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Eric Carlson
Einstein’s Equation
Relates the shape of spacetime
to the stuff that’s in it
Curvature
G  8 GT
of
spacetime



jx

T 
 jy

 jz
jx
S xx
jy
S xy
S yx
S zx
S yy
S zy
jz 

S xz 
S yz 

S zz 
Energy density (and other quantities
in T) need to include quantum effects


i
T  x      x        x      x      x  

2
The problem: We don’t have
a quantum theory of gravity
G  8 GT
Semi-Classical Gravity
G  8 GTT

•Calculate T including quantum
effects in curved spacetime
•Replace T by its expectation value
•Find the shape of spacetime from
Einstein’s equation (semi-classical
version)
•Repeat until it converges
Why it
might
make
sense
•There are lots
of particles we
know how to
do quantum
mechanics on
Particle
Higgs
symbols spin
H
0
Electron
e
Electron neutrino e
Up quark
uuu
Down quark
ddd
Muon

Muon neutrino 
Up quark
ccc
Down quark
sss
Tau

Tau neutrino

Top quark
ttt
Bottom quark
bbb

d.o.f.
1
½
½
½
½
½
½
½
½
½
½
½
½
4
2
12
12
4
2
12
12
4
2
12
12
2
16
6
3
2
Photon
Gluon
W-boson
Z-boson
gggggggg
W
Z
1
1
1
1
Graviton
h
2
118
2
Spin ½ fields near wormholes
x
r
x=0
r=r0 “throat”
What does the asymptotic
energy density look like?
•Wormholes connect distant
points in space
•Wormholes require negative
energy density
•It is possible (likely) that
wormholes would have
negative energy density
•Naive use of the “analytic
approximation” predicted that the
energy density would fall as 1/r6
at large r
•Other arguments predicted 1/r5
The Method
1. Convert classical equations for free fields to curved spacetime
    0
    0

2. Solve Green’s function equations in curved spacetime
   S  x, x   4  x  x '
3. Use Green’s functions to calculate expectation value of T
  )  S E  S Ec 
  
   
T  x  unren   14 lim Im Tr  ( 
x x '
  g  ' '  S E  S Ec  I  x, x '    


  
4. Renormalize to get rid of infinities
 
T  x 
ren
 T  x 
modes
 T  x 
WKBfin
 T  x 
analytic
Computational approach
1. Solve lots of coupled differential equations


G , j 
x
j  12
F , j 
G , j
r  x
f  x

F 
x

do i=1,imax
h=htot/nseq(i)
zold=z
znew=z+h*dzdx
xx=x+h
1
twoh=h+h
2
do j=2,nseq(i)
, j
, j
, j
call rf(xx,r,f)
swap=zold+twoh*(ell*(1.q0-znew**2)/r
+
-2*omega*znew/sqrt(f))
zold=znew
znew=swap
xx=xx+h
enddo
call rf(xx,r,f)
zold=half*(zold+znew+h*(ell*(1.q0-znew**2)/r
+
-2*omega*znew/sqrt(f)))
do i=1,ihi
a1=l*omega/r(i)*(zp(i)+zq(i))/(zp(i)-zq(i))
a2=l*sqrt(f(i))*l/r(i)**2*(1-zp(i)*zq(i))/(zp(i)-zq(i))
w=sqrt(omega**2*r(i)**2+l**2*f(i))
a1w=t10(i)*l*omega**2/w
a2w=t20(i)*qfloat(l)**3/w
do k=1,lev
do j=1,2*k
a1w=a1w+l*t1(i,k,j)*l*omega**(2*j)*l/w**(2*j+2*k+1)
a2w=a2w+l*t2(i,k,j)*l*omega**(2*j)*l/w**(2*j+2*k+1)
enddo
enddo
f  x
G 
j
F
r  x
2. Add together all the modes
3. Integrate over frequency
4. Add other terms
What you get
R. Chainani, 9/21/09
0.03
0.025
0.02
Tt t
0.015
T
2r5Tµ/b
0.01
Trr
0.005
0
1
2
3
4
5
6
7
8
-0.005
-0.01
-0.015
-0.02
2
dr
2
2
2
2
2
ds 2  dt 2 

r
d


r
sin

d

1  b2 r 2
9
10
r /b
What you need to do this research
Undergraduates:
•Strong Mathematical Background
•Computer Skills Helpful
•Maple or Mathematica Experience
Graduates:
•Graduate Quantum Mechanics
•General Relativity – must be arranged
•Quantum Field Theory - must be arranged