Die Methode der nichtlokalen effektiven Wirkung in höherdimensionalen Raumzeitmodellen [Elektronische Ressource] / vorgelegt von Andreas Rathke

Andreas Rathke

Sujets

• Physik

Informations

• Publié par: albert-ludwigs-universitat_freiburg
• Publié le: 01 janvier 2004
• Nombre de lectures: 6
• Langue: English

Extrait

Die Methode der
nichtlokalen e ektiv en Wirkung
in h oherdimensionalen
Raumzeitmodellen
INAUGURAL-DISSERTATION
zur
Erlangung des Doktorgrades
der
Fakult at fur Mathematik und Physik
der
Albert-Ludwigs-Universit at Freiburg im Breisgau
vorgelegt von
Andreas Rathke
aus Hemer
Juni 2003Dekan: Prof. Dr. Rolf Schneider
Leiter der Arbeit: Prof. Dr. Hartmann R omer
Referent: Prof. Dr. R omer
Korreferent: Prof. Dr. Jochum van der Bij
Tag der mundlic hen Prufung: 23.10.2003The Method of the Nonlocal E ectiv e Action
in Higher-Dimensional Spacetime Models
Andreas RathkeContents
1 E ectiv e methods in braneworld dynamics 7
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 The role of the nonlocal e ectiv e action . . . . . . . . . . . . 10
1.1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 The Randall-Sundrum models . . . . . . . . . . . . . . . . . . . . . 12
1.3 E ectiv e actions for warped braneworlds . . . . . . . . . . . . . . . . 14
1.3.1 Kaluza-Klein description . . . . . . . . . . . . . . . . . . . . . 15
1.3.2 Holographic . . . . . . . . . . . . . . . . . . . . . 18
1.3.2.1 One-brane e ectiv e gravity from AdS/CFT . . . . . 18
1.3.2.2 The AdS/CFT interpretation of the two-brane model 22
2 The nonlocal braneworld action 25
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 The e ectiv e action of brane-localized elds and the methods of its
calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 The structure of the braneworld e ectiv e action . . . . . . . . 26
2.2.2 The role of radion elds . . . . . . . . . . . . . . . . . . . . . 29
2.3 Two-brane Randall-Sundrum model: the nal answer for the two- eld
braneworld action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 The e ectiv e equations of motion . . . . . . . . . . . . . . . . . . . . 34
2.5 The recovery of the braneworld e ectiv e action . . . . . . . . . . . . 39
2.6 Green functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6.1 The low-energy limit | recovery of Einstein theory . . . . . 43
2.6.2 Low-energy derivative expansion . . . . . . . . . . . . . . . . 45
2.7 The particle content of the two- eld braneworld action . . . . . . . . 47
2.7.1 The graviton sector . . . . . . . . . . . . . . . . . . . . . . . 48
2.7.2 Problems with the scalar sector of the theory . . . . . . . . . 49
2.7.3 Large interbrane distance . . . . . . . . . . . . . . . . . . . . 51
2.8 The reduced e ectiv e action . . . . . . . . . . . . . . . . . . . . . . . 53
2.8.1 Small interbrane distance . . . . . . . . . . . . . . . . . . . . 55
2.8.2 Large in and Hartle boundary conditions . 57
2.9 The nonlocal action for the RS one-brane model . . . . . . . . . . . 59
3 From nonlocal action to other methods 63
3.1 E ectiv e action of brane-localized elds vs. Kaluza-Klein reduction . 63
3.2 The recovery of the Kaluza-Klein tower . . . . . . . . . . . . . . . . 64
56
3.2.1 The particle interpretation of the transverse-traceless sector . 64
3.2.2 The eigenmode expansion of the Green function . . . . . . . 65
3.2.3 The spectrum and the eigenmodes of the e ectiv e action . . 67
3.2.4 The graviton e ectiv e action in the diagonalization approxi-
mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3 Phenomenological digression: radion-induced graviton oscillations . . 73
3.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.2 Gravitational waves on the -brane . . . . . . . . . . . . . 75+
3.3.3 Quantum oscillations | an analogy . . . . . . . . . . . . . . 76
3.3.4 Gravitational-wave oscillations on the -brane . . . . . . . 78+
3.3.5 High-amplitude RIGO’s on the -brane from M-theory . . . 79+
3.3.6 Graviton oscillations on the -brane . . . . . . . . . . . . . 82
3.3.7 RIGO’s in bi-gravity models . . . . . . . . . . . . . . . . . . . 84
3.3.8 Summary on RIGO’s . . . . . . . . . . . . . . . . . . . . . . . 86
3.4 Holographic interpretation . . . . . . . . . . . . . . . . . . . . . . . . 87
3.4.1 The RS one-brane model . . . . . . . . . . . . . . . . . . . . 87
3.4.2 The RS two-brane model . . . . . . . . . . . . . . . . . . . . 88
4 Conclusions and Outlook 93
A Anti-deSitter space and the geometrical setting of the RS models
99
B Diagonalization of the kinetic and mass terms 101
Bibliography 1051
E ectiv e methods in braneworld
dynamics
1.1 Introduction
Among the most promising candidates for a uni ed description of quantum theory
and gravitation is superstring theory [1]. It predicts the existence of multidimensional
stable objects called \branes". On these branes open strings can end, which also
means that the gauge elds associated with the ends of a string reside on the branes.
Gauge elds, rather than spreading through the entire ten-dimensional spacetime
of superstring theory, are thus con ned to these lower-dimensional objects. This led
to the proposal that the universe we inhabit could be a three-dimensional brane
(\3-brane") on which all matter consisting of gauge elds is trapped [2, 3, 4]. In
string theory, gravity is mediated by the exchange of closed strings and therefore is
not restricted to the branes but propagates through the whole higher-dimensional
spacetime, commonly called bulk. The shape of the bulk can be constrained because
no deviations from four-dimensional gravity have yet been observed (see however
[5]).
Such a \braneworld" model can be constructed from compacti ed at higher di-
mensions, producing a Kaluza-Klein theory of gravity [6]. This results in one massless
graviton mode responsible for the observed behavior of gravity and a tower of mas-
sive graviton states, only observable at higher energies. It is imaginable that the
compacti cation is large enough in order to produce e ects at accelerators of the
next generation [4].
In a di eren t scenario [7, 8]. The higher-dimensional manifold does no longer
1have to \factorize" into our four-dimensional world and the additional dimensions.
In particular, one can assume that the bulk space has an anti-de Sitter (AdS) struc-
ture transverse to the branes. The brane showing the behavior of e ectiv e four-
2dimensional gravity (\our Universe") is put at an Z -orbifold point in the bulk2
1The term non-factorizable geometry is used in the context of higher-dimensional spacetime
models di eren tly from its actual geometrical meaning. A geometry is called
factorizable if its metric can be put in a block diagonal form | with one of these blocks being
the four-dimensional metric | in which the blocks do not depend on coordinates belonging to a
di eren t block of the full metric. cf. p. 16, Sec. 1.3.1.
2An orbifold is a coset space M=H, where H is a group of discrete symmetries of a manifold
78 1 Effective methods in braneworld dynamics
space. Consequently, the bulk metric has a re ection-symmetry with respect to the
position of the brane. The metric of AdS space yields an e ectiv e potential for the
linearized gravitational modes with high barriers on both sides of the brane. This
\volcano potential" leads to a zero-mass graviton state trapped at the position of
the brane and a continuous spectrum of massive modes, which are exponentially
suppressed on the brane [9] but can propagate through the whole bulk. The most
prominent models of this class are the Randall-Sundrum two-brane model [7] (com-
monly called RS1) and the Randall-Sundrum one-brane model [8] (called RS2).
There have been similar earlier proposals (e. g. [10]). The two Randall-Sundrum
(RS) models will form the framework for our present investigation.
The insight that even non-compact extra dimensions allow realistic e ectiv e four-
dimensional gravity has stired a lot of activity in the investigation of braneworlds.
The Realization of zero-mode localization in one-brane models has been explored in
[11]. In particular it was found that the appearance of a massless four-dimensional
graviton in the theory is not necessarily tied to an anti-de Sitter bulk but can also
be achieved in a deSitter bulk which seems more favorable from the point of view of
having a viable higher dimensional cosmic evolution [11, 12]. It has also been realized
that one can have a viable model of four-dimensional gravity even without an e ec-
tive massless graviton. In certain setups a tower of extremely light massive gravitons
may be indistinguishable from a massless graviton [13, 14]. Even gravity mediated by
a massless and an extremely light massive graviton at roughly equal strengths may
be viable and hard to distinguish from massless four-dimensional gravity [15, 16].
In these setups the Veltman-van-Dam-Zakharov discontinuity (VvDZ discontinuity)
[17] (for a recent analysis of VvDZ discontinuity at the classical level see [18]) is
either unobservable due to the setup [14] or cured by non-linear e ects [19], though
it is still under dispute if these models exhibit a realistic behavior in the limit of
strong gravitational elds [20].
The understanding of gravity in brane models can, however, only be considered
preliminary because the analysis is usually done studying only the zero-mode, ne-
glecting the continuous spectrum [21, 22]. A remarkable deviatio