Are Local Lorentz Transformations Possible with Varying ##\vec{x}##?

[Tio Barnabe]

We are always taught in books that a Lorentz transformation is possible as long as the Lorentz matrices $\Lambda$ in $\vec{x}{\ }' = \Lambda \vec{x}$ are not function of $\vec{x}$. The reason for this is obvious, since in this way the relation $t^2 - x^2 - y^2 - z^2 = t'^2 - x'^2 - y'^2, - z'^2$ is true.

Nevertheless, I wonder if it's possible to exist a transformation $\vec{x}{\ }' = \Lambda (\vec{x}) \vec{x}$ as long as we do something else. Is it possible?

 

[dextercioby]

Yes, there exists such a transformation, but it's no longer called Lorentz transformation. It can be a GCT, a general coordinate transformation, i.e. pedantically a diffeomorphism of the space-time manifold on itself.

However, if you take a pseudo-orthogonal matrix and make its Lie-group parametrization to be coordonate dependent, then you are "gauging the Lorentz transformations" at a point in spacetime, and this was first properly accomplished by Utyiama. "Invariant Theoretical Interpretation of Interaction,", The Physical Review, 1956, section 4.

 

[Tio Barnabe]

Interesting.
What does it mean for a Lorentz transformation to have $\Lambda$ constant (aside from the fact that the metric is preserved)? Would it mean that it is a global transformation? Is not that against Einstein's principle of locality? Because if we give a value for $\Lambda$ it will have the same value for all $x$, instantaneously.

 

[haushofer]

See also

https://www.physicsforums.com/insights/general-relativity-gauge-theory/

Your remark about locality is not clear to me. Locality does not forbid global coordinate transformations.

 

[haushofer]

These local Lorentz transformations are crucial in defining spinors in curved spacetime. Spinor-representations are defined for the Lorentzgroup, and for that you have to go to the tangent spacetime. This is the essence of the vielbein formulation of GR.