Re: Symmetries of general relativity

On Mon, 27 Oct 2003, Marc Nardmann wrote:

> You say that the theory of general relativity (whatever you mean
> precisely here) is invariant under the whole "general covariance
> group" (whatever you mean by this). To say that some thing X is
> invariant under some group G makes only sense after some action of G
> on some thing Y has been specified and after it has been made clear
> how X is related to Y (is X a subset of Y? or what?).  Tell us what X
> and Y are, mathematically precise.

Let me try to do it although it can't be guaranteed that you will be
satisfied.

X is the space of all spacetime configurations of GR, i.e. all functions
from M^d - the spacetime manifold - to R^n where "n" is the total number
of components of all fields in your theory. (One could use a more abstract
language, but let me not do so.) The group G consists of all possible
diffeomorphisms which are continuous invertible maps M from M^d to M^d, x
-> x'. The functions - fields - "f" transform as

        f(x) -> T { f[M(x)] }

which means that the value of the tensor fields at the point f(x) in the
configuration M(C) is given by the value of the tensor fields in the
original configuration at the point x, and they are transformed as
tensors. For each index of the tensor, you write down a different index,
and contract it with the partial derivative   d x'^i / d x^j.  The basic
rules of tensor calculus can be found in many books, and I can give you
some references.

The fact that the theory (its defining equations) is invariant under G
means that if you focus on the subset "x" of "X" that includes the
configurations that satisfy the equations of motion, the subset "x" will
be mapped onto "x" by the elements of "G".

> Give a mathematical definition of G (maybe you mean the group of all
> diffeomorphisms of R^4? or diffeomorphisms of what?).

I hope that I have said it. G is made of all continuous differentiable
invertible maps from M^d to M^d, which is the spacetime manifold.

> Write down a formula for the action of G on Y.

I've essentially done that, too - with the reduction of the notation of
indices that was forced by the ASCII environment here.

> If X is a subset of Y, tell us what you mean by "invariant":

I've explained it. It means that if a configuration C satisfies the
equations of motion, then g(C) must satisfy them as well.

> Do you mean that each g in G maps X to X, or do you mean that each g in
> G maps each element of X to itself?

No, this is not the right interpretation of the phrase "a theory is
invariant". The phrase "theory is invariant" means that transforming a
configuration/history that satisfies the laws of physics by an element of
the symmetry group produces another configuration/history that satisfies
the same laws - but in general, it is - of course - a different
configuration/history! Otherwise the notion of symmetry/invariance would
be meaningless. Symmetries are only interesting if they can be used to
create *new* solutions.

> Maybe you didn't want to say "invariant".

I definitely wanted to say "invariant". If you find the statement "General
relativity is INVARIANT under transformations of the diffeomorphism group"
difficult, be sure that it is your problem only, and I can find you
thousands of textbooks, articles, pages etc. that contain exactly this
sentence, for example:

        http://philsci-archive.pitt.edu/archive/00000834/
        http://www.iop.org/EJ/article/0264-9381/13/11/001/cq13011l1.html

> Maybe you wanted to say "the whole general covariance group is a
> symmmetry group of the theory of general relativity".

It's the same statement. The statements "AB is invariant under XY" and "XY
is the symmetry group of AB" are identical.

Best wishes
Lubos
______________________________________________________________________________
E-mail: [EMAIL PROTECTED]   fax: +1-617/496-0110   Web: http://lumo.matfyz.cz/
        phone:          work: +1-617/496-8199  home: +1-617/868-4487
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
      Superstring/M-theory is the language in which God wrote the world.