Hello everyone, here I come with a question about inertial frames as defined in General Relativity, and how to prove that the general definition is consistent with the particular case of Special Relativity.(adsbygoogle = window.adsbygoogle || []).push({});

So to contextualize, I have found that one can define inertial frames in General Relativity, as follows: given a 4-dimensional lorentzian manifold with metric tensor g and signature (-1,1,1,1), aframe fieldis defined to be a set of four vector fields e0,e1,e2,e3 such that:

g(ei,ej) = ηij (η being the Minkowski matrix)

Further, one says that the frame field in question isinertial and nonrotatingif the following condition is satisfied:

∇_{e0}ei = 0

where ∇ is the Levi-Civita connection on the manifold (let me call this "Definition A"). All this straight out of Wikipedia, and quite elegant and nice.

Now, my question is how does this correspond to the definition of inertial frames in Special Relativity as usually found in textbooks. So now assume that g is the Minkowski metric and that we are in Minkowski spacetime, where there is a given global coordinate chart (t,x,y,z) (I am setting c=1 for simplicity), so that:

g = -dt⊗dt + dx⊗dx + dy⊗dy + dz⊗dz

Then it is straightforward to see that the vector fields ∂/∂t, ∂/∂x, ∂/∂y, ∂/∂z are indeed an inertial nonrotating frame field in the abstract sense of Definition A. But in Minkowski spacetime one hasDefinition B(the one usually found in textbooks), where we are given an initial "precursor" inertial frame (the one used by the scientist writting the textbook), which in this context is a coordinate chart with certain physical properties, namely that there are no inertial forces (i.e. the Christoffel symbols vanish identically), and can of course be identified with the coordinate chart (t,x,y,z) I mentioned earlier. And then we are told that "inertial frames" are all those coordinate charts related to this precursor coordinate chart by a Lorentz transformation, which is a linear coordinate transformation Λ such that η = transpose(Λ) * η * Λ (this guarantees that the spacetime interval is preserved).

So, here comes my question: take anarbitrary nonrotating inertial framein the sense of Definition A, that is a set of four vector fields e0,e1,e2,e3 satistying the requisites of that abstract definition. Is it the case, then, that there exists a coordinate chart x0,x1,x2,x3 and a suitable Lorentz transformation Λ from (t,x,y,z) (the "precursor coordinate chart" inherent to the construction of Minkowski spacetime) to (x0,x1,x2,x3) such that:

ei = ∂/∂xi for i=0,1,2,3 ?

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# A Inertial frames: from GR to SR

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