? Eric Carlson

Einstein’s EquationRelates the shape of spacetime

to the stuff that’s in it

8G GT Curvature

of spacetime

x y z

x xx xy xz

y yx yy yz

z zx zy zz

j j jj S S S

Tj S S Sj S S S

Energy density (and other quantitiesin T) need to include quantum effects

The problem: We don’t havea quantum theory of gravity

2iT x x x x x

8G GT

8G GT

Semi-Classical Gravity

•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

T

Why it might make sense

Particle symbols spin d.o.f.Higgs H 0 1

Electron e ½ 4Electron neutrino e ½ 2Up quark uuu ½ 12Down quark ddd ½ 12Muon ½ 4Muon neutrino ½ 2Up quark ccc ½ 12Down quark sss ½ 12Tau ½ 4Tau neutrino ½ 2Top quark ttt ½ 12Bottom quark bbb ½ 12

Photon 1 2Gluon gggggggg 1 16W-boson W 1 6Z-boson Z 1 3

Graviton h 2 2

118

2

•There are lots of particles we know how to do quantum mechanics on

Spin ½ fields near wormholes

r

r=r0 “throat”x=0

x

•Wormholes connect distant points in space

•Wormholes require negative energy density

•It is possible (likely) that wormholes would have negative energy density

What does the asymptotic energy density look like?

•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 Method1. Convert classical equations for free fields to curved spacetime

0

2. Solve Green’s function equations in curved spacetime

3. Use Green’s functions to calculate expectation value of T

0

4, 'S x x x x

4. Renormalize to get rid of infinities

ren modes WKBfin analytic

T x T x T x T x

)1

(4unren '''

lim Im Tr, '

cE E

cx xE E

S ST x

g S S I x x

Computational approach1. Solve lots of coupled differential equations

12

, , ,

12

, , ,

j j j

j j j

jG F Gx r xf x

jF G Fx r xf x

2. Add together all the modes

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

do i=1,imax h=htot/nseq(i) zold=z znew=z+h*dzdx xx=x+h twoh=h+h do j=2,nseq(i) 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)))

4. Add other terms3. Integrate over frequency

What you get

/r b

2 r5 T

µ /bTtt

R. Chainani, 9/21/09

22 2 2 2 2 2 2

2 2 sin1drds dt r d r db r

TrrT

What you need to do this researchUndergraduates:

•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