Is Schwarzschild 'spacetime' time-independent?

[jk22]

I'm a bit confused about GR : what is more significant about the considered spacetime, the metric, which is time-independent, or the embedding (there are already some posts on PF about it), which describes the shape of a manifold, but is time-dependent ?

 

[PeterDonis]

what is more significant

That depends on what you consider "significant".

the metric, which is time-independent

More precisely, the spacetime geometry admits a family of timelike worldlines (which can represent a family of observers) along which the spacetime geometry does not change (and therefore the spacetime geometry "looks the same" at all times to that family of observers).

the embedding (there are already some posts on PF about it), which describes the shape of a manifold, but is time-dependent ?

What embedding are you referring to? Links to the PF posts you mention would help.

 

[jk22]

On PF :

https://www.physicsforums.com/threads/schwarzschild-metric-as-induced-metric.412388/

On arXiv :

https://arxiv.org/abs/1202.1204

 

[PeterDonis]

On PF :

https://www.physicsforums.com/threads/schwarzschild-metric-as-induced-metric.412388/

On arXiv :

https://arxiv.org/abs/1202.1204

These are talking about embeddings of Schwarzschild spacetime, as a 4-dimensional manifold, in higher dimensional spaces. That has nothing to do with GR.