Frame fields in general relativity

A frame field in general relativity (also called a tetrad or vierbein) is a set of four pointwise-orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime. The timelike unit vector field is often denoted by and the three spacelike unit vector fields by . All tensorial quantities defined on the manifold can be expressed using the frame field and its dual coframe field. Frame were introduced into general relativity by Albert Einstein in 1928 and by Hermann Weyl in 1929. The index notation for tetrads is explained in tetrad (index notation). Frame fields of a Lorentzian manifold always correspond to a family of ideal observers immersed in the given spacetime; the integral curves of the timelike unit vector field are the worldlines of these observers, and at each event along a given worldline, the three spacelike unit vector fields specify the spatial triad carried by the observer. 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. In general, the worldlines of these observers need not be timelike geodesics. If any of the worldlines bends away from a geodesic path in some region, we can think of the observers as test particles that accelerate by using ideal rocket engines with a thrust equal to the magnitude of their acceleration vector. 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. It is important to recognize that frames are geometric objects.