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As I understood spacetime is just spacetime inside and out of a black hole, except it is extremely curved inside. But the structure remains the same.

I have read this question:

What's the proper distance from the event horizon to the singularity?

Inside the horizon, we can't have a ruler at rest. The spacetime inside the horizon is not static.

In GR, is 'Static' the same as 'Time-symmetric'?

A stationary spacetime is one that has a timelike Killing vector. There is also a notion of an asymptotically stationary spacetime, which is what some authors mean by "stationary."Although a stationary spacetime does not have a uniquely pre- ferred time, it does prefer some time coordinates over others. In a stationary spacetime, it is always possible to find a “nice” t such that the metric can be expressed without any t-dependence in its components. A static spacetime is one that is not only stationary but also has the property that coordinates exist in which it is diagonal. (Coordinates will also exist in which it is not diagonal.)

I need some clarification as to what static means here. Does this mean that the dimensions change, is it the same as GWs stretching and squeezing spacetime itself?

Is the non-static spacetime one where the tidal forces change the distances between events?

What do the off-diagonal elements of the metric tensor represent?

Non-diagonal elements of the Schwarzchild metric

As I understand based on the comments, when spacetime is not static, this means that the metric is not diagonal. When the metric is not diagonal, does that mean that it is like when spacetime is changing, like stretching/squeezing always, like if GWs would pass by always?

For example, the Kerr metric is not diagonal, because of the rotation, that is a physical phenomenon. I am asking for other physical phenomena inside the BH, that are described by the not diagonal metric, like stretching and squeezing.

Question:

  1. What do we mean when we say that spacetime is not static inside a black hole?
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    $\begingroup$ "A static spacetime is one that is not only stationary but also has the property that coordinates exist in which it [the metric] is diagonal." is a perfectly fine definition. Can you explain what's unclear to you about it? $\endgroup$
    – ACuriousMind
    Jan 28 '20 at 21:53
  • $\begingroup$ I'm voting to close this question as off-topic because, as @Acuriousmind has observed, the question already contains its own answer. $\endgroup$
    – WillO
    Jan 29 '20 at 0:31
  • $\begingroup$ @ACuriousMind "In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero" en.wikipedia.org/wiki/Metric_tensor_(general_relativity) I do understand that the metric is not diagonal inside the black hole. I will edit because I would like clarification on what spacetime is like when the metric is not diagonal. $\endgroup$ Jan 29 '20 at 1:10
  • $\begingroup$ @safesphere can you please elaborate on what asymptotically static is? $\endgroup$ Jan 29 '20 at 1:15
  • $\begingroup$ 1. Please do not leave comments asking for reasons of downvotes - if the voters would have wanted to leave feedback on your post, they would already have, and the comment does not notify the voters anyway. 2. It is still not clear to me what you want to know. GR is a mathematical theory and it is not evident why would would expect the mathematical definition of "static" to be expressible in any other than those mathematical terms. $\endgroup$
    – ACuriousMind
    Jan 29 '20 at 1:18
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Static means that there is a family of observers, covering spacetime (which means that you can find one such observer at each point of space and time), such that spacetime doesn't change from their perspective.* In other words, as far as these observers are concerned, spacetime today is physically indistinguishable from spacetime yesterday.

Observers with these properties only exist outside a black hole. Inside, every observer sees change, because in particular every observer eventually hits the singularity. Spacetime looks like a weird expanding/contracting universe, not like the inside of a static sphere.


* Actually, there is one more technical requirement to distinguish static from stationary, but that doesn't matter here.

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  • $\begingroup$ thank you do much. Is it the expansion/contraction that is what is this not static spacetime? So basically it is like if GWs would pass by always? $\endgroup$ Jan 29 '20 at 1:12
  • $\begingroup$ @ÁrpádSzendrei Not exactly like a gravitational wave. I don't really know how to explain it in words (not even to myself), but the point is that spacetime and the gravitational field change for all observers, and everyone eventually reaches the singularity. $\endgroup$
    – Javier
    Jan 29 '20 at 1:40

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