Classification of the Einstein field equations.

[Eric Gisse]

I understand the -simple- way of classifying partial differential
equations when they are linear, and the conditions therein. However, I
have been trying to understand how the Einstein field equations get
classified or even -how- to classify them. It is not clear - to me -
how one can classify the differential equations.

I frequently see them referred to as elliptic-hyperbolic, at least in
connection with the 3+1 decomposition for numerical relativity which
is my current topic of interest. This is useful information because it
allows me to honestly believe that suitable initial / boundary data
makes this a solvable problem, but I hesitate to trust such a
declaration implicitly.

Does anyone have a useful reference or helpful paragraph associated
with any of this?

[Igor]

On Oct 13, 12:37Â*pm, Eric Gisse <jowr...@gmail.com> wrote:
> I understand the -simple- way of classifying partial differential
> equations when they are linear, and the conditions therein. However, I
> have been trying to understand how the Einstein field equations get
> classified or even -how- to classify them. It is not clear - to me -
> how one can classify the differential equations.
>
> I frequently see them referred to as elliptic-hyperbolic, at least in
> connection with the 3+1 decomposition for numerical relativity which
> is my current topic of interest. This is useful information because it
> allows me to honestly believe that suitable initial / boundary data
> makes this a solvable problem, but I hesitate to trust such a
> declaration implicitly.
>
> Does anyone have a useful reference or helpful paragraph associated
> with any of this?


Maybe this could offer some assistance:

http://en.wikipedia.org/wiki/Petrov_classification

[Jonathan Thornburg [remove -animal to reply]]

In sci.physics.research Eric Gisse <jowr.pi@gmail.com> wrote:
> I understand the -simple- way of classifying partial differential
> equations when they are linear, and the conditions therein. However, I
> have been trying to understand how the Einstein field equations get
> classified or even -how- to classify them. It is not clear - to me -
> how one can classify the differential equations.
>
> I frequently see them referred to as elliptic-hyperbolic, at least in
> connection with the 3+1 decomposition for numerical relativity which
> is my current topic of interest. This is useful information because it
> allows me to honestly believe that suitable initial / boundary data
> makes this a solvable problem, but I hesitate to trust such a
> declaration implicitly.
>
> Does anyone have a useful reference or helpful paragraph associated
> with any of this?

Depending on what you're looking for, you might find either/both of
the following papers interesting:

Alan D. Rendall
"Theorems on Existence and Global Dynamics for the Einstein Equations"
Living Reviews in Relativity 8 (2005), 6.
http://relativity.livingreviews.org/Articles/lrr-2005-6/
Abstract:
This article is a guide to theorems on existence and global
dynamics of solutions of the Einstein equations. It draws attention
to open questions in the field. The local-in-time Cauchy problem,
which is relatively well understood, is surveyed. Global results
for solutions with various types of symmetry are discussed. A
selection of results from Newtonian theory and special relativity
that offer useful comparisons is presented. Treatments of global
results in the case of small data and results on constructing
spacetimes with prescribed singularity structure or late-time
asymptotics are given. A conjectural picture of the asymptotic
behaviour of general cosmological solutions of the Einstein
equations is built up. Some miscellaneous topics connected with
the main theme are collected in a separate section.

Oscar A. Reula
"Hyperbolic Methods for Einstein's Equations"
Living Reviews in Relativity 1 (1998), 3.
http://relativity.livingreviews.org/Articles/lrr-1998-3/
Abstract:
I review evolutionary aspects of general relativity, in particular
those related to the hyperbolic character of the field equations
and to the applications or consequences that this property entails.
I look at several approaches to obtaining symmetric hyperbolic
systems of equations out of Einstein's equations by either removing
some gauge freedoms from them, or by considering certain linear
combinations of a subset of them.

ciao,

--
-- From: "Jonathan Thornburg [remove -animal to reply]" <jthorn@astro.indiana-zebra.edu>
Dept of Astronomy, Indiana University, Bloomington, Indiana, USA
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam

[Eric Gisse]

On Oct 14, 11:32Â*am, Igor <thoov...@excite.com> wrote:
> On Oct 13, 12:37Â*pm, Eric Gisse <jowr...@gmail.com> wrote:
>
> > I understand the -simple- way of classifying partial differential
> > equations when they are linear, and the conditions therein. However, I
> > have been trying to understand how the Einstein field equations get
> > classified or even -how- to classify them. It is not clear - to me -
> > how one can classify the differential equations.
>
> > I frequently see them referred to as elliptic-hyperbolic, at least in
> > connection with the 3+1 decomposition for numerical relativity which
> > is my current topic of interest. This is useful information because it
> > allows me to honestly believe that suitable initial / boundary data
> > makes this a solvable problem, but I hesitate to trust such a
> > declaration implicitly.
>
> > Does anyone have a useful reference or helpful paragraph associated
> > with any of this?
>
> Maybe this could offer some assistance:
>
> http://en.wikipedia.org/wiki/Petrov_classification

Thanks but no, as the Petrov classifcation stuff is just a way of
classifying the _solutions_ based on algebraic structure.

[Igor]

On Oct 16, 2:17*pm, Eric Gisse <jowr...@gmail.com> wrote:
> On Oct 14, 11:32*am, Igor <thoov...@excite.com> wrote:
>
> > On Oct 13, 12:37*pm, Eric Gisse <jowr...@gmail.com> wrote:
>
> > > I understand the -simple- way of classifying partial differential
> > > equations when they are linear, and the conditions therein. However, I
> > > have been trying to understand how the Einstein field equations get
> > > classified or even -how- to classify them. It is not clear - to me -
> > > how one can classify the differential equations.
>
> > > I frequently see them referred to as elliptic-hyperbolic, at least in
> > > connection with the 3+1 decomposition for numerical relativity which
> > > is my current topic of interest. This is useful information because it
> > > allows me to honestly believe that suitable initial / boundary data
> > > makes this a solvable problem, but I hesitate to trust such a
> > > declaration implicitly.
>
> > > Does anyone have a useful reference or helpful paragraph associated
> > > with any of this?
>
> > Maybe this could offer some assistance:
>
> >http://en.wikipedia.org/wiki/Petrov_classification
>
> Thanks but no, as the Petrov classifcation stuff is just a way of
> classifying the _solutions_ based on algebraic structure.

Actually, the Petrov classification is not so much a classification
scheme for solutions, but is based on the general symmetries of the
Weyl tensor and on similar symmetries of the Riemann curvature in
particular. These obviously have a bearing on the outcome of the
Einstein equations. But as far as actually classifying the
differential equations themselves, I'm not really aware of any such
general scheme for nonlinear equations. Progress in this area has
always been sluggish at best. But good luck in your search and
hopefully you'll find something that relates to what you're looking
for.