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### Transcript of 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

-

-

-

-

-

p = 10

6

e = 0.01

x = 10

6

p = 10

6

e = 0.2

x = 0.96 106

log10

p

22,+tr

() residuals

log10

p

22,tr

() residuals

G

`m

() = 8

p

M

(`m+ 1)(` 1)`(`+ 1)(`+ 2)

s

(2`+ 1)(`m)!(m+ `)!

P

m

`+1

(

h

2 + 2ei(m 1) 2ei(m+ 1) e+O e2i

x

3/2

+

h

1 +

e

i(4 2m) + 2ei(m+ 2) e+O e2

i

x

5/2+O x7/2

)

x

3/2

R

`mn

() = C

`mn

R

`mn

()

p = 10

e = 0.5

Chris Kavanagh

Erik Forseth

Charles Evans

Institut des Hautes tudes Scientifiques - November 12, 2015

• An electron in free fall will radiate

e-

e-

e-

An electron is a non-local object

• Outline

Orbits

Homogeneous solutions

Particular solutions

Black Hole Perturbation Theory

Formalism

Analytic PN solutions

Numeric PN solutions

• Outline

Orbits

Homogeneous solutions

Particular solutions

Black Hole Perturbation Theory

Formalism

Analytic PN solutions

Numeric PN solutions

• In black hole perturbation theory, we alternate

between correcting fields and particle motion

Field equations Equations of motion

0

1

2

Order

G

= 0

dE

dt

, New parameters, 7PN

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521 10100

dE

dt

lm

=

X

n

!

2

64

(l + 2)!

(l 2)! |C+lmn

|2

L4 = 1(1 e2)15/2

3231055494673178375200

+

232597

4410

E 1369126

2+

39931

294

log(2) 473851568

log(3)

+

12841239813723543520

+

4923511

2940

E 104549252

2 343177252

log(2) +

55105839

15680

log(3)

e

2

+

98148075481851725427001600

+

142278179

17640

E 1113487504

2+

762077713

5880

log(2) 259529759171680

log(3)

15869140625903168

log(5)

e

4

+

87459039028769912713500800

+

318425291

35280

E 881501336

2 9076298532163504

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 = 16285504

1

(1 e2)7

1 + a2e2+ a4e

4+ a6e

6+

dE

dt

=

dE

dt

3PN

+

32

5

M

2x

5

x

7/2L7/2 + x4

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 + log2(x)L6L2

+ x

13/2

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

+ x

7

L7 + log(x)L7L + log2(x)L7L2

G

= 0

Du

d

= 0

dE

dt

, New parameters, 7PN

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521 10100

dE

dt

lm

=

X

n

!

2

64

(l + 2)!

(l 2)! |C+lmn

|2

L4 = 1(1 e2)15/2

3231055494673178375200

+

232597

4410

E 1369126

2+

39931

294

log(2) 473851568

log(3)

+

12841239813723543520

+

4923511

2940

E 104549252

2 343177252

log(2) +

55105839

15680

log(3)

e

2

+

98148075481851725427001600

+

142278179

17640

E 1113487504

2+

762077713

5880

log(2) 259529759171680

log(3)

15869140625903168

log(5)

e

4

+

87459039028769912713500800

+

318425291

35280

E 881501336

2 9076298532163504

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 = 16285504

1

(1 e2)7

1 + a2e2+ a4e

4+ a6e

6+

dE

dt

=

dE

dt

3PN

+

32

5

M

2x

5

x

7/2L7/2 + x4

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 + log2(x)L6L2

+ x

13/2

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

+ x

7

L7 + log(x)L7L + log2(x)L7L2

g

= g

BH

+ p

2p

+ 2R

p

= 16T

G

= 0

Du

d

= 0

dE

dt

, New parameters, 7PN

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521 10100

dE

dt

lm

=

X

n

!

2

64

(l + 2)!

(l 2)! |C+lmn

|2

L4 = 1(1 e2)15/2

3231055494673178375200

+

232597

4410

E 1369126

2+

39931

294

log(2) 473851568

log(3)

+

12841239813723543520

+

4923511

2940

E 104549252

2 343177252

log(2) +

55105839

15680

log(3)

e

2

+

98148075481851725427001600

+

142278179

17640

E 1113487504

2+

762077713

5880

log(2) 259529759171680

log(3)

15869140625903168

log(5)

e

4

+

87459039028769912713500800

+

318425291

35280

E 881501336

2 9076298532163504

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 = 16285504

1

(1 e2)7

1 + a2e2+ a4e

4+ a6e

6+

g

= g

BH

+ p

2p

+ 2R

p

= 16T

G

= 0

Du

d

= 0

dE

dt

, New parameters, 7PN

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521 10100

dE

dt

lm

=

X

n

!

2

64

(l + 2)!

(l 2)! |C+lmn

|2

L4 = 1(1 e2)15/2

3231055494673178375200

+

232597

4410

E 1369126

2+

39931

294

log(2) 473851568

log(3)

+

12841239813723543520

+

4923511

2940

E 104549252

2 343177252

log(2) +

55105839

15680

log(3)

e

2

+

98148075481851725427001600

+

142278179

17640

E 1113487504

2+

762077713

5880

log(2) 259529759171680

log(3)

15869140625903168

log(5)

e

4

+

87459039028769912713500800

+

318425291

35280

E 881501336

2 9076298532163504

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 = 16285504

1

(1 e2)7

1 + a2e2+ a4e

4+ a6e

6+

Du

d

= F

self [p ]

Du

d

= 0

g

= g

BH

+ p

2p

+ 2R

p

= 16T

G

= 0

dE

dt

, New parameters, 7PN

dE

dt

22

= 3.24778951144055583601917 . . . 50 more digits . . . 0413022224171085141009521 10100

dE

dt

lm

=

X

n

!

2

64

(l + 2)!

(l 2)! |C+lmn

|2

L4 = 1(1 e2)15/2

3231055494673178375200

+

232597

4410

E 1369126

2+

39931

294

log(2) 473851568

log(3)

+

12841239813723543520

+

4923511

2940

E 104549252

2 343177252

log(2) +

55105839

15680

log(3)

e

2

+

98148075481851725427001600

+

142278179

17640

E 1113487504

2+

762077713

5880

log(2) 259529759171680

log(3)

15869140625903168

log(5)

e

4

+

87459039028769912713500800

+

318425291

35280

E 881501336

2 9076298532163504

log(2) +

31649037093

1003520

log(3)

+

10089048828125

1625702