New variables for brane-world gravity László Á. Gergely University of Szeged, Hungary Albert Einstein Century Internatonal Conference, Paris, 2005 Early Universe and Theoretical Cosmology In collaboration with Zoltán Kovács, Max Planck Institut für Astronomie, Heidelberg

Gravitation acts in 5D (the bulk) Gravitation acts in 5D (the bulk) according to the Einstein-equation according to the Einstein-equation Standard model fields live in 4D Standard model fields live in 4D (on the brane) (on the brane) the brane has a tension >0, which is the brane has a tension >0, which is fine-tuned to the bulk cosmological fine-tuned to the bulk cosmological constantto give a small constantto give a small (vanishing) cosmological constant on the (vanishing) cosmological constant on the brane. brane. the Lanczos equation relates brane the Lanczos equation relates brane matter to the jump in the extrinsic matter to the jump in the extrinsic curvature: curvature: 1. Brane-new-world Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely

sources of Einstein gravity sources of Einstein gravity brane projection brane projection Geometry T 2 term of bulk sources = electric part = electric part gravitation of the bulk gravitation of the bulk Weyl-curvature Weyl-curvature dark matter? for cosmological brane brane LÁ Gergely Phys. Rev. D 68, (2003) dark energy / accelerated expansion 2. The effective Einstein equation source term from the asymmetric embedding New! modified early cosmology Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely

symmetric embedding boundary of space-time asymmetric embeddings BH-s with different masses 3. Asymmetric embedding no BH on the right moving domain wall Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely 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 (canonical gravitational dynamics on the brane) LÁ Gergely, Z Kovács (canonical gravitational dynamics on the brane) submitted, gr-qc/ (no Swiss-cheese universe on the brane) LÁ Gergely (no Swiss-cheese universe on the brane) Phys. Rev. D 71, (2005) LÁ Gergely, R Maartens (asymmetric Friedmann branes with induced gravity) Phys. Rev. D 71, (2005) LÁ Gergely, E Leeper, R Maartens (radiating Friedmann branes – asymmetric embedding) (radiating Friedmann branes – asymmetric embedding) Phys. Rev. D 70, (2004) (generalized Kantowski-Sachs homogeneous brane) LÁ Gergely (generalized Kantowski-Sachs homogeneous brane) Class. Quantum Grav. 21, (2004) (generalized Friedmann brane – asymmetric embedding) LÁ Gergely (generalized Friedmann brane – asymmetric embedding) Phys. Rev. D 68, (2003) (generalized Einstein brane) LÁ Gergely, R. Maartens (generalized Einstein brane) Class. Quantum Grav. 19, (2002 ) Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely

Meant for initial value problem and canonical gravitational dynamics on the brane. Double foliation (first timelike, second containing the brane) (s+2)-metric lapse N lapse N shift N a shift N a No off-brane component of the shift (trajectories of the standard model particles are confined to the brane Frobenius theorem gives a constraint, fulfilled with this choice) 5. Gravitational dynamics on the brane Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely l

6. First fundamental forms Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely lmanifolds: induced metrics: covariant derivatives:

7. Second fundamental forms Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely Extrinsic curvature of the constant time leaves: extrinsic curvature of w.r.to n extrinsic curvature of w.r.to n normal fundamental form normal fundamental scalar Extrinsic curvature of the brane: extrinsic curvature of w.r.to l with with

8. Evolution equations I. Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely temporal:off-brane: only K ab, K i and K are dynamical curvatures of n and l (accelerations):

9. Jump in the extrinsic curvature across the brane Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely The Lanczos equation:, Projections od the Lanczos equation: from all dynamical quantities only K i is discontinuous! for perfect fluid brane all dynamical variables are continuous!

10. Intrinsic curvatures Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely The Gauss equation: Twice contracted Gauss-equation: or:

11. Decomposition of the Riemann tensor Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely + the Gauss-equation.

12. Decomposition of the Ricci tensor Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely

13. Decomposition of the Einstein tensor Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely

14. Evolution equations II. Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely

15. Gravitational dynamics Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely

Maeda-Sasaki-Nakamura-Mijama decomposition formalism for stationary and axisymmetric spacetimes relies on the use of a factor space with respect to the rotational Killing vector the rotational Killing vector the induced metric is defined with a more complicated formalism, hardly applicable for braneworlds 16. Comparison with alternative formalisms Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely

Gourgoulhon – Bonazzola decomposition formalism for stationary and axisymmetric spacetimes evolutions along Killing vectors evolutions along Killing vectors 17. Comparison with alternative formalisms (continued) Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely

Maartens decomposition formalism with respect to the brane normal andthe brane normal and fluid 4-velocity ufluid 4-velocity u Brief comparison: Maartensour formalism Maartensour formalism time evolution along u∂/∂t induced metric defined in the hypersurface ┴ to uin the hypersurface ┴ n rather than to ∂/∂t extrinsic curvatures absent extrinsic curvatures absent ( K i, K, K ab ) 18. Comparison with alternative formalisms (continued) Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely

Results: The s+1+1 decomposition of the space-time New gravitational variables on the brane, with clear geometrical meaning Evolution equations for these variables Junction conditions in terms of these variables Work in progress: Action principle in terms of the new variables and algebra of constraints Connection between traditional brane- world variables and ours 19. Summary and Outlook Albert Einstein Century International Conference, Paris, 2005 László Á. Gergely