Vorticity in the Acoustic Analogue of Gravity. Part 1 [PDF]

School of Mathematical and Computing Sciences Te Kura Pangarau, Rorohiko

Vorticity in the acoustic analogue of gravity

Matt Visser

Text School of Mathematical and Computing Sciences Te Kura Pangarau, Rorohiko

And I cherish more than anything else the Analogies, my most trustworthy masters.

They know all the secrets of Nature, and they ought least to be neglected in Geometry.

--- Johannes Kepler

Text

Abstract:

Sound waves propagating in a ﬂowing ﬂuid can be described by a wave equation on an “acoustic geometry”.

This geometry is described by an “effective metric” that leads to a “curved spacetime” qualitatively similar to that of general relativity.

In particular, it is relatively easy to set up “dumb holes” which trap sound in the same way that the black holes of general relativity trap light.

rTelativext ely easy to set up "dum b holes" which trap sound in the same way that the black holes of gener al relativ ity trap light.

Vorte x ﬂow create s an "acou stic geom etry" that is qualit atively simila r to the equat orial plane of a rotati ng Kerr black hole. There is an acous tic analog y to the gneral relativ ity notio n of an "ergo spher e" as well as to the gener al relativ ity notio n of "horiz on".

I will ﬁrst qualit atively explai n these ideas, and descri be why they are useful, and then make a quanti tative comp arison betwe en the analog acous tic and gener al relativ istic sytem s. Abstract:

Vortex ﬂow creates an “acoustic geometry” that is qualitatively similar to the equatorial plane of a rotating Kerr black hole.

There is an acoustic analogy to the general relativity notion of an “ergosphere” as well as to the general relativity notion of “horizon”.

I will ﬁrst qualitatively explain these ideas, and describe why they are useful, and then make a quantitative comparison between the analog acoustic and general relativistic sytems. Key idea:

Consider sound waves in a flowing fluid.

If the fluid is moving faster than sound, then the sound waves are swept along with the flow, and cannot escape from that region.

This sounds awfully similar to a black hole in general relativity --- is there any connection?

Yes!

(Of course the devil is in the details...)

Key results:

Acoustic propagation in ﬂuids can be described in terms of Lorentzian differential geometry.

The acoustic metric depends algebraically on the ﬂuid ﬂow.

Acoustic geometry shares kinematic aspects of general relativity, but not the dynamics.

Einstein equations versus Euler equation.

In particular: Acoustic black holes divorce kinematic aspects of black hole physics from the speciﬁc dynamics due to the Einstein equations. Advanced features:

There are also other “analogue models” of general relativity, apart from the acoustic models.

Acoustic black holes have Hawking radiation without black hole entropy.

Hawking radiation is a purely kinematic effect that exists independent of whether or not the Lorentzian geometry obeys any particular geometrodynamics.

You do not need the Einstein equations to get Hawking radiation.

April 2000 Sponsored by:

Photonics Technology News Send News to: [email protected]

Water down the Drain Leads to Black Holes in the Lab

Archimedes, the famous story goes, cried, "Eureka!" and leapt from his overflowing bathtub, having discovered the principle of buoyancy. But the drain has much to offer, too. Physicists Ulf Leonhardt and Paul Piwnicki of the Royal Institute of Technology suggest in the Jan. 31 issue of Physical Review Letters that vortices or other rapid flows analogous to the swirl of water down the drain may enable researchers to model the relativistic effects Sponsoredof black holes.by: April 2000 AprilSponsored 2000 by: Leonhardt explained that, as early as 1818, Augustin-Jean Fresnel predicted from theories of the luminiferous ether that a flowing medium will drag light with it. "For quite a long time we Photonics Technology Photonics Technology were rather puzzled and could not understand what is happening in moving media if one takes them seriously," said Leonhardt. "Then, News News suddenly, the picture emerged, and we realized the connection to Send News to: Send News to: gravity and curved space-time. This has come as a surprise." [email protected] The researchers demonstrated that, if a medium swirls more [email protected] quickly than the speed of light through it, the vortex it forms can Advertising decelerate and trap light in much the same way as a black hole. If Information they could be created in the lab, these optical black holes would Water down the Drain Leads enable physicists to test such controversial concepts as Hawking Water down the Drainradiation, the emissions Leads by black holes theorized by Stephen W. Hawking. The problem is that no known materials display to Black Holes in the LabApril 2000superluminal rotation without also formingSponsored a hollow core by: that is to Black Holes in thelarger than Lab the critical optical event horizon. But recent efforts by Lene Vestergaard Hau of Harvard Archimedes, the famous story goes, cried, "Eureka!" and leapt University in Cambridge, Mass., have dramatically slowed light from his overflowing bathtub,Archimedes, having discovered the famous the principle story goes, of cried,with Bose-Einstein "Eureka!" condensates and leapt (see Photonics Spectra, April buoyancy. But the drain has frommuch histo offer, overflowing too. Physicists bathtub,Photonics Ulf having 1999, discoveredTechnology pages 26-27), the andprinciple they could of produce the 1-cm/s speed limit Leonhardt and Paul Piwnicki of the Royal Institute of Technology required by Leonhardt and Piwnicki’s model. Leonhardt also buoyancy. But the drainNews has much to offer,suggested too. that Physicists rotating alkali Ulf vapor could create a "weak" black suggest in the Jan. 31 issue ofLeonhardt Physical Reviewand Paul Letters Piwnicki that of the Royalhole that Institute only traps of light Technology moving against its current, as may other vortices or other rapid flows suggestanalogous in tothe the Jan. swirl 31 of Sendissue water of Newsdown Physical to: Review Letters that the drain may enable researchersvortices to model or other the [email protected] flows effects analogous to the swirl of water down of black holes. Leonhardt explained that, theas early drain as may 1818, enable Augustin-Jean researchers to model the relativistic effects Fresnel predicted from theoriesof blackof the holes.luminiferous ether that a flowing medium will drag light Leonhardt with it. "For explained quite a long that, time as weearly as 1818, Augustin-Jean were rather puzzled and couldFresnel not understand predicted what from is theorieshappening of in the luminiferous ether that a moving media if one takes themflowing seriously," medium said will Leonhardt. dragWater light "Then, with it. "Fordown quite a long timethe we Drain Leads suddenly, the picture emerged,were and rather we realized puzzled the and connection could not to understand what is happening in gravity and curved space-time.moving This has media come if asone a surprise." takesto them Black seriously," said Holes Leonhardt. "Then, in the Lab The researchers demonstratedsuddenly, that, if thea medium picture swirls emerged, more and we realized the connection to quickly than the speed of lightgravity through and it, curvedthe vortex space-time. it forms can This has come as a surprise." Advertising decelerate and trap light in much The the researchers same way as demonstrated a blackArchimedes, hole. Ifthat, if the a medium famous swirls story more goes, cried, "Eureka!" and leapt Information they could be created in the lab,quickly these than optical the black speed holes of light would through it, the vortex it forms can enable physicists to test such controversial concepts fromas Hawking his overflowing bathtub, having discovered the principle of decelerate and trap light in much the same way as a black hole. If radiation,Advertising the emissions by black holes theorized by buoyancy.Stephen W. But the drain has much to offer, too. Physicists Ulf Hawking.Information The problem is thatthey no knowncould bematerials created display in the lab, these optical black holes would superluminal rotation withoutenable also forming physicists a hollow to testLeonhardt core such that controversial is and Paul concepts Piwnicki as Hawking of the Royal Institute of Technology larger than the critical opticalradiation, event horizon. the emissions suggest by black in holes the theorized Jan. 31 by issue Stephen of W.Physical Review Letters that But recent efforts by LeneHawking. Vestergaard The Hau problem of Harvard is that no known materials display University in Cambridge, Mass.,superluminal have dramatically rotation slowed vorticeswithout light also or forming other arapid hollow flows core that analogous is to the swirl of water down with Bose-Einstein condensateslarger (see than Photonics the critical Spectrathe optical, April drain event may horizon. enable researchers to model the relativistic effects 1999, pages 26-27), and they could But recentproduce efforts the 1-cm/s byof Lene speedblack Vestergaard limit holes. Hau of Harvard required by Leonhardt and Piwnicki’s model. Leonhardt also University in Cambridge, Leonhardt Mass., have dramatically explained slowed that, lightas early as 1818, Augustin-Jean suggested that rotating alkaliwith vapor Bose-Einstein could create a condensates"weak" black (see Photonics Spectra, April hole that only traps light moving1999, against pages its 26-27), current, and Fresnelas maythey other could predicted produce the from 1-cm/s theories speed limit of the luminiferous ether that a required by Leonhardtflowing and Piwnicki’s medium model. will Leonhardt drag alsolight with it. "For quite a long time we suggested that rotatingwere alkali rathervapor could puzzled create anda "weak" could black not understand what is happening in hole that only traps light moving against its current, as may other moving media if one takes them seriously," said Leonhardt. "Then, suddenly, the picture emerged, and we realized the connection to gravity and curved space-time. This has come as a surprise." The researchers demonstrated that, if a medium swirls more quickly than the speed of light through it, the vortex it forms can Advertising decelerate and trap light in much the same way as a black hole. If Information they could be created in the lab, these optical black holes would enable physicists to test such controversial concepts as Hawking radiation, the emissions by black holes theorized by Stephen W. Hawking. The problem is that no known materials display superluminal rotation without also forming a hollow core that is larger than the critical optical event horizon. But recent efforts by Lene Vestergaard Hau of Harvard University in Cambridge, Mass., have dramatically slowed light with Bose-Einstein condensates (see Photonics Spectra, April 1999, pages 26-27), and they could produce the 1-cm/s speed limit required by Leonhardt and Piwnicki’s model. Leonhardt also suggested that rotating alkali vapor could create a "weak" black hole that only traps light moving against its current, as may other Sequencing Week of Feb. 5, 2000; Vol. 157, No. 6 Human Genome Frontiers of Sci Black hole recipe: Slow light, swirl atoms Technology Math Trek Recorded We A Minimal By P. Weiss Winter’s Tale Physicists may soon create Food for artificial black holes in the Browse a Sci Thought laboratory, analogous to the ones News photo coll Sickening Food expected to lurk in distant space. A new study by a pair of theorists in Science Safari Sweden describes how swirling Beams of Our clouds of atoms could slug down Subscribe to Sc Lives all nearby light, making them as News in spoken Computer-generated plot black as their astronomical format. shows paths of light rays TimeLine cousins. 70 Years Ago in sucked into optical black hole. (Leonhardt and Science News Called optical black holes, these Piwnicki/Physical Review A) eddies could provide an extraordinary test-bench for the theory of general relativity, which gave rise to the concept of gravitational black holes, the researchers say. Ulf Leonhardt and Paul Piwnicki of the Royal Institute of Technology in Stockholm find that the same mathematics describes both the terrible tug of an astronomical Visit our onl black hole on light and the gentle corralling of rays by an atom bookstore Sequencing vortex. Human Genome Week of Feb. 5, 2000; Vol. 157, No. 6 "We were quite surprised that it worked that well," Piwnicki Frontiers of Sci says. "We’re still working on it to understand it more deeply," he Black hole recipe: Slow light, swirl atoms Technology adds. The researchers report their findings in the Jan. 31 Physical Math Trek Recorded We Review Letters and the December 1999 Physical Review A. A Minimal By P. Weiss Winter’s Tale The laboratory analogy goes only so far, however. Black holes Order Science Physicists may soon create out in space are massive remnants of collapsed stars that pull in on CD-RO Food forM artificial black holes in the Browse a Sci not just light but everything else in their vicinity. By contrast, the Thought laboratory, analogous to the ones News photo coll proposed atomic whirlpools would have too little gravity to Sickening Food swallow any matter. Tiny tornadoes within wispy clouds of gas, expected to lurk in distant space. A they would snag photons through their remarkable ability to slow new study by a pair of theorists in light pulses. Science Safari Sweden describes how swirling Buy aBeams Science of Our clouds of atoms could slug down Subscribe to Sc The proposed mechanism by which such a vortex would capture PangaeaLives Mu all nearby light, making them as News in spoken Computer-generated plot light rests on principles discovered in the 1800s. Many black as their astronomical format. substances, such as water or glass, retard light as it passes shows paths of light rays TimeLine cousins. through them. Consequently, a fluid flow can drag light along 70 Years Ago in sucked into optical black hole. (Leonhardt and Science News Called optical black holes, these Piwnicki/Physical Review A) eddies could provide an extraordinary test-bench for the theory of general relativity, which gave rise to the concept of gravitational black holes, the researchers say. Ulf Leonhardt and Paul Piwnicki of the Royal Institute of Technology in Stockholm find that the same mathematics describes both the terrible tug of an astronomical Visit our onl black hole on light and the gentle corralling of rays by an atom bookstore vortex.

"We were quite surprised that it worked that well," Piwnicki says. "We’re still working on it to understand it more deeply," he adds. The researchers report their findings in the Jan. 31 Physical Review Letters and the December 1999 Physical Review A.

The laboratory analogy goes only so far, however. Black holes Order Science out in space are massive remnants of collapsed stars that pull in on CD-ROM not just light but everything else in their vicinity. By contrast, the proposed atomic whirlpools would have too little gravity to swallow any matter. Tiny tornadoes within wispy clouds of gas, they would snag photons through their remarkable ability to slow light pulses. Buy a Science The proposed mechanism by which such a vortex would capture Pangaea Mu light rests on principles discovered in the 1800s. Many substances, such as water or glass, retard light as it passes through them. Consequently, a fluid flow can drag light along

This Issue | Back Issues | Editorial Staff

Physicswatch

Optical black holes could be made in the lab This Issue | Back Issues | Editorial Staff Black holes could be created in the laboratory, claim physicists from Sweden and Scotland. Physicswatch Their calculations show that, thanks to some This Issue | Back Issues | Editorial Staff recent advances in condensed matter physics, it is theoretically possible to create an optical black hole to attract and trap specific colours of light in Optical black justholes the same could way as be an astronomicalmade in black the hole lab Physicswatch Optical black hole attracts and consumes matter. The researchers unearthed a 1923 paper in which Walter Gordon, startingBlack from holes Einstein’s could be idea created of gravity in the mediated laboratory, by changes in the metric ofclaim space physicists and time, realizedfrom Sweden that space-time and Scotland. is effectively a medium and Theirthat consequently, calculations any show moving that, dielectric thanks to some Optical black holes could be made in the lab medium acts on light as an effective gravitational field. However, to see effects as dramatic as a blackrecent hole, advances the velocity in condensed of light in the matter medium physics, it is must be low compared withtheoretically the velocity possibleof the medium. to create an optical black Black holes could be created in the laboratory, Recent results from US holephysicists to attract working and with trap Bose-Einstein specific colours of light in claim physicists from Sweden and Scotland. condensates (BEC) suggest just that the this same is feasible: way as quantized an astronomical vortices haveblack hole Their calculations show that, thanks to some beenOptical generated black in ahole BEC ofattracts rubidium and atoms, consumes while matter. light has been recent advances in condensed matter physics, it is slowed down inside a BEC toThe a mere researchers 50 cm/s, unearthed and even 1 a cm/s 1923 may paper in soon be achieved. theoretically possible to create an optical black which Walter Gordon, starting from Einstein’s idea of gravity mediated by A changesBEC vortex, in the swirling metric faster of space than theand light time, can realized move, would that space-time drag the is hole to attract and trap specific colours of light in lighteffectively into its centre,a medium where and it wouldthat consequently, eventually be absorbedany moving by the dielectric gas. An event horizon - a radius of no return - would form about the vortex, just the same way as an astronomical black hole medium acts on light as an effective gravitational field. However, to see attracts and consumes matter. just like the Schwarzschild radius of an astronomical black hole. Optical black hole effects Then, someas dramatic spectacular as a effectsblack ofhole, general the velocityrelativity ofcould light be in seen the in medium a The researchers unearthed a 1923 paper in terrestrialmust be lowlaboratory, compared perhaps with demonstrating the velocity anof analoguethe medium. of Hawking which Walter Gordon, starting from Einstein’s idea of gravity mediated radiation Recent from results black from holes US (obscured physicists by theworking cosmic with background Bose-Einstein by changes in the metric of space and time, realized that space-time is radiationcondensates and so(BEC) far never suggest observed that inthis astronomy) is feasible: and quantized even exploring vortices have effectively a medium and that consequently, any moving dielectric prototypebeen generated theories in of a quantum BEC of gravity.rubidium atoms, while light has been slowed However, down the insideexperimental a BEC difficulty to a mere in 50generating cm/s, and sufficiently even 1 cm/s fast may medium acts on light as an effective gravitational field. However, to see and durable BEC vortices means that a home-grown black hole is effects as dramatic as a black hole, the velocity of light in the medium soon be achieved. probably A BEC still vortex, about swirlingfive years faster away. than AIP the light can move, would drag the must be low compared with the velocity of the medium. light into its centre, where it would eventually be absorbed by the gas. Recent results from US physicists working with Bose-Einstein An event horizon - a radius of no return - would form about the vortex, condensates (BEC) suggest that this is feasible: quantized vortices have just like the Schwarzschild radius of an astronomical black hole. been generated in a BEC of rubidium atoms, while light has been Then, some spectacular effects of general relativity could be seen in a slowed down inside a BEC to a mere 50 cm/s, and even 1 cm/s may terrestrial laboratory, perhaps demonstrating an analogueArticle 10of Hawkingof 20. Copyright © IOP radiation from black holes (obscured by the cosmic background soon be achieved. Publishing Ltd 1998, A BEC vortex, swirling faster than the light can move, would1999, drag 2000. the All rights radiation and so far never observed in astronomy) and even exploring light into its centre, where it would eventually be absorbed byreserved the gas. prototype theories ofPrevious quantum article gravity. | Next article An event horizon - a radius of no return - would form about the vortex, However, the experimental difficulty in generating sufficiently fast just like the Schwarzschild radius of an astronomical black hole. and durable BEC vortices means that a home-grown black hole is probably still about five years away. AIP Then, some spectacular effects of general relativity could be seen in a terrestrial laboratory, perhaps demonstrating an analogue of Hawking radiation from black holes (obscured by the cosmic background radiation and so far never observed in astronomy) and even exploring prototype theories of quantum gravity. However, the experimental difficulty in generating sufficientlyCopyright fast © IOP Article 10 of 20. and durable BEC vortices means that a home-grown blackPublishing hole is Ltd 1998, probably still about five years away. AIP 1999, 2000. All rights reserved Previous article | Next article

Geometrical acoustics:

In a ﬂowing ﬂuid, if sound moves a distance d!x in time dt then

d!x !v dt = cs dt. || − || Write this as

(d!x !v dt) (d!x !v dt) = c2dt2. − · − s Now rearrange a little:

(c2 v2) dt2 2 !v d!x dt + d!x d!x = 0. − s − − · · Notation — four-dimensional coordinates:

xµ = (x0; xi) = (t; !x). Then you can write this as µ ν gµν dx dx = 0. With an eﬀective acoustic metric (c2 v2) . !v − s − − gµν(t, !x)   . ∝ · · · · · · · · · · · · .· · · · · · ·  !v . I   −  5 Geometrical acoustics:

In a ﬂowing ﬂuid, if sound moves a distance d!x in time dt then

d!x !v dt = cs dt. || − || Write this as

(d!x !v dt) (d!x !v dt) = c2dt2. Geometrical− · acoustics:− s Now rearrange a little:

(c2 v2) dt2 2 !v d!x dt + d!x d!x = 0. − s − − · · Notation — four-dimensional coordinates:

Eikonals:

The sound paths of geometrical acoustics are the null geodesics of this eﬀective metric.

Geometrical acoustics, by itself, does not give you enough information to ﬁx an overall mul- tiplicative factor (conformal factor).

Note: This also works for geometrical optics in a ﬂowing ﬂuid, with cs c/n; replace the speed of sound by the spe→ed of light in the medium (speed of light divided by refractive index).

This is already enough to give you some very powerful results:

Fermat’s principle is now a special case of geodesic propagation.

Sound focussing can be described by the Riemann tensor of this eﬀective metric.

But there is a lot more hiding in the woodwork. 6 Eikonals:

The sound paths of geometrical acoustics are the null geodesics of this eﬀective metric.

Geometrical acoustics, by itself, does not give you enough information to ﬁx an overall mul- tiplicative factor (conformal factor).

Note: This also works for geometrical optics in a ﬂowing ﬂuid, with cs c/n; replace the →Geometrical speed of sound by the speedacoustics:of light in the medium (speed of light divided by refractive index).

This is already enough to give you some very powerful results:

Fermat’s principle is now a special case of geodesic propagation.

Sound focussing can be described by the Riemann tensor of this eﬀective metric.

But there is a lot more hiding in the woodwork. 6 Physical acoustics:

The Acoustic metric:

Suppose you have a non-relativistic ﬂowing ﬂuid, governed by the Euler equation plus the continuity equation.

Suppose the ﬂuid ﬂow is barotropic, irrotational, and inviscid.

Suppose we look at linearized ﬂuctuations.

ThenThisthe thenline aleadsrized toﬂuc a watuavtionse equation...(aka sound waves, aka phonons) are described by a massless minimally coupled scalar ﬁeld propagating in a (3+1)-dimensional acoustic metric

(c2 v2) . "v ρ − − − gµν(t, "x)   . ≡ c · · · · · · · · · · · · .· · · · · · ·  "v . I   −  Proof: Unruh81, Visser93, Unruh94, Visser97. 7 The Acoustic metric:

Suppose you have a non-relativistic ﬂowing ﬂuid, governed by the Euler equation plus the continuity equation.

Suppose the fluid flow is barotropic, irrotational, and inviscid. Physical Suppose we look at linearizedacoustics:fluctuations.

Then the linearized fluctuations (aka sound Q: wHavowes,doakaacousticphonons)distare urdebascribncedes bpyropagata mass- e in alenon–homss minimallyogecouplneoused sflocalawringfieldfluid?propagating in a (3+1)-dimensional acoustic metric A: If the fluid is barotropic and inviscid, and (c2 v2) . "v the flow is irrotational,ρ − the− equation−of motion for thegµvνelo(t, "xcit) y potential can be put in. the ≡ c · · · · · · · · · · · · .· · · · · · ·  "v . I  (3 + 1)–dimensional form− 

Proof: Unruh81,1 Visser93, µUνnruh94, Visser97. ∆ψ ∂µ √ g g ∂νψ = 0. ≡ √ g − 7 − ! "

The acoustic metric is: • (c2 v2) . $v ρ − − − gµν(t, $x)   ≡ c · · · · · · · · · · · · .· · · · · · ·  $v . I .  − 

The underlying ﬂuid dynamics is Newtonian, • non–relativistic, in ﬂat space + time.

The ﬂuctuations (sound waves) are gov- • erned by a Lorentzian spacetime geometry! Physical acoustics:

Fundamental ﬂuid dynamics:

Continuity —

∂ ρ + (ρ#v) = 0. t ∇ ·

Euler (inviscid) —

ρ [∂ #v + (#v )#v] = p ρ φ. t · ∇ −∇ − ∇

Barotropic — p dp 1 ρ = ρ(p); ζ(p) = # ; ζ = p. !0 ρ(p#) ∇ ρ∇

Standard Manipulations:

Euler + Barotropic —

1 2 ∂t#v = #v ( #v) ( v + ζ + φ). × ∇ × − ∇ 2 Irrotational — #v = 0; #v = ψ. ∇ × −∇ Euler + Barotropic + Irrotational —

1 2 ∂tψ + ζ + ( ψ) + φ = 0. − 2 ∇ Fundamental ﬂuid dynamics:

Continuity —

∂ ρ + (ρ#v) = 0. t ∇ ·

Euler (inviscid) —

ρ [∂ #v + (#v )#v] = p ρ φ. t · ∇ −∇ − ∇

Barotropic — Physical p acoustics:dp 1 ρ = ρ(p); ζ(p) = # ; ζ = p. !0 ρ(p#) ∇ ρ∇

Standard Manipulations:

Euler + Barotropic —

1 2 ∂t#v = #v ( #v) ( v + ζ + φ). × ∇ × − ∇ 2 Irrotational — #v = 0; #v = ψ. ∇ × −∇ Euler + Barotropic + Irrotational —

1 2 ∂tψ + ζ + ( ψ) + φ = 0. − 2 ∇ Linearize:

ρ ρ0 + "ρ1; p p0 + "p1; ψ ψ0 + "ψ1; φ φ0; ≡ ≡ ≡ ≡

∂ρ ζ ζ0 + "ζ1 = ζ0 + "(p1/ρ0); ρ1 = p1. ≡ ∂p

Sound linearized ﬂuctuations. • ≡ Physical acoustics: Continuity —

Linearize∂:tρ1 + (ρ1'v0 + ρ0'v1) = 0. ∇ ·

ρ ρ0 + "ρ1; p p0 + "p1; ψ ψ0 + "ψ1; φ φ0; Eule≡ r — ≡ ≡ ≡ p ∂ ψ + 1 'v ψ = 0. ∂ρ ζ ζ0 +t "ζ11 = ζ0 + "(p01/ρ0);1 ρ1 = p1. ≡ − ρ0 − · ∇ ∂p

p1 = ρ0(∂tψ1 + 'v0 ψ1). Sound linearized ﬂuctuat· ∇ions. • ≡ Substitute the linearized Euler equation into Continuit• y — the linearized continuity equation. This gives a wave equat∂ ρ ion+ for(ρthe'v scala+ ρ r'v p)otential.= 0. t 1 ∇ · 1 0 0 1

Euler — p1 ∂tψ1 + 'v0 ψ1 = 0. − ρ0 − · ∇ p = ρ (∂ ψ + 'v ψ ). 1 0 t 1 0 · ∇ 1

Substitute the linearized Euler equation into • the linearized continuity equation. This gives a wave equation for the scalar potential. LineaLinearize:rize:

ρ ρ0 + "ρ1; p p0 + "p1; ψ ψ0 + "ψ1; φ φ0; ρ ρ0 +≡"ρ1; p p0≡+ "p1; ψ ψ≡0 + "ψ1; φ ≡φ0; ≡ ≡ ≡ ≡ ∂ρ ζ ζ0 + "ζ1 = ζ0 + "(p1/ρ0); ρ1 =∂ρ p1. ζ ζ0 +≡ "ζ1 = ζ0 + "(p1/ρ0); ρ1 =Physical∂pp1. ≡ acoustics:∂p Sound linearized ﬂuctuations. SoundLinearized• line≡ equations:arized ﬂuctuations. • ≡ Continuity — Continuity — ∂ ρ + (ρ 'v + ρ 'v ) = 0. t 1 ∇ · 1 0 0 1 ∂ ρ + (ρ 'v + ρ 'v ) = 0. t 1 ∇ · 1 0 0 1 Euler — p1 Euler — ∂tψ1 + 'v0 ψ1 = 0. − ρ0 − · ∇ p1 ∂tψ1 + 'v0 ψ1 = 0. Rearrange:− p1 =ρρ0(−∂tψ1·+∇'v0 ψ1). 0 · ∇

p1 = ρ0(∂tψ1 + 'v0 ψ1). Substitute the linearized·Eul∇ er equation into • the linearized continuity equation. This gives Subsa titutewave ethequatlineionaforirzethed Eulscalaer requationpotential.into • the linearized continuity equation. This gives a wave equation for the scalar potential. Physical acoustics:

Wave equation:

∂ρ ∂t ρ0 (∂tψ1 + $v0 ψ1) − ∂p · ∇ ! " ∂ρ + ρ0 ψ1 ρ0$v0 (∂tψ1 + $v0 ψ1) = 0. ∇ · ∇ − ∂p · ∇ ! "

Coeﬃcients ρ , $v , and 1/c2 (∂ρ/∂p) can 0 0 ≡ have arbitrary time and space dependencies.

Deﬁne: . j 1 . v0 µν ρ0 − − f 2   ≡ c · · · ·i· · .· ·2· · ·ij· · · · ·i· ·j·  v0 . (c δ v0v0) .  − − 

Then: µν ∂µ(f ∂νψ1) = 0.

That’s it! Everything else is now technical ﬁddling. Wave equation:

∂ρ ∂t ρ0 (∂tψ1 + $v0 ψ1) − ∂p · ∇ ! " ∂ρ + ρ0 ψ1 ρ0$v0 (∂tψ1 + $v0 ψ1) = 0. ∇ · ∇ − ∂p · ∇ ! "

Coeﬃcients ρ , $v , and 1/c2 (∂ρ/∂p) can 0 0 Physical≡ have arbitrary time and spaceacoustics:dependencies.

Deﬁne: . j 1 . v0 µν ρ0 − − f 2   ≡ c · · · ·i· · .· ·2· · ·ij· · · · ·i· ·j·  v0 . (c δ v0v0) .  − − 

Then: µν ∂µ(f ∂νψ1) = 0.

That’s really it! That’Everythings it! elseEvery is minorthing technicalelse is finoddling.w technical fiddling. Physical acoustics: µν IntroducRewriing thete theacousticwave equationmetric g as:, defined by

1 µν ∆ψ ∂µ √ g g ∂νψ = 0. 1 f≡µ√ν =g√ g−gµν; g = (35) Introducing the acoustic metric− gµ−ν!, defined by " det(gµν) — Then 1 Introductheingwatvhee equaacoustictioµνn (33metric√) correspµgνµν,odendsfinedto thebynormal d’Alembertian wave equa- f = g g ; g = µν (35) tion in a curved-space:g−= det(f µν) = det(ρ4/cg2. ) − 0 µν µν 1 the wave equation f(33) is=co independentrresp√ gondsg of;to thethe numbergno=rmal of2 spaced’A dimensions.lembertTian2wave equa(3- 5) 2/(d 1) 1/c µν "v /c —µνAnd ρ− − − det(g ) tion in a curved-space: depends on the number− of space dimensions.− g = . 2 j T 2 , (36) c 1 . "v/c v Id d "v "v /c the wave equation (33) correspo1nds −to the2# −normal−Td−0’A2le×mb−ertian wave&equa- ρ gµ2ν/!(d 1)" 1/c "v /c gµν = − ≡−ρ c −· · · · · · · · · ·−· · · · · · · · ·  , (36) tion in a curved-space: 0 vi 2 . (c2δij vi$vTj ) 2 % where d is thec dimension# of"vsp/c0 ace (noIdt spad c"veti0"v0me/c.). & ! " −− − × −−  The covariant acoustic metr2 ic is T 2 2/(d 1) 1/c $ "v /c % where d isµνthe dimensionρThis− is−aof(3+spac1)–dime−e (not spansionalcetimespac−). etime metric g = 2/(d2 1) 2 2 T 2 T , (36) The covariantcaincousticADM metrform.icρis"v/c − Id(cd "vv "v) /c "v gµν =# − − − × −− | & . (37) ! " c "v Id d ρ 2/(d 1) 2 2 T × −! " 2(c '2 $v ). "v j | % ( where d is the dimensiongµν = of space (no(−tc spav−c0eti) me. |). v0 . (37) c ρ0 − −"v −Id d gµν ' × ( The co3.va1riantda=coustic3 !metr" ≡ icc is · · · · · · · · · · · · · | · · · · · ·  vi . δij  − 0 . 3.1 d = 3 2/(d 1)  2 2 T The acoustic line-eleρ men− t for (tchree vspace) and"v one time dimension reads gµν = − − | . (37) The acoustic line-elemenc t for three space"vand 2one time2Id dimensiond T reads ! " ' ρ (c |v ) × ("v gµν = 2 −2 − T | . (38) ρ (cc v ) "v "v I3 3 3.1 d = 3 gµν = −! "−' | .| × ( (38) c "v I3 3 This is the primary!case" ' of interest in| this× article.( The Thisacousticis theline-eleprimarymencaset foofr inthreeterestspacein thisandarticle.one time dimension reads 3.2 d = 2 ρ (c2 v2) "vT 3.2 d = 2 gµν = − − | . (38) c "v I3 3 The acoustic line-elemen' t for two space| and× o(ne time dimension reads The acoustic line-elemen!t fo"r two space and one time dimension reads This is the primary case of interest in this2 article.2 2 T 2 ρ 2 2 (c vT ) "v gρµν = (c v−) −"v | . (39) gµν = −c − |"v . I2 2 (39) c "v' I2 2 | × ( 3.2 d = 2 ! " ' ! " | × ( This situaThistionsituawouldtion bweouldapproprbe aiate,pproprfor iate,instancfore,instawhenncdee,alingwhenwithdeasurlingfacewith surface The wacaovusticeswoarvexcitline-elees oratioexcitmenns confinedatiot fonsr tconfinedwtoo aspaceparticulartoanda parosubsneticulartimetrate.subsdimenstratioe.n reads ρ 2 (c2 v2) "vT 3.3 3.d3= 1d =gµν1= − − | . (39) c "v I2 2 ' | × ( The naivThee naivformeofforthem !ofacoustic"the acousticmetric inmetric(1+1)indimens(1+1io)nsdimensis ill-defineions isd, ill-defined, Thisbsituaecausebtionectheausewconforouldthebmalconfore approprfactormalisiate,factorraisedforistoinstaraiseda forncmallye,towhena infiniteformallydealingpoinfinitewerwith— sthispurofwaerce — this waves or excitations confined to a particular substrate. 12 12 3.3 d = 1 The naive form of the acoustic metric in (1+1) dimensions is ill-defined, because the conformal factor is raised to a formally infinite power — this

12 Introducing the acoustic metric gµν, deﬁned by

Introducing the acoustic metric gµν, deﬁned by1 µν √ µν f = Physicalg g acoustics:; g = µν (35) − det(1g ) f µν = √ g gµν; g = (35) − det(gµν) the wave equatioInn (33(d+1)) co dimensions:rresponds to the normal d’Alembertian wave equation inthea curvwaveed-space:equation (33) corresponds to the normal d’Alembertian wave equation in a curvInved-space:erse metric: 2 T 2 2/(d 1) ρ 1/c 2 "vT /c2 µν − −2/(d 1) − 1/c −"v /c g = µν ρ − − − 2 − T 2 , (36) g c= "v/c 2 Id d "v "vT /c2 , (36) c # #− "v/c − Id× d −"v "v /c & & ! "! " − − × − $$ % % where dwhereis thed isdimensionthe dimensionof spofacspeac(noe (not tspaspaccetietimeme)). The coThevariancovtaMetric:rianacoustict acousticmetrmetric isic is

2/(d 1) 2 2 22 TT ρ 2ρ/(d 1)− (c(c vv )) "v"v gµν = − − − | . (37) gµν = c − −"v | Id d . (37) c ! " ' "v | Id× d( ! " ' | × ( 3.1 d = 3 3.1 d = 3 The acoustic line-element for three space and one time dimension reads The acoustic line-element for three space and one time dimension reads ρ (c2 v2) "vT gµν = − − | . (38) c 2 "v 2 I3 T3 ρ ' (c v ) | "v× ( gµν = ! "− − | . (38) This is the primary casecof interest in"v this article.I3 3 ! " ' | × ( This is the primary case of interest in this article. 3.2 d = 2 3.2 Thed =ac2oustic line-element for two space and one time dimension reads ρ 2 (c2 v2) "vT The acoustic line-elemengµtνfo=r two space− a−nd one| time dimens. ion reads(39) c "v I2 2 ! " ' | × ( 2 2 T This situation would beρappropr2 iate,(c for vinsta) nce, "vwhen dealing with surface gµν = − − | . (39) waves or excitations confinedc to a par"vticular subsIt2rat2e. ! " ' | × ( This situa3.3tiond w=ould1 be appropriate, for instance, when dealing with surface waves or excitations confined to a particular substrate. The naive form of the acoustic metric in (1+1) dimensions is ill-defined, because the conformal factor is raised to a formally infinite power — this 3.3 d = 1 12 The naive form of the acoustic metric in (1+1) dimensions is ill-defined, because the conformal factor is raised to a formally infinite power — this

12 Introducing the acoustic metric gµν, defined by 1 f µν = √ g gµν; g = (35) − det(gµν) the wInatroveducequaingtiotnhe(33ac)ousticcorrespmetricondsgtoµν,thedefinednormalby d’Alembertian wave equation in a curved-space: µν µν 1 f = √ g g ; 2 g = T 2 (35) 2/(d 1)− 1/c det("vgµν/c) µν ρ − − − − g = 2 T 2 , (36) c # "v/c Id d "v "v /c & the wave equation!(33") corresponds−to the no− rmal× d−’Alembertian wave equation in a curved-space: $ % where d is the dimension of space (not spacetime). 2 T 2 The covariant acoustic2/(dmetr1) ic is1/c "v /c µν ρ − − − − g = 2 T 2 , (36) c 2/(d 1) "v/c 2 I2d d "v "vT /c ρ #− − (c − v )× − "v & gµ!ν =" − − | . (37) c "v $ Id d % where d is the dimension! of" space (no' t spacetime)|. × ( The covariant acoustic metric is 3.1 d = 3 2/(d 1) 2 2 T ρ − (c v ) "v The acoustic line-elegµνPhmen= ysicalt for t hreeacoustics:space− −and one| time dimension. reads(37) c "v Id d ! " ' | × ( ρ (c2 v2) "vT To be specific: gµν = − − | . (38) 3.1 d = 3 c "v I3 3 ! " ' | × ( The acoustic line-element for three space and one time dimension reads This is the primary case of interest in this article. ρ (c2 v2) "vT gµν = − − | . (38) 3.2 d = 2 c "v I3 3 ! " ' | × ( TheThisaiscotheusticprimline-eleary casementoffoinrttewreostspacein thisandarticle.one time dimension reads

ρ 2 (c2 v2) "vT 3.2 d = 2 gµν = − − | . (39) c "v I2 2 ' | × ( The acoustic line-element !for"two space and one time dimension reads ThisThersituae tionis aw fouldormalbe a pproprdifficultyiate, for ininsta (1+1)nce, whendimensions.dealing with surface wisaavessideoreffexcitect ofatiothenswconfinedell-knownρ to2confoa parrmal(c2ticularinvv2a)riancesubs"voftTratthee. Laplacian in 2 gµν = − −µν | . (39) dimensions.The confTheormalwave e qfactoruationc in isterms raisedof"vf contot inanuesI2 infit2o maknitee go pood sensewer. — it is only the step from f µ!ν to" the' effective metric| tha×t br( eaks down. 3.This3Notsituaedthat=tion1thiswouldissuebonlye approprpreseniate,ts a diforffiinstacultyncfore,phwhenysicadel systemsaling withthatsurface Thearweavinesnaivtrinsicoreexcitaforllymatioone-dimeofnstheconfinednsioacousticnal.toAametricparthreeticulardimensin (1subs+1ionalt)ratdimenssysteme. iowitnsh isplaill-definene d, bsymmetryecause theis pconforerfectlymalwellfactorbehavedis araiseds in thetocasae ford =mally3 abovinfinitee. power — this 3.3 d = 1 4The Thenaive forKerm ofr theequaacoustictor metric12 in (1+1) dimensions is ill-defined, Tboeccompausearethetheconforvortexmalacousticfactorgeometryis raisedtototheaphforysicalmallyKerrinfinitegeomeptorywerof a— this rotating black hole, consider the equatorial slice θ = π/2 in Boyer–Lindquist coordinates [19, 20]: 12 2m dr2 (ds2) = dt2 + (dt a dφ)2 + +(r2 +a2) dφ2. (40) (2+1) − r − 1 2m/r + a2/r2 − We would like to put this into the form of an “acoustic metric”

2 mn ρ c g vn vn vj gµν = −{ − } − . (41) c vi gij ! " # − $ If we look at the 2-d r-φ plane, the metric is

dr2 2ma2 (ds2) = + r2 + a2 + dφ2. (42) (2) 1 2m/r + a2/r2 r − % & Now it is well-known that any 2-d geometry is locally conformally ﬂat, though this fact is certainly not manifest in these particular coordinates. Introduce a new radial coordinate r˜ such that: dr2 2ma2 + r2 + a2 + dφ2 = Ω(r˜)2 [dr˜2 + r˜2 dφ2]. (43) 1 2m/r + a2/r2 r − % & This implies 2ma2 r2 + a2 + = Ω(r˜)2 r˜2, (44) r % & and dr2 = Ω(r˜)2 dr˜2, (45) 1 2m/r + a2/r2 −

13 Physical acoustics:

There are two distinct metrics.

Call it a bi-metric theory.

The physical metric is the ﬂat Minkowski metric.

Fluid particles couple only to the physical metric.

Acoustic perturbations do not “see” the physical metric --- they couple only to the acoustic metric.

The acoustic metric inherits several nice properties from the underlying physical metric. 1 Introduction

It is by now well-known that the propagation of 1soundInintraoductimovingonfluid can be described in terms of an effective spacetime geometry. (See, for instance, [1, 2, 3, 4, 5], and references therein). It is by now well-known that the propagation of sound in a moving fluid can be described in terms of an effective spacetime geometry. (See, for in- In the geometrical acoustics approximation, thisstance,emerg[1, 2,es3,in4,a5],strandaighreferenct- es therein). • forward manner by considering the way the “sound cones” are dragged Comments: In the geometrical acoustics approximation, this emerges in a straight- along by the fluid flow, thereby obtaining the confor• forwmalardclassmannerof mebytricsconsidering the way the “sound cones” are dragged (see,Comments:for instance, [5]): along by the fluid flow, thereby obtaining the conformal class of metrics There are two distinct metri(see, forcs.instaThence, [5])physic: al metric is 2 mn c h vn vn vj 2 Theretheare flatgtµwν o distM−ink{iComments:nct−owsmetrikics.metri} The− c:physic. ηµν al me(diag[tr(1ic) is c2 ,mn1, 1, 1])µν. ∝ vi hij 2 c h vn vn vj the flat Minkows! ki metric: ηµν (diag[" c ,≡1, 1g,µ1])ν µν−.−{ ∞− } − . (1) (c = speed of− light.) ∝ vi hij (c = s∞peed of lightThe.) re are two≡distinct−metri∞ cs. The! physic− al metric is" Here c∞is the velocity of sound, v is the velocity of the fluid, and hij is 2 the flat MinkowskiHeremetric is thec: vηelocity o(diag[f sound, v cis the, 1v,elo1,cit1])y of the. fluid, and hij is the 3-metric of the ordinary Euclidean space of Newtonian mecµhaν nics µν FluidFluidparticlepas couplrtic(lecesonly=couplspeetodetheofonlylightphysicalthe.)3t-metro theicmetricof thephysical≡ordinaη .ry Euclidean−metric∞ space ofηµNeνwto. nian mechanics (possibly in curvilinear coordinates). µν FluidFluidmotionmisotioncomple∞istecomplely non–retlately(pivossiblyisticnon–rein—curvlatilinearivcoisticordinates).— Fluid particles couple only to the physical metric η . In thev0physicalvc0 a.cousticsc a.pproximation we can goInstheomewhatphysicalfurther:acoustics approximation we can µgoν somewhat further: • || || $ ∞ A wave-equa||tion||for$sound∞ canFluidbe derivmotioned by islinearcomple• Aizingwave-atndequaelycotionnon–rembiningfor soundlatcanivisticbe deriv—ed by linearizing and combining Acoustic perturbationsv0 docnot. “see”thetheEulephysir equacaltion, methe cto-ntinuity equation, and a barotropic equation the Euler equaActionoust, theiccopntinerturbations|| uit||y$equat∞ion, anddoa baronottropic“sequaee”tionthe physical met- of striatec. [1The, 2, y3, couple4]. This onlyprocesstonothew spacousticecifies theof smotvateertall[1ric, co2,gnf3,µνo4]..rmalThis process now specifies the overall conformal ric. They coupleAcousticonlyperturbationsto factother andacousticdothenotresult“singee”macousticethetricmetricphysig incal.(3+1me) dimenst- ions is (see, for factor and the resulting acoustic metric in (3+1) dimensions is (see, for µν The acoustic geometryric. Theinhey couplerits someonlyinstance,ktoey[5]):thepropacousticerties metric gµν. instance, [5]): from Thethe underacousticlying flatgeometryphysical metricinhe. rits some key2 propmn erties The acoustic geometry inherits ρsome ckeyh propvn vertien vsj ρ c2 hmn v v v gµν = −{ − } − . (2) from the4 underfrom lyingthe undernflatn lyingphysicj flat physical metrical metricc . . vi hij Topologygµν—= Phwit−ysical{h p− acoustics:ossibly a} few− regi. ons exci(2#s)e$d! − " %c vi hij (due to imposed# $ b!oundary− conditions)4 Here."ρ is the density of the fluid. Sound is then described by a massless Simple topology ---T opology4 — with possibly a few regions excised Here ρ is theTopdensitologyy of the fluid.— Soundwitis thehn%pdesossiblyminimallycribed bycoupledaamasslesfewscalas rregifield proonspagatingexciin sthisedacoustic geometry. Stable causality —(due%to imposed b(Aoundatechnicryal complicaconditions)tion arises. in that deriving such a simple wave minimally(with(duecoupled maybetscalao a impfrewfield excisedosedpropaga regionsbtingounda induethis tory boundaries.)acocusticonditions)geometry. . µν equation requires the velocity flow "v to be irrotational — no such re- (A tgechnic( aµlt)(complicaνt) =tion a1rises/(ρ0inc)tha< t0.deriving such a simple wave ∇ ∇ − Stable causality —striction applies in the geometrical acoustics limit [6].) µν equation StablereStquiresable causality:thecausalitvelocitgy flo( wyµ"vt—)(to bνet)irro=tational1/(ρ0—c)no

History:

Recent: Unruh 81 Visser 93 Jacobson Barcelo, Liberati, Sonego Stone, Perez-Bergliaffa, Hibberd Fischer, Fedichev Garay, Cirac, Zoller, Anglin Laughlin, Santiago, Chapline Schutzhold Volovik, Leonhardt, Piwnicki Novello, Salim, Klippert, Weinfurtner Older: Gordon, Madelung, ...

Example: Draining bathtub geometry

A (2 + 1) dimensional ﬂow with a sink.

ExaUsempleconst: Draiantningdebatnshtubity, cgeontometryinuity, conservation of angular momentum: A (2 + 1) dimensional flow with a sink. Implies pressure p and speed of sound cs are Use: constant density, constant sound speed, zero Usealsoconstcantonstantdensitty,hrcoughoutontinuity,theconserfluivationd flow. of angulatorquer .m (Yomeou willnt um:need an external force.) The velocity of the fluid flow is Implies pressure p and speed of sound cs are also constant throughout the fluid flow. (A rˆ+ B θˆ) !v = . The velocity of the fluid flow isr

Streamlines are(Aerˆquiangula+ B θˆ) r spirals. !v = . r The acoustic metric is Streamlines are equiangular spirals.

2 2 The acoustic2 2metric2 is A B ds = cs dt + dr dt + r dθ dt . − ! − r " ! − r " 2 2 2 2 2 A B 14 ds = cs dt + dr dt + r dθ dt . − ! − r " ! − r " 14

What is the most general acoustic metric that can [even in principle] be • constructed for the most general line vortex geometry? This question should be investigated in both the regimes of geometrical acoustics and physical acoustics. What is the most general acoustic metric that can [even in principle] be • constructedCan the equatfororiathel slicmoestofgeneralKerr thelinen bveorputtex ingeotometry?this form?ThisIfquestionnot, how • shouldclose cabne oinnevestigaget? ted in both the regimes of geometrical acoustics and physical acoustics. We shall now explore these issues in some detail. Can the equatorial slice of Kerr then be put into this form? If not, how • close can one get? 2 Vortex flow We shall now explore these issues in some detail. 2.1 General framework The2 bacVkgorroutexnd fluidflofloww [on which the sound waves are imposed] is governed by three key equations: The continuity equation, the Euler equation, and a bar2.1otroGenerpic equaaltionMostfrofamewstate: generalork line vortex: The backgrouThend backgrfluid flooundw [on ∂flwhicρow,h whichthe so unddetermineswaves are imposed] is governed + (ρ #v) = 0. (3) by three keytheequa “acoustictions: The metric”co∂tntin is∇uit g·oyvequatioerned n,by thetheEuler equation, and a barotropic equacontintionuityof, Eulerstate:, and barotropic equations: ∂#v ρ ∂+ρ (#v )#v = p + f#. (4) ∂t + · ∇(ρ #v) =−∇0. (3) ! ∂t ∇ · " p = p(ρ). (5) ∂#v Here we have includedρfor generalit+ (#v y)#van =arbitrarp +y fexternal#. force f#, possibly(4) ∂t · ∇ −∇ magneto-hydrodynamic !in origin, that "we can in principle think of imposing on the fluid flow to shape it in sopme= pde(ρ)sired. fashion. From an engineering(5) pHereerspwective haevtheEngineeringe includedEuler equafo perspectivr generalittion is byeste:anrearraarbitrarngedy externalas force f#, possibly magneto-hydrodynamic in origin, that we can in principle think of imposing ∂#v on the fluid flow to shapf#e=itρin some+ (#dev sired)#v fa+shiop,n. From an engineering(6) ∂t · ∇ ∇ perspective the Euler equation!is best rearranged" as with the physical interpretation∂b#veing that f# is now telling you what external f# = ρ + (#v )#v + p, (6) force you would need in order t∂ot set up· ∇a specified∇ fluid flow. ! " with the physical interpretation being that f# is now telling you what external force you would need in order to set up a specified fluid flow.

4 Example: Draining bathtub geometry

A (2 + 1) dimensional ﬂow with a sink.

Use constant density, continuity, conservation of angular momentum:

Implies pressure p and speed of sound cs are also constant throughout the ﬂuid ﬂow.

Example: Draining bathtub geometry The velocity of the ﬂuid ﬂow is

Acoustic metric: (A rˆ+ B θˆ) = !v . 2 2 2 2 r 2 A B ds = cs dt + dr dt + r dθ dt . − ! − r " ! − r " Streamlines are equiangular spirals. Example: DrPhaiysicalning batacoustics:htubThege Drainingometryacoust bathtubic event horizon forms once the ra- The acoustic mdialetriccompis onent of the ﬂuid velocity exceeds Acoustic metric: the speed of sound, that is at

A 2 B 2 ds2 = c2dt2 +2 dr dt +2 r dθ A dt . 2 2 2 s A B r = | |. ds = cs dt + d−r dt !+ r−dθr "dt hor!. izon − cr " − ! − r " ! − r " The acoustic horizon forms once the radial velocity14 The acoustic event hoexceedsrizonSupfo rmstheers speedononceic ﬂo theofw sound.setsra- in outside the event hori- dial component of the ﬂuidzon,velwhenocitytheexcemagnitudeeds of the velocity equals An ergo-surface theformssp eedonceof thesound. speed exceeds the the speed of sound, that is at speed of sound.

A A2 + B2 rhorizon = | |. rergo surface = . c − # c

16 Supersonic ﬂow sets in outside the event horizon, when the magnitude of the velocity equals the speed of sound.

A2 + B2 rergo surface = . − # c

16 Key points:

— The signature is ( , +, +, +). − — There are two distinct metrics: the physical spacetime metric, and the acoustic metric .

Event horizons and ergo-regions:

Event horizon: the boundary of the region from which null geodesics (phonons; sound rays) cannot escape.

At the event horizon, the inward normal component of ﬂuid velocity equals the speed of sound.

Ergo surface: the boundary of the region of supersonic ﬂow.

In general relativity this is important for spin- ning black holes. 13 Surface gravity:

For a static geometry, we can apply all of the standard tricks for calculating the “surface gravity” developed in general relativity.

The surface gravity is a useful characterization of general properties of the event horizon and is given in terms of a normal derivative by

1∂(c2 v2 ) ∂(c v ) g = − = c − ⊥ . H 2 ∂n ⊥ ∂n

The surface gravity is essentially the acceleration of the ﬂuid as it crosses the horizon.

Non static geometries are not too bad.

(Thanks to the second metric: It gives you an unambiguous background for making comparisons.)

23 Physical acoustics: Surface gravity

The surface gravity is essentially the acceleration of the ﬂuid as it crosses the horizon.

Even non-static geometries are not too bad.

The second background metric simpliﬁes some of the technical complications encountered in general relativity.

(The second metric gives you an unambiguous background for making comparisons.) There is a reason that for the importance of surface gravity, patience.... Back to geometric acoustics: EikGeometrional limit:c Acoustics: Take the short wavelength/high frequency limit. Sound rays (phonons) follow the null geodesics of the acoustic metric. Null geodesics are insensitive to any overall conformal factor in the metric. Simplify life by considering . j (c2 v2) . v − − 0 − 0 hµν ≡ . ! · · · · · ·v·i· · · · · .· · ·δ·ij· · · " − 0 . In the geometric acoustics limit, sound propagation is insensitive to the density of the fluid. It depends only on the local speed of sound and the velocity of the fluid. The density of the medium is important only for specifically wave related properties.

Sanity check: Parameterize Xµ(t) (t, #x(t)). — The null condition implies ≡ dXµ dXν hµν = 0 dt dt dxi dxi dxi (c2 v2) 2vi + = 0 ⇐⇒ − − 0 − 0 dt dt dt d#x #v0 = c. ⇐⇒ dt − # # # # # # The norm is tak#en in the# ﬂat physical metric. Interpretation: the ray travels at the local speed of sound relative to the moving medium. Geometric Acoustics:

Take the short wavelength/high frequency limit. Sound rays (phonons) follow the null geodesics of the acoustic metric. Null geodesics are insensitive to any overall conformal factor in the metric. Simplify life by considering . j (c2 v2) . v − − 0 − 0 hµν ≡ . ! · · · · · ·v·i· · · · · .· · ·δ·ij· · · " − 0 . In the geometric acoustics limit, sound propagation is insensitive to the density of the fluid. It depends only Backon the to geometriclocal speed acoustics:of sound and the velocity of the fluid. The density of the medium is important only for specifically wave related properties.

Stationary geometry:

µ Let X (s) (t(s); !x(s)) be some null path from !x1 to ≡ !x2 parameterized in terms of physical arc length (i.e. d!x/ds 1). || || ≡ The condition for the path to be null (though not yet necessarily a null geodesic) is

dt 2 dxi dt (c2 v2) 2vi + 1 = 0. − − 0 ds − 0 ds ds ! " ! " ! " — Solve the quadratic —

i dxi 2 2 i dxi 2 dt v0 + c v0 + v0 = − ds − ds . ds %c2 v2 ! " # $ − 0 # $ The total time taken to traverse the path is

!x2 T [γ] = (dt/ds)ds &!x1 = (c2 v2)ds2 + (vi dxi)2 vi dxi /(c2 v2). { − 0 0 − 0 } − 0 &γ % Extremizing the total time taken is Fermat’s principle. Cf• p 262 Landau and Lifshitz. Stationary geometry:

dt 2 dxi dt (c2 v2) 2vi + 1 = 0. − − 0 ds − 0 ds ds ! " ! " ! " — Solve the quadratic —

i dxi 2 2 i dxi 2 dt v0 + c v0 + v0 = − ds − ds . From geometricds acoustics to F%cermat:2 v2 ! " # $ − 0 # $ The total time taken to traverse the path is

!x2 T [γ] = (dt/ds)ds &!x1 = (c2 v2)ds2 + (vi dxi)2 vi dxi /(c2 v2). { − 0 0 − 0 } − 0 &γ % Extremizing the total time taken is Fermat’s principle. Cf• p 262 Landau and Lifshitz.

We have now come full circle: Geometric acoustics ==> physical acoustics ==> wave equation ==> eikonal ==> geometric acoustics ==> Why bother?

If you are a general relativist, this acoustic analogy gives you simple concrete physical models for curved spacetime.

If you are a ﬂuid mechanic (or more generally a condensed matter physicist) the differential geometry of curved spacetimes gives you a whole new way of looking at sound. Of course these analogue models can be greatly generalized: all you really need are well-deﬁned characteristic speeds.

Simple example: Linearize any Lagrangian ﬁeld theory. Example 1:

Example 1: ExampleLagraEx1:ngiamplean: 1:

LagrLaagrngiangian:an: (∂µφ, φ). Lagrangian analysis:L Lagrangian: ( ) (∂∂µµφφ,,φφ).. LL Convention: (∂µφ, φ). Convention: L ConvConention:vention: ∂µφ = (∂tφ ; ∂iφ) = (∂tφ ; φ). ∂µφ = (∂tφ ; ∂iφ) = (∂tφ ; φ∇). Convention: ∂µφ = (∂ φ ; ∂ φ) = (∂ φ ; ∇ φ). t i t ∇ Action:∂AcµAction:φtion:= (∂ φ ; ∂ φ) = (∂ φ ; φ). t i t ∇ Action: d+1 S[φ] = d x (∂µφ, φ). d+1 L S[φ] = !dd+1 x (∂µφ, φ). S[φ] = d x L(∂µφ, φ). Action: ! L EulerEule-Lagrange:r–Lagrange !equations: Euler–LagrangeS[φ] = dequatd+1xions(∂: φ, φ). Euler–Lagrange equat∂ ions: ∂µ ∂µ L L L = 0. ! "∂(∂µφ)# − ∂φ ∂∂ ∂∂ ∂µ L L = 0. ∂µ ( L ) − L = 0. Euler–LagrangeLinearize theequat""fi∂∂eld(∂∂iµonsaφφr)ound##: − a∂∂φsolution:φ $2 φ(t, #x) = φ (∂t, #x)+$φ (t,∂#x)+ φ (t, #x)+O($3). LineaLinearizerize∂µthethe0fifieldeldL arroundound1 Laa=solution:solution:2 02. "∂(∂µφ)# − ∂φ 2 $ 2 4 φ(t, #x) = φ (t, #x)+$φ (t, #x)+ $φ (t, #x)+O($3).3 φ(t, #x) = φ00(t, #x)+$φ11(t, #x)+2 φ2 2(t, #x)+O($ ). Linearize the field around a solution:2 4 $2 4 φ(t, #x) = φ (t, #x)+$φ (t, #x)+ φ (t, #x)+O($3). 0 1 2 2

4 Example 1:

Lagrangian:

(∂µφ, φ). L

Convention:

∂µφ = (∂ φ ; ∂ φ) = (∂ φ ; φ). t i t ∇

Action: d+1 S[φ] = d x (∂µφ, φ). ! L

Euler–Lagrange equations: ∂ ∂ ∂µ L L = 0. "∂(∂ φ)# − ∂φ Lagrangian analysis: µ

Linearize the ﬁeld around a solution: $2 Exφ(t,ample#x) = φ1:(t, #x)+$φ (t, #x)+ φ (t, #x)+O($3). 0 1 2 2

Linearized action 4

S[φ] = S[φ0] 2 2 " d+1 ∂ + d x L ∂µφ1 ∂νφ1 2 ! "#∂(∂µφ) ∂(∂νφ)$ ∂2 ∂2 + L ∂µ L φ1 φ1 % ∂φ ∂φ − #∂(∂µφ) ∂φ$& ' + O("3). Linear pieces [O(")] vanish by equations of motion.

Quadratic in φ ﬁeld-theory normal modes. 1 ⇒

Linearized equations of motion: ∂2 ∂µ L ∂νφ1 %#∂(∂µφ) ∂(∂νφ)$ & ∂2 ∂2 L ∂µ L φ1 = 0. − %∂φ ∂φ − #∂(∂µφ) ∂φ$& Formally self-adjoint. 5 Example 1: Example 1: Linearized action LineaExamplerized1:action S[φ] = S[φ0] 2 2 SLinea[φ] rize= dS"action[φ0] d+1 ∂ + 2 d x 2 L ∂µφ1 ∂νφ1 S[φ] = S"[φ2 ] d+1 "#∂(∂∂µφ) ∂(∂νφ)$ + 0!d x L ∂µφ1 ∂νφ1 2 2 ( 2 ) (2 ) " ! d+1∂ "#∂ ∂∂µφ ∂∂∂νφ $ + d x L ∂µφ1 ∂νφ1 2+ ∂2 L ∂(∂∂µφ) ∂∂(2∂ φ)L φ1 φ1 ! % ∂φ ∂"#φ − µ #∂(∂νµφ)$ ∂φ$& ' + 2 L ∂µ 2 L φ1 φ1 % ∂∂3φ ∂φ − #∂∂(∂µφ) ∂φ$& ' ++O(" )L. ∂µ L φ1 φ1 + O(%"∂3φ).∂φ − #∂(∂µφ) ∂φ$& ' Lagrangian analysis: Linea+r pOiec("3es). [O(")] vanish by equations of Lineamotiron.pieces [O(")] vanish by equations of motLineaion.r pieces [O(")] vanish by equations of motion.

Quadratic in φ1 field-theory normal modes. Quadratic in φ1 ⇒field-theory normal modes. Quadratic in φ1 ⇒field-theory normal modes. ⇒ Linearized equations of motion: LineaLinearizedd equationsequationsofofmmotion:otion: 2 2 ∂∂2 ∂ ∂∂µµµ LLL ∂ν∂φν1φ∂1νφ1 %#%#%#∂∂(∂(∂∂(µµ∂φφµ))φ∂∂)(∂(∂ν∂φ(ν∂)φ$ν)φ$)$ & & & ∂∂22∂2 ∂2∂2∂2 LLL ∂µ∂µ∂µ L L L φ1 =φ10=φ. 10=. 0. −−−%%∂%∂φφ∂∂φ∂φφ∂−φ−−#∂#(∂∂#(µ∂φµ)(φ∂∂)µφφ$&∂)φ$&∂φ$& FFFooorrrmamamallllyllyysseeslf-adjlf-adjelf-adjoint.oint.oint. 5 5 5 Example 1: Example 1: LagrangianGeometriExample analcalysis:1: interpretation: Geometrical interpretation: Geometri[cal∆(gi(ntφe))rpretatV (iφon:)] φ = 0. 0 − 0 1 [∆(g(φ0)) V (φ0)] φ1 = 0. [∆(g(φ0))− V (φ0)] φ1 = 0. Metric: − Metric: 2 Metric: µν µν ∂ √ g g f 2L . − ≡ ≡ !∂(∂µφ)∂ ∂(∂νφ)"# √ g gµν f µν 2L #φ0 . µν µν ∂ # √− g g ≡ f ≡ !∂(∂µφ) ∂L(∂νφ)"# #φ . − ≡ ≡ !∂(∂µφ) ∂(∂νφ)#"# #0 # #φ0 Potential: # # # # Potential: 1 ∂2 ∂2 # PVotential:(φ0) = 2L ∂µ 2L . √ 1g $∂φ∂ ∂φ − !∂(∂µ∂φ) ∂φ"% V (φ0) = −1 ∂2L ∂µ ∂2L . V (φ0) = √ g $∂φ ∂Lφ − ∂µ!∂(∂µφ)L∂φ"% . √− g $∂φ ∂φ − !∂(∂µφ) ∂φ"% And − And linearization metric; And ⇒ linearization metric; hyperboliclinearizapseudo-tion⇒Riemannian;metric; ⇒ ⇒ hyperbolic pseudo-Riemannian; hyperpabrabolicolic⇒ pseudo-degenerRiemannian;ate; ⇒ parabolic⇒ degenerate; elpalipticrabolic⇒Riedegemanniannera.te; ⇒ elliptic ⇒Riemannian . 6 elliptic⇒ Riemannian. ⇒ 6 6 Example 1:

Geometrical interpretation: [∆(g(φ )) V (φ )] φ = 0. 0 − 0 1

Metric: ∂2 √ g gµν f µν L . − ≡ ≡ !∂(∂µφ) ∂(∂νφ)"# #φ0 # # # Potential: 1 ∂2 ∂2 V (φ0) = L ∂µ L . √ g $∂φ ∂φ − !∂(∂µφ) ∂φ"% Lagrangian analysis:−

AndKey results: linearization metric; ⇒ hyperbolic pseudo-Riemannian; ⇒ parabolic degenerate; ⇒ elliptic Riemannian. ⇒ This strongly suggests that the “analogue gravity” 6 phenomena is generic to almost any linearization. ExExExaamplempleample2:2:2:

BaBaBarotropirotropirotropicc icirrotatrrotatirrotationalionalionalinviinviinviscidscidscidfluidfluiddydydynamnamnamiicsicscs Lagrangian analysis: barotropic, irrotational, inLaLaviscidLagrgragrangi flngiauid:ngiaannan(t(tw(twowoofiefiefields):lds):lds): ρρ 111 222 ===ρρ∂ρ∂ttθ∂θtθ ρρ((ρ( θθ))θ) ddρρ## #hhh(((ρρρ##))#)... LL −− −− 2 ∇∇ −− 00 L − − 22 ∇ − !!0! p(ρ) pp((ρρ) dpdp# hh((hρρ())ρ=)==hh[[hpp[((pρρ()])]ρ)]=== ## ... 00 ρρ((pp)) !!0! ρ(p##)# VVaaVryryaryρρ ρ BernoulBernoulBernoullililieequatquatequationionion(Eule(Eule(Eulerreeequaquaquation)tion)tion)... ⇒⇒⇒ 11 +1 ( 22)2 + ( ) = 0 ∂∂tθ∂θt+θ+ (( θθ))θ ++ hhh((ρρ)ρ = 0... t 222∇∇∇ Vary θ continuity equation. VVaaryry θθ ⇒continuicontinuittyy equationequation.. ⇒⇒ ∂ ρ + (ρ θ) = 0. ∂∂tρρt++ ∇((ρρ ∇θθ)) == 0. t ∇∇ ∇∇ Use the Bernoulli equation to algebraically UseUsettheheBernoulBernoullili eequatquationion ttoo algebraicraicaalllyly eleliminatiminate e :ρ: eliminate ρρ: 1 1 11 2 ρ = h1− (z) = h−1 ∂tθ ( θ)2 . ρ = h−1 (z) = h−1 −∂tθ −12( ∇θ)2 . ρ = h− (z) = h− "−∂tθ − 2(∇θ) #. "− − 2 ∇ # " # 7 7 7 Example 2:

Barotropic irrotational inviscid ﬂuid dynamics

Lagrangian (two ﬁelds): ρ 1 2 = ρ ∂tθ ρ( θ) dρ# h(ρ#). L − − 2 ∇ − !0 p(ρ) dp h(ρ) = h[p(ρ)] = # . !0 ρ(p#) Vary ρ Bernoulli equation (Euler equation). ⇒ 1 ∂ θ + ( θ)2 + h(ρ) = 0. t 2 ∇ Vary θ continuity equation. ⇒ Lagrangian analysis: ∂barρotr+opic(ρ, irrotational,θ) = 0. inviscid ﬂuid: t ∇ ∇

Use the Bernoulli equation to algebraically eliminate ρ: ExExaamplemple 2:2: 1 ρ = h 1(z) = h 1 ∂ θ ( θ)2 . − − − t − 2 ∇ ReduReducedced LLagragraangian:ngian: " # But then: 7 ρ(z() ) (z) = z ρ(z) ρ zdρ h(ρ ). (z) = z ρ(z) d"ρ" h(" ρ"). L − !0 L − !0 ButFur: thermore: But: ρ p(ρ) = ρ h(ρ) dρρ" h(ρ"). p(ρ) = ρ h(ρ)− !0 dρ" h(ρ"). − !0 Proof: Diﬀerentiate Prd[RHoof:S]Diﬀeredhntiate dh dp d[RHS] = ρ dh+ h(ρ) h(ρ) = ρ(z) d=h .dp dρ = ρ dρ + h(ρ)− h(ρ) = ρ(zd)ρ =dρ . dρ dρ − dρ dρ Finally:

Finally: 1 1 2 = p(ρ(z)) = p h− ∂tθ ( θ) . L − − 2 1∇ = p(ρ(z)) = p" h 1" ∂ θ ( θ##)2 . L − − t − 2 ∇ This reduces the Lagrangian" " to the form##of Example 1. This reduces the Lagrangian to the form of ExTheamplere is 1.a metric hiding here waiting to be found... There is a metric hiding here waiting8 to be found... 8 Example 2:

Reduced Lagrangian: ρ(z) (z) = z ρ(z) dρ" h(ρ"). L − !0 But: ρ p(ρ) = ρ h(ρ) dρ" h(ρ"). − !0

Proof: Diﬀerentiate d[RHS] dh dh dp Lagrangian =analρysis: bar+ otrh(ρopic) , irhr(otational,ρ) = ρ( z) = . d d − d d inviscidρ ﬂuid: ρ ρ ρ

FinallyFinally:: 1 1 2 = p(ρ(z)) = p h− ∂tθ ( θ) . L " "− − 2 ∇ ## ThisThis rreduceseduces evertheythingLagrangian to a Lagrangianto dependingthe form of Exona onlmpley a single1. scalar ﬁeld.

TheUse retheis precedinga met single-ﬁric hidingeld Lagrangianhere w analaitiysis.ng to be found... There is a metric hiding here waiting to be found...8 ExExampleample2:2:

ApplApply ythethereresultsultofofExExamamplple e1:1: Example 2: 2∂2 √ g µgνµν µfνµν ∂ L . Example√ 2:−g g ≡f ≡ ( )L ( ) . Lagrangian −analysis:≡ barotr≡opic!!∂, (∂ir∂rµ∂otational,φµ)φ∂(∂∂ν∂φν )φ"#"#φ Apply the result of Example 1: #φ#0 0 # # inviscid ﬂuid: # # 2 # # Apply the resultµν of µExν ample ∂1: SomeUse: Use:boring√ manipulations:g g f L . − ≡ ≡ !∂(∂µφ) ∂(∂νφ)"# 2 #φ0 ∂z ∂ # µ∂νz µν µ µ i #i √ g g f== (1;(1; θ)θ) ==L(1;(1; θ#)θ.). . − ∂(∂∂(∂≡φµ)φ) −≡− !∇∂∇(∂µφ)−∂−(∂ν∇φ∇)"# Use: µ #φ0 # ∂z # µ i # And: = (1; θ) = (1; θ). Use:And: ∂(∂µφ) − ∇ − ∇ 2∂2z ∂ z = ijij ∂z µ = δ δ. . i And: ∂=(∂∂(µ∂φ(1;µ)φ)∂(∂∂(θν∂)φν)φ=) −−(1; θ). ∂(∂µφ) − ∂2∇z − ∇ = δij. ∂(∂µφ) ∂(∂νφ) − TherTheTheeforreforefoe: re:re: And: 2 µν d2dp p µ ν dpdp ij Theµrefofν =r=e: (1;(1;2 )θµ) (1;(1; )θν) ijδ . f d2 2∂ z ∇θ ∇θ − d δ . dz z2 ∇ =∇ ij− dz z µν d p µ δν .dp ij f ∂=(∂µφ)(1;∂(∂θν)φ)(1; −θ) δ . dz2 ∇ ∇ − dz

Therefore: 9 9 d2p dp 9 f µν = (1; θ)µ (1; θ)ν δij. dz2 ∇ ∇ − dz

9 ExaExmpleample2: 2:

ButBut dp Example 2: Lagrangian analysis: barotropic=d, pirρr;otational, dz = ρ; inviscid ﬂuid: dz while But Continwhileued boring manipulations: d2p dρ dρ dp d =2 = = 2 p 2d p dρ dρ dpρ cs− . 2 = ρ; dz d=z dp=dz = ρ c− . dz dz2 dz dp dz s while CollectingCollecti ngterms:terms: 2 Collecting terms: d p dρ dρ dp . i = = = 2 1 . θ 2 µν 2ρ cs− −. . −∇ i dz dz f dp=dzρ cs−  . . . . 1 ...... θ.  . µν− 2 j −.· 2 ij i−∇j f = ρ c− θ. . .. . c δ . . . . .θ. . . .θ...... − s −∇ s· − ∇ ∇    jθ . c2 δij iθ  jθ  s  Collecting terms: This should by now be a familiar−∇ result... − ∇ ∇  This is equivalent to the standard (d+1) . dimensi1This. onais equil “acousticvaleintθ tmoethetric”.standaUse rd (d+1) µν 2 − −∇ f = ρ cs−  . . . .dimensi. .ona. . .l.“acoustic...... me.tric”. Use − j .· 2 ij i j θ . cs δµν θ 1/θ(d 1) µν  −∇ g =− ∇detf∇− − f .  | |  gµν = detf 1/(d 1) f µν. | |− − This is equivalent tAndo thenotste andaan ovrdera(d+1)ll minus sign is irrelevant. dimensional “acoustic metric”. Use And note an overall minus sign is irrel10evant. 10 gµν = detf 1/(d 1) f µν. | |− −

And note an overall minus sign is irrelevant. 10 Where do we go from here? Four routes to better physics: 1. Adding vorticity (rather non-trivial).

2. Multiple ﬁelds (generalized Fresnel equation).

3. More physics examples: --- Acoustic horizons (experimental/observational); --- BECs (a special type of ﬂuid); --- Laval nozzles; --- Kerr geometry (a special type of vortex?).

4. Quantum physics in curved spacetimes: --- Hawking radiation.