Frame fields in general relativity
The index notation for tetrads is explained in tetrad (index notation).
These fields are required to write the Dirac equation in curved spacetime.
In particular, the vector fields in the frame can be expressed this way:
In "designing" a frame, one naturally needs to ensure, using the given metric, that the four vector fields are everywhere orthonormal.
Alternatively, the metric tensor can be specified by writing down a coframe in terms of a coordinate basis and stipulating that the metric tensor is given by
Local Lorentz indices are raised and lowered with the Lorentz metric in the same way as general spacetime coordinates are raised and lowered with the metric tensor. For example:
The vierbein field enables conversion between spacetime and local Lorentz indices. For example:
A few more examples: Spacetime and local Lorentz coordinates can be mixed together:
The local Lorentz coordinates transform differently from the general spacetime coordinates. Under a general coordinate transformation we have:
Coordinate basis vectors can be null, which, by definition, cannot happen for frame vectors.
Given a Lorentzian manifold, we can find infinitely many frame fields, even if we require additional properties such as inertial motion. However, a given frame field might very well be defined on only part of the manifold.
It will be instructive to consider in some detail a few simple examples. Consider the famous Schwarzschild vacuum that models spacetime outside an isolated nonspinning spherically symmetric massive object, such as a star. In most textbooks one finds the metric tensor written in terms of a static polar spherical chart, as follows:
More formally, the metric tensor can be expanded with respect to the coordinate cobasis as
To see that this coframe really does correspond to the Schwarzschild metric tensor, just plug this coframe into
The frame dual is the coframe inverse as below: (frame dual is also transposed to keep local index in same position.)
The components of various tensorial quantities with respect to our frame and its dual coframe can now be computed.
For example, the tidal tensor for our static observers is defined using tensor notation (for a coordinate basis) as
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:
Notice that there is simply no way of defining static observers on or inside the event horizon. 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.