Undergrad Solving Boundary Conditions in 4D Spacetime Volume

[BookWei]

Hi, my classmate asks me an interesting question: For a finite 4D volume in spacetime, its boundary is a 3D close surface. If the 4D volume is a 4D rectangular, the boundary consists of eight 3D surfaces. The boundary condition is specified on these eight 3D surface. Please explain the physical meaning of the boundary conditions on each of these eight 3D surface. We do not know how to explain this question in physics or mathematics. Can someone help us to attack this problem?
Thanks for all responses.

 

[Ibix]

These aren't boundary conditions so much as boundaries. I'd suggest drawing a Minkowski diagram, a 1+1 dimensional spacetime. Draw a region bounded by four straight lines which you get by sequentially holding one coordinate constant at some pair of maximum and minimum values and varying the other. Describe those boundaries. Then sketch a 2+1 dimensional spacetime. You get six planes bounding a cuboidal region by sequentially holding one coordinate constant at some pair of maximum and minimum values and varying the others. Describe those boundaries. Then think about a 3+1 dimensional spacetime in the same way.

A couple of things to think about. Are your descriptions coordinate dependent or not? Does the above procedure generalise to curved spacetime?

 

[sweet springs]

Say E(t,x,y,z) is a field entity, boundary conditions are written as

t=0 All E(0,x,y,z) are given.
t=T All E(T,x,y,z) are given.

x=0 All E(t,0.y,z) are given
x=X All E(t,X.y,z) are given

similarly for y and z . Total eight lines.

where
0<t<T,0<x<X,0<y<Y,0<z<Z.