USNO Circular 179 - U.S. Naval Observatory

2 RELATIVITY
In 1991, the IAU made a series of recommendations concerning how the theory of relativity could
best be incorporated into positional astronomy. These recommendations and their implications were
studied by several working groups in the 1990s and some deficiencies were noted. As a result, a
series of new recommendations was proposed and discussed at IAU Colloquium 180 (Johnston et
al. 2000). The new recommendations were passed by the IAU General Assembly in 2000. It is
these recommendations that are described briefly in this chapter.
In special relativity, the Newtonian idea of absolute time in all inertial reference systems is
replaced by the concept that time runs differently in different inertial systems, in such a way that
the speed of light has the same measured value in all of them. In both Newtonian physics and
special relativity, inertial reference systems are preferred: physical laws are simple when written in
terms of inertial coordinates. In general relativity, however, time (and even space-time) is influenced
not only by velocity but also by gravitational fields, and there are no preferred reference systems.
One can use, in principle, any reference system to model physical processes. For an infinitely small
space-time region around an observer (considered to be a massless point), one can introduce socalled
locally inertial reference systems where, according to the Einstein’s equivalence principle, all
physical laws have the same form as in an inertial reference system in special relativity. Such locally
inertial reference systems are used to describe observations taken by the point-like observer. In
general-relativistic reference systems of finite spatial extent, the geometry of space-time is defined
by a metric tensor, a 4×4 matrix of mathematical expressions, that serves as an operator on two 4vectors.
In its simplest application, the metric tensor directly yields the generalized (4-dimensional)
distance between two neighboring space-time events. The metric tensor effectively determines the
equations through which physics is described in the reference system.
Time in general relativity can be understood as follows. As a particle moves through space-time,
each point (a space-time event) on the path that it follows can be characterized by a set of four
numbers. These four numbers are the values of the four coordinates in four-dimensional space-time
for a given coordinate system. For the same path in a different coordinate system, the numbers will,
in general, be different. Proper time is simply the time kept by a clock co-moving with the particle,
in whatever trajectory and gravity field it finds itself. Proper time is always measurable if a clock is
available that can travel with the particle. Coordinate time is one of the four independent variables
used to characterize a space-time event. Coordinate time is not measurable. The coordinate time of
a reference system is the independent argument of the equations of motion of bodies in that reference
system. The IAU resolutions on relativity passed in 2000 are concerned with two coordinate frames,
one barycentric and one geocentric, and the coordinate times used in each one.
1.2 The BCRS and the GCRS
In res. B1.3 of 2000, the IAU defined two coordinate frames for use in astronomy, one with its origin
at the solar system barycenter and one with its origin at the geocenter. In current astronomical
usage these are referred to as reference systems. (The astronomical distinction between reference
systems and reference frames is discussed in Chapter 3.) The two systems are the Barycentric
Celestial Reference System (BCRS) and the Geocentric Celestial Reference System (GCRS). Harmonic
coordinates are recommended for both systems (i.e., the harmonic gauge is used). The
resolution provides the specific forms of the metric tensors for the two coordinate systems and the
4-dimensional transformation between them. (The latter would reduce to a Lorentz transformation
for a fictitious Earth moving with constant velocity in the absence of gravitational fields.) The general
forms of the gravitational potentials, which appear in the metric tensors, are also presented. In