Wolfram Cloud Document

This post is the second foundation to “Testing the Speed of Gravity with Black Hole Ringdown”.
First foundation: https://community.wolfram.com/groups/-/m/t/2988459
ORIGINAL ARTICLE: Sergi Sirera Lahoz, Testing the Speed of Gravity with Black Hole Ringdown, arXiv: 2301.10272.
DOI: https://doi.org/10.48550/arXiv.2301.10272
GITHUB

In this notebook we derive the master equations governing the evolution of perturbations of a Schwarzschild black hole in Horndeski gravity.Compared to previous literature, we introduce a transformation on the metric perturbations at the level of the action which completely uncouples it from the scalar perturbations. After the demixing transformation, the metric sector looks exactly like GR. Therefore, we derive the following 3 master equations: 1. Odd gravitational sector - Regge-Wheeler equation 2. Even gravitational sector - Zerilli equation 3. Even scalar sector - Scalar equation of motion (massive Klein-Gordon equation)Note that the method to obtain the Regge-Wheeler and Zerilli equations is exactly the same as in GR, therefore the same as in the notebook BH-perturbations-GR (also in the repository).This notebook has hugely benefitted from resources provided by: - xAct ( http://www.xact.es/ ) - Johannes Noller - Reggie Bernardo’s public git-hub repository ( https://github.com/reggiebernardo/notebooks/tree/main/thesis_notebooks ). - Public notebooks by Vitor Cardoso et al. ( https://centra.tecnico.ulisboa.pt/network/grit/files/ringdown/ ).This notebook follows closely Tattersall et. al. ( https://arxiv.org/pdf/1804.08950.pdf , https://arxiv.org/pdf/1904.05112.pdf ). We provide here a more straightforward way to obtain the results.If you use any of the results or techniques in this notebook please cite the repository and [2301.10272] ( https://arxiv.org/abs/2301.10272 ).If you have any comments or feedback on this notebook, please let me know. Thanks!Sergi Sirera LahozInstitute of Cosmology and Gravitation - University of Portsmouth

0. Setup and definitions


1. Quadratic action

Action

We start with the Horndeski action, which has the following Lagrangians.
Note that we use geometric units:

G=c=1

.

In[]:=

L2=G2[phi[],X[]];L3=-G3[phi[],X[]]CD[b]@CD[-b]@phi[];L4=G4[phi[],X[]]RicciScalarCD[]+G4X[phi[],X[]](

2

(CD[b]@CD[-b]@phi[])

-CD[c]@CD[d]@phi[]CD[-c]@CD[-d]@phi[]);L5=G5[phi[],X[]]EinsteinCD[-b,-c]CD[b]@CD[c]@phi[]-

1

6

G5X[phi[],X[]](

3

(CD[b]@CD[-b]@phi[])

-3CD[e]@CD[c]@phi[]CD[-e]@CD[-c]@phi[]CD[d]@CD[-d]@phi[]+2CD[-f]@CD[-k]@phi[]CD[f]@CD[j]@phi[]CD[k]@CD[-j]@phi[]);LT=Sqrt[-Detg[]](L2+L3+L4+L5)

Out[]=

-g

G

2

[ϕ,X]+

G

4

[ϕ,X]R-

G

3

[ϕ,X]

b

∇

b

ϕ+

G


b	c

G

5

[ϕ,X]

b

∇

c

∇

ϕ+

G

4 X

[ϕ,X]

2



b

∇

b

ϕ

-

c

∇

d

∇

ϕ

∇

d

∇

c

ϕ-

1

6

G

5 X

[ϕ,X]

3



b

∇

b

ϕ

-3

d

∇

d

ϕ

∇

e

∇

c

ϕ

e

∇

c

∇

ϕ+2

f

∇

j

∇

ϕ

∇

k

∇

f

ϕ

k

∇

j

ϕ

We also define here the metric perturbations

In[]:=

DefMetricPerturbation[g,h,ϵ];Unprotect[IndexForm];(*thisselectsacolourtodifferentiateindicesreferringtotheorderinperturbationsfromotherindices*)IndexForm[LI[x_]]:=ColorString[ToString[x],RGBColor["#00a898"]];Protect[IndexForm];IndexForm[LI[n]];DefTensorPerturbation[δϕ[LI[order]],phi[],Md];

The following rules will take care of selecting contributions only from first order in the perturbations, applying differentiation by parts, specifying Ricci flat and non-hairy spacetimes.

In[]:=

(*keeponlyfirstorder*)onlyfirst={h[LI[2],x_,y_]:>0,δϕ[LI[2]]->0}(*integrationbypartsrules*)IBPrule1=XX_*h[LI[1],aa_,bb_]CD[cc_]@CD[dd_]@h[LI[1],ee_,ff_]->-CD[cc]@(XX*h[LI[1],aa,bb])*CD[dd]@h[LI[1],ee,ff]IBPrule2=XX_*CD[cc_]@CD[dd_]@YY_->-CD[cc]@XX*CD[dd]@YY(*Ricciidentity*)RicciID=XX_*CD[-cc_][h[LI[1],-bb_,-dd_]]CD[dd_][h[LI[1],bb_,cc_]]->Scalar[XX*h[LI[1],bb,cc]h[LI[1],dd,e]RiemannCD[-cc,-dd,-bb,-e]+XX*CD[dd][h[LI[1],-dd,-bb]]CD[-cc][h[LI[1],cc,bb]]-XX*h[LI[1],cc,bb]h[LI[1],-bb,e]RicciCD[-cc,-e]](*Ricciflatspacetimes*)Ricciflat={RicciCD[-a_,-b_]0,RicciScalarCD[]0,EinsteinCD[x_,y_]->0}(*nohair*)DefConstantSymbol[φ];GoBald=phi[]φ;(*scalarfieldkineticterm*)Xtoϕ=X[]-

1

2

CD[-b][phi[]]CD[b][phi[]]

Out[]=

{

h

2	x	y

0,

δϕ

2

0}

Out[]=

h

1	aa	bb

XX_

cc

∇

dd

∇

h

1	ee	ff

(-

h

1	aa	bb

cc

∇

XX-XX

cc

∇

h

1	aa	bb

)

dd

∇

h

1	ee	ff

Out[]=

XX_

cc

∇

dd

∇

YY_-

cc

∇

XX

dd

∇

YY

Out[]=

XX_

∇

cc

h

1
	bb	dd

dd

∇

h

1	bb	cc

-XX

h

1		e
	bb

h

1	cc	bb

R


cc	e

+(XX

h

1	bb	cc

h

1	dd	e

R


cc	dd	bb	e

)+XX

∇

cc

h

1	cc	bb

dd

∇

h

1
	dd	bb



Out[]=

{

R


a	b

0,R0,

G

x	y

0}

Out[]=

X-

1

2

∇

b

ϕ

b

∇

ϕ

Here we perturb the Lagrangian to second order and then we specify that our solution has no hair and is Ricci flat.

In[]:=

LT/.Xtoϕ;

1

2

Perturbation[%,2]//ExpandPerturbation//NoScalar//Expand;%/.GoBald;%//ToCanonical//Simplify//NoScalar;%/.onlyfirst//ContractMetric//Simplification//ToCanonical//NoScalar;%/.IBPrule1//ToCanonical//NoScalar;%//NoScalar//Simplification//NoScalar;δL=%/.Ricciflat//NoScalar

Out[]=

1

8

-g



G

2

[φ,0]-2

h

1
	b	c

h

1	b	c

+

h

1	b
		b

h

1	c
		c

-2

G

4

[φ,0]-

∇

c

h

1	d
		d

c

∇

h

1	b
		b

+2

c

∇

h

1	b
		b

∇

d

h

1		d
	c

-2

∇

c

h

1
	b	d

d

∇

h

1	b	c

+

∇

d

h

1
	b	c

d

∇

h

1	b	c

+42

G

4 X

[φ,0]

∇

b

∇

δϕ

1

∇

c

∇

δϕ

1

-

G

3

[φ,0]

∇

b

h

1	c
		c

b

∇

δϕ

1

-2

b

∇

δϕ

1

∇

c

h

1		c
	b

-2

h

1	b	c

∇

c

∇

b

δϕ

1

+

h

1	b
		b

∇

c

∇

δϕ

1

-

G

5

[φ,0]

∇

c

∇

b

h

1	d
		d

c

∇

b

∇

δϕ

1

-2

G

4 X

[φ,0]

∇

c

∇

b

δϕ

1

c

∇

b

∇

δϕ

1

+2

G

5

[φ,0]

c

∇

b

∇

δϕ

1

∇

d

∇

c

h

1		d
	b

-

G

5

[φ,0]

∇

b

∇

δϕ

1

∇

d

∇

c

h

1	c	d

-

G

5

[φ,0]

c

∇

b

∇

δϕ

1

∇

d

∇

h

1
	b	c

+

G

5

[φ,0]

∇

b

∇

δϕ

1

∇

d

∇

h

1	c
		c

-

∇

b

δϕ

1

b

∇

δϕ

1

(0,1)

G

2

[φ,0]+

h

1	b
		b

δϕ

1

(1,0)

G

2

[φ,0]-2

δϕ

1

∇

b

∇

δϕ

1

(1,0)

G

3

[φ,0]+2

δϕ

1

∇

c

∇

b

h

1	b	c

(1,0)

G

4

[φ,0]-2

δϕ

1

∇

c

∇

h

1	b
		b

(1,0)

G

4

[φ,0]+

2

δϕ

1

(2,0)

G

2

[φ,0]

We can now exploit the symmetries of our background to simplify the quadratic Lagrangian above in different ways. Doing that, we obtain the following simplified Lagrangian

In[]:=

δLsimp=δL/.{G2[φ,0]->0,Derivative[1,0][G2][φ,0]->0,G3[φ,0]->0,G4X[φ,0]->0,G5[φ,0]->0}//Simplify

Out[]=

1

4

-g



G

4

[φ,0]

∇

c

h

1	d
		d

c

∇

h

1	b
		b

-2

c

∇

h

1	b
		b

∇

d

h

1		d
	c

+2

∇

c

h

1
	b	d

-

∇

d

h

1
	b	c



d

∇

h

1	b	c

-2

∇

b

δϕ

1

b

∇

δϕ

1

(0,1)

G

2

[φ,0]+2

δϕ

1

-2

∇

b

∇

δϕ

1

(1,0)

G

3

[φ,0]+2

∇

c

∇

b

h

1	b	c

(1,0)

G

4

[φ,0]-2

∇

c

∇

h

1	b
		b

(1,0)

G

4

[φ,0]+

δϕ

1

(2,0)

G

2

[φ,0]

In the subsections below we show the arguments that allow these simplifications. Note that while we call this a simplified Lagrangian, this is still the full quadratic action for perturbations of a Schwarzschild black hole in Horndeski gravity.

G
2
=0=
G
2ϕ

To show this, we find the background equations of motion from the linear action (i.e. Euler-Lagrange equations)

In[]:=

Perturbed[LT/.Xtoϕ,1]//ExpandPerturbation//NoScalar//Expand;%/.GoBald/.Ricciflat;linaction=%//ToCanonical//Simplify//NoScalar

Out[]=

1

2

-g



G

2

[φ,0]2+ϵ

h

1	b
		b

-ϵ

g

b	c

2

G

3

[φ,0]

∇

c

∇

b

δϕ

1

+

G

4

[φ,0]

∇

c

∇

b

h

1	d
		d

-2

∇

d

∇

c

h

1		d
	b

+

∇

d

∇

h

1
	b	c

+2ϵ

δϕ

1

(1,0)

G

2

[φ,0]

In[]:=

VarD[δϕ[LI[1]],CD][linaction]==0;%/.{VarD[δϕ[LI[1]],CD][ϵ,AA_]->0,CD[aa_][ϵ]->0,delta[-LI[1],LI[1]]->1}

This one is more difficult to prove, as it involves the use of Bianchi identities in addition to multiple integrations by parts. The use of Bianchi identities to simplify expressions in xAct is, as far as we know, not yet well implemented. We therefore provide the check by hand as we have not yet dedicated the time to write this nicely in xAct. Let me know if you have any suggestions.
Below we show the relevant term and simplify it a bit

From this point we continue the calculation by hand as shown below

Demixing the action

Finally, we apply a transformation to the metric perturbation such that it decouples from the scalar perturbation

Looking at the quadratic action above we highlight that metric perturbations h and scalar perturbations δϕ are completely uncoupled. In addition, the metric sector looks exactly like GR. Hence, from that part of the action we obtain the original Regge-Wheeler and Zerilli equations for the odd and even sectors respectively. The notebook BH-perturbations-GR does exactly this calculation but we include it here as well for completeness. For the scalar sector (which belongs to the even polarisations) we will show that the action above results in a massive Klein-Gordon equation where the effective mass μ is given by a combination of the Horndeski G-functions on the background.

2. Quadratic action in components

Then we input the component values in the previous covariant expression. This is the full quadratic Lagrangian in terms of the perturbation functions.

3. Odd sector

We now use the bookkeeping constants ζ and η to select the odd terms.

Combination

To obtain the Regge-Wheeler equation, we rewrite h1EOM using all these rules.

4. Even sector

We now move to the even sector, which is considerably more involved, as can be seen from the quadratic action:

K variation

Combination

Lets start by displaying the equations of motion.

Step 1

Step 2

Finally, to obtain the Zerilli equation we take the algebraic equation and rewrite it using the rules above.

Scalar equation

To find the scalar equation of motion, we can vary directly the quadratic action with respect to δϕ:

Hence, we recover the expected equation for the scalar.

5. Results

Here we summarize the results of this notebook

CITE THIS NOTEBOOK

Testing Speed of Gravity with Black Hole Ringdown: equations governing the perturbations evolution
by Sergi Sirera Lahoz
Wolfram Community, STAFF PICKS, December 20, 2023
https://community.wolfram.com/groups/-/m/t/3088057

0. Setup and definitions

1. Quadratic action

Action

G2=0=G2ϕ

Demixing the action

2. Quadratic action in components

3. Odd sector

Combination

4. Even sector

K variation

Combination

Step 1

Step 2

Scalar equation

5. Results

CITE THIS NOTEBOOK

0. Setup and definitions


G
2
=0=
G
2ϕ