Given a curved 4-dimensional space with a real-valued metric tensor and real-valued coordinate vectors (e.g., a Minkowski space), is it always possible to describe the same space using coordinate vectors having one imaginary component and three real components, by appropriately changing the metric tensor?
The background of this question is that the use of the (ict,x1,x2,x3) coordinate vector has fallen out of favor in descriptions of relativistic 4-space, and has been replaced by (ct, x1,x2,x3) plus the Minkowski metric tensor whose values on the diagonal are -1,1,1,1 and zero off-diagonal. In a flat space the two approaches are exactly equivalent. However, the (ict,x1,x2,x3) approach is almost never used for curved spaces. Why? Is there a reason it can't be used for curved spaces? The only important thing that seems to distinguish between the two approaches is the number of free parameters in a 4 x 4 complex metric tensor: 10 in the first approach and and perhaps 16 in the second approach.