analogue gravity: an overview of recent experimentsone-day relativity workshop - ift, 2019 analogue...

one -day re lat i v i ty work shop - i f t , 2 0 19

Analogue Gravity: an overview of

recent experiments

Maurıcio Richartz ([email protected])Universidade Federal do ABC (UFABC), Brasil

(Collaborators: S. Weinfurtner, A. Geelmuyden, ...)November 21, 2019


Motivation

– Some processes in the Universe(specially when gravity and quan-tum physics are both important)are impossible (or extremely hard)to observe.

– Can we set up experiments to re-build the Universe in the labora-tory?

– Can we study fundamental pro-cesses (like Hawking radiation andcosmic inflation) using analogueclassical or quantum simulators?

Maurıcio Richartz, UFABC — Analogue Gravity: an overview of recent experiments — November 21, 2019

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Experimental black hole evaporation?


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What are analogue models of gravity?

– Short answer: analogue models are systems that can mimickcurved spacetime effects (both classical and quantum), like Hawk-ing radiation and superradiance, leading naturally to the ques-tion: “Can we see black hole evaporation and black hole super-radiance in a laboratory?”

Analogue model ↔ Gravity

– They were first proposed by Unruh in 1981. In 1980, however,Moncrief had already used similar ideas to study accretion pro-cesses in black holes. In 1993, unaware of Unruh’s paper, MattVisser rediscovered the analogy.


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Simplest case: sound wavesBill Unruh, when trying to explain for a broad audience what a blackhole is, came up with the following pictorial analogy:

Ref.: Unruh (Found Phys, 2014)


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In 1980, when teaching a Fluid Mechanics course, he realized howgood his analogy was by showing that sound waves propagate as ifthey were in a curved spacetime. Using the basic equations for aninviscid, barotropic, and irrotational flow he showed that pressureperturbations (i.e. sound waves) propagate as a massless field in acurved spacetime.In other words: velocity perturbations (which are related to a scalarfield ψ) satisfy the Klein-Gordon equation in a curved spacetimewhose metric is

ds2 ∝ −(c2 − v2

)dt2 + 2vdrdt + r2dΩ2,

where v is the velocity of the fluid and c denotes the speed of sound.


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Analogue black holes

Compare the acoustic metric

ds2 ∝ −(c2 − v2

)dt2 + 2vdrdt + dr2 + r2dΩ2.

with the Schwarzschild black hole metric in the Painleve-Gullstrandcoordinates:

ds2 = −(

1− 2M

r

)dt2 ± 2

√2M

rdrdt + dr2 + r2dΩ2.

Recall that the coordinate singularity r = 2M is the event horizonof a black hole (or of a white hole). Following the same steps, onecan show that the points where |v | = c represent an analogue eventhorizon.


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Following Hawking’s derivation of black hole evaporation step-by-step, one is able to conclude that acoustic black holes should emitHawking-like radiation with a temperature of

T =~

2πckB

∂

∂r

(c2 − v2

)∣∣∣∣|v |=c

There is, however, a problem with Hawking’s original derivation:when traced back in time, outgoing Hawking radiation modes haveinfinitely high frequencies (which is inconsistent with the semiclas-sical approximation). The first success of analogue models was toprovide a possible solution to this problem.


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Recent experiments (Nottingham)


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Recent experiments in Nottingham

– Superradiant scattering

– Characteristic oscillations

– Hawking radiation (ongoing)


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Based on the propagation of surface waves on the air-water interfaceof a rectangular tank.

Ref.: Euve et al (1806.05539, 2018)

The main advantage of using surface waves (in comparison withsound waves) is that the propagation speed of the waves can beeasily controlled by changing the depth of the fluid since cwaves =√gh. These velocities can be made much smaller than the sound

speed in water (∼ 1400 m/s), thus requiring slower flows to set upan event horizon.


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Set up an analogue rotating black hole


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• Water tank: ∼ 1,5 m × 3,0 m.

• Closed circuit, water pumped from auxilliary tank (below maintank). Stationary rotating flow obtained.

• Wave generator on one side, absorption beach on another.

• Cameras and detector placed above the tank (close to theceiling).




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Background flow

– Flow: constant density, inviscid, irrotational, and axisymmetric.

∇ · ~v = 0, ρ(∂t + ~v · ∇)~v = −∇P, ∇× ~v = 0

– Free boundary problem: depth h is unknown a priori.

– The simplest solution is the Irrotational vortex (“Drainingbathtub”): ~v = −D

r r + Cr φ

– If surface waves propagate with velocity c(r), an analogue eventhorizon will be present where c(r) = D/r .


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Wave propagation

– Surface waves with frequency ω and wavenumber k satisfy thedispersion relation(

ω − ~v · ~k)2

=

(gk +

σ

ρk3

)tanh(kh).

– For long wavelengths (kh << 1), no surface tension (σ = 0) andzero flow (~v = 0), the dispersion relation is linear ω =

√ghk,

and we can see that c =√gh is the wave speed.

Analogy (linear dispersion):

(∂t + ~v · ∇)2 δh = gh∇2δh ↔ ∂µ(√−ggµν∂ν∂ψ

)= 0


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Detection methods

How do we measure the flow velocity?

– Particle Image Velocimetry (PIV)

How do we measure the height perturbations (i.e. waves on thesurface)?

– Depending on the experiment, we can use two differentmethods: “Enshape sensor” and Fast CheckerboardDemodulation (FCD).


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The velocity field can be measured by monitoring particles on the surface

Particle Image Velocimetry (PIV)

Particles

The velocity field can be measured by monitoring particles on the surface

Particle Image Velocimetry (PIV)


E.g. of background flow (from PIV)

0.1 0.15 0.2

0.1

0.15

0.2

Green dots correpond to the averaged (over time, angle andexperiments) angular velocity.Maurıcio Richartz, UFABC — Analogue Gravity: an overview of recent experiments — November 21, 2019

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Detection methods: Enshape


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Reconstructed vortex:


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Detection methods: FCD

A periodic pattern composed of two orthogonal sinusoidal waves isplaced at the bottom of the tank. Deformations of this pattern dueto free surface fluctuations are recorded using a high speed camera(usually at a frame rate of 40 fps–200 fps) for a pre-defined amountof time (usually 10s–30s). For each of the pictures of the deformedpattern, we reconstruct the free surface in the form of an arrayh(tk , xi , yj) giving the height of the water at the 1600×1600 pointson the free surface (xi , yj) at every time step tk . We choose thecentre of our coordinate system to be the centre of the hole andconvert the signal from cartesian to polar coordinates. We end upwith h(tk , ri , θj).


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Camera

The Fast Checkerboard Demodulation (FCD) method

Initial pattern

Camera


Deformed patternInitial pattern

Reconstructed surface Deformed patternInitial pattern



Experiment 1: superradiance

Superradiance is a special type of scattering in which the reflectedwave has larger amplitude than the incident wave. We call it rota-tional superradiance when the energy extracted is rotational energy.

Occurs when 0 < ω < mΩ. Was discovered by Zel’dovich (1971).Only waves corotating with the cylinder (m > 0) are amplified.


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Zel’dovich (1971)


Solve the wave equation near the event horizon and far away fromit. Impose the appropriate boundary condition at the event horizon.Calculate the reflection coefficient:

|Rω|2 = 1−(ω −mΩ

ω

)|Tω|2

No evidence/measurements of superradiance in Astrophysics.Can we see the effect in analogue models? Basak, Majumdar(2002).


• Plane wave (created by the wave generator) propagates andscatters on the vortex. We work in the frequency range 2,9 Hz- 4,1 Hz. The fluid is 6.25 cm deep.

• The observed pattern is a mixture: incident + scattered waves.

Data analysis for superradiance

– The data acquired by the detector gives us the surface of the fluidas a function of time and position. We fourier transform the sig-nal in time and filter out the excitation frequency f0. Higherharmonics (2f0, 3f0, ...) and background flow (f = 0) are elimi-nated.

– We decompose the signal into azimuthal waves:

ϕm(r) =

√r

2π

∫ 2π

0φ(r , θ)e imθdθ

For each m, we write radial profile as: ϕm = Ainme−ikr +Aout

m e+ikr

– By comparing the ingoing and outgoing parts, we extract thereflection coefficients.

x(m)

y(m) E

-0.5 0 0.5

-0.4

-0.2

0

0.2

0.4

x(m)

y(m) A

-0.5 0 0.5

-0.4

-0.2

0

0.2

0.4x(m)

y(m) B

-0.5 0 0.5

-0.4

-0.2

0

0.2

0.4

x(m)

y(m) C

-0.5 0 0.5

-0.4

-0.2

0

0.2

0.4x(m)

y(m) D

-0.5 0 0.5

-0.4

-0.2

0

0.2

0.4

x(m)

y(m) F

-0.5 0 0.5

-0.4

-0.2

0

0.2

0.4

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Radial profiles

-200 -100 0 100 200

m =

-2

0

0.1

0.2

-200 -100 0 100 200

m =

-1

0

0.1

0.2

-200 -100 0 100 200

m =

0

0

0.1

0.2

-200 -100 0 100 200

m =

1

0

0.1

0.2

-200 -100 0 100 200

m =

2

0

0.1

0.2

0.2 0.25 0.3 0.35 0.4-1

0

1

0.2 0.25 0.3 0.35 0.4-1

0

1

0.2 0.25 0.3 0.35 0.4-2

0

2

0.2 0.25 0.3 0.35 0.4-2

0

2

0.2 0.25 0.3 0.35 0.4-2

0

2

Right side

• Dots: signal

• Lines: fit to

Aine−ikr +Aoute

ikr

Left side

• Fourier profiles

Observed spectrum

f (Hz)2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2

R

0

0.5

1

1.5without vortexwith vortex, m= +2with vortex, m= +1with vortex, m= 0with vortex, m= -1with vortex, m= -2


Reflection coefficient for f = 3,7 Hz

m-2 -1 0 1 2

R

0

0.2

0.4

0.6

0.8

1

1.2

1.4

with vortexwithout vortex

Vortex flow:

• m ≤ 0 modes areabsorbed (|R| < 1);

• m = 1, 2 modes areamplified (|R| > 1).

• Amplification for m = 2modes was 14%± 8%:superradiance,arXiv:1612.06180

Without vortex:

• No scattering: |R| < 1.


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Experiment 2: characteristic modes

– Rotating BHs are simple objects (in the sense that they dependonly on two parameters: mass M and angular momentum J).

– Once perturbed, a BH will relax through the emission of charac-teristic waves which are independent of the initial perturbation.


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– BH spectroscopy: to identify a black hole by observing the spec-trum of its characteristic waves (their oscillation frequencies anddecay times).

– Vortex spectroscopy: can we identify a hydrodynamical vortexby observing the spectrum of its characteristic waves?


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Theoretical predictions for the

characteristic modes– The characteristic spectrum can be approximated using the prop-

erties of lightrings (LRs):

ωQNM(m) ≈ ω?(m)− iΛ(m)

(n +

1

2

),

where ω?(m) = 2πf?(m) is the angular frequency of an m-modeorbiting on the LR, Λ(m) is the Lyapunov exponent of the orbitfor this specific m-mode, and n is the overtone number.

– The LR properties can be deduced from the hamiltonian

H = ω − ~v · ~k = ±

√(gk +

σ

ρk3

)tanh(kh)

by looking for critical points: H = 0, ∂H∂r = 0, ∂H∂kr = 0.


– We set up a vortex flow out of equilibrium to observe the emissionof characteristic modes during its relaxation. Our experimentwas conducted in a 3 m long and 1.5 m wide rectangular tankwith a 2 cm-radius sink hole at the centre. Water is pumpedcontinuously from one corner at a flow rate of 15± 1 `/min.

– The sink-hole is covered until the water raises to a height of10.00± 0.05 cm. Water is then allowed to drain, leading to theformation of a vortex. We recorded the perturbations of the freesurface when the flow was in a quasi-stationary state at a waterdepth of 5.55± 0.05 cm. The entire procedure was repeated 25times.

– The data acquired by the detector gives us the surface of thefluid as a function of time and position.

Data Analysis

– The resulting vortex is axisymmetric to a good approximation,allowing us to perform an azimuthal decomposition to study thecharacteristic modes:

δh(t, r , θ) = Re

[∑m∈Z

δhm(t, r)e imθ

]

– We select specific azimuthal modes by performing a polar Fouriertransform and we extract the associated radial profiles δhm(t, r).Azimuthal modes with m > 0 are co-rotating with the flow whilemodes with m < 0 are counter-rotating with the flow.

– By calculating the time Fourier transform of δhm(t, r), we es-timate the Power Spectral Density (PSD) of each m-mode forr ∈ [7.4 cm, 25 cm].

PSDs in a single experiment

0.1 0.15 0.2 0.250

5

10

15

0.1 0.15 0.2 0.250

5

10

15

0.1 0.15 0.2 0.250

5

10

15

0.1 0.15 0.2 0.250

5

10

15

0.1 0.15 0.2 0.250

5

10

15

0.1 0.15 0.2 0.250

5

10

15

0.1 0.15 0.2 0.250

5

10

15

0.1 0.15 0.2 0.250

5

10

15

0.1 0.15 0.2 0.250

5

10

15

0.1 0.15 0.2 0.250

5

10

15

0 0.2 0.4 0.6 0.8 1

Averaged PSDs

The PSDs are finally averaged over the radius in order to look atthe r-independent frequency content, i.e. the oscillation frequencyof the LR modes. For each averaged PSD, corresponding to adifferent m, the location of the peak, fpeak(m), is obtained.

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2


Characteristic spectrum


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Characteristic spectrum

-25 -20 -15 -10 -5 00.5

1

1.5

2

2.5

3

3.5

4

4.5

5


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-25 -20 -15 -10 -5 00.5

1

1.5

2

2.5

3

3.5

4

4.5

5

The frequency spectrum fpeak(m), extracted from the experimentaldata and represented by green dots, is compared with the theoret-ical prediction for the light-ring frequencies, f?(m). The error barsindicate the standard deviation over 25 experiments. The dashedorange curve is the predicted spectrum, f PIV? (m), computed forC = 151 cm2/s and D = 0 cm2/s. These flow parameters wereobtained via an independent flow measurement, in our case Par-ticle Imaging Velocimetry (PIV). The two spectra agree, confirm-ing the detection of light-ring mode oscillations. The solid blackcurve, f BM

? (m), is the non-linear regression of the experimentaldata to the draining bathtub vortex model - arXiv:1811.07858and 1905.00356.


Vortex spectroscopy

50 100 150

50

100

150

200

250

300

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1140 150 160

10

20

30

40

The intensity of the background image represents the normalisedweighted sum of squared residuals between the experimental spec-trum, fpeak(m), and the theoretical prediction for the light-ring fre-quencies, f?(m), as a function of the flow parameters. The red curverepresents the family of possible values for C and D that best matchthe experimental data.


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Vortex spectroscopy

50 100 150

50

100

150

200

250

300

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1140 150 160

10

20

30

40


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Conclusions

– Analogue models of gravity have been around for more than 35years, but only in the past 10 years people have started doingexperiments with them. Most experiments are based on 1D sys-tems, so we want to improve and explore more possibilities in 2Dsystems.

– We have been able to observe superradiant scattering and char-acteristic oscillations of a vortex.

– Next steps: match superradiance data with theoretical predic-tion, observe the characteristic decay times of the vortex, bettervisualization of the core, measurement of Hawking radiation.


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analogue gravity: an overview of recent experimentsone-day relativity workshop - ift, 2019 analogue...

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