CC BY
36
UDC 530.1; 539.1
Creation of baby-universe E. V. Palesheva
Department of Physics and Mathematics, Tomsk State Pedagogical University, Kievskava Street 60, Tomsk 634061,
Russia.
E-mail: [email protected]
We introduce a metric in a 4-dimensional spacetime which describes creation of baby-universe from initially flat spacetime. Such metric may be used for construction of a transient intra-universe wormhole.
Keywords: baby-universe, wormhole.
1 Introduction
The problem of creation and existence of wormholes has attracted the attention of researchers for many years. A wormhole is a compact region of spacetime with nontrivial topology. According to Visser fl] there are permanent (or quasipermanent) and transient wormholes. The (quasi-)permanent intrauniverse wormhole is a compact region which topologically equivalent to R x E, where S is a spacelike hypersurface of nontrivial topology. Such wormholes are essentially three-dimensional objects. The transient wormhole is created and destructed without having R x E topological structure. These wormholes are intrinsically four-dimensional objects. In this case if we consider spacelike hypersurface x0 = const then wormhole is absent. Such hypersurface is either simply connected or consist of two disconnected regions. Creation of transient intra-universe wormhole may be thought of as the creation of baby-universe, it’s separation from parent-universe at some point of spacetime and pasting both universe at the another point of spacetime. At the interval from separation to pasting, both universes are moving in their own time trajectories. Such construction was considered in [2] and the energy of transient wormhole creation was estimated (see also [3]). However above mentioned transient wormhole metrics are not obtained.
The aim of the present paper is to construct the metric tensor of spacetime such that there is a creation of baby-universe.
2 Set of O-like curves
In Ref. [4] a three-parametric set of Q-like curves Q(x, u; A, m, v) = 0 was received. It describes as the sphere S1 separates from initially flat one-dimensional space R. In other words, this set can be thought as one-dimensional model of baby-universe creation. Let us rewrite the set of Q-like curves to one parametric
form. It is a simple performed procedure, if we assume that A M and v are sufficiently smooth functions of t. We accept that these dependencies have forms presented in the table 1. Here A0, m1 and k are positive constants such that m1 > A0 and k > 1. Moreover, first and second derivatives of functions ^(t)> v (t) and A (t) are equal to zero at the points tl5t2, i^d t = 0. The functions M(t) and A(t) increase, and A(t) is a decreasing function. Herewith M(0) = 0 M(t1) = M^
^(tl) = A2/VA0 + Ml! A(t2) = A0/VMl - A0> A(t2) =
Ao, A(t3) = 0.
So we receive that above mentioned set of curves is determined by equation Q(x,u; t) = 0, where
Q
G(x, u; t), S 1(x, u; t),
if 0 < u < v, x > 0,
u > v,
(1)
G( —x,u; t), if 0 < u < v, x < 0,
S1 ( x, u; t)
(m2 + A2)2v4 - A8 4m2A2v3
-R2
and G(x, u; t) = (A2 — m2)u2 + 2Amxu — A4. Here radius R( t)
R2
(A4 + (m2 — A2)v2)2 ((m2 — A2)2v4 — A8)
4m2A2
16m4 A4
.
The sphere S1(x,u; t) = 0 tangents to the hyperbola G(x, u; t) = 0 in a point Mv = (x^M(v),v), where xv(v) = A4 — (A2 — m2)v2/2mAv. This point may be also described by condition u = v. Functions A(t), M(t), v (t) evolve according to table 1.
t
t
curve. It will be denoted by Q. While t G [0, t1] a protuberant spherical region § is formed from line u = A0.
2
u
t t < 0 t G [0, ti] t G (ti,t2) t G [t2,t3] t > t3
M(t) 0 M(t) Mi Mi Mi
v (t) Ao *0 a/ ^0+M2 A(t) k\2 j Ml-^2 0
A(t) Ao Ao Ao A(t) 0
Table 1: Parameters A, ^ and v dependent from t.
As soon as t G (ti, t2) a throat is created on formed „ _ _ -(r. t)
u u ( i . t),
protuberant region. When t G [t2, t3J the throat is constricted to a point. At the completion of process the sphere will separate after the throat will have constricted to a point.
if
= u+(r; t),
if
r G [ro, +œ). t G (t*,t3],
r G (ro, r6]. t G (t*, t3],
(10)
(H)
3 3-dimensional surface
Thus, we receive that in moment t G [0, t*] a threedimensional metric tensor on H(t) corresponds to
If we have rotated Q(t) at every fixed point t then we obtain a non-stationary three-dimensional sur- ,
222 2 2 2 1 it \! \ 2 \
face E(t). At every point t, it is described by sur- ds _ r cos +r + ((u )t) j
faces H(x, y, z, u; t) _ 0, if 0 < u < v(t), and
S3(x,y, z,u; t) _ 0^f u > v(t), joined together on r n ni
some 2-dimensional surface u _ v(t). Let us introd uce ^ G [ , ^ G_ 2, 2. , r G [Г6, TO)"
two three-dimensional non-stationary metrics on surfaces H and S3 at every moment t.
dr2
And in every point t G (t*,t3] a three-dimensional
A first metric"tensor i‘s deteTmined on the spherical metric tensor on H(t) is tw0 three-dimensional metrics
pasted together on the two-dimensional surface r _ r0. One of them is
part S3 of surface E. We have ds2 = R2 cos2 n (cos2 0d^>2 + d02) + R2dn2,
^ G [0, 2n], 0 G
n n
2 , 2J
n G (^arcsin a, —
(3)
(4)
ds2
where
A8 - (M2 - A2)2v4
r2 cos2 0d^>2 + r2d02 + ^1 + ((u )r)2 j r G [r0, +to)
dr2
4Ru2A2v 3
R was defined % (2). The boundary of the surface S3 is given by formula n _ arcsin a.
A second metric tensor 7^(H) is determined on the hyperbolic part H of the surface E. Let
^ G [0, 2n], 0 G
and another is
n n
2 , 2J
ds2
r2 cos2 0d^>2 + r2d02 + ( 1 + ((u+)r) ) dr
n n
2 , 2J
r G (ro, r6) .
u±(r
(r;t) =
Am
uo(t) and rb(t) =
M2 - A2 A2
r
ro(t)
A4 - (A2 - M2)v2
2^Av
Then u(r) is determined by equations
A2
u = — ,
2r
if
r G [rb, +to) t = t',
= u- ( r; t) ,
if
r G [rb, +to) , t G [0,t') U (t',t*],
(5)
(6)
(7)
(8) (9)
Without any restrictions we can introduce an united note for every considering metric:
ds2
r2 cos2 0d^>2 + r2d02 + ^1 + (u)2j dr
^ G [0, 2n], 0 G
n n
2 , 2J
(12)
(13)
r
- 11). Herewith uT is a derivative of u(r; t) with re-rH formula r _ rb.
t
E(t)
metric on S3(t) and the metric on H(t). These metrics is identified on the two-dimensional surface.
2
2
2
o
2
u
4 Four-dimensional metric tensor
Now we can construct a four-dimensional metric tensor gik on the surface E by using Gaussian normal coordinates. There are two expression
ds2
dt2 — r2d^2 — (l + (u'r )2)
dr2
(14)
(15)
ds2 _ dt2 — R2 cos2 ndO2 — R2dn2.
We used denote d^2 _ cos2 0d^2 — d02.
If t G [0, t*] then we paste together (15) and (14), where u _ u- mid r G [rb, +to). If t G (t*,t3] then three four-dimensional metric tensors are pasted together. The fist metric is (14) with u _ u- and r G [r0, +ro), the second metric is (14) with u _ u+ and r G (r0,rb), and the third metric is (15). The metric tensor (15) describes created baby-universe. If
we find the energy-momentum tensor of whole spacetime, then we receive the violation of energy condition in some points. This violation exist in the region t G (t*,t3] and corresponds to negative extrinsic curE.
5 Conclusion
We construct metric tensor of spacetime at which the baby-universe is created. Further we can construct a transient intra-universe wormhole with closed timelike curve and without it. The energy conditions are violated in the receiving spacetime. At first moment we have a fiat spacetime ds2 _ dt2 — r2d^2 — dr2, at the end there are a baby-universe and a parent-universe pasted together in some point. At the end the parent-universe becomes flat again.
References
[1] Visser M. Lorentzian wormholes: from Einstein to Hawking. 1995. NY: AIP Press.
[2] Guts A. K. Soviet Physics J. 1982. 5 P. 396-399.
[3] Palesheva E. V. Mathematical Structures and Modelling. 2005. 15 P. 92-101.
[4] Palesheva E. V. Mathematical Structures and Modelling. 2009. 20 P. 74-77.
Received 01.10.2012
E. В. Палешева СОЗДАНИЕ ДОЧЕРНЕЙ ВСЕЛЕННОЙ
Рассматривается метрика 4-мериого пространства-времени, которая описывает создание и отрыв сферы й13 из изначально плоского пространства-времени. Данный отрыв части 3-мерного пространства можно рассматривать как создание дочерней вселенной. Приведенная метрика может быть использована для построения 4-мерной кротовой норы, топологическая структура которой не эквивалентна К х £, где £ — пространственно-подобная гиперповерхность нетривиальной топологии.
Ключевые слова: кротовая нора, отрыв шара.
Палешева Е. В., кандидат физико-математических наук. Томский государственный педагогический университет.
Ул. Киевская, 60, Томск, Россия, 634061.
E-mail: [email protected]
