I doubt that having a -1 in the metric is the defining characteristic of a time dimension, I could have a static universe without time and still a -1 in the met

I doubt that having a -1 in the metric is the defining characteristic of a time dimension, I could have a static universe without time and still a -1 in the metric. As I said, I am not sure what truly makes time time, but the sign in the metric seems not sufficient, not even sure if it is necessary. Are you sure that it is impossible to build something consistent on top of denying the law of identity?