Hamiltonian vs stability in alternative theories of...

Introduction Causal cones & Hamiltonian Stable black hole in a Horndeski theory Conclusions Appendix: cgrav.

Hamiltonian vs stabilityin alternative theories of gravity

Gilles Esposito-Farese

GRεCO, Institut d’Astrophysique de Paris,98bis boulevard Arago, F-75014 Paris, France

[E. Babichev, C. Charmousis, GEF, A. Lehebel,Phys. Rev. Lett. 120 (2018) 241101,

Phys. Rev. D 98 (2018) 104050]

Rencontres de MoriondMarch 29th, 2019

Hamiltonian vs stability in alternative theories of gravity • La Thuile, March 29th, 2019 Gilles Esposito-Farese, IAP, CNRS


Introduction: (A⇒ B) does not mean (¬A⇒ ¬B)

• Known theorem

Hamiltonian density bounded by below⇒ lowest energy state is stable

• Converse statement often assumed

Hamiltonian density unbounded from below“⇒” unstable solution

Actually used several times in the literature:

[H. Ogawa, T. Kobayashi, T. Suyama, Phys. Rev. D 93 (2016) 064078][K. Takahashi, T. Suyama, T. Kobayashi, Phys. Rev. D 93 (2016) 064068][K. Takahashi, T. Suyama, Phys. Rev. D 95 (2017) 024034][R. Kase, M. Minamitsuji, S. Tsujikawa, Y.L. Zhang, JCAP 1802 (2018) 048]



Causal cones

In alternative theories of gravity, ∃ generically different causal conesfor perturbations around a given background

Simplest example: k-essence

L = −14 f (X), with X ≡ gµν∂µϕ∂νϕ

Let ϕ︸︷︷︸full field

= ϕ︸︷︷︸background

+ χ︸︷︷︸perturbation

t

x

mattercone

scalarcone

Second-order expansion of Lagrangian:

L2 = −12Gµν∂µχ∂νχ, with Gµν︸︷︷︸

effective metric

= f ′(X)gµν + 2 f ′′(X)∇µϕ∇νϕ



Causal cones (continued)

L2 = −12

Gµν︸︷︷︸effective metric

∂µχ∂νχ

t

x

t

xboost

mattercone

scalarcone

mattercone

scalarcone



Hamiltonian

L2 = −12

Gµν︸︷︷︸effective metric

∂µχ∂νχ

Conjugate momentum:

p ≡ ∂L2

∂χ= −G00χ− G0i∂iχ

t

x

mattercone

scalarcone

Hamiltonian density:

H2 = p χ− L2 = − 12G00

(p + G0i∂iχ

)2+

12Gij∂iχ∂jχ

Hamiltonian bounded by below

H2 ≥ 0 ⇔ G00 < 0 and Gij positiveCompare to

Hyperbolicity in the (t, x) subspace

Gµν =

(G00 G0x

G0x Gxx

)with determinant D ≡ G00Gxx −

(G0x)2< 0



Hamiltonian for a tilted scalar causal cone

Hyperbolicity in the (t, x) subspace

Gµν =

(G00 G0x

G0x Gxx

)with determinant D ≡ G00Gxx −

(G0x)2< 0

Let Gµν denote the inverse effective metric [6= gµλgνρGλρ!]

In the (t, x) subspace:(G00 G0xG0x Gxx

)=

(Gxx −G0x

−G0x G00

)/D

If time axis outside scalar cone:G00 dt dt > 0⇒ Gxx < 0

⇒ Hamiltonian density unbounded by below!

H2 = − 12G00

(p + G0i∂iχ

)2+

12Gij∂iχ∂jχ

t

x

mattercone

scalarcone



Hamiltonian is coordinate dependent

H2 ≥ 0 H2 unbounded by below

t

x

t

xboost

mattercone

scalarcone

mattercone

scalarcone

This is strictly the same theory, viewed by a moving observer.

⇒Must be stable, in spite of Hamiltonian unboundedness!



Superluminal case

boost

t

x

t

x

mattercone

mattercone

scalarcone

scalarcone

If x axis inside scalar cone: Gxx dx dx < 0⇒ G00 > 0

because(G00 G0xG0x Gxx

)=

(Gxx −G0x

−G0x G00

)/D

⇒ Hamiltonian density also unbounded by below

H2 = − 12G00

(p + G0i∂iχ

)2+

12Gij∂iχ∂jχ



Relative orientations of two causal conest

x

t

x

t

x

t

x

t

x

t

x

t

x

t

x

t

x

t

x

t

x

t

x

! ! ! !

✓

✓

✓

✓

✓

✓

✓

✓



Conserved quantities

L2 = −12G

µν∂µχ∂νχ is diffeomorphism-invariantbecause Gµν is a tensor

−Tνµ ≡δL2

δ(∂νχ)∂µχ− δνµ L2 is thus conserved:

∂νTνµ = 0 ⇔ ∂0T0µ + ∂iT i

µ = 0 [flat-spacetime case to simplify]

Let Pµ ≡ −∫∫∫

V T0µ d3x ⇒ ∂tPµ = 0

In particular, energy conservation ∂tP0 = 0where −T0

0 = on-shell value of the Hamiltonian density

Coordinate change P′λ = (∂xµ/∂x′λ)Pµ

Even if P0 ≥ 0, this is not always so for P′0 = (∂xµ/∂x′0)Pµ

[= (P0 + vPx)/√

1− v2 for a mere boost]

But all P′µ are conserved, and the linear combination

P0 = (∂x′µ/∂x0)P′µ is bounded by below ⇒ stable!



Relative orientations of two causal conest

x

t

x

t

x

t

x

t

x

t

x

t

x

t

x

t

x

t

x

t

x

t

x

! ! ! !

✓

✓

✓

✓

✓

✓

✓

✓



Conditions on effective metric Gµν for stability

Choose coordinates such that gµν =

(−1 00 1

)in the (t, x) subspace

Stability needs

Hyperbolicity:D ≡ G00Gxx −

(G0x)2

< 0

∃ consistent time and space coordinates:

G00 < Gxx and/or |G00 + Gxx| < 2|G0x|

⇒ off-diagonal component G0x should be large enough

whereas sign of Hamiltonian density does not depend on it:

H2 = − 12G00

(p + G0i∂iχ

)2+

12Gij∂iχ∂jχ



Covariant conditions for stability

Weak Energy Condition in general relativity∀ timelike Uµ, TµνUµUν ≥ 0

t

x

singlecausalcone

Uμ

But what does “timelike” meanwhen there are different causal cones?

Here, we need:All effective metrics (gµν , Gµν , . . . ) are of (−,+,+,+) signature∃ contravariant vector Uµ and covariant vector uµ such that

gµνUµUν < 0, GµνUµUν < 0(∃ common interior to all causal cones: dx0 in the direction of Uµ)

gµνuµuν < 0, Gµνuµuν < 0(∃ spatial hypersurface exterior to all cones: uµdxµ = 0)and TνµUµuν ≥ 0(positivity of Hamiltonian in a good coordinate system)



Covariant conditions for stability

Weak Energy Condition in general relativity∀ timelike Uµ, TµνUµUν ≥ 0

t

x

Uμorthogonalto u

μBut what does “timelike” meanwhen there are different causal cones?

Here, we need:All effective metrics (gµν , Gµν , . . . ) are of (−,+,+,+) signature∃ contravariant vector Uµ and covariant vector uµ such that

gµνUµUν < 0, GµνUµUν < 0(∃ common interior to all causal cones: dx0 in the direction of Uµ)

gµνuµuν < 0, Gµνuµuν < 0(∃ spatial hypersurface exterior to all cones: uµdxµ = 0)and TνµUµuν ≥ 0(positivity of Hamiltonian in a good coordinate system)



Example of k-essence

Simplest example: k-essence

L = −14 f (X), with X ≡ gµν∂µϕ∂νϕ

Well-posed Cauchy problem (hyperbolic equations) and stability

⇔ f ′(X) > 0 and 2X f ′′(X) + f ′(X) > 0

(But no condition on f ′′(X) alone:f ′′(X) < 0 infraluminal scalar,f ′′(X) > 0 superluminal scalar)

[Y. Aharonov, A. Komar, L. Susskind, Phys. Rev. 182 (1969) 1400][C. Armendariz-Picon, T. Damour, V. Mukhanov, Phys. Lett. B 458 (1999) 209][E. Babichev, V. Mukhanov, A. Vikman, JHEP 0609 (2006) 061][J.P. Bruneton, Phys. Rev. D 75 (2007) 085013][J.P. Bruneton & GEF, Phys. Rev. D 76 (2007) 124012][E. Babichev, V. Mukhanov, A. Vikman, JHEP 0802 (2008) 101]



A simple self-tuning model

“John” ∈ “Fab Four” model ⊂ Horndeski class

S =

∫ √−g d4x

ζ(R− 2Λbare)− η (∂µϕ)2 + β Gµν∂µϕ∂νϕ︸︷︷︸“John”

Exact Schwarzschild-de Sitter solution:

ds2 = −A(r) dt2 +dr2

A(r)+ r2 (dθ2 + sin2 θ dφ2)

A(r) = 1− 2Gmr− Λeff

3r2, with Λeff = − η

β

ϕ = q

[t −∫ √

1− A(r)

A(r)dr

], with q2 =

η + β Λbare

η βζ

[E. Babichev & C. Charmousis, JHEP 1408 (2014) 106](generalized in [E. Babichev & GEF, Phys. Rev. D 95 (2017) 024020])



Graviton causal cone

Diagonalize the spin-2 and spin-0 kinetic termsaround an arbitrary background (“Einstein frame”)?

Well-known for f (R) theories

S =

∫d4x√−g f (R)⇔

∫d4x√−g{

f (φ) + (R− φ) f ′(φ)}

Let g∗µν ≡ f ′(φ)gµν and ϕ ≡√

32 ln f ′(φ) ⇒ standard scalar-tensor theory

Quadratic + cubic Galileon

S =1

4πG

∫d4x√−g{

R4− 1

2(∂µϕ)2 − k3

2(∂µϕ)2 �ϕ

}Redefine h∗µν ≡ hµν + 4k3

[∂µϕ∂νϕ− 1

2 gµν (∂λϕ)2]χ

[E. Babichev & GEF, Phys. Rev. D 87 (2013) 044032]

For the “John” model, we did not find any covariant diagonalization!⇒ Study odd-parity perturbations⇒ effective metric for gravitons

[H. Ogawa, T. Kobayashi, T. Suyama, Phys. Rev. D 93 (2016) 064078]



Matter and graviton causal cones

r

t'

r = 1.6 ·10 - 2

r

t'

r = 2.8 ·10 - 2

r

t'

r = 8.1 ·10 - 2

r

t'

r = 1.3 ·10 - 1

r

t'

r = 3.0 ·10 - 1r

t'

r = 1.4

r

t'

r = 1.6

r

t'

r = 1.8

r

t'

r = 2.2

mattercone

gravitoncone

Close toblack-hole

horizon

Close tocosmological

horizon



Scalar causal cone

But this does not suffice to prove stability:The scalar causal cone should also be consistentwith the graviton causal cone

⇒ Study ` = 0 even-parity modes, to extract theeffective metric for the scalar degree of freedom

In the asymptotic de Sitter Universe, the solution is stable if

either η > 0, β < 0 andΛbare

3< − η

β︸︷︷︸Λeff

< Λbare

or η < 0, β > 0 and Λbare <

︷︸︸︷− ηβ< 3Λbare

What happens in the vicinity of the black hole?



Graviton, scalar and matter causal cones

r

t'

r = 1.6 ·10 - 2

r

t'

r = 2.8 ·10 - 2

r

t'

r = 8.1 ·10 - 2

r

t'

r = 1.3 ·10 - 1

r

t'

r = 3.0 ·10 - 1r

t'

r = 1.4

r

t'

r = 1.6

r

t'

r = 1.8

r

t'

r = 2.2

mattercone

gravitoncone

scalarconeClose to

black-holehorizon


horizon



Conclusions

A Hamiltonian density which is unbounded from belowdoes not always imply an instability!

3-momentum is also conserved, and can be combinedwith Hamiltonian to give a bounded-by-below quantity(= Hamiltonian in another coordinate system).

Simplest way of analyzing a generic case: Plot the causalcones for each degree of freedom. There should exist both acommon interior and a common exterior spacelike hypersurface.

∃ generalization of the Weak Energy Condition of generalrelativity to encompass the case of several causal cones.

An exact Schwarzschild-de Sitter solution of a Horndeskitheory is stable for a given range of its parameters,contrary to a claim in the literature.



Graviton speed

In the asymptotic de Sitter solution, cgravity 6= clightInconsistent with GW170817 event

[P. Creminelli & F. Vernizzi, Phys. Rev. Lett. 119 (2017) 251302][J.M. Ezquiaga & M. Zumalacarregui, Phys. Rev. Lett. 119 (2017) 251304][T. Baker et al., Phys. Rev. Lett. 119 (2017) 251301]

Couple matter to the disformal metric

gµν = gµν −β

ζ +β

2(∂λϕ)2

∂µϕ∂νϕ =

(1 +

Λbare

Λeff

)Ggravitonµν

i.e., write Smatter [matter fields; gµν ]

r

t'

r = 2.2


horizon



Graviton speed

In the asymptotic de Sitter solution, cgravity 6= clightInconsistent with GW170817 event

[P. Creminelli & F. Vernizzi, Phys. Rev. Lett. 119 (2017) 251302][J.M. Ezquiaga & M. Zumalacarregui, Phys. Rev. Lett. 119 (2017) 251304][T. Baker et al., Phys. Rev. Lett. 119 (2017) 251301]

Couple matter to the disformal metric

gµν = gµν −β

ζ +β

2(∂λϕ)2

∂µϕ∂νϕ =

(1 +

Λbare

Λeff

)Ggravitonµν

i.e., write Smatter [matter fields; gµν ]

r

t'

r = 2.2

matter &graviton

conesClose to

cosmologicalhorizon



cgravity = clight everywhere!

r

t'

r = 1.6 ·10 - 2

r

t'

r = 2.8 ·10 - 2

r

t'

r = 8.1 ·10 - 2

r

t'

r = 1.3 ·10 - 1

r

t'

r = 3.0 ·10 - 1r

t'

r = 1.4

r

t'

r = 1.6

r

t'

r = 1.8

r

t'

r = 2.2

matter &graviton

cones

matter &graviton

cones

Close toblack-hole

horizon


horizon



Graviton and scalar causal cones

r

t'

r = 1.6 ·10 - 2

r

t'

r = 2.8 ·10 - 2

r

t'

r = 8.1 ·10 - 2

r

t'

r = 1.3 ·10 - 1

r

t'

r = 3.0 ·10 - 1r

t'

r = 1.4

r

t'

r = 1.6

r

t'

r = 1.8

r

t'

r = 2.2

gravitoncone

scalarconeClose to

black-holehorizon


horizon


Hamiltonian vs stability in alternative theories of...

Documents

Transcript of Hamiltonian vs stability in alternative theories of...