New Brans-Dicke wormholes
Feng He*Department of Science Education, Ewha Womans University, Seoul 120-750, Korea, Department of Physics, Beijing Normal University, Beijing 100875, People’s Republic of China, and Department of Physics, Xiang Tan Normal University, Xiang Tan 411201, People’s Republic of China
Sung-Won Kim†
Department of Science Education, Ewha Womans University, Seoul 120-750, Korea 共Received 4 May 2001; published 1 April 2002兲
Two new classes of exact solutions in vacuum Brans-Dicke theory are obtained, each of which is a two-way traversable wormhole for the coupling parameter⬍⫺2 or ⫺2⬍⭐0, respectively. Each of the two new classes of exact solutions satisfies not only the general constraints given by Morris and Thorne关Am. J. Phys.
56, 395共1988兲兴, as concluded earlier, but also the constraints from a trip through a wormhole. It also follows
that the scalar field plays the role of exotic matter violating the weak energy condition.
DOI: 10.1103/PhysRevD.65.084022 PACS number共s兲: 04.20.Gz, 04.50.⫹h
Wormholes are topological handles in space-time linking widely separated regions of a single universe or ‘‘bridges’’ joining two different space-times. Interest in these configu-rations dates back at least as far as 1916关1兴 with punctuated revivals of activity following both the work of Einstein and Rosen in 1935关2兴 and the later series of works initiated by Wheeler in 1955关3兴. Recently wormhole physics has gradu-ally drawn people’s attention 关4–9兴 as has the analysis of classical traversable wormholes for interstellar travel per-formed by Morris and Thorne关4兴. If an advanced civilization could construct such wormholes, they could be used as a galactic or intergalactic transportation system and they might also be usable for backward time travel 关4兴. Morris and Thorne, and other people, have found traversable wormhole solutions within the framework of Einstein’s general relativ-ity theory共GRT兲. It is known that GRT can be recovered in the limiting case→⬁ of the Brans-Dicke theory 共BDT兲. It seems natural that in the context of wormhole physics, one looks for wormhole solutions of BDT. The search for static wormhole geometry in BDT has been initiated by Agnese and La Camera关5兴. They show that a static spherically sym-metric Brans-Dicke 共BD兲 solution, obtained in a certain gauge by Krori and Bhattacharjee 关6兴, supports a two-way traversable wormhole for ⬍⫺2 and one way for ⬎
⫺3/2. Nandi et al. 关7兴 also show that three of the I–IV
classes of Brans-Dicke共BD兲 solutions obtained by Brans 关8兴 support a two-way traversable wormhole for⬍⫺2 and 0
⬍⬍⬁. However, they only show that these three solutions
satisfy the general constraints on the shape function b(R) and the redshift function ⌽(R), but not the constraints on them from a trip through the wormhole关4兴.
In this paper, we wish to obtain two new classes of exact solutions in vacuum BD theory and show that each of the two new classes of exact solutions satisfies not only the gen-eral constraints on the shape function b(R) and the redshift function⌽(R), as concluded earlier, but also the constraints
on them from a trip through a wormhole, to represent a two-way traversable wormhole. It will also be apparent that the presence of the BD scalar field cannot prevent weak en-ergy condition violation, showing that it is not a consequence of the GRT alone.
The BD field equations are
共;兲 ;⫽ 8 3⫹2T , 共1兲 R⫺1 2gR⫽⫺ 8 T⫺ 2
冋
;;⫺ 1 2g; ;册
⫺1关;;⫺g共;兲;兴, 共2兲 where T is the matter energy-momentum tensor excluding the field, is a dimensionless coupling parameter. The general metric, in spherically symmetric coordinates, is given by (G⫽c⫽1)d2⫽⫺e2␣(r)dt2⫹e2(r)dr2⫹e2(r)r2关d2⫹sin2d2兴.
共3兲
Our solutions are subject to the gauge ⫺⫽0. From Eqs.
共1兲 and 共2兲, we get the field equations in spherically
symmet-ric coordinates: e⫺2
冋
2⬙⫹
4⬘
r ⫹共⬘兲
2册
⫽e⫺2冋
⬘
␣⬘⫺
共⬘
兲2 2册
⫹8冉
T0 0⫺ T 2⫹3冊
, 共4兲 e⫺2冋
2␣⬘
⬘⫹共
⬘
兲2⫹2共␣⬘
⫹⬘兲
r册
*Email address: fhedoc@hotmail.com †Email address: sungwon@mm.ewha.ac.kr
PHYSICAL REVIEW D, VOLUME 65, 084022
⫽e⫺2
冋
⬙
⫺⬘
⬘
⫹共⬘
兲2 2册
⫹8冉
T1 1⫺ T 2⫹3冊
, 共5兲 e⫺2冋
␣⬙⫹共
␣⬘兲
2⫹␣⬘⫹
⬘
r ⫹⬙
册
⫽e⫺2冋
⬘
⬘
⫹⬘
r ⫺ 共⬘兲
2 2册
⫹8冉
T2 2⫺ T 2⫹3冊
共6兲 e⫺2冋
⬙⫹
⬘
冉
␣⬘⫹
⬘⫹
2 r冊册
⫽ 8T 2⫹3 共7兲 T␣; ⫽0, 共8兲 where ␣⬘
, ␣⬙
, ⬘
and ⬙
, respectively, denote d␣/dr, d2␣/dr2, d/dr and d2/dr2. For the vacuum case, T␣ 
⫽0. Equations 共4兲–共8兲 are resultant coupled nonlinear
dif-ferential equations for which it is very difficult to obtain exact solutions. Fortunately, solving these equations, we ob-tain two new classes of exact solutions.
New class I solutions are given by
␣共r兲⫽␣0, 共9兲 共r兲⫽0⫺2C arctan
冉
r B冊
⫺ln冉
r2 r2⫹B2冊
, 共10兲 共r兲⫽0e2C arctan(r/B), 共11兲 C2⬅⫺冉
⫹2 2冊
⫺1 ⬎0, 共12兲where ␣0,0, B, C and 0 are integration constants. The constants ␣0 and 0 are determined by the asymptotic flat-ness condition as follows:
␣0⫽0, 0⫽C. 共13兲 In order to investigate whether a given solution represents a wormhole geometry, it is convenient to cast the metric共3兲 into Morris-Thorne canonical form:
d2⫽⫺e2⌽(R)dt2⫹
冋
1⫺b共R兲 R册
⫺1
dR2⫹R2关d2
⫹sin2d2兴, 共14兲
where⌽(R) and b共R兲 are called the redshift and shape func-tions, respectively. These functions are required to satisfy some constraints, given by Morris and Thorne关4兴, in order to represent a wormhole. However it is important to stress that the choice of coordinates by Morris-Thorne is purely a mat-ter of convenience and not a physical necessity关9兴. Redefin-ing the radial coordinate as
R⫽r
冉
1⫹B 2 r2冊
exp冋
1⫺ 2 arctan冉
r B冊册
0, 共15兲comparing Eq. 共3兲 and Eq. 共14兲, we obtain the functions
⌽(R) and b(R) as ⌽共R兲⫽␣0⫽0, 共16兲 b共R兲⫽R
再
1⫺冋
1⫺2B关Cr共R兲⫹B兴 r2共R兲⫹B2册
2冎
. 共17兲The throat of the wormhole occurs at R⫽R0 such that b(R0)⫽R0. This gives minimum allowed r-coordinate radii as
r0⫾⫽BC
冋
1⫾冉
1⫹ 1 C2冊
1/2
册
. 共18兲The values R0⫾can be obtained from Eq.共15兲 using this r0⫾. Noting that R→⬁ as r→⬁, we find that b(R)/R→0 as R
→⬁. The function ⌽(R) is zero everywhere, so no horizon
exists, and ⌽(R)→0 as R→⬁. In order for a wormhole to be two-way traversable, we should have r0⫾⬎0. Therefore, we have BC⬎0 for r0⫹⬎0 or BC⬍0 for r0⫺⬎0. Equation
共12兲 requires ⬍⫺2.
The axially symmetric embedded surface z⫽z(R) shap-ing the wormhole’s spatial geometry is obtained from
dz dR⫽⫾
冋
R b共R兲⫺1册
⫺1/2 . 共19兲At the value R⫽R0 共the wormhole throat兲 Eq. 共19兲 is diver-gent, which means that the embedded surface is vertical there. For a coordinate-independent description of wormhole physics, one may use proper length l instead of R such that
l⫽⫾
冕
R 0 ⫾ R dR 关1⫺b共R兲/R兴1/2. 共20兲 In the present case,l⫽⫾
冕
r0⫾ r
e(r)dr. 共21兲
It can be seen that l→⫾⬁ as r→⫾⬁.
Consider a spaceship traveling radially through the worm-hole with its propulsion power shut off关4兴, beginning at rest in a space station in the lower universe, at l⫽⫺l1, and end-ing at rest in a space station in the upper universe, at l⫽
⫹l2. From Eq.共16兲, we get ⌽⫽0 everywhere, which corre-sponds to precisely zero tidal force as seen by stationary observers.
The constraints that gravity is weak at⫺l1 and l2are关4兴 b/RⰆ1,⌬/⫽e⫺⌽⫺1⬇兩⌽兩Ⰶ1, g⫽⫺
冉
1⫺b R冊
1/2d⌽ dR⬇冏
d⌽ dR冏
⭐g丣 共22兲 at l⫽⫺l1 and l⫽l2,FENG HE AND SUNG-WON KIM PHYSICAL REVIEW D 65 084022
where represents wavelength, g丣⫽980 cm/s2 共in cgs units兲 ⬇1/0.97(1y) 共in units G⫽c⫽1). Because 兩⌽兩Ⰶ1 at the stations, the proper time ticked by clocks there is equal to coordinate time t; cf. the space-time metric共14兲. Notice that
兩⌽兩⫽0Ⰶ1 and g⫽⫺(1⫺b/R)1/2d⌽/dR⫽0Ⰶ1. The con-straints in Eq.共22兲 are all satisfied if we locate the two space stations at large enough radii that the factor 关1⫺b(R)/R兴 differs from unity by only 1%. That is corresponding to l1
⫽l2⬇104R0, where we take the radial location of the two stations to be R1⫽R2⫽104R0.
The acceleration that the traveler in the spaceship feels is
关4兴 a⫽⫿
冉
1⫺b R冊
1/2 e⫺⌽ d dR共␥e ⌽兲d dl共␥e ⌽兲, 共23兲where␥⬅(1⫺2)⫺1/2共in our units G⫽c⫽1). The traveler does not feel an acceleration larger than about 1 Earth grav-ity, which corresponds to
冏
e⫺⌽ d dl共␥e ⌽兲冏
⭐g 丣⬇ 1 0.97共1y兲. 共24兲 Equation 共23兲 tells us that, since travelers in the spaceship feel no acceleration 共since a⫽0兲, the spaceship must travel with constant ␥e⌽. For the zero-tidal-force solutions (⌽⫽0), this corresponds to constant␥⫽(1⫺2)⫺1/2and hence to constant speed ⫽dl/dt as measured by stationary ob-servers,
⫽dldt⫽const, for an unpowered spaceship. 共25兲
The acceleration constraint 共24兲 is trivially satisfied since
⌽⫽0 and␥ keeps constant for the trip.
The constraints of radial tidal acceleration and lateral tidal acceleration can be written, respectively, as
冏
冉
1⫺b R冊
冋
⫺ d2⌽ dR2⫹ Rdb/dR⫺b 2R共R⫺b兲 d⌽ dR⫺冉
d⌽ dR冊
2册
冏
⭐g丣 2m⬇ 1 共1010 cm兲2, 共26兲冏
␥2 2R2冋
2冉
db dR⫺ b R冊
⫹2共R⫺b兲 d⌽ dR册
冏
⭐2mg丣 ⬇ 1 共1010 cm兲2, 共27兲where 2m is the size of the traveler’s body. The radial tidal constraint共26兲 is satisfied as ⌽⫽0 everywhere. We are thus left with constraint 共27兲 limiting the tidal forces associated with motion through the tunnel:
␥22 2R2
冏
db dR⫺ b R冏
⭐ 1 共1010 cm兲2. 共28兲Substituting Eqs. 共15兲 and 共17兲 into the above equation yields 2␥22 r3
冏
BC⫹ 2B2 r ⫺ B3C r2冏
e ⫺2C[1⫺(2/)arctan(r/B)] ⫻冉
1⫹B 2 r2冊
⫺4 ⭐ 1 共1010 cm兲2. 共29兲This constraint is most severe for the smallest radius r⫽r0⫾
共at the throat兲. We choose r⫽r0⫹ to estimate the magnitude of travel velocity . Supposing 兩C兩⬎1, Eq. 共18兲 gives r0⫹
⬇2BC. Computing Eq. 共29兲 approximately gives
␥⭐2⫻10⫺10A, A2⬅B2C2
冉
1⫹ 1 4C2冊
4 e2C[1⫺(2/)arctan(2C)] ⫻冉
1⫹ 3 4C2冊
⫺1 , 共30兲where A is measured in units of cm. In the limit that the motion is nonrelativistic (␥⬇1) we obtain 共in cgs units兲
⭐6A cm/s. 共31兲
Correspondingly, the total time lapse共in cgs units兲 for travel from station 1 to station 2 关4兴 is the same 共since ␥⬇1, ⌽
⫽0) for clocks ticking in the stations and on board the
spaceship: ⌬T⬇⌬t⬇
冕
⫺l1 l2 dl ⬇2⫻104 R0 ⬇2⫻10 4 3冉
1⫹ 1 4C2冊
⫺1冉
1⫹ 3 4C2冊
1/2 s. 共32兲The energy densityof the scalar field is obtained by computing the Einstein tensor G00 such that
G00⫽⫺ 1 8共T兲00⫽ 8⫽ 1 R2 db dR. 共33兲 From Eqs.共15兲 and 共17兲, we obtain
db dR⫽⫺
4B2r2共C2⫹1兲
共r2⫹B2兲2 . 共34兲 From the above equation, we obtain db/dR⭐0. This implies ⭐0, witheverywhere non-negative. This shows that the scalar field plays the role of exotic matter at the wormhole throat and there is consequently a violation of the weak en-ergy condition.
New class II solutions are given by
␣共r兲⫽␣0, 共35兲
NEW BRANS-DICKE WORMHOLES PHYSICAL REVIEW D 65 084022
e(r)⫽e0共1⫹B/r兲2
冉
1⫺B/r 1⫹B/r冊
1⫺C , 共36兲 共r兲⫽0冉
1⫺B/r 1⫹B/r冊
C , 共37兲 C2⬅冉
⫹2 2冊
⫺1 ⬎0, 共38兲where ␣0,0,B,C and 0 are integration constants. The constants ␣0 and 0 are determined by the asymptotic flat-ness condition as follows:
␣0⫽0, 0⫽0. 共39兲 With the same method as in the new class I solutions, in order to investigate whether the new class II solutions repre-sent a wormhole geometry, it is convenient to cast the metric
共3兲 into Morris-Thorne canonical form 关see Eq. 共14兲兴.
Rede-fining the radial coordinate as
R⫽re0共1⫹B/r兲2
冉
1⫺B/r1⫹B/r
冊
1⫺C, 共40兲
comparing Eq. 共3兲 and Eq. 共14兲, we obtain the functions
⌽(R) and b(R) as ⌽共R兲⫽␣0⫽0, 共41兲 b共R兲⫽R
再
1⫺冋
r 2共R兲⫺2BCr共R兲⫹B2 r2共R兲⫺B2册
2冎
. 共42兲The throat of the wormhole occurs at R⫽R0 such that b(R0)⫽R0. This gives minimum allowed r-coordinate radii as
r0⫾⫽BC
冋
1⫾冉
1⫺ 1 C2冊
1/2
册
. 共43兲The values R0⫾can be obtained from Eq.共40兲 using this r0⫾. Noting that R→⬁ as r→⬁, we find that b(R)/R→0 as R
→⬁. The function ⌽(R) is zero everywhere, so no horizon
exists, and ⌽(R)→0 as R→⬁. In order for a wormhole to be two-way traversable, we should have r0⫾⬎0. If r0⫾⬎0,
then R0⫾⬎0. From r0⫾⬎0 and using Eq. 共43兲, we obtain that C2⭓1 and BC⬎0. Substituting C2⭓1 into Eq. 共38兲 yields
⫺2⬍⭐0.
The energy densityof the scalar field is obtained by computing the Einstein tensor G00 such that
G00⫽⫺ 1 8共T兲00⫽ 8⫽ 1 R2 db dR. 共44兲 From Eqs.共42兲 and 共40兲, we obtain
db dR⫽⫺
4r2B2
共r2⫺B2兲2共C
2⫺1兲. 共45兲
From the above equation, we obtain db/dR⭐0. This implies ⭐0, with everywhere non-negative. This also shows that the scalar field plays the role of exotic matter at the wormhole throat, which violates the weak energy condition. Using the same method as in the new class I solutions, we can easily see that the new class II solutions satisfy all the other constraints proposed by Morris and Thorne关4兴.
It was shown in the foregoing that two new classes of exact solutions to vacuum BD field equations are obtained and each of them gives rise to a two-way traversable worm-hole provided the constants are chosen appropriately. Each of the two new classes of exact solutions satisfies not only the general constraints on the shape function b(R) and the red-shift function⌽(R), but also the constraints on them from a trip through a wormhole, and represents a two-way travers-able wormhole. It was also shown that the presence of the BD scalar field cannot prevent the weak energy condition violation.
ACKNOWLEDGMENTS
The authors would like to thank Young-Joo Moon and Dr. Hyunjoo Lee for their kind help. This research was supported partially by the Basic Research Program of the Korea Sci-ence and Engineering Foundation under Grant No. R01-2000-00015, mostly by National Research Program 共00-B-WB-06-A-04兲 for Women’s University, by National Natural Science Foundation of China, by Natural Science Foundation of Hunan Province, and by Education Committee of Hunan Province.
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FENG HE AND SUNG-WON KIM PHYSICAL REVIEW D 65 084022
