Tuesday 15 November 2016

general relativity - Can we always locally split the metric as $- N^2dt^2+g_{ij}dx^idx^j$?


In the book by Y. Choquet-Bruhat, General Relativity and the Einstein Equations, the following technical lemma is found on page 9:



A Lorentzian metric can always be written in a small enough neighborhood by a change of coordinates under the form $$-N^2dt^2+g_{ij}dx^idx^j.$$



The proof (I think that's what it's supposed to be) she gives makes little sense:




Indeed, under a change of coordinates $(x'^\alpha)\mapsto (x^\beta)$ with $x^0=x'^0$ we have $$g'_{i0}=\frac{\partial x^j}{\partial x'^i}\left(g_{j0}+g_{jh}\frac{\partial x^h}{\partial x'^0}\right),$$ we make $g_{i0}'=0$ by solving the linear first-order system $g_{j0}+g_{jh}\frac{\partial x^h}{\partial x'^0}=0$ for the functions $x^h(x'^i,x'^0)$.



The reason this is problematic is that we assumed $x^0=x'^0$, so that in fact $x^h$ cannot be a function of $x'^0$, and the linear system falls apart.


Am I misinterpreting what she's saying? Can the proof be salvaged or is this a bad typo? Is the result true?




No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...