# Local translations in curved spacetime

A global Poincare transformation on a scalar field induces

$$\delta(a, \lambda)\phi(x) = [a^{\mu}+\lambda^{\mu\nu}x_{\nu}]\partial_{\mu}\phi(x). \tag{11.46}$$

In curved spacetime we replace $a^{\mu} \rightarrow \xi^{\mu}(x)$, but I read that in fact this new spacetime dependent parameter $\xi^{\mu}$ eats up the effect of the "orbital part of global Lorentz", so that in fact $\xi^{\mu}(x)=a^{\mu}(x)+\lambda^{\mu\nu}(x)x_{\nu}$, and we effectively have to treat $\xi^{\mu}$ and $\lambda^{ab}$ (the spin part of Lorentz) as the basis for gauge transformations.

I don't understand why this should be, any comments that might help? Why does the new local translation parameter include the effect of what globally used to be rotations/boosts?

References:

1. D.Z. Freedman & A. Van Proeyen, SUGRA, 2012; p. 225.

1. Ref. 1 defines a local translation on spacetime $M$ as a diffeomorphism.

2. Note that the words local and global in this physics context mean point-dependent (=$x$-dependent) and point-independent (=$x$-independent), respectively. [Be aware that mathematicians in other contexts typically use the words local and global to refer to objects defined in a local neighborhood $U\subseteq M$, or globally on the whole manifold $M$, respectively. Ref. 1 does not dwell on this latter distinction, and implicitly assumes that the definitions of objects can be extended/restricted appropriately, if needed.]

3. An infinitesimal diffeomorphisms $\xi^{\mu}(x)$ can be identified with a vector field.

4. Note that in flat Minkowski spacetime the affine transformations are global$^1$ Poincare transformations. However, Ref. 1 concerns GR and curved spacetime. Global$^1$ Lorentz transformations, global$^1$ Poincare transformations, and affine transformations become obsolete/meaningless notions in GR. Such transformations are all just special cases of local translations.

5. On the other hand, in the terminology of Ref. 1, the local Lorentz transformations do not denote (and should not be confused with) a subset of diffeomorphisms. Instead a local Lorentz transformation $\Lambda^a{}_b(x)$ is a transformations between choices of frames in the tangent bundle. In other words, it transforms the flat indices $a,b,\ldots$, (but not the curved indices $\mu,\nu,\ldots$), of various tensors, e.g. the vielbein $e^a_{\mu}(x)$.

6. Together the local translations and local Lorentz transformations form the local Poincare transformations.

7. See also this & this related Phys.SE posts.

References:

1. D.Z. Freedman & A. Van Proeyen, SUGRA, 2012; p. 225.

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$^1$ We repeat that the word global means here $x$-independent, cf. pt. 2.