When are two spacetimes the same

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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?
 
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smallphi said:
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.
 

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