(1) where: components of the metric. g: A Greek index assumes the range 0, 1, 2, 3 and a Latin index assumes the range 1, 2, 3. A repeated index implies summation on that index. The metric depends upon the gravitational potentials and upon the angular velocity and linear acceleration of the reference frame. Upon a transformation of the coordinates, the space-time interval remains invariant. Thus the metric transforms as a second g order covariant tensor. The general expression for the relationship between proper time and the coordinates of the chosen coordinate system, comprising the coordinate time and the spatial coordinates is by: x0 ct xi, (2) where: : the proper time. Thus = for a clock at rest in an inertial frame of reference, for which = 0 and = 1, = dt d dxi g00 g0 j 0, and = The elapsed coordinate time corresponding to the measured proper time as gi j i j. registered by a clock along a path between points and is: A B (3) For an electromagnetic signal, the interval is: (4) The speed of light is in every inertial frame of reference. The elapsed coordinate time of c propagation along a path between points and is: A B (5) The expression + 0 0 / (–g 00) represents the metric of space and i j g i j g i g j represents the increment of three-dimensional distance. Time scales Atomic time scales The fundamental scale of time based on atomic clocks is International Atomic Time (TAI), which is calculated at the BIPM from a weighted average of the readings of atomic clocks in timing laboratories distributed around the world. It is a continuous reference timescale without steps. The atomic timescale for civil timekeeping is Coordinated Universal Time (UTC), which differs from TAI by an integral number of seconds. In 2011, UTC = TAI 34 s. UTC is disseminated – every month in BIPM in the form of the differences from individual laboratory Circular T realizations UTC(k). Coordinate time scales Geocentric Coordinate Time (TCG) is the coordinate time in a coordinate system with origin at the center (ECI or ECEF).
(1) where: g: components of the metric. A Greek index assumes the range 0, 1, 2, 3 and a Latin index assumes the range 1, 2, 3. A repeated index implies summation on that index. The metric depends upon the gravitational potentials and upon the angular velocity and linear acceleration of the reference frame. Upon a transformation of the coordinates, the space-time interval remains invariant. Thus the metric g transforms as a second order covariant tensor. The general expression for the relationship between proper time and the coordinates of the chosen coordinate system, comprising the coordinate time x0 ct and the spatial coordinates xi, is by: (2) where: : the proper time. Thus dt = d for a clock at rest in an inertial frame of reference, for which dxi = 0 and g00 = 1, g0 j = 0, and gi j = i j. The elapsed coordinate time corresponding to the measured proper time as registered by a clock along a path between points A and B is: (3) For an electromagnetic signal, the interval is: (4) The speed of light is c in every inertial frame of reference. The elapsed coordinate time of propagation along a path between points A and B is: (5) The expression i j g i j + g 0 i g 0 j / (–g 00) represents the metric of space and represents the increment of three-dimensional distance. Time scales Atomic time scales The fundamental scale of time based on atomic clocks is International Atomic Time (TAI), which is calculated at the BIPM from a weighted average of the readings of atomic clocks in timing laboratories distributed around the world. It is a continuous reference timescale without steps. The atomic timescale for civil timekeeping is Coordinated Universal Time (UTC), which differs from TAI by an integral number of seconds. In 2011, UTC = TAI – 34 s. UTC is disseminated every month in BIPM Circular T in the form of the differences from individual laboratory realizations UTC(k). Coordinate time scales Geocentric Coordinate Time (TCG) is the coordinate time in a coordinate system with origin at the center (ECI or ECEF).