CC BY
38
Chelyabinsk Physical and Mathematical Journal. 2019. Vol. 4, iss. 1. P. 118-124.
DOI: 10.24411/2500-0101-2019-14111
GRAVITATIONAL LENSING
BY MORRIS — THORNE TYPE WORMHOLE
R.F. Lukmanova", I.V. Obukhovb, M.M. Tayupovc
M. Akmullah Bashkir State Pedagogical University, Ufa, Russia "r.lukmanova@list.ru, bigor.obukhov.1992@mail.ru, cmin4elcman@yandex.ru
Wormholes are exact solutions of Einstein's equations and such objects are physically more likely to exist as a result of some high energy process. Using Will — Bodenner method an exact formula for light deflection is obtained. Some special features like energetics and tidal forces of the considered solution are pointed out.
Keywords: wormhole, gravitational lensing, tidal forces.
Introduction
Wormholes are objects of research in the frontier of theoretical physics. Einstein's gravitational field equations predict not only black holes, but also wormholes and neither of these has yet been ruled out by observations. Wormhole physics has recently gained renewed impetus following the pioneering work by Morris and Thorne [1] and require for their construction what is called "exotic mater", which violates the null energy condition (NEC) of general theory of relativity. A traversable wormhole interesting physically as it might be an observable object.
Gravitational lensing is an important and effective window to look for signatures of peculiar astrophysical objects such as black holes. This field of activity has lately attracted a lot of interest among the physics community. There is however another exiting possibility that has not received enough attention to date: It is lensing by stellar size traversable wormholes which are just as interesting objects as black holes are. The idea of this kind of matter has received further justification in the notion of "phantom field" or "dark matter" invoked to interpret the observed galactic flat rotation curves or the present acceleration of the Universe. Unfortunately, work on wormhole lensing is still relatively scarce though observables in wormhole lensing have the potential to serve a dual purpose: They would establish not only the wormhole itself but also throw light on the existence of classical exotic matter. This fact provides the basic motivation for the present theoretical investigation.
1. Wormhole geometry and energetics of wormhole solution
Consider a static and spherically symmetric Morris — Thorne traversable wormhole in the Schwarzschild coordinates given by
ds2 = -e2Hr)dt2 + + r2dQ2 (1)
1 _ bir) ' v '
The reported study was funded by the Russian Foundation of Basic Research according to research project no. 18-32-00377.
which describes a wormhole geometry with two identical, asymptotically flat regions joined together at the throat ro > 0. $(r) and b(r) are arbitrary functions of the radial coordinate r, denoted as the redshift function and the shape function, respectively. The radial coordinate has a range that increases from a minimum value at r0, corresponding to the wormhole throat, to ro.
To avoid the presence of event horizons, $(r) is impose to be finite throughout the coordinate range. At the throat r0, one has b(r0) = r0, and a fundamental property is the flaring-out condition given by (b'r — b)/b2 < 0, which is provided by the mathematics of embedding [1].
In particular, for a constant redshift function, $,r(r), static observers are geodesic. Thus, the convention used is that $,r(r) is positive for an inwardly gravitational attraction, and negative for an outward gravitational repulsion.
Recently, Harko et al. proposed a wormhole solution with redshift function
$(r) = — r0 (2)
r
and the shape function
b(r) = r0 + Yr0 (1 — r^j , (3)
where 0 < 7 < 1. This solution is considered throughout this work. The mass function is of used solution is
m(r) = f (1 - ^ . (4)
Using the Einstein field equation, G= 8nTMV, the (orthonormal) stress-energy tensor components in the bulk are given by
2
Yr°
P
8nr4'
r0{r(r - 2r0) - Y(r - r0)(r + 2r0)}
Pt
8nr5 '
ro {r3(Y - 1) + 27rg + r2ro (y + 5) - 2rrQ(3Y +1)}
16nr6
Here p is the energy density, pr and pt are the radial pressures measured in the orthogonal direction to the radial direction. We thus obtain the NEC condition
p + Pr = ro{2ro(Yro - r) - r2(Y - 1)} < 0 Vr,
8nr5
which shows null energy condition violating. Using Eq. (1), one has
00
1 db Yro ^ n n 1 f 2A Yro n
p = 8nrodr = 8nro4 >0, ^WEC = W prdr = ^ >0,
r 0
where HWEc is the total amount of Weak Energy Condition (WEC) violating matter, which, in this case, is positive. Let us consider the Lynden-Bell, Katz and Biscak [2; 3] form of gravitational energy EG defined by EG = HWEC - EM, where EM is the sum of
other forms of energy like rest energy, kinetic energy, internal energy etc. and is defined by
oo
Em = 1 f p(grr)1/2r2dr.
In the present case, we find that To^Y arctan
E
M
8n
> 0 ^ EG = To(Y arCtan ^) > 0,
16n
which implies that, even though the mass in Eq. (4) is positive, the gravitational energy Eg is positive, hence repulsive in nature.
2. Tidal forces and influence of 7
We start with the general form of a static spherically symmetric physical metric
, 2 F(r) , 2 dr2 dr2 = dt2 + —
G (r) F (r)
+ R2(r)[dd2 + sin2 0dp2].
(5)
For a traveller in a static orthonormal basis, we shall denote the only nonvanishing components of the Riemann curvature tensor as Rom, Ro2o2, Ro3o3, R1212, R1313, and R2323. Radially freely falling observers with conserved energy E are connected to the static orthonormal frame by a local Lorentz boost with an instantaneous velocity given by
v =
1
F GE
1/2
Then the nonvanishing Riemann curvature components in the Lorentz boosted frame with velocity v are (k = 2, 3):
R
0101
R
R R
0303
1313
0313
R0k0k + (R0k0k + R1k1k) sinh2 01., R1k1k + (R0k0k + R1k1k) sinh2 OL-, (Rokok + R1k1k) sinh a cosh a,
where sinh a = v/V 1 — v2. The relative tidal acceleration Aaq between two parts of the
traveller's body in his orthonormal basis is given by Aa-j = -Rofjôj^C11, where £ is the
vector separation between the two parts [4]. Thus the curvature components contributing to tidal force on the traveller in the Lorentz boosted frame are Rom, Rolce, and R (Components in the coordinate basis are not required here). The components R<3202 are given by
D303-
R
0202
1
R
R (E2sG' - F') - (R"G - RG
E2
o(s) 1 D' r0202 + R
(ex) 0202,
where
2 F F E2 = — + —
G G V 1 - v
Es2 + Eex,
where E^ represents the value of E2 in the static frame and E^ represents the enhancement in E^ due to geodesic motion.
2
v
2
It is easy to verify that the term |R0202l actually represents the curvature component in the static frame, viz., R0202 = R>202. Thus, only the term R02o2(= sinh6 a(R0202 + Ri2i2) represents overall enhancement in curvature in the Lorentz-boosted frame over the static frame. It is this part that needs to be particularly examined as the observer approaches the throat. Note also that the energy E2 is finite (it can be normalized to unity) and so are E^ and E^. As the throat is approached, (F/G) ^ 0, v ^ 1 such that E2 ^ Ee2x.
From Eqs. (2), (3), (5) we find
F (r) G(r) =
1 - ro
r
1 + Y( 1 -^
R(r)
r.
Eqs. (6), (7)-(9) imply that
! _ If 1+ y (! -
r l V r
2ro
e r
(7)
R
(s) ro A ro\ A ro oooo = -3 1 - - 1 - Y-
and
R(ex) =
Roooo = 2r3
In the limit r ^ ro we have
ro
1 - y - 2- + 2Y — rr
ro
v
1- v2
lim R
r^r o
(s)
o2o2
0, lim RoOoO
r^r o
Y - 1 2ro2
1 — v2
Here the second limit tends to zero as ro ^ 0. These mean, that excess tidal force in the geodesic frame near the throat becomes arbitrarily large, i. e. Ro<2o2 ^ ^ when Y ^ 0. In contrast, however, depending on the values of y, the tidal forces may become arbitrarily .small when y is increasing. What is of interest here is the possibility of having large or small excesses in curvature by controlling Y.
3. Bending angle in weak deflection limit
Using Harko solution (1)-(3), it can be verified that rps = r0 is the photon sphere, which is also the throat radius rth = r0. Following Bodenner and Will [5], the light path equation in the equatorial plane (0 = n/2) is given by (u = 1/r), when expanded in powers of u, gives:
dOu + = ro(1 - Y) rpY + d^2 +U = 2b2 b2 U +
— (1 + Y) -2 (1 + Y) b2
u2 + O(u3).
:iq)
where b is the impact parameter. Defining
M = ^ (1 — y) , a =--^ •
4( Y)' (1 — Y)2 '
we find that the path equation can be re-written as
d2u 2M r ., ^ . ^
+ u = — [1 + a (Mu) + O (Mm)2]
d^
b2
2
2
v
This equation formally resembles the Schwarzschild path equation in isotropic coordinates [5]. With the dimensionless parameter Mu0 = M/R assumed to be small, we take an ansatz u = u0 [(Mu0)0 cos ^ + (Mu0) 8u + (Mu0)2 8u2 + .. . ] , then putting it in Eq. (10) and collecting terms of equal powers of Mu0, we get
d2Uo
d^2
d2 d^2 d2 #u2 d^2
+ uo = 0,
+ iu1
+
2
(uob)2' 2a cos (uob)2
The solution is
u uo
2Mu0 a (Mu0)2 ^ sin ^
cos +
+
2
11)
(U06) (U06)2
The minimum of r is the closest approach distance R0 and it is the maximum of u,
, um,
i.e., um, which occurs at ^ = 0 giving (um = 1/R0):
2Mu0
Um
1 +
M)2
Restoring the value of M in the above and inverting, we obtain
(Y - 1)r0
Uo = Um +
2b2
'12)
From the definition of b [5], we get
b
1 1 ro
ro +--^ Um = -fZ- ^ T + 77.
Um Ro b b2
'13)
Putting the value of b in Eq. (12) and expanding it in powers of um, we get
(Y - 1) r0_
Uo = Um +
Um + O(Um).
'14)
At ^ = n/2 + 8 (where the one-way bending 8 is small so that cos 8 =1, sin 8 = 8), we have u ^ 0 and we get from Eq. (11): A^ = 28 = 4Mu0 + an (Mu0)2 . Putting u0 from Eq. (14) in the above, restoring M, and expanding in powers of um, we get
ro2
= (1 - Y) roUm + (y - Y2 - 0 Um =
(1 - y ) rn r
b
+ # (Y - Y2 - 0 .
where we have used from Eq. (13), um ~ 1/b. We can write the bending also in terms of the closest approach distance using Eq. (13), um = 1/R0, which gives
(1 - y) ro r a = A^ =----+
Ro
2Ro
- Y2 - 1) .
The bending angle as a function of the ratio between the closest approach distance and the throat radius of the wormhole has been obtained. It is clear from obtained lens equation that both the first and the second terms affected by crucial parameter 7. In particular increasing of 7 decreases the deflection angle. As 7 ^ 0 we get a massless wormhole solution.
2
2
Conclusion
Gravitational lensing by wormholes and investigating of their properties may help in interpreting several outstanding problems in astrophysics. Lensing phenomena in the wormhole environment offers a good possibility that one might detect the presence not only of a wormhole, which is by itself interesting, but also of the presence of naturally occurring exotic mater much advocated on galactic or cosmological scales [6; 7].
In this paper we examined gravitational energy of the wormhole solution described by Harko et al. [8] and using Horovitz and Ross algorithm showed that the tidal forces may become arbitrarily small when y is increasing. What is of interest here is the possibility of having large or small excesses in curvature by controlling y in considered solution.
Finally, an exact formula of the bending of light passing by positive mass mouth using Will — Bodenner method is obtained and influence of y is shown.
References
1. Morris M.S., Thorne K.S. Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity. American Journal of Physics, 1988, vol. 56, no. 5, pp. 395-412.
2. Lynden-Bell D., Katz J., Bicak J.N. Energy and angular momentum densities of stationary gravitational fields Physical Review D, 2007, vol. 75, no. 2, p. 024040.
3. Katz J., Lynden-Bell D., Bicak J.N. Gravitational energy in stationary spacetimes. Classical and Quantum Gravity, 2006, vol. 23, no. 23, p. 7111.
4. Horowitz G.T., Ross S.F. Naked black holes. Physical Review D, 1997, vol. 56, no. 4, pp. 2180-2187.
5. Bodenner J., Will C.M. Deflection of light to second order: A tool for illustrating principles of general relativity. American Journal of Physics, 2014, vol. 71, no. 8, p. 770.
6. Bhattacharya A., Bagchi B., Garipova R. et al. Modeling by autonomous Hamiltonian system: fixing the sign of a parameter. Indian Journal of Physics, 2012, vol. 86, no. 6, pp. 463-469.
7. Lukmanova R., Kulbakova A., Izmailov R., et al. Gravitation microlensing by Ellis wormhole: second order effects. International Journal of Theoretical Physics, 2016, vol. 55, no. 11, pp. 4723-4730.
8. Harko T., Kovacs Z., Lobo F.S.N. Electromagnetic signatures of thin accretion disks in wormhole geometries. Physical Review D, 2008, vol. 78, no. 8, p. 084005.
Accepted article received 30.03.2018 Corrections received 23.10.2018
Челябинский физико-математический журнал. 2019. Т. 4, вып. 1. С. 118-124.
УДК 530.12 DOI: 10.24411/2500-0101-2019-14111
ГРАВИТАЦИОННОЕ ЛИНЗИРОВАНИЕ КРОТОВОЙ НОРОЙ ТИПА МОРРИСА — ТОРНА1
Р. Ф. Лукманова", И. В. Обухов6, М. М. Таюповс
1 Башкирский государственный педагогический университет им. М. Акмуллы, Уфа, Россия
"r.lukmanova@list.ru, bigor.obukhov.1992@mail.ru, cmin4elcman@yandex.ru
Кротовые норы являются точными решениями уравнения Эйнштейна. Физически подобные объекты могут существовать как результат некоторых высокоэнергетических процессов. C использованием метода Уилла — Боденнера получена точная формула для угла отклонения света. Рассмотрены и проанализированны некоторые физические особенности, такие как гравитационная энергия и приливные силы.
Ключевые слова: кротовая нора, гравитационное линзирование, приливные силы.
Поступила в редакцию 30.03.2018 После переработки 23.10.2018
Сведения об авторах
Лукманова Регина Фануровна, аспирант физико-математического факультета, Башкирский государственный педагогический университет им. М. Акмуллы, Уфа, Россия; e-mail: r.lukmanova@list.ru.
Обухов Игорь Валерьевич, студент физико-математического факультета, Башкирский государственный педагогический университет им. М. Акмуллы, Уфа, Россия; e-mail: igor.obukhov.1992@mail.ru.
Таюпов Мансаф Масхутович, студент физико-математического факультета, Башкирский государственный педагогический университет им. М. Акмуллы, Уфа, Россия; e-mail: min4elcman@yandex.ru.
1 Исследование поддержано Российским фондом фундаментальных исследований, грант № 18-
32-00377.
