My limited understanding of metrics comes from Cartan. From there, I understand that a metric is something invariant under certain transformations, e.g. Lorentz in special relativity. But with the metric varying from point to point (event-to-event), what is the meaning of the metric?
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1$\begingroup$ The metric is diffeomorphism invariant $\endgroup$– John RennieCommented Mar 26, 2014 at 17:46
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5$\begingroup$ @JohnRennie - A metric is diffeomorphism covariant, not invariant right? $\endgroup$– PraharCommented Mar 26, 2014 at 17:54
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2$\begingroup$ @Prahar: Depends on whether you're a mathematician or a physicist :P Physicists talk about objects with indices as if they transform (which they do). Mathematicians prefer to contract all indices with the corresponding basis and talk about "invariants". $\endgroup$– SivaCommented Mar 26, 2014 at 18:49
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3$\begingroup$ Right. I presume when I say metric I think of the "matrix" $g_{\mu\nu}$ whereas mathematicians think of $g = g_{\mu\nu} {\text d} x^\mu \otimes {\text d} x^\nu$. $\endgroup$– PraharCommented Mar 26, 2014 at 18:51
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1$\begingroup$ Given a diffeomorphism $\phi : M \to N$between (semi-)Riemannian manifolds $M, N$ with metrics $g, g'$ there need not be any relation between $g(x)$ and $g'(\phi(x))$. If they are equal, the diffemorphism is an isometry and $M, N$ describe the same physical spacetime. (Strictly speaking, $\phi$ should be proper and ortochronous, i.e., preserve space orientation and the direction of time, too.) $\endgroup$– Robin EkmanCommented Apr 20, 2014 at 18:25
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2 Answers
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I don't like to think of general relativity as allowing more coordinate systems or transformations than special relativity, or even Newtonian physics. You can do Newtonian physics in any strange, silly coordinate system you can cook up. You can do special relativity in any strange coordinate system you want, too. However, you will discover that you need to introduce machinery that is usually introduced in the context of general relativity to do so. Those extra terms in the vector Laplacian in curvilinear coordinates are precisely the Christoffel symbols -- non-zero, even though space(time) is flat.
Coordinate independence is a basic, fundamental requirement of a sensible physical theory. The generalization of general relativity from special relativity lies not in that it allows more coordinate systems but in that it allows more geometries and for geometry to be a dynamical quantity. Galilean space and Minkowski spacetime come with privileged coordinate systems, inertial frames, a spacetime in general does not.
answered Apr 20, 2014 at 18:43
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In general relativity, the metric is covariant under diffeomorphisms. Infinitesimally, we may write a transformation as a shift by a vector field, wherein,
$$X_a \to X_a + \xi_a$$
By definition, the metric changes by a Lie derivative with respect to the field, i.e.
$$g_{ab}\to g_{ab} + \mathcal{L}_\xi \, g_{ab}$$
where $\mathcal{L}_\xi \, g_{ab}=2\nabla_{(a}\xi_{b)}$. By performing a diffeomorphism, we induce a perturbation in the metric, but this is not a physical perturbation - this distinction is important when looking for instabilities of a particular solution. From the quantum field theory perspective, the invariance of the action,
$$S = \frac{1}{16\pi G} \int \! \mathrm{d}^d x \sqrt{-g}\, R$$
under local diffeomorphisms reflects a redundancy in our description of the system. For the case of global diffeomorphisms, these are honest symmetries; infinitesimally simply spacetime translations, which give rise via Noether's theorem to conserved currents, namely the energy-momentum tensor.
Other theories also possess a diffeomorphism symmetry, but with restrictions. In the special case where a coordinate transformation,
$$X_a \to X'_a =\tilde{X}_a (X)$$
changes the metric solely by a factor, i.e.
$$g_{ab} \to g'_{ab}=\Omega^2 (X)g_{ab}$$
the transformation is said to be a 'conformal transformation.' A notable conformal field theory is the Polyakov action, which describes a relativistic string with an auxiliary field.
