1. Local Conservation Law and Dark Radiation in Brane Models
Masato Minamitsuji
Osaka University (D1)
In collaboration with Misao Sasaki (YITP)
October 30th _ 31st, 2003
@Tokyo Institute of Technology

2. §1.Introduction (a) For the case of static bulk Static Bulk;
Effective cosmology on the brane ;
(b) For the case of dynamical bulk
In this case, we need to generalize the to the
locally conserved quantity as introduced.
We investigate the bulk geometry and brane cosmology with the energy inflow from the brane, in terms of the local conservation law.

3. §2. Locally Conservation Law in the dynamical Bulk
Bulk = space-time with the maximally symmetric 3-space,
5D Einstein system with a negative cosmological constant as
The flux vector field ;
The locally conservation law ; 33

4. The locally conserved quantity;
the matter part + the cosmological constant part
We will focus on the matter part.
The matter part is related to the local energy-momentum tensor by using the Einstein equation as,
Weyl tensor (has only one non-trivial component)
the locally conserved quantity part + the local matter part

5. §2.2 Brane cosmology in the dynamical bulk
Brane = a singular boundary of the dynamical space-time.
The brane moves along the trajectory parameterized by
The junction condition symmetry
The effective Friedmann equation on the brane is given by
◎The projected Weyl tensor
Local conservation quantity part (“dark radiation”)
+ Remaining bulk energy-momentum part
uniquely decomposed.

6. §3. Collapse of the Bulk - Null dust model
In this section, we analyze the backreaction problem of the bulk geometry. For the first step, we approximate the bulk matter as null dust .
§3-1. Set up
Null dust:
To satisfy the energy conservation law,
and are determined as
　The energy variation rate along the null directions is given as
where and are determined by the Einstein equations.

7. §3-2. Apparent horizons
The and are related to the out- and in-going expansion of the congruence of the null geodesics, respectively.
out-going expansion rate
in-going expansion rate
(i). Out-going apparent horizon
(ii). In-going apparent horizon
Einstein equations

8. The space-time is classified into 3 regions
[I] “ Normal region ”.
[II] “ Black Hole region ” .
[III] “ White Hole region ” .
We discuss the condition of the formation of apparent horizons for the following two cases about the condition that the energy flux is turned on.
(a) (b)
For the situation (a) and (b), we can obtain the results about the possibility of formation of apparent horizons.

9. [I]. (a) (b).
[II]. (a) (b).
[III]. (a) 　(b).

10. §3-3. The case for the single null dust
For the case that single in-falling null dust exists, the dynamical collapse of the bulk space-time is analyzed.
The in-falling null dust
Generated by the decay or collisions of matter on the brane
　 Transmitted along the constant lines .
The bulk geometry
exact solution
Locally conserved quantity;
In this case, we can analyze the bulk geometry and the effective cosmology on the brane.
BH ?

11. Schematic View of Single Null Dust Collapse
○Location of apparent horizon ;
○Trajectory of apparent horizon ;
The apparent horizon is space-like and becomes null for
For example, considering null dust collapse as follows,
(a) AdS :
(b) AdS-Vaidya :
(c) AdS-Sch. :

12. From the point of view of the bulk
The proper time on the brane
the advanced time in the bulk
Now, we require that the brane trajectory is time-like and
(i).
(ii).
(iii).
(iv).
The existence possibility of the brane in the region [I] ~ [III];
[I] The expanding and contracting
can exist.
[II] The contracting brane can exist Limitation on
[III] No time-like brane can exist.

13. From the point of view of the brane
The dynamics on the brane is determined by the equations
If the intensity is given or related to the quantities on brane,
brane trajectory
energy density on the brane
locally conserved quantity
self-consistently determined.

14. §3-4. The case for the double null dust
It is worth mentioned that if the both of the null dust exists.
In-falling flux exponentially enhanced by the out-going flux
Out-going flux exponentially suppressed by the in-going flux.
cross flow region ;
Enhancement factor of in-falling flux
Suppression of the outgoing null flux

15. for the AdS cosmological constant,
Perturbative analysis
for the AdS cosmological constant,
For the closed chart ( ) case ,
For the constant intensity out-flow, we
evaluate the enhancement factor near the
future Cauchy horizon ( ) as ;
No problem in the perturbative level.

16. §4. Summary and Future Works
We would like to investigate the bulk geometry and brane cosmology in terms of the local conservation law.
(a). For the bulk with maximally symmetric 3-space, we can exactly decompose the locally conserved quantity
and the remaining bulk energy momentum.
(b). For the case of the null dust collapse,
we confirm the formation of apparent horizons and BH.
(c).　For single null dust, the exact solution of bulk geometry exists and the dynamics on the brane is determined self-consistently .
(d). For double null dust, the in-falling flux is enhanced by out-one.
No problem for the perturbative level.
The enhancement at dynamical level have to be
analyzed.
Future Works ;
(a). Analysis of bulk collapse for the ingoing scalar wave and
the dynamics on the brane .
(b). Quantum effects with respect to the bulk BHs .