Causality Hyperbolicity Shock formation Lovelock Theories

with H. S. Reall, B. Way [DAMTP] arXiv: 1406.3379, 1409.3874

S. Ohashi [KEK] arXiv: 16xx.xxxxx

&

in

Norihiro Tanahashi [Osaka U]

CV

Apr 26, 2016 Seminar @ Particle Physics Theory Group, Osaka U 2

– 2010: PhD, Kyoto U with T. Tanaka

2010 – 2013: UC Davis with N. Kaloper

2013 – 2014: IPMU with S. Mukohyama

2014 – 2016: DAMTP with H. S. Reall

2016 – : Osaka U with K. Hashimoto

General relativity and its applications

Gauge/gravity correspondence

Modified gravity and cosmology

Apr 26, 2016 Seminar @ Particle Physics Theory Group, Osaka U 3

– 2010: PhD, Kyoto U with T. Tanaka

2010 – 2013: UC Davis with N. Kaloper

2013 – 2014: IPMU with S. Mukohyama

2014 – 2016: DAMTP with H. S. Reall

2016 – : Osaka U with K. Hashimoto

General relativity and its applications

Gauge/gravity correspondence

Modified gravity and cosmology

CV

Causality, Hyperbolicity & Shock formation in Lovelock Theories

• Lovelock Theories

= GR + (higher-curvature corrections)

EoM up to 2nd derivatives Avoids ghost instability

From string theory?

• GR: Gravity propagate at c

• Lovelock: Faster/slower propagation than c

Causality in Lovelock theories?

Does EoM remain hyperbolic?

Shock formation of gravitational wave?

• Causality in Lovelock theories? – Can we define causality in this theory?

– Can GW escape from BH?

• Does EoM remain hyperbolic? – Hyperbolic EoM = Wave equation

– Determined by principal part of EoM

GR: Guaranteed to be hyperbolic

Lovelock: ?

• Shock formation due to variable sound speed of GW?

Causality, Hyperbolicity & Shock formation in Lovelock Theories

Contents

1. Introduction – Lovelock theories

– Characteristics

2. Questions – Can GW escape from black hole interior?

– GW propagation on plane wave solutions

– GW propagation around black holes

– Shock formation?

3. Summary

Apr 26, 2016 Seminar @ Particle Physics Theory Group, Osaka U 6

Contents

1. Introduction – Lovelock theories

– Characteristics

2. Questions – Can GW escape from black hole interior?

– GW propagation on plane wave solutions

– GW propagation around black holes

– Shock formation?

3. Summary

Apr 26, 2016 Seminar @ Particle Physics Theory Group, Osaka U 7

Introduction: Lovelock theories

• Lovelock theories in d dimensions ( p ≤ (d−1)/2 )

• EoM = Einstein eq. + correction

where

8

Seminar @ Particle Physics Theory Group,

Introduction: Characteristics

• Wave propagation in flat space

Seminar @ Particle Physics Theory Group,

t

t = 0

ψ

x

Introduction: Characteristics

Seminar @ Particle Physics Theory Group,

t

t = 0

• t =const. surface : spacelike • normal vector (dt)μ : timelike

Introduction: Characteristics

Seminar @ Particle Physics Theory Group,

t

t = 0

• t’=const. surface : null • normal vector (dt’)μ : null

Introduction: Characteristics

EoM of ψ :

: is uniquely determined

usual time evolution of ψ

: is not fixed by EoM

can be non-unique

•

•

Seminar @ Particle Physics Theory Group,

Introduction: Characteristics

13

t’ = const. surface = wave front

“characteristic surface”

Apr 26, 2016

: can be non-unique •

t

t = 0

: is uniquely determined •

Introduction: Characteristics

14 Apr 26, 2016

t

t = 0

• Characteristics in GR

Characteristic det P = 0

Introduction: Characteristics

15 Apr 26, 2016

t

t = 0

• Characteristics in Lovelock theories

Characteristic det P = 0 ??

[Aragone ‘87] [Choquet-Bruhat’88]

Gravitational wave in Lovelock theories

16

1. Can GW escape from black hole interior?

2. GW propagation on plane wave solutions

3. GW propagation around black holes

Apr 26, 2016 Seminar @ Particle Physics Theory Group, Osaka U

• Does it obey causality? • Is hyperbolicity maintained?

1. Can GW escape from black hole interior?

GR: Null surface is characteristic BH horizon is a characteristic surface GW cannot escape from BH

GR + Gauss-Bonnet correction: BH horizon shown to be characteristic

Lovelock: ?

[Izumi ‘14]

Does GW propagate across the horizon?

Is the BH horizon characteristic for any mode?

GR: Null surface is characteristic BH horizon is a characteristic surface GW cannot escape from BH

GR + Gauss-Bonnet correction: BH horizon shown to be characteristic

Lovelock: Solve characteristic eq. det P = 0 on BH BH horizon is always characteristic GW cannot escape from BH

[Izumi ‘14]

1. Can GW escape from black hole interior?

Does GW propagate across the horizon?

Is the BH horizon characteristic for any mode?

2. GW Propagation on plane wave solutions

Rlilj aij

Other components = 0

19 Apr 26, 2016

Seminar @ Particle Physics Theory Group,

: null w.r.t. GI

Characteristic cones tangent to

Nested characteristic cones

Causality w.r.t. the largest cone 20

Proposition: Characteristic surfaces are null w.r.t. “effective metrics”:

2. GW Propagation on plane wave solutions

• Spherically symmetric black holes

• Proposition: Characteristic surfaces are null w.r.t. “effective metrics”:

A : Tensor, Vector, Scalar modes

cA(r): (Propagation speed)2 in Σ directions 21

3. GW Propagation around BH

BH

3. GW Propagation around BH

metric:

Effective metric:

Compare “gravity cone” with light cone

22

r

@ r = 4rh @ r = 3rh @ r = 1.5rh

: Photon cone : Tensor : Vector : Scalar

(d=7, k2=−1/4, rh=1)

r

3. GW Propagation around BH

r

for d=7, k2=−1/4, rh=1

: Photon cone : Tensor : Vector : Scalar

r

d=5

cscalar(r)<0 near horizon

d=6

ctensor(r)<0 near horizon

: Photon cone : Tensor : Vector : Scalar

c < 0 near horizon of small BH?

BH

3. GW Propagation around BH

metric:

Effective metric:

• Light cone and Gravity cone

– coincide in r direction – deviate in Ω direction

• c(r) > 1 superluminal GW

• c(r) < 0 near small BH

Hyperbolicity violation?

• Small BH c(r) < 0 near horizon

violation of hyperbolicity

Angular mode number l

ω2 − l2 unstable mode exp( l t )

There is perturbation that is zero at t = 0 but blows up at t > 0

with •

•

Solution is not continuous w.r.t. initial perturbation

Solutions do not exist generically

(Initial value problem not well-posed) 27

Answers

28

1. Can GW escape from black hole interior?

No: BH horizon is characteristic surface

2. GW propagation on plane wave solutions

Characteristics = Null w.r.t. effective metrics

Causality w.r.t. the largest cone

3. GW propagation around black holes

Characteristics = Null w.r.t. effective metrics

Hyperbolicity violation near small BH horizons

Apr 26, 2016 Seminar @ Particle Physics Theory Group, Osaka U

Contents

1. Introduction – Lovelock theories

– Characteristics

2. Questions – Can GW escape from black hole interior?

– GW propagation on plane wave solutions

– GW propagation around black holes

– Shock formation?

3. Summary

Apr 26, 2016 Seminar @ Particle Physics Theory Group, Osaka U 29

Shock formation

• Sound speed ≠ const.

• Waveform distortion Shock formation?

ex. 1) Burgers’ equation:

x

u(t,x)

ex. 2) Relativistic perfect fluid Shock formation from smooth data [Christodoulou ‘07]

Shock formation • Propagation of discontinuity in ∂2gab

GR: Amplitude obeys linear eq.

Lovelock: Amplitude obeys nonlinear eq.

Amplitude blow up?

31

• EoM in Lovelock

( rcd: eigenvector s.t. P ∙ r = 0 )

(discontinuous part)

• Transport equation of amplitude Π(xi)

• Take discontinuous part

Apr 26, 2016 33

• GR: N = 0 Π(s) finite unless e‒Φ diverges

• Lovelock: N ≠ 0 Π(s) diverges at finite s for large |Π(0)|

Shock formation

Seminar @ Particle Physics Theory Group, Osaka U

Shock formation

• Ricci-flat type N spacetimes

– Along : N = 0 No shock

– Along other directions: N ≠ 0 Shock formation

Apr 26, 2016 34

Shock formation

Does a shock formation imply pathology?

• Violation of weak cosmic censorship?

Shock = Naked singularity

• Stability of Minkowski in Lovelock theories?

Flat background N = 0, stable?

• Shock = Weak solution in Lovelock theory

Shock formation from smooth initial data?

Apr 26, 2016 Seminar @ Particle Physics Theory Group, Osaka U 35

Summary

• Can GW escape from black hole? No

• GW propagation on plane wave & BH backgrounds

Characteristics = Null w.r.t. effective metrics

Causality w.r.t. the largest cone

Hyperbolicity violation near small BH horizons

?: Shock formation in more general theories?

ex.) Lovelock theory in higher dimensions

~ Scalar-tensor theories in 4 dimensions.

Do they suffer from shock formation etc.?

Characteristics in Lovelock theories

Shock formation in Lovelock theories nonlinear term, shock forms for large initial data

[Ohashi & NT, in prep]