CC BY
6Scientific Notes of Taurida National V. I. Vernadsky University
Series : Physics and Mathematics Sciences. Volume 27 (66). 2014. No. 2. P. 50-64
UDK 539. 391+514. 764.2
DYNAMICS OF A TEST NULL STRING IN THE GRAVITATIONAL FIELD OF A CLOSED "THICK" NULL STRING MOVING IN THE PLANE Lelyakov A. P., Karpenko A. S., Babadzhan R.-D. A.
Taurida National V. I. Vernadsky University, 4 Vernadsky Ave., Simferopol 295007, Crimea, Russia
E-mail: lelyako crimen. edu
The dynamics of a test null string moving in the gravitational field of a closed "thick" null string radially expanding or collapsing in the plane is considered, provided that the former string does not rotate initially. Keywords: "thick" null string, gravitational field.
PACS: 98.80. ± k
INTRODUCTION
The idea of that the researches of multidimensional objects, including strings, may form a basis for our understanding of the Nature has already been expressed rather neatly in modern physics. One of the directions of those researches in the string theory deals with the role of such objects in cosmology. The gauge theories of Grand Unification, which are based on the idea of a spontaneous symmetry breaking, predict a possibility of the formation of one-dimensional topological defects in the course of phase transitions in the early Universe. Those objects were called space strings [1-7].
In work [8], it was shown that the presence of such objects in the Universe does not contradict the existence of the observed microwave relic radiation. Null strings realize the zero-tension limit in the string theory [5, 7]. Therefore, since the tension is measured in units of the Plank mass, M , scale, the zero-tension limit corresponds, from the physical viewpoint, to the asymptotically large energy scale, E >> M. From this viewpoint, the null strings, which realize a high-temperature phase of strings, could arise at the Big Bang moment and, hence, affect the observed structure of the Universe. In particular, in work [9], it was demonstrated that, by considering the gas of null strings as a dominant source of the gravitation in ^-dimensional Friedman Robertson Walker spaces with k = 0, one can describe the inflation mechanism typical of those spaces.
In a number of works, the gas of relic null strings is considered as one of the probable candidates for the role of a carrier of the so-called "dark" matter, the existence of which in the Universe can be regarded as an proved fact. Although, the object of research in the quoted examples is not a separate null string, but a gas of null strings, the properties of this gas still remain unclear. In our opinion, the first step to understanding the properties of the gas of null strings may consist in the solution of the problems concerning the gravitational field generated by a null string moving along different trajectories, as well as the dynamics of a test null string in such gravitational fields.
For instance, let the equations of motion for a test null string have solutions that can be interpreted as moving test null strings with time-independent shapes determined by
50
initial conditions. At the same time, the trajectory of this null string is similar to that of the null string generating the gravitational field. Then, we may say that there exist a state (a phase) of the ideal gas consisting of identical null strings. The existence of such a gas may form a basis for the formulation of various multistring problems.
In this work, the dynamics of a test null string in a gravitational field generated by a closed "thick" null string that radially expanding or collapsing in the plane z = 0 is considered. In this research, we are interested first of all in the presence of solutions for the equations of motion that would give rise to the possibility of the existence of a state (a phase) of the ideal gas consisting of identical null strings in this gravitational field. We are also interested in the features of the interaction between the test null string and the string generating the gravitational field.
1. EQUATION OF MOTION FOR A NULL STRING
The quadratic form describing the gravitational field of a closed "thick" null string radially expanding or collapsing in the plane z = 0 can be presented as follows [10, 11]:
dS2 = e2v
((dt)2 -(dp)2)-B(dOf -e2^(dz)2
(1)
Here,
e
2v(],z) _
u
a(]) + U(]) f (z)
urn1 um
V (1-/)
B], z)=
a(]) + U(]) f (z)
1/ (1-/)
j
2м(v,z) _
( f z )2
f y-V4-2/
a(]) + U(]) f (z)
\V (1-/)
(m )■
]] = t + p.
The functions and a(]) are coupled by the relation
U(]) = (1-a(]))/f^ fo = const,
(2)
(3)
(4)
(5)
(6)
U =
dU(]) _df ( z)
d]
fz = "J v ', / = 8nG, the functions a(]) and f (z) are finite and, at
,z dz
every T] e (-to; and z e (-to; , acquire values within the intervals
51
0<a(,)< 1, 0<f(z)<fo. The limiting cases are [10, 11]
^ 0, *>(v) ^0^
fo
f (z)l.
i(-œ;-Az )u(+Az:
,+œ)^ f0, f ( Z ) 1 ^ 0 '
(8) (9)
where Aq and Az are small positive constants that determine the "thickness" of the "thick" null string generating the gravitational field (A// 1.Az 1). In the limiting case of the contraction into a one-dimensional object (a null string), the following conditions (at Aq ^ 0 and Az ^ 0) have also to be satisfied:
a.
a
^ œ.
iJ^-0
f ( z )
^ 0,
a.
L
^
a
(,) f ( z )
^0. (10)
As an example, the following functions [10, 11] satisfy conditions (10):
a
(,) = exp
1
S f ( z ) = f0exP j-/
1 - exp< -
1
& )
(11)
Here, the constants i; and C determine the size (the "thickness") of the "thick" null string that generates the gravitational field (depending on q and z , respectively), and the positive constant s and ¿u provide the satisfaction of conditions (10) at Aq ^ 0 and Az ^ 0, namly,
u^®, s^ 0. (12)
The dynamics of a null string in the pseudo-Riemannian space is governed by the following system of equations:
+rmpq (x) x^ = 0, (13)
gmnKK= 0, gp„x; x"a= 0, (14)
where gpn and Tpq are the metric tensor and the Cristoffel symbols, respectively, of the external space; xmT = 8xp / 8t and x"a = dx" / da, the indices p, n, p and q take integer values from 0 to 3; the functions xp (t,ct) determine the trajectory of motion of the null
string; t and a are parameters on the world surface of the null string; a is the spacelike parameter that marks the points on the null string, and t is time-like parameter. In the cylindrical coordinate system,
52
x0 = t, x1 = p, x2 = O, x3 = z,
and the functions xm , which determine the trajectories of motion for the null string generating the gravitational field and are considered in this work, have the following form: t = t , p = -T, O = a, z = 0, re(-®,0], ^e[0,2^]. (15)
From Eqs. (15), it follows that the null string that generates gravitational field is in the plane z = 0 and has an infinitely large radius at the initial time moment. As the time t grows, the null string, remaining in the same plane z = 0, only decreases its radius, i.e. it radially collapses in the plane z = 0 .
]],TT+ 2V]T= 0, (16)
+ 2^q,Tz,T + e~2vB] (Ot)2 + 2e2(M-v^] (z,tf = 0, (17)
-2u
. >. e 2 2
z,tt+ e2(F-^V,2V],tz,t--— B,z (O,t ) + U,z (z,t) = 0, (18)
BT
O,zz+~ltO,z= 0, (19)
B
e2vr,qr - B (Ot ) 2 - e2* (z,r ) 2 = 0, (20)
2e22a+ 9])-BOjOa -e2zjzta = 0, (21)
where
q = t-p. (22)
When integrating Eq. (16), the following two cases have to be considered:
]],t = 0, ^ ] = ]», (23)
T]T ^ 0, ^ ]] = T](r,a) . (24)
2. SOLUTION OF THE EQUATIONS OF MOTION FOR THE TEST NULL STRING IN THE CASE T]T = 0
In case (23), Eq. (20) looks like
B (Ot )2 + e2*(zT )2 = 0. (25)
Since the functions B = B (], z) and e2*(],z) are positive at every T] e (-to; +to) and z e (-to; +to) , it follows from Eq. (20) that
53
z,T = 0, ^ z = z (ct) , (26)
O,T= 0,^O = O(CT). (27)
Under conditions (23), (26), and (27), Eqs. (16), (18), and (19) are satisfied identically, and Eqs. (17) and (21) take the forms
q,TT= 0, (28)
q,],v= 0, (29)
respectively. Integrating Eq. (28), we obtain
q,T = Pq CT) , ^ q (t, ct) = 90 CT) + Pq (?) T , (30)
where q0 (ct) and Pq (ct) are integration "constants". One should pay attention that
Pq (ct)* 0, (31)
because, otherwise, we have q = q0 (ct) . The latter together with Eqs. (23), (26), and
(27) means the realization of a static solution for the null string, which is impossible. Under conditions (23) and (30), Eq. (29) takes the form
]CTPq (ct) = 0. (32)
From whence, taking Eqs. (5) and (31) into account, we have
]]CT= 0,= t + p = const. (33)
The solution described by Eqs. (26), (27), (30), and (33) means that, under condition (23), the closed test null string moves in the same direction as the null string generating the gravitational field, i.e. it collapses. At every fixed time moment t, all points of the closed test null string are equidistant from the axes z. Moreover, as follows from equality (26), the test null string is not localized in the single plane z in the general case. In other words, the obtained solution describes a closed test null string that, at every fixed time moment, is completely
localized between two planes, z = z^ = const and z = z2 = const, where zx = min z (ct) and z2 = max z (ct) , where ct e [0; 2n], on the surface of a cylinder with the radius p = -t + const. At the same time, if we fix z (ct) = z0 = const in Eq. (27), this case describes the radial collapse of the test null string completely remaining in the plane z = z0 at every time moment and preserving the circular shape.
Hence, it follows from the obtained solution that there may exist a state for the gas of null strings, in which closed circular null strings located in parallel planes z = const (the polarization effect) collapse simultaneously preserving their shape, i.e. without interaction (the phase of ideal gas of null strings).
54
3. SOLUTION OF THE EQUATIONS OF MOTION FOR A TEST NULL STRING IN THE CASE q ^ 0
Integrating Eq. (19), we obtain
0(t,a) = 00 (a) + Pe (a){(B)_1 dT, (34)
where the functions 00 (a) and P (a) (the integration "constants") determine, with the
respect to the variable 0, the positions and the velocities, respectively, of null string points at the initial time moment. From equality (34), it follows that, in the case where P (a) = 0 at the initial time moment, i.e. the closed test null string does not rotate, its further dynamics will also evolve without rotation, so that
0 = 0(a). (35)
In this work, we have found a solution of the equations of motion for the closed test null string in case (24) and under the condition that its rotation is absent at the initial time moment, i.e. provided that
q = q( t, a), 0 = 0(a). (36)
In this case, the variable q depends on the parameter t (it changes in time). Therefore, Eq. (36) describes the motion of the test null string "toward" the null string that generates the gravitational field. However, the polar angle corresponding to every point of the test null string does not vary in time.
If the test null string moves "toward" the null string that generates the gravitational field, the q - value only increases. Therefore,
q t > 0. (37)
The case q T < 0 describes the motion of a test null string in the same direction as the
null string generating the gravitational field, but at a higher velocity, i.e. at a velocity higher than the speed of light, which is imposible. Under conditions (36), Eq. (20) looks like
e2 Vq, t q,T = e2u( z T T )2. (38)
From whence, taking Eq. (37) and the positive definiteness of metric functions into account, it follows that
q,T ^ 0. (39)
In the case
q, =0 t ^ q=q0 (a),
55
(40)
where q0 (ct) is the integration "constant", Eq. (38) gives rise to
z,T= 0, ^ z = z (ct) , (41)
where z0 (ct) is the integration "constant". Under conditions (36), (40) and (41), the
equations of motion (17)-(20) for the test null string are satisfied identically, and Eq. (21) takes the form
],tV,ct= 0. (42)
From whence, taking Eqs. (22), (37) and (40) into account, we have
q = t - p = q0 = const. (43)
To summarize, Eqs. (36) and (40) describe the motion of a closed test null string with arbitrary shape "toward" the null string generating the gravitational field. At every fixed time moment t, all points of closed test null string are equidistant from the axes z, and the shape of the test null string given by the functions z0 (ct) and O0 (ct) remains invariant. If the test null string is completely located in the plane z = z0 = const at the initial time moment, its further dynamics evolves in this plane. The only possible shape for it is the circle. The radius of this circle can only increase in time (the closed test null string radially expands in the plane z = z0).
Hence, requirement (40) brings about a solution testifying to the possibility for the gas of null strings to exists in a state composed of two non-interacting subsystems. In each subsystem, the closed circular null strings are located in parallel planes z = const (the polarization effect). The null strings radially expand in one subsystem and, simultaneously, radially collapse in another one without changing their shape, i.e. without interaction.
Under conditions (36) and (37), the first integral of Eq. (16) looks like
]t= P (ct) e, (44)
where
P (ct)> 0 (45)
is the integration "constant". One can show that, for the case
q,T > 0, (46)
and taking Eq. (38) into account, the first integrals of Eqs. (17) and (18) take the form
\f,*r\ = PCTvAtt (47)
56
q,=
?M}2 _AL p H J {mj-
(48)
where the function P2 (a) (the integration "constant") determines the z-velocities of test null string points at the initial time moment; and, as follows from Eqs. (37), (45) and (47)
P2 (a) > 0 . (49)
From Eqs. (47) and (48), it follows that, in the case of (36) and (46), the variables q , q and z, which determine the position of the test null string at every fixed time moment, are no more independent, but interrelated.
From Eqs. (8) and (9), it follows that, for Eqs. (47) and (48), the whole region of variation for the variables q and z can be divided into four domains depending on the
sign of derivatives of the functions h(q) and f (z):
(I) q e (-»;0) and z e (0; +(»), in which fz > 0 and > 0;
(II) q e (-»;0) and z e (—»;0), in which fz < 0 and > 0;
(III) q e (0; +<») and z e (0; +<»), in which fz > 0 and < 0; and
(IV) q e (0; +<») and z e (—»;0), in which fz < 0 and < 0.
Integrating Eq. (48) firstly at q < 0 (regions I and II; Aq> 0) and then at q> 0 (regions III and IV; Aq< 0), and matching the obtained solutions across the boundary
q = 0 (using X(q) ^ = ( f(}) !), we have: in regions I and II (q < 0),
q = q0 (ct) + 2f0 in regions III and IV ( q > 0),
P (*)J IP (*)
(my
(50)
q=qo
P2 (*) P (").
(51)
where q0 is the integration "constant".
In each region, two possible directions of motion of the test null string along the axis z can be realized, zT > 0 and zT < 0. Therefore, the solution of Eq. (47) in each region can be presented in the form
2
57
fL = U'L + )_1, (52)
where the subscript L takes values I to IV and corresponds to the number of the region, in which the found solution is realized; the superscript i takes values 0 (the case zT > 0, the
test null string moves in the positive direction of the axis z); the constants y'L equal
0 11 0 1 1 0 0 1 1 /J--,*.
Y0 = Yn = yui = yV = -1, Yi = Yn = Ym =YIv =1, (53)
the functions U'L (ct) , in view of the continuity of the obtained solution across the boundary j = 0, look like
U°i=FA°), U)=Fx{ct), Ujj = F2 (<t) , Uln=F2(cr),
lfm=F,{<r)-2f0^\, U]n=F,{<T) + 2f0^\, (54)
P1 (CT) P1 (CT)
U°r=F2(a) + 2f0U]r=F2(a)~
and the functions /*,' (cr). I<\ (cr). F2(g) and F2(g) are integration "constants".
From equality (52), it follows that the size, i.e. the radius, of the moving test null string is strictly related to its position with respect to the null string generating the gravitational field; i.e. it depends on the variable r . Analogously, since the function
f (z) on the left-hand side of equality (52) is finite and the function in the
denominator of the right-hand side of this equality, in accordance with Eq. (8), tends to zero for je(-®;-Aj)u(+Aj; +<»), any choice of integration "constants" is always associated with a certain confined region symmetric in r , where equality (52) is satisfied. However, in this case, since there are no restrictions on the test null string coordinates z and t (in the general case, j e (-<x>; +<»)), only those test null string that are located in this
narrow region (the "interaction zone") are "visible", i.e they interact with the null string generating the gravitational field. The same test null string located at this moment beyond this zone, in accordance with Eq. (52), remain "invisible" for the null string that generates the gravitational field. Here, we cannot say that they do not interact, because, in the framework of the general theory of relativity, the absence of interaction manifests itself in the null string preservation without changing the trajectory of its motion. Whereas, in our case, it is impossible to say anything about the trajectory of motion of the test null string beyond this region. However, one cannot exclude that, at a certain time moment, such an "invisible" null string will enter this region, and its subsequent dynamics will be determined, at least until the time moment, when the test null string leaves it. In other words, the test null string, when entering this narrow "interaction zone", already has a
58
prehistory, and its dynamics in this zone depends on the size, location, and direction of its motion along the axis z (it moves in the positive or negative direction of the axis z, i.e. z z > 0 or z z < 0), being determined by equality (52).
Under conditions (44) and (50)-(52), Eq. (21) takes the following form in each region determined by the subscript L:
% (CT) + 2/o
^ v P CT)
P2 (ct) .
f'l (CT = 0. (55)
p1 (ct) lv ka
Here, the indices i and L have the same interpretation and accept the same values as in Eq. (52), and the functions F'L (ct) are
Ff=F%=~FM), rf = Fjn = Fx (ct) , F»=F?v=F2(cJ), Fln = F]v = -F2(ct). (56)
The functions Pk (ct) , k = 1,2, determine the initial momenta of the test null string points. As follows from equalities (52), the requirement
Pk(<r),Fk(<T),Fk(<r) = const, k = 1,2, (57)
describes the case where the test null string shape is not changed (remains to be a circle) in the course of motion, and the variations of the radius of the test null string and its position on the axis z are determined by the form of the functions /(z) and U(j). Note that, under condition (57), Eq. (55) is reduced to a single requirement,
%0 (ct),ct = 0, ^ %0 (ct) = %0 = const. (58)
From Eqs. (50) and (51), it follows that the constant q0 defines the surface, on which the test null string and the null string generating the gravitational field "meet", while moving "toward" each other.
Using Eq. (2), let us express Eq. (44) in the form
Ul (a(j) + U(j) / ( z) )2-^ j I ,jl( (1) }')rL\))-= p CT, (59)
since the function 0<a« + U«/(z)< 1 at any j and z, and the constant
X = 8nGU 1, so that the difference 24-2^ « 0, Eq. (59) can be presented in the form
<1,-^-r = P>(ct) .
59
Integrating this equation, we obtain the following relations between the variable 77 and the parameters r and c on the world surface of the test null string:
in regions I and II (7 < 0),
(A(7))_1 =7o(c) - P(c)r; (60)
in regions III and IV (77 > 0),
(^))"1=%(i7) + JP1(i7)r. (61)
Here, the integration "constant" //„ (cr) and //„ (cr) determine the value of parameter
r, at which the test null string moving "toward" the null string generating the gravitational field meets the latter on the same surface. For instance, under condition (57), by fixing
% {?) = % (?) = /0 = comt (62)
in Eqs. (60) and (61), we obtain that, at 77 = 0, the parameter r = 0. Moreover, in regions I and II (7 <0) at 77 e(—»;0), the parameter r e(-<»;0); in regions III and IV (77 >0) at 77e(0; +<»), the parameter re(0; +<»). Under conditions (57), (58) and (62), the variables 7 and q determined by equalities (50), (51), (60) and (61) depend only on the parameter r by means of the relations
77 = A(/0+^T), (63)
q=q0 + /0
P2 (c)Y (P2 (c)) 2
+V 2V, V r, (64)
P1 (c), p1 (c) ' 1 '
where the choice of the sign in Eq. (63) is related to the region of the test null string location (the sign "-" at 77 <0 and "+" at j]>0), and the function A(/n + I\t) is
determined by the explicit form of the function ^(7); for example, for expression (11),
A( /0+ Pr) = +- Jln
111 f ^ 1 -1 - Jo
1 —
I /0+^
— G
J
Note that equality (52) put restrictions on the values of parameter r, i.e. they determine the boundaries of the region, in which the moving test null string becomes "visible" for the null string generating the gravitational field and interacts with it.
60
4. EXAMPLES OF TEST NULL STRING MOTION
In the case z z > 0, the solutions of the equations of motion for the test null string in regions I and III look like
t + p = + -Jln 1 P #
(
1--
fo
\
-s
t-p = qo + fo
fo+Pi^ ^2 (a) Y ( P (a))2
P (a)) P (a)
P
f ( z ) = Fi - P fo+ P2 H.
(65)
(66)
(67)
In equalities (65) and (69), the upper sign is selected for region I (re(-»;0)), and the lower one for region III (,e(0; +<»)). The interaction zone boundaries in those regions determine the minimum possible value of the right-hand side in equality (67) in region I, which equals zero and is reached at || = F](P2) 1 -/0(p) 1 >0 (the leftmost boundary of the interaction zone), and the maximum possible value of the right-hand side in equality (67) in region III, which equals / and is reached at
, ^ /0 ((P2) 1 + (p) 1) - F1 (P2) 1 > 0 (the rightmost boundary of interaction zone).
Fig. 1. Plots of the functions t(,), p(,) and z(,) in region I in the case zT >0 at /0 = 100, P = 1, P2 = /0-1, % = C = V = 5, s = 107, q0 = -10 and for various F = 100 (z (,)), 90 (z2 (,)), and 50 (z3 (,)).
61
1 t,p z Z2C0 Z3CO
J T
1000 2 000 3 000 4000 5 000 6000 7 000 SOOO 900CT
Fig. 2. Plots of the functions t(r) , p(r) and z(r) in region III in the case zT > 0 at / = 100, p = 1, P2 = f-1, g = £ = p = 5, s = 107, q0 =-10 and for various F = 100 ( Z (t) ), 90 (Z2 (r) ), and 50 (Z3 (r)).
In Figs. 1 (for region I) and 2 (for region III), the functions t(r), p(r) and z(r) are plotted for the case z T > 0, at the certain fixed values of constants p and P2, and for three different values of constant F1. The figures demonstrate that the test null string, when approaching the right boundary of the interaction zone (Fig. 2), becomes always pushed out by the gravitational field (the variable z) to the infinity within a very short time interval.
From the given examples of the test null string motion, it follows that, in the case where the initial momenta of the test null string points along the axis z differ from zero (P2 (ct) ^ 0), every test null string in the "interaction zone" is always either pushed out to
the infinity (Fig. 2) or attracted to the plane, where the null string generating the gravitational field is located, irrespective of how far it is, by the gravitational field (the variable z) within a very short time interval. The specific scenario depends on the test null string position with respect to the plane, in which the null string generating the gravitational field is located, and the direction of the test null string motion along the axis z. In our opinion, the presence of trajectory sections with this anomalous behavior for every test null string in the "interaction zone" may indirectly testify that the ability to inflate can be an internal property of the gas of null strings. However, this statement requires an additional research.
CONCLUSIONS
By analyzing the results of this work, we may suppose that, since separate regions in the gas of null strings are causally independent at the initial time moment, there may appear a domain structure in this gas. In other words, there may exist a large number of separated regions, in which the null strings radially collapse in parallel planes (i.e. they are strictly polarized). The spatial orientation of those planes is random in every domain, without any correlation between neighbor domains. The conditions for such domains to
62
emerge and exist, as well as the physical processes in the interdomain regions, can be a subject of further researches of the gas of null strings.
References
1. P. J. E. Peebles, Principles o/physical cosmology (Prinston University Press, 1994).
2. A. D. Linde, Partical Physics and Inflationary Cosmology (Harwood, Chur, 1990).
3. T. Vashaspati and A. Vilenkin, Phys. Rev. D 30, 2036 (1984).
4. A.Vilenkin and E. P. S. Shellard, Cosmic String and Other Topological De/ects (Cambridge Univ. Press, Cambridge, 1994).
5. T. W. B. Kibble and M. B. Hindmarsh, arXiv:hep-ph/9411342v1.
6. D. P. Bennet, "High resolution simulations of cosmic string evolution: Numerics and long string evolution,'' in Formation and Evolution o/ Cosmic Strings (Cambridge University Press, Cambridge, 1990; Ed. by G. Gibbons, S. Hawking and T. Vachaspati).
7. A. Schild, Phys. Rev. D 16, 1722 (1977).
8. C. T. Hill, D. N. Schramm, and J. N. Fry, Commun. Nucl. Part. Phys. 19, 25 (1999).
9. S. N. Roshchupkin and A. A. Zheltukhin, Class. Quant. Grav. 12, 2519 (1995).
10. A. P. Lelyakov and A. S. Karpenko, Scientific Notes o/Taurida National V. I. Vernadsky University, Ser. Physics and Mathematics Sciences 24 (63), No 2, 3 (2011).
11. A. P. Lelyakov, A. S. Usachov, and R. A. Babadzhan, Scientific Notes o/ Taurida National V. I. Vernadsky University, Ser. Physics and Mathematics Sciences 26 (65), No 2, 18 (2013).
Леляков О. П. Динамжа пробно!" нуль-струни в гравггацшному полi замкнено! «розмазано!» нуль-струни, що прямуе в площиш / О. П. Леляков, А. С. Карпенко, Р.-Д. О. Бабаджан // Вчет записки Тавршського национального утверситету iMeHi В. I. Вернадського. Серш : Фiзико-математичнi науки. - 2014. - Т. 27 (66), № 2. - С. 50-64.
У робота розглянута динамжа пробно! нуль-струни в гравггацшному пот замкнено! «розмазано!» нуль-струни, що радiально розширюеться або радiально колапсуе в площиш , за умови, що початкове обертання пробно! нуль-струни було вщсутне. KnwHoei слова: «розмазана» нуль-струна, гравггацшне поле.
Леляков А. П. Динамика пробной нуль-струны в гравитационном поле замкнутой «размазанной» нуль-струны движущейся в плоскости / А. П. Леляков, А. С. Карпенко, Р.-Д. А. Бабаджан // Ученые записки Таврического национального университета имени В. И. Вернадского. Серия : Физико-математические науки. - 2014. - Т. 27 (66), № 2. - С. 50-64. В работе рассмотрена динамика пробной нуль-струны в гравитационном поле замкнутой «размазанной» нуль-струны радиально расширяющейся и радиально коллапсирующей в плоскости, при условии, что начальное вращение пробной нуль-струны отсутствует. Ключевые слова: «размазанная» нуль-струна, гравитационное поле.
Список литературы
1. Peebles P. J. E. Ppinciples of physical cosmology / P. J. E. Peebles. - Prinston University Press, 1994. -534 p.
2. Linde A. D. Partical Physics and Inflationary Cosmology / A. D. Linde. - Harwood, Chur, 1990. - 380 p.
3. Vachaspati T. Formation and Evolution of Cosmic Strins / T. Vachaspati and A. Vilenkin // Phys. Rev. D. - 1984. - Vol. 30. - P. 2036.
4. Vilenkin A. Cosmic Strings and Other Topological Defects / A. Vilenkin, E. P. S. Shellard. - Cambridge Univ. Press, Cambridge, 1994. - 534 p.
5. Kibble T. W. B. Cosmic strings / T. W. B. Kibble and M. B. Hindmarsh // arXiv.org. - Режим доступа : arXiv:hep-ph/9411342v1.
63
6. Bennet D. P. High resolution simulations of cosmic string evolution: Numerics and long string evolution / D. P. Bennet ; G. Gibbons, S. Hawking and T. Vachaspati, eds. // Formation and Evolution of Cosmic Strings. - Cambridge University Press, Cambridge, 1990.
7. Schild A. Classical null strings / A. Schild // Phys. Rev. D. - 1977. - Vol. 16. - P. 1722.
8. Hill C. T. Cosmological Structure Formation From Soft Topological Defects / C. T. Hill, D. N. Schramm, and J. N. Fry // Commun. Nucl. Part. Phys. - 1999. - Vol. 19. - P. 25.
9. Roshchupkin S. N. Friedman Universes and exact solutions in string cosmology / S. N. Roshchupkin, A. A. Zheltukhin // Class. Quantum. Grav. - 1995. - Vol. 12. - P. 2519.
10. Lelyakov A. P. Scalar field potential distribution for a "thick" collapsing null string / A. P. Lelyakov, A. S. Karpenko // Scientific Notes of Taurida National V. I. Vernadsky University. - Series: Physics and Mathematics Sciences. - 2011. - Vol. 24(63), No 2 - P. 3-12.
11. Lelyakov A. P. Scalar Field Potential Distribution for a Closed Radially Expanding Null String in Plane z = 0 / A. P. Lelyakov, A. S. Usachov, and R. A. Babadzhan // Scientific Notes of Taurida National V. I. Vernadsky University. - Series : Physics and Mathematics Sciences. - 2013. - Vol. 26 (65), No 2. - P. 69-78.
Received 19 August 2014.
64
