Frame fields were introduced into general relativity by Albert Einstein in 1928[1] and by Hermann Weyl in 1929.
The triad may be thought of as defining the spatial coordinate axes of a local laboratory frame, which is valid very near the observer's worldline.
Alternatively, if our observer is attached to a bit of matter in a ball of fluid in hydrostatic equilibrium, this bit of matter will in general be accelerated outward by the net effect of pressure holding up the fluid ball against the attraction of its own gravity.
Other possibilities include an observer attached to a free charged test particle in an electrovacuum solution, which will of course be accelerated by the Lorentz force, or an observer attached to a spinning test particle, which may be accelerated by a spin–spin force.
Once a signature is adopted, by duality every vector of a basis has a dual covector in the cobasis and conversely.
For example: The vierbein field enables conversion between spacetime and local Lorentz indices.
Except in locally flat regions, at least some Lie brackets of vector fields from a frame will not vanish.
The resulting baggage needed to compute with them is acceptable, as components of tensorial objects with respect to a frame (but not with respect to a coordinate basis) have a direct interpretation in terms of measurements made by the family of ideal observers corresponding to the frame.
Particularly in vacuum or electrovacuum solutions, the physical experience of inertial observers (who feel no forces) may be of particular interest.
The mathematical characterization of an inertial frame is very simple: the integral curves of the timelike unit vector field must define a geodesic congruence, or in other words, its acceleration vector must vanish: It is also often desirable to ensure that the spatial triad carried by each observer does not rotate.
The criterion for a nonspinning inertial (NSI) frame is again very simple: This says that as we move along the worldline of each observer, their spatial triad is parallel-transported.
Given a Lorentzian manifold, we can find infinitely many frame fields, even if we require additional properties such as inertial motion.
Consider the famous Schwarzschild vacuum that models spacetime outside an isolated nonspinning spherically symmetric massive object, such as a star.
This is the frame that models the experience of static observers who use rocket engines to "hover" over the massive object.
Using tensor notation for a tensor field defined on three-dimensional euclidean space, this can be written The reader may wish to crank this through (notice that the trace term actually vanishes identically when U is harmonic) and compare results with the following elementary approach: we can compare the gravitational forces on two nearby observers lying on the same radial line: Because in discussing tensors we are dealing with multilinear algebra, we retain only first order terms, so
Here, we are referring to the obvious frame obtained from the polar spherical chart for our three-dimensional euclidean space: Plainly, the coordinate components
The result will be a frame which we can use to study the physical experience of observers who fall freely and radially toward the massive object.
By appropriately choosing an integration constant, we obtain the frame of Lemaître observers, who fall in from rest at spatial infinity.
(This phrase doesn't make sense, but the reader will no doubt have no difficulty in understanding our meaning.)
In the static polar spherical chart, this frame is obtained from Lemaître coordinates and can be written as Note that
"leans inwards", as it should, since its integral curves are timelike geodesics representing the world lines of infalling observers.
If our massive object is in fact a (nonrotating) black hole, we probably wish to follow the experience of the Lemaître observers as they fall through the event horizon at
(This is a remarkable and rather special property of the Schwarzschild vacuum; most spacetimes do not admit a slicing into flat spatial sections.)
On the other hand, the Lemaître observers are not defined on the entire exterior region covered by the static polar spherical chart either, so in these examples, neither the Lemaître frame nor the static frame are defined on the entire manifold.
direction by an undetermined parameter (depending on the radial coordinate), compute the acceleration vector, and require that this vanish in the equatorial plane
The new Hagihara frame describes the physical experience of observers in stable circular orbits around our massive object.
In the static polar spherical chart, the Hagihara frame is which in the equatorial plane becomes The tidal tensor
turns out to be given (in the equatorial plane) by Thus, compared to a static observer hovering at a given coordinate radius, a Hagihara observer in a stable circular orbit with the same coordinate radius will measure radial tidal forces which are slightly larger in magnitude, and transverse tidal forces which are no longer isotropic (but slightly larger orthogonal to the direction of motion).
The principal reason why is easy to spot: in this frame, each Hagihara observer keeps his spatial vectors radially aligned, so
As before, frames can be specified in terms of a given coordinate basis, and in a non-flat region, some of their pairwise Lie brackets will fail to vanish.