Electromagnetic Self Force · 2013. 7. 22. · Electromagnetic Self Force Circular Orbits in...
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Electromagnetic Self Force
Circular Orbits in Schwarzschild Spacetime
Patrick Nolan, University College Dublin
Tuesday 16 July 13
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Motivation
• Working in Electromagnetism is a good warm-up for gravitational self-force
• Has its own physical motivation
• Circular orbits can be straightforwardly extended to eccentric ones
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Field Equations
• The Self Force is given by the formula
with Field Tensor
• Calculate this using the field equations for a spin one field, in Lorenz Gauge:
Fµ = �qFµ⌫u⌫
Fµ⌫ = rµA⌫ �r⌫Aµ
⇤Aµ �Rµ⌫A⌫ = 0
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Simplifying the problem: 1
• To solve these PDEs, decompose fields into angular and radial components:
• Current decomposed similarly.
+
Jµ
Aµ =
0
BB@
R1(r)Y lm(✓,�)R2(r)Y lm(✓,�)R3(r)Zlm
✓ (✓,�)R3(r)Zlm
� (✓,�)
1
CCA
0
BB@
00
R4(r)X lm✓ (✓,�)
�R4(r)X lm� (✓,�)
1
CCAei!t
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Simplifying the problem: 2
• Separating Even and Odd modes in equations leaves 3+1 ODEs to be solved
• We can use the gauge equation to eliminate one of our even sector fields:
• System now has one decoupled, 2 coupled fields
R3(r) ⇠ f(R1(r) +R2(r))
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• We use a series expansion to approximate the fields at the boundaries:
• To match these solutions at the particle’s orbit, we impose matching conditions
Solving for the Fields
INNER:
OUTER:
n1X
n=0
ainrn
nHX
n=0
bin(r � 2M)n
r = 1
r = 2M
r0
5 10 15 20
0.2
0.4
0.6
0.8
1.0ei!r⇤
e�i!r⇤
R1
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Construct the Self Force
• Having solved for the l-mode fields, it is now straightforward to construct the l-mode Faraday Tensor, and hence the self force:
Fµ =1X
l=0
lX
m=�l
F lmµ⌫
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Regularised Self Force:
1 2 5 10 20 5010-20
10-16
10-12
10-8
10-4
1
{
F lr
Electromagnetic self-force
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Self Force EM
• 0.0012098217906065(1) (circular orbit with r0=10, M=1)
• Much more accurate than current EM data
• Successfully applies the new regularisation parameters
Fr =
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Comparing to Gravity
• Method mostly extends directly to gravity
• coupling in both sectors
• static mode complications
EM Gravity
3+1 fields 7+3 fields1+0 gauge 3+1 gauge
2+1 to solve 4+2 to solve
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Static Terms
• EM static (m=0) modes are known analytically
• Only known for gravity odd sector
• even sector requires asymptotic expansion
• Outgoing ansatz must be changed:
R
iinf =
n1X
n=0
a
in + b
inlog(r)
r
n
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10.05.02.0 3.01.5 7.0l
10-7
10-5
0.001
0.1
Frl
Gravity Regularised:
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Gravitational Self Force
• Very close to making use of new regularisation parameters
• Expect to have more accurate data than is currently available
Fr = 0.013389(5)
Ft = �0.000091907(6)
(circular orbit with r0=10, M=1)
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To-Do List
• Need to extend gravity data to include higher l-modes
• Want to check data against independent calculation
Regge-Wheeler Comparison!
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