Cauchy surface

In the mathematical field of Lorentzian geometry, a Cauchy surface is a certain kind of submanifold of a Lorentzian manifold. In the application of Lorentzian geometry to the physics of general relativity, a Cauchy surface is usually interpreted as defining an "instant of time"; in the mathematics of general relativity, Cauchy surfaces are important in the formulation of the Einstein equations as an evolutionary problem. They are named for French mathematician Augustin-Louis Cauchy (1789-1857) due to their relevance for the Cauchy problem of general relativity. Although it is usually phrased in terms of general relativity, the formal notion of a Cauchy surface can be understood in familiar terms. Suppose that humans can travel at a maximum speed of 20 miles per hour. This places constraints, for any given person, upon where they can reach by a certain time. For instance, it is impossible for a person who is in Mexico at 3 o'clock to arrive in Libya by 4 o'clock; however it is possible for a person who is in Manhattan at 1 o'clock to reach Brooklyn by 2 o'clock, since the locations are ten miles apart. So as to speak semi-formally, ignore time zones and travel difficulties, and suppose that travelers are immortal beings who have lived forever. The system of all possible ways to fill in the four blanks in "A person in (location 1) at (time 1) can reach (location 2) by (time 2)" defines the notion of a causal structure. A Cauchy surface for this causal structure is a collection of pairs of locations and times such that, for any hypothetical traveler whatsoever, there is exactly one location and time pair in the collection for which the traveler was at the indicated location at the indicated time. There are a number of uninteresting Cauchy surfaces. For instance, one Cauchy surface for this causal structure is given by considering the pairing of every location with the time of 1 o'clock (on a certain specified day), since any hypothetical traveler must have been at one specific location at this time; furthermore, no traveler can be at multiple locations at this time.