Because time is incorporated as a fourth dimension and because space is curved, a general relativistic coordinate system centered on a spherically symmetric mass is more complicated than a three-dimensional Euclidean coordinate system. The azimuthal angle φ can still be defined as the ratio of the arc length to the circumference on an imaginary circle because the spherically symmetric gravitational mass, M, is located at the origin. However, the radial coordinate is not defined as the physical distance from the center because sometimes a particle cannot reach the center. Rather, it is calculated using a path that circumnavigates the central mass:
The time coordinate is defined using a wristwatch located far from the center of attraction. Note the nonlocal character of the (r, φ, t) spacetime coordinates. The wristwatch worn by the surveyor circumnavigating the mass used to measure r is not the time used to record events at that value of r.
This (r, φ, t) spacetime coordinate system is known as Schwarzschild coordinates, and is a universal bookkeeping device that enables us to translate observations from one reference frame to another. The Schwarzschild coordinates give rise to a metric, known as the Schwarzschild metric, that enables us to calculate the four-dimensional distance between adjacent spacetime events. This metric is given by
if we are exterior to the spherically symmetric mass M.
There are two options in how to run these programs:
| Run Locally | Run via Browser |
|---|---|
| Download the osp_gr.jar. Double-click the “osp_gr.jar” file. When the jar file opens, select the “Programs” button at the bottom of the splash screen. Navigate to the “Schwarzschild Metric” folder where you will find the programs. | To run via your web browser, simply select one of the "Programs" links. |
You may directly access (browse) the binary directory containing jar files. Note that the binary directory also contains code libraries (jar files) that are not designed to run as stand-alone applications.