Difference Between Locally Flat & Locally Inertial?

[cianfa72]

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”

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]

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) ?

Yes, with the additional restriction that the spacelike vector fields do not rotate along the timelike integral curves

 

[vanhees71]

To define a general rotation free (non-inertial) frame of reference along an arbitrary time-like curve you use Fermi-Walker transport of the three space-like vectors of a tetrad with the time-like vector being, of course, the tangent vector on this time-like curve.

If the time-like curve is a geodesic this defines a local inertial frame along the time-like curve, and the Fermi-Walker transport of the space-like vectors of the tetrad is identical with parallel transport.

 

[PAllen]

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.

I agree and would add some further distinctions:

- inertial motion is a characteristic of world line (and of a body to the extent it can be approximated by a world line). In SR and GR, this corresponds to the world line being a timelike geodesic.

- an inertial frame or coordinate system is one whose (t,0,0,0) world line is inertial and the rest is built by a standard procedure from the origin world line.

- locally flat in GR is simply the property that in small regions and short times, SR is a good approximation to GR. Or, mathematically, that all curvature effects (e.g. triangle angular defect/surplus) go to zero as size goes to zero. Curvature at a point is realized by normalizing by the area. Without normalization, the effect goes to zero as the square of the size. This has nothing to do with the coordinates or motion, and is simply part ot the definition of a Riemannian or pseudo-Riemannian manifold.