Another paper gives us: ds2 = dz2 – dt2 + dx2 – 2 ex/a dt dy – (ex/a dy)2.
The Wikipedia article has much insightful information on this metric but ignores what seems to me to be the most significant property; the metric is not locally Minkowskian! A Minkowski space has a metric whose signature is <– + + +>. The signature of the metric above is <– + + –>. The big black Gravitation book (ISBN=0716703440) uses "Lorentz" as an adjective for manifolds with a signature (–1, 1, 1, 1) (Page 311). It also says that manifolds must be Lorentz to be physical.
With terms off the diagonal the claim above is not manifest, but consider map from a circle to a small loop of directions from event <0, 0, 0, 0>. For 0 ≤ θ ≤ 2π: (cos θ) dt + (sin θ) dz. These directions go continuously from a forward time arrow (θ=0) to a backward time arrow (θ=π) while ds2 remains negative. In a proper Lorentz metric (of signature <– + + +>) there is no path between such arrows with ds2 < 0. The metric thus does not divide the future from the past, even locally! It is peculiar locally, as well as globally. Gödel found a metric with uniform signature <– – + +> that solves Einstein’s equations. I wonder if there is a solution with signature <+, +, +, +>.
I have heard from physicists that several well known physicists have tried to make sense of two time coordinates without success. Even 11 dimensional string theory sticks to exactly one time dimension. In the above expression of the metric there is nothing to tell which of t and z is the time coordinate, except the connotation of ‘t’.
Compare
ds2 = –(dt + exdz)2 + dx2 + dy2＋(exdz)2/2.
with
ds2 = –(dz + exdt)2 + dx2 + dy2＋(exdt)2/2.
They are the same manifold; only the names are changed.
Indeed any affine transformation of the t-z space results in a reparameterization of the manifold.
The signature is an invarient property of the manifold and excludes it, I think, from those that Einstein considered.
I need to consider this page to see if it refutes my theses, and if not include some of my information.