spacetime in educational contexts. After he had realized the validity of this geometric analogy, it took Einstein a further three years to find the missing cornerstone of his theory: the equations describing how matter influences spacetime's curvature. Having formulated what are now known as Einstein's equations (or, more precisely, his field equations of gravity), he presented his new theory of gravity at several sessions of the Prussian Academy of Sciences in late 1915, culminating in his final presentation on November 25, 1915. Geometry and gravitation Paraphrasing John Wheeler, Einstein's geometric theory of gravity can be summarized thus: spacetime tells matter how to move; matter tells spacetime how to curve. What this means is addressed in the following three sections, which explore the motion of so-called test particles, examine which properties of matter serve as a source for gravity, and, finally, introduce Einstein's equations, which relate these matter properties to the curvature of spacetime. Probing the gravitational field In order to map a body's gravitational influence, it is useful to think about what physicists call probe or test particles: particles that are influenced by gravity, but are so small and light that we can neglect their own gravitational effect. In the absence of gravity and other external forces, a test particle moves along a straight line at a constant speed. In the language of spacetime, this is equivalent to saying that such test particles move along straight world lines in spacetime. In the presence of gravity, spacetime is non-Euclidean, or curved, and in curved spacetime straight world lines may not exist. Instead, test particles move along lines called geodesics, which are "as straight as possible", that is, they follow the shortest path between starting and ending points, taking the curvature into consideration. A simple analogy is the following: In geodesy, the science of measuring Earth's size and shape, a geodesic (from Greek "geo", Earth, and "daiein", to divide) is the shortest route between two points on the Earth's surface. Approximately, such a route is a segment of a great circle, such as a line of longitude or the equator. These paths are certainly not straight, simply because they must follow the curvature of the Earth's surface. But they are as straight as is possible subject to this constraint. The properties of geodesics differ from those of straight lines. For example, on a plane, parallel lines never meet, but this is not