The above equation comes from Wikipedia. Another paper gives us:
ds² = dz² − dt² + dx² − 2 e^x/a dt dy − (e^x/a dy)².

I believe neither of these for the coefficients of the spacial differentials (squared) must always be positive!! The cross term doesn’t ameliorate things! The signature must be (1, 3), not (2, 2). Well actually you can't tell without putting the metric in a diagonal form. See below.

With terms off the diagonal the claim above is shaky 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 from a forward time arrow (θ=0) to a backward time arrow (θ=π) and ds² is always negative. In a proper Lorentz metric (of signature (− , +, +, +)) there is no path between such arrows with ds² < 0. The metric 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 (+, +, +, +).

If we hold t and y constant we get a hyperbolic metric for the resulting 2D subspace:
>ds² = (e^xdz)² + ...