What are the local Lorentz transformations in general relativity?

Question

What is the exact form of local Lorentz transformations (from the point of view of the metric) in a curved spacetime background like in general relativity? It should deviate substantially from ordinary Lorentz transformations in Minkowski space.

Answer 1

Yes, there is such a thing in GR. These are the rotations of tetrads.

At each point of a space define an orthonormal basis: $$ (\vec e_{(a)}, \vec e_{(b)}) = \eta_{ab} $$ Where $a, b$ - denote the indices, corresponding to the local frame, in constract to the Greek spacetime indices $\mu, \nu$. The coordinate basis is related to the local basis, by some invertible $4 \times 4$ matrix : $$ \vec e_{\mu} = e_{\mu}^{a} \vec e_a $$ The metric in the coordinate space is expressed therefore, as: $$ g_{\mu \nu}= e_{\mu}^{a} e_{\nu}^{b} \eta_{ab} $$ Local Lorentz transformations can be made at any point: $$ \vec e_a = \Lambda_a^{b} (x) \vec e_b $$ Where the matrix of transformation depends on the point $x$.

For a good reference I recommend - https://arxiv.org/abs/1106.2037.