When are two spacetimes the same

[smallphi]

Suppose you have two spacetimes described by two metrics.

If I prove that they have foliations with the same geometrical properties, would that prove the spacetimes are actually the same only in different coordinates?

Has anyone seen a source with theorems of this type?

 

[MeJennifer]

Suppose you have two spacetimes described by two metrics.

If I prove that they have foliations with the same geometrical properties, would that prove the spacetimes are actually the same only in different coordinates?

Has anyone seen a source with theorems of this type?

The first thing to distinguish is a metric from a coordinate chart. Different coordinate charts can refer to the same metric.