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Holographic Noise
Craig Hogan
Fermilab and U. Chicago
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Are time and space infinitely smooth?
Einsteins theory assumes spacetime is a classical manifold,infinitely divisible
This may be just an approximate behavior Can we measure the minimum interval of time?
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The smallest interval of time
Quantum gravity suggests a minimum (Planck) time,
~ particle energy 1016 TeV
seconds
ma
ss
length
quantum
gravity
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Two approaches to the Planck scale
mass
length
quantum
gravity
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position
momentum
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Best microscopes vs best microphones
CERN/Fermilab: TeV-1~10-18 m: particle interactions
LIGO/GEO600: ~10-18 m, coherent over
~103 m baseline: Positions of massive
bodies
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A new phenomenon?: holographic noise
The minimum interval of time may affect interferometers Transverse uncertainty much larger than Planck scale in
holographic theories
precise, zero-parameter prediction of Holographic Noise
Planck diffraction limit at L
is >> Planck length
x ~ L
6
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Spatial frequency limit causes transverse indeterminacy:transverse position wavefunction at distance L
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L
L
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Indeterminacy in difference of orthogonal transverse positions
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GEO-600 (Hannover): best displacement sensitivity
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Mystery Noise in GEO600
Prediction: CJH, arXiv:0806.0665
(Phys Rev D.78.087501)
Data: S. Hild (GEO600)
Total noise: not fitted
zero-parameter prediction for
holographic noise in GEO600(equivalent GW strain)
tPlanck
/
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Measurement of holographic noise
Holographic wave geometry predicts a new detectable effect:"holographic noise
Not the same as zero-point field mode fluctuations Spectrum and distinctive spatial character of the noise is predicted
with no parameters
It may already be detected An experimental program is motivated
CJH: arXiv:0806.0665 Phys Rev D.78.087501 (2008)
CJH: arXiv:0712.3419 Phys Rev D 77, 104031 (2008)
CJH and M. Jackson, Phys. Rev. D in press, arXiv:0812.1285
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This is what we found out about Naturesbook keeping system: the data can be written
onto a surface, and the pen with which the
data are written has a finite size.
-Gerard t Hooft
Everything about the
3D world can be
encoded on a 2D null
surface at Planckresolution
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Holographic Theories of Everything
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Holographic geometry: a phenomenological layer
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Fundamental theory (Matrix, string, loop,)
Holographic geometry (paraxial waves, diffraction, transverse
spacetime wavefunction, holographic uncertainty)
Observables in classical apparatus (effective beamsplitter
motion, holographic noise in interferometer signals)
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Holographic Quantum Geometry: theory
Black holes: entropy=area/4
Black hole evaporationEinstein's equations from heat flowClassical GR from surface theoryUniversal covariant entropy boundExact state counts of extremal holes in large DAdS/CFT type dualities: N-1 dimensional dualsMatrix theoryAll suggest theory on 2+1 dimensional null surfaceswith Planck frequency bound
Beckenstein, Hawking, Bardeen et al.,
'tHooft, Susskind, Bousso, Srednicki,
Jacobson, Padmanabhan, Banks,
Fischler, Shenker, Unruh14AEI, May 2009
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Holography 2: Black Hole Evaporation
Hawking (1975): black holes radiate ~thermal radiation, loseenergy and disappear
evaporated quanta carry off degrees of freedom (~1 perparticle) as area decreases
States on 2D event horizon completely account for informationof evaporated states, assembly histories
Information of evaporated particles=entropy of hole= A/4
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black hole evaporation can obey quantum mechanics if
distant, nearly flat space has transverse indeterminacy
If the quantum states of the evaporated particles allowed relativetransverse position observables with arbitrary angular precision, at
large distance they would contain more information than the hole
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~ One particle evaporates per Planck areaposition recorded on film at distance Lwavelength ~ hole size Rstandard position uncertaintyParticle images on distant film: must have fewer pixels than holeRequires transverse uncertainty at distance L independent of R
Uncertainty of flat spacetime independent of black hole massSimilarly for number of position states of an interferometer
x > L
Holographic uncertainty and black hole evaporation
(L /x)2 < (R /)2
x > R
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Holography 3: nearly-flat spacetime
Unruh (1976): Hawking radiation seen by accelerating observer Appears with any event horizon, not just black holes: identify
entropy of thermal radiation with missing information
Jacobson (1995): Einstein equation derived fromthermodynamics (~ equation of state)
Classical GR from 2+1D null surface (Padmanabhan 2007)
Jacobson: points=2D surfaces19AEI, May 2009
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Holography 4: Covariant (Holographic) Entropy Bounds
't Hooft (1985): black holes are quantum systems 't Hooft, Susskind et al. (~1993): world is "holographic",
encoded in 2+1D at the Planck scale
Black hole is highest entropy state (per volume) and setsbound on entropy of any system (includes quantum degrees of
freedom of spacetime)
All physics within a 3D volume can be encoded on a 2Dbounding surface ("holographic principle")
Bousso (2002): holographic principle generalized to "covariantentropy bound" based on causal diamonds: entropy of 3D lightsheets bounded by area of 2D bounding surface in Planck units
Suggests that 3+1D geometry emerges from a quantum theoryin 2+1D: light sheets
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Holography 5: Exact dual theories in N-1 dimensions
Maldacena, Witten et al. (1997): AdS/CFT correspondence N dimensional conformal field "boundary" theory exactly maps
onto (is dual to) N+1 dimensional "bulk" theory with gravity and
supersymmetric field theory
Is nearly flat 3+1 spacetime described as a dual in 2+1?
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Holography 6: string/M theory
Strominger, Vafa (1996): count degrees of freedom ofextremal higher-dimension black holes using duality
All degrees of freedom appear accounted for Agrees with Hawking/Beckenstein thermodynamic count Unitary quantum system Strong indication of a minimum length ~ Planck length What do the degrees of freedom look like in a realistic system? Matrix theory: wavefunctions of transverse position Matrix
Hamiltonian (CJH& M. Jackson)
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Holographic geometry implements holographic entropy
bound in emergent 3+1D spacetime
3+1D spacetime from 2+1Dbuilt on light sheets: covariant formulationfewer independent modes than field theoryindependent pixels in 3D volume~ area of 2D null surface elementbandwidth limit of spacetime states
t
z
1
2
1
2
z
x
k = lP
k
y
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Theories with holographic noise
Two conditions are sufficient:
1. Maximum Planck frequency in any frame2. Planck wavelength resolution on light sheets
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1
2
y
x
t
3
25
1D segment of length L on
null wavefront
Sweeps out 2D surface:
independent position
degrees of freedom
Position variance in 2D
x2 Ll
P
(L /x)2 L / l
P
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Example: Matrix theory
Banks, Fischler, Shenker, & Susskind 1997: a candidate theoryof everything
Fundamental objects are 9 N x N matrices, describing N D0branes (particles)
Dual relationship with string theory Gives rise to 10 space dimensions, 1 compact, plus time
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R=size of Mdimension
D0 branes= KK modes
9 larger dimensions
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3+1D spacetime
emerges from
2+1D: lightsheet with z=t
2
1 2z
t
z
1
x
y
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Only 2 of the 9 space dimensions survive to be macroscopicThe third space dimension is virtual, swept out by 2D null sheet
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Holographic spacetime: wave theory from M theory
N D0 branes, N x N matrices Xi, , i= 1 to 9, compact Mdimension with radius R ~ Planck length
Hamiltonian from Banks, Fischler, Shenker, & Susskind:
Notions of position, distance emerge on scales >>R local in 2+1 D, incompressible on Planck scale: holographic Center of mass position of macroscopic body, x= trX Macroscopic longitudinal position encoded by first (kinetic)
term,conjugate momenta to position matrices
CJH & M. Jackson, arXiv:0812.1285AEI, May 2009 28
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Macroscopic wave equation from M theory
M Hamiltonian stripped to macroscopic essentials
substitute
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tr2 h22/x2,
H ih/z+,
H=R
2htr2
R k1 = /2
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Macroscopic wave equation from M theory
becomes
Schrodinger equation, with z+as time dimension Quantum mechanics without Plancks constant
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2u
x2+
4i
u
z+= 0
H= R2h
tr2
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Solutions of wave equation mix dimensions
Solutions display diffusion, diffraction:
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2
ux2
+ 4i
uz+
= 0
u(x, z+) =
k
Ak expi[k+z+ kx]
k =
4k+/
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New uncertainty principle: widths of wavepackets
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2ux2
+ 4i
uz+
= 0
x2 > L+/2
x2k2 162
L+ (4/)(2/k2)
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Nonlocal modes connect longitudinal and transverse positions
Wave solutions: Holographic geometry Transverse gaussian beam solutions from wave optics New macroscopic behavior, not the same as field theory limit
x
z,tAEI, May 2009 33
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Wave Theory of Spacetime
Adapt wave optics to theory ofspacetime wavefunctions
transverse indeterminacy fromdiffraction of Planck waves
Allows calculation of holographicnoise with no parameters
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Paraxial wave equation
phasors in wavefronts: wavefunction relative to carrier wave equation in each transverse dimension x
Basis of laser wave optics Solutions display diffraction: e.g. laser cavities reinterpret as a position wavefunction
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2u
x2 4i
u
z = 0
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Gaussian Beam solutions
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Rayleigh range and uncertainty of rays
Aperture D, wavelength : angular resolution /DSize of diffraction spot at distance L: L/Dpath is determined imprecisely by wavesMinimum uncertainty at given L whenaperture size =spot size, or
( )D L/D
L
D = L37AEI, May 2009
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Indeterminacy of a Planckian path
Classical spacetime manifold defined by paths and eventspath~ ray approximation of waveIndeterminacy of geometry reflects limited information contentof band-limited waves
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holographic approach to the classical limit
Angles are indeterminate at the Planck scale, and becomebetter defined at larger separations:
But uncertainty in relative transverse positionincreases atlarger separations:
Not the classical limit of field theory Indeterminacy and nonlocality persist to macroscopic scales
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Holographic Noise in Interferometers
Nonlocality: relative positions at km scale Fractional precision: angle < 10-21, > "halfway to Planck" Transverse position measured in Michelson layout Heavy proof masses, small Heisenberg uncertainty (SQL):
positions measure spacetime wavefunction holographic noise appears in signal
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Measurement of holographic geometry requires coherenttransverse position measurement over macroscopic distance
CERN/FNAL: TeV-1
~10-18
m
LIGO/GEO600: ~10-18 m
over ~103 m baseline
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Signal phase~ difference ofintegrated distance along two
orthogonal arms
Beamsplitter
Beamsplitter and signal in Michelson interferometer
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Signal: random phase differenceof reflection events from
indeterminate position differenceof beamsplitter at the two events
reflection
events at two
timesseparated by
2L/c
Holographic noise in the signal of a Michelson interferometer
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Quantum uncertainty of transverse positions of beamsplitter
Position wavefunctionwidths of beamsplittter at
reflection events given byGaussian beamwidth
apparent arm lengthdifference is a random
variable, with variance
this is a new effect predicted with no parameters44AEI, May 2009
L/
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Wavefunction and wavefronts
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In an optical cavity of any size, theholographic transverse uncertainty is
smaller than the beam waist by a factor
P
/laser
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Interferometer with Planck radiation
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Beamsplitter mass limited to Planck surface densityNo better measurement is possible
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Power Spectral Density of Shear Noise
At f=c/2L, shear fluctuations with power spectral density
Uncertainty in angle ~ dimensionless shear
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Universal Holographic Noise
flat power spectral density ofshearperturbations:
general property of holographic quantum geometryPrediction of spectrum with no parametersPrediction of spatial shear character: only detectable innonlocal relative transverse position observablesDefinitively falsifiableBetter estimate at low frequencies in interferometers:
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h(f) = N1/L2 = N12
tP/ = N
12.6 1022/
Hz
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Holographic noise does not carry energy or information
~ classical gauge mode (flat space, no classicalspacetime degrees of freedom excited)~sampling or pixelation noise, not thermal noiseBandwidth limit of spacetime relationshipsNecessary so the number of distinguishable positionstates does not exceed holographic bound ondegrees of freedom
No curvatureno strain, just shearno detectable effect in a purely radial measurement
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Normal incidence optics: phase signal does notrecord the transverse position of a surface
But phase of beam-split signal is sensitive to transverseposition of surface
( )
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GEO-600 (Hannover)
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Large power
cycles through
beamsplitter,
adds transverseholographic
noiseK.Strain
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Noise in GEO600 over time
H. Lck, S. Hild, K. Danzmann, K. Strain
K.Strain
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S. Hild, GEO600, May 2008
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S. Hild, GEO600
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h =
tP/ = 1.3 1022/
Hz
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Mystery Noise in GEO600
Prediction: CJH, arXiv:0806.0665
(Phys Rev D.78.087501)
Data: S. Hild (GEO600)
Total noise: not fitted
zero-parameter prediction for
holographic noise in GEO600(equivalent GW strain)
tPlanck /
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Why doesn't LIGO detect holographic noise?
LIGO design is not as sensitive to transverse displacementnoise as GEO600
relationship of holographic to gravitational wave depends ondetails of the system layout
Transverse position
measurement is not
made in FP cavities
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LIGO noise (astro-ph/0608606)
Measured LIGO noise spectrum (GW strain
equivalent, rms power spectral density)
(if shear=strain)
holographic noise
spectrum (shear)
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2.6 1022
/
Hz
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holographic noise prediction for LIGO: reduced by
~arm cavity finesse
about 100 times less
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N12.6 1022/Hz
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Beamsplitter position indeterminacy inserts holographicnoise into signal
system with GEO600 technology can detectholographic noise if it exists
Signatures: spectrum, spatial shear
Interferometers can detect quantum
indeterminacy of holographic geometry
CJH: Phys. Rev. D 77, 104031 (2008); arXiv:0806.0665
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Current experiments: summary
Most sensitive device, GEO600, sees noise compatible withholographic spacetime indeterminacy requires testing and confirmation! H. Lck: "...it is way too early to claim we might have seen
something.
But GEO600 is operating at holographic noise limit LIGO: current system not sensitive enough, awaits upgrade Proof: new apparatus, coherence of adjacent systems
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Dedicated holographic noise experiments:
beyondGW detectors
f ~100 to 1000 Hz with GW machinesf ~ 3 MHz possible with new apparatus on ~40m scaleEasier suspension, isolation, optics, vacuum, smallerscale
Correlated holographic noise in adjacent paths:signature of holographic effect
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Two ~40m Michelsoninterferometers incoincidence
~1000 W cavity
holographic noise= laserphoton shot noise in ~5
minutes (1 sigma)
Conceptual Design from Rai Weiss
Currently discussing: Fermilab (CJH, Chou, Wester, Steffen,
Ramberg, Gustafson, Stoughton, Tomlin, Ruan, Bhat), MIT (Weiss,
Waldman), Caltech (Whitcomb), UC (Meyer)
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Status of the Fermilab Holographic Interferometer
Team so far: Fermilab (CJH, Chou, Wester, Steffen,Ramberg, Gustafson, Stoughton, Tomlin, Ruan, Bhat),
MIT (Weiss, Waldman), Caltech (Whitcomb), UC (Meyer)
Building tabletop prototype Planning around Weiss design Sites on site available and surveyed: ~40m arms possible
(partially outdoors), seismically acceptable
Invited by Director Oddone to move forward Internal R&D proposal in preparation, decision in ~June
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Candidate site on old neutrino beamline
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Science of Holographic Noise
Measure fundamental interval of time Measure all physical degrees of freedom: explore physics
from above
Study holographic relationship between space and time,emergence of spatial dimensions
Precisely compare noise spectrum with Planck time derivedfrom Newtons G: test fundamental theory
Test predictions for spectrum and spatial correlations:properties of holographic geometry
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Projects in phenomenology
Calculate spectrum via a different argument
Exact numerics of black hole/flat space system: normalizationof value of effective lambda to black hole physics
Full quantum wave model of apparatus, spacetime, signal Numerically evaluate displacement spectrum at all f Numerically evaluate signal spectrum from displacement
correlation function for various devices
Develop theory of cross correlation for arbitrary interferometeroffsets and orientations, numerical predictions
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Projects in Theory
Use Matrix theory interpretation to bridge to strings
Secure lambda normalization to black hole entropy Generalize to curved spacetime backgrounds What happens inside black holes Relation to field theory Effect on inflationary modes (scalar, vector, tensor) Effect on quantum field modes (zero point energy) Cosmological observables (CMB, DE) Corrections to quantum processes Effect for masses less than M_P (atom interferometers)
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Holographic geometry: part of new dark energy physics?
Holographic blurring is ~0.1mm at the Hubble length
~(0.1mm)^-4 is the dark energy density Nonlocality length for dark energy is holographic
displacement uncertainty, scaled to Hubble length
(literature on holographic dark energy centers on samenumerology)
Does not explain dark energy!
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Items to discuss at the Hannover workshop
What is the status of the GEO600 mystery noise? What are the prospects for GEO600 to test the holographic
noise hypothesis?
Are the theoretical arguments strong enough to motivate anew, dedicated high-frequency experiment, independent of the
results from GEO600?
What are the optimal design choices? (configuration, size,power,...)
Will there be two experiments? (Fermilab and Hannover?) If so, what will be the similarities and differences betweenthem? What about LIGO?
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