# Characteristic problem Usually in classical physics, equations of motion are second order, meaning that, if we specify the value of physical quantities and their first time-derivatives at some time, we can use the equation of motion to evolve the state to the future (and the past). When the solution to the evolution equations exists, is unique, and does not change too much when the initial data changes slightly, then we say that the theory has a *well-posed* initial value problem. In [[0554 Einstein gravity|General Relativity]], where space and time are interweaved, one can nevertheless artificially recover the notion of evolution using for example the [[0490 The 3+1 decomposition|ADM decomposition]], where space and time again become orthogonal in some sense. Specifying the metric and its first time derivative at some spatial hypersurface then leads to a well-posed initial value problem, when formulated in some appropriate gauge. However, one can ask whether there exist other formulations of the physics of evolution in General Relativity. For example, one can obtain a null surface by taking the limit of a sequence of spacelike surfaces. It is then interesting to ask whether one can specify initial data on this null surface. The answer turns out to be yes, but taking the limit is subtle. For example, on the null surface with a null generator $l^a$, we have $l_al^a=0$, so $l^a$ is both parallel and orthogonal to the surface. The notion of specifying time-derivatives in this case is therefore not a simple limit of the spacelike case. That's why physicists and mathematicians are interested in formulating this rigorously and cleanly. The problem of formulating the physics of evolution by specifying data on a null surface is called the **characteristic problem**. ## Refs - original: to be added - covariant formulation: [[2022#Mars, Sanchez-Perez]]