The conventional physics way of explaining this is as follows:
One way of asking “what is the current state of the universe?” is to pick a Cauchy surface. This is just a “slice” of the entire universe at a given time. There is a lot of freedom in the choice of slice: In Minkowsky space for example, there are slices corresponding to every choice of rest frame, and many more besides those. We just need to make sure that no points on the surface lie within each-other’s light cones (Δs2<0).
The information (about field values & derivatives) lying on any particular Cauchy surface is enough to predict the future and past from that surface. Pick any two Cauchy surfaces, and there’s a unitary operator mapping one to the other. This is the relativistic version of a time-evolution operator.
Some Cauchy surfaces are entirely later in time than other Cauchy surfaces. (Though some pairs of Cauchy surfaces are partially later and partially earlier than each other.) We’ll say that for Cauchy surfaces A,B, that A<B exactly when for all points a∈A,b∈B, either a is spacelike separated from b or b is in the future lightcone of a.
Let S() be a function that measures the entropy on a given Cauchy surface. The second law of thermodynamics then says that if A<B then S(A)≤S(B).
The conventional physics way of explaining this is as follows:
One way of asking “what is the current state of the universe?” is to pick a Cauchy surface. This is just a “slice” of the entire universe at a given time. There is a lot of freedom in the choice of slice: In Minkowsky space for example, there are slices corresponding to every choice of rest frame, and many more besides those. We just need to make sure that no points on the surface lie within each-other’s light cones (Δs2<0).
The information (about field values & derivatives) lying on any particular Cauchy surface is enough to predict the future and past from that surface. Pick any two Cauchy surfaces, and there’s a unitary operator mapping one to the other. This is the relativistic version of a time-evolution operator.
Some Cauchy surfaces are entirely later in time than other Cauchy surfaces. (Though some pairs of Cauchy surfaces are partially later and partially earlier than each other.) We’ll say that for Cauchy surfaces A,B, that A<B exactly when for all points a∈A,b∈B, either a is spacelike separated from b or b is in the future lightcone of a.
Let S() be a function that measures the entropy on a given Cauchy surface. The second law of thermodynamics then says that if A<B then S(A)≤S(B).