Braneworld with induced axial symmetry

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 h_5µ-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, h_5µ¹ 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 g_AB = h_AB+h_AB where A,B = 0,1,2,3,5, h_AB = diag(-1,1,1,1,1) and h_AB small gravity fluctuations. The 5D line element is

The above spacetime take the axially symmetric form

where x^(A) = x^B, with the given by

where P_µ = h_5µ, 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)), g_b = 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 h_AB satisfies ¶^Bh_AB = 0, = 0, which means that

The the "prime" represents ¶_z. The field equations are

where = h^AB¶_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 h_mn 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)y_s(z), with ¶^a¶_a (x) = m (x). Then, (9) implies

with V_s(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 V_v(z) = -5(f'²+f"), for y_v defined by _µ(x,z) = (x)y_v(z), ¶^a¶_{a m}(x) = (x),and _µ(x,z) = e^f(z)P_µ(x,z). For RS warp, the potential is V_v(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 y_v = 25a_v(k|z|+1), where a_v is a constant. At z = 0, (17) implies, a_v = 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, p^mc^m = 0

Graviton. To obtain the graviton potential we take c_µ = 0 and = 0. Then, the eq. (11) implies

where V_g = -(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

Publication Dates

• Publication in this collection
  16 Jan 2006
• Date of issue
  Dec 2005

History

• Received
  13 Oct 2005