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I tried to understand the importance of conformal transformations in general relativity, but I failed. I didn't see that conformal transformations help to simplify the metrics, and also I didn't see that some physical metric (i.e., metric which describes geometry of some physical system) with conformal scale factor.

Can you give some examples when conformal transformations are physically useful?

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Conformal transformations are critically important in any analysis of the global structure of a spacetime--the strategy is usually to use a conformal structure to make a topologically open spacetime conformally equivalent to one with a finite area in coordinate space, and then to look at the boundary of this conformally equivalent spacetime. –  Jerry Schirmer Nov 28 '13 at 16:59
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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.

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