The discussion clarifies the distinction between "locally flat" and "locally inertial" in the context of pseudo-Riemannian manifolds. Locally flat spacetime indicates that tidal forces diminish faster than first order, applicable to all spacetimes regardless of curvature. In contrast, locally inertial refers to a specific reference frame where the metric is expressed as ds^2=-c^2 dt^2 + dx^2 + dy^2 + dz^2, which is not universally valid across all coordinate systems. The conversation emphasizes that while every locally flat spacetime allows for a locally inertial frame, the two concepts are not interchangeable.
This discussion is beneficial for physicists, mathematicians, and students of general relativity who seek a deeper understanding of the nuances between locally flat and locally inertial frames in the context of spacetime geometry.
Do you mean that a tetrad/verbatin can be said inertial when its unit timelike vector field is indeed inertial (i.e. its timelike integral curve results in a geodesic of underlying spacetime) ?Dale said:Ah, I see. You are using “frame” to mean something different from “coordinate chart”. Possibly you mean something like a tetrad? So you could describe an inertial tetrad in terms of a polar coordinate basis.
I was using “frame” to mean “coordinate chart”
Yes, with the additional restriction that the spacelike vector fields do not rotate along the timelike integral curvescianfa72 said:Do you mean that a tetrad/verbatin can be said inertial when its unit timelike vector field is indeed inertial (i.e. its timelike integral curve results in a geodesic of underlying spacetime) ?
I agree and would add some further distinctions:Dale said:No. Because “flat” is a term that describes a spacetime and “inertial” is a term that describes a reference frame, worldline, or coordinate chart. You wouldn’t say “this is an inertial manifold” and you wouldn’t say “this is a flat reference frame”, so the terms are not interchangeable.