# Metrics on a manifold, gravity waves, gauge freedom

## Main Question or Discussion Point

Suppose I have a manifold. I say that it can support a certain configuration of gravity field described by metric tensor \gamma. I do not write \gamma_{\mu\nu}, because that would immediately imply a reference to a particular chart. A tensor field, however, exists on a manifold unrelated to this manifold's parameterisation. It is a box, in which you deposit two vectors, and which then spits out a scalar.

Can I always assume that, for any twice-covariant tensor \gamma, there exists (not necessarily global) a chart, for which this tensor acts as a metric?

Let me now express my question a bit more carefully. We always introduce a manifold via local charts. Suppose some patch of the manifold is parametrisable by some grid z^\mu with metric w_{\mu\nu}. Can I always introduce on this patch a new grid x^\alpha, whose metric is exactly the desired \gamma? (whom I shall now call \gamma_{\alpha\beta} ) Please be mindful that I am talking not about an infinitesimal neighbourhood, but about a finite patch.

I suspect that the answer to my question is negative, though I am not 100% sure. Here is my argument. Suppose that such a coordinate chart can always be assembled. I take a "background metric" \gamma and build the appropriate grid x^\alpha. I take a "perturbed metric" g and construct an appropriate chart y^\alpha. Then, if g and \gamma are "close", their difference being h, I shall say that the difference between g and \gamma is solely due to a coordinate transformation. Hence the gravitational perturbation h is always a gauge effect.

If h can be an actual physical variation, then the answer to my above question must be negative.

This logic, if correct, presents a physical answer to the question.
Still, I would love to see a mathematical argument.

Great many thanks,

Michael

Related Special and General Relativity News on Phys.org
Dear atyy,
Thanks for the interesting link. While it is indeed relevant, it still does not provide an immediate answer to the question: given an arbitrary twice-covariant tensor field, is it possible to build, on a *finite* patch, a coordinate grid, for which this tensor field act as a metric?
Thanks again,
Michael

dx
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What do you mean "act as a metric"? Whether a given tensor can be a metric or not doesnt depend on any coordinates. The only condition is that it must be symmetric, nondegenerate, and have a Lorentz signature.