Yi-Zen Chu Particle Astrophysics/Cosmology Seminar, ASU Wednesday, 6 October 2010

Yi-Zen ChuYi-Zen Chu

Particle Astrophysics/Cosmology Seminar, ASUParticle Astrophysics/Cosmology Seminar, ASU

Wednesday, 6 October 2010Wednesday, 6 October 2010

Don’t ShakeDon’t ShakeThat SolenoidThat Solenoid

Too Hard:Too Hard:Particle Production Particle Production

From Aharonov-BohmFrom Aharonov-Bohm

K. Jones-Smith, H. Mathur, and T. Vachaspati, Phys. Rev. D K. Jones-Smith, H. Mathur, and T. Vachaspati, Phys. Rev. D 8181:043503,2010:043503,2010Y.-Z.Chu, H. Mathur, and T. Vachaspati, Phys. Rev. D Y.-Z.Chu, H. Mathur, and T. Vachaspati, Phys. Rev. D 8282:063515,2010:063515,2010

Aharonov and Bohm Aharonov and Bohm (1959)(1959)• Quantum Mechanics:Quantum Mechanics:

• Vector potential AVector potential Aμμ is not merely is not merely computational crutch but computational crutch but indispensable for quantum indispensable for quantum dynamics of charged particles.dynamics of charged particles.

• (2010) Quantum Field Theory:(2010) Quantum Field Theory:• Spontaneous pair production of Spontaneous pair production of charged particles just by shaking a charged particles just by shaking a thin solenoid.thin solenoid.

SetupSetup

BB

xx

yy

zz

ShakeShakethethe

Solenoid!Solenoid!

ztv

,0,sin0

BB

xx

yy

zz

e+e+

e-e-

e+e+

e-e-

Pop!Pop!

ztv

,0,sin0SetupSetup

SetupSetup

ieAD

mD

FFxS

222

4QED

||||

4

1

d

ieAD

mDi

FFxS

4

1

d4QED

flux Magnetic

dd~

2 sheet world

_QED

xxFS

SSS • Effective theory of Effective theory of magnetic flux tube: magnetic flux tube: Alford and Wilczek Alford and Wilczek (1989).(1989).• Bosonic or fermionic Bosonic or fermionic quantum quantum electrodynamics (QED)electrodynamics (QED)

Bosonic QEDBosonic QED Fermionic QEDFermionic QED

I: Time dependent HI: Time dependent H• The gauge potential AThe gauge potential Aμμ around a around a moving solenoid is time-dependent.moving solenoid is time-dependent.

• Hamiltonian of QFT is explicitly Hamiltonian of QFT is explicitly time-dependent: Htime-dependent: Hii = ∫d = ∫d33x Ax AμμJJμμ..• Zero particle state (in the Zero particle state (in the Heisenberg picture) at different times Heisenberg picture) at different times not the same vector – i.e. particle not the same vector – i.e. particle creation occurs.creation occurs.

0],[)]([,0,0 ytxA x

II: Aharonov-Bohm II: Aharonov-Bohm InteractionInteraction• However thin the flux tube is, particle However thin the flux tube is, particle

production will occur – purely quantum production will occur – purely quantum process.process.• Pair production rate contains Pair production rate contains topological aspect.topological aspect.

BB',' xt xt,

xtq

xtq

ii qDxAieqqiSxtxt

][

']'[

0 ]d],[exp[',',

ABABQMQM

• QM: Amp. for paths belonging to QM: Amp. for paths belonging to different classes will differ by different classes will differ by exp[i(integer)eexp[i(integer)eФФ]]

• However thin the flux tube is, particle However thin the flux tube is, particle production will occur – purely quantum production will occur – purely quantum process.process.• Pair production rate contains Pair production rate contains topological aspect.topological aspect.

BB',' xt xt,

• Expect: Pair production rate to have Expect: Pair production rate to have periodic dependence on AB phase eperiodic dependence on AB phase eФФ..

II: Aharonov-Bohm II: Aharonov-Bohm InteractionInteraction

xtq

xtq

ii qDxAieqqiSxtxt

][

']'[

0 ]d],[exp[',',

ABABQMQM

Moving framesMoving frames• Expand scalar operator Expand scalar operator φφ and Dirac and Dirac operator operator ψψ in terms of the instantaneous in terms of the instantaneous eigenstates of the Hamiltonian of the eigenstates of the Hamiltonian of the field equations.field equations.

• i.e. Solve the mode functions for the i.e. Solve the mode functions for the stationary solenoid problem and shift stationary solenoid problem and shift them by them by ξξ, location of moving , location of moving solenoid.solenoid.

kkkk

kCH

kkk

kkkk

kCH

kkk

uEHutxtvtbtxtuta

fEHftxthtbtxtfta

),)](,[][)](,[][(

,)])(,[][)](,[][(

..

..

Moving frames Moving frames φφ results results

1 mod2

,2

8

sin

lengthunit per of productionpair of Rate

022

222

022

2220

220

0

*

ekkk

k

k

k

k

k

kkv

mkkdkdk

c

c

c

c

zz

• Rate carries periodic dependence on Rate carries periodic dependence on eeФФ• Non-relativisticNon-relativistic

Moving frames Moving frames ψψ results results

1 mod2

'

'8

sin

''

lengthunit per ee of productionpair of Rate

22'00

22'00

22

220

''00

'

00

-

e

k

k

k

kkkkm

kk

v

kkkkdkdkkdkdkk

z

zzzz

• Rate carries periodic dependence on Rate carries periodic dependence on eeФФ• Non-relativisticNon-relativistic

RelativisticRelativisticeeФФ << 1 << 1

φφ**

φφ

222

4QED

||||

4

1

d

mD

FFxS

mDi

FFxS 4

1

d4QED

sheet world

_QED

dd~

2

xxFS

SSS

Small AB phase: eSmall AB phase: eФФ << 1 << 1

φφ**

φφ

][

worldsheet

0

2

1dd]exp[

2

1~

~]20[

IIxxxipS

IIJp

eSJp

pp

eM

o Valid for any flux tube trajectory.Valid for any flux tube trajectory.


φφ**

φφ

x

xz p

pvpJppy

II

IIJp

eSJp

pp

eM

000

2

0

)]([][)2(ˆ

)solenoid Moving(

~]20[


φφ**

φφ

x

xz p

pvpJppy

II

IIJp

eSJp

pp

eM

000

2

0

)]([][)2(ˆ

)solenoid Moving(

~]20[

• Spins of eSpins of e++ee-- anti-correlated along anti-correlated along direction determined by their direction determined by their momenta and Imomenta and I+ + x Ix I--..

Sm

all

AB

ph

ase

S

mall

AB

ph

ase

re

sult

sre

sult

s ][][

)(

4

space phaseunit per length unit per

* of productionpair of Rate

'00

'02

12'

00

2'2

2

kkkkvp

J

pkk

kke

zzx

x

yy

][][

2

)()()(

4

space phaseunit per length unit per

ee of productionpair of Rate

'00

'02

12'

00

2'2'22

2

-

kkkkvp

J

pkk

kkkke

zzx

x

yyxx

eeФФ<<1: Total Power<<1: Total Power

o Similar plot for bosons.Similar plot for bosons.

vv00 = 0.001 = 0.001

vv00 = 0.1 = 0.1

vv00 = 1 = 1

eeФФ<<1, <<1, ΩΩ>>m, v>>m, v00~1, k~1, kzz=0, =0, kkxyxy=k’=k’xyxy

Cosmic String LoopsCosmic String Loopso Motivated by astrophysical Motivated by astrophysical observations of anomalous excess of observations of anomalous excess of e+e-, some recent particle physics e+e-, some recent particle physics models involve mixing of photon models involve mixing of photon U(1) vector potential with some U(1) vector potential with some “dark” sector (spontaneously “dark” sector (spontaneously broken) U(1)’ vector potential.broken) U(1)’ vector potential.o This allows emission of electrically This allows emission of electrically charged particle-antiparticle pairs charged particle-antiparticle pairs from cosmic strings via the AB from cosmic strings via the AB interaction.interaction.o Consider: Kinky and cuspy loops.Consider: Kinky and cuspy loops.

eeФФ << 1: Kinky Cosmic << 1: Kinky Cosmic String LoopString Loop

o Like solenoid, there are infinite # of harmonicsLike solenoid, there are infinite # of harmonicso Both bosonic and fermionic emission show similar Both bosonic and fermionic emission show similar linear-in-N behavior.linear-in-N behavior.o Cut-off determined by finite width of cosmic string Cut-off determined by finite width of cosmic string loop.loop.

t

RLX

])[][(2

1

Kinky loop Kinky loop configurationconfiguration

eeФФ<<1: Cuspy Cosmic String <<1: Cuspy Cosmic String LoopLoop

o Like solenoid, there are infinite # of harmonicsLike solenoid, there are infinite # of harmonicso Both bosonic and fermionic emission show Both bosonic and fermionic emission show similar constant-in-ℓ behavior.similar constant-in-ℓ behavior.o Cut-off determined by finite width of cosmic Cut-off determined by finite width of cosmic string loop.string loop.

t

RLX

])[][(2

1

Cuspy loop Cuspy loop configurationconfiguration

Gravitational “AB” Gravitational “AB” interactioninteraction• Spacetime metric far from a straight, Spacetime metric far from a straight,

infinite cosmic string is Minkowski, with infinite cosmic string is Minkowski, with a deficit angle determined by string a deficit angle determined by string tension.tension.• Shaking a cosmic string would Shaking a cosmic string would generate a time-dependent metric and generate a time-dependent metric and induce gravitationally induced induce gravitationally induced production of all particle spieces.production of all particle spieces.

• Scalars, Scalars, Photons, Photons, FermionsFermions

The N-Body Problem The N-Body Problem in in

General RelativityGeneral Relativityfrom from

Perturbative (Quantum) Field Perturbative (Quantum) Field TheoryTheory

Y.-Z.Chu, Phys. Rev. D Y.-Z.Chu, Phys. Rev. D 7979: 044031, 2009: 044031, 2009arXiv: 0812.0012 [gr-qc]arXiv: 0812.0012 [gr-qc]

Yi-Zen ChuYi-Zen Chu

Particle Astrophysics/Cosmology Seminar, ASUParticle Astrophysics/Cosmology Seminar, ASU

Wednesday, 6 October 2010Wednesday, 6 October 2010

• System of n ≥ 2 System of n ≥ 2 gravitationally bound compact gravitationally bound compact objects:objects:

• Planets, neutron stars, Planets, neutron stars, black holes, etc.black holes, etc.

• What is their effective What is their effective gravitational interaction?gravitational interaction?

?...||2

1

2

1

1

2

n

ba ba

baNn

aaa xx

MMGvML

eff

• Compact objects ≈ point Compact objects ≈ point particlesparticles• n-body problem: Dynamics for n-body problem: Dynamics for the coordinates of the point the coordinates of the point particlesparticles• Assume non-relativistic Assume non-relativistic motionmotion

• GR corrections to GR corrections to Newtonian gravity: an Newtonian gravity: an expansion in (v/c)expansion in (v/c)22Nomenclature: O[(v/c)Nomenclature: O[(v/c)2Q2Q] = Q PN] = Q PN

?...||2

1

2

1

1

2

n

ba ba

baNn

aaa xx

MMGvML

eff

• Note that General Relativity Note that General Relativity is non-linear.is non-linear.

• Superposition does not Superposition does not holdhold• 2 body lagrangian is not 2 body lagrangian is not sufficient to obtain n-body sufficient to obtain n-body lagrangianlagrangian

Nomenclature: O[(v/c)Nomenclature: O[(v/c)2Q2Q] = Q PN] = Q PN

• n-body problem known up to n-body problem known up to O[(v/c)O[(v/c)22]: ]:

• Einstein-Infeld-Hoffman Einstein-Infeld-Hoffman lagrangianlagrangian• Eqns of motion used regularly to Eqns of motion used regularly to calculate solar system dynamics, calculate solar system dynamics, etc.etc.

• Precession of Mercury’s Precession of Mercury’s perihelion begins at this orderperihelion begins at this order

• O[(v/c)O[(v/c)44] only known partially.] only known partially.• Damour, Schafer (1985, 1987)Damour, Schafer (1985, 1987)• Compute using field theory? Compute using field theory? (Goldberger, Rothstein, 2004)(Goldberger, Rothstein, 2004)

• Solar system probes of GR Solar system probes of GR beginning to go beyond O[(v/c)beginning to go beyond O[(v/c)22]: ]:

• New lunar laser ranging observatory New lunar laser ranging observatory APOLLO; Mars and/or Mercury laser APOLLO; Mars and/or Mercury laser ranging missions?ranging missions?• ASTROD, LATOR, GTDM, BEACON, ASTROD, LATOR, GTDM, BEACON, etc.etc.• See e.g. Turyshev (2008)See e.g. Turyshev (2008)

• n-body Ln-body Leffeff gives not only dynamics but gives not only dynamics but also geometry.also geometry.

• Add a test particle, M->0: it moves Add a test particle, M->0: it moves along geodesic in the spacetime metric along geodesic in the spacetime metric generated by the rest of the n massesgenerated by the rest of the n masses• Metric can be read off its LMetric can be read off its Leffeff

...2

1

2

1

2 000

2

test

eff

dt

dz

dt

dzg

dt

dzggM

dt

zdMM

dt

dz

dt

dzgML

ji

ij

i

i

gg

• Gravitational wave observatories may Gravitational wave observatories may need the 2 body Lneed the 2 body Leffeff beyond O[(v/c) beyond O[(v/c)77]:]:

• LIGO, VIRGO, etc. can track gravitational LIGO, VIRGO, etc. can track gravitational waves (GWs) from compact binaries over waves (GWs) from compact binaries over O[10O[1044] orbital cycles.] orbital cycles.• GW detection: Raw data integrated against GW detection: Raw data integrated against theoretical templates to search for theoretical templates to search for correlations.correlations.• Construction of accurate templates Construction of accurate templates requires 2 body dynamics.requires 2 body dynamics.• Currently, 2 body dynamics known up to Currently, 2 body dynamics known up to O[(v/c)O[(v/c)77], i.e. 3.5 PN], i.e. 3.5 PN• See e.g. Blanchet (2006).See e.g. Blanchet (2006).

• Starting at 3 PN, O[(v/c)Starting at 3 PN, O[(v/c)66], GR ], GR computations of 2 body Lcomputations of 2 body Leffeff start to start to give divergences – due to the point give divergences – due to the point particle approximation – that were particle approximation – that were eventually handled by dimensional eventually handled by dimensional regularization.regularization.• Perturbation theory beyond Perturbation theory beyond O[(v/c)O[(v/c)77] requires systematic, ] requires systematic, efficient methods.efficient methods.

• Renormalization & Renormalization & regularizationregularization• Computational algorithm – Computational algorithm – Feynman diagrams with Feynman diagrams with appropriate dimensional appropriate dimensional analysis.analysis.

QFTQFTOffersOffers::

• GR: Einstein-HilbertGR: Einstein-Hilbert• n point particles: any scalar functional of n point particles: any scalar functional of geometric tensors, geometric tensors, dd-velocities, etc. integrated -velocities, etc. integrated along world linealong world line

a

a

a

a

a

aaa

a

n

aapp

ddplGR

ppGR

Npldpl

ds

dxu

dt

dx

dt

dxgdtds

uuuuRRc

RRcRRcdsMS

RgxdMS

SSS

GMM

hg

,

...

1

2

32,

3

2

21

1

2

2/1

1)2/(

a

a

a

a

a

aaa

a

n

aapp

ddplGR

ppGR

Npldpl

ds

dxu

dt

dx

dt

dxgdtds

uuuuRRc

RRcRRcdsMS

RgxdMS

SSS

GMM

hg

,

...

1

2

32,

3

2

21

1

2

2/1

1)2/( • – –M∫ds describes M∫ds describes structureless point structureless point particleparticle


a

a

a

a

a

aaa

a

n

aapp

ddplGR

ppGR

Npldpl

ds

dxu

dt

dx

dt

dxgdtds

uuuuRRc

RRcRRcdsMS

RgxdMS

SSS

GMM

hg

,

...

1

2

32,

3

2

21

1

2

2/1

1)2/(

• Non-minimal terms Non-minimal terms encode information on the encode information on the non-trivial structure of non-trivial structure of individual objects.individual objects.


a

a

a

a

a

aaa

a

n

aapp

ddplGR

ppGR

Npldpl

ds

dxu

dt

dx

dt

dxgdtds

uuuuRRc

RRcRRcdsMS

RgxdMS

SSS

GMM

hg

,

...

1

2

32,

3

2

21

1

2

2/1

1)2/(

• GR: Einstein-HilbertGR: Einstein-Hilbert• n point particles: any scalar functional of n point particles: any scalar functional of geometric tensors, geometric tensors, dd-velocities, etc.integrated -velocities, etc.integrated along world linealong world line

• Coefficients {cCoefficients {cxx} have to be } have to be

tuned to match physical tuned to match physical observables from full description observables from full description of objects.of objects.• E.g. n non-rotating black holes.E.g. n non-rotating black holes.

a

a

a

a

a

aaa

a

n

aapp

ddplGR

ppGR

Npldpl

ds

dxu

dt

dx

dt

dxgdtds

uuuuRRc

RRcRRcdsMS

RgxdMS

SSS

GMM

hg

,

...

1

2

32,

3

2

21

1

2

2/1

1)2/(

• GR: Einstein-HilbertGR: Einstein-Hilbert• n point particles: any scalar functional of n point particles: any scalar functional of geometric tensors, geometric tensors, dd-velocities, etc.integrated -velocities, etc.integrated along world linealong world line

• Point particle approximation Point particle approximation gives us computational control.gives us computational control.• Infinite series of actions Infinite series of actions truncated based on desired truncated based on desired accuracy of theoretical prediction. accuracy of theoretical prediction.

a

a

a

a

a

aaa

a

n

aapp

ddplGR

ppGR

Npldpl

ds

dxu

dt

dx

dt

dxgdtds

uuuuRRc

RRcRRcdsMS

RgxdMS

SSS

GMM

hg

,

...

1

2

32,

3

2

21

1

2

2/1

1)2/(

• For non-rotating For non-rotating compact objects, up to compact objects, up to O[(v/c)O[(v/c)88], only minimal ], only minimal terms -Mterms -Maa∫ds∫dsaa needed needed


• Expand GR and point particle action in Expand GR and point particle action in powers of graviton fields hpowers of graviton fields hμνμν … …

jaiaij

iai

dplaa

n

aa

ddpl

dpl

aaa

d

xxhxhhMxdtM

RgxdMS

M

hg

xxxLL

SSiDhLdti

0002/12

1

2

1)2/(

effeff

classical

gf

1

0eff

21

2

,...}],,[{

diagrams treeconnectedFully exp

exp exp

• Expand GR and point particle action in Expand GR and point particle action in powers of graviton fields hpowers of graviton fields hμνμν … …

jaiaij

iai

dplaa

n

aa

ddpl

dpl

aaa

d

xxhxhhMxdtM

RgxdMS

M

hg

xxxLL

SSiDhLdti

0002/12

1

2

1)2/(

effeff

classical

gf

1

0eff

21

2

,...}],,[{

diagrams treeconnectedFully exp

exp exp

• ∞ ∞ terms just from terms just from Einstein-Hilbert and Einstein-Hilbert and -M-Maa∫ds∫dsaa..

• … … but some dimensional analysis but some dimensional analysis before computation makes before computation makes perturbation theory much more perturbation theory much more systematicsystematic• The scales in the n-body problemThe scales in the n-body problem

• rr – typical separation between n – typical separation between n bodies.bodies.• vv – typical speed of point particles – typical speed of point particles• r/vr/v – typical time scale of n-body – typical time scale of n-body systemsystem 1

000

~

~][~

vrxd

r

vxx

dx

d

dd

• Lowest order effective actionLowest order effective action

n

bad

ba

badN

n

aaa xx

MMGxMdtS

3

1)2/(

1

2

||2

1 0

• Schematically, conservative part Schematically, conservative part of effective action is a series:of effective action is a series:

23

1)2/(

321)2/(20

04

42

20

~

~~~

~...,~

vr

MG

MvrrMGdtMvdtS

SSvSvSSS

d

dN

ddN

n

eff

• Virial theoremVirial theorem

• Look at Re[Graviton propagator], Look at Re[Graviton propagator], non-relativistic limit:non-relativistic limit:

2

2

][||

]2/)3[(

8

0],[],[0Re

;

0032/)1(

;

00

dP

xxxx

diP

xxhxxhT

dd

cv

][||

]2/)3[(

8

0],[],[0Re

0032/)1(

;

00

xxxx

diP

xxhxxhT

dd

cv

• Look at Re[Graviton propagator], Look at Re[Graviton propagator], non-relativistic limit:non-relativistic limit:

2/32/0

2/12/

2/12/1

~

~

~

vrh

vrh

vrh

d

di

d

• n-graviton piece of -Mn-graviton piece of -Maa∫ds∫dsaa with with χχ powers of velocities scales aspowers of velocities scales as• n-graviton piece of Einstein-Hilbert n-graviton piece of Einstein-Hilbert action with action with ψψ time derivatives scales as time derivatives scales as

222/10

nn vS

422/10

nn vS

222/0

)()()( wvw nNnn vS

• With nWith n(w)(w) world line terms -M world line terms -Maa∫ds∫dsaa,,• With nWith n(v)(v) Einstein-Hilbert action terms, Einstein-Hilbert action terms,• With N total gravitons,With N total gravitons,• Every Feynman Every Feynman diagram scales asdiagram scales as

• n-graviton piece of -Mn-graviton piece of -Maa∫ds∫dsaa with with χχ powers of velocities scales aspowers of velocities scales as• n-graviton piece of Einstein-Hilbert n-graviton piece of Einstein-Hilbert action with action with ψψ time derivatives scales as time derivatives scales as

222/10

nn vS

422/10

nn vS

222/0

)()()( wvw nNnn vS

=1 =1 for classical for classical

problemproblem

Q PNQ PN

• With nWith n(w)(w) world line terms -M world line terms -Maa∫ds∫dsaa,,• With nWith n(v)(v) Einstein-Hilbert action terms, Einstein-Hilbert action terms,• With N total gravitons,With N total gravitons,• Every Feynman Every Feynman diagram scales asdiagram scales as• Know exactly Know exactly which terms in which terms in action & diagrams action & diagrams are necessary.are necessary.

PN2

0 ,

)(

220

)(

w

n

n

vS w

• Limited form of superposition holdsLimited form of superposition holds• At Q PN, i.e. O[(v/c)At Q PN, i.e. O[(v/c)2Q2Q], max number of ], max number of distinct point particles in a given diagram is distinct point particles in a given diagram is Q+2Q+2• 1 PN, O[(v/c)1 PN, O[(v/c)22]: 3 body problem]: 3 body problem• 2 PN, O[(v/c)2 PN, O[(v/c)44]: 4 body problem]: 4 body problem• 3 PN, O[(v/c)3 PN, O[(v/c)66]: 5 body problem]: 5 body problem

• Every Every Feynman Feynman diagram scales diagram scales asas

||

)2(

]2/)1[(2

2

1

2

1

1,3

12

2/1

82

5

2

1

PN) (0eff

baab

n

baba

dab

ba

d

N

d

a

n

aa

xxR

R

MMG

d

dvML

2 body2 bodydiagramsdiagrams


Einstein-Infeld-HoffmanEinstein-Infeld-Hoffmandd-spacetime dimensions-spacetime dimensions



Einstein-Infeld-HoffmanEinstein-Infeld-Hoffmandd-spacetime dimensions-spacetime dimensionsRelativistic Relativistic

correctionscorrections

Relativistic Relativistic correctionscorrections



Einstein-Infeld-HoffmanEinstein-Infeld-Hoffmandd-spacetime dimensions-spacetime dimensionsGravitationalGravitational

1/r1/r22 potentials potentials

GravitationalGravitational1/r1/r22 potentials potentials



Einstein-Infeld-HoffmanEinstein-Infeld-Hoffmandd-spacetime dimensions-spacetime dimensions• Non-linearities of GRNon-linearities of GR

• Three body forceThree body force

• Non-linearities of GRNon-linearities of GR• Three body forceThree body force



Einstein-Infeld-HoffmanEinstein-Infeld-Hoffmandd-spacetime dimensions-spacetime dimensions• Time derivative of Time derivative of

Runge-Lenz vector Runge-Lenz vector gives perihelion gives perihelion precession of precession of planetary orbits.planetary orbits.

• Time derivative of Time derivative of Runge-Lenz vector Runge-Lenz vector gives perihelion gives perihelion precession of precession of planetary orbits.planetary orbits.

No gravitonNo gravitonverticesvertices

GravitonGravitonverticesvertices





212121

2121232322

||||||

|||| ~

zxzxzy

yxyxzdydI

encj

bmaimnij

Work in progress (with Roman Buniy)Work in progress (with Roman Buniy)

• N-body problem for rotating N-body problem for rotating masses, multi-pole moments, masses, multi-pole moments, gravitational radiation, etc.gravitational radiation, etc.• Higher PN computation:Higher PN computation:

• Different field variables: ADM, Different field variables: ADM, Kol-Smolkin-Kaluza-Klein.Kol-Smolkin-Kaluza-Klein.• Different gravitational Different gravitational lagrangian: Bern-Grant.lagrangian: Bern-Grant.• Are there recursion relations for Are there recursion relations for off-shell gravitational amplitudes?off-shell gravitational amplitudes?• Software development.Software development.

Documents

Yi-Zen Chu Particle Astrophysics/Cosmology Seminar, ASU Wednesday, 6 October 2010