Gravitation below a millimetre

P. J. S. Watson

June 16th, 2004

Gravity: Ancient History

F = Gm₁m₂ 
      R²

modified by GR gives us

No known conflicts between GR and experiment, so maybe we should quit while we are ahead .........

Extra-dimensional theories (intended to fix hierarchy problem): see Nima Arkani-Hamedi TU-A2)
If we live in a 3+n dimensional space, gravitational force goes as 

F = G3+n m₁m₂
         Rn+2

but we don't! However, if we have the extra dimensions only showing up at short distances r < r0, it can work: i.e. 

F = G₃ m₁m₂      r>r0
       R2
F = G3+n m₁m₂    r<r0
         Rn+2
So obviously 
G3+n = G₃ r0n

Major consequences:

  1. Kalusza-Klein excitations of graviton GKK exist : M(k) = 2πk/r0
  2. Short-range behaviour of gravity will deviate from 1/r2 for r<r0. Roughly:
    n (Number of Extra Dimensions r0 Size of Extra Dimension KK scale
    1 1013 m (so ruled out)
    2 1 mm
    3 10-9 m 400 eV
    4 10-11 m 0.2 MeV
  3. Gravity starts being very strong at ~ 1TeV, so new phenomena at LHC, in particular (n+3)-D black holes can be formed τ∼ 10-26s! (see Das WE-P10-1)
  4. Gravity exists in 3+n dimensions: matter particles live on 3-D brane.
  5. Gravity is confined to smaller # of dimensions at large distances

    Picture from Greg Landsberg

LEP has (negative) searches for KK states: 

e+e- ⇒ γ GKK
Hence it is very interesting to look at gravity at distances of microns.
Assume: 
V =  G'm₁m₂e-(r/λ)
        r
Washington experiment: confirms Newton down to ~ 150μ

How do we get to shorter distances?

Need system with very low interaction energy (comparable to gravitational PE)

Ultra-Cold Neutrons

UCN's with energies E<10-6 eV scatter coherently off the atomic nuclei in a lattice. Bulk interaction gives rise to a potential of the form 

Us = 2πħ²Nb
       m

b = scattering length, N = number density of nuclei. This gives rise to a critical velocity vc below which neutrons are totally reflected from the surface. For Be

Anomalous interaction gives extra potential 
VS = -4π G'm ρλ²

roughly equivalent to a work-function. Must have 

VS + Us > 0 giving

G'= a ,   a = ħ²Nb
    λ²       2m²ρ

If we could measure vc for Pb to (say) .1 ms-1, this limit could be improved by about 100.
Terrible limit (see later) but only game in town!

A direct experiments: the bouncing neutron, a bound state of the neutron produced by the hard surface and gravity.

Theory:

Bounce eigenfunctions Zn(z) 
 - ħ² ∂²u(z) + gzu(x) = Eu(z)
  2m  ∂z²
Solve for neutron,
n En zn
1 1.41 peV 13.7 μ
2 2.46 peV 24.0 μ
3 3.32 peV 32.5 μ
4 4.08 peV 39.9 μ

Observation: the Nesvizhevsky Experiment

Proof of existence
Neutron absorber is lowered, extinguishing signal. Classical prediction is for signal to vanish smoothly, Q.M. is for sudden cutoff at height h
Compare classical prediction (UCN's of all bounce heights, but M-B distrib) to quantum prediction (no bounce height ≤ 13 μ)
Blow-up of lowest part of plot

Would expect the state to be sensitive to λ ∼ 10-5μ.

Putting slab of dense material below apparatus would shift energy levels:
Extra interaction is 
δV(z) = 2πG'ρmN λ2e-z/λ
This is for G' = 1. i.e. not a useful result: it would require measuring the levels to accuracy of 10-18 eV(!)

Oscillating slab (amplitude z0, frequency ω0) to produce time varying potential ⇒ transitions between states

Could probably set limits for G' ∼ 10-5 for λ ∼ 10 μ

better than anyone else, but not useful for G' ∼ G.
Limited by neutron half-life.

Quantum Pendulum

Atom-surface interactions 
Veff =  a  -  b 
       z9    z³
For He Cs, the relevant parameters are 
V0 = -.45 meV, x0 = .5 nm
which supports exactly 1 bound state .6 nm above surface: BE ∼ 80 μeV.
Hence can form quantum "stringless pendulum" 
V(r) = mgr²
       2R₀
with corresponding energies 
En = (2n+1)1.45 peV (R₀ in μ)
            R₀½
Static perturbations are extremely small, but can resonantly excite n=1 state to n=2 state.
for G' = G, excitation times are ∼ months

In principle, this lets us get a limit on G' well below 1μ. Graph shows:
  1. Hoyle et al limit
  2. Cheiverini et al. limit (not competitive but only direct measurement below 100 μ
  3. UCN limit: lousy but only limit below 1μ
  4. Quantum pendulum limit (assuming excitation time of 106 s)
However: we have ignored a number of effects:


Conclusions: any experiment will be very hard, but very important.


References: