Graduate Merging timelike and spacelike surfaces

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The discussion revolves around a proposed method for determining boundary length by smoothly merging timelike and spacelike surfaces that are homologous to the boundary and possess a well-defined first derivative at the merging point. A key question raised is whether this merging procedure can be applied in the context of thermal AdS, particularly when the boundary is located at infinity. Participants are exploring the implications of this approach for understanding the geometry of spacetime. The feasibility of this method in thermal AdS could provide insights into the behavior of boundaries in such geometrical frameworks. Overall, the conversation highlights the intersection of mathematical theory and physical applications in the study of spacetime surfaces.
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How to merge spacelike and timelike surfaces in the case of thermal AdS?
In the following paper, there is a statement which says:

We propose to determine the boundary length by merging smoothly the timelike and spacelike surfaces in such a way that they are homologous to the boundary and have a well-defined first derivative at the merging point.

Can this procedure be done in the case of thermal AdS, where the boundary is at infinity?


[1]: https://arxiv.org/abs/2404.01393
 

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