# General Relativity and the Schwarzschild Metric

# General Relativity and the Schwarzschild Metric

Previously, we saw that variations in the time part of the spacetime metric cause free particles to accelerate, thus unifying gravity and relativity; and that orbits trace those accelerations, which follow the inverse‐square law around spherical source masses. But a metric that empirically models orbits is not enough; we want to understand how any arrangement of mass *determines* the metric in the surrounding spacetime. This chapter describes thinking tools, especially the frame‐independent idea of spacetime curvature, that helped Einstein develop general relativity. We describe the Einstein equation, which determines the metric given a source or set of sources. Solving that equation for the case of a static spherical mass (such as the Sun) yields the Schwarzschild metric. We compare Schwarzschild and Newtonian predictions for precession, the deflection of light, and time delay of light; and we contrast the effects of variations in the time and space parts of the metric.

*Keywords:*
general relativity, Einstein equation, spacetime curvature, curvature of space, Schwarzschild metric, deflection of light, time delay of light, Mercury, precession

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