of Brans’ or Einstein minimally coupled solutions are important in their own right, we shall here consider, for illustrative purposes, only one of them (Ellis I) as seed solution. The same procedure can be easily adopted in other three classes.
The general motivation in the present paper is the following: The rotational picture is more complicated but it often provides more enriching information. We want to properly frame an algorithm for generating singularity-free asymptotically flat rotating wormhole solutions from the Ellis seed solutions and to investigate the role of new parameters in the extended solutions. The analyses also answer a certain long standing query on the wormhole solutions inthe Brans-Dicke theory.
In this article, using a slightly modified algorithm of Matos and Nunez [7; 8], we shall provide a method for generating wormhole solutions from the known static seed solutions belonging to Einstein minimally coupled scalar field theory. The solutions can then be transferred to those of Brans-Dicke theory via inverse Dicke transformations. For illustration of the method, only Ellis I seed solution is considered here, others are left out because they can be dealt with similarly. The Brans-Dicke solutions can be further rephrased as solutions of the vacuum 4-dimensional low energy string theory (© = -1) and the section II shows how to do that. In sections II-IV, we shall analyze and compare the behavior Ellis III and the Wick rotated Ellis I solution pointing out
certain interesting differences in these geometries. Finally, in section V, we shall summarize the results. Throughout the article, we take the signature (-,+,+,+) and units such that 8nG = c = 1 unless restored specifically. Greek indices run from 0 to 3 while Roman indices run from 1 to 3.
II. The new algorithm. The ansatz we take is the following:
ds2 = -f (l)(dt + a cos0dx^ )2 +
+f-1 (l) dl2 + (l2 +102 )(d02 + sin2 0oty2), (1)
where l0 is an arbitrary constant, the constant a has been interpreted in [7; 8] as a rotational parameter of the wormhole. We call it the Matos-Nunez parameter. The ansatz in (7) is actually a subclass of the more general class of stationary metrics given by [13; 14]:
ds2 = - f (dt - ro.dx‘) + f-1 h.dx‘dx1, (2)
where the metric function f, the vector potential ©i and the reduced metric h.. depend only on space coordinates x‘. We shall see below that the parameter a can be so adjusted as to make a symmetric, traversable wormhole out of an asymmetric, nontraversable one. Note also that the replacement of a ^ -a does not alter the field equations.
The field equations are given by:
R,v =-«V,v; (3)
(4)
The function f(l) of the ansatz (1) is then a solution of the field equations (3) and (4), if it satisfies the following:
(i!
a2f n
+ ^-T = 0;
l2 '2
(
f j + 4lQ + a2f2 f J (i2+10 )2
- 2<p12 = 0,
(5)
(6)
where the prime denotes differentiation with respect to l.
Algorithm:
Let f = 1 (i; p,q;a = 0) and ^ = (l; p, q;a =0)
be a known seed solution set of the static configuration in which p, q are arbitrary constants in the solution interpreted as the mass and scalar charge of the configuration. Then the new generated (or extended) solution set f ^)is:
f (I; p, q;a )= 2 npqb fo ^ (/; p, q; a ) = ^,(7)
a2 + nS2 f0 2
where n is a natural number and the parameters p, q are specific to a given seed solution set (/0A ), while 5 is a free parameter allowed by the generated solution in the sense that it cancels out of the nonlinear field equations. The scalar field ^0 is remarkably given by the same static solution of the massless Klein-Gordon equation ^ = 0. The seed solution (a = 0), following from eqs. (5) and (6) gives 5 = 2pg. For the generated solution (a ^ 0), the value of 5 may be fixed either by the condition of asymptotic flatness or via the matching conditions at specified boundaries. Eq. (7) is the algorithm we propose. This is similar to, but not quite the same as, the Matos-Nunez [7; 8] algorithm. The difference is that they defined the free parameter as 5 = *J~D. The difficulty in this case is that, for our seed solution set (/q,^0), below, the field eqs. (5) and (6) identically fix 52 = D = 0, giving f = 0, which is obviously meaningless. The other difference is that we have introduced a real number n that now designates each seed solution f0 and likewise the corresponding new solution f. With the known parameters n, p and q plugged into the right side of eq. (7), the new solution set (f ,^ ) identically satisfies eqs. (5) and (6). One also sees that the algorithm can be applied with the set (f ,^ ) as the new seed solution and the process can be indefinitely iterated to generate any number of new solutions. This is a notable generality of the algorithm.
III. Ellis I solution and its geometry. The study of the solutions of the Einstein minimally coupled scalar field system has a long history. Static spherically symmetric solutions have been independently discovered in different forms by many authors and their properties are well known [16; 17]. We start from the following form of Class I wormhole solution, due to Buchdahl [18], of Einstein minimally coupled theory:
ds2 =-
1 -
m
2r
\2 P
m
2r j
1 -
m
2 r
2(1-P)
m j 2 r j
2(1+p)
[dr
2 ■ -2d02
+ r
2 • 2 + r sin
2 (P2 -1)
^ (r )=\l—------- ln
a
1 -
2r
1 +
m
2r
(8)
where m and P are two arbitrary constants. The same solution, in harmonic coordinates, has been obtained and analyzed also by Bronnikov [19]. Because of
m2
the occurrence of naked singularity at l = r + ,
4r
solutions (8) and (9) can be viewed as: ds2 = - f0 (l )dt2 +
1
+-
f)(/)
dl2 + (l2 + m2
)(d02 + sin2
0 2)],
% (l ) =
f0(l )=l 7 i
l - m
m
P2 -1
2
ln
l - m l + m
(10)
(11)
Because of the occurrence of naked singularity at r = m / 2, the wormhole is not traversable and so Visser [3] called it a “diseased” wormhole.
Iv. Ellis III solution via Wick rotation. One procedure to remove the aforementioned singularities is to analytically continue the Ellis I solutions (f0, ^0) by means of Wick rotation of the parameters while maintaining the real numerical value of the throat radius. In the solution set (J0,^0), we choose
m ^ -im, P ^ i'P, (12)
so that /0 = mP is invariant in sign and magnitude.
A. a = 0:
Then the metric resulting from the seed eq. (10) is and it is our redefined seed solution:
ds2 = f0 (l )dt2 +
1 Vdl2 + (l2 + m2 )(d02 + sin2 0 d^ 2)] ;(13)
r l j
f0 (l )= exp
-2P arc cot
v m
(14)
^0
(l )=rv^V1+P
arc cot
- I. (15)
m
This is no new solution but is just the Ellis III solution [19] which can be obtained in the original form by using the relation [20]
arc cot (x)+ arctan (x )=+ n, x > 0; (16)
=----, x < 0
2
and the function on the left shows a finite jump (of magnitude n) at x = 0. Thus, we get from eqs. (8) and (9), two branches, the +ve sign corresponds to the side l > 0 and the -ve sign to l < 0 [21]:
r Ellis
J 0±
(l )= exp
-2P j ±n - arctan f—
n r l
±-------arctan I —
2 I m
; (18)
. (19)
We might study the solutions (14) and (15) per se, or equivalently, study the two restricted branches taken together, while allowing for a discontinuity at the origin l = 0. Otherwise, we might disregard (14), (15) and treat each of the ± set in Eqs. (18), (19) as independently derived exact solution valid in the unrestricted range of l, with no discontinuity at l = 0. The two alternatives do not appear quite the same. In fact, each of the individual branches represents a geodesically complete, asymptotically flat traversable wormhole (termed as "drainholes" by Ellis) having different masses, one positive and the other negative, on two sides respectively. The known Ellis III solution is the +ve branch which is continuous over the entire interval l £ (-®; +«). The -ve branch is also equally good. What we have shown here is that the Ellis solutions I and III are not independent solutions of the Einstein minimally coupled scalar field theory as one can be obtained from the other.
It is of interest to compare the behaviors of the Ellis III solutions (23) with the Wick rotated Ellis I solutions (19):
(i) The Ellis III metric function f+is (le-JIp -> 1 as l ^ +®, but fEl's (l) ^ e^ ^1 as l ^ -®. These two limits correspond to a Schwarzschild mass M at one mouth and -Me™P at the other. There is no discontinuity at the origin because f™is (l )^ e~n|P -»1 as l ^ ±0. In the solution (19), on the other hand, there is a discontinuity at the origin because f0 (l) ^ e±JIp as l ^ ±0, while there is no asymptotic mass jump since f0 (l )^ 1 as l ^ ±®. The curvature scalars for both (19) and (23) are formally the same and given by
2m2 (1 + P2) (l2 + m2 )2
exp
-2P arc cot I — m
„ 2m2 (1 + P2)
RE (i)=- (>2 v 2)/exp
(l + m2)
(20)
. (21)
which go to zero as l ^ ±®. That means that spacetime is flat on two sides for both the solutions.
Next, we verify what happens to these scalars at the singular coordinate radius (r = m / 2) that has now
m
been shifted to the origin l = r--------= 0.
4r
(ii) The Ellis curvature scalar
2 (1 + |32)
K
Ellis
o-nP
as l ^ ±0, whereas the
m
curvature scalar R0 exhibits a finite jump from -2(1 + P2)c-, to -as l ^ ±0.
m
m
(iii) The area radius ^f0+1(Ellis) (l2 + m2) as l ^ ±0, whereas p0 ()=f¥ + m2)
a finite jump from k m^e-Ip as l ^ ±0.
These show that while Ellis wormhole (23) is traversable, but jumps at the origin in the Wick rotated solution (19) prevent traversability. The behaviors of (23) and (19) are different at the origin except that, for both the solutions, the throat appears at the same radius /0 = M = m P . Similar considerations apply for the -ve branch.
Ellis III wormholes (23) can be straightforwardly extended to the rotating form via the algorithm (7) and they would be likewise traversable, as has been shown by Matos and Nunez [7; 8]. Thus we do not deal with it here. Henceforth, we would rather concentrate on the Wick rotated Ellis I solution and ask: Can we somehow remove the discontinuities in (19)? That is exactly where the new parameter a comes in. Let us see what happens when a 4 0.
B. a 4 0:
The minimum area radius of the extended solution is obtained from the equation d~^— = 0, where
dl
p'(l )=fW + m2) is the area radius. From numerical study of the resulting equation, we found that the minimum area occurs at l < mp and it decreases with the increase of a for fixed values of m and p. We also notice that the finite jump persists in the area radius of the extended solution f' (l; m, P, a), 5 being
still given by 5 =
2M ± V4M2 - a~
. Surprisingly
however, when a = 2mp, the area function p'(l) decreases from +w to the minimum value at the throat, then increases to a finite value at l = 0, undergoes no jump at l = 0, but passes continuously, though not with C2, smoothness, across l = 0 , on to - ro. The
Ricci scalar R for f' (l; m, P, a
8p (1 + P2)(2mp +J4m2p2 - a2 )
exp
R' = -
is given by
2P arc cot
1 + ~~ 2 | a2 + (2mP + J4m2P2 - a2 ) exp 4P arc cot f— m J V / I m
(22)
and it approaches the value
4 (1+p 2 )enP (1 + e)
m
that is, no jump in it.
The area behavior shows that, for the extreme case a = 2mp, we do have a traversable wormhole with a single metric covering both the asymptotically flat universes (l ^ ±0), connected by a finite wedgelike protrusion in the shape function at l = 0. This wedge prevents C2 continuity across l = 0 in the area function but sews up two exactly symmetrical asymptotically flat universes on both sides. The numerical values of the free parameters m and p can always be suitably controlled to make the tidal force humanly tolerable and the travel safer.
For a 4 2mp, such a single coordinate chart is not possible as the area has a jump at l = 0. However, we can artificially circumvent this discontinuity and connect the two disjoint universes by multiple metric choices on different segments. We can get a cue for this construction from the static case. Consider the metric form (10) on one segment (AB) and the Wick rotated metric (13) on the other side (BC) so that the areas match at a radial point l = l1. The radius l = l1 is a root of the equation (area from right (AB) = area from left (BC))
l-m l + m
- (m2 +12)x Exp
2P arc cot I — m
. (23)
By numerical computation, it turns out that 0 < l1 < m such that the two otherwise disjoint universes, one represented by the branch AB and the other by BC, can be connected at (B(l = l1)). At the joining point B, there is continuity in the area function (again not C2) and the tidal forces can be shown to be finite throughout the generator curve ABC. Exactly similar arguments hold in the rotating case. Branch AB belongs to the Wick rotated solution (f), while the sector BC belongs to the original solution (f). Numerical calculations show that the matching occurs at either of the two points B(l1) or B(l2) such that -m < l1, l2 < m.
Traversable wormholes can also be constructed by employing the "cut-and-paste" procedure [3]. One takes two copies of the static wormholes and joins them at a radius l = lb > l0. The interface between the two copies will then be described by a thin shell of exotic matter. The shape functions on both sides will be symmetric. However, when rotation is introduced, numerical calculation shows that the throat radius decreases from the static value while the flaring out occurs faster. It is of some interest to note that Crisostomo and Olea [20; 21] developed
m
a Hamiltonian formalism to obtain the dynamics of a massive rotating thin shell in (2 + 1) dimensions. There, the matching conditions are understood as continuity of the Hamiltonian functions for an ADM foliation of the metric. Of course, this procedure can be trivially extended to deal with axially symmetric solutions in (3 + 1) dimensions.
v. Summary. Asymptotically flat rotating solutions are rather rare in the literature, be they wormholes or naked singularities. The algorithm (7), together with some operations, provides a method to generate new traversable wormhole solutions in the Einstein minimally coupled and then in the Brans-Dicke theory. The present study opens up possibilities to explore in more detail new solutions in other theories too. For instance, the string solutions are just the Brans-Dicke solutions with © = -1. As we saw, the asymptotically flat wormhole solutions admit two arbitrary parameters a and 5. The Matos-Nunez parameter a has been interpreted by the authors (ref. [7; 8]) as a rotation parameter. Here we have shown how the parameters can be adjusted for obtaining traversability.
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