New variables for brane-world gravityNew variables for brane-world gravity

László Á. GergelyLászló Á. GergelyUniversity of Szeged, HungaryUniversity of Szeged, Hungary

Albert Einstein CentAlbert Einstein Century Internatonalury Internatonal Conference, Conference, Paris, Paris, 20052005

Early Universe and Theoretical Cosmology Early Universe and Theoretical Cosmology

In collaboration with In collaboration with Zoltán KovácsZoltán Kovács, Max Planck Institut für Astronomie, , Max Planck Institut für Astronomie, HeidelbergHeidelberg

• Gravitation acts in 5D (the bulk)Gravitation acts in 5D (the bulk) according to the according to the Einstein-equationEinstein-equation• Standard model fields live in 4D Standard model fields live in 4D (on the brane)(on the brane)• the brane has a tension the brane has a tension >0>0 , which is , which is fine-tuned to the bulk cosmologicalfine-tuned to the bulk cosmological constantconstant to give a smallto give a small (vanishing) cosmological constant on the(vanishing) cosmological constant on the brane. brane. • the the Lanczos equationLanczos equation relates brane relates brane matter to the jump in the extrinsicmatter to the jump in the extrinsic curvature:curvature:

1. Brane-new-world

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

sources of Einstein gravitysources of Einstein gravity brane brane

projection projection GeometryGeometry TT22 term term of bulk of bulk

sourcessources == electric part electric part

gravitationgravitation of the bulk of the bulk Weyl-curvature Weyl-curvature

dark dark matter?matter?

forfor cosmologicalcosmological

branebrane

LÁ Gergely Phys. Rev. D 68, 124011-1-13 (2003)

dark energy / acceleratdark energy / accelerated expansioned expansion

2. The effective Einstein equation

source term from the source term from the asymmetricasymmetric embeddingembedding

New!New!

modified early modified early cosmologycosmology

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

symmetric embedding symmetric embedding boundary of space-timeboundary of space-time

asymmetric embeddingsasymmetric embeddings

BHBH--s with different massess with different masses

3. Asymmetric embedding

no BH on the rightno BH on the right

moving domain wallmoving domain wall

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

asymmetry source term:asymmetry source term:

4. Results on brane-worlds

• LÁ Gergely, Z Keresztes, R Maartens (Friedmann branes absorbing Hawking radiation from the bulk BH)(Friedmann branes absorbing Hawking radiation from the bulk BH)

in preparation• LÁ Gergely, Z Kovács (canonical gravitational dynamics on the brane)(canonical gravitational dynamics on the brane)

submitted, gr-qc/0507020• LÁ Gergely (no Swiss-cheese universe on the brane)(no Swiss-cheese universe on the brane)

Phys. Rev. D 71, 084017-1-5 (2005) • LÁ Gergely, R Maartens

(asymmetric Friedmann branes with induced gravity)(asymmetric Friedmann branes with induced gravity)Phys. Rev. D 71, 024032-1-7 (2005)

• LÁ Gergely, E Leeper, R Maartens (radiating Friedmann branes – asymmetric embedding)(radiating Friedmann branes – asymmetric embedding)

Phys. Rev. D 70, 104025-1-5 (2004) • LÁ Gergely (generalized Kantowski-Sachs homogeneous brane)(generalized Kantowski-Sachs homogeneous brane)

Class. Quantum Grav. 21, 935-940 (2004) • LÁ Gergely (generalized Friedmann brane – asymmetric embedding)(generalized Friedmann brane – asymmetric embedding)

Phys. Rev. D 68, 124011-1-13 (2003) • LÁ Gergely, R. Maartens (generalized Einstein brane)(generalized Einstein brane)

Class. Quantum Grav. 19, 213-221 (2002)

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

Meant for Meant for initial value probleminitial value problem and and canonical gravcanonical gravitationalitational dynamics dynamics on the brane.on the brane.

Double foliationDouble foliation (first timelike, second (first timelike, second containing the brane)containing the brane)

(s+2)-metric(s+2)-metric

lapse Nlapse N

shift Nshift Naa

No off-brane component of the shiftNo off-brane component of the shift (trajectories of the standard model (trajectories of the standard model particles are confined to the brane particles are confined to the brane

Frobenius theorem gives a Frobenius theorem gives a constraint, fulfilled with this constraint, fulfilled with this choice)choice)

5. Gravitational dynamics on the brane

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

ll

6. First fundamental forms

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

ll

manifolds:manifolds:

induced metrics:induced metrics:

covariant derivatives:covariant derivatives:

7. Second fundamental forms

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

Extrinsic curvature of the constant time leaves:Extrinsic curvature of the constant time leaves:

extrinsic curvature of w.r.to extrinsic curvature of w.r.to nn

normal fundamental formnormal fundamental form

normal fundamental scalarnormal fundamental scalar

Extrinsic curvature of the brane:Extrinsic curvature of the brane:

extrinsic curvature of w.r.to extrinsic curvature of w.r.to ll

withwith

withwith

8. Evolution equations I.

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

temporal:temporal: off-brane:off-brane:

only only KKabab, , KKii and and KK are dynamical are dynamical

curvatures of curvatures of nn and and ll (accelerations): (accelerations):

9. Jump in the extrinsic curvature across the brane

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

The Lanczos equation:The Lanczos equation:

,,

Projections od the Lanczos equation:Projections od the Lanczos equation:

from all dynamical quantities only from all dynamical quantities only KK ii is discontinuous! is discontinuous!

for perfect fluid brane for perfect fluid brane allall dynamical variables are dynamical variables are continuous!continuous!

10. Intrinsic curvatures

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

The Gauss equation:The Gauss equation:

Twice contracted Gauss-equation:Twice contracted Gauss-equation:

or:or:

11. Decomposition of the Riemann tensor

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

+ the Gauss-equation.+ the Gauss-equation.

12. Decomposition of the Ricci tensor

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

13. Decomposition of the Einstein tensor

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

14. Evolution equations II.

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

15. Gravitational dynamics

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

Maeda-Sasaki-Nakamura-Mijama 2+1+1 decomposition formalism for Maeda-Sasaki-Nakamura-Mijama 2+1+1 decomposition formalism for stationary and axisymmetric spacetimesstationary and axisymmetric spacetimes

relies on the use of a factor space with respect torelies on the use of a factor space with respect to

the rotational Killing the rotational Killing vectorvector

the induced metric is defined with the induced metric is defined with

a more complicated formalism, hardly applicable for braneworldsa more complicated formalism, hardly applicable for braneworlds

16. Comparison with alternative formalisms

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

Gourgoulhon – Bonazzola 2+1+1 decomposition formalism for stationary and Gourgoulhon – Bonazzola 2+1+1 decomposition formalism for stationary and axisymmetric spacetimesaxisymmetric spacetimes

evolutions alongevolutions along Killing vectors Killing vectors

17. Comparison with alternative formalisms (continued)

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

Maartens 3+1+1 decomposition formalism with respect to Maartens 3+1+1 decomposition formalism with respect to

• the brane normal and the brane normal and

• fluid 4-velocity ufluid 4-velocity u

Brief comparison:Brief comparison:

MaartensMaartens our formalismour formalism

time evolution along utime evolution along u ∂/∂t∂/∂t

induced metric defined in the hypersurfaceinduced metric defined in the hypersurface ┴┴ to uto u in the hypersurfacein the hypersurface ┴┴ n n

rather than to ∂/∂trather than to ∂/∂t

extrinsic curvaturesextrinsic curvatures absentabsent (K i , K , Kab )

18. Comparison with alternative formalisms (continued)

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely

ResultsResults:: The The s+1+1 decompositions+1+1 decomposition of the space- of the space-

timetime

New gravitational variablesNew gravitational variables on the brane, on the brane, with clear geometrical meaning with clear geometrical meaning

Evolution equationsEvolution equations for these variables for these variables

Junction conditionsJunction conditions in terms of these in terms of these variablesvariables

Work in progress:Work in progress:AAcction principletion principle in terms of the new in terms of the new

variablesvariables and and algebra of constraintsalgebra of constraints

ConnectionConnection between traditional brane- between traditional brane-world variables and oursworld variables and ours

19. Summary and Outlook

Albert Einstein Century International Conference, Albert Einstein Century International Conference, Paris, Paris, 20052005László Á. GergelyLászló Á. Gergely