Near-horizon Expansion of Second-order BH Perturbations

Kei Yamada (山田 慧生) Kyoto U. with A. Pound & T. Tanaka

20th Capra meeting @UNC, Chapel Hill

• The vacuum field equations to the second order are where & are linear & quadratic in .

• Solutions in FD should diverge near the horizon of BH.

• Once we identify the secularly growing piece , solutions to the following equations do NOT diverge:

Summary of today’s talk

�Gµ⌫ [h] �2Gµ⌫ [h, h] h

�Gµ⌫ ["h

(1) + "2h(2)] = ��2Gµ⌫ ["h

(1), "h(1)],

�Gµ⌫ ["h

(1) + "2h(2)] = ��2Gµ⌫ ["h

(1)]� �Gµ⌫ ["h

(1)sec].

h(1)sec

Outlines

• Introduction

• Near-horizon expansion

• Static/secularly growing pieces

• Summary

Outlines

• Introduction

• Near-horizon expansion

• Static/secularly growing pieces

• Summary

Extreme Mass Ratio Inspirals

• Expand equations in the mass ratio:

• Consider the two time-scale expansion.

• Orbital “fast time”:

• Inspiral “slow time”:

" ⌘ µ/M ⌧ 1,

v

v

• We expand equations in the mass ratio:

• The field equations to the second-order are where & are linear & quadratic in .

IR divergence around boundaries

gµ⌫ = gBGµ⌫ + "h(1)

µ⌫ + "2h(2)µ⌫ + · · · .

h

�Gµ⌫ ["h

(1) + "2h(2)] = ��2Gµ⌫ ["h

(1), "h(1)]

• The field equations to the second-order are

�Gµ⌫ ["h

(1) + "2h(2)] = ��2Gµ⌫ ["h

(1), "h(1)],

�Gµ⌫ [h] �2Gµ

⌫ [h, h]

•

• Schematically, the following integral diverges around boundaries.

✓ @infinity use the PN/PM results (cf. [Pound 2015]).

• We discuss the near-horizon expansion.

IR divergence around boundaries

h(2)!`m =

Z 1

rh

G!`m(r, r0)�2G(r0)dr0,

�Gµ⌫ ["h

(1) + "2h(2)] = ��2Gµ⌫ ["h

(1), "h(1)]

• The field equations to the second-order are

�Gµ⌫ ["h

(1) + "2h(2)] = ��2Gµ⌫ ["h

(1), "h(1)].

• The origins of secular growth are the “stationary” parts of .

• Almost all of the “stationary” parts should be balanced with “stationary” solutions .

• Let us decompose as

• Focus on the “stationary” piece, because does not cause the divergence.

Decomposition of

Oscillatory piece “Stationary” pieceh(i)µ⌫ = h(i)

µ⌫v�dep(v, v, r, ✓,�) + h(i)

µ⌫sta(v, r, ✓).

h(i)sta

h(i)

h(i)

h(i)µ⌫

v�dep

��2Gµ⌫

Outlines

• Introduction

• Near-horizon expansion

• Static/secularly growing pieces

• Summary

The Eddington-Finkelstein coordinates

• The Schwarzschild background metric is in the ingoing Eddington-Finkelstein coordinates, where .

✓Regularity of metric perturbations on the horizon.

✓On the horizon : does not appear in the Lorenz gauge.

grr = 0d2/dr2

ds2 = �fdv2 + 2dvdr + r2d✓2 + r2 sin2 ✓d�2

f = 1� 2M/r

Near-horizon expansion

• The metric perturbations are

• Expand the perturbations near the horizon where is a slow-time variable.

✓ in is not necessary because of the absence of .

gµ⌫ = gBGµ⌫ + "h(1)

µ⌫ + "2h(2)µ⌫ + · · · .

d2/dr2

v ⌘ "v

h(i)staµ⌫ (v, r, ✓) = h(i)sta

0µ⌫ (v, ✓) + fh(i)sta1µ⌫ (v, ✓) + O(f2),

O(f2) h(i)staµ⌫ (v, r, ✓)

The Lorenz gauge conditions

• The Lorenz gauge conditions are formally in the near-horizon limit , where .

✓Under the Lorenz gauge conditions, we can eliminate for 4 components of .

f ! 0 I = ✓,�

h(i)µ⌫

;⌫ = 0

h(i)1 �Gµ

⌫

h(i)1vv = F (i)

v [h0], h(i)1I

I = F (i)r [h0],

h(i)1v✓ = F (i)

✓ [h0], h(i)1v� = F (i)

� [h0],

First order in

• At the first order in , 4 components of NOT containing are

• Although these do not vanish in general,

• Non-trivial solutions for the zero modes.

�Gµ⌫ h(1)

1

"

"

sin ✓�G(1)sta rv = �@✓

sin ✓

8M2

⇣h(1)sta0v✓ + @✓h

(1)sta0vv

⌘�,

...

Zsin ✓ �G(1)sta r

v d✓ d� =

Zsin ✓ �G(1)sta r

� d✓ d� = 0.

First order in

• The first-order field equations do NOT determine the zero modes of .

• The second order does, because appears in the second order.

• The solutions are secularly growing.

"

Zsin ✓ �G(1)sta r

v d✓ d� =

Zsin ✓ �G(1)sta r

� d✓ d� = 0.

h(1)sta

@vh(1)sta

Energy & angular momentum fluxes

• Ingoing GW’s & across the horizon are where .

• These components of the second-order source terms for zero modes of correspond to & of first-order ingoing GWs.

E L

E L

h(1)sta

E ⌘ 1

8⇡

Z p�g

���2Gr

v

�d✓ d�,

L ⌘ 1

8⇡

Z p�g

���2Gr

�

�d✓ d�,

˙⌘ @/@v

Outlines

• Introduction

• Near-horizon expansion

• Static/secularly growing pieces

• Summary

Second-order Einstein tensor

• At the second order in , 4 components of NOT containing are

�Gµ⌫

"

h(2)1

sin ✓ �G(2)sta rv =

sin ✓

8M@v

⇣4h(1)sta

0vv + h(1)sta0✓✓ + h(1)sta

0��

⌘

� @✓

sin ✓

8M2

⇣h(2)sta0v✓ + @✓h

(2)sta0vv � 4M@vh

(1)sta0v✓

⌘�,

sin ✓ �G(2)sta r� =

sin2 ✓

2@v

⇣4h(1)sta

0v� + h(1)sta0r�

⌘

+ @✓

� sin3 ✓

4M@✓

✓1

sin ✓h(2)sta0v�

◆+ sin2 ✓ @vh

(1)sta0✓�

�,

...

Abbott & Deser’s quantities

• We findwhere

1

8⇡

Z p�g sin ✓ �G(2)sta r

v d✓ d�

����r=rh

= �@v�MAD

���r=rh

,

1

8⇡

Z p�g sin ✓ �G(2)sta r

� d✓ d�

����r=rh

= �@v�LAD

���r=rh

,

MAD =1

2

ZF (v)↵� d⌃↵� , LAD =

1

2

ZF (�)↵� d⌃↵� ,

F (v/�)µ⌫ ⌘ � 1

8⇡

h⇠(v/�)↵h(1)sta

↵[µ;⌫] + ⇠(v/�)↵;[µh(1)sta⌫]↵ + ⇠(v/�)µ h(1)sta

⌫]↵;↵i.

Physical secular growth

• & components of determine the secular growth of zero modes.

• The secular growth => the secular change of the BH’s mass/spin & .

Z p�g �Gµ

⌫ d✓ d� =

Z p�g

���2Gµ

⌫

�d✓ d�,

{r�}{rv}

�M = M v �a = av

Identification of the secular pieces

• Calculating deviations due to , , we obtain which reproduces the zero modes.

• Therefore, the effective source term, can be integrated without any divergences.

�M �a

�Gµ⌫ [h(1)sec] =

0

BBBBBBB@

2

r2@v �M(v) 0 0 � (M + r) sin2 ✓

r2@v �a(v)

0 0 0sin2 ✓

r@v �a(v)

0 0 0 0

� (M + r) sin2 ✓

r2@v �a(v)

sin2 ✓

r@v �a(v) 0 0

1

CCCCCCCA

,

��2Gµ⌫ ["h

(1), "h(1)]� �Gµ⌫ ["h

(1)sec],

Outlines

• Introduction

• Near-horizon expansion

• Static/secularly growing pieces

• Summary

Summary

• Need the second-order metric perturbations for EMRI observations by LISA.

• IR divergences appear around boundaries.

• By the near-horizon expansion, we found

• the 2nd-order eqs. determine the zero modes of .

• the secular growth => & due to & .

• Extension to the Kerr case is straightforward.

• How to tame secular growth of pure gauge modes…?

E L

h(1)sta

@vMAD @vL

AD

THANK YOU FOR YOUR ATTENTION