CC BY
38
UDC 530.1; 539.1
COMPLEX COORDINATE TRANSFORMATIONS TO GET EXACT AXIALLY SYMMETRIC
SOLUTIONS IN F(R)-GRAVITY
M. De Laurentis1'2, O. Luongo2'3'4
1 Tomsk State Pedagogical University, 634061 Tomsk and Tomsk State University, 634050 Tomsk, Russia. 2 Dipartimento di Física, Universitá di Napoii "Federico IF', Via Cinthia, 1-80126, Napoli, Italy. 3Istituto Nazionale di Física Nucleare (INFN), Sez. di Napoli, Via Cinthia, 1-80126, Napoli, Italy. 4Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, DF 04510, México, Mexico.
E-mail: felicia@na.infn.it, luongo@na.infn.it
We present a strategy to get axially symmetric solutions in f (R) gravity by starting from spherically symmetric space-times. To do so, we assume the validity of a complex coordinate transformation, which acts on the spherically symmetric metric
f(R)
particular emphasis to define a class of compatible axially symmetric solutions, which fairly well describe the motion in cylindrical geometries in the field of f (R), in two different classes of coordinates. We demonstrate that our approach is general and may be applied for several cases of interest. We also show that our treatment is compatible with the standard approach of general relativity, evaluating the motion of a freely falling particle in the context of our metric.
f(R)
1 Introduction
Modified theories of gravity have been introduced to address several inconsistencies that general relativity seems to be unable to explain [1]. In particular, the issue of understanding in which manners exact solutions may be modified, involving alternative theories of gravity has become a highly relevant topic [2]. Moreover, the need of understanding whether those theories may reproduce the general relativity results is essential to check their viabilities. Among all the possibilities, we circumscribe our attention to f (R)-gravity, where f (R) represents an analytic function the R
principle and candidate as a serious alternative to standard Einstein's gravity, especially for describing both dark matter and dark energy effects at different cosmic scales [3]. The great disadvantage of those f(R) f(R)
their shapes with present-time data has become essential for their determinations. Here, we investigate some classes of exact solutions. Their importance lies on the fact that in order to describe astrophysical compact objects, e.g. black holes or active g alactic nuclei and so forth, those solutions may represent a
f(R)
theories should be consistent with general relativity-outcomes, reproducing standard exact solutions, i.e. Schwarzschild spherically symmetric solution or Kerr axially symmetric solution, etc. In addition, getting new suitable solutions may be of great physical interest,
f(R)
effects at high energy astrophysical regimes, fixing additional bounds on the f (R) determination. Thus, strategies to find out either exact or approximate solutions in modified gravities become profoundly
f(R)
gravities. To better clarify this fact, we may start from recent developments spanning from spherically f(R)
static spherically symmetric formulations in presence of perfect matter fluid in metric formalism picture [5,6]. Keeping in mind those results, we extend the standard formalism by proposing a more complete treatment to find out axially symmetric solutions, by means of a complex coordinate transformation acting on the spherically symmetric metric [7,8]. In particular, interior solutions have not so far obtained to characterize a whole exact solution. The problem lies on the loss of a symmetry degrees which makes the corresponding derivation of any solution highly-complicated. Among all techniques, we employ the Newman and Janis treatment, which seems to alleviate the problem of losing symmetry degrees. They argued that one may obtain an axially symmetric solution, i.e. a Kerr-like metric, by considering a complex transformation on the spherical solution [9,10]. In [11] a self consistent and rigorous proof that the Kerr metric can be effectively determined from a complex transformation on the Schwarzschild solution, has been given. Here, we extend such a formalism, showing that the complex transformation may be framed in the f(R)
as follows. In Sec. II, we describe the method and we highlight its fundamental properties. To do so, we
TSPU Bulletin. 2014. 12 (153)
consider the general treatment and we specialize it to the case of pure spherically symmetric solutions. We therefore obtain the corresponding modifications to the standard Kerr metric in the context of f (R) gravity and we describe some dynamical properties of this solution, by means of circular orbits in the framework of the Hamiltonian formalism. We therefore demonstrate that our strategy is general and may be extended to the case of fourth order gravities without stability problems. In Sec. Ill, we summarize our results and we propose possible perspectives of our method.
2 From spherical symmetry to axially symmetric solutions in f (R) gravity
In the framework of f (R) gravity, the action takes
the simple form S = f d4xy—g
f (R) + XLm
By
f»v - f '(R);»v + nf '(R) = X Tf
»v ■
f»v = f '(R)R»v - 2 f (R)9^v ,
^ 3mf'(R) + f '(R)R - 2f (R) = XT,
ds2 = gtt(t, r)dt2 — grr(t, r)dr2 — r2dQ ,
ds2
(a + fir)dt2 — -
1 fir
2 a + fir
-dr2 - r2 dn.
procedure, extending their treatment in the context f(R)
usage of using the Newman-Janis procedure in general relativity only. To this end, as we already stressed before, we employ the existence of Noether symmetries f(R)
corresponding field equations. For our purposes, let us recast the spherically symmetric metric as ds2 = e2*(r)dt2 — e2X(r)dr2 — r2dn, with gtt(t,r) = e2*(r) and grr (t,r) = e2X(r\ Hereafter, our convention is to refer to time-like components as tt or 00, whereas space-like as rr or ii, with ¿running from i = 0 to i = 3.
Considering the suitable Eddington-Finkelstein coordinates, i.e. (u,r,6,$), which represent a viable choice for our coordinate representation, after simple algebra, we definitively get ds2 = e2^(r^du2 ± 2eA(r)+^(r)dudr — r2dQ. Thus, the matrix associated to the metric is rewritable in terms of a null tetrad as:
varying it, in terms of the metric gMV, one argues the corresponding field equations:
(1)
where TMV represents the standard energy-momentum tensor for dust-like matter, which can be expressed in
the form: Tuv = rj,^ cons^an^ %
contains the gravitational constant G, sinee X = —-—,
c4
g
Our formalism involves the use of spherically-symmetric space-time as starting point. In fact, we set up our treatment by assuming the most general spherically symmetric space-time below:
g»v = i^nv + ivnM — mMmv — mv m ,
where lM, nM, m^d m» should satisfy
lMlM = mMmM = nMnM = 0, l»n» = —m»m» = 1, l»m» = n»m» = 0 ,
(4)
(5)
(6) (7)
where we assumed the bars as indication of the complex conjugation.
In our case, a generic space-time event becomes
c» ^ x» = x» + iy»(xa),
(8)
(2)
in which we notice that y^(xa) are functions of the real coordinates xa. Analogously, the null tetrad vectors Z£ = , m^, mM), with a = 1, 2, 3, 4, should
satisfy-
in which dQ represents the solid angle. The basic demands consists in employing on it a transformation that maps Eq. (2), providing that the off-diagonal terms vanish. Hence, the spherically symmetric spacetime may be obtained by assuming that Eq. (2) satisfies particular cosmic symmetries. Here, we consider the Noether symmetries and so, after several calculations, we can write down the simplest spherically symmetric space-time as:
~ - dx»
z» ^ zn:(x ) = zp —.
(9)
All this procedure provides a net effect which consists in generating a new metric. The component of such a space-time are real and depend upon complex variables. We have:
(3)
where we assumed a as a combination of auxiliary-constants, e.g. So and ^d fi = k\ [12].
Here, we demonstrate how it is possible to get an axially symmetric solution adopting the Newman-Janis
g»v ^ g»v : x x x ^ R, where we consider:
Z*(x°)|x=x = Z»(x°).
(10)
(11)
From the transformed null tetrad vectors, a new metric is therefore obtained. So, assuming the covariant form,
we can list the corresponding metric components as:
20(f ,6) 9oo =e V( ' ),
g01 =e^(r66) ,
903 =ae^(f ,6)[eA(f ,6) - e*(f gi3 = - ae^6)+A(f,6) sin2 9,
922 = - E2 ,
)] sin2 9,
933
- [E2 + a2 sin2 9e*(f ,6)(2eA(r ,6) - e^(f ,6))] sin2 9.
9oo =
9o3 =
911 = -
r(a + $r) + a2$ cos2 9 " E ,
a(-2ar - 2$E2 + V2$E3/2) sin2 9
2E
$E2
2ar + $(a2 + r2 + E2) 922 = - E2 ,
933 = -
2 a2 (ar + $E2 - V2$E3/2)sin2 9
E2
E
sin2 9.
H
Pi9
0i
9
oo
+
Pi9
oi
9
oo
m2 + PiPj gi:
9
oo
1/2'
dH
dpi
dpi dt
dH dxi '
which permit to numerically obtain the requested orbits. In particular, in the equatorial plane, which corresponds to the case 9 = 9 = 0
conventionally employ a = 1 and $ = 2, without losing generality and we consider the dependence on ^ and on the conjugate momentum p^, which represents an integral of motion. As a consequence, we find out that the coupled equations for {r, 9, ^,pr ,p6} may be numerically integrated, giving compatible trajectories with respect to the ones inferred from the standard Kerr space-time. To better clarify this statement, we explicitly report below the geodesic equations:
Where we assumed that all the other components, i.e. the components that we did not report above, are zero. This procedure is circumscribed to the use of the particular choice of coordinates. However, one can also perform the Newman-Janis algorithm on any static spherically symmetric solutions, by means of the more practically Boyer-Lindquist coordinates. So, evaluating the same steps performed above and the analogous strategy to get the tetrad null vectors in the case of axially symmetric space-time, we simply obtain:
"dX
dH
dpM
= 9MVpv = pM ,
(14)
dpM dH 1 ~
IX = - dX^ = - 2 SX^3 = 9 aPaPfi , (15)
and also the corresponding reduced Hamiltonian:
2ap0 (-2r3 + r2 - 1)
H
a2 (-2(r - 1)r2 - 1) + r5
+ {A(a,p0,r)B(a,p0,r)
1
x (C(a,p0,r) -D(a,p0,r) -pr + 1)}2 , where
A(a,p0, r) = 4a2p^ (-2r3 + r2 - 1)2 -a2 (-2(r - 1)r2 - 1) - r5 , B = (a2 (r2(r(2r - 3)(2r + 1) + 6) - 2) + (2r + 1)r4) and
C(a,p0,r) =
-p0(2r + 1)
,2 (r2(r(2r - 3)(2r + 1) + 6) - 2) + (2r + 1)r4
Again all components, which do not appear above, are zero.
As in standard general relativity, our treatment should be compatible with the motion of a freely falling particle. Hence, we can treat a physical example which accounts for a freely falling particle moving in our so-obtained metric. To do so, we make extensive use of the Hamiltonian formalism, which has the advantage not to show any sign ambiguity which may come from turning points in the orbits [13]. The reduced Hamiltonian, linearly reported in terms of momenta, is:
D(a,p0, r)
pr (a2 + r2 + r) + p 6
(12)
providing H = -p0 and even satisfying the following motion equations:
(13)
we
Soon, it is evident that H is independent from ^ and, as already above stressed, the conjugate momentum p^ is an integral of motion. Finally, the numerical results may be found in [12].
3 Final outlooks and perspectives
In this paper, we considered the framework of f (R) gravity to describe a technique able to get axially symmetric solutions from spherical ones. This treatment has been extensively described by Newman-Janis in a precise algorithm, which takes into account complex transformations. In particular, assuming a spherically symmetric expression for the space-time, we demonstrated that it is possible to extend the complex transformations in the context of f (R) gravity. To do so, we evaluated the null tetrad associated to this method in two different classes of coordinates and we found out the corresponding axially symmetric metrics. In order to understand if the thus obtained space-time works well in the field of particle motion,
4
r
2
TSPU Bulletin. 2014. 12 (153)
we considered a freely falling particle and we showed that its motion is perfectly compatible with the expected standard Kerr metric, which corresponds to the simplest axially symmetric solution in general relativity. Further investigations will be carried forward in order to describe different symmetries by means of the Newman-Janis strategy. In particular, measuring
possible corrections due to f (R) around compact objects, e.g. evaluating possible discrepancies from the standard cases of accretion disks, one would constrain f(R)
f(R)
reconstructions.
References
[1] Capozziello S. and De Laurentis M. 2011 Phys. Kept. 509 167.
[2] Bamba К., Capozziello S., Nojiri S. and Odintsov S. D. 2012 Astrophys. Space Sei. 342 155-228.
[3] Capozziello S., De Laurentis M., Luongo 0. and Ruggeri A. C. 2013 Galaxies 1 216.
[4] Multamaki T. and Vilja I. 2006 Phys. Rev. D 74 064022.
[5] Multamaki T. and Vilja I. 2007 Phys. Rev. D 76 064021.
[6] Kainulainen K., Piilonen J., Reijonen V., and Sunhede D. 2007 Phys. Rev. D 76 024020.
[7] McManus D. 1991 Class. Quant. Grav. 8 863.
[8] Kransinski A. 1978 /Inn. Phys. 112 22.
[9] Newman E. T. and Janis A. I. 1965 J. Math. Phys. 6 915.
[10] Newman E. T., Couch E., Chinnapared K., Exton A., Prakash A. and Torrence R. 1965 J. Math. Phys. 6 918.
[11] Schiffer M. M., Adler R. J., Mark J. and Sheffield C. 1973 J. Math. Phys. 14 52.
[12] Capozziello S., De laurentis M. and Stabile A. 2010 Class. Quant. Grav. 27 165008.
[13] Berti E. 2014 ArXiv[gr-qc]:1410.4481.
Received 16.11.2014
M. Де Лаурентис, О. Луонго
КОМПЛЕКСНЫЕ ПРЕОБРАЗОВАНИЯ КООРДИНАТ ДЛЯ ПОЛУЧЕНИЯ ТОЧНЫХ
f(R)
Мы описываем стратегию получения точных аксиально-симметричных решений в f (г)-гравитации начиная со сферически симметричного пространства-времени. Для этого мы предполагаем справедливость комплексных преобразований координат, действующих в сферически-симметричной метрике и допускающих введение соответствующей f (R) модификации. Описаны следствия такого подхода, в частности, подчеркивается возможность получения класса совместимых аксиально-симметрических решений, которые довольно хорошо описывают движение в иоле f (R) цилиндрической геометрии в двух различных классах координат. Мы показываем, что наш подход является общим и применим в различных случаях. Мы также показываем, что наш метод совместим со стандартным подходом общей теории относительности при рассмотрении свободно падающей частицы в контексте нашей метрики.
f(R)
Де Лаурентис М., доктор.
Томский государственный педагогический университет.
Ул. Киевская, 60, 634061 Томск, Россия. Университет Неаполя «Федерико II». Via Cinthia, 1-80126, Napoli, Италия. Томский государственный университет.
Пр. Ленина, 36, 634050 Томск, Россия. E-mail: felicia@na.infn.it
Луонго О., доктор.
Университет Неаполя «Федерико II».
Via Cinthia, 1-80126, Napoli, Италия. Национальный институт ядерной физики. Sez. di Napoli, Via Cinthia, 1-80126, Napoli, Италия. Национальный автономный университет Мехико.
DF 04510, México, Мексика. E-mail: luongo@na.infn.it
