Our universe is described as having three physical dimensions, plus one time dimension, where the eigenvalues for the physical dimensions are all the same, but the eigenvalue for time is opposite (x, y, z, t: +, +, +, -) or (x, y, z, t: -, -, -, +)
A universe where the eigenvalue of time is equal to that of the spatial dimensions (x, y, z, t: +, +, +, +) has been explored in depth by Greg Egan in his Orthogonal series of books.
However, what would happen if the eigenvalues of the physical dimensions were not identical, e.g.: (x, y, z, t: +, +, -, -)? or even, in a 4-physical-dimensional universe, (w, x, y, z, t: -, +, +, +, -)?
In the universe I'm interested in, there is one time dimension with a negative eigenvalue, and 3 or 4 physical dimensions, but one of the physical dimensions has an eigenvalue equivalent to that of the time dimension, while the others have the opposite eigenvalue as is the case in our universe.
How would the physics and chemistry of such a universe be different to our own? How would movement in the negative-eigenvalue spatial dimension work? Would life - or even matter - be able to exist?