The Path Integral, Perturbation Theory and Complex Actions

atter free solution to the Einstein equations for the gr
ials ic field for is point given sources. by the Indeed, Minkowski themetric
empty space,
eld • is the given Minkowski by the Minkowski metric 0 is metric
1 0matt
0
Einstein equations0for the1gravitational 0 0 10metric 0 field 1 is 0
1 [g 0 [g µ⌫ ]=[⌘ µ⌫ ]=
[g µ⌫ ]=[⌘ µ⌫ ]= B 0
B 0 µ⇥ ]=[
1 0 µ⇥ ]= ⇧
0 ⇤
@ 1 0
C
0 0 1 C0
@ 0 1 A . A . ⇥
1 0 0 0
0 0
0 0
0
1 0
1
0
[g
Now µ⇥ ]=[
adding 0a µ⇥ ]= ⇧
⌃
small, 0 ⇤ perturbing 0 0 1 1source 0 yields ⌅ .
a small,
mall, • perturbing and
perturbing
we expect source a
source
small yields perturbation
yields
0 0 0 1
g
g
g µ⌫ = µ⌫ = ⌘
⌘ µ⌫ + µ⌫ + h
h µ⌫ . µ⌫ . µ⇥ = µ⇥ + h µ⇥
dding a small, perturbing source yields
where
T µν = p µ(t)p ν (t)
E(t)
• then we have
δ 3 (⃗x − ⃗r(t)) ≈ mβ µ (t)β ν (t)δ 3 (⃗x − ⃗r(t))
5
5
g µ⇥ = µ⇥ + h µ⇥
0 0 1
2h µ⇥ =4⇥GS µ⇥
and S µ⇥ = T µ⇥ (1/2)T µ⇥
In the slow moving approximation this yield
⇥ = T µ⇥ (1/2)T µ⇥
Tuesday, 2 April, 13
2h µ⇥ =4⇥GS µ⇥
⇤g(⇤x, t +
t) =
GM ˆr(
r(t) 2