When introducing the vielbein formalism in general relativity, I came across the use of an infinitesimal general transformation, or Einstein transformation. The latter term seems to not be covered on standard references, so my asking here.
I quote the lectures in which I found this term: Horatiu Nastase - Introduction to Supergravity, p.12, right above (2.2).
On both the metric and the vielbein we have also general coordinate transformations. We can check that an infinitesimal general coordinate transformation ("Einstein" transformation) $\delta x^\mu = \xi^\mu$ acting on the metric gives $$ \tag{2.2} (\delta_\xi g)_{\mu\nu} (x) = (\xi^\rho \partial_\rho) g_{\mu\nu} + (\partial_\mu \xi^\rho) g_{\rho\nu} + (\partial_\nu \xi^\rho) g_{\rho \nu} $$ where the first term corresponds to a translation (the linear term in the Fourier expansion of a field), but there are extra terms. Thus the general coordinate transformations are the general relativity version, i.e. the local version of the (global) $P_\mu$ translations in special relativity (in special relativity we have a global parameter $\xi^\mu$, but now we have a local $\xi^\mu(x)$).
- What is meant by an Einstein transformation?
- What is meant with $\delta_\xi$, and how is the relation (2.2) obtained?
- The following transformation for the vielbein is given: $$ \tag{2.3} (\delta_\xi e)_\mu^a(x) = (\xi^\rho \partial_\rho) e_\mu^a + ( \partial_\mu \xi^\rho) e_\rho^a. $$ Why is this relation different from (2.2)? Why is the last term missing?