Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 23; Problem 23.1.
A static, plane-symmetric spacetime is one in which spacetime is independent of time (static) and is composed of a set of planes, where each plane is labelled by a coordinate . Within each plane, points are labelled by coordinates and and because the spacetime is static, the distance between two points depends only on these two coordinates:
where the subscript denotes the plane with coordinate .
If the basis vector is everywhere perpendicular to and (and ), then the spatial off-diagonal components of the metric are zero
The general metric between any two spacetime points is then
Because the spacetime is static, a displacement forward in time by should give the same separation as a displacement backwards by the same amount . Because of this symmetry, the term should remain unchanged when is replaced by . However, since the metric is independent of time, , so the only way we can satisfy the symmetry requirement is if . Thus the plane-symmetric metric is symmetric:
Further, can depend at most on alone.
To work out the consequences of this metric, we need to evaluate the Christoffel symbols and Ricci tensor. The Christoffel symbol worksheet is:
In this case and
Thus the only nonzero symbols will be those involving , since all other derivatives are zero. These are
[We can use the total derivative rather than partial because depends only on .]
From the Ricci tensor worksheet, the only nonzero components of are those involving or only, so we see that
with all other . In flat space, all components satisfy so these two components both give the same condition on :
To examine the structure of the spacetime, we need the full Riemann tensor, which is defined in terms of the Christoffel symbols:
Next, we’ll identify the index combinations that give (potentially) nonzero values for components of in each term, using the fact that only and are nonzero, and that only the derivative with respect to (index 1) is nonzero.
- Term 1:
- Term 2:
- Term 3:
- Term 4:
- Term 5:
- Term 6:
From these tables, we see that there are 7 unique index combinations that can potentially give nonzero Riemann tensor components . We have (remember that the Christoffel symbols are symmetric in their lower 2 indices: ):
Thus only the last 4 can potentially be nonzero. To go further, we need the derivative terms:
To get the Riemann tensor with all 4 indices lowered, we multiply by the metric:
Here, the only two metric components we need are and so
Note that in this lowered form, the symmetries of the Riemann tensor are obeyed: .
Finally, if we impose the condition 15 in the form , we find that all four of these components are zero, thus making the entire Riemann tensor zero, indicating that spacetime is completely flat. [There are a lot of indices flying about here, so I’m hoping I got them all right…]