Near Horizon Geometries as Tangent Spacetimes Sean Stotyn

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

Near Horizon Geometries as Tangent Spacetimes Sean Stotyn, University of Calgary CCGRRA-16, Vancouver, BC July 7 2016

• Coinciding horizon limit of Schwarzschild-de Sitter: finite 4 -volume between horizons in

• 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, Ad. S/CFT) Sean Stotyn, CCGRRA 16, July 2016 Outline

Ginsparg-Perry Limit Consider the black hole near extremality: Perform the following diffeomorphism: Sean Stotyn,

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

Ginsparg-Perry Limit Things to notice: 1. There are two non-degenerate horizons 2. The static

Ginsparg-Perry Limit Things to notice: 1. There are two non-degenerate horizons 2. The static patch between the original horizons remains static in the limit Sean Stotyn, CCGRRA 16, July 2016 Grinding through the calculation, one ends up with a patch of d. S 2 × S 2

ü Each non-degenerate horizon in Sd. S maps to a non-degenerate horizon in Nariai.

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

ü Each non-degenerate horizon in Sd. S maps to a non-degenerate horizon in Nariai.

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

Limits of Spacetimes (Geroch) Sean Stotyn, CCGRRA 16, July 2016 The notion of “the”

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

The notion of “the” limit of a spacetime is ill-conceived. Certain properties of spacetimes

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 Limits of Spacetimes (Geroch)

The notion of “the” limit of a spacetime is ill-conceived. Certain properties of spacetimes

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 Limits of Spacetimes (Geroch)

The notion of “the” limit of a spacetime is ill-conceived. Certain properties of spacetimes

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 Limits of Spacetimes (Geroch)

There is no meaningful way in which these horizons are identified because the KV

There is no meaningful way in which these horizons are identified because the KV generating the horizons in Sd. S 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 Ginsparg-Perry Limit problem

Another Approach to NHGs 2. Expand the metric around the degenerate horizon. 3. Keep

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

Sean Stotyn, CCGRRA 16, July 2016 Another Approach to NHGs Notice: There is no

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

Mapping geometrical objects Horizon generators are tangent to spacelike KV and the degenerate horizon

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

Sean Stotyn, CCGRRA 16, July 2016 Mapping geometrical objects

Sean Stotyn, CCGRRA 16, July 2016 Mapping geometrical objects

• Need to be very careful about interpretation of how geometrical objects transform

• 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 Sd. S; it is the NHG of extremal Sd. S and has zero temperature wrt the “correct” Killing vector. Sean Stotyn, CCGRRA 16, July 2016 Conclusions

• Extremal black hole entropy: these results suggest entropy calculated via global properties

• 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. • Ad. S/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 Further Implications

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