1. ? Eric Carlson

2. Einstein’s Equation Relates the shape of spacetime to the stuff that’s in it Curvature of spacetime

3. Energy density (and other quantities in T  ) need to include quantum effects The problem: We don’t have a quantum theory of gravity

4. 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

5. Why it might make sense Particlesymbolsspind.o.f. HiggsH01 Electrone½4 Electron neutrino e ½2 Up quarkuuu½12 Down quarkddd½12 Muon  ½4 Muon neutrino  ½2 Up quarkccc½12 Down quarksss½12 Tau  ½4 Tau neutrino  ½2 Top quarkttt½12 Bottom quarkbbb½12 Photon  12 Gluon gggggggg116 W-bosonW16 Z-bosonZ13 Gravitonh22 118 2 There are lots of particles we know how to do quantum mechanics on

6. Spin ½ fields near wormholes r r=r 0 “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/r 6 at large r Other arguments predicted 1/r 5

7. The Method 1. Convert classical equations for free fields to curved spacetime 2. Solve Green’s function equations in curved spacetime 3. Use Green’s functions to calculate expectation value of T  4. Renormalize to get rid of infinities

8. Computational approach 1. Solve lots of coupled differential equations 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 terms 3. Integrate over frequency

9. What you get  2 r 5 T  µ /b TttTtt R. Chainani, 9/21/09 TrrTrr T  

10. 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