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Tetrads in General Relativity

VI. Physical Geometry

Local Flatness

   The crux of Einstein's equivalence principle is that, locally, the gravitational force can be negated by a suitable acceleration of the reference frame. Or equivalently, an observer undergoing constant acceleration may, as far as all laws of nature are concerned, consider itself at rest in a homogeneous gravitational field. Over extended distances, however, objects in free fall within a realistic gravitational field converge towards one other. This tidal effect is accounted for in general relativity as a consequence of the curvature of spacetime; see section X.

   Thus, Einstein’s equivalence principle is a local principle that is made more precise in the following theorem:

Local Flatness Theorem [S. C. Fletcher, 2023]

Given a Lorentzian spacetime ($\mathcal{M}_4,{g_{\mu \nu }}$), any embedded curve $\gamma $ therein, and any point $P$ . Then, there exists, within some neighborhood containing the point $P$, a flat metric ${\eta _{\mu \nu }}$, such that on the curve $\gamma $ :
  1. ${g_{\mu \nu }} = {\eta _{\mu \nu }}$
  2. $D = \nabla $, where $D$ and $\nabla $ are the Levi-Civita derivative operators associated with ${g_{\mu \nu }}$ and ${\eta _{\mu \nu }}$, respectively.

The theorem does not necessitate that the curve be a geodesic or even timelike.

   Given a flat metric with the aforementioned properties, one can invariably find an isometry from the region where the metric is defined to a region of Minkowski spacetime, subsequently employing that isometry to introduce standard Minkowski coordinates. Thus, a geodesic (non-spinning) frame in $\mathcal{M}_4$ is the geometric representation of a local observer frame of reference.

   The construction cannot be done in the whole of spacetime but only in a neighborhood of the worldline of the observer small enough to ignore tidal effects. It can be shown that this imposes the heuristic condition ${\cal{a}} \cdot {\cal{l}}/{c^2} \ll 1$ , where ${\cal{a}}$ is a measure for the local acceleration and ${\cal{l}}$ the extension of the local neighborhood. For example, if the acceleration is of the order of 10 m/s2 then the length is of the order of one light year, which is big enough to encompass all laboratories and observatories in the solar system.