Measuring the Metric, and Curvature versus Acceleration

These notes show how observers can set up a coordinate system and measure the spacetime geometry using clocks and lasers. The approach is similar to that of special relativity, but the reader must not be misled. Spacetime diagrams with rectilinear axes do not imply flat spacetime any more than flat maps imply a flat earth. Cartography provides an excellent starting point for understanding the metric. Terrestrial maps always provide a scale of the sort One inch equals 1000 miles. If the map is of a sufficiently small region and is free from distortion, one scale will suffice. However, a projection of the entire sphere requires a scale that varies with location and even direction. The Mercator projection suggests that Greenland is larger than South America until one notices the scale difference. The simplest map projection, with latitude and longitude plotted as a Cartesian grid, has a scale that depends not only on position but also on direction. Close to the poles, one degree of latitude represents a far greater distance than one degree of longitude. The map scale is the metric. The spacetime metric has the same meaning and use: it translates coordinate distances and times ( one inch on the map ) to physical ( proper ) distances and times. The terrestrial example also helps us to understand how coordinate systems can be defined in practice on a curved manifold. Let us consider how coordinates are defined on the Earth. First pick one point and call it the north pole. The pole is chosen along the rotation axis. Now extend a family of geodesics from the north pole, called meridians of longitude. Label each meridian by its longitude φ. We choose the meridian going through Greenwich, England, and call it the prime meridian, φ = 0. Next, we define latitude λ as an affine parameter along each meridian of longitude, scaled to π/2 at the north pole and decreasing linearly to −π/2 at the point where the meridians intersect