Abstract
We take arbitrary gravitational perturbations of a 5d spacetime and reduce it to the form an axially symmetric warped braneworld. Then, we write the filed equations for the linearized gravity perturbations. We obtain the equations that describes the graviton, gravivector and the graviscalar fluctuations and analyse the effects of the Schrödinger potentials that appear in these equations.
Braneworld with induced axial symmetry
Edgard Casal de Rey Neto
Instituto Tecnológico de Aeronáutica - Divisão de Física Fundamental, Praça Marechal Eduardo Gomes 50, São José dos Campos, 12228-900 SP, Brazil
ABSTRACT
We take arbitrary gravitational perturbations of a 5d spacetime and reduce it to the form an axially symmetric warped braneworld. Then, we write the filed equations for the linearized gravity perturbations. We obtain the equations that describes the graviton, gravivector and the graviscalar fluctuations and analyse the effects of the Schrödinger potentials that appear in these equations.
I. INTRODUCTION
In the almost all works on the Randall-Sundrum (RS) braneworlds [1], the axial gauge is used to derive the linearized gravity dynamics. In this gauge there are no fluctuations transverse to the brane, and the scenario is axially symmetric. An other important gauge is the harmonic (de Donder) gauge for which, in 5D, the h55-graviscalar and the h5µ-gravivector 4D fluctuations can be non zero, breaking the axial symmetry. However, as pointed out in [2], a new coordinate frame in 5D can be found, where the metric becomes axially symmetric even with h55, h5µ¹ 0. By following [2], we call such coordinate frame the local frame. Here, we analyse the field equations for the vacuum fluctuations that arise in the local frame, using the 5D de Donder gauge.
II. AXIALLY SYMMETRIC BRANEWORLD
The 5D metric is expanded as gAB = hAB+hAB where A,B = 0,1,2,3,5, hAB = diag(-1,1,1,1,1) and hAB small gravity fluctuations. The 5D line element is
The above spacetime take the axially symmetric form
where x(A) = xB, with the given by
where Pµ = h5µ, j = h55, y = x(5) and e2 = -1,1. The physical 4D metric can be given by g(µ)(n), or (µ)(n) = e-2fg(µ)(n). If f = f(y), we have
Then, we assume that (5) satisfies the action
where g = det(g(A)(B)), gb = det(g(µ)(n)) and k = M*-3. The vacuum solution gives f'(y)2 = -k2L5(y)/12 and f"(y) = k2s(y)/12.
III. LOCAL FRAME GRAVITATIONAL FLUCTUATIONS
We work in the conformal frame where
The equations for the gravitational fluctuations are derived with g(µ)(n) = ¶B (...), with [(A), (B)] ¹ 0. The hAB satisfies ¶BhAB = 0, = 0, which means that
The the "prime" represents ¶z. The field equations are
where = hAB¶A¶B. The local frame equations depends on the comutator of partial derivatives of the warp function. The equation for the scalar do not changes [3].
Extended KK-gravity: L5 = s = 0. The system (9)-(11) decouples to
The scenario is an extended Kaluza-Klein gravity with a no compact extra dimension. The gauge conditions enable us to write the 4D tensor hmn in terms of spin-2, spin-1 and spin-0 fluctuations. The vacuum is flat.
IV. WARPED GEOMETRY FLUTUATIONS
Graviscalar. Consider T = 0. The j is re-scaled to j(x,z) = e-3f(z)/2 (x,z) and we look for solutions (x,z) = (x)ys(z), with ¶a¶a (x) = m (x). Then, (9) implies
with Vs(z) = -( f" - f'2). The vacuum solutions for j are of no interest, because the compatibility condition between (9) and (10) force us to set j = 0 [3].
Gravivector. With j = 0, eq. (10) in vacuum gives
where Vv(z) = -5(f'2+f"), for yv defined by µ(x,z) = (x)yv(z), ¶a¶a m(x) = (x),and µ(x,z) = ef(z)Pµ(x,z). For RS warp, the potential is Vv(z) = -10kd(z). Then, we have masive solutions with = -25 k2, whereas = -36k2 in the coordinate frame [3].
The compatibility condition between (10) and (11) is
If f = Log(k|z|+1), we have
For |z| > 0, (17) is satisfied by yv = 25av(k|z|+1), where av is a constant. At z = 0, (17) implies, av = 0. With the smooth warp, f(z) = Log(k2z2+1), eq. (16) becomes
At z = 0, we have y'v(0) = 0. A solution of (15) that satisfy (18), is the massive mode described by
with mass = -3k2( 7 + 25k2z2 ) / ( 1 - 2k2z2 )2.
The Fig. 1 shows the variation of with the extra coordinate. It is almost constant for small z and diverges for z ® ± z* = ±1/ . On the |z| = 0 3-brane, µ (x) = where p2 = (0) = -21k2 and, pmcm = 0
Graviton. To obtain the graviton potential we take cµ = 0 and = 0. Then, the eq. (11) implies
where Vg = -(f'2 - f"). This potential reproduces the RSII result for f(z) = Log(k|z|+1).
V. CONCLUSIONS
In the no warped scenario, the vacuum fluctuations are described by three independent wave equations which describes the 4D scalar, vector and tensor fluctuations on the |z| = 0 3-brane. In warped scenarios, there are no scalar propagation on the 3-brane vacuum. For the RS warp, there are no gravivector on the 3-brane. For the smoothed warped braneworld, we obtain a tachyonic mass solution for the gravivector, that also satisfies the compatibility condition. This solution becomes a massless spin-1 fluctuation if L5® 0.
VI. ACKNOWLEDGEMENTS
I thank Dr. R. M. Marinho Jr. This work supported by brazilian agency CNPQ (gr. 150854/2003-0) and ITA.
References
[1] L. Randall and R. Sundrum, Rev. 83, 4690 (1999) [hep-th/9906064]; Ibd. 83, 3370 (1999) [hep-ph/9905221].
[2] J. Ponce de Leon, Grav. Cosmol. 8 (2002) 272-284. [gr-qc/0104008]; Int. J. Mod. Phys. D11, 1355 (2002) [gr-qc/0105120].
[3] Y. S. Myung, [hep-th/0010208].
Received on 13 October, 2005
- [1] L. Randall and R. Sundrum, Rev. 83, 4690 (1999) [hep-th/9906064];
- [2] J. Ponce de Leon, Grav. Cosmol. 8 (2002) 272-284. [gr-qc/0104008];
- Int. J. Mod. Phys. D11, 1355 (2002) [gr-qc/0105120].
Publication Dates
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Publication in this collection
16 Jan 2006 -
Date of issue
Dec 2005
History
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Received
13 Oct 2005