Improved constrained scheme for the Einstein equations: an approach to the

uniqueness issue

in collaboration with

Pablo Cerdá-Durán

Harald Dimmelmeier

José Luis Jaramillo

Jérôme Novak

Eric Gourgoulhon

Isabel Cordero-CarriónDepartment of Astronomy and

AstrophysicsUniversity of Valencia

Salamanca, 2008

Cordero-Carrión et al., gr-qc/0809.2325

Outline:

- 3+1, Fully Constrained Formalism (FCF) and Conformally Flat Condition (CFC).

- Maximum principle.

- Einstein equation in CFC and new approach.

- Generalization to FCF.

- Conclusions.

3+1 formalism:- Spatial hypersurfaces , and decomposition

- Spatial metric:

- Metric of the space-time:

- Extrinsic curvature: .

- Dirac gauge, , and maximal slicing, .

- Decomposition of the conformal metric:

FCF formalism (Bonazzola et al., 2004):- Flat metric , conformal factor , conformal metric

CFC scheme (Isenberg, 1979/2008; Wilson and Mathews, 1989):

- The spatial metric is conformally flat, . It corresponds in the FCF formalism to set to zero.

FCF CFC with more sources + evolution:

- Einstein equations display in elliptic equations for

+

Hydrodynamical conserved variables:

(energy density)

(trace stress tensor)

(momentum density)

We are going to consider an elliptic equation of the form (see O. Rinne’s Monday talk):

Sign of the exponent p the sign of the proper function h

we cannot use the maximum principle

to prove local uniqueness of solutionsProtter and Weinberger, Maximum principles in Differential Equations

(1967).York, Sources of Gravitational Radiation (1979). Analytical simple examples (Baumgarte et al., 2006) of scalar equations of this type as well as numerical examples in the Einstein equations (Pfeiffer and York, 2005) have pointed out this pathology. Numerical simulations show these problems in the FCF case (Cordero-Carrión et al., 2008).

The key in the elliptic equations is in the sign…

Einstein equations in

CFC:

First equation inherit the local non-uniqueness problems due to the extrinsic curvature term.

Necessity of knowing the conformal factor in order to compute due to the dependence on the pressure.

The approaches to solve these problems can not be applied in simulations with very strong gravity. 1D case is not a problem (Shapiro and Teukolsky, 1980):

Conformal transverse traceless decomposition of the conformal extrinsic curvature (Lichnerowicz, 1944):

New approach:

From the momentum constrained,

an equation for can be derived and it can be obtained:

can be computed from the previous vector.

The elliptic equation for decouples and maximum principle can be applied:

The elliptic equation for has no problems in computing the source term, it decouples and maximum principle can be applied:

Finally, from , and taking divergence, an elliptic equation for is obtained:

From the numerical point of view, it has been possible to successfully perform both the migration test and the collapse of a neutron star to a black hole (figures) in the CFC case in a consistent way.

Generalization to FCF: Decomposition of the tensors and in longitudinal

and transverse traceless (TT) parts: is already TT (gauge), and

that is equivalent to the elliptic equation for the vector :

The TT ones are evolved in time, and .

The longitudinal part of is computed, like in the CFC case, from the momentum constraint,

The elliptic equation for has no problems in computing the source term, it decouples and maximum principle can be applied:

And finally, can be obtained from the following elliptic vector equation:

The tensors and are completely known and we can follow exactly the same scheme as in the CFC case. The elliptic equation for decouples and maximum principle can be applied:

Conclusions:- FCF is a natural generalization of CFC, including the gravitational radiation.

- New approach in the CFC (includes Shapiro and Teukolsky work). From the theoretical point of view:

· One more elliptic vector equation.

· Correct signs in the powers of the elliptic equations: it solves the local uniqueness problems.

· Decoupled system.

- Numerical simulations show the good behavior: collapse of unstable neutron stars to black hole.

- Generalization to FCF (full Einstein) recovering the CFC ideas for the elliptic equations.