The Lie derivative of a tensor is a tensor of the same rank and type. But it is connection independent meaning it can be expressed in terms of covariant or partial derivatives. Since the Strong Equivalence Principle (SEP) requires the connection, at a point on a Lorentzian manifold, to vanish, spacetime is locally Minkowskian. At that point, the connection in Lie derivative when expressed in terms of its covariant derivatives, vanishes as well. So it seems the Lie derivative always satisfies the SEP and local Lorentz invariance. However, in Gravity and Strings by Ortin, it says on P374: the Lie derivative is not covariant under local Lorentz transformations and its action on Lorentz tensors is frame-dependent. So, does the Lie derivative of a tensor satisfy the SEP and local Lorentz invariance or not?

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    $\begingroup$ Can you clarify what you mean by a mathematical operation obeying (or failing to obey) a physical principle? $\endgroup$ – J. Murray Mar 7 at 17:37
  • $\begingroup$ I am looking for a clarification on whether or not the Lie derivative of a tensor is locally Lorentz invariant. ie. given the fact that normal coordinates exist which make the connection vanish, it seems to me a Lie derivative always is locally Lorentz invariant and satisfies the SEP. Perhaps the statement by Ortin that he is using a Killing vector in the Lie derivative is why he is saying the Lorentz invariance is broken. I am not restricting the Lie derivative to be along a Killing vector. $\endgroup$ – Kolten Mar 7 at 17:52
  • $\begingroup$ The Lie derivative of a tensor is along the flow of a vector field - it only depends on the properties of the manifold. On the other hand, the covariant derivative requires addition structure - namely a connection. In general, the Lorentz Lie group is invariant along the flow. I would argue it's the action of the string which is breaking Lorentz invariance. The Killing vector is a symmetry of the metric. $\endgroup$ – Cinaed Simson Mar 7 at 21:41
  • $\begingroup$ I agree the Lie derivative should be invariant along the flow. So I assume you would agree the Lie derivative of a tensor is locally Lorentz invariant. $\endgroup$ – Kolten Mar 7 at 21:50
  • $\begingroup$ I agree - the vector fields use local coordinates. $\endgroup$ – Cinaed Simson Mar 8 at 0:57

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