Download - Finding self-force quantities in a post-Newtonian expansion

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Page 1: Finding self-force quantities in a post-Newtonian expansion

Finding self-force quantities in a post-Newtonian expansion

Seth Hopper

Eccentric orbits on a Schwarzschild background

π�

π�π�

�π

-�

-�

-�

-�

-�

p = 10

6

e = 0.01

x = 10

�6

p = 10

6

e = 0.2

x = 0.96⇥ 10

�6

log10

p

22,+tr

(�) residuals

log10

p

22,�tr

(�) residuals

¯

G

`m

(�) = 8

p⇡

µ

M

(`�m+ 1)

(`� 1)`(`+ 1)(`+ 2)

s

(2`+ 1)(`�m)!

(m+ `)!

P

m

`+1

⇥(

h

� 2 +

2e

i�

(m� 1)� 2e

�i�

(m+ 1)

e+O �

e

2�

i

x

3/2

+

h

1 +

e

�i�

(4� 2m) + 2e

i�

(m+ 2)

e+O �

e

2�

i

x

5/2+O �

x

7/2�

)

x

3/2

R

±`mn

(�) = C

±`mn

ˆ

R

±`mn

(�)

p = 10

e = 0.5

Chris Kavanagh

Adrian Ottewill

Erik Forseth

Charles Evans

Institut des Hautes Études Scientifiques - November 12, 2015

Page 2: Finding self-force quantities in a post-Newtonian expansion

An electron in free fall will radiate

e-

e-

e-

An electron is a non-local object

Page 3: Finding self-force quantities in a post-Newtonian expansion

Outline

Orbits

Homogeneous solutions

Particular solutions

Black Hole Perturbation Theory

Formalism

Analytic PN solutions

Numeric PN solutions

Page 4: Finding self-force quantities in a post-Newtonian expansion

Outline

Orbits

Homogeneous solutions

Particular solutions

Black Hole Perturbation Theory

Formalism

Analytic PN solutions

Numeric PN solutions

Page 5: Finding self-force quantities in a post-Newtonian expansion

In black hole perturbation theory, we alternate

between correcting fields and particle motion

Field equations Equations of motion

0

1

2

Order

G

µ⌫

= 0

dE

dt

, New parameters, 7PN

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10

�100

dE

dt

lm

=

X

n

!

2

64⇡

(l + 2)!

(l � 2)!

|C+lmn

|2

L4 =

1

(1� e

2)

15/2

�323105549467

3178375200

+

232597

4410

�E � 1369

126

2+

39931

294

log(2)� 47385

1568

log(3)

+

�128412398137

23543520

+

4923511

2940

�E � 104549

252

2 � 343177

252

log(2) +

55105839

15680

log(3)

e

2

+

�981480754818517

25427001600

+

142278179

17640

�E � 1113487

504

2+

762077713

5880

log(2)� 2595297591

71680

log(3)

� 15869140625

903168

log(5)

e

4

+

�874590390287699

12713500800

+

318425291

35280

�E � 881501

336

2 � 90762985321

63504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

16257024

log(5)

e

6

+ d8e8+ d10e

10+ d12e

12+ d14e

14+ d16e

16+ d18e

18+ d20e

20+ d22e

22+ d24e

24+ d26e

26

+ d28e28

+ d30e30

+ d32e32

+ d34e34

+ d36e36

+ d38e38

+ d40e40

+ · · ·�

L7/2 = �16285⇡

504

1

(1� e

2)

7

1 + a2e2+ a4e

4+ a6e

6+ · · · �

dE

dt

=

dE

dt

3PN

+

32

5

µ

M

⌘2x

5

x

7/2L7/2 + x

4⇣

L4 + log(x)L4L

+ x

9/2⇣

L9/2 + log(x)L9/2L

+ x

5⇣

L5 + log(x)L5L

+ x

11/2⇣

L11/2 + log(x)L11/2L

+ x

6⇣

L6 + log(x)L6L + log

2(x)L6L2

+ x

13/2⇣

L13/2 + log(x)L13/2L

+ x

7⇣

L7 + log(x)L7L + log

2(x)L7L2

G

µ⌫

= 0

Du

µ

d⌧

= 0

dE

dt

, New parameters, 7PN

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10

�100

dE

dt

lm

=

X

n

!

2

64⇡

(l + 2)!

(l � 2)!

|C+lmn

|2

L4 =

1

(1� e

2)

15/2

�323105549467

3178375200

+

232597

4410

�E � 1369

126

2+

39931

294

log(2)� 47385

1568

log(3)

+

�128412398137

23543520

+

4923511

2940

�E � 104549

252

2 � 343177

252

log(2) +

55105839

15680

log(3)

e

2

+

�981480754818517

25427001600

+

142278179

17640

�E � 1113487

504

2+

762077713

5880

log(2)� 2595297591

71680

log(3)

� 15869140625

903168

log(5)

e

4

+

�874590390287699

12713500800

+

318425291

35280

�E � 881501

336

2 � 90762985321

63504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

16257024

log(5)

e

6

+ d8e8+ d10e

10+ d12e

12+ d14e

14+ d16e

16+ d18e

18+ d20e

20+ d22e

22+ d24e

24+ d26e

26

+ d28e28

+ d30e30

+ d32e32

+ d34e34

+ d36e36

+ d38e38

+ d40e40

+ · · ·�

L7/2 = �16285⇡

504

1

(1� e

2)

7

1 + a2e2+ a4e

4+ a6e

6+ · · · �

dE

dt

=

dE

dt

3PN

+

32

5

µ

M

⌘2x

5

x

7/2L7/2 + x

4⇣

L4 + log(x)L4L

+ x

9/2⇣

L9/2 + log(x)L9/2L

+ x

5⇣

L5 + log(x)L5L

+ x

11/2⇣

L11/2 + log(x)L11/2L

+ x

6⇣

L6 + log(x)L6L + log

2(x)L6L2

+ x

13/2⇣

L13/2 + log(x)L13/2L

+ x

7⇣

L7 + log(x)L7L + log

2(x)L7L2

g

µ⌫

= g

BHµ⌫

+ p

µ⌫

2pµ⌫

+ 2R

↵µ�⌫

p

↵�

= �16⇡T

µ⌫

G

µ⌫

= 0

Du

µ

d⌧

= 0

dE

dt

, New parameters, 7PN

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10

�100

dE

dt

lm

=

X

n

!

2

64⇡

(l + 2)!

(l � 2)!

|C+lmn

|2

L4 =

1

(1� e

2)

15/2

�323105549467

3178375200

+

232597

4410

�E � 1369

126

2+

39931

294

log(2)� 47385

1568

log(3)

+

�128412398137

23543520

+

4923511

2940

�E � 104549

252

2 � 343177

252

log(2) +

55105839

15680

log(3)

e

2

+

�981480754818517

25427001600

+

142278179

17640

�E � 1113487

504

2+

762077713

5880

log(2)� 2595297591

71680

log(3)

� 15869140625

903168

log(5)

e

4

+

�874590390287699

12713500800

+

318425291

35280

�E � 881501

336

2 � 90762985321

63504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

16257024

log(5)

e

6

+ d8e8+ d10e

10+ d12e

12+ d14e

14+ d16e

16+ d18e

18+ d20e

20+ d22e

22+ d24e

24+ d26e

26

+ d28e28

+ d30e30

+ d32e32

+ d34e34

+ d36e36

+ d38e38

+ d40e40

+ · · ·�

L7/2 = �16285⇡

504

1

(1� e

2)

7

1 + a2e2+ a4e

4+ a6e

6+ · · · �

g

µ⌫

= g

BHµ⌫

+ p

µ⌫

2pµ⌫

+ 2R

↵µ�⌫

p

↵�

= �16⇡T

µ⌫

G

µ⌫

= 0

Du

µ

d⌧

= 0

dE

dt

, New parameters, 7PN

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10

�100

dE

dt

lm

=

X

n

!

2

64⇡

(l + 2)!

(l � 2)!

|C+lmn

|2

L4 =

1

(1� e

2)

15/2

�323105549467

3178375200

+

232597

4410

�E � 1369

126

2+

39931

294

log(2)� 47385

1568

log(3)

+

�128412398137

23543520

+

4923511

2940

�E � 104549

252

2 � 343177

252

log(2) +

55105839

15680

log(3)

e

2

+

�981480754818517

25427001600

+

142278179

17640

�E � 1113487

504

2+

762077713

5880

log(2)� 2595297591

71680

log(3)

� 15869140625

903168

log(5)

e

4

+

�874590390287699

12713500800

+

318425291

35280

�E � 881501

336

2 � 90762985321

63504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

16257024

log(5)

e

6

+ d8e8+ d10e

10+ d12e

12+ d14e

14+ d16e

16+ d18e

18+ d20e

20+ d22e

22+ d24e

24+ d26e

26

+ d28e28

+ d30e30

+ d32e32

+ d34e34

+ d36e36

+ d38e38

+ d40e40

+ · · ·�

L7/2 = �16285⇡

504

1

(1� e

2)

7

1 + a2e2+ a4e

4+ a6e

6+ · · · �

Du

µ

d⌧

= F

µ

self [p↵� ]

Du

µ

d⌧

= 0

g

µ⌫

= g

BHµ⌫

+ p

µ⌫

2pµ⌫

+ 2R

↵µ�⌫

p

↵�

= �16⇡T

µ⌫

G

µ⌫

= 0

dE

dt

, New parameters, 7PN

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10

�100

dE

dt

lm

=

X

n

!

2

64⇡

(l + 2)!

(l � 2)!

|C+lmn

|2

L4 =

1

(1� e

2)

15/2

�323105549467

3178375200

+

232597

4410

�E � 1369

126

2+

39931

294

log(2)� 47385

1568

log(3)

+

�128412398137

23543520

+

4923511

2940

�E � 104549

252

2 � 343177

252

log(2) +

55105839

15680

log(3)

e

2

+

�981480754818517

25427001600

+

142278179

17640

�E � 1113487

504

2+

762077713

5880

log(2)� 2595297591

71680

log(3)

� 15869140625

903168

log(5)

e

4

+

�874590390287699

12713500800

+

318425291

35280

�E � 881501

336

2 � 90762985321

63504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

16257024

log(5)

e

6

+ d8e8+ d10e

10+ d12e

12+ d14e

14+ d16e

16+ d18e

18+ d20e

20+ d22e

22+ d24e

24+ d26e

26

+ d28e28

+ d30e30

+ d32e32

+ d34e34

+ d36e36

+ d38e38

+ d40e40

+ · · ·�

L7/2 = �16285⇡

504

1

(1� e

2)

7

1 + a2e2+ a4e

4+ a6e

6+ · · · �

2p2 = (1 + p1)T [z + �z] + (rp1)2

Du

µ

d⌧

= F

µ

self [p↵� ]

Du

µ

d⌧

= 0

g

µ⌫

= g

BHµ⌫

+ p

µ⌫

2pµ⌫

+ 2R

↵µ�⌫

p

↵�

= �16⇡T

µ⌫

G

µ⌫

= 0

dE

dt

, New parameters, 7PN

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10

�100

dE

dt

lm

=

X

n

!

2

64⇡

(l + 2)!

(l � 2)!

|C+lmn

|2

L4 =

1

(1� e

2)

15/2

�323105549467

3178375200

+

232597

4410

�E � 1369

126

2+

39931

294

log(2)� 47385

1568

log(3)

+

�128412398137

23543520

+

4923511

2940

�E � 104549

252

2 � 343177

252

log(2) +

55105839

15680

log(3)

e

2

+

�981480754818517

25427001600

+

142278179

17640

�E � 1113487

504

2+

762077713

5880

log(2)� 2595297591

71680

log(3)

� 15869140625

903168

log(5)

e

4

+

�874590390287699

12713500800

+

318425291

35280

�E � 881501

336

2 � 90762985321

63504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

16257024

log(5)

e

6

+ d8e8+ d10e

10+ d12e

12+ d14e

14+ d16e

16+ d18e

18+ d20e

20+ d22e

22+ d24e

24+ d26e

26

+ d28e28

+ d30e30

+ d32e32

+ d34e34

+ d36e36

+ d38e38

+ d40e40

+ · · ·�

Page 6: Finding self-force quantities in a post-Newtonian expansion

Eccentric orbits on Schwarzschild have two

frequencies

Tr

'(t)

'(Tr)

�'(t)

t

0 t Tr

rmin ! rmin

1

Tr

'(t)

'(Tr)

�'(t)

t

0 t Tr

rmin ! rmin

1

r

=

2⇡

T

r

'

=

'

p

(T

r

)

T

r

�'(t) = '

p

(t)� ⌦

'

t

"d

2

dr

2⇤+ !

2 � V

`

(r)

#R

`!

(r) = 0

f

d

dr

✓f

d

dr

◆+ !

2 � V

l

(r)

�X

lm!

(r) = 0

7

Tr

1

Tr

'(t)

t

1

x =

�M⌦

'

�2/3

T

r

'

p

(t)

'(T

r

)

�'(t)

X

1

=

M

r

X

2

= (!r)

2

4

x =

�M⌦

'

�2/3

T

r

'

p

(t)

'

p

(T

r

)

�'(t)

X

1

=

M

r

X

2

= (!r)

2

4

Page 7: Finding self-force quantities in a post-Newtonian expansion

The radial frequency is oscillatory, but the azimuthal

frequency is not

Tr

1

Tr

'(t)

t

1

Tr

'(t)

'(Tr)

�'(t)

t

1

r

=

2⇡

T

r

'

=

'(T

r

)

T

r

�'(t) = '

p

(t)� ⌦

'

t

"d

2

dr

2⇤+ !

2 � V

`

(r)

#R

`!

(r) = 0

f

d

dr

✓f

d

dr

◆+ !

2 � V

l

(r)

�X

lm!

(r) = 0

7

Tr

1

Tr

'(t)

t

1

x =

�M⌦

'

�2/3

T

r

'

p

(t)

'(T

r

)

�'(t)

X

1

=

M

r

X

2

= (!r)

2

4

x =

�M⌦

'

�2/3

T

r

'

p

(t)

'

p

(T

r

)

�'(t)

X

1

=

M

r

X

2

= (!r)

2

4

r

=

2⇡

T

r

'

=

'

p

(T

r

)

T

r

�'(t) = '

p

(t)� ⌦

'

t

"d

2

dr

2⇤+ !

2 � V

`

(r)

#R

`!

(r) = 0

f

d

dr

✓f

d

dr

◆+ !

2 � V

l

(r)

�X

lm!

(r) = 0

7

Page 8: Finding self-force quantities in a post-Newtonian expansion

Darwin’s parametrization puts the orbit into a

“Newtonian-like” form

'

=

1

M

1� e

2

p

!3/2

"1 + 3

e

2

p

+O⇣p

�2

⌘#

r

p

(�) =

pM

1 + e cos�

x =

�M⌦

'

�2/3

T

r

'(t)

'(T

r

)

�'(t)

1

pM

C

±=

1

WT

r

ZT

r

0

"1

f

p

X

⌥G(t)+

2M

r

2

p

f

2

p

X

⌥ � 1

f

p

dX

⌥dr

!F (t)

#e

i!t

dt

'

p

(�)

C

±=

1

WT

r

Z2⇡

0

"1

f

p

X

⌥¯

G+

2M

r

2

p

f

2

p

X

⌥ � 1

f

p

dX

⌥dr

F

#e

in⌦

r

t

dt

p

d�

d�

�'(�)

r

p

(�) =

hM �M cos(�)e +O

⇣e

2

⌘ ix

�1

+

h2e

2

M � 2e

3

M cos(�) +O⇣e

4

⌘ ix

0

+O⇣x

1

⌘.

(p, e) = (10, 0.5)

0 � 2⇡ () 0 t T

r

pM

C

±=

1

WT

r

ZT

r

0

"1

f

p

X

⌥G(t)+

2M

r

2

p

f

2

p

X

⌥ � 1

f

p

dX

⌥dr

!F (t)

#e

i!t

dt

'

p

(�)

C

±=

1

WT

r

Z2⇡

0

"1

f

p

X

⌥¯

G+

2M

r

2

p

f

2

p

X

⌥ � 1

f

p

dX

⌥dr

F

#e

in⌦

r

t

dt

p

d�

d�

�'(�)

r

p

(�) =

hM �M cos(�)e +O

⇣e

2

⌘ ix

�1

+

h2e

2

M � 2e

3

M cos(�) +O⇣e

4

⌘ ix

0

+O⇣x

1

⌘.

dt

p

d�

=

p

2

M

(p� 2� 2e cos�)(1 + e cos�)

2

"(p� 2)

2 � 4e

2

p� 6� 2e cos�

#1/2

'

p

(�) =

✓4p

p� 6� 2e

◆1/2

F

✓�

2

�����4e

p� 6� 2e

'

=

1

M

1� e

2

p

!3/2

"1 + 3

e

2

p

+O⇣p

�2

⌘#

r

p

(�) =

pM

1 + e cos�

1

t

p

(�) =

Z�

0

p

2

M

(p� 2� 2e cos�

0)(1 + e cos�

0)

2

"(p� 2)

2 � 4e

2

p� 6� 2e cos�

0

#1/2

d�

0

One radial period

Page 9: Finding self-force quantities in a post-Newtonian expansion

The metric perturbation in Regge-Wheeler gauge is

found by solving a wave equation

"� @

2

@t

2

+

@

2

@r

2⇤� V

`

(r)

#

`m

(t, r) = S

`m

(t, r)

�'(�) = '

p

(�)� ⌦'

t

p

(�)

'

p

(�) = �+

h3� + sin(�)e + 3�e

2

+O⇣e

3

⌘ix+O

⇣x

2

�'(�) =

2e sin(�)� 3

4

e

2

sin(2�) +O⇣e

3

⌘�x

0

+

4e sin(�)� 3

4

e

2

sin(2�) +O⇣e

3

⌘�x

1

+O⇣x

2

TD master equation

"d

2

dr

2⇤+ !

2 � V

`

(r)

#R

`mn

(r) = Z

`mn

(r)

! ⌘ !

mn

= m⌦

'

+ n⌦

r

`m

(t, r) =

1X

n=�1R

`mn

(r) e

�i!t

S

`m

(t, r) =

1X

n=�1Z

`mn

(r) e

�i!t

FD master equation

`m

(t, r) =

1X

n=�1R

`mn

(r) e

�i!t

S

`m

(t, r) =

1X

n=�1Z

`mn

(r) e

�i!t

"� @

2

@t

2

+

@

2

@r

2⇤� V

`

(r)

#

`m

(t, r) = S

`m

(t, r)

�'(�) = '

p

(�)� ⌦'

t

p

(�)

'

p

(�) = �+

h3� + sin(�)e + 3�e

2

+O⇣e

3

⌘ix+O

⇣x

2

! ⌘ !

mn

= m⌦

'

+ n⌦

r

`m

(t, r) =

1X

n=�1R

`mn

(r) e

�i!t

S

`m

(t, r) =

1X

n=�1Z

`mn

(r) e

�i!t

"� @

2

@t

2

+

@

2

@r

2⇤� V

`

(r)

#

`m

(t, r) = S

`m

(t, r)

�'(�) = '

p

(�)� ⌦'

t

p

(�)

Fourier Series

Doubly periodic

Page 10: Finding self-force quantities in a post-Newtonian expansion

Exponential convergence is achieved through

extended homogeneous solutions

10-13

10-11

10-9

10-7

10-5

10-3

10-1

3 4 5 6 7 8 9 10 11 12 13 14

r/M

log

!

|Ψstd

22

(nm

ax)−

Ψstd

22

(N)|"

rmin rmax

N = 3

N = 5

N = 8

N = 10

N = 12

N = 14

10-13

10-11

10-9

10-7

10-5

10-3

10-1

3 4 5 6 7 8 9 10 11 12 13 14

r/M

log

!

|ΨE

HS

22

(nm

ax)−

ΨE

HS

22

(N)|"

rmin rmax

N = 3

N = 5

N = 8

N = 10

N = 12

N = 14

R

±`mn

(r) ⌘ C

±`mn

ˆ

R

±`mn

(r)

±`m

(t, r) =

X

n

C

±`mn

ˆ

R

±`mn

(r) e

�i!t

r > 2M

`m

(t, r) =

+

`m

(t, r)✓

⇥r � r

p

(t)

⇤+

�`m

(t, r)✓

⇥r

p

(t)� r

"d

2

dr

2⇤+ !

2 � V

`

(r)

#R

`mn

(r) = Z

`mn

(r)

! ⌘ !

mn

= m⌦

'

+ n⌦

r

R

±`mn

(r) ⌘ C

±`mn

ˆ

R

±`mn

(r)

±`m

(t, r) =

X

n

C

±`mn

ˆ

R

±`mn

(r) e

�i!t

r > 2M

`m

(t, r) =

+

`m

(t, r)✓

⇥r � r

p

(t)

⇤+

�`m

(t, r)✓

⇥r

p

(t)� r

"d

2

dr

2⇤+ !

2 � V

`

(r)

#R

`mn

(r) = Z

`mn

(r)

! ⌘ !

mn

= m⌦

'

+ n⌦

r

Time domain EHS

Page 11: Finding self-force quantities in a post-Newtonian expansion

Normalization coefficients are found by integrating a

radially-periodic function

C

±`mn

=

1

W

`mn

T

r

ZT

r

0

"1

f

p

ˆ

R

⌥`mn

(r

p

)G

`m

(t)+

2M

r

2p

f

2p

ˆ

R

⌥`mn

(r

p

)� 1

f

p

d

ˆ

R

⌥`mn

(r

p

)

dr

!F

`m

(t)

#e

i!t

dt

'

p

(�)

C

±=

1

WT

r

Z 2⇡

0

"1

f

p

X

⌥¯

G(t) +

✓2M

r

2p

f

2p

X

⌥ � 1

f

p

dX

dr

◆¯

F (t)

#e

in⌦r

t

dt

p

d�

d�

�'(�)

r

p

(�) =

hM � eM cos(�) +O �e2�

ix

�1

+

h2e

2M � 2e

3M cos(�) +O �e4�

ix

0+O �x1� .

¯

G

`m

(t) ⌘ G

`m

(t) e

im⌦

'

t

,

¯

F

`m

(t) ⌘ F

`m

(t) e

im⌦

'

t

±`m

(t, r) =

X

n

R

±`mn

(r) e

�i!t

! = m⌦

'

+ n⌦

r

a, b, . . . 2 {t, r}

A,B, . . . 2 {✓,'}

(t, r, ✓,') ! ⇥t

p

(�), r

p

(�), ✓

p

(�),'

p

(�)

±`m

(t, r) =

X

n

R

±`mn

(r) e

�i!t

! = m⌦

'

+ n⌦

r

a, b, . . . 2 {t, r}

A,B, . . . 2 {✓,'}

(t, r, ✓,') ! ⇥t

p

(�), r

p

(�), ✓

p

(�),'

p

(�)

Radially periodic

S

`m

(t, r) = G

`m

(t) �[r � r

p

(t)] + F

`m

(t) �

0[r � r

p

(t)]

¯

±22

(�)

±22

(�)

C

±`mn

=

1

W

`mn

T

r

ZT

r

0

"1

f

p

ˆ

R

⌥`mn

(r

p

)

¯

G

`m

(t)+

2M

r

2p

f

2p

ˆ

R

⌥`mn

(r

p

)� 1

f

p

d

ˆ

R

⌥`mn

(r

p

)

dr

F

`m

(t)

#e

in⌦r

t

dt

¯

G

`m

(t) ⌘ G

`m

(t) e

im⌦

'

t

,

¯

F

`m

(t) ⌘ F

`m

(t) e

im⌦

'

t

Master equation source term

Homogeneous solutions

C

±`mn

=

1

W

`mn

T

r

ZT

r

0

"1

f

p

ˆ

R

⌥`mn

(r

p

)

¯

G

`m

(t)+

2M

r

2p

f

2p

ˆ

R

⌥`mn

(r

p

)� 1

f

p

d

ˆ

R

⌥`mn

(r

p

)

dr

F

`m

(t)

#e

in⌦r

t

dt

¯

G

`m

(t) ⌘ G

`m

(t) e

im⌦

'

t

,

¯

F

`m

(t) ⌘ F

`m

(t) e

im⌦

'

t

±`m

(t, r) =

X

n

R

±`mn

(r) e

�i!t

! = m⌦

'

+ n⌦

r

a, b, . . . 2 {t, r}

A,B, . . . 2 {✓,'}

Manifestly radially periodic

Page 12: Finding self-force quantities in a post-Newtonian expansion

Removing the azimuthal frequency dependence from

the master function makes it periodic

±`m

(t, r) =

X

n

R

±`mn

(r) e

�i!t

! = m⌦

'

+ n⌦

r

a, b, . . . 2 {t, r}

A,B, . . . 2 {✓,'}

(t, r, ✓,') ! ⇥t

p

(�), r

p

(�), ✓

p

(�),'

p

(�)

-�

-�

�π�

π�π�

�π

¯

±22

(�)

±22

(�)

C

±`mn

=

1

W

`mn

T

r

ZT

r

0

"1

f

p

ˆ

R

⌥`mn

(r

p

)

¯

G

`m

(t)+

2M

r

2p

f

2p

ˆ

R

⌥`mn

(r

p

)� 1

f

p

d

ˆ

R

⌥`mn

(r

p

)

dr

F

`m

(t)

#e

in⌦r

t

dt

¯

G

`m

(t) ⌘ G

`m

(t) e

im⌦

'

t

,

¯

F

`m

(t) ⌘ F

`m

(t) e

im⌦

'

t

±`m

(t, r) =

X

n

R

±`mn

(r) e

�i!t

p = 10

e = 0.5

'

,⌦

r

E ,Jr

p

(�), r

p

(�), t

p

(�), �'(�)

ˆ

R

e/o,±`mn

(�)

@

r

ˆ

R

e/o,±`mn

(�)

@

2r

ˆ

R

e/o,±`mn

(�)

¯

G

e/o

`m

(�),

¯

F

e/o

`m

(�)

C

e/o,±`mn

R

e/o,±`mn

(�)

@

r

R

e/o,±`mn

(�)

@

2r

R

e/o,±`mn

(�)

¯

e/o,±`m

(�)

@

t

¯

e/o,±`m

(�), @

2t

¯

e/o,±`m

(�)

@

r

¯

e/o,±`m

(�), @

2r

¯

e/o,±`m

(�)

.

.

.

¯

±22

(�)

±22

(�)

C

±`mn

=

1

W

`mn

T

r

ZT

r

0

"1

f

p

ˆ

R

⌥`mn

(r

p

)

¯

G

`m

(t)+

2M

r

2p

f

2p

ˆ

R

⌥`mn

(r

p

)� 1

f

p

d

ˆ

R

⌥`mn

(r

p

)

dr

F

`m

(t)

#e

in⌦r

t

dt

¯

G

`m

(t) ⌘ G

`m

(t) e

im⌦

'

t

,

¯

F

`m

(t) ⌘ F

`m

(t) e

im⌦

'

t

±`m

(t, r) =

X

n

R

±`mn

(r) e

�i!t

-�

�π�

π�π�

�π

h

`m,±t

=

f

2

@

r

r

±`m

h

`m,±r

=

r

2f

@

t

±`m

¯

h

`m,±t

=

f

2

@

r

r

¯

±`m

¯

h

`m,±r

=

r

2f

@

t

¯

±`m

� im⌦

'

¯

±`m

¯

±`m

(t, r) =

±`m

(t, r)e

im⌦'t

¯

±`m

(t, r) ⌘ ±`m

(t, r)e

im⌦'t

=

X

n

R

±`mn

(r) e

�in⌦rt

h

21r

(�)

¯

h

21r

(�)

C

±`mn

=

1

W

`mn

T

r

Z 2⇡

0

"

1

f

p

ˆ

R

⌥`mn

(�)

¯

G

`m

(�)+

2M

r

2p

f

2p

ˆ

R

⌥`mn

(�)� 1

f

p

d

ˆ

R

⌥`mn

(�)

dr

!

¯

F

`m

(�)

#

e

in⌦rtp(�) dtp

d�

d�

e

9

e

5

p = 10

6

e = 0.01

x = 10

�6

h

`m,±t

=

f

2

@

r

r

±`m

h

`m,±r

=

r

2f

@

t

±`m

¯

h

`m,±t

=

f

2

@

r

r

¯

±`m

¯

h

`m,±r

=

r

2f

@

t

¯

±`m

� im⌦

'

¯

±`m

¯

±`m

(t, r) =

±`m

(t, r)e

im⌦'t

¯

±`m

(t, r) ⌘ ±`m

(t, r)e

im⌦'t

=

X

n

R

±`mn

(r) e

�in⌦rt

h

21r

(�)

¯

h

21r

(�)

C

±`mn

=

1

W

`mn

T

r

Z 2⇡

0

"

1

f

p

ˆ

R

⌥`mn

(�)

¯

G

`m

(�)+

2M

r

2p

f

2p

ˆ

R

⌥`mn

(�)� 1

f

p

d

ˆ

R

⌥`mn

(�)

dr

!

¯

F

`m

(�)

#

e

in⌦rtp(�) dtp

d�

d�

e

9

e

5

p = 10

6

e = 0.01

x = 10

�6

Page 13: Finding self-force quantities in a post-Newtonian expansion

“Barred” versions of the metric perturbation

amplitudes pick up counter terms

h

`m,±t

=

f

2

@

r

�r

±`m

�, h

`m,±r

=

r

2f

�@

t

±`m

¯

h

`m,±t

=

f

2

@

r

�r

¯

±`m

�,

¯

h

`m,±r

=

r

2f

�@

t

¯

±`m

� im⌦

'

¯

±`m

¯

±`m

(t, r) =

±`m

(t, r)e

im⌦

'

t

¯

±`m

(t, r) ⌘

±`m

(t, r)e

im⌦

'

t

=

X

n

R

±`mn

(r) e

�in⌦

r

t

X

1

! M

r

p

X

2

! (!r

p

)

2

π�

π�π�

�π

-���

-���

-���

���

���

h

`m,±t

=

f

2

@

r

r

±`m

h

`m,±r

=

r

2f

@

t

±`m

¯

h

`m,±t

=

f

2

@

r

r

¯

±`m

¯

h

`m,±r

=

r

2f

@

t

¯

±`m

� im⌦

'

¯

±`m

¯

±`m

(t, r) =

±`m

(t, r)e

im⌦'t

¯

±`m

(t, r) ⌘ ±`m

(t, r)e

im⌦'t

=

X

n

R

±`mn

(r) e

�in⌦rt

C

±`mn

=

1

W

`mn

T

r

Z 2⇡

0

"

1

f

p

ˆ

R

⌥`mn

(�)

¯

G

`m

(�)+

2M

r

2p

f

2p

ˆ

R

⌥`mn

(�)� 1

f

p

d

ˆ

R

⌥`mn

(�)

dr

!

¯

F

`m

(�)

#

e

in⌦rtp(�) dtp

d�

d�

e

9

e

5

p = 10

6

e = 0.01

x = 10

�6

p = 10

6

e = 0.2

x = 0.96⇥ 10

�6

h

`m,±t

=

f

2

@

r

r

±`m

h

`m,±r

=

r

2f

@

t

±`m

¯

h

`m,±t

=

f

2

@

r

r

¯

±`m

¯

h

`m,±r

=

r

2f

@

t

¯

±`m

� im⌦

'

¯

±`m

¯

±`m

(t, r) =

±`m

(t, r)e

im⌦'t

¯

±`m

(t, r) ⌘ ±`m

(t, r)e

im⌦'t

=

X

n

R

±`mn

(r) e

�in⌦rt

h

21r

(�)

¯

h

21r

(�)

C

±`mn

=

1

W

`mn

T

r

Z 2⇡

0

"

1

f

p

ˆ

R

⌥`mn

(�)

¯

G

`m

(�)+

2M

r

2p

f

2p

ˆ

R

⌥`mn

(�)� 1

f

p

d

ˆ

R

⌥`mn

(�)

dr

!

¯

F

`m

(�)

#

e

in⌦rtp(�) dtp

d�

d�

e

9

e

5

p = 10

6

e = 0.01

x = 10

�6

π�

π�π�

�π

-���

-���

���

���

h

`m,±t

=

f

2

@

r

r

±`m

h

`m,±r

=

r

2f

@

t

±`m

¯

h

`m,±t

=

f

2

@

r

r

¯

±`m

¯

h

`m,±r

=

r

2f

@

t

¯

±`m

� im⌦

'

¯

±`m

¯

±`m

(t, r) =

±`m

(t, r)e

im⌦'t

¯

±`m

(t, r) ⌘ ±`m

(t, r)e

im⌦'t

=

X

n

R

±`mn

(r) e

�in⌦rt

C

±`mn

=

1

W

`mn

T

r

Z 2⇡

0

"

1

f

p

ˆ

R

⌥`mn

(�)

¯

G

`m

(�)+

2M

r

2p

f

2p

ˆ

R

⌥`mn

(�)� 1

f

p

d

ˆ

R

⌥`mn

(�)

dr

!

¯

F

`m

(�)

#

e

in⌦rtp(�) dtp

d�

d�

e

9

e

5

p = 10

6

e = 0.01

x = 10

�6

p = 10

6

e = 0.2

x = 0.96⇥ 10

�6

h

`m,±t

=

f

2

@

r

r

±`m

h

`m,±r

=

r

2f

@

t

±`m

¯

h

`m,±t

=

f

2

@

r

r

¯

±`m

¯

h

`m,±r

=

r

2f

@

t

¯

±`m

� im⌦

'

¯

±`m

¯

±`m

(t, r) =

±`m

(t, r)e

im⌦'t

¯

±`m

(t, r) ⌘ ±`m

(t, r)e

im⌦'t

=

X

n

R

±`mn

(r) e

�in⌦rt

h

21r

(�)

¯

h

21r

(�)

C

±`mn

=

1

W

`mn

T

r

Z 2⇡

0

"

1

f

p

ˆ

R

⌥`mn

(�)

¯

G

`m

(�)+

2M

r

2p

f

2p

ˆ

R

⌥`mn

(�)� 1

f

p

d

ˆ

R

⌥`mn

(�)

dr

!

¯

F

`m

(�)

#

e

in⌦rtp(�) dtp

d�

d�

e

9

e

5

p = 10

6

e = 0.01

x = 10

�6

p = 10

e = 0.5

'

,⌦

r

E ,Jr

p

(�), r

p

(�), t

p

(�), �'(�)

ˆ

R

e/o,±`mn

(�)

@

r

ˆ

R

e/o,±`mn

(�)

@

2r

ˆ

R

e/o,±`mn

(�)

¯

G

e/o

`m

(�),

¯

F

e/o

`m

(�)

C

e/o,±`mn

R

e/o,±`mn

(�)

@

r

R

e/o,±`mn

(�)

@

2r

R

e/o,±`mn

(�)

¯

e/o,±`m

(�)

@

t

¯

e/o,±`m

(�), @

2t

¯

e/o,±`m

(�)

@

r

¯

e/o,±`m

(�), @

2r

¯

e/o,±`m

(�)

.

.

.

Page 14: Finding self-force quantities in a post-Newtonian expansion

Quantities evaluated on the worldline are strictly

periodic in the radial motion

p

tt

(t, r, ✓,') =

X

`,m

h

`m

tt

(t, r)Y

`m

(✓,')

p

tt

(t

p

, r

p

, ✓

p

,'

p

) =

X

`,m

¯

h

`m

tt

(t

p

, r

p

)e

�im⌦

'

t

Y

`m

(✓

p

, 0)e

im⌦

'

t

p

tt

(t

p

, r

p

, ✓

p

,'

p

) =

X

`,m

¯

h

`m

tt

(t

p

, r

p

)

¯

Y

`m

¯

Y

`m

(✓

p

, 0)Y

`m

(✓

p

, 0)

h

`m,±t

=

f

2

@

r

�r

±`m

�, h

`m,±r

=

r

2f

�@

t

±`m

Dependent on four variables

p

tt

(t, r, ✓,') =

X

`,m

h

`m

tt

(t, r)Y

`m

(✓,')

p

tt

(�) =

X

`,m

h

`m

tt

(t

p

, r

p

)Y

`m

(✓

p

,'

p

)

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t

p

, r

p

)e

�im⌦'t⇤ · ⇥Y `m

(✓

p

, 0)e

im�'(t)e

im⌦'tp⇤

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t, r

p

)

¯

Y

`m

(✓

p

,'

p

)

¯

Y

`m

(✓

p

,'

p

) = Y

`m

(✓

p

,'

p

)e

�im⌦'t

h

`m,±t

=

f

2

@

r

r

±`m

h

`m,±r

=

r

2f

@

t

±`m

¯

h

`m,±t

=

f

2

@

r

r

¯

±`m

¯

h

`m,±r

=

r

2f

@

t

¯

±`m

� im⌦

'

¯

±`m

¯

±`m

(t, r) =

±`m

(t, r)e

im⌦'t

¯

±`m

(t, r) ⌘ ±`m

(t, r)e

im⌦'t

=

X

n

R

±`mn

(r) e

�in⌦rt

h

21r

(�)

¯

h

21r

(�)

(t, r, ✓,') ! [t

p

(�), r

p

(�), ✓

p

(�),'

p

(�)]

p

tt

(t, r, ✓,') =

X

`,m

h

`m

tt

(t, r)Y

`m

(✓,')

p

tt

(�) =

X

`,m

h

`m

tt

(t

p

, r

p

)Y

`m

(✓

p

,'

p

)

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t

p

, r

p

)e

�im⌦'t⇤ · ⇥Y `m

(✓

p

, 0)e

im�'(t)e

im⌦'tp⇤

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t, r

p

)

¯

Y

`m

(✓

p

,'

p

)

¯

Y

`m

(✓

p

,'

p

) = Y

`m

(✓

p

,'

p

)e

�im⌦'t

h

`m,±t

=

f

2

@

r

r

±`m

h

`m,±r

=

r

2f

@

t

±`m

¯

h

`m,±t

=

f

2

@

r

r

¯

±`m

¯

h

`m,±r

=

r

2f

@

t

¯

±`m

� im⌦

'

¯

±`m

¯

±`m

(t, r) =

±`m

(t, r)e

im⌦'t

¯

±`m

(t, r) ⌘ ±`m

(t, r)e

im⌦'t

=

X

n

R

±`mn

(r) e

�in⌦rt

h

21r

(�)

¯

h

21r

(�)

Evaluated at the particle

p

tt

(t, r, ✓,') =

X

`,m

h

`m

tt

(t, r)Y

`m

(✓,')

p

tt

(t) =

X

`,m

⇣¯

h

`m

tt

(t, r

p

)e

�im⌦

'

t

⌘⇣Y

`m

[✓

p

,�'(t)]e

im⌦

'

t

p

tt

(t) =

X

`,m

¯

h

`m

tt

(t, r

p

)

¯

Y

`m

(✓

p

,'

p

)

¯

Y

`m

(✓

p

,'

p

) = Y

`m

(✓

p

,'

p

)e

�im⌦

'

t

h

`m,±t

=

f

2

@

r

�r

±`m

�, h

`m,±r

=

r

2f

�@

t

±`m

(t, r, ✓,') ! ⇥t

p

(�), r

p

(�), ✓

p

(�),'

p

(�)

p

tt

(t, r, ✓,') =

X

`,m

h

`m

tt

(t, r)Y

`m

(✓,')

p

tt

(�) =

X

`,m

h

`m

tt

(t, r

p

)e

�im⌦

'

t

i·hY

`m

(✓

p

, 0)e

im�'(t)

e

im⌦

'

t

i

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t, r

p

)

¯

Y

`m

(✓

p

,'

p

)

¯

Y

`m

(✓

p

,'

p

) = Y

`m

(✓

p

,'

p

)e

�im⌦

'

t

�π�

π�π�

�π

�����

�����

�����

�����Re

hp

22

tt

(�)

i

Im

hp

22

tt

(�)

i

t

p

(�) =

Z�

0

p

2

M

(p� 2� 2e cos�

0)(1 + e cos�

0)

2

"(p� 2)

2 � 4e

2

p� 6� 2e cos�

0

#1/2

d�

0

p = 10

e = 0.5

'

,⌦

r

E ,Jr

p

(�), r

p

(�), t

p

(�), �'(�)

ˆ

R

e/o,±`mn

(�)

@

r

ˆ

R

e/o,±`mn

(�)

@

2r

ˆ

R

e/o,±`mn

(�)

¯

G

e/o

`m

(�),

¯

F

e/o

`m

(�)

C

e/o,±`mn

R

e/o,±`mn

(�)

@

r

R

e/o,±`mn

(�)

@

2r

R

e/o,±`mn

(�)

¯

e/o,±`m

(�)

@

t

¯

e/o,±`m

(�), @

2t

¯

e/o,±`m

(�)

@

r

¯

e/o,±`m

(�), @

2r

¯

e/o,±`m

(�)

.

.

.

Page 15: Finding self-force quantities in a post-Newtonian expansion

Outline

Orbits

Homogeneous solutions

Particular solutions

Black Hole Perturbation Theory

Formalism

Previous work

Analytic PN solutions

Previous work

Numeric PN solutions

Page 16: Finding self-force quantities in a post-Newtonian expansion

There has been a lot of previous work, but here are a

couple highlights

Mano, Suzuki & Takasugi

Ganz, Hikida, Nakano, Sago &

Tanaka

Sago & Fujita

1996 1997 2007 2015

Mano & Takasugi

General orbits on Kerr

4PN & 𝑒6

Page 17: Finding self-force quantities in a post-Newtonian expansion

Outline

Orbits

Homogeneous solutions

Particular solutions

Black Hole Perturbation Theory

Formalism

Previous work

Analytic PN solutions

Previous work

Numeric PN solutions

Page 18: Finding self-force quantities in a post-Newtonian expansion

The fundamental frequencies can be expanded in the

gauge invariant PN parameter 𝑥

T

r

= 2⇡M

p

1� e

2

◆3/2

1 +

3 (1� e

2)

p

+O �

p

�2�

T

r

=

2⇡M

x

3/2+

6⇡M

(1� e

2)

px

+O �

x

1/2�

(t, r, ✓,') ! [t

p

(�), r

p

(�), ✓

p

(�),'

p

(�)]

p

tt

(t, r, ✓,') =

X

`,m

h

`m

tt

(t, r)Y

`m

(✓,')

p

tt

(�) =

X

`,m

h

`m

tt

(t

p

, r

p

)Y

`m

(✓

p

,'

p

)

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t

p

, r

p

)e

�im⌦'t⇤ · ⇥Y `m

(✓

p

, 0)e

im�'(t)e

im⌦'tp⇤

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t, r

p

)

¯

Y

`m

(✓

p

,'

p

)

¯

Y

`m

(✓

p

,'

p

) = Y

`m

(✓

p

,'

p

)e

�im⌦'t

h

`m,±t

=

f

2

@

r

r

±`m

h

`m,±r

=

r

2f

@

t

±`m

¯

h

`m,±t

=

f

2

@

r

r

¯

±`m

¯

h

`m,±r

=

r

2f

@

t

¯

±`m

� im⌦

'

¯

±`m

¯

±`m

(t, r) =

±`m

(t, r)e

im⌦'t

¯

±`m

(t, r) ⌘ ±`m

(t, r)e

im⌦'t

=

X

n

R

±`mn

(r) e

�in⌦rt

Z 2⇡

0

dt

p

d�

d�

T

r

= 2⇡M

p

1� e

2

◆3/2

1 +

3 (1� e

2)

p

+O �

p

�2�

T

r

=

2⇡M

x

3/2+

6⇡M

(1� e

2)

px

+O �

x

1/2�

(t, r, ✓,') ! [t

p

(�), r

p

(�), ✓

p

(�),'

p

(�)]

p

tt

(t, r, ✓,') =

X

`,m

h

`m

tt

(t, r)Y

`m

(✓,')

p

tt

(�) =

X

`,m

h

`m

tt

(t

p

, r

p

)Y

`m

(✓

p

,'

p

)

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t

p

, r

p

)e

�im⌦'t⇤ · ⇥Y `m

(✓

p

, 0)e

im�'(t)e

im⌦'tp⇤

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t, r

p

)

¯

Y

`m

(✓

p

,'

p

)

¯

Y

`m

(✓

p

,'

p

) = Y

`m

(✓

p

,'

p

)e

�im⌦'t

h

`m,±t

=

f

2

@

r

r

±`m

h

`m,±r

=

r

2f

@

t

±`m

¯

h

`m,±t

=

f

2

@

r

r

¯

±`m

¯

h

`m,±r

=

r

2f

@

t

¯

±`m

� im⌦

'

¯

±`m

¯

±`m

(t, r) =

±`m

(t, r)e

im⌦'t

Integrate order-by-order

⌦'

=1

M

1� e

2

p

!3/2 "1 + 3

e

2

p

+O⇣p

�2⌘#

x =�M⌦

'

�2/3

T

r

'(t)

'(Tr

)

�'(t)

X1 =M

r

X2 = (!r)2

1

⌦' =1

M

1� e2

p

!3/2 "1 + 3

e2

p+O

⇣p�2⌘#

Tr

'(t)

'(Tr)

�'(t)

X1 =M

rX2 = (!r)2

X1 ⇠ X2 ⌧ 1

1

𝑥 as an expansion in 1/𝑝

p =

1� e

2

x

+ 2e

2

+O⇣x

1

r

=

x

3/2

M

1� 3

1� e

2

x +O(x

2

)

'

=

x

3/2

M

dt

p

d�

=

p

2

M

(p� 2� 2e cos�)(1 + e cos�)

2

"(p� 2)

2 � 4e

2

p� 6� 2e cos�

#1/2

1

Invert the expansion

r

=

x

3/2

M

1� 3

1� e

2

x +O(x

2

)

'

=

x

3/2

M

dt

p

d�

=

p

2

M

(p� 2� 2e cos�)(1 + e cos�)

2

"(p� 2)

2 � 4e

2

p� 6� 2e cos�

#1/2

'

p

(�) =

✓4p

p� 6� 2e

◆1/2

F

✓�

2

�����4e

p� 6� 2e

◆1

r

=

x

3/2

M

1� 3

1� e

2

x +O(x

2

)

'

=

x

3/2

M

dt

p

d�

=

p

2

M

(p� 2� 2e cos�)(1 + e cos�)

2

"(p� 2)

2 � 4e

2

p� 6� 2e cos�

#1/2

'

p

(�) =

✓4p

p� 6� 2e

◆1/2

F

✓�

2

�����4e

p� 6� 2e

1

Fundamental frequencies

Page 19: Finding self-force quantities in a post-Newtonian expansion

Homogeneous solutions are found by expanding in

two separate limits

Tr

'(t)

'(Tr)

�'(t)

X1 =M

rX2 = (!r)2

X1 ⇠ X2 ⌧ 1

t

0 t Tr

rmin ! rmin1

Expansion variables Example, 𝑙 = 2, through 1PN

Tr

'(t)

'(Tr)

�'(t)

X1 =M

rX2 = (!r)2

X1 ⇠ X2 ⌧ 1

R+ = � i

16X2c2 � i(10X1 +X2)

96X2c0 +O

⇣c�2

1R� =

i

384X41pX2

c9 � ipX2

5376X41

c7 +O⇣c5⌘

t

0 t Tr

rmin ! rmin

` = 2

⌦r =2⇡

Tr

⌦' ='(Tr)

Tr

�'(t) = '(t)� ⌦'t

2

"d2

dr2⇤+ !2 � V`(r)

#R`!(r) = 0

fd

dr

✓fd

dr

◆+ !2 � Vl(r)

�Xlm!(r) = 0

3

Odd parity, homogenous

equation

Page 20: Finding self-force quantities in a post-Newtonian expansion

Use expressions for the position and the

frequency, expanded in both 𝑥 and 𝑒

'

p

(�)

C

±=

1

WT

r

Z 2⇡

0

"1

f

p

X

⌥¯

G(t) +

✓2M

r

2p

f

2p

X

⌥ � 1

f

p

dX

dr

◆¯

F (t)

#e

in⌦r

t

dt

p

d�

d�

�'(�)

r

p

(�) =

hM � eM cos(�) +O �

e

2� i

x

�1

+

h2e

2M � 2e

3M cos(�) +O �

e

4� i

x

0+O �

x

1�.

(p, e) = (10, 0.5)

! =

(m + n)

M

x

3/2

+

"�3n

M

� 3ne

2

M

+O⇣e

3

⌘#x

5/2

+O⇣x

3

⌘.

R

+

= � i

16!

2

r

2

p

c

2 � 1

96

i

10M

!

2

r

3

p

+ 1

!c

0

+O⇣c

�2

R

�=

ir

3

p

384M

4

!

c

9 � i!r

5

p

5376M

4

c

7

+O⇣c

5

R

+=

� i

16(m + n)

2� i cos(�)e

8(m + n)

2+O �e2�

�x

�1

+

"�i

�m

3+ 3m

2n + 3mn

2+ 10m + n

3+ 46n

96(m + n)

3� ie(5m + 17n) cos(�)

16(m + n)

3+O �e2�

#x

0+O �x1�

Spectrum from source’s Fourier seriesR

+

= � i

16!

2

r

2

p

c

2 � 1

96

i

10M

!

2

r

3

p

+ 1

!c

0

+O⇣c

�2

R

�=

ir

3

p

384M

4

!

c

9 � i!r

5

p

5376M

4

c

7

+O⇣c

5

R

+=

� i

16(m + n)

2� i cos(�)e

8(m + n)

2+O �e2�

�x

�1

+

"�i

�m

3+ 3m

2n + 3mn

2+ 10m + n

3+ 46n

96(m + n)

3� ie(5m + 17n) cos(�)

16(m + n)

3+O �e2�

#x

0+O �x1�

R

±`mn

(r) ⌘ C

±`mn

ˆ

R

±`mn

(r)

±`m

(t, r) =

X

n

C

±`mn

ˆ

R

±`mn

(r) e

�i!t

r > 2M

R

�=

i

384(m+ n)

� i cos(�)e

128(m+ n)

+O �e

2��

x

�9/2

+

"�i

(m+ n)

3 � 42n

5376(m+ n)

2+

i

�5m

3+ 15nm

2+ 15n

2m+ 5n

3 � 126n

�cos(�)e

5376(m+ n)

2+O �

e

2�#x

�7/2+O

⇣x

�5/2⌘

¯

G

o

2m =

p5⇡µ

3M

(m� 3)

s(2�m)!

(2 +m)!

P

m

3

(h2 +

�e

i�

(2� 2m) + 2e

�i�

(m+ 1)

�e+O �

e

2� i

x

3/2

+

h� 1 +

�2e

�i�

(m� 2)� 2e

i�

(m+ 2)

�e+O �

e

2� i

x

5/2+O

⇣x

7/2⌘)

R

+=

� i

16(m+ n)

2� i cos(�)e

8(m+ n)

2+O �

e

2��

x

�1

+

"�i

�m

3+ 3m

2n+ 3mn

2+ 10m+ n

3+ 46n

96(m+ n)

3� ie(5m+ 17n) cos(�)

16(m+ n)

3+O �

e

2�#x

0+O �

x

1�

R

±`mn

(r) ⌘ C

±`mn

ˆ

R

±`mn

(r)

±`m

(t, r) =

X

n

C

±`mn

ˆ

R

±`mn

(r) e

�i!t

r > 2M

Homogeneous solutions, odd parity, 𝑙=2, generic 𝑚,𝑛

Page 21: Finding self-force quantities in a post-Newtonian expansion

Normalized frequency domain solutions are found for

generic 𝑚 and 𝑛R

+2mn = µ

s(2�m)!

(2 +m)!

sin(n⇡)p5⇡

(�8 ((m� 3)P

m3 )

3n

+

✓8(m� 3)(2m� n)

3 (n

2 � 1)

� 16(m� 3) cos(�)

3n

◆P

m3 e+O �e2�

�px

+

"� 2 ((m� 3) (5m

2+ 10nm+ 5n

2+ 98)P

m3 )

63n

+

2(m� 3) (10m

3+ 33nm

2+ 36n

2m+ 511m+ 13n

3 � 287n)

63 (n

2 � 1)

+

4(m� 3) (3m

2+ 6nm+ 3n

2 � 56) cos(�)

21n

!P

m3 e+O �e2�

#x

3/2+O �x2

�)

C

+2mn = µ

s(2�m)!

(2 +m)!

sin(n⇡)p5⇡

(�128i(m� 3)(m+ n)

2P

m3

3n

+

128i(m� 3)(2m� n)(m+ n)

2P

m3 e

3 (n

2 � 1)

+O �e2��x

3/2

+

"32i(m� 3) (3m

4+ 12nm

3+ 2 (9n

2+ 7)m

2+ 4n (3n

2+ 49)m+ n

2(3n

2+ 182))P

m3

21n

� 32i(m� 3)(m+ n) (6m

4+ 9nm

3 � (9n

2+ 77)m

2 � 7n (3n

2 � 44)m� n

2(9n

2+ 119))P

m3 e

21 (n

2 � 1)

+O �e2�#x

5/2+O �x3

�)

R

±`mn

(�) = C

±`mn

ˆ

R

±`mn

(�)

p = 10

e = 0.5

'

,⌦

r

E ,Jr

p

(�), r

p

(�), t

p

(�), �'(�)

ˆ

R

e/o,±`mn

(�)

@

r

ˆ

R

e/o,±`mn

(�)

@

2r

ˆ

R

e/o,±`mn

(�)

¯

G

e/o

`m

(�),

¯

F

e/o

`m

(�)

C

e/o,±`mn

R

e/o,±`mn

(�)

@

r

R

e/o,±`mn

(�)

@

2r

R

e/o,±`mn

(�)

¯

e/o,±`m

(�)

@

t

¯

e/o,±`m

(�), @

2t

¯

e/o,±`m

(�)

@

r

¯

e/o,±`m

(�), @

2r

¯

e/o,±`m

(�)

.

.

.

R

+2mn = µ

s(2�m)!

(2 +m)!

sin(n⇡)p5⇡

(�8 ((m� 3)P

m3 )

3n

+

✓8(m� 3)(2m� n)

3 (n

2 � 1)

� 16(m� 3) cos(�)

3n

◆P

m3 e+O �e2�

�px

+

"� 2 ((m� 3) (5m

2+ 10nm+ 5n

2+ 98)P

m3 )

63n

+

2(m� 3) (10m

3+ 33nm

2+ 36n

2m+ 511m+ 13n

3 � 287n)

63 (n

2 � 1)

+

4(m� 3) (3m

2+ 6nm+ 3n

2 � 56) cos(�)

21n

!P

m3 e+O �e2�

#x

3/2+O �x2

�)

C

+2mn = µ

s(2�m)!

(2 +m)!

sin(n⇡)p5⇡

(�128i(m� 3)(m+ n)

2P

m3

3n

+

128i(m� 3)(2m� n)(m+ n)

2P

m3 e

3 (n

2 � 1)

+O �e2��x

3/2

+

"32i(m� 3) (3m

4+ 12nm

3+ 2 (9n

2+ 7)m

2+ 4n (3n

2+ 49)m+ n

2(3n

2+ 182))P

m3

21n

� 32i(m� 3)(m+ n) (6m

4+ 9nm

3 � (9n

2+ 77)m

2 � 7n (3n

2 � 44)m� n

2(9n

2+ 119))P

m3 e

21 (n

2 � 1)

+O �e2�#x

5/2+O �x3

�)

FD extended homogeneous solutions, odd parity, 𝑙=2, generic 𝑚,𝑛

C

±`mn

=

1

W

`mn

T

r

Z 2⇡

0

"

1

f

p

ˆ

R

⌥`mn

(�)

¯

G

`m

(�)+

2M

r

2p

f

2p

ˆ

R

⌥`mn

(�)� 1

f

p

d

ˆ

R

⌥`mn

(�)

dr

!

¯

F

`m

(�)

#

e

in⌦rtp(�) dtp

d�

d�

e

9

e

5

p = 10

6

e = 0.01

x = 10

�6

p = 10

6

e = 0.2

x = 0.96⇥ 10

�6

log10

p

22,+tr

(�) residuals

log10

p

22,�tr

(�) residuals

¯

G

`m

(�) = 8

p⇡

µ

M

(`�m+ 1)

(`� 1)`(`+ 1)(`+ 2)

s

(2`+ 1)(`�m)!

(m+ `)!

P

m

`+1

⇥(

h

� 2 +

2e

i�

(m� 1)� 2e

�i�

(m+ 1)

e+O �e2�i

x

3/2

+

h

1 +

e

�i�

(4� 2m) + 2e

i�

(m+ 2)

e+O �e2�i

x

5/2+O �x7/2

)

x

3/2

Page 22: Finding self-force quantities in a post-Newtonian expansion

The finite-order 𝒆 expansion truncates the sum over

harmonics

¯

±`m(t, r) =

X

n

R

±`mn(r) e

�in⌦rt

n R

+2mn(�)

�3 0

�2 �µ3

q

⇡(2�m)!5(2+m)! (m� 3) (4m

2 �m� 2)P

m3 e

2

�1 �4µ3

q

⇡(2�m)!5(2+m)! (m� 3)(�2m� 1)P

m3

h

e+ 2e

2cos(�)

i

0 �8µ3

q

⇡(2�m)!5(2+m)! (m� 3)P

m3

n

1 + 2e cos(�) + e

2⇥

2�m

2+ cos

2(�)

o

1 �4µ3

q

⇡(2�m)!5(2+m)! (m� 3)(2m� 1)P

m3

h

e+ 2e

2cos(�)

i

2 �µ3

q

⇡(2�m)!5(2+m)! (m� 3) (4m

2+m� 2)P

m3 e

2

3 0

R

+2mn = µ

s

(2�m)!

(2 +m)!

sin(n⇡)p5⇡

(

�8 ((m� 3)P

m3 )

3n

+

8(m� 3)(2m� n)

3 (n

2 � 1)

� 16(m� 3) cos(�)

3n

P

m3 e+O �e2�

�px

+

"

� 2 ((m� 3) (5m

2+ 10nm+ 5n

2+ 98)P

m3 )

63n

+

2(m� 3) (10m

3+ 33nm

2+ 36n

2m+ 511m+ 13n

3 � 287n)

63 (n

2 � 1)

+

4(m� 3) (3m

2+ 6nm+ 3n

2 � 56) cos(�)

21n

!

P

m3 e+O �e2�

#

x

3/2+O �x2

)

C

+2mn = µ

s

(2�m)!

(2 +m)!

sin(n⇡)p5⇡

(

�128i(m� 3)(m+ n)

2P

m3

3n

+

128i(m� 3)(2m� n)(m+ n)

2P

m3 e

3 (n

2 � 1)

+O �e2��

x

3/2

+

"

32i(m� 3) (3m

4+ 12nm

3+ 2 (9n

2+ 7)m

2+ 4n (3n

2+ 49)m+ n

2(3n

2+ 182))P

m3

21n

� 32i(m� 3)(m+ n) (6m

4+ 9nm

3 � (9n

2+ 77)m

2 � 7n (3n

2 � 44)m� n

2(9n

2+ 119))P

m3 e

21 (n

2 � 1)

+O �e2�#

x

5/2+O �x3

)

p

`m,±µ⌫

(�)

@

t

p

`m,±µ⌫

(�)

@

r

p

`m,±µ⌫

(�)

¯

±`m

(�) =

X

n

R

±`mn

(�) e

�in⌦rtp(�)

n R

+2mn

(�)

�3 0

�2 �µ

3

q

⇡(2�m)!5(2+m)! (m� 3) (4m

2 �m� 2)P

m

3 e

2

�1 �4µ3

q

⇡(2�m)!5(2+m)! (m� 3)(�2m� 1)P

m

3

h

e+ 2e

2cos(�)

i

0 �8µ3

q

⇡(2�m)!5(2+m)! (m� 3)P

m

3

n

1 + 2e cos(�) + e

2⇥

2�m

2+ cos

2(�)

o

1 �4µ3

q

⇡(2�m)!5(2+m)! (m� 3)(2m� 1)P

m

3

h

e+ 2e

2cos(�)

i

2 �µ

3

q

⇡(2�m)!5(2+m)! (m� 3) (4m

2+m� 2)P

m

3 e

2

3 0

R

+2mn

= µ

s

(2�m)!

(2 +m)!

sin(n⇡)p5⇡

(

�8 ((m� 3)P

m

3 )

3n

+

8(m� 3)(2m� n)

3 (n

2 � 1)

� 16(m� 3) cos(�)

3n

P

m

3 e+O �e2��p

x

+

"

� 2 ((m� 3) (5m

2+ 10nm+ 5n

2+ 98)P

m

3 )

63n

+

2(m� 3) (10m

3+ 33nm

2+ 36n

2m+ 511m+ 13n

3 � 287n)

63 (n

2 � 1)

+

4(m� 3) (3m

2+ 6nm+ 3n

2 � 56) cos(�)

21n

!

P

m

3 e+O �e2�#

x

3/2+O �x2

)

Page 23: Finding self-force quantities in a post-Newtonian expansion

The retarded metric perturbation is computed only once

for each 𝑙 mode

'

,⌦

r

E ,Jr

p

(�), r

p

(�), t

p

(�), �'(�)

ˆ

R

e/o,±`mn

(�)

@

r

ˆ

R

e/o,±`mn

(�)

@

2r

ˆ

R

e/o,±`mn

(�)

¯

G

`m

(�),

¯

F

`m

(�)

C

e/o,±`mn

R

e/o,±`mn

(�)

@

r

R

e/o,±`mn

(�)

@

2r

R

e/o,±`mn

(�)

¯

±`m

(�)

@

t

¯

±`m

(�), @

2t

¯

±`m

(�)

@

r

¯

±`m

(�), @

2r

¯

±`m

(�)

.

.

.

p

`m

µ⌫

(�)

@

t

p

`m

µ⌫

(�)

@

r

p

`m

µ⌫

(�)

Orbit quantities

'

,⌦

r

E ,Jr

p

(�), r

p

(�), t

p

(�), �'(�)

ˆ

R

e/o,±`mn

(�)

@

r

ˆ

R

e/o,±`mn

(�)

@

2r

ˆ

R

e/o,±`mn

(�)

¯

G

`m

(�),

¯

F

`m

(�)

C

e/o,±`mn

R

e/o,±`mn

(�)

@

r

R

e/o,±`mn

(�)

@

2r

R

e/o,±`mn

(�)

¯

±`m

(�)

@

t

¯

±`m

(�), @

2t

¯

±`m

(�)

@

r

¯

±`m

(�), @

2r

¯

±`m

(�)

.

.

.

p

`m

µ⌫

(�)

@

t

p

`m

µ⌫

(�)

@

r

p

`m

µ⌫

(�)

Norm. coeffs.

'

,⌦

r

E ,Jr

p

(�), r

p

(�), t

p

(�), �'(�)

ˆ

R

e/o,±`mn

(�)

@

r

ˆ

R

e/o,±`mn

(�)

@

2r

ˆ

R

e/o,±`mn

(�)

¯

G

`m

(�),

¯

F

`m

(�)

C

e/o,±`mn

R

e/o,±`mn

(�)

@

r

R

e/o,±`mn

(�)

@

2r

R

e/o,±`mn

(�)

¯

±`m

(�)

@

t

¯

±`m

(�), @

2t

¯

±`m

(�)

@

r

¯

±`m

(�), @

2r

¯

±`m

(�)

.

.

.

p

`m

µ⌫

(�)

@

t

p

`m

µ⌫

(�)

@

r

p

`m

µ⌫

(�)

FD particular sols. Metric perturbation

'

,⌦

r

E ,Jr

p

(�), r

p

(�), t

p

(�), �'(�)

ˆ

R

e/o,±`mn

(�)

@

r

ˆ

R

e/o,±`mn

(�)

@

2r

ˆ

R

e/o,±`mn

(�)

¯

G

e/o

`m

(�),

¯

F

e/o

`m

(�)

C

e/o,±`mn

R

e/o,±`mn

(�)

@

r

R

e/o,±`mn

(�)

@

2r

R

e/o,±`mn

(�)

¯

e/o,±`m

(�)

@

t

¯

e/o,±`m

(�), @

2t

¯

e/o,±`m

(�)

@

r

¯

e/o,±`m

(�), @

2r

¯

e/o,±`m

(�)

.

.

.

p

`m,±µ⌫

(�)

@

t

p

`m,±µ⌫

(�)

@

r

p

`m,±µ⌫

(�)

TD particular sols.

'

,⌦

r

E ,Jr

p

(�), r

p

(�), t

p

(�), �'(�)

ˆ

R

e/o,±`mn

(�)

@

r

ˆ

R

e/o,±`mn

(�)

@

2r

ˆ

R

e/o,±`mn

(�)

¯

G

e/o

`m

(�),

¯

F

e/o

`m

(�)

C

e/o,±`mn

R

e/o,±`mn

(�)

@

r

R

e/o,±`mn

(�)

@

2r

R

e/o,±`mn

(�)

¯

e/o,±`m

(�)

@

t

¯

e/o,±`m

(�), @

2t

¯

e/o,±`m

(�)

@

r

¯

e/o,±`m

(�), @

2r

¯

e/o,±`m

(�)

.

.

.

p

`m,±µ⌫

(�)

@

t

p

`m,±µ⌫

(�)

@

r

p

`m,±µ⌫

(�)

'

,⌦

r

E ,Jr

p

(�), r

p

(�), t

p

(�), �'(�)

ˆ

R

e/o,±`mn

(�)

@

r

ˆ

R

e/o,±`mn

(�)

@

2r

ˆ

R

e/o,±`mn

(�)

¯

G

`m

(�),

¯

F

`m

(�)

C

e/o,±`mn

R

e/o,±`mn

(�)

@

r

R

e/o,±`mn

(�)

@

2r

R

e/o,±`mn

(�)

¯

±`m

(�)

@

t

¯

±`m

(�), @

2t

¯

±`m

(�)

@

r

¯

±`m

(�), @

2r

¯

±`m

(�)

.

.

.

p

`m

µ⌫

(�)

@

t

p

`m

µ⌫

(�)

@

r

p

`m

µ⌫

(�)

Homog. sols.

Source terms

'

,⌦

r

E ,Jr

p

(�), r

p

(�), t

p

(�), �'(�)

ˆ

R

e/o,±`mn

(�)

@

r

ˆ

R

e/o,±`mn

(�)

@

2r

ˆ

R

e/o,±`mn

(�)

¯

G

e/o

`m

(�),

¯

F

e/o

`m

(�)

C

e/o,±`mn

R

e/o,±`mn

(�)

@

r

R

e/o,±`mn

(�)

@

2r

R

e/o,±`mn

(�)

¯

e/o,±`m

(�)

@

t

¯

e/o,±`m

(�), @

2t

¯

e/o,±`m

(�)

@

r

¯

e/o,±`m

(�), @

2r

¯

e/o,±`m

(�)

.

.

.

p

`m,±µ⌫

(�)

@

t

p

`m,±µ⌫

(�)

@

r

p

`m,±µ⌫

(�)

Page 24: Finding self-force quantities in a post-Newtonian expansion

Metric perturbation expressions are remarkably simple

p

22,+tt

=

µ

M

3

4

+

✓3e

�i�

8

+

3e

i�

8

◆e +O �

e

2��

x

+

µ

M

�149

56

+

✓�229

112

e

�i� � 437e

i�

112

◆e +O �

e

2��

x

2+O �

x

3�

p

22,�tt

=

µ

M

3

4

+

✓3e

�i�

8

+

3e

i�

8

◆e +O �

e

2��

x

+

µ

M

�149

56

+

✓�509

112

e

�i� � 157e

i�

112

◆e +O �

e

2��

x

2+O �

x

3�

�π�

π�π�

�π

�����

�����

�����

�����

Re

hp

22

tt

(�)

i

Im

hp

22

tt

(�)

i

t

p

(�) =

Z�

0

p

2

M

(p� 2� 2e cos�

0)(1 + e cos�

0)

2

"(p� 2)

2 � 4e

2

p� 6� 2e cos�

0

#1/2

d�

0

r

p

(�) =

hM �M cos(�)e +O

⇣e

2

⌘ ix

�1

+

h2e

2

M � 2e

3

M cos(�) +O⇣e

4

⌘ ix

0

+O⇣x

1

⌘.

(p, e) = (10, 0.5)

p = 10

e = 0.5

x = 0.08

p =

1� e

2

x

+ 2e

2

+O⇣x

1

1

Page 25: Finding self-force quantities in a post-Newtonian expansion

We see convergence in 𝑥 for small values of 𝑒

π�

π �π�

�π

-��

-��

-�

-�

-�

-��� �����

�����

p = 1000

e = 0.01

x ⇡ 0.001

log

10

hp

22

tt

res.

i

p

22

tt

(�)

Re

hp

22

tt

(�)

i

Im

hp

22

tt

(�)

i

t

p

(�) =

Z�

0

p

2

M

(p� 2� 2e cos�

0)(1 + e cos�

0)

2

"(p� 2)

2 � 4e

2

p� 6� 2e cos�

0

#1/2

d�

0

p = 1000

e = 0.01

x ⇡ 0.001

log

10

hp

22

tt

residuals

i

p

22

tt

(�)

Re

hp

22

tt

(�)

i

Im

hp

22

tt

(�)

i

t

p

(�) =

Z�

0

p

2

M

(p� 2� 2e cos�

0)(1 + e cos�

0)

2

"(p� 2)

2 � 4e

2

p� 6� 2e cos�

0

#1/2

d�

0

Page 26: Finding self-force quantities in a post-Newtonian expansion

Sometimes the convergence with 𝑒 stalls

p = 10

6

e = 0.2

x = 0.96⇥ 10

�6

log10

p

22,+tr

residuals

log10

p

22,�tr

residuals

¯

G

`m

(�) = 8

p⇡

µ

M

(`�m+ 1)

(`� 1)`(`+ 1)(`+ 2)

s

(2`+ 1)(`�m)!

(m+ `)!

P

m

`+1

⇥(

h

� 2 +

2e

i�

(m� 1)� 2e

�i�

(m+ 1)

e+O �

e

2�

i

x

3/2

+

h

1 +

e

�i�

(4� 2m) + 2e

i�

(m+ 2)

e+O �

e

2�

i

x

5/2+O �

x

7/2�

)

x

3/2

R

±`mn

(�) = C

±`mn

ˆ

R

±`mn

(�)

p = 10

e = 0.5

'

,⌦

r

E ,Jr

p

(�), r

p

(�), t

p

(�), �'(�)

������������ ������ � � � � � � � � � ��

π�

π �π�

�π

-�

-�

-�

p = 10

6

e = 0.01

x = 10

�6

p = 10

6

e = 0.2

x = 0.96⇥ 10

�6

log10

p

22,+tr

(�) residuals

log10

p

22,�tr

(�) residuals

¯

G

`m

(�) = 8

p⇡

µ

M

(`�m+ 1)

(`� 1)`(`+ 1)(`+ 2)

s

(2`+ 1)(`�m)!

(m+ `)!

P

m

`+1

⇥(

h

� 2 +

2e

i�

(m� 1)� 2e

�i�

(m+ 1)

e+O �

e

2�

i

x

3/2

+

h

1 +

e

�i�

(4� 2m) + 2e

i�

(m+ 2)

e+O �

e

2�

i

x

5/2+O �

x

7/2�

)

x

3/2

R

±`mn

(�) = C

±`mn

ˆ

R

±`mn

(�)

p = 10

e = 0.5

π�

π�π�

�π

-�

-�

-�

-�

-�

p = 10

6

e = 0.01

x = 10

�6

p = 10

6

e = 0.2

x = 0.96⇥ 10

�6

log10

p

22,+tr

(�) residuals

log10

p

22,�tr

(�) residuals

¯

G

`m

(�) = 8

p⇡

µ

M

(`�m+ 1)

(`� 1)`(`+ 1)(`+ 2)

s

(2`+ 1)(`�m)!

(m+ `)!

P

m

`+1

⇥(

h

� 2 +

2e

i�

(m� 1)� 2e

�i�

(m+ 1)

e+O �

e

2�

i

x

3/2

+

h

1 +

e

�i�

(4� 2m) + 2e

i�

(m+ 2)

e+O �

e

2�

i

x

5/2+O �

x

7/2�

)

x

3/2

R

±`mn

(�) = C

±`mn

ˆ

R

±`mn

(�)

p = 10

e = 0.5

e

9

e

5

p = 10

6

e = 0.01

x = 10

�6

p = 10

6

e = 0.2

x = 0.96⇥ 10

�6

log10

p

22,+tr

(�) residuals

log10

p

22,�tr

(�) residuals

¯

G

`m

(�) = 8

p⇡

µ

M

(`�m+ 1)

(`� 1)`(`+ 1)(`+ 2)

s

(2`+ 1)(`�m)!

(m+ `)!

P

m

`+1

⇥(

h

� 2 +

2e

i�

(m� 1)� 2e

�i�

(m+ 1)

e+O �

e

2�

i

x

3/2

+

h

1 +

e

�i�

(4� 2m) + 2e

i�

(m+ 2)

e+O �

e

2�

i

x

5/2+O �

x

7/2�

)

x

3/2

R

±`mn

(�) = C

±`mn

ˆ

R

±`mn

(�)

e

9

e

5

p = 10

6

e = 0.01

x = 10

�6

p = 10

6

e = 0.2

x = 0.96⇥ 10

�6

log10

p

22,+tr

(�) residuals

log10

p

22,�tr

(�) residuals

¯

G

`m

(�) = 8

p⇡

µ

M

(`�m+ 1)

(`� 1)`(`+ 1)(`+ 2)

s

(2`+ 1)(`�m)!

(m+ `)!

P

m

`+1

⇥(

h

� 2 +

2e

i�

(m� 1)� 2e

�i�

(m+ 1)

e+O �

e

2�

i

x

3/2

+

h

1 +

e

�i�

(4� 2m) + 2e

i�

(m+ 2)

e+O �

e

2�

i

x

5/2+O �

x

7/2�

)

x

3/2

R

±`mn

(�) = C

±`mn

ˆ

R

±`mn

(�)

Page 27: Finding self-force quantities in a post-Newtonian expansion

For small values of 𝑒 the convergence continues

π�

π�π�

�π

-��

-��

-�

������������ ������ � � � � � � � � � ��

π�

π �π�

�π

-��

-��

-��

-�

p = 10

6

e = 0.01

x = 10

�6

p = 10

6

e = 0.2

x = 0.96⇥ 10

�6

log10

p

22,+tr

residuals

log10

p

22,�tr

residuals

¯

G

`m

(�) = 8

p⇡

µ

M

(`�m+ 1)

(`� 1)`(`+ 1)(`+ 2)

s

(2`+ 1)(`�m)!

(m+ `)!

P

m

`+1

⇥(

h

� 2 +

2e

i�

(m� 1)� 2e

�i�

(m+ 1)

e+O �

e

2�

i

x

3/2

+

h

1 +

e

�i�

(4� 2m) + 2e

i�

(m+ 2)

e+O �

e

2�

i

x

5/2+O �

x

7/2�

)

x

3/2

R

±`mn

(�) = C

±`mn

ˆ

R

±`mn

(�)

p = 10

e = 0.5

p = 10

6

e = 0.01

x = 10

�6

p = 10

6

e = 0.2

x = 0.96⇥ 10

�6

log10

p

22,+tr

(�) residuals

log10

p

22,�tr

(�) residuals

¯

G

`m

(�) = 8

p⇡

µ

M

(`�m+ 1)

(`� 1)`(`+ 1)(`+ 2)

s

(2`+ 1)(`�m)!

(m+ `)!

P

m

`+1

⇥(

h

� 2 +

2e

i�

(m� 1)� 2e

�i�

(m+ 1)

e+O �

e

2�

i

x

3/2

+

h

1 +

e

�i�

(4� 2m) + 2e

i�

(m+ 2)

e+O �

e

2�

i

x

5/2+O �

x

7/2�

)

x

3/2

R

±`mn

(�) = C

±`mn

ˆ

R

±`mn

(�)

p = 10

e = 0.5

p = 10

6

e = 0.01

x = 10

�6

p = 10

6

e = 0.2

x = 0.96⇥ 10

�6

log10

p

22,+tr

(�) residuals

log10

p

22,�tr

(�) residuals

¯

G

`m

(�) = 8

p⇡

µ

M

(`�m+ 1)

(`� 1)`(`+ 1)(`+ 2)

s

(2`+ 1)(`�m)!

(m+ `)!

P

m

`+1

⇥(

h

� 2 +

2e

i�

(m� 1)� 2e

�i�

(m+ 1)

e+O �

e

2�

i

x

3/2

+

h

1 +

e

�i�

(4� 2m) + 2e

i�

(m+ 2)

e+O �

e

2�

i

x

5/2+O �

x

7/2�

)

x

3/2

R

±`mn

(�) = C

±`mn

ˆ

R

±`mn

(�)

p = 10

e = 0.5

Page 28: Finding self-force quantities in a post-Newtonian expansion

Outline

Orbits

Homogeneous solutions

Particular solutions

Black Hole Perturbation Theory

Formalism

Previous work

Analytic PN solutions

Previous work

Numeric PN solutions

Page 29: Finding self-force quantities in a post-Newtonian expansion

Here are specific numerical highlights

Mano, Suzuki & Takasugi

1996 1997

Mano & Takasugi

2014

Shah

Kerr fluxes

Shah, Friedman, & Whiting

Redshift invariant

Page 30: Finding self-force quantities in a post-Newtonian expansion

Outline

Orbits

Homogeneous solutions

Particular solutions

Black Hole Perturbation Theory

Formalism

Previous work

Analytic PN solutions

Previous work

Numeric PN solutions

Page 31: Finding self-force quantities in a post-Newtonian expansion

Our new code can solve the first order field equations

to hundreds of digits

Mathematica for arbitrary precision

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10

�100

dE

dt

lm

=

X

n

!

2

64⇡

(l + 2)!

(l � 2)!

|C+lmn

|2

L4 =

1

(1� e

2)

15/2

�323105549467

3178375200

+

232597

4410

�E � 1369

126

2+

39931

294

log(2)� 47385

1568

log(3)

+

�128412398137

23543520

+

4923511

2940

�E � 104549

252

2 � 343177

252

log(2) +

55105839

15680

log(3)

e

2

+

�981480754818517

25427001600

+

142278179

17640

�E � 1113487

504

2+

762077713

5880

log(2)� 2595297591

71680

log(3)

� 15869140625

903168

log(5)

e

4

+

�874590390287699

12713500800

+

318425291

35280

�E � 881501

336

2 � 90762985321

63504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

16257024

log(5)

e

6

+ d8e8+ d10e

10+ d12e

12+ d14e

14+ d16e

16+ d18e

18+ d20e

20+ d22e

22+ d24e

24+ d26e

26

+ d28e28

+ d30e30

+ d32e32

+ d34e34

+ d36e36

+ d38e38

+ d40e40

+ · · ·�

L7/2 = �16285⇡

504

1

(1� e

2)

7

1 + a2e2+ a4e

4+ a6e

6+ · · · �

dE

dt

=

dE

dt

3PN

+

dE

dt

N

L7/2x7/2

+ x

4⇣

L4 + log(x)L4L

+ x

9/2⇣

L9/2 + log(x)L9/2L

+ x

5⇣

L5 + log(x)L5L

+ x

11/2⇣

L11/2 + log(x)L11/2L

+ x

6⇣

L6 + log(x)L6L + log

2(x)L6L2

+ x

13/2⇣

L13/2 + log(x)L13/2L

+ x

7⇣

L7 + log(x)L7L + log

2(x)L7L2

p = 1000

p = 10

20

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10

�100

dE

dt

lm

=

X

n

!

2

64⇡

(l + 2)!

(l � 2)!

|C+lmn

|2

L4 =

1

(1� e

2)

15/2

�323105549467

3178375200

+

232597

4410

�E � 1369

126

2+

39931

294

log(2)� 47385

1568

log(3)

+

�128412398137

23543520

+

4923511

2940

�E � 104549

252

2 � 343177

252

log(2) +

55105839

15680

log(3)

e

2

+

�981480754818517

25427001600

+

142278179

17640

�E � 1113487

504

2+

762077713

5880

log(2)� 2595297591

71680

log(3)

� 15869140625

903168

log(5)

e

4

+

�874590390287699

12713500800

+

318425291

35280

�E � 881501

336

2 � 90762985321

63504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

16257024

log(5)

e

6

+ d8e8+ d10e

10+ d12e

12+ d14e

14+ d16e

16+ d18e

18+ d20e

20+ d22e

22+ d24e

24+ d26e

26

+ d28e28

+ d30e30

+ d32e32

+ d34e34

+ d36e36

+ d38e38

+ d40e40

+ · · ·�

L7/2 = �16285⇡

504

1

(1� e

2)

7

1 + a2e2+ a4e

4+ a6e

6+ · · · �

dE

dt

=

dE

dt

3PN

+

dE

dt

N

L7/2x7/2

+ x

4⇣

L4 + log(x)L4L

+ x

9/2⇣

L9/2 + log(x)L9/2L

+ x

5⇣

L5 + log(x)L5L

+ x

11/2⇣

L11/2 + log(x)L11/2L

+ x

6⇣

L6 + log(x)L6L + log

2(x)L6L2

+ x

13/2⇣

L13/2 + log(x)L13/2L

+ x

7⇣

L7 + log(x)L7L + log

2(x)L7L2

p = 1000

p = 10

20

e = 0.1

(1)

"d

2

dr

2⇤+ !

2 � V

`

(r)

#R

`mn

(r) = Z

`mn

(r)

! ⌘ !

mn

= m⌦

'

+ n⌦

r

`m

(t, r) =

1X

n=�1R

`mn

(r) e

�i!t

S

`m

(t, r) =

1X

n=�1Z

`mn

(r) e

�i!t

Page 32: Finding self-force quantities in a post-Newtonian expansion

Energy flux from an eccentric binary is known to 3PN

8

Once the C

±lmn

are determined, the energy fluxes at infinity can be calculated using

dE

dt

=X

lmn

!

2

64⇡

(l + 2)!

(l � 2)!|C+

lmn

|2, (3.26)

given our initial unit normalization of the modes X±lmn

.

IV. CONFIRMING ECCENTRIC ORBIT FLUXES THROUGH 3PN RELATIVE ORDER

The average energy and angular momentum fluxes from an eccentric binary are known to 3PN relative order[49–51] (see also the review by Blanchet [20]). The expressions are given in terms of three parameters; e.g., the

gauge-invariant post-Newtonian compactness parameter x ⌘ (M⌦'

)2/3, the eccentricity, and the symmetric massratio ⌫ = m

1

m

2

/(m1

+m

2

)2 ' µ/M . In this paper we ignore contributions that are higher order in the mass ratiothan O(⌫2), as these would require a second-order GSF calculation to reach. The energy flux is composed of a set ofeccentricity-dependent coe�cients. Its form through 3PN relative order is

dE

dt

3PN

=32

5

µ

M

2

x

5

I0

+ x I1

+ x

3/2 K3/2

+ x

2 I2

+ x

5/2 K5/2

+ x

3 I3

+ x

3 K3

. (4.1)

The In

are instantaneous flux functions [of eccentricity and (potentially) log(x)] that have known closed-form ex-pressions (summarized below). The K

n

coe�cients are hereditary, or tail, contributions (without apparently closedforms). In a subsequent section, we derive new expansions for these hereditary terms, which are useful at minimumfor BHP/PN comparisons.

A. Known instantaneous energy flux terms

For later reference and use, we list here the instantaneous energy flux functions, expressed in modified harmonic(MH) gauge [20, 49, 51] and in terms of e

t

, a particular definition of eccentricity (“time eccentricity”) used in thequasi-Keplerian (QK) representation [52] of the orbit (see also [20, 49–51, 53–55])

I0

=1

(1� e

2

t

)7/2

1 +73

24e

2

t

+37

96e

4

t

, (4.2)

I1

=1

(1� e

2

t

)9/2

�1247

336+

10475

672e

2

t

+10043

384e

4

t

+2179

1792e

6

t

, (4.3)

I2

=1

(1� e

2

t

)11/2

�203471

9072� 3807197

18144e

2

t

� 268447

24192e

4

t

+1307105

16128e

6

t

+86567

64512e

8

t

+1

(1� e

2

t

)5

35

2+

6425

48e

2

t

+5065

64e

4

t

+185

96e

6

t

,

(4.4)

I3

=1

(1� e

2

t

)13/2

2193295679

9979200+

20506331429

19958400e

2

t

� 3611354071

13305600e

4

t

+4786812253

26611200e

6

t

+21505140101

141926400e

8

t

� 8977637

11354112e

10

t

+1

(1� e

2

t

)6

�14047483

151200+

36863231

100800e

2

t

+759524951

403200e

4

t

+1399661203

2419200e

6

t

+185

48e

8

t

+1712

105log

"

x

x

0

1 +p

1� e

2

t

2(1� e

2

t

)

#

F (et

),

(4.5)

where the function F (et

) in Eqn. (4.5) has the following closed-form [49]

F (et

) =1

(1� e

2

t

)13/2

1 +85

6e

2

t

+5171

192e

4

t

+1751

192e

6

t

+297

1024e

8

t

. (4.6)

Peters & Mathews, 1963

8

Once the C

±lmn

are determined, the energy fluxes at infinity can be calculated using

dE

dt

=X

lmn

!

2

64⇡

(l + 2)!

(l � 2)!|C+

lmn

|2, (3.26)

given our initial unit normalization of the modes X±lmn

.

IV. CONFIRMING ECCENTRIC ORBIT FLUXES THROUGH 3PN RELATIVE ORDER

The average energy and angular momentum fluxes from an eccentric binary are known to 3PN relative order[49–51] (see also the review by Blanchet [20]). The expressions are given in terms of three parameters; e.g., the

gauge-invariant post-Newtonian compactness parameter x ⌘ (M⌦'

)2/3, the eccentricity, and the symmetric massratio ⌫ = m

1

m

2

/(m1

+m

2

)2 ' µ/M . In this paper we ignore contributions that are higher order in the mass ratiothan O(⌫2), as these would require a second-order GSF calculation to reach. The energy flux is composed of a set ofeccentricity-dependent coe�cients. Its form through 3PN relative order is

dE

dt

3PN

=32

5

µ

M

2

x

5

I0

+ x I1

+ x

3/2 K3/2

+ x

2 I2

+ x

5/2 K5/2

+ x

3 I3

+ x

3 K3

. (4.1)

The In

are instantaneous flux functions [of eccentricity and (potentially) log(x)] that have known closed-form ex-pressions (summarized below). The K

n

coe�cients are hereditary, or tail, contributions (without apparently closedforms). In a subsequent section, we derive new expansions for these hereditary terms, which are useful at minimumfor BHP/PN comparisons.

A. Known instantaneous energy flux terms

For later reference and use, we list here the instantaneous energy flux functions, expressed in modified harmonic(MH) gauge [20, 49, 51] and in terms of e

t

, a particular definition of eccentricity (“time eccentricity”) used in thequasi-Keplerian (QK) representation [52] of the orbit (see also [20, 49–51, 53–55])

I0

=1

(1� e

2

t

)7/2

1 +73

24e

2

t

+37

96e

4

t

, (4.2)

I1

=1

(1� e

2

t

)9/2

�1247

336+

10475

672e

2

t

+10043

384e

4

t

+2179

1792e

6

t

, (4.3)

I2

=1

(1� e

2

t

)11/2

�203471

9072� 3807197

18144e

2

t

� 268447

24192e

4

t

+1307105

16128e

6

t

+86567

64512e

8

t

+1

(1� e

2

t

)5

35

2+

6425

48e

2

t

+5065

64e

4

t

+185

96e

6

t

,

(4.4)

I3

=1

(1� e

2

t

)13/2

2193295679

9979200+

20506331429

19958400e

2

t

� 3611354071

13305600e

4

t

+4786812253

26611200e

6

t

+21505140101

141926400e

8

t

� 8977637

11354112e

10

t

+1

(1� e

2

t

)6

�14047483

151200+

36863231

100800e

2

t

+759524951

403200e

4

t

+1399661203

2419200e

6

t

+185

48e

8

t

+1712

105log

"

x

x

0

1 +p

1� e

2

t

2(1� e

2

t

)

#

F (et

),

(4.5)

where the function F (et

) in Eqn. (4.5) has the following closed-form [49]

F (et

) =1

(1� e

2

t

)13/2

1 +85

6e

2

t

+5171

192e

4

t

+1751

192e

6

t

+297

1024e

8

t

. (4.6)

“Enhances” flux from Hulse-Taylor pulsar (𝑒=0.62) by factor of 12

Page 33: Finding self-force quantities in a post-Newtonian expansion

We confirm previously known PN parameters through

3PN

�80

�70

�60

�50

�40

�30

�20

�10

0

10

0 0.02 0.04 0.06 0.08 0.1

log

(P

Nresiduals)

eccentricity e

Peters-Mathews Flux

hEi � Peters-Mathews = O(1PN)

hEi � 1PN = O(1.5PN)

hEi � 1.5PN = O(2PN)

hEi � 2PN = O(2.5PN)

hEi � 2.5PN = O(3PN)

hEi � 3PN = O(3.5PN)

This was hard

p = 10

20

p

µ⌫

(�),

@p

µ⌫

@t

(�),

@p

µ⌫

@r

(�)

0 ` 10

4 PN, e

10

Z 2⇡

0

dt

p

d�

d�

T

r

= 2⇡M

p

1� e

2

◆3/2

1 +

3 (1� e

2)

p

+O �

p

�2�

T

r

=

2⇡M

x

3/2+

6⇡M

(1� e

2)

px

+O �

x

1/2�

(t, r, ✓,') ! [t

p

(�), r

p

(�), ✓

p

(�),'

p

(�)]

p

tt

(t, r, ✓,') =

X

`,m

h

`m

tt

(t, r)Y

`m

(✓,')

p

tt

(�) =

X

`,m

h

`m

tt

(t

p

, r

p

)Y

`m

(✓

p

,'

p

)

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t

p

, r

p

)e

�im⌦'t⇤ · ⇥Y `m

(✓

p

, 0)e

im�'(t)e

im⌦'tp⇤

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t, r

p

)

¯

Y

`m

(✓

p

,'

p

)

¯

Y

`m

(✓

p

,'

p

) = Y

`m

(✓

p

,'

p

)e

�im⌦'t

h

`m,±t

=

f

2

@

r

r

±`m

h

`m,±r

=

r

2f

@

t

±`m

Page 34: Finding self-force quantities in a post-Newtonian expansion

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10

�100

dE

dt

lm

=

X

n

!

2

64⇡

(l + 2)!

(l � 2)!

|C+lmn

|2

L4 =

1

(1� e

2)

15/2

�323105549467

3178375200

+

232597

4410

�E � 1369

126

2+

39931

294

log(2)� 47385

1568

log(3)

+

�128412398137

23543520

+

4923511

2940

�E � 104549

252

2 � 343177

252

log(2) +

55105839

15680

log(3)

e

2

+

�981480754818517

25427001600

+

142278179

17640

�E � 1113487

504

2+

762077713

5880

log(2)� 2595297591

71680

log(3)

� 15869140625

903168

log(5)

e

4

+

�874590390287699

12713500800

+

318425291

35280

�E � 881501

336

2 � 90762985321

63504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

16257024

log(5)

e

6

+ d8e8+ d10e

10+ d12e

12+ d14e

14+ d16e

16+ d18e

18+ d20e

20+ d22e

22+ d24e

24+ d26e

26

+ d28e28

+ d30e30

+ d32e32

+ d34e34

+ d36e36

+ d38e38

+ d40e40

+ · · ·�

L7/2 = �16285⇡

504

1

(1� e

2)

7

1 + a2e2+ a4e

4+ a6e

6+ · · · �

dE

dt

=

dE

dt

3PN

+

32

5

µ

M

⌘2x

5

x

7/2L7/2 + x

4⇣

L4 + log(x)L4L

+ x

9/2⇣

L9/2 + log(x)L9/2L

+ x

5⇣

L5 + log(x)L5L

+ x

11/2⇣

L11/2 + log(x)L11/2L

+ x

6⇣

L6 + log(x)L6L + log

2(x)L6L2

+ x

13/2⇣

L13/2 + log(x)L13/2L

+ x

7⇣

L7 + log(x)L7L + log

2(x)L7L2

dE

dt

=

dE

dt

3PN

+

32

5

µ

M

⌘2x

5

x

7/2L7/2 + x

4⇣

L4 + log(x)L4L

+ x

9/2⇣

L9/2 + log(x)L9/2L

+ x

5⇣

L5 + log(x)L5L

+ x

11/2⇣

L11/2 + log(x)L11/2L

+ x

6⇣

L6 + log(x)L6L + log

2(x)L6L2

+ x

13/2⇣

L13/2 + log(x)L13/2L

+ x

7⇣

L7 + log(x)L7L + log

2(x)L7L2

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10

�100

dE

dt

lm

=

X

n

!

2

64⇡

(l + 2)!

(l � 2)!

|C+lmn

|2

L4 =

1

(1� e

2)

15/2

�323105549467

3178375200

+

232597

4410

�E � 1369

126

2+

39931

294

log(2)� 47385

1568

log(3)

+

�128412398137

23543520

+

4923511

2940

�E � 104549

252

2 � 343177

252

log(2) +

55105839

15680

log(3)

e

2

+

�981480754818517

25427001600

+

142278179

17640

�E � 1113487

504

2+

762077713

5880

log(2)� 2595297591

71680

log(3)

� 15869140625

903168

log(5)

e

4

+

�874590390287699

12713500800

+

318425291

35280

�E � 881501

336

2 � 90762985321

63504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

16257024

log(5)

e

6

+ d8e8+ d10e

10+ d12e

12+ d14e

14+ d16e

16+ d18e

18+ d20e

20+ d22e

22+ d24e

24+ d26e

26

+ d28e28

+ d30e30

+ d32e32

+ d34e34

+ d36e36

+ d38e38

+ d40e40

+ · · ·�

L7/2 = �16285⇡

504

1

(1� e

2)

7

1 + a2e2+ a4e

4+ a6e

6+ · · · �

dE

dt

=

dE

dt

3PN

+

32

5

µ

M

⌘2x

5

x

7/2L7/2 + x

4⇣

L4 + log(x)L4L

+ x

9/2⇣

L9/2 + log(x)L9/2L

+ x

5⇣

L5 + log(x)L5L

+ x

11/2⇣

L11/2 + log(x)L11/2L

+ x

6⇣

L6 + log(x)L6L + log

2(x)L6L2

+ x

13/2⇣

L13/2 + log(x)L13/2L

+ x

7⇣

L7 + log(x)L7L + log

2(x)L7L2

dE

dt

=

dE

dt

3PN

+

32

5

µ

M

⌘2x

5

x

7/2L7/2 + x

4⇣

L4 + log(x)L4L

+ x

9/2⇣

L9/2 + log(x)L9/2L

+ x

5⇣

L5 + log(x)L5L

+ x

11/2⇣

L11/2 + log(x)L11/2L

+ x

6⇣

L6 + log(x)L6L + log

2(x)L6L2

+ x

13/2⇣

L13/2 + log(x)L13/2L

+ x

7⇣

L7 + log(x)L7L + log

2(x)L7L2

We found new parameters through 7PN

L7/2 = �16285⇡

504

1

(1� e

2)

7

1 + a2e2+ a4e

4+ a6e

6+ · · · �

dE

dt

=

dE

dt

3PN

+

dE

dt

N

L7/2x7/2

+ x

4⇣

L4 + log(x)L4L

+ x

9/2⇣

L9/2 + log(x)L9/2L

+ x

5⇣

L5 + log(x)L5L

+ x

11/2⇣

L11/2 + log(x)L11/2L

+ x

6⇣

L6 + log(x)L6L + log

2(x)L6L2

+ x

13/2⇣

L13/2 + log(x)L13/2L

+ x

7⇣

L7 + log(x)L7L + log

2(x)L7L2

p = 1000

p = 10

20

p

µ⌫

(�),

@p

µ⌫

@t

(�),

@p

µ⌫

@r

(�)

0 ` 10

4 PN, e

10

Z 2⇡

0

dt

p

d�

d�

T

r

= 2⇡M

p

1� e

2

◆3/2"

1 +

3

1� e

2�

p

+O �

p

�2�

#

T

r

=

2⇡M

x

3/2+

6⇡M

(1� e

2)

px

+O⇣

x

1/2⌘

(t, r, ✓,') ! [t

p

(�), r

p

(�), ✓

p

(�),'

p

(�)]

p

tt

(t, r, ✓,') =

X

`,m

h

`m

tt

(t, r)Y

`m

(✓,')

p

tt

(�) =

X

`,m

h

`m

tt

(t

p

, r

p

)Y

`m

(✓

p

,'

p

)

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t

p

, r

p

)e

�im⌦'t

⇤ ·h

Y

`m

(✓

p

, 0)e

im�'(t)e

im⌦'tp

i

Circular orbit limit Guessing the correct singular behavior extends range of validity

Page 35: Finding self-force quantities in a post-Newtonian expansion

We spanned the two dimensional space of orbits

𝑒 = 0 𝑒 = 0.1

𝑝 = 1010

𝑝 = 1035

1683 orbits

33 eccentricities

51 radii

Page 36: Finding self-force quantities in a post-Newtonian expansion

Each orbit involves solving for hundreds of modes

�300

�250

�200

�150

�100

-100 -50 0 50 100

log

D E1 `m

n

E

Harmonic number n

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10

�100

dE

dt

lm

=

X

n

!

2

64⇡

(l + 2)!

(l � 2)!

|C+lmn

|2

L4 =

1

(1� e

2)

15/2

�323105549467

3178375200

+

232597

4410

�E � 1369

126

2+

39931

294

log(2)� 47385

1568

log(3)

+

�128412398137

23543520

+

4923511

2940

�E � 104549

252

2 � 343177

252

log(2) +

55105839

15680

log(3)

e

2

+

�981480754818517

25427001600

+

142278179

17640

�E � 1113487

504

2+

762077713

5880

log(2)� 2595297591

71680

log(3)

� 15869140625

903168

log(5)

e

4

+

�874590390287699

12713500800

+

318425291

35280

�E � 881501

336

2 � 90762985321

63504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

16257024

log(5)

e

6

+ d8e8+ d10e

10+ d12e

12+ d14e

14+ d16e

16+ d18e

18+ d20e

20+ d22e

22+ d24e

24+ d26e

26

+ d28e28

+ d30e30

+ d32e32

+ d34e34

+ d36e36

+ d38e38

+ d40e40

+ · · ·�

L7/2 = �16285⇡

504

1

(1� e

2)

7

1 + a2e2+ a4e

4+ a6e

6+ · · · �

dE

dt

=

dE

dt

3PN

+

dE

dt

N

L7/2x7/2

+ x

4⇣

L4 + log(x)L4L

+ x

9/2⇣

L9/2 + log(x)L9/2L

+ x

5⇣

L5 + log(x)L5L

+ x

11/2⇣

L11/2 + log(x)L11/2L

+ x

6⇣

L6 + log(x)L6L + log

2(x)L6L2

+ x

13/2⇣

L13/2 + log(x)L13/2L

+ x

7⇣

L7 + log(x)L7L + log

2(x)L7L2

p = 1000

p = 10

20

e = 0.1

(1)

Page 37: Finding self-force quantities in a post-Newtonian expansion

PN parameters involve sums of trancendentals

L4 =

1

(1� e

2)

15/2

�323105549467

3178375200

+

232597

4410

�E � 1369

126

2+

39931

294

log(2)� 47385

1568

log(3)

+

�128412398137

23543520

+

4923511

2940

�E � 104549

252

2 � 343177

252

log(2) +

55105839

15680

log(3)

e

2

+

�981480754818517

25427001600

+

142278179

17640

�E � 1113487

504

2+

762077713

5880

log(2)� 2595297591

71680

log(3)

� 15869140625

903168

log(5)

e

4

+

�874590390287699

12713500800

+

318425291

35280

�E � 881501

336

2 � 90762985321

63504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

16257024

log(5)

e

6

+ d8e8+ d10e

10+ d12e

12+ d14e

14+ d16e

16+ d18e

18+ d20e

20+ d22e

22+ d24e

24+ d26e

26

+ d28e28

+ d30e30

+ d32e32

+ d34e34

+ d36e36

+ d38e38

+ d40e40

+ · · ·�

L7/2 = �16285⇡

504

1

(1� e

2)

7

1 + a2e2+ a4e

4+ a6e

6+ · · · �

dE

dt

=

dE

dt

3PN

+

dE

dt

N

L7/2x7/2

+ x

4⇣

L4 + log(x)L4L

+ x

9/2⇣

L9/2 + log(x)L9/2L

+ x

5⇣

L5 + log(x)L5L

+ x

11/2⇣

L11/2 + log(x)L11/2L

+ x

6⇣

L6 + log(x)L6L + log

2(x)L6L2

+ x

13/2⇣

L13/2 + log(x)L13/2L

+ x

7⇣

L7 + log(x)L7L + log

2(x)L7L2

p = 1000

p = 10

20

p

µ⌫

(�),

@p

µ⌫

@t

(�),

@p

µ⌫

@r

(�)

0 ` 10

4 PN, e

10

Thanks to Nathan Johnson-McDaniel

L4 =

1

(1� e

2)

15/2

�323105549467

3178375200

+

232597

4410

�E � 1369

126

2+

39931

294

log(2)� 47385

1568

log(3)

+

�128412398137

23543520

+

4923511

2940

�E � 104549

252

2 � 343177

252

log(2) +

55105839

15680

log(3)

e

2

+

�981480754818517

25427001600

+

142278179

17640

�E � 1113487

504

2+

762077713

5880

log(2)� 2595297591

71680

log(3)

� 15869140625

903168

log(5)

e

4

+

�874590390287699

12713500800

+

318425291

35280

�E � 881501

336

2 � 90762985321

63504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

16257024

log(5)

e

6

+ d8e8+ d10e

10+ d12e

12+ d14e

14+ d16e

16+ d18e

18+ d20e

20+ d22e

22+ d24e

24+ d26e

26

+ d28e28

+ d30e30

+ d32e32

+ d34e34

+ d36e36

+ d38e38

+ d40e40

+ · · ·�

L4 =

1

(1� e

2)

15/2

�323105549467

3178375200

+

232597

4410

�E � 1369

126

2+

39931

294

log(2)� 47385

1568

log(3)

+

�128412398137

23543520

+

4923511

2940

�E � 104549

252

2 � 343177

252

log(2) +

55105839

15680

log(3)

e

2

+

�981480754818517

25427001600

+

142278179

17640

�E � 1113487

504

2+

762077713

5880

log(2)� 2595297591

71680

log(3)

� 15869140625

903168

log(5)

e

4

+

�874590390287699

12713500800

+

318425291

35280

�E � 881501

336

2 � 90762985321

63504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

16257024

log(5)

e

6

+ d8e8+ d10e

10+ d12e

12+ d14e

14+ d16e

16+ d18e

18+ d20e

20+ d22e

22+ d24e

24+ d26e

26

+ d28e28

+ d30e30

+ d32e32

+ d34e34

+ d36e36

+ d38e38

+ d40e40

+ · · ·�

L7/2 = �16285⇡

504

1

(1� e

2)

7

1 + a2e2+ a4e

4+ a6e

6+ · · · �

dE

dt

=

dE

dt

3PN

+

dE

dt

N

L7/2x7/2

+ x

4⇣

L4 + log(x)L4L

+ x

9/2⇣

L9/2 + log(x)L9/2L

+ x

5⇣

L5 + log(x)L5L

+ x

11/2⇣

L11/2 + log(x)L11/2L

+ x

6⇣

L6 + log(x)L6L + log

2(x)L6L2

+ x

13/2⇣

L13/2 + log(x)L13/2L

+ x

7⇣

L7 + log(x)L7L + log

2(x)L7L2

p = 1000

p = 10

20

p

µ⌫

(�),

@p

µ⌫

@t

(�),

@p

µ⌫

@r

(�)

0 ` 10

4 PN, e

10

Found with PSLQ algorithm

Page 38: Finding self-force quantities in a post-Newtonian expansion

New PN parameters are valid well outside the regime

where they were found

Higher order PN residuals

0.60

0.90

�1.3

�1.6

�2.5

�2.2

�4

�4.3

�4.2

�4.5

�5.5

�5.2

�7.1

�7.4

�7.2

�7.5

�8.5

�8.2

�10.6

�10.9

�10.2

�10.5

0 0.2 0.4 0.6

Eccentricity e

0 0.2 0.4 0.6

�11.5

�11.2

Eccentricity e

I0 + O(x)

hEi/hEiC

xI1 + O(x3/2)

0

BBBBBBBBBBBBB@

hEi � I0

1

CCCCCCCCCCCCCA

/hEiC

x

3/2K3/2 + O(x2)

0

BBBBBBBBBBBBB@

hEi � hEi1PN

1

CCCCCCCCCCCCCA

/hEiC

x

2I2 + O(x5/2)

0

BBBBBBBBBBBBB@

hEi � hEi1.5PN

1

CCCCCCCCCCCCCA

/hEiC

x

5/2I5/2 + O(x3)

0

BBBBBBBBBBBBB@

hEi � hEi2PN

1

CCCCCCCCCCCCCA

/hEiC

x

3I3 + O(x7/2)

0

BBBBBBBBBBBBB@

hEi � hEi2.5PN

1

CCCCCCCCCCCCCA

/hEiC

x

7/2L7/2 + O(x4)

0

BBBBBBBBBBBBB@

hEi � hEi3PN

1

CCCCCCCCCCCCCA

/hEiC

x

4L4 + O(x9/2)

0

BBBBBBBBBBBBB@

hEi � hEi3.5PN

1

CCCCCCCCCCCCCA

/hEiC

x

9/2L9/2 + O(x5)

0

BBBBBBBBBBBBB@

hEi � hEi4PN

1

CCCCCCCCCCCCCA

/hEiC

x

5L5 + O(x11/2)

0

BBBBBBBBBBBBB@

hEi � hEi4.5PN

1

CCCCCCCCCCCCCA

/hEiC

x

11/2L11/2 + O(x6)

0

BBBBBBBBBBBBB@

hEi � hEi5PN

1

CCCCCCCCCCCCCA

/hEiC

x

6L6

0

BBBBBBBBBBBBB@

hEi � hEi5.5PN

1

CCCCCCCCCCCCCA

/hEiC

p = 1000

p = 10

20

p

µ⌫

(�),

@p

µ⌫

@t

(�),

@p

µ⌫

@r

(�)

0 ` 10

4 PN, e

10

Z 2⇡

0

dt

p

d�

d�

T

r

= 2⇡M

p

1� e

2

◆3/2

1 +

3 (1� e

2)

p

+O �

p

�2�

T

r

=

2⇡M

x

3/2+

6⇡M

(1� e

2)

px

+O �

x

1/2�

(t, r, ✓,') ! [t

p

(�), r

p

(�), ✓

p

(�),'

p

(�)]

p

tt

(t, r, ✓,') =

X

`,m

h

`m

tt

(t, r)Y

`m

(✓,')

p

tt

(�) =

X

`,m

h

`m

tt

(t

p

, r

p

)Y

`m

(✓

p

,'

p

)

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t

p

, r

p

)e

�im⌦'t⇤ · ⇥Y `m

(✓

p

, 0)e

im�'(t)e

im⌦'tp⇤

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t, r

p

)

¯

Y

`m

(✓

p

,'

p

)

¯

Y

`m

(✓

p

,'

p

) = Y

`m

(✓

p

,'

p

)e

�im⌦'t

Page 39: Finding self-force quantities in a post-Newtonian expansion

New PN parameters have been confirmed with a

separate code

Flux residuals after subtracting new PN parameters

Code by Thomas Osburn

Page 40: Finding self-force quantities in a post-Newtonian expansion

These are the main points

π�

π�π�

�π

-�

-�

-�

-�

-�

p

µ⌫

,

@p

µ⌫

@t

,

@p

µ⌫

@t

0 ` 10

4 PN, e

10

Z 2⇡

0

dt

p

d�

d�

T

r

= 2⇡M

p

1� e

2

◆3/2

1 +

3 (1� e

2)

p

+O �

p

�2�

T

r

=

2⇡M

x

3/2+

6⇡M

(1� e

2)

px

+O �

x

1/2�

(t, r, ✓,') ! [t

p

(�), r

p

(�), ✓

p

(�),'

p

(�)]

p

tt

(t, r, ✓,') =

X

`,m

h

`m

tt

(t, r)Y

`m

(✓,')

p

tt

(�) =

X

`,m

h

`m

tt

(t

p

, r

p

)Y

`m

(✓

p

,'

p

)

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t

p

, r

p

)e

�im⌦'t⇤ · ⇥Y `m

(✓

p

, 0)e

im�'(t)e

im⌦'tp⇤

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t, r

p

)

¯

Y

`m

(✓

p

,'

p

)

¯

Y

`m

(✓

p

,'

p

) = Y

`m

(✓

p

,'

p

)e

�im⌦'t

h

`m,±t

=

f

2

@

r

r

±`m

h

`m,±r

=

r

2f

@

t

±`m

p

µ⌫

,

@p

µ⌫

@t

,

@p

µ⌫

@t

0 ` 10

4 PN, e

10

Z 2⇡

0

dt

p

d�

d�

T

r

= 2⇡M

p

1� e

2

◆3/2

1 +

3 (1� e

2)

p

+O �

p

�2�

T

r

=

2⇡M

x

3/2+

6⇡M

(1� e

2)

px

+O �

x

1/2�

(t, r, ✓,') ! [t

p

(�), r

p

(�), ✓

p

(�),'

p

(�)]

p

tt

(t, r, ✓,') =

X

`,m

h

`m

tt

(t, r)Y

`m

(✓,')

p

tt

(�) =

X

`,m

h

`m

tt

(t

p

, r

p

)Y

`m

(✓

p

,'

p

)

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t

p

, r

p

)e

�im⌦'t⇤ · ⇥Y `m

(✓

p

, 0)e

im�'(t)e

im⌦'tp⇤

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t, r

p

)

¯

Y

`m

(✓

p

,'

p

)

¯

Y

`m

(✓

p

,'

p

) = Y

`m

(✓

p

,'

p

)e

�im⌦'t

h

`m,±t

=

f

2

@

r

r

±`m

h

`m,±r

=

r

2f

@

t

±`m

p

µ⌫

(�),

@p

µ⌫

@t

(�),

@p

µ⌫

@r

(�)

0 ` 10

4 PN, e

10

Z 2⇡

0

dt

p

d�

d�

T

r

= 2⇡M

p

1� e

2

◆3/2

1 +

3 (1� e

2)

p

+O �

p

�2�

T

r

=

2⇡M

x

3/2+

6⇡M

(1� e

2)

px

+O �

x

1/2�

(t, r, ✓,') ! [t

p

(�), r

p

(�), ✓

p

(�),'

p

(�)]

p

tt

(t, r, ✓,') =

X

`,m

h

`m

tt

(t, r)Y

`m

(✓,')

p

tt

(�) =

X

`,m

h

`m

tt

(t

p

, r

p

)Y

`m

(✓

p

,'

p

)

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t

p

, r

p

)e

�im⌦'t⇤ · ⇥Y `m

(✓

p

, 0)e

im�'(t)e

im⌦'tp⇤

p

tt

(�) =

X

`,m

¯

h

`m

tt

(t, r

p

)

¯

Y

`m

(✓

p

,'

p

)

¯

Y

`m

(✓

p

,'

p

) = Y

`m

(✓

p

,'

p

)e

�im⌦'t

h

`m,±t

=

f

2

@

r

r

±`m

h

`m,±r

=

r

2f

@

t

±`m

dE

dt

, New parameters, 7PN

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521⇥ 10

�100

dE

dt

lm

=

X

n

!

2

64⇡

(l + 2)!

(l � 2)!

|C+lmn

|2

L4 =

1

(1� e

2)

15/2

�323105549467

3178375200

+

232597

4410

�E � 1369

126

2+

39931

294

log(2)� 47385

1568

log(3)

+

�128412398137

23543520

+

4923511

2940

�E � 104549

252

2 � 343177

252

log(2) +

55105839

15680

log(3)

e

2

+

�981480754818517

25427001600

+

142278179

17640

�E � 1113487

504

2+

762077713

5880

log(2)� 2595297591

71680

log(3)

� 15869140625

903168

log(5)

e

4

+

�874590390287699

12713500800

+

318425291

35280

�E � 881501

336

2 � 90762985321

63504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

16257024

log(5)

e

6

+ d8e8+ d10e

10+ d12e

12+ d14e

14+ d16e

16+ d18e

18+ d20e

20+ d22e

22+ d24e

24+ d26e

26

+ d28e28

+ d30e30

+ d32e32

+ d34e34

+ d36e36

+ d38e38

+ d40e40

+ · · ·�

L7/2 = �16285⇡

504

1

(1� e

2)

7

1 + a2e2+ a4e

4+ a6e

6+ · · · �

dE

dt

=

dE

dt

3PN

+

32

5

µ

M

⌘2x

5

x

7/2L7/2 + x

4⇣

L4 + log(x)L4L

+ x

9/2⇣

L9/2 + log(x)L9/2L

+ x

5⇣

L5 + log(x)L5L

+ x

11/2⇣

L11/2 + log(x)L11/2L

+ x

6⇣

L6 + log(x)L6L + log

2(x)L6L2

+ x

13/2⇣

L13/2 + log(x)L13/2L

+ x

7⇣

L7 + log(x)L7L + log

2(x)L7L2

Questions?