Selected Formulas

The Schwarzschild metric describes the separation between two neighboring events in the vicinity of a spherically symmetric, nonrotating center of gravitational attraction. This is equation [10] on page 2-19.

• dx is the wristwatch time between the two events as measured on a wristwatch that moves directly from one event to the other.

• dt is the time between the events measured on a clock far from the center (page

• r is the reduced circumference: circumference divided by 271 (page 2-7).

• M is the mass of the center of attraction measured in units of meters (page 2-13).

Equation [A] is called the timelike version of the Schwarzschild metric, useful when a clock can be carried between the two events at less than the speed of light. If this is not possible, then we use the spacelike version, equation [11] on page 2-19.

Here da is the proper distance between the two events: the distance between them recorded by a measuring rod moving in such a way that the two events occur at the same time in its rest frame.

Equation [C] relates the time lapse dfSheii between two ticks of a shell clock (a clock at rest on a stationary spherical shell of radius r concentric to the center of attraction) to the time lapse dt between the same two ticks measured by a far-away clock. This is equation [19] on page 2-23.

shell -

Equation [D] relates the radial distance drshen measured directly by the shell observer between two events that lie along the same radial direction and the radial separation dr calculated by a far-away observer. This is equation [12] on page 2-22.

The following approximation is used often in this book. The approximation is accurate for positive, negative, or fractional values of n.

(1 + df = 1 + nd provided \d\ « 1 and \nd\ « 1 [E]

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