Conformal transformations are used in order to analyze the structure of a given spacetime. One can map a formally infinite spacetime to a compact interval, and study its properties there. This process is referred to as "conformal compactification", and enables one to draw Penrose diagrams. They serve to identify and classify horizons, infinities and singularities and are popular for various spacetimes such as Minkowski space and Schwarzschild black holes.

For example, Minkowski space, given by

$$ds^2=-dt^2+dr^2+r^2d\Omega^2,$$

can be conformally compactified by a change of coordinates from $(r,t)$ to $(u,v)$ by the transformations

$$u=\arctan(t-r),$$
$$v=\arctan(t+r),$$

which leads to

$$ds^2=\frac{1}{4\cos^2u\,\cos^2v}\left(-4dudv+\sin^2(v-u)d\Omega^2\right).$$

Due to the nature of the $\arctan$ function, the coordinates will now take on values on the interval $(-\pi/2,\pi/2)$ and are hence compact.

Another part of GR where conformal invariance plays a role regards curvature: the traceless part of the Riemann tensor, i.e. the Weyl tensor, is conformally invariant.

Further applications of conformal invariance related to general relativity include string theory, which is conformally invariant on the string worldsheet, and the AdS/CFT correspondence, where a string theory on a 5-dimensional AdS space is equivalent to a 4-dimensional (supersymmetric) conformal field theory. In certain coupling limits of this duality, the string theory part reduces to supergravity, which again reproduces standard general relativity once one breaks supersymmetry. Such models are used for example to describe duals of QCD-like theories within holography.

## Best Answer

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 geometriesand forgeometry to be a dynamical quantity. Galilean space and Minkowski spacetime come with privileged coordinate systems, inertial frames, a spacetime in general does not.