Near Horizon Geometries as Tangent Spacetimes

Sean Stotyn, University of CalgaryCCGRRA-16, Vancouver, BC

July 7 2016

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)

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Ginsparg-Perry LimitStart with non-extremal Schwarzschild-de Sitter black hole:

Consider the black hole near extremality:

Perform the following diffeomorphism:

ds2 = − 1− 2Mr−Λ3r2

#

$%

&

'(dt2 +

dr2

1− 2Mr−Λ3r2

#

$%

&

'(+ r2 dθ 2 + sin2θdφ 2( )

r+ = r0 (1−ε)

rc = r0 (1+ε)

r = r0 +ερ t = τε

9M 2Λ =1−3ε 2 r0 =1Λ1− 16ε 2

#

$%

&

'(

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Ginsparg-Perry LimitGrinding through the calculation, one ends up with a patch of dS2 × S2

Things to notice:1. There are two non-degenerate horizons2. The static patch between the original horizons remains

static in the limit

ds2 = − 1− ρ2

r02

"

#$

%

&'dτ 2 +

dρ2

1− ρ2

r02

"

#$

%

&'

+ r02 dθ 2 + sin2θdφ 2( )

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Ginsparg-Perry Limit

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

ü The static patch in non-extremal SdS maps to static patch in Nariai.

ü Extremal SdS is the same as Nariai.

Standard story sounds airtight, right?

Sean

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Ginsparg-Perry Limit

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

ü The static patch in non-extremal SdS maps to static patch in Nariai.

ü Extremal SdS is the same as Nariai.

Standard story sounds airtight, right?

Sean

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Limits of Spacetimes (Geroch)The notion of “the” limit of a spacetime is ill-conceived. Take Schwarzschild as an example:

ds2 = − 1− 2ε3r

"

#$

%

&'dt2 +

dr2

1− 2ε3r

"

#$

%

&'+ r2 dθ 2 + sin2θdφ 2( )

ds2 = −dt2 + dx2 + dρ2 + ρ2dφ 2 ds2 = 2!rdτ 2 − !r

2d!r 2 + !r 2 dρ2 + ρ2dφ 2( )

r = x −ε−4 θ = ερ r = !r ε θ = ερt = ετ

Minkowski Kasner

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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 HereditaryRab=0 Topology (homology, homotopy)Cabcd=0 Existence of singularitiesSpinor structure existence Spinor structure non-existenceAbsence of CTCs Presence of CTCs

Sean

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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

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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! **

r = r0 +ερ t = τε

∂t →ε∂τ

ds2 = − 1− ρ2

r02

"

#$

%

&'dτ 2 −

dρ2

1− ρ2

r02

"

#$

%

&'

+ r02 dθ 2 + sin2θdφ 2( )

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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!

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Another Approach to NHGs1. Take the canonical extremal limit:

2. Expand the metric around the degenerate horizon.

3. Keep lowest order terms only.

ds2 =V (τ )dσ 2 −dτ 2

V (τ )+τ 2 dθ 2 + sin2θdφ 2( )

τ = r0 + !τ

9M 2Λ =1

V (τ ) = τ + 2r03τ

!

"#

$

%&τr0−1

!

"#

$

%&

2

ds2 =!τ +3r03(r0 + !τ )!

"#

$

%&!τr0

!

"#

$

%&

2

dσ 2 −d !τ 2

!τ +3r03(r0 + !τ )!

"#

$

%&!τr0

!

"#

$

%&

2 + (r0 + !τ )2 dθ 2 + sin2θdφ 2( )

ds2 ≈!τ 2

r02 dσ

2 −r02

!τ 2d !τ 2 + r0

2 dθ 2 + sin2θdφ 2( ) This is dS2 x S2!

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Another Approach to NHGsds2 =V (τ )dσ 2 −

dτ 2

V (τ )+τ 2 dθ 2 + sin2θdφ 2( ) V (τ ) = τ + 2r0

3τ!

"#

$

%&τr0−1

!

"#

$

%&

2

Notice: There is no static patch in extremal SdS. Horizons are not bifurcate.Regions sandwiched by dashed lines approximately static

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Mapping geometrical objectsNHG 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

ξ a = ∂σ

ds2 = − 1+ y2

r022

"

#$

%

&'dt2 +

dy2

1+ y2

r022

"

#$

%

&'

+ r022 dθ 2 + sin2θdφ 2( )

ds2 =!τ 2

r02 dσ

2 −r02

!τ 2d !τ 2 + r0

2 dθ 2 + sin2θdφ 2( )

!τ = 0

!τ = −r0e−t/r0 1− y

2

r02

σ =yet/r0

1− y2

r02

!τAbove coordinates only valid for small enough

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Mapping geometrical objects

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Conclusions• Need to be very careful about interpretation of how

geometrical objects transform when taking spacetimelimits (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.

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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

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