Finding self-force quantities in a post-Newtonian expansion

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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

    Adrian Ottewill

    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