1. Near Horizon Geometries as Tangent Spacetimes
Sean Stotyn, University of Calgary
CCGRRA-16, Vancouver, BC
July

2. Outline
Coinciding horizon limit of Schwarzschild-de Sitter: finite 4-volume between horizons in extremal limit
Coordinate patches and Killing horizons in limit
Subtleties of spacetime limits (Geroch, 1969)
A new approach: mapping geometrical data from bulk to near horizon geometry (horizons, Killing vectors, etc.)
Consequence: near horizon geometries are tangent spacetimes, valid in an open coordinate neighbourhood
Further implications (extremal BH entropy, AdS/CFT)
Sean Stotyn, CCGRRA-16, July 2016

3. Ginsparg-Perry Limit
Start with non-extremal Schwarzschild-de Sitter black hole: Consider the black hole near extremality: Perform the following diffeomorphism:
Sean Stotyn, CCGRRA-16, July 2016

4. Ginsparg-Perry Limit
Grinding through the calculation, one ends up with a patch of dS2 × S2
Things to notice:
There are two non-degenerate horizons
The static patch between the original horizons remains static in the limit
Sean Stotyn, CCGRRA-16, July 2016

5. Ginsparg-Perry Limit
Each non-degenerate horizon in SdS maps to a non-degenerate horizon in Nariai.
The static patch in non-extremal SdS maps to static patch in Nariai.
Extremal SdS is the same as Nariai.
Standard story sounds airtight, right?
Sean Stotyn, CCGRRA-16, July 2016

6. Ginsparg-Perry Limit
Each non-degenerate horizon in SdS maps to a non-degenerate horizon in Nariai.
The static patch in non-extremal SdS maps to static patch in Nariai.
Extremal SdS is the same as Nariai.
Standard story sounds airtight, right?
Sean Stotyn, CCGRRA-16, July 2016

7. Limits of Spacetimes (Geroch)
The notion of “the” limit of a spacetime is ill-conceived. Take Schwarzschild as an example:
Sean Stotyn, CCGRRA-16, July 2016
Minkowski
Kasner

8. Limits of Spacetimes (Geroch)
The notion of “the” limit of a spacetime is ill-conceived. Certain properties of spacetimes are hereditary, while others are not. Hereditary Not Hereditary Rab=0 Topology (homology, homotopy) Cabcd=0 Existence of singularities Spinor structure existence Spinor structure non-existence Absence of CTCs Presence of CTCs
Sean Stotyn, CCGRRA-16, July 2016

9. Limits of Spacetimes (Geroch)
The notion of “the” limit of a spacetime is ill-conceived. Certain properties of spacetimes are hereditary, while others are not. Dimension of isometry group increases or remains the same.
Sean Stotyn, CCGRRA-16, July 2016

10. Limits of Spacetimes (Geroch)
The notion of “the” limit of a spacetime is ill-conceived. Certain properties of spacetimes are hereditary, while others are not. Dimension of isometry group increases or remains the same. Killing vectors need not have a smooth limit (this is key!) Under the diffeomorphism , the Killing vector which is singular in the limit where ε vanishes. ** The horizons under consideration are Killing horizons! **
Sean Stotyn, CCGRRA-16, July 2016

11. Ginsparg-Perry Limit problem
There is no meaningful way in which these horizons are identified because the KV generating the horizons in SdS does not map smoothly to the KV generating the horizons in Nariai.
This is a subtle point that has obscured what is really going on!
Sean Stotyn, CCGRRA-16, July 2016

12. Another Approach to NHGs
1. Take the canonical extremal limit:
2. Expand the metric around the degenerate horizon.
3. Keep lowest order terms only.
Sean Stotyn, CCGRRA-16, July 2016
This is dS2 x S2!

13. Another Approach to NHGs
Sean Stotyn, CCGRRA-16, July 2016
Notice: There is no static patch in extremal SdS.
Horizons are not bifurcate.
Regions sandwiched by dashed lines approximately static

14. Mapping geometrical objects
NHG is given by Horizon generators are tangent to spacelike KV and the degenerate horizon is located at . Extend the coordinate chart to the standard Nariai chart via
Sean Stotyn, CCGRRA-16, July 2016
Above coordinates only valid for small enough

15. Mapping geometrical objects
Sean Stotyn, CCGRRA-16, July 2016

16. Conclusions
Need to be very careful about interpretation of how geometrical objects transform when taking spacetime limits (dates back to Geroch in 1969)
The interpretation of 4-volume between degenerating horizons remaining finite in the extremal limit is called into question. Killing horizons not preserved.
Nariai is NOT the same as extremal SdS; it is the NHG of extremal SdS and has zero temperature wrt the “correct” Killing vector.
Sean Stotyn, CCGRRA-16, July 2016

17. Further Implications
Extremal black hole entropy: these results suggest entropy calculated via global properties of the NHG are measuring “something else.”
Extend analysis to degenerate BH horizons (work in progress): the “infinite throat” only maps to an open neighbourhood around a degenerate null hypersurface in the NHG.
AdS/CFT: any calculation relying on global properties of NHG is suspect when making a connection to the full spacetime.
Sean Stotyn, CCGRRA-16, July 2016