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While reading article "Entanglement Entropy of Extremal BTZ" I saw a phrase:

In the more general case where the entangling interval does not lie in a single time slice of the boundary, the Ryu–Takayanagi formula is generalized to a covariant prescription [1].


In the article [1] they consider case of dynamical bulk and boundary theories. However they only formulate this new proposal for boundary(entangling) intervals belonging to constant time slice.
So my questions are:

1)I was thinking that one can only define entropy for a given Cauchy surface(given time slice).Is it correct? Yet it seems to be in contradiction with aforementioned quote.
2) Is it possible to generalize RT formula for space-like boundary regions which do not completely belong to single time slice? Any links to articles where it is done will be wellcomed.

EDITED
Thanks to Bianchira I was able to sharpen my question a bit.
Suppose we have $AdS_{3}/CFT_{2}$ In this case we may use well known Ryu-Takayanagi formula:

enter image description here
Now let us do some coordinate transformation in order to obtain different foliation of the same manifold:

$$\tilde{\tau} = \cos \alpha \cdot \tau - \sin \alpha \cdot \phi$$ $$\tilde{\theta} = \sin \alpha \cdot \tau + \cos \alpha \cdot \theta$$

enter image description here

So this is what we will get.
Now my question is whether we can still use Ryu-Takayanagi formula for new blue region?

If this is indeed the case then it follows that I can take any geodesic in the bulk(which starts and ends on the boundary) and obtain its interpretation from the boundary point of view by aforementioned algorithm. Only some of these geodesics (which belong to signle time slice) will have nice interpretation in terms of entanglement entropy though.
Are these statements in a way trivial and are already implied in the statement of RT proposal?

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    $\begingroup$ In my head the words Cauchy slice and time slice are synonymous - if I foliate my geometry into cauchy slices I would call each slice a time slice. So I don't understand your question here. $\endgroup$
    – zzz
    Commented Nov 1, 2016 at 4:49
  • $\begingroup$ May be I could be a bit more specific. As I understand it when we consider RT formula(let us consider $AdS_3/CFT_2$ case) we take boundary region which completely belongs to the same time slice. Yet is it possible to take boundary region which starts at $t_1$ and ends at $t_2$? How could one calculate entropy for such a region? Is it even meaningful to speak about such set up? Yet it seems to me that in above quote they are talking about such case. This is my contradiction. P.S. To my mind Cauchy slice and time slice are synonymous as well. $\endgroup$ Commented Nov 1, 2016 at 8:29
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    $\begingroup$ It's still a bit of a vague question so I'll make some comments. Re: your last point about taking ANY geodesic - you need to be careful there. What RT tells you that your geodesic would compute a boundary entanglement entroy if it happens to be a geodesic on a leaf in a spacelike foliation of the bulk. For example, if your geodesic is not everywhere spacelike, this will fail. $\endgroup$
    – zzz
    Commented Nov 12, 2016 at 2:25

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I will summarize here what I believe to be correct direction of thinking about this problem. I do not write another comment because it will be too large and I want to add couple of pictures.
Let us first consider RT formula for some boundary region A in empty $AdS_3$ in global coordinates: $$ds^2 =l^2_{AdS}(-\text{cosh}^2 \rho d\tau^{2} +d\rho^2 + \text{sinh}^2d \theta^2)$$

enter image description here

Now let us consider RT formula for another boundary region B now in Poincare coordinates(note that these coordinates cover only part of AdS spacetime) $$ds^2=l^2_{AdS} \frac{(dz^2 -dt^2 +dx^2)}{z^2}$$

enter image description here

Region B lies in constant time slice(t=const)in Poincare coordinates. According to RT formula it has interpretation in terms of corresponding spacelike(in Poincare coordinates) geodesic. However if we take this result and transform it into global coordinates we will see that original region B does not lie in single time slice ($\tau = const$). Yet it still has certain interpretation in terms of bulk geodesic.
This result seemed to me a bit fishy. However now I believe that this is indeed how it works and I just didn't fully understand formulation of RT proposal so far.

Note:

I do agree with Bianchira that boundary regions of interest should contain only those points which are spacelike separated. Otherwise RT proposal will not hold for them.

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