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Physics
Special and General Relativity
Constructing Bondi Coordinates on General Spacetimes
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[QUOTE="leo., post: 6056063, member: 447865"] [USER=252793]@martinbn[/USER] I thought the same when I've read section 2 of Sachs' paper the first time. But notice that Strominger points out that [I]any geometry can be locally written in these coordinates with that metric tensor[/I]. I actually have the impression that it is true. My problem is that if [I]any geometry admits such local description[/I], then there must be a construction in an arbitrary geometry which yields these coordinates. Take the Riemann normal coordinates for example. It is locally available on any geometry. And there is such a cosntruction: take one point ##z\in M##, build one orthonormal tetrad ##e_a## on the point. There is a neighborhood of the origin on which the exponential map is a diffeomorphism, let ##N## be the image, so that ##\exp_z^{-1}## is well defined on ##N##. Let ##q\in N## then ##\exp_z^{-1}(q)## can be expanded on ##e_a##. Define ##\exp_z^{-1}(q) = x^{a}(q)e_a##. The map ##x(q) = (x^a(q))## is a coordinate chart because ##\exp_z## is a diffeomorphism on the region of interest. So you see: there is mathematical proof that we can construct the system, and the procedure which yields the coordinate functions. Another example would be angular coordinates on a spherically symmetric spacetime like Schwarzschild spacetime. There is a derivation that shows how the existence of the angular coordinates follows from spherical symmetry. I'm looking for what yields these Bondi coordinates on any spacetime. A construction which shows what is the domain of validity of the coordinates and the respective ranges. Unfortunately, up to this point I didn't find this. [/QUOTE]
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Constructing Bondi Coordinates on General Spacetimes
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