The Black Hole Graviton Laser TIME BOMB
Éric Dupuis and Manu Paranjape Groupe de physique des particules, Département de physique, Université de Montréal
Graviton Laser arXiv:1604.02762 Gravitation and Quantum Mechanics • There has not been very much work on the interplay of gravity and quantum mechanics. • Here I do not mean quantum gravity, but the effect of gravity on a quantum mechanical system. • The reason is probably that the effects are very weak. The gravitational coupling constant is G c4 • This combination has units of inverse Newtons, and in the 45 MKS system it is numerically of the order 10 ⇠ • However the gravitational potential is also proportional to the product of the two masses involved. • The interaction of gravitation with a quantum mechanical system in the lab has only recently been observed. Q-bounce • The Q-bounce experiment, which stands for quantum bouncer, was proposed and carried out in the last ten years • Here a system of ultra cold neutrons were observed. • Ultra cold neutrons are normally defined to have a kinetic energy of less that 300 neV, and they are unable to penetrate into the solid material walls of a vessel, they bounce off the walls, and are in fact contained. • If they are further distilled in energy so that the kinetic energy is in the few peV range, then they start to feel the gravitational potential due to the earth. • The energy levels of the Schrödinger equation are easily found. Schrodinger equation is: ~2 d2 2 E(z)+mN gz E(z)=E E(z). (1) 2mN dz The normalizable solutions are simply Airy functions, Ai( z ↵) is a eigenfunction with z0 3 ~2 energy mN gz0↵ where z0 = 2 .Becauseoftheinfiniteenergybarrieratz =0,ourwave 2gmN functions must vanish there. The Airy functions are non-zero for positive argument and have an infinite set of discrete zeros for negative argument at z/z0 = ↵n, n =1, 2, 3 . Schrodinger equation is: Schrodinger equation is: ··· Schrodinger equation is: 2 2 ThereforeSchrodinger~ d equation the energy is: eigenfunctions which2 satisfy2 the boundary condition are simply n(z)= E(z)+mN gz E(z)=E E(z). ~ d (1) 2 2 Schrodinger2 equationSchrodinger2 is:2 equation is: E(z)+mN gz E(z)=E E(z). (1) ~ d 2mNz dz ~ d 2m dz2 nAi( ↵n) where n Eis(2z)+ thed2 mN appropriategz E(z)=EN E2(z normalization). 2 [7(1)]: E(z)+mN gz E(z)=E E(z). z0 (1) 2m dz2 ~ z d 2 The normalizable solutions are simply Airy functions,N Ai( E(z↵)+) ismN agz eigenfunction E(~z)=E E(z) with. z (1) 2m dz N TheN normalizablez20 solutions are simply AiryE(z)+ functions,mN gz EAi(z()=E↵ )Eis(z a). eigenfunction with (1) N 2mN dz z 2 z0 The normalizable2 solutions are simply Airy functions, Ai ( 2mN↵dz) is a eigenfunction with 3 ~ 1z02 z 1 energyzmN gz0↵ where z = 2The.Becauseoftheinfiniteenergybarrierat normalizable solutions are simply Airy functions,3 ~z =0Ai(,ourwave↵) is a eigenfunction with 0 2gm 2 m gz ↵ z = z0 z z =0 The normalizable solutions are simply Airy functions, Ai( ↵) is a eigenfunctionN with3 energy~ Nn =0 where 0 2gm2 .Becauseoftheinfiniteenergybarrierat = Ai( ↵) ,ourwave (2) z0 energy mN gz0↵ where z0 = 2gmThe2 .Becauseoftheinfiniteenergybarrierat normalizable2 solutions areN simply Airyz =0,ourwave functions, z is a eigenfunction with N 3 ~ 2 z Ai ( ↵ ) 0 energy mN gz0↵ where z0 =N2gm2 .Becauseoftheinfiniteenergybarrierat1 p 0z =00,ourwaven 2 functions must vanish there. The Airy functions are non-zeroN forz0 positiveAi argument(⌘2 ) d⌘ and 3 ~ Schrodingerfunctions equation must is:• vanishThe Schrödinger there.functions The equation Airy must functions is: vanish are there. non-zero↵3n The~ for Airy positive functions argument are andnon-zero for positive argument and energy mN gz0↵ where z0 = 2gm2 .Becauseoftheinfiniteenergybarrieratz =0,ourwave energy mN gz0↵ where z0 = 2gm2 .Becauseoftheinfiniteenergybarrieratz =0functions,ourwave must vanish there. The Airy functions are non-zeroN for positive argument and N have an infinite set of discrete zeros for negative2 2 argument at z/z0 = ↵n, n =1, 2, 3 . z/z = ↵ n =1, 2, 3 have an infinite set of discretehaved zeros an infinite for negativeq set of argument discrete at zerosz/z0 for= negative↵···n, n =1 argument, 2, 3 . at 0 n, . The energyhave an of infinite~ this setfunctions eigenstate of discrete zeros must is for vanish negative ofR course there. argument ThemN at Airygzz/z 00 ↵= functionsn.↵n, Then =1··· are,↵2, 3n non-zeroare. known for positive numerically argument··· and to E(z)+mN gz E(z)=E E (z()z.)= ··· (1) functions must vanish there. The Airy functions areTherefore non-zero the for energy positive eigenfunctionsTherefore argument the energy which eigenfunctions satisfy andTherefore the2 boundary which the satisfy energy condition the eigenfunctions boundary are simply condition whichn are satisfy simply the n boundary(z)= condition are simply n(z)= ThereforeSchrodinger 2 them energyN dzhave eigenfunctions equation an infinite is: which set satisfy of discrete the boundary zeros condition for negative are simply argument n(z)= at z/z0 = ↵n, n =1, 2, 3 . z arbitrarilyz high accuracy,z however, the Bohr-Sommerfeld approximation [5, 8 ]issurprisingly··· nAi( z ↵n) where nnAiis( z the↵ appropriaten) wherez n normalizationis thenAi appropriate( ↵n [)7]:where normalization2 n 2is the [7]: appropriatez normalization [7]: have an infinite set of discrete zeros for negative argument at0 z/z0 = ↵n, n0 =1nAi•,(The2, 3 energy↵n) where. eigenfunctionsnz0is the appropriate are: normalizationd [7]: N The normalizableNN solutionsN z0 N areN simplyThereforeN Airy the energy functions,~ N eigenfunctionsAi( z which↵) is satisfy a eigenfunction the boundary condition with are simply n(z)= ··· E(z)+0 mN gz E(z)=E E(z). (1) 1 1 1 1 2 1 1 1 1 accurate and useful2n = as it yieldsz an2m= analyticN dz formula: (2) n = 3 ~ = n = = (2) (2) m gz ↵ z =(z)=N 2 nAi( ↵n)2 wherepz0nnAi=is0 ( the↵n) appropriate normalization= z =0 [7]: (2) Therefore the energy eigenfunctions which satisfy the boundaryenergy conditionN are0N simplywhere 0 n 2gm 2 .BecauseoftheinfiniteenergybarrieratzN0pz1z00AiAi0((⌘) ↵dn⌘) 2 pz0Ai0 ( ↵n) z ,ourwave z 1 Ai(⌘)NdN⌘ ↵n z0 1 Ai(⌘)NNd⌘ 1 /3 1 2 pz0Ai0 ( ↵n) 0 The↵ normalizable solutions↵n are simply Airyz0 functions, Ai(⌘) d⌘Ai( ↵) is a eigenfunction with n 2 2 2 ↵n z0 2/3 z 1 1 9mqN Rh gq 21 1 Ai( ↵ ) functions mustThe energy vanishqThe of this there. energy eigenstate of Thethis eigenstate is Airy of course functions is of3mR courseN gz0~↵nm are.N gz The0 non-zero↵nn↵.=n Theare↵ knownn forare12 known positivenumerically numerically= argument to to and (2) n z n where n is the appropriate normalizationThe energy [7]: of this eigenstate is ofR course mN gz0↵n. The ↵n are knownq numerically to 0 energyEn = mNThegz0↵ energywhere ofz0 thisn= 2 eigenstategm2 .BecauseoftheinfiniteenergybarrieratN is ofR course10 meVN gz=120↵n., 69 Thepz0Ai↵nn0 (are↵ knownzn)=0,ourwavepeV numerically to (3) N N N z0 1 Ai(⌘) d⌘ arbitrarily higharbitrarily accuracy, high however, accuracy,32 the however, Bohr-Sommerfeld the Bohr-Sommerfeld 4 approximation approximation⇥ [5↵,n8]issurprisingly [5, 8]issurprisingly 4 have an infinite set of discrete zeros for negative✓ argument◆ ! at z/z 0 = ↵n, n =1✓ , 2, 3 ◆. 1 arbitrarily1 high accuracy, however, the•functionsIn Bohr-Sommerfeldnumbers,arbitrarily mustthe energy vanish levels high approximation there. accuracy,are approximately The however, [5 Airy, 8]issurprisingly given functionsq the by: Bohr-Sommerfeld are non-zero approximation for positive argument [5, 8]issurprisingly and n = = accurate andaccurate useful and as it useful(2) yieldsThe as an it analyticyields energy an formula: of analytic this formula: eigenstate is ofR course m N gz0↵n. The ↵n ···are known numerically to 1/3 N 2 pzaccurate0Ai0 (Therefore and↵n useful) asthe it energy yields an eigenfunctions analytichave an formula: infiniteaccurate which set and2 of satisfy useful discrete2 as1 the/3 itzeros yields boundary for an negative analytic condition2/ argument3 formula:2/3 are at simplyz/z0 = n↵(zn,)=n =1, 2, 3 . z0 1 Ai(⌘) d⌘ 9m h2g2 9m h21g2 1 1 1 ↵n 1N/3 arbitrarilyN high accuracy,12 however,12 the1 Bohr-Sommerfeld/3 approximation [5, 8]issurprisingly··· En = 2 En = n n 10 eV 10=12 /3,eV69 =1n , 692 n peV peV (3) (3) 2/3 z 9mGRAVITATIONALh2g2 1 32 PERTURBATION32 4 ⇥4 9m ⇥1h2g2 1 4 4 1 (z)= nAi( ↵Nn) where Thereforen is the appropriate the12 energy✓! eigenfunctions normalization◆ ! N which [ satisfy7]: ✓ the boundary◆ 12 condition are simply n q En =z0 n 10accurate✓ eV◆=1 andEn, 69= usefuln as it yieldspeVn✓ an analytic◆ (3)10 formula: eV =1, 69 n peV (3) The energy of this eigenstate is ofR course mN gz0↵n. The ↵Nn are known 32 numerically N4 z to⇥ 324 4 ⇥ 4 n!Ai( ↵n) where n is the appropriate✓ normalization◆ ! 1/3 [7]: ✓ ◆ ✓ ◆ z0 1 ✓ ◆ 2 2 1 2 2/3 GRAVITATIONALGRAVITATIONALN PERTURBATION PERTURBATIONN 9mN h g 1 12 1 arbitrarily high accuracy, however, the Bohr-Sommerfeld approximation [5,Spherical8]issurprisingly perturbationn = En = = 1 n 101 eV =1, 69 n(2) peV (3) N 2 p32z0Ai0 ( ↵4n) ⇥ 4 GRAVITATIONAL PERTURBATION GRAVITATIONALz0 1 Ai(⌘) nd=⌘ PERTURBATION✓ ◆ =! ✓ ◆ (2) ↵n N 2 pz0Ai0 ( ↵n) Spherical perturbationSpherical perturbation z0 1 Ai(⌘) d⌘ accurate and useful as it yields an analytic formula: ↵n q 1/3 The energy of this eigenstate is ofR course mN gz0q↵n. The ↵n are known numerically to 2 Spherical perturbation2/3The gravitationalThe energyGRAVITATIONAL perturbationSpherical of this eigenstate perturbation that is PERTURBATION of weR course imaginemN gz the0↵n. system The ↵n isare subjected known numerically to, corresponds to to 2 2 The gravitationalThe gravitational perturbation perturbation that we imagine that we the imagine system the is system subjected is subjected to, corresponds to, corresponds to to 9mN h g 1 12 1 En = n 10 eV =1, 69arbitrarilyn highpeV accuracy,the effarbitrarilyect however, of a(3) macroscopic high the accuracy, Bohr-Sommerfeldmass, M,broughtascloseaspossibletothesystemofneutrons, however, the Bohr-Sommerfeldapproximation [ approximation5, 8]issurprisingly [5, 8]issurprisingly 32 4 ⇥ 4 thethe eff eectffect of a of macroscopic a macroscopic mass, M,broughtascloseaspossibletothesystemofneutrons, mass, M,broughtascloseaspossibletothesystemofneutrons, ! The gravitational perturbationand that subjected we imagine to oscillatoryThe theSpherical gravitational system motion is perturbation at subjected exactly perturbation the to, frequency corresponds that corresponding we imagine to the to a system resonance. is subjected to, corresponds to ✓ ◆ accurate✓ and◆and useful subjected as to itaccurate oscillatory yields anand motion analytic useful at exactly as it formula: yields the frequency an analytic corresponding formula: to a resonance. and subjected to oscillatorythe effect of a motion macroscopic at mass, exactlyM,broughtascloseaspossibletothesystemofneutrons, the frequency corresponding to a resonance. the effect of a macroscopic mass,ItM is,broughtascloseaspossibletothesystemofneutrons, simplest to imagine the mass1/3M as a spherical body,2 1/3 of density ⇢ and radius ⇣0,which 2/3 It is simplest to imagine2 2 the mass M2 as a spherical2 2 body, of density ⇢ and radius2⇣/03,which 9m h g 1The gravitational9mN h g perturbation1 that we12 imagine1 the system1 is subjected to, corresponds to is broughtN to a positionE =⇣ on the z axis, aboven12 the system of10 neutrons. eV =1 Its, height69 n varies peV GRAVITATIONAL PERTURBATION and subjected to oscillatoryE = motion at exactlynand then subjected frequency to corresponding oscillatory10 M eV motion=1 to, a69 resonance. at exactlyn the frequencypeV corresponding⇢ to a resonance.(3) ⇣ Itis broughtn is simplest to a position to imagine⇣ on the z the axis, abovemass32 the system as4 a of! spherical neutrons.⇥ Its body, height varies of density 4 (3)and radius 0,which as a 32⇣(t)=⇣0 + ⇣the+4 e⇣ffcosect(! oft),inthiswayitjustgrazestheultracoldneutronsatits a⇥ macroscopic✓ ◆ mass, M,broughtascloseaspossibletothesystemofneutrons, 4 ✓ ◆ It is simplest! to imagine the mass M as a spherical body, of density ⇢ and radius ⇣0,which It is simplest to imagineas a the⇣(t)= mass⇣0 M+ as⇣ + a spherical✓⇣cos(!t),inthiswayitjustgrazestheultracoldneutronsatits◆ body, of density ⇢ and radius ⇣✓0,which ◆ is broughtminimum to a height. position Theand distance subjected⇣ on from the a to neutron oscillatoryz axis, at position motion abovez is of at the course exactly system⇣(t) thez. frequency Then of neutrons.the corresponding Its height to a resonance. varies is brought to a positionminimum⇣ on the height.z axis, The distance aboveis brought the from system a neutron to a of position at neutrons. position⇣ onz Itsis the of heightz courseaxis, varies⇣ above(t) z. the Then system the of neutrons. Its height varies Spherical perturbation perturbingGRAVITATIONAL potential is: PERTURBATION It is simplest to imagine the mass M as a spherical body, of density ⇢ and radius ⇣0,which GRAVITATIONALasperturbing a ⇣(t)= potential PERTURBATION⇣0 is:+ as⇣ + a ⇣ (t)=⇣cos⇣0 (+! t)⇣,inthiswayitjustgrazestheultracoldneutronsatits+ ⇣cos(!t),inthiswayitjustgrazestheultracoldneutronsatits as a ⇣(t)=⇣0 + ⇣ + ⇣cos(!t),inthiswayitjustgrazestheultracoldneutronsatitsGmN M W (t, z)= W1(z) ⇣W2(z)cos(!t) (4) is broughtGmN M ⇣ to(t) a positionz ⇡ ⇣ on the z axis, above the systemz of neutrons.⇣(t) z Its height varies minimum height. Theminimum distance from height. a neutronSphericalW (t, The zminimum)= at position perturbation distance height.z Wis 1 from of( Thez) course distance a⇣W neutron⇣2((tz))cos fromz(!.t a Then) at neutron position the at positionz (4)is ofis ofcourse course ⇣(t) .z Then. Then the the GmN M⇣(t) z ⇡ Gm N M where W1(z)= and W2(z)= 2 (to first order in ⇣). The first term can The gravitational perturbation that we imagine the system is subjected to, corresponds to(⇣0+as ⇣ az)⇣( t)=⇣0 + (⇣0⇣ z+) ⇣cos(!t),inthiswayitjustgrazestheultracoldneutronsatits Spherical perturbationGmN M perturbing potentialGmN M is: perturbing potential is: W (z)= W (z)= 2 ⇣ where 1 (⇣0+ ⇣ z) and 2 (⇣0 z) (to first order in ). The first term can perturbingbe treated potential by time-independent is: perturbation theory while the second term needs the time- M The gravitationalminimum perturbation height. The distance that weGm imagine fromN M a neutron the system at positionis subjectedz is to, of coursecorresponds⇣(t) toz. Then the the effect of a macroscopic mass, ,broughtascloseaspossibletothesystemofneutrons,be treated byGm time-independentN M perturbation theoryW while(t, z)= the second termW needs1(z) the time-⇣W2(z)cos(!t) (4) W (t, z)=dependent theory.W1( Thez) relevant ⇣W2 time(z)cos dependent(!t) perturbation theory(4) is that which computes GmNMM ⇣(t) z ⇡ The gravitationaldependent theory.⇣ perturbation(t)the Thez e⇡ff relevantect ofperturbing that a timeW macroscopic( wet, dependent z imagine)= potential mass, perturbation the is: system,broughtascloseaspossibletothesystemofneutrons, theoryW is is( subjectedz that) which ⇣W computes to,(z corresponds)cos(!t) to and subjected to oscillatory motion at exactly the frequency corresponding to a resonance. GmN M 1 GmN M 2 (4) GmN M GmN Mwhere W1(z)= (⇣ ⇣+( t⇣)3z) andz W2(z)= (⇣ z)2 (to first order in ⇣). The first term can W (z)= W (z)= 2 0 ⇣ ⇡ GmN M 0 where 1 (⇣0+ ⇣ z) and 2 and( subjected⇣0 z) (to to first oscillatory order in motion ). The at exactly first term the can frequency corresponding to a resonance. the effect of a macroscopic mass, M,broughtascloseaspossibletothesystemofneutrons,3 W (t, z)= W1(z) ⇣W2(z)cos(!t) (4) It is simplest to imagine the mass M as a spherical body, of density ⇢ and radius ⇣0,whichGmbeN treatedM by time-independentGm perturbationN M⇣(t) z ⇡ theory while the second term needs the time- be treated by time-independentwhere W perturbation1(zIt)= is simplest theory to while imagineand theW the second2( massz)= termM as needs a spherical2 the(to time- first body, order of density in ⇢ and⇣). radius The⇣0 first,which term can (⇣0+ ⇣ z) GmN M (⇣0 z) GmN M and subjected to oscillatory motion atW exactly(z)= the frequency W corresponding(z)= 2 to a resonance. ⇣ dependentwhere theory.1 The(⇣0+ relevant⇣ z) and time2 dependent(⇣0 perturbationz) (to first theoryorder in is that). which The computes first term can is brought to a position ⇣ on the z axis, above the systemdependent of theory. neutrons. Thebe relevant treated Its height time byis dependent brought varies time-independent perturbation to a position theory⇣ perturbationon the is that z axis, which above theory computes the systemwhile ofthe neutrons. second Its term height needs varies the time- It is simplest to imagine the massbeM treatedas a by spherical time-independent body, of perturbation density ⇢ and theory radius while⇣ the0,which second term needs the time- as a ⇣(t)=⇣0 + ⇣ + ⇣cos(!t),inthiswayitjustgrazestheultracoldneutronsatits3 as a ⇣(t)=⇣0 + ⇣ + ⇣cos(!t),inthiswayitjustgrazestheultracoldneutronsatitsdependent theory.3 Thedependent relevant theory. time The dependent relevant time perturbation dependent perturbation theory theory is that is that which which computes computes is brought to a positionminimum⇣ on the height.z axis, The above distance the from system a neutron of neutrons. at position z Itsis of height course ⇣ varies(t) z. Then the minimum height. The distance from a neutron at position z is of course ⇣(t) z. Then the 3 as a ⇣(t)=⇣0 + ⇣ + perturbing⇣cos(!t) potential,inthiswayitjustgrazestheultracoldneutronsatits is: 3 perturbing potential is: GmN M minimum height. The distance from a neutronW (t, z at)= position z isW1 of(z) course ⇣W⇣2(zt)cos(z!.t) Then the (4) ⇣(t) z ⇡ GmN M W (t, z)= W1(z) ⇣W2(z)perturbingcos(!t) potential is: (4) GmN M GmN M W (z)= W (z)= 2 ⇣ where 1 (⇣0+ ⇣ z) and 2 (⇣0 z) (to first order in ). The first term can ⇣(t) z ⇡ GmN M GmN M GmN M Wbe(t, treated z)= by time-independentW1(z) perturbation ⇣W2(z)cos theory(!t) while the second term needs(4) the time- W (z)= W (z)= 2 ⇣ where 1 (⇣0+ ⇣ z) and 2 (⇣0 z) (to first order in ). The first term can⇣(t) z ⇡ dependent theory. The relevant time dependent perturbation theory is that which computes GmN M GmN M W (z)= W (z)= 2 ⇣ be treated by time-independent perturbation theory while thewhere second1 term needs(⇣0+ ⇣ thez) and time- 2 (⇣0 z) (to first order in ). The first term can 3 dependent theory. The relevant time dependent perturbation theorybe treated is that by time-independent which computes perturbation theory while the second term needs the time- dependent theory. The relevant time dependent perturbation theory is that which computes 3 3 Trapping UCN‘s in the earth‘s gravitational field
Schrödinger equation: En En 1st state 1.41peV 1.41peV 2 2 mgz (z) E (z) 2nd state 2.46peV 2.56peV 2m z 2 n n n 3rd state 3.32peV 3.97peV boundary conditions:
n (0) 0 with 2nd mirror at height l
n (l) 0 solutions: Airy-functions scales: energies: peV length: m
V [peV] neutron mirror
Tobias Jenke, 7 th International UCN Workshop, St. Petersburg, 13.06.2009 5 Graviton Laser • The idea of a laser which amplifies gravitational waves is very intriguing. • For this possibility, one needs to believe in the existence of gravitons. • For a laser, one needs a “lasing medium”, a quantum mechanical system which can exist for long enough time, in an excited state in the bulk, where the population is inverted. • An ensemble of very cold UCN’s could satisfy these conditions, and could be imagined as providing the lasing medium. • A laser works by the principle that in a lasing medium, spontaneously emitted “gravitons” can subsequently stimulate the emission of “gravitons” that are coherent with the original graviton. The process can cascade multiple times, creating in principle a the end, an avalanche of coherent gravitons. Lasing medium • If the gain in the lasing medium is sufficiently high, a single pass is adequate to obtain a significant amplification, there is no need for reflecting mirrors. • With photons, there exist many examples of single pass lasing media, nitrogen gas lasers for example. • Very cold UCN’s, organize themselves according to the Schrödinger energy levels of a particle in a linear (gravitational) potential. • Neutrons in an excited state can only decay through their interactions. They interact with the phonons in the base, and they interact with gravity. • A neutron in an excited state can disintegrate, emit a phonon or emit a graviton. We can compute the rate of spontaneous (and stimulated) emission of a graviton for an excited neutron. systemsystem by by mechanically mechanically vibrating vibrating the the base base at at the the appropriate appropriate frequency, frequency, approximately approximately 600Hz. The neutrons can be lifted to say the third excited level using the mechanical 600Hz. The neutrons can be lifted to say the third excited level using the mechanical pumping method. Pumping at the resonant frequency for the transition between the first and pumping method. Pumping at the resonant frequency for the transition between the first and third levels asymptotically will populate each of these levels equally. But then there will be a third levels asymptotically will populate each of these levels equally. But then there will be a population inversion with respect to the second excited level. Thus spontaneous emission of a populationgraviton inversion from a transition with respect from the to thirdthe second to the secondexcited level, level. and Thus the subsequent spontaneous stimulated emission of a gravitonemission from of a gravitons transition could from yield the significant, third to the laser second type level, amplification and the of subsequent the gravitational stimulated emissionwave. of gravitons could yield significant, laser type amplification of the gravitational wave.
QUANTUM GRAVITONS AND THE AMPLITUDE FOR SPONTANEOUS AND STIMULATED EMISSION QUANTUM GRAVITONS AND THE AMPLITUDE FOR SPONTANEOUS AND
STIMULATEDIn a zero order EMISSION approximation, gravitons correspond to the non-interacting excitations of the quantized, linearized Einstein equations. These equations describe the free field Indynamics a zero orderof a massless approximation, spin 2 field. gravitons The quantization correspond is trivial to the and non-interacting well understood. excitations The of thequantum quantized, gravitational linearized field correspondsEinsteinGravitons equations. to the tensor Thesehµ⌫, which equations can be describe decomposed the intofree field dynamicsits Fourier of a modes massless spin 2 field. The quantization is trivial and well understood. The • A graviton corresponds to the non-interacting, free field µ µ ikµx ikµhx quantum gravitationalquantum field excitation corresponds3 ofe the linearized to the tensor Einsteine µ⌫ ,equations: which can be decomposed into hµ⌫ = d k A(~k)✏µ⌫ + ✏⇤ A†(~k) (1) pV pV µ⌫ its Fourier modes ✓ ◆ gµ⌫Z = ⌘µ⌫ + hµ⌫ @ @ hµ⌫ =0 µ µ ~ where ✏µ⌫ is the polarization tensor forik masslessµx spin twoik particlesµx and the operators A†(k) 3 e ~ e ~ ~ hµ⌫ = d k A(k)✏µ⌫ + ✏µ⇤⌫A†(k) (1) and A(k) satisfy the simple algebra ofp annihilationV and creationpV operators Z ✓ ◆ 2 ~ ~ 2⇡~c 3 ~ ~ where ✏µ⌫ is the polarization tensorA(k),A for†(k massless0) = spin (k twok particles0). and the operators(2)A†(~k) for a gravitational wave propagating in the x direction! polarized in the + sense in Minkowski A(~k) h i and satisfy the simple algebra of annihilation and creation operators ~ space-time.These In operators the• The weak respectively weak field field linear metric create approximation, andis given destroy by: one the free combined graviton of metric momentum is k. The ~ 2 frequency satisfies k0 = k which is appropriate2 for⇡~c a massless3 particle. 2 2 | | 2AGM(~k),A†(~k20) 2= 2(~kGM~k0). 2 2 (2) c d⌧ = 1 c dt 1+ (~x dx~ ) /r The gravitational field of the2 graviton, interacts! with the 2 system of ultra-cold neutrons h c r i c r · ✓ ◆ ✓2 ◆ 2 2 2 by distorting the space-time in whichh( thex neutronct)dy is+ immersed.h(x ct) Thedz ambientr d⌦ space-time. ~k is (5) These operators respectively create and destroy one free graviton of momentum . The that which corresponds to the gravitational field of the earth. This space-time is simply the frequency satisfies k0 = ~k which is appropriate for a massless particle. The classicalSchwarzschild dynamics geometry of the| | neutrons, apart from the reflection from the base, is governed The gravitational field of the graviton, interacts with the system of ultra-cold neutrons by the geodesic equation: 1 2 2 2GM2 i 2 2 µ 2⌫GM 2 2 2 by distorting the space-timec d⌧ = 1 in whichd x thec dt neutrondx1 dx is immersed. dr Ther d⌦ ambient space-time(3) is c2r + i c2=0r ✓ 2◆ µ⌫ ✓ ◆ (6) that which corresponds to the gravitationald⌧ fieldd⌧ ofd⌧ the earth. This space-time is simply the The connection is given to first order by 3=(1/2)⌘ (@µh ⌫ + @⌫h µ @ hµ⌫).Wefollow Schwarzschild geometry µ⌫ closely the exposition by Bertschinger in [4], decomposing the1 gravitational field as 2 2 2GM 2 2 2GM 2 2 2 c d⌧ = 1 2 c dt 1 2 dr r d⌦ (3) h00 = 2 c,hr 0i = wi ,hij = c r2 ij +2s ij (7) ✓ ◆ ✓ ◆ ij 3 where sij =0,wehavewi =0, = GM /r, = GM /3r and sij = sij + sij⇡ with 3 2 sij =(2GM /r )(xixj (r /3) ij) and szz⇡ = syy⇡ = h(x t) where we now add the use the symbol for the gravitational wave. ⇡ The Hamiltonian for the system is given by
i ij 2 1/2 H(x , ⇡j)=(1+ )E(pj)whereE(pj)=( pipj + mN ) (8)
ij 2 1/2 j with pi =(1+ )⇡i ( ⇡i⇡j + m ) wi s ⇡j which in our case becomes just pi = N i j (1 + )⇡i s ⇡j.As⇡i and pi differ only by first order terms, we get the Hamiltonian to i first order ij ij i ( s )⇡i⇡j H(x , ⇡j)=E(⇡j)+ + E(⇡j). (9) E(⇡j) Expanding to second order in the canonical momentum gives 2 ij ij 2 i ~⇡ ( s )⇡i⇡j ~⇡ H(x , ⇡j)=mN + | | + + (mN + | | ). (10) 2mN mN 2mN From this expression we extract our basic Hamiltonian of the neutron interacting with the gravitational field of the earth 2 i ⇡ H0(x , ⇡j)= | | + mN (11) 2mN subtracting off the rest mass, and the perturbation, which we split into two terms 2 ij ij i ~⇡ ( s )⇡i⇡j H (x , ⇡j)= | | + (12) 2mN mN ij i s⇡ ⇡i⇡j H⇡(x , ⇡j)= . (13) mN
4 for a gravitational wave propagating in the x direction polarized in the + sense in Minkowski for a gravitational wave propagating in the x direction polarized inspace-time. the + sense In in the Minkowski weak field linear approximation, the combined metric is
2 2 2GM 2 2 2GM 2 2 for aspace-time. gravitationalfor a gravitational In wave the weak propagating wave field propagating linear in the approximation,x x direction in the x polarizeddirection the combined polarizedin the + metric sense in the is in + Minkowski sensec ind⌧ Minkowski= 1 c dt 1+ (~x dx~ ) /r for a gravitational wave propagating in thex direction polarized in the + sense in Minkowski 2 2 for a gravitational wave propagatingfor a gravitational in the wavedirection propagating polarized in in the thex +direction sense in polarized Minkowski in the + sense in Minkowskic r c r · 2 2 2GM 2 2 2GM 2 2 ✓ ◆ ✓ ◆ space-time.space-time.space-time. In In the the weak weak In field the field linear weak linear approximation, field approximation, linear approximation, the the combined combined the metric combined metric is~ is metric is 2 2 2 2 space-time. In the weak fieldc d⌧ linearfor= a gravitational1 approximation,2 wavec dt the propagating combined1+ in metric2 the x(direction is~x dx) /r polarized in the + sense in Minkowskih(x ct)dy + h(x ct)dz r d⌦ . (5) space-time.2GM In thec r weak field linear2GM approximation,c r · the combined metric is 2 2 2 2 2 2✓GM2 2 222GM◆2 22✓GM2 2GM◆~2 22 2 2 2 2c dc2⌧d⌧==1space-time.c21GMd⌧ = 21 Inc 2dtc thedt weak 1+2 field1+cGMdt2 linear 1+ approximation,(~x(~xdx2~ dx)22 /r) /r(2~x thedx~2 combined) /r metric is 2 2 2 h2 (x2 ct)dy2GM +2 2h(x 2~ ct2)dz2 r 2dGM⌦ . 2 (5)2 c d⌧ = 1 2 c rc rc dt c r1+ 2 c rc r(~x dx·) c·/rr The classical· dynamics~ of the neutrons, apart from the reflection from the base, is governed ✓ c r c◆d ⌧ = 1✓ c r 2 ·c◆ dt 1+ 2 (~x dx) /r ✓ ✓◆ 2 ✓2 ◆2 c r2GM✓◆ 22 2 2 ◆ 2 c2GMr 2 2 ✓ ◆ ✓ 2 ◆ 2 2 2 2 2 2 2 · ~ h(hx(x ctc )ctd2dy⌧)✓dyh=+(x+h1(hxct(x)dyct2◆)2ctdz+)dzhc2(xdtr 2dr✓ct⌦byd)⌦dz.1+ the. geodesicr2 d◆⌦ .(~x equation:(5)dx(5)) /r (5) The classical dynamics of theh( neutrons,x ct) dy apart+ h(x from ct) thedz c r reflectionr d ⌦ . 2 from thec base,r (5)2 is· governed2 2 2 i µ ⌫ Gravitational ✓ hInteraction(x ◆ ct) dy +✓ h(x ct)dz◆ r d⌦ . d x (5) dx dx 2 2 2 2 + i =0 (6) TheThe classicalby classical theThe geodesic dynamics dynamics classical equation: of dynamics of the the neutrons, neutrons, of the apart neutrons, apart from from the apart the reflection reflection fromh the(x from reflection fromct) thedy the base,+ base,h from(x is isgoverned thect) governeddz base,r isd⌦ governed. d⌧ 2 (5) µ⌫ d⌧ d⌧ The classical dynamics of the neutrons,• Classically apart 2the fromi neutrons the reflection interactµ ⌫ with from the the gravitational base, is governed field The classical dynamicsd x of thei dx neutrons,dx apart from the reflection from the base, is governed essentially through+ the geodesic=0 equation. The connection is given(6) to first order by =(1/2)⌘ (@µh ⌫ + @⌫h µ @ hµ⌫).Wefollow byby theby the the geodesic geodesic geodesicby equation: the equation: equation: geodesic equation:The classical dynamics2 µ of⌫ the neutrons, apart from the reflection from the base, is governed µ⌫ by the geodesic2 2i equation:i d⌧ 2µ iµ ⌫d⌧⌫ d⌧ µ ⌫ 2 di dx x µidxddxx⌫dxdx i dx dx by thed x geodesici +dxi equation: dx =02 i µ closely⌫ the exposition by Bertschinger in [4], decomposing the gravitational field as The connection is given to first+2 order +2 µ⌫ byµ⌫ 2 =0+=(1 =0µ⌫/d2)x⌘ (@µih=0 ⌫dx+ dx@⌫h µ @ hµ⌫).Wefollow(6)(6) (6) d2 ⌧d⌧ µ⌫ dd⌧⌧dµ⌧⌫d⌧d⌧ d⌧ 2d⌧i µ ⌫ (6) d⌧ d⌧ d⌧ 2d+x µ⌫ i dx dx=0 (6) d⌧ 2 + dµ⌫⌧ d⌧ =0 (6) TheThe connectionclosely connectionThe the is connection exposition is given given to to first is by first given Bertschingerorder order to by first by order=(1 in=(1 [4/ by],2) / decomposing⌘ 2)⌘(=(1@µ(@hµ ⌫dh/⌧2) ⌫+⌘+@ the⌫@h(⌫ @ gravitationalhµµd h⌧µ ⌫@d +⌧@h @µh⌫⌫)µh.Wefollow⌫ )µ.Wefollow field@ ashµ⌫).Wefollowh00 = 2 ,h0i = wi,hij = 2 ij +2sij (7) The connection is given to first order by µ⌫ µ=(1⌫ µ⌫ /2)⌘ (@µµ⌫ h ⌫ + @⌫h µ @ hµ⌫) .Wefollow The connection is given to first order by µ⌫ =(1 /2) ⌘ ( @µh ⌫ + @⌫h µ @ hµ⌫) .Wefollow The connection is given to first order by µ⌫ =(1/2)⌘ (@µh ⌫ + @⌫h µ @ hµ⌫).Wefollow closelycloselyclosely the the the exposition expositionclosely exposition the by by exposition Bertschinger by Bertschinger Bertschinger by in Bertschinger in [4], in [4 decomposing], [4 decomposing], decomposing in [4], the decomposing the gravitational the gravitational gravitational the field gravitational field as fieldij as as field as h00 = 2 ,h0i = wi,hij = 2 ij +2wheresij sij =0,wehave(7) wi =0, = GM /r, = GM /3r and sij = sij + sij⇡ with closelyclosely the exposition the exposition by Bertschinger by Bertschinger in [ in4], [4 decomposing], decomposing the the gravitational gravitational field field as as 3 2 ij h00h00==•2For 2,h h the,h00 0=wavei 0=i =w2 in iw,h,h ai ,hSchwarzschildij0i =ij==w2i,h 2 ij background, ij+2ij =+2sijs2ij s ijij we+2=(2 have:sijGM /r )((7)xi(7)xj (r /3)(7) ij) and szz⇡ = syy⇡ = h(x t) where we now add the use the s =0h00 = 2 ,hw 0i=0= wi,h =ijGM= 2/r ij +2=sGMij /3r s = s(7) + s where ij ,wehave i , h00 = 2 ,,h 0i = w i,hijand= 2ij ij +2ij sij ij⇡ with (7) h00 = 2 ,h0i = wi,hij = 2 ij +2sij (7) ij ij 3 2 symbol for the gravitational wave. sij =(2GM /rij )(xixj (r /3) ij) and s⇡ = s⇡ = h(x t) where we now⇡ add the use the wherewherewhere ijsij sij=0swhereij=0,wehave=0,wehave ,wehavesij =0wiw,wehave=0iwij=0i ,=0 , = , =wGMi=GM=0GM/rzz,,/r /r, == , GMGM=yy=GMGM//r3r,/ 3and/r3=andr sGMandij s=ijs/sij=3 r=+sandij ss⇡+ij +withsij⇡sij⇡with= withsij + sij⇡ with ij ij ij wherewhere sij =0sij,wehave=0 ,wehavewi =0wi =0, ,= GM= GM/r,/r The, = = HamiltonianGMGM/3/r3randand forssijij the== s systemsijij ++ sij⇡ iswithwith given by 3 3 3 2 23 2 2 s sij =(2s =(2=(2GMsymbolGMGM/rs /r)(=(2/rx)(forx)(GMixx thejix(jr gravitational/r(/r3)(/r)( 3)x/ 3))ixij j)ijand)(sandr wave.s/=zz⇡3)s⇡ =ijs=) ands=yy⇡sh⇡=s(⇡x=h(=hxt()xst⇡) twhere)=whereh(x we wet now) where now add add we the the now use use add the the the use the ij ij ij i j ij and zz⇡ 3 zz 3yy⇡ yy2zz 2 whereyy we now add the use the ⇡ s ij =(sij 2=(GM 2/rGM)(x/r ix )(j x ix(jr /3)( r /ij 3)) andij) andszz⇡ s=zz⇡ =syy⇡ s=(1yy⇡ =/h2)(xh(xt) wheret) where we now we now add add the use the i ij 2 1/2 symbolsymbolsymbolforTheforsymbol thefor Hamiltonianthe gravitationalthe gravitational gravitationalfor the for wave. gravitational the wave. wave. system is wave. given by H(x , ⇡j)=(1+ )E(pj)whereE(pj)=( pipj + mN ) (8) ⇡⇡⇡ ⇡ use thethe symbol symbol forfor the the gravitational gravitational wave. wave. ⇡⇡ TheTheThe Hamiltonian Hamiltonian HamiltonianThe for Hamiltonian for the for the system thei system system for is given is the is given system given by by by is given by ij 2 1/2 ij 2 1/2 j H(x , ⇡j)=(1+The Hamiltonian )E(pj)where for the systemE(pj)=( is given pi bypjwith+ m p)i =(1+ )⇡i (8)( ⇡i⇡j + mN ) wi s i⇡j which in our case becomes just pi = The Hamiltonian for the system is given by N j i i i i ij ij ij 2 1/22(1ij21/ +21/2 )⇡ 2 s1/2⇡ ⇡ p HH(xH(,x⇡(,x⇡)=(1+,j⇡)=(1+)=(1+H(x ij),E ⇡)( Ep)=(1+)(E)wherepj()wherep )where2 1E/)2Ei(pE(p()=(Ep)wherej()=(pj )=(p p Epi+(pppjmp+)=(+m) Nm ) p) p + mi )ij i j.As(8) 2 1i/2and i differ only by first order terms, we get the Hamiltonian to j j j j j iH(x , ⇡jj)=(1+j j i )jE(pijj)wherejN N Ei (jpj)=( ijN p(8)ipj + (8)m2N )1/2 (8) (8) with pi =(1+ )⇡i ( ⇡i⇡j +H(mxN,)⇡j)=(1+wi s i ⇡)jEwhich(pj)where in ourE(p casej)=( becomes pipj + justmN )pi = (8) first order j ij ij 2 12/21/2 j j j j j p(1=(1+ + )⇡i )s⇡ ⇡j.As( ⇡ij⇡⇡i and+ mpiij)di2 ff1erw/2 onlys2 by⇡1/ij2 first order2 terms,1/2 we get the Hamiltonianp = to ij ij withwithwithpi i=(1+pi =(1+with )⇡pi =(1+)ii⇡(i ⇡(i ⇡ jwithi ⇡)+⇡ji⇡imjpN+i ()=(1+ NmN⇡w)ii⇡j i w+s)i⇡mii⇡ijjNis)which(ji ⇡whichjw⇡iwhichi⇡j in+2s our inmi1⇡ in/N ourj2) casewhich ourw casei becomes casej ins becomesi⇡ becomes ourj which just case justp in justi becomes= ouri pi case= just becomesi pi = just pi = ( s )⇡i⇡j with pi =(1+ )⇡ i ( ⇡i⇡j + mN ) wi s i⇡j which in our caseH becomes(x , ⇡j)= justE(p⇡ij)+= + E(⇡j). (9) j j j (1 + )⇡ s ⇡j ⇡ j p E(⇡j) (1 +(1 +)⇡first i )i⇡si order(1i⇡ +ijs.Asji ⇡.As)j⇡.Asi⇡i andis⇡andii⇡andpj(1i.Asdii +pffdii er⇡ffdi)i⇡er onlyjffiander onlys only bypi⇡i byj firstdi.As byff firster order first⇡ onlyi orderand order terms, bypi terms,di first terms,ffer we order only we get we get by the terms, get first the Hamiltonian the order Hamiltonian we Hamiltonian get terms, the to we Hamiltonian to get to the Hamiltonian to to (1 + )⇡i s i⇡ j.As⇡i and piji differij only by first order terms, we get the Hamiltonian to i ( s )⇡i⇡j Expanding to second order in the canonical momentum gives firstfirstfirst order order orderfirst order Hfirst(x order, ⇡j)=E(⇡j)+ + E(⇡j). (9) first order ij ij ijij ij ijE(⇡jij) ij ( ij sij)⇡ ⇡ 2 ij ij 2 i i ( ( ( s )s⇡is⇡)⇡j()i ⇡⇡ j⇡ s )⇡ ⇡ij ij i j ~⇡ ( s )⇡i⇡j ~⇡ i i H(x , ⇡j)=i jE(⇡j)+( i j s )⇡ ⇡ + E(⇡j)i. (9) H(HxH(,x⇡(,jx)=⇡,j⇡)=jE)=(HE⇡j((E)+x⇡(j,⇡)+⇡jj)+)= E( ⇡i j )+ + +E + (⇡E j)(E.⇡(j)⇡.j).+ E(⇡i jj). (9)H(9)(x(9), ⇡j)=m(9)N + | | + + (mN + | | ). (10) Expanding to second order in the canonicalHE((x⇡E,()⇡⇡ momentumjj)=) E(⇡j)+ gives E(⇡j) + E(⇡j). 2m (9) m 2m j E(⇡j) E(⇡j) E(⇡ ) N N N Expanding to second2 orderij in theij canonical momentumj 2 gives ExpandingExpandingExpanding toExpanding to second to second second order order to order ini second in the in the canonical the order canonical canonical in~⇡ momentum the momentum canonical momentum( gives momentums gives gives)⇡i⇡j givesFrom~⇡ this expression we extract our basic Hamiltonian of the neutron interacting with the HExpanding(x , ⇡j)=m toN second+ | | order+ in the canonical2 + momentum (mijN + ij| | gives). (10)2 2 2 22mijN iji ijij2 ij ijmNij ~⇡ ij ( 2 2 s22)m⇡iN⇡j 2 ~⇡ i i ~⇡ ~⇡ ~⇡ ( (H (( x ~⇡,s⇡j)s)=⇡is⇡)⇡jm()i ⇡⇡N j⇡+ 2| s| )+⇡i⇡~⇡ ijj ~⇡ ~⇡ij +~⇡ (mN + | 2| ). (10) HH(x(,x⇡, ⇡)=i )=mm+ i+ + + +i~⇡ +j(m (m+( + )gravitational.s )).⇡i⇡j field of the~⇡ earth H(xj ,j⇡j)=HN (Nmx N,|⇡+|j|)=| |mN+i + | | + 2+mN N(Nm|N |+|+|| m(|Nm)N. + | (10)| )(10). (10)2mN (10) From this expression we extract2m2Nm H our(x , basic⇡jm)=Nm HamiltonianmN + | | + of the2m neutronN2 m interacting+ (mN + with| | the). (10) 2NmN 2mNNmN 2m mN m2NmN 2mN 2m 2 From this expression we extract ourN basic HamiltonianN of the neutron interactingN withi the ⇡ From this expression we extract our basic Hamiltonian of the neutron interacting with the H0(x , ⇡j)= | | + mN (11) FromFrom thisgravitational this expressionFrom expression this field we expression we of extract extract the earth our we our extract basic basic Hamiltonian our Hamiltonian basic Hamiltonian of of the the neutron neutron of the interacting interacting neutron with interacting with the the with the 2mN Fromgravitational this expression field we of extract the earth our basic Hamiltonian of the neutron interacting with the gravitationalgravitational field field of of the the earth earth ⇡ 2 gravitationalgravitational field of the field earth of the earth i subtracting2 off the rest mass, and the perturbation, which we split into two terms gravitational fieldH of0(x the, ⇡j earth)= | | + mN ⇡ (11) 2 2 i ⇡ 22mN H02(x , ⇡j)= | | + mN (11) 2 ij ij i i ⇡ ⇡ ⇡ 22m i ~⇡ ( s )⇡i⇡j H0H(x0(,x⇡j,)=⇡i j)=| || |+i m+Nm N i ⇡ N (11)(11) H (x , ⇡ )= + H0(x , ⇡j)=H0(x| ,|⇡j)=+ mN| | + mN (11) (11) j | | (12) subtracting off the rest mass, and the2m perturbation,2NmN H0(x which, ⇡j)= we| split| + intomN two terms (11)2mN mN subtracting off the rest2m mass,N and2m theN perturbation,2mN which we split into two terms 2 ij ij ij subtracting off the rest mass, and the perturbation, ~⇡ which( we splits into)⇡ 2 two⇡ termsij ij i s⇡ ⇡i⇡j subtracting off the rest mass, and thei perturbation, which we split ~⇡ intoi j two( termss )⇡ ⇡ H⇡(x , ⇡ )= . subtractingsubtracting off the restsubtracting off mass,the rest andH o mass,ff( thexthe, ⇡ perturbation,j andrest)= mass,the| | perturbation,+ and which thei perturbation, we which split into we splittwo which terms into we two spliti j terms into(12) two terms j (13) 2 2 ij ijH ij(x ,ij⇡j)= | | + (12) mN ~⇡ ~⇡ (2 2m ( N sij 2s) ⇡i⇡)ij⇡jm⇡Nij 2m ij m i i ~⇡ ( i j 2 sN )⇡iji⇡j ijN H H( x(,x⇡j,)=⇡i j)=| || +|i~⇡ + (ij i s )⇡~⇡i⇡j ( s )⇡(12)i⇡j (12) H (x , ⇡j)=s⇡ ⇡|i⇡|j + ij (12) H (x , ⇡j2)=im2NmN| | +H m(xN,m ⇡Nji)= | | s+⇡ ⇡i⇡j (12) (12) H⇡(x , ⇡j)=2mN 2Hm⇡N(.xm,N⇡ )=2m mN . m (13) 4 ij ij j N N (13) i s⇡ s⇡⇡i⇡⇡jij⇡ mN ij ij mN i i j s⇡ ⇡i⇡j H⇡H(⇡x(,x⇡j,)=⇡i j)= is⇡ .⇡i⇡. j i s⇡ ⇡i⇡j (13)(13) H⇡(x , ⇡j)=H⇡m( x , ⇡j)=H⇡ (.x , ⇡j)=. . (13) (13) (13) NmN m mN 4 N mN 4 4 4 4 4 4 2 Particle Motion and Gauge Dependence
We begin by studying an analogue of general relativity, the motion of a charged particle. The covariant action for a particle of mass2m Particleand charge Motionq has two and terms: Gauge one for Dependence the free particle and another for its coupling to electromagnetism: It happensWe that begin while by studying the Christo an analogueffel connection of general relativity, is not the gauge- motioninvariant, of a charged in particle. linearized gravity (but notThe in covariant general)1/2 action the Riemann for a particle tensor of mass ism gauge-invaand chargeriant.q has two Thus terms: one one way for to the form dxµ dxν dxµ S[xµ(τ)] = gauge-invariantm η free quantities particle anddτ is+ another to replaceqA for its equation couplingdτ . to (4) electromagnet by th(1)e geodesicism: deviation equation, µν dτ dτ µ dτ ! − "− # ! µ ν 1/2 µ µ 2 µ dx dx dx S[xd(τ()]∆ =x) m µηµν α β dτν+ qAµ dτ . (1) − "− dτ dτ # dτ Varying the trajectory and requiring it to be stationary, with τ 2being! = anR aαβνffineV parameterV (∆x) ! (7) µ µ µ ν dτ such that V = dx /dτ is normalized η VVaryingV = the trajectory1, yields and the requiring equation it to of be motion stationary, with τ being an affine parameter µνν −µ µ µ ν where (∆x) issuch the that infinitesimalV = dx /dτ is separation normalized η vectorµνV V between= 1, yields a the pair equation of geodesi of motioncs. While dV µthisq tells us all about the local environment of a freely-fal− ling observer, it fails to tell us = F µ V ν ,F = ∂ A ∂ AdV µ. q (2) ν µν µ ν ν µ = F µ V ν ,F = ∂ A ∂ A . (2) dτ wherem the observer goes. In− mostdτ applicationsm ν weµν needµ toν − knowν µ the trajectories. Thus we will have to find other strategies for coping with the gauge problem. Regarding gravity as a weak (linearized)Regarding field on gravity flat spacetim as a weakLagrangiane, (linearized) the action field for on aflat spacetime, the action for a particle of mass m also has two terms, one for the free particle and another for its particle of mass m also has two terms, one for• We the obtain free particlethe equation and of another motion from for itsa Lagrangian: coupling to gravity: 3 Hamiltoniancoupling to gravity: Formulation and Gravitomagnetism µ ν 1/2 µ ν µ ν −1/2 µ dx dx m dx dx dx dx µ ν 1/2 S[x (τ)] = µm νηµν µ dτν +−1/2 hµν ηµν dτ . µ dxSomedx aid in solvingm the gauge!dx− dx" problem− dτ d comesτdx# dx if we! abandon2 dτ d manifτ "− estd covarianceτ dτ # and use S[x (τ)] = m ηµν dτ + hµν ηµν dτ . (3) − "− dτ dτ0 # 2 dτ dτ "− dτ dτ # ! t = x to parameterizeThis! result trajectories comes from using instead the free-field of the proper action with time metricdτ.Thisyieldstheaddedg = η + h and This comes from linearizing the free field(3) action constructedµν µν µν benefit of highlightinglinearizing• the in the similarities small quantities betweenhµν. Note linearized that for thisgravity to be valid, and electromagnetism. two requirements This result comes from usingIn particular, the free-field itmust illustrates action be satisfied:with with metric: the First, metric phenomenon the curvaturegµν = of scalesηµ gravitomagnν + givenhµν byand theetism. eigenvalues of the Ricci tensor linearizing in the small quantitiesChanginghµν. Note the(which parameterizationthat• haveChanging for units this of to the inverse be parametrization in valid, equation length two squared) requirements(1) from from must τ be to large t and and compa performingperformingred with the a length Legendre must be satisfied: First, the curvature scalesscales given under by consideration the eigenva (e.g.lues one of the must Ricci be far tensor from the Schwarzschild radius of any transformationblack gives holes). thea Legendre Hamiltonian Second, transformation the coordinates mustyields be the nearly Hamiltonian. orthonormal. One cannot, for (which have units of inverse length squared)example, must use be spherical large compa coordinates;red with Cartesian the coordinates length are required. (While this i 2 2 1/2 i scales under consideration (e.g. one mustHsecond( bex , farπ conditioni,t from)= the canp + be Schwarm relaxed,zschild it+ makesqφ ,p radius the analysis ofπi any muchqAi simp, φler. If theA0 first. condition (8) holds, then coordinates can always be found such≡ that− the second ≡−condition holds also.) black holes). Second, the coordinates must be nearly! orthonor"mal. One cannot, for example, use spherical coordinates;Here we denote Cartesian theRequiring coordinates conjugate the gravitational momentum are required. action by to beπ (Whilei stationaryto distinguish this yields the it equation from the of motion mechanical i µ i second condition can be relaxed,momentum it makesp the. (Note analysis thatdV muchp and simp1 π ler.are If the the components first condition of 3-vectorsµ in Euclidean space, = ηµνi (∂ h + ∂ h ∂ h ) V αV β = Γ V αV β . (4) holds, then coordinates canso always that theirbe found indices such may thatd beτ the raised− second2 orαcondition loweredνβ β αν without holds− ν αβ also.) change.)− Itαβ is very important to Requiring the gravitationaltreat action the Hamiltonian to beThe stationary object as multiplying a functionyields t thehe of equation 4-velocities the conjugate of on motion the momentum right-hand sideand is just not the the linearized mechanical Christoffel connection (with ηµν rather than gµν used to raise indices). µ momentum, because only in this way do Hamilton’s equations givethecorrectequations dV 1 µν Equations (2)α andβ (4) areµ veryα similar,β as are the actions from which they were = η (of∂α motion:hνβ + ∂βhανderived.∂νhαβ Both) VF V and=Γµ Γareαβ tensorsV V under. Lorentz(4) transformations. This fact ensures dτ −2 − µν −αβ i thati equations (2) and (4) hold in any Lorentz frame. Thus, in the weak field limit it is dx p i dπi j 2 2 m The object multiplying the 4-velocities= onstraightforward thev right-hand, to= analyzeq side arbitrary∂iφ is+ justv relativistic∂i theAj linearized,E motions ofp the+ sourcesm = and test particles,. (9) dt Eas long≡ as all thedt components− of the Lorentz-transformed≡ field, h = Λµ Λ√ν 1h arev2 small Christoffel connection (with ηµν rather than gµν used to raise! indices). " # µ¯ν¯ µ¯ ν¯ µ−ν Equations (2) and (4) areCombining very similar, these gives as are the the familiar actions form from ofwhich the Lorentz2 they were force law, derived. Both F and Γµ are tensors under Lorentz transformations. This fact ensures µν αβ dpi that equations (2) and (4) hold in any Lorentz frame.= q (E Thus,+ v inB) the,E weak fieldφ limit∂ itA is,B= A (10) dt i t straightforward to analyze arbitrary relativistic motions of the× sources and≡−∇ test particles,− ∇ × where underscores denote standard 3-vectorsµ ν in Euclidean space. The dependence of the as long as all the components of the Lorentz-transformed field, hµ¯ν¯ = Λ µ¯Λ ν¯hµν are small fields on the potentials ensures that the equation of motion is still invariant under the gauge transformation φ φ ∂ Φ, A A + Φ. 2 → − t → ∇ Now we repeat these steps for gravity, starting from equation (3). For convenience, we first decompose hµν as h = 2φ ,h= w ,h= 2ψδ +2s , where sj = δijs =0. (11) 00 − 0i i ij − ij ij j ij 4 subtracting off the rest mass, and the perturbation, which we split into two terms 2 ij ij i ~⇡ ( s )⇡i⇡j H (x , ⇡j)= | | + (12) 2mN mN ij i s⇡ ⇡i⇡j H⇡(x , ⇡j)= . (13) mN The Schrödinger equation resulting from Eqn. (11)isessentiallyfreeinthex and y directions, but in the z direction it is ~2 d2 2 + mN gz (z,t)=E (z,t). (14) 2mN dz The Schrödinger equationSchrödinger resulting✓ from Eqn. Equation◆ (11)isessentiallyfreeinthe and the x and y the perturbation from Eqn.(12)inducessomeperturbativechangesintheenergiesandwave- directions, butThe in Schrödinger the z directioni equation itInteraction is resulting from Hamiltonian Eqn. (11)isessentiallyfreeinthex and y functions of H0(x , ⇡j),however,thesechangesarestaticanddonotgiverisetoanytran- • The Schrödinger equation governing the “free” part of the directions, but in the z direction2 2 it is sitions, spontaneousdynamics or~ stimulated,d is given between by: levels. The time dependent perturbation that + mN gz (z,t)=E (z,t). (14) 2m dz2 2 2 pG is relevant, from Eqn.(13N )(replacing~ d h(x ct) 2 h(x ct) which is the canonically ✓ + m◆N gz (z,t)=c E (z,t). (14) 2m dz2 ! normalized metric perturbation)✓ N is given by ◆ the perturbation from Eqn.(•12While)inducessomeperturbativechangesintheenergiesandwave- the interaction is given by: the perturbationi from Eqn.(12)inducessomeperturbativechangesintheenergiesandwave-p 2 2 H (x , ⇡ ) i G h(x ct)(⇡z ⇡y) functions of 0 j ,however,thesechangesarestaticanddonotgiverisetoanytran-⇡ i H (x , ⇡j)= 2 (15) functions of H0(x , ⇡j),however,thesechangesarestaticanddonotgiverisetoanytran- c mN sitions, spontaneous or stimulated between levels. The time dependent perturbation that sitions, spontaneous• orHaving stimulated rescaled between the field by: levels. The time dependent perturbation that which is written for a gravitational wave thatp isG propagating exclusively in the x direction. is relevant, from Eqn.(13)(replacingh(x ct) 2 hp(Gx ct) the canonically normalized 2 c is relevant, from Eqn.(13)(replacing h(x !ct) c2 h (x ct) the canonically normalized We can in principle drop the ⇡y term as it will not! contribute in the matrix element of the rel- metric perturbation) is • to get a canonically normalized field. metricevant states perturbation) of the neutron. is For a direction of propagation kˆ =(sin✓, 0, cos ✓) corresponding • However, this is one forp a graviton moving2 in2 the plane. i i G hp(Gxh(xct)⇡ctz)⇡z H⇡(x , ⇡⇡ )= . to a polar angle ✓ (taking it forH j( simplicityx , ⇡j)=2 in the2 the (x, z) plane),. we get (15)(15) c c mNmN ˆ ˆ ˆ 2 2 ˆ 2 2 Then amplitudei forp spontaneousG h(k ~x ct or)(( stimulatedkz⇡x + k emissionx⇡z) ⇡ ofy) a gravitonpG h is(k proportional~x ct)sin to✓⇡ thez Then amplitudeH⇡(x for, ⇡ spontaneous)= or· stimulated emission of a graviton= is proportional to the. j 2 2 2 · 2 c ⇡z mN c mN matrix element ⇡ zn n which, using Eqn.(14), gives 0 mN matrix element n0 hm | n which,| i using Eqn.(14), gives (16) h | N | i 2 Then the amplitude for spontaneous2 ⇡z or stimulated emission of a graviton is proportional to ⇡n n = n (2mN gz 2En) n z0 2 0 n h | ⇡ z mn N=| i n h(2m|N gz 2 En) |n i the matrix element 0 n n which,0 using Eqn.(14), gives h | 0 mNm|N i h | | i h | | i = n 2mN gz n . (16) h 0 | | i 2= n 2mN gz n . (16) ⇡z h 0 | | i n0 n = n0 (2mN gz 2En) n The matrix element is easilyh | computed mN | i usingh the| exact eigenfunctions | i and the integral
The matrix element is easily computed using the= exactn0 2mN eigenfunctionsgz n . and the integral (17) 1 h | 2Ai| 0( i ↵m)Ai0( ↵n) dyyAi(y ↵m)Ai(y ↵n)= (17) (↵ ↵ )2 The matrix1 elementZ0 is easily computed using the2 exactAi0( eigenfunctions↵m) Ai0n( ↵n) and the integral dyyAi(y ↵m)Ai(y ↵n)= 2 (17) which gives0 1 (2↵Aim 0( ↵↵nm))Ai0( ↵n) Z dyyAi(y ↵ )Ai(y ↵ )= m n 4m gz 2 (18) 0 2m gz = N (0↵m. ↵n) which gives Z n0 N n 2 (18) h | | i (↵n0 ↵n) which gives 4mN gz 0 The rate, per unit time andn 2m volume,N gz ofn spontaneous= 4 orm stimulatedgz. emission of gravitons(18) is h 0 | | i (↵ ↵ N)2 0 n0 2mN gz n =n0 n 2 . (19) then is given by, modifying a calculationh | in| Baym,i ( [5↵]n0 ↵n) The rate, per unit time and volume, of spontaneous or stimulated emission of gravitons is 2 2 emm. G 4⇡ c 5 then is given by, modifying ad calculation n n = in Baym,(Nk +1) [5] 0! c4 ckV ⇥ 2 2 2 n0 2mN gz n (En0 En ~ck) (19) emm. G 4⇥⇡ |ch | | i| d n n = (Nk +1) 0! c4 ckV ⇥ where Nk is the number of photons involved2 in the stimulated emission, and when Nk =0 n0 2mN gz n (En0 En ~ck) (19) we get the rate for spontaneous⇥ |h | emission.| (Thei| whole system is imagined in a box of volume 3 3 3 V .) There are Vd k/(2⇡) states in the volume d k of phase space and as Ek = ~c k where Nk is the number of photons involved in the stimulated emission, and when| N| k =0 3 we get the rate for spontaneous emission. (Thed k whole2 dE systemkd⌦ is imagined in a box of volume V 3 = VEk 3 (20) 3 3 (2⇡) 3 (2⇡~c) V .) There are Vd k/(2⇡) states in the volume d k of phase space and as Ek = ~c k | | 3 5 d k 2 dEkd⌦ V = VEk (20) (2⇡)3 (2⇡~c)3
5 subtracting off the rest mass, and the perturbation, which we split into two terms subtracting off the rest mass, and the perturbation, which we split2 into twoij termsij i ~⇡ ( s )⇡i⇡j subtracting off the rest mass, and the perturbation, which we split into2 H two (ijx terms, ⇡j)=ij | | + (12) i ~⇡ ( s )⇡2im⇡jN mN 2 Hij (x , ⇡ijj)= | | + ij (12) i ~⇡ ( s )⇡i⇡j2mN i mN s⇡ ⇡i⇡j H (x , ⇡j)= | | + ij H⇡(x , ⇡j)= (12) . (13) 2mN miN s⇡ ⇡i⇡j mN ij H⇡(x , ⇡j)= . (13) i s⇡ ⇡i⇡j mN H⇡(x , ⇡j)= The. Schrödinger equation resulting from(13) Eqn. (11)isessentiallyfreeinthex and y mN The Schrödinger equationdirections, resulting but in the fromz direction Eqn. (11 it)isessentiallyfreeinthe is x and y The Schrödinger equationdirections, resulting but in from the Eqn.z direction (11)isessentiallyfreeinthe it is x and y ~2 d2 z + mN gz (z,t)=E (z,t). (14) directions, but in the direction it is 2 d2 2m dz2 ~ ✓ N ◆ 2 2 2 + mN gz (z,t)=E (z,t). (14) ~ d 2mN dz + mN gz the ( perturbationz,t✓ )=E (z,t from). Eqn.(◆ 12)inducessomeperturbativechangesintheenergiesandwave-(14) 2m dz2 N i the✓ perturbation from◆functions Eqn.(12)inducessomeperturbativechangesintheenergiesandwave- of H0(x , ⇡j),however,thesechangesarestaticanddonotgiverisetoanytran- the perturbation from Eqn.(12)inducessomeperturbativechangesintheenergiesandwave-i functions of H0(x , ⇡j)sitions,,however,thesechangesarestaticanddonotgiverisetoanytran- spontaneous or stimulated, between levels. The time dependent perturbation that i functions of H0(x , ⇡j),however,thesechangesarestaticanddonotgiverisetoanytran- pG sitions, spontaneous oris stimulated, relevant, from between Eqn.( levels.13)(replacing The timeh( dependentx ct) perturbation2 h(x ct) thatwhich is the canonically ! c sitions, spontaneous or stimulated, between levels. The time dependent perturbationpG that is relevant, from Eqn.(normalized13)(replacing metrich( perturbation)x ct) c2 ish given(x ct by) which is the canonically pG ! is relevant, from Eqn.(13)(replacingh(x ct) 2 h(x ct) which is the canonically normalized metric perturbation)! c is given by 2 2 i pG h(x ct)(⇡z ⇡y) ⇡ normalized metric perturbation) is given by H (x , ⇡j)= 2 (15) 2 2 c mN i pG h(x ct)(⇡z ⇡y) H⇡(x , ⇡j)= (15) p 2 2 2 i G whichh(x ct is)( written⇡z ⇡y for) ac gravitationalmN wave that is propagating exclusively in the x direction. ⇡ H (x , ⇡j)= 2 (15) c mN 2 which is written for aWe gravitational can in principle wave drop that the is propagating⇡y term as it exclusively will not contribute in the x indirection. the matrix element of the rel- which is written for a gravitational wave that is propagatingInteraction2 exclusively for arbitrary in the x direction. gravitons ˆ We can in principle dropevant the states⇡y term of asthe it neutron. will not contribute For a direction in the of matrix propagation elementk =(sin of the rel-✓, 0, cos ✓) corresponding 2 We can in principle dropevant the ⇡y statesterm as of it the will neutron. notto• For a contribute polar a For graviton a angle direction in moving the✓ (taking matrix of in propagation a direction it element for simplicity ofkˆ the=(sin rel- in the✓, 0, thecos ✓(x,) corresponding z) plane), we get evant states of the neutron. For a direction of propagation kˆ =(sin✓, 0, cos ✓) corresponding to a polar angle ✓ (taking it for simplicity in theˆ the (x, z) plane),ˆ ˆ we get2 2 ˆ 2 2 i pG h(k ~x ct)(( kz⇡x + kx⇡z) ⇡y) pG h(k ~x ct)sin ✓⇡z H⇡(x , ⇡ )= · = . to a polar angle ✓ (taking it for simplicity in the the (x, zj) plane),2 we get 2 · ˆ ˆ c ˆ 2 2 mN ˆ 2 c 2 mN i pG h(k ~x ct)(( kz⇡x + kx⇡z) ⇡y) pG h(k ~x ct)sin ✓⇡z H⇡(x , ⇡ )= · = . (16) ˆ j ˆ 2ˆ 2 2 ˆ 2 2 2 · i pG h(k ~x ct)(( kz⇡ x +c kx⇡z) ⇡y) mpNG h(k ~x ct)sin ✓⇡ z c mN ⇡ · H (x , ⇡j)= 2 Then the amplitude= 2 for spontaneous· or stimulated. emission of a graviton is proportional to c mN c mN (16) 2 ⇡z (16) Then the amplitude forthe spontaneous matrix element or stimulated n0 m emission n which, of a using graviton Eqn.( is14 proportional), gives to h | N | i 2 Then the amplitude for spontaneous or stimulated emission⇡z of a graviton is proportional2 to the matrix element n0 n which, using Eqn.(⇡14z ), gives 2 mN ⇡z h | | i n0 n = n0 (2mN gz 2En) n the matrix element n n which, using Eqn.(14), gives h | m | i h | | i 0 mN 2 N h | | i ⇡z 2 n0 n = n0 (2mN gz 2En=) n n0 2mN gz n . (17) ⇡z h | mN | i h | | h i | | i n0 n = n0 (2mN gz 2En) n h | mN | i h | | =i n 2mN gz n . (17) The matrix element ish easily0 | computed| i using the exact eigenfunctions and the integral = n 2mN gz n . (17) h 0 | | i The matrix element is easily computed using1 the exact eigenfunctions and2 theAi0( integral↵m)Ai0( ↵n) dyyAi(y ↵m)Ai(y ↵n)= 2 (18) The matrix element is easily computed using the exact eigenfunctions0 and the integral (↵m ↵n) 1 Z 2Ai0( ↵ )Ai0( ↵ ) dyyAi(y ↵ )Ai(y ↵ )= m n which gives m n 2 (18) 1 0 2Ai 0( ↵m)Ai0 ( ↵n) (↵m ↵n) dyyAi(y ↵m)Ai(y Z ↵n)= 2 (18) 4mN gz0 0 (↵m ↵n) n 2mN gz n = . (19) Z which gives h 0 | | i (↵ ↵ )2 n0 n which gives 4mN gz0 n 2mN gz n = . (19) 4m gz0 2 h N | 0 | i (↵n0 ↵n) 5 n0 2mN gz n = 2 . (19) h | | i (↵n0 ↵n) 5 5 The Schrödinger equation resulting from Eqn. (11)isessentiallyfreeinthex and y directions, but in the z direction it is
~2 d2 + mN gz (z,t)=E (z,t). (14) 2m dz2 ✓ N ◆ the perturbation from Eqn.(12)inducessomeperturbativechangesintheenergiesandwave- The Schrödinger equation resulting from Eqn. (11)isessentiallyfreeinthex and y i functions of H0(x , ⇡j),however,thesechangesarestaticanddonotgiverisetoanytran- directions, but in the z direction it is sitions, spontaneous or stimulated between levels. The time dependent perturbation that 2 d2 h(~x ct) pG h(x ct) is relevant, from Eqn.(13)(replacing 2 + mNc2gz (z,t)=theE canonically(z,t). normalized (14) 2mN dz ! metric perturbation) is ✓ ◆ the perturbation from Eqn.(12)inducessomeperturbativechangesintheenergiesandwave-2 i pG h(x ct)⇡z H⇡(x , ⇡ )= . i j 2 (15) functions of H0(x , ⇡j),however,thesechangesarestaticanddonotgiverisetoanytran- c mN Then amplitudesitions, for spontaneous spontaneous or or stimulated stimulated between emission levels. of a graviton The time is dependent proportional perturbation to the that 2 ⇡z pG matrix elementis relevant, n from Eqn.(n which,13)(replacing using Eqn.(h(x14),ct gives) 2 h(x ct) the canonically normalized h 0 | mN | i ! c metric perturbation) is⇡2 z = (2m gz 2E ) 2 n0 n in0 N pG h(xn ctn )⇡z h | mN | i ⇡h | | i H (x , ⇡j)= 2 . (15) c mN = n 2mN gz n . (16) h 0 | | i Then amplitude forSpontaneous spontaneous or stimulated emission emission of a graviton is proportional to the 2 ⇡z The matrixmatrix element element is easily n computed n usingwhich, the using exact Eqn.( eigenfunctions14), gives and the integral • The amplitudeh 0 | formN spontaneous| i emission of a graviton will be proportional to: 1 ⇡2 2Ai0( ↵m)Ai0( ↵n) dyyAi(y ↵ m)Ai(y z ↵ n)== (2 m gz 2 E ) (17) n0 n n0 (↵ N ↵ )2 n n Z0 h | mN | i h | m n | i = n 2mN gz n . (16) which gives • which gives: h 0 | | i 4mN gz0 n 2mN gz n = . (18) The matrix elementh is0 easily| computed| i using(↵ the↵ exact)2 eigenfunctions and the integral n0 n The rate,The per rate, unit per• time unitThen time and the rate and volume,1 of volume, spontaneous of of spontaneous spontaneous or stimulated or or stimulated stimulated emission,2Ai emission modifying0( emission↵m) ofAi gravitons a0( of↵ gravitonsn) is is dyyAi(y ↵m)Ai(y ↵n)= 2 (17) then is given by,calculation modifying in a Baym, calculation is given in Baym, by: [5 ] (↵m ↵n) then is given by, modifying aZ0 calculation in Baym, [5] 2 2 emm. G 4⇡ c 4 2 d n n = (Nk +1)sin2 2 ✓ n0 2mN gz n (En0 En ~ck) (20) which gives0! c4 ckV G 4⇡ c |h | | i| d emm. = (N +1) n0 n 4 k 4mN gz0 where Nk is the number! of gravitonsc ckV involved in the⇥ stimulated emission, and when Nk =0 • This can be used to compute n0 2m theN gzabsorption n = cross section 2 . (18) h | 2| i (↵n0 ↵n) we get the rate for spontaneous emission.n0 2mN gz (The n whole (E systemn0 E isn imagined ~ck) in a box of volume (19) ⇥ |h | | i| The rate, per3 unit3 time and volume, of spontaneous or stimulated emission of gravitons is V .) There are Vd k/(2⇡) states in the volume of phase space and as Ek = ~c k | | N 3 N =0 where k isthen the is number given by, of modifying photons involved ad calculationk in the2 dE ink stimulatedd Baym,⌦ [5] emission, and when k V = VEk (21) (2⇡)3 (2⇡~c)3 we get the rate for spontaneous emission. (TheG 4⇡ whole2c2 system is imagined in a box of volume states per unit energy. Thend integrating emm. = Eqn.(20)overtheenergyoftheemittedgraviton(N +1) 3 3 n0 n 4 3 k V .) There are Vd k/(2⇡) states in the! volumec ckVd k of phase⇥ space and as Ek = ~c k we get | | 2 2 G (E E )d⌦ n0 2mN gz n 4 m(Engz0 En ~ck) (19) emm. n0 3 ⇥n |h | 4| i| N 0 d n n = d k (Nk +1)sin2 dEkd⌦✓ (22) 0! c4 2⇡ 2c (↵ ↵ )2 V ~ 3 = VEk 3 n0 n (20) (2⇡2 ) (2⇡ c) ✓ ◆ 3 ~ ~ SincewhereEn =NmkNisgz0 the↵n where numberz0 = of photons2 and integrating involved over in the directions, stimulated we get emission, and when Nk =0 2gmN we get the rate for spontaneous256 G emission. (Them wholegz 3 system is imagined in a box of volume emm. = (5N +1) N 0 n0 n 2 5 k 3 ! 3 15 ~ c ↵n0 ↵3n V .) There are Vd k/(2⇡) states in the volume✓ 2 d2 k◆of phase space and as Ek = ~c k 128 G mN g ~ | | = 2 5 (Nk +1) 3 15 ~ c 3 (↵n ↵n) d k 0 2 dEkd⌦ 2 128 G V 3 m=NVEg k 3 (20) = (Nk +1)(2⇡) (2⇡~c) (23) 15 c5 (↵ ↵ )3 n0 n 5 POPULATION INVERSION
We can obtain a population inversion by mechanically vibrating the base at the resonant frequency corresponding to the transition between two levels [2]. Resonant pumping of a system has the property that the probability of inducing a transition from level n to level n0 is equal to the probability of inducing a transition in the opposite direction. The long time behaviour of such a system is governed by the simple equations, with n(t) the occupation N number of the neutrons in the nth level,
˙ (t)= (t)+ (t) Nn0 Nn0 Nn
˙ n(t)= n(t)+ n (t) (24) N N N 0 where is the effective rate of transitions per unit time. The solution of this system is 2 t 2 t ( n0 (0)(1 + e )+ n(0)(1 e )) n (t)= N N (25) N 0 2 2 t 2 t ( n(0)(1 + e )+ n0 (0)(1 e )) n(t)= N N (26) N 2
6 The Schrödinger equation resulting from Eqn. (11)isessentiallyfreeinthex and y directions, but in the z direction it is
~2 d2 + mN gz (z,t)=E (z,t). (14) 2m dz2 ✓ N ◆ 24 the earth.the perturbation This yields from an Eqn.( additional12)inducessomeperturbativechangesintheenergiesandwave- factor of 10 . Coupled with a pair of excited levels, it is i not impossiblefunctions of thatH0(x there, ⇡j),however,thesechangesarestaticanddonotgiverisetoanytran- could be a lasing amplification of spontaneously emitted gravitons. Neutronsitions, stars spontaneous are supposed or stimulated to have between an atmosphere levels. The of time carbon, dependent only perturbation3cm thick that and of mass pG is relevant, from Eqn.(13)(replacingh(x ct) 2 h(x ct) the canonically normalized density around 3530 kg/m3, the density of diamond.! c This corresponds to a number density metric perturbation) is 27 29 2 of carbon =3530/(12 1.67 10 i)=1.75pG10h(x carbonct)⇡z atoms per cubic meter, which for N ⇥ ⇥ H⇡(x , ⇡j)= ⇥ . (15) c2 m simplicity we will assume all are in an excited state.N We assume the carbon atoms replace the earth. This yields an additionalThen amplitude factor for spontaneous of 1024 or. Coupled stimulated emission with a of pair a graviton of excited is proportional levels, to the it is 2 the ultra cold neutrons of⇡z the terrestrial considerations, the carbon atoms dropping down matrix element n n which, using Eqn.(14), gives not impossible that there could be a lasingh 0 | m amplificationN | i of spontaneously emitted gravitons. from excited states would spontaneously2 emit gravitons. The amplification factor per unit ⇡z n0 n = n0 (2mN gz 2En) n emm. Neutron stars are supposedlength then to is have carbon an.Tocalculate atmosphereh | mN | ofi (for carbon,h | details only see Baym3cm| i thick [5])we mustand of divide mass n n by N 0! = n0 2mN gz n . (16) density around 3530thekg/m flux3, the density of diamond. Thish | corresponds| i to a number density emm. The matrix element is easily computed24 using = the exact/ eigenfunctions and the integral the earth. This yields27 an additional factor29 of 10 . Coupledn0 withn a pair of excited levels, it is (30) of carbon =3530/(12 1.67 10 )=1.75 10 carbon atoms! per cubic meter, which for N ⇥not impossible⇥ that there could1 be a⇥ lasing amplification of spontaneously2Ai0( ↵m emitted)Ai0( gravitons.↵n) dyyAi(y ↵m)Ai(y ↵n)= (17) where is the graviton flux (gravitons per unit area(↵ per↵ unit)2 time). The expression for simplicity we will assumeNeutron all stars are are in supposed anZ0 excited to have an state. atmosphere We of carbon, assume only the3cmm thick carbonn and of atoms mass replace emm. should be integrated3 over gravitons with24 a wave vector in a given direction with a ndensity0 n which around givesthe3530 earth.kg/m This, the yields density an additional of diamond. factor This of corresponds10 . Coupled to a with number a24 pair density of excited levels, it is the ultra cold neutrons! of the terrestrial considerations,the earth. This yields an the additional carbon factor atoms of 10 . Coupled dropping with a pairdown of excited levels, it is 27 29 1 4mN gz0 2 3 of carbon =3530not/(12 impossible1.67d that⌦10 there)=1 could.752m be10gz a lasingcarbon= amplification atoms perd cubic⌦ of. spontaneously meter,! d!/ which(2⇡c emitted for) gravitons. narrowN angular spread⇥ ⇥ not. Thus impossiblen0 we⇥N replace that theren V could~k be a lasing2 amplification of spontaneously, which(18) then emitted gives gravitons. h | | i (↵n0 !↵n) from excited states wouldsimplicity spontaneously we willNeutron assume stars all are are emitNeutron supposedin an gravitons. excited stars to have are state. supposed an We atmosphere The assume to amplification have the of an carbon,carbon atmosphere atoms only 3 replacecm factor of carbon,thick and per only of massunit3cm thick and of mass The rate, per unit2 time and volume,2 of spontaneousP or stimulatedR emission of gravitons is emm. ! d! 2⇡~c N3k 2⇡ G 4 2 the ultra= coldd⌦density neutrons around of the3530 terrestrialdensitykg/m around considerations,, the density3530 kg/msin of the diamond.3✓, the carbon density2 This atomsm of correspondsgz diamond. dropping downto( ThisE a number correspondsEemm density. to!) a. number density length then is carbon.Tocalculaten0 n 3(for details see4 Baym [5n])we0 N mustn dividen0 n ~by (31) !then is given by,(2⇡ modifyingc) a! calculation27 inc Baym,| [h5]29 | | i| n0 n N from excited statesof carbon would=3530 spontaneously/(12 1.67 emit10 gravitons. ~ )=1.75 The10 amplificationcarbon27 atoms factor per29 per cubic unit meter, which! for ZN ✓of carbon⇥ =3530⇥◆ /(12 1.67⇥ 10 )=1.75 10 carbon atoms per cubic meter, which for N 2 2 ⇥ ⇥ ⇥ emm. the flux length then is simplicity carbon.Tocalculate we willemm assume. (for allG 4 are details⇡ c in an see excited Baym state. [5])we We must assume divide the n carbonn by atoms replace N d simplicity= we will( assumeN +1) all are in an excited state.0! We assume the carbon atoms replace We identify the energy fluxn0 pern unit4 area perk unit frequency as the flux the ultra cold neutrons! ofc theckV terrestrial considerations,⇥ the carbon atoms dropping down the ultraemm. cold neutrons of the2 terrestrial considerations, the carbon atoms dropping down = /n emm2m. N gz 4n (En E2n ~ck) (19)(30) n0 n= 0 / ! 2⇡0 !c Nk (30) from excited states would⇥!| spontaneouslyh n|0 n | emiti| gravitons.24~ The amplification factor per unit the earth. Thisfrom yieldsI excited(! an)= additional states!d⌦ would factor spontaneously of 10 . Coupled emit with gravitons. a pair of excited The amplification levels, it is factor(32) per unit (2⇡c)4 !2 emm. length then is carbon.Tocalculate (for details see Baym [5])we must divide n n by emm. wherewhere is theNk gravitonisnot the impossible number flux (gravitonsNlength that of photons there then per could is involvedunit becarbon area a lasing in.Tocalculate per the amplification unit✓ stimulated time). The of(for emission, spontaneously◆ expression details seeand for Baym emitted when [5N gravitons.])wek0!=0 must divide n n by where is the graviton flux (gravitons per unit areaN per unit time). The expression for 0! emm. should bethe integrated flux over gravitons with a wave vector in a given direction with a andn0 thenwe correspondingget the rateNeutron for spontaneous stars gravitonRatethe are flux supposed emission. numberof tograviton have (The flux an whole atmosphere (number system emission of of carbon,is imaginedgravitons only in3cm aper boxthick unit of and volume area of mass per unit emm. ! emm. 1 = 2 / 3 emm. (30) should be integrated over gravitons3 3 with a3 wave vector3n0 n in a given direction with a n0 n narrowV angular.) There spread aredensityVd•d⌦Then around.k/ Thus(2 the⇡)3530 we numberstates replacekg/m in ofV, the the states~k volume density ind⌦ ofd ! k diamond. !d of!of/ phase(2phase⇡c) This, space space which= corresponds n and thenisn / as givesE tok = a number~c k density (30) ! time) ! 0! | | 27 29 where!of2d !carbonis2 the⇡=3530c graviton2N /(1212⇡ fluxG1.67 (gravitonsP310 )=1 perR.75 unit2 10 areacarbon per unit atoms3 time). per cubic The meter, expression which for for d⌦emm. N ~ wherek ⇥is the⇥d4 k gravitond⌦1 flux!dE⇥d (gravitonsk!d⌦2/(2⇡c per) unit area per unit time). The expression for narrow angular spread .= Thusd⌦ we replace V ~ksin ✓ n 2mN2gz n (En En, which!). (31) then gives n0 n emm. 3 4 V = =0 VEk d!I(!). 0 ~ (20) (33) ! (2simplicity⇡cshould) we be! will integratedemm assume. ~ c over all! are gravitons|h3 in| an excited with| ai state.| wave3 We vector assume in a the given carbon direction atoms with replace a Z n0 n ✓ ◆ (2⇡) ! (2⇡~c) ! n0 n should be integrated~ over gravitons with a wave vector in a given direction with a 2 2 ! Z 1 2 3 We identify thenarrow energythe• ultra angular fluxIntegrating cold per spread unit neutrons areaoverdP⌦. perof Thusthe the unit energy we terrestrial frequency replace Rof the considerations,V as emitted~k d⌦ graviton! thed!1 carbon/(2 gives:⇡c) atoms, which dropping2 then gives down3 ! d! 2⇡ c Nk 2⇡ Gnarrow angular spread d⌦. Thus! we replace ~ d⌦ ! d!/(2⇡c) , which then gives emm. Then the total~ absorption cross4 section is 5 2 V k ! = d⌦ from excited statessin2 would✓ spontaneouslyn2 2mN gz emit n gravitons. (E Then amplificationEn ! factor). per(31) unit n0 n 3 4 ! d! 2⇡!4c N0 2⇡ 2!⇡cG2N P R 0 ~ ! (2⇡c) ! emm. c |h~ |k !2~d! k2⇡|4 c2Ni| 2⇡ G P2 R n n =~d⌦ I(!)=emmd⌦. sin ~✓ nk0 2mN gz n4 (En0 (32)En ~!)2. emm(31). length0! then is carbon3 2⇡.Tocalculate=cd2⌦⇡4 G 2 4 (for details see Baymsin [✓5])we must2m gz divide (E by E !). Z ✓ ◆ (2⇡cn)0 n (2⇡c!) ~! 34c |h | 4 | i| 2 n0 N n n0 nn0 n ~ (31) Z N=!✓ ✓ (2◆sin⇡c) ✓ ◆ !n 2mN~gzc n |h | | i| ! (34) the flux !2 Zc4 ✓|h 0 | ◆ | i| We identify the energyand flux the corresponding perWe unit identify• area gravitonWhich the energyper should number unit flux be flux per integrated frequency~ (number unit area ofover per gravitons unit asall energies frequency per unit in as a area narrow per unit We identify the energy flux peremm unit. area per unit frequency as angular spread. Then: = n n/ (30) time) 4 0! 2 with ✓ = ⇡/2 and ! = En En from the delta! function.2⇡~!c Nk The4 matrix2 element is given in ~ 04 1 I(!)=d2⌦ ! 2⇡~!c Nk (32) where• With is the ! graviton = 2 flux⇡ d (gravitons!Ic(!)N.(2k⇡ perc)4 I unit(!)= area!2d⌦ per unit time).(33) The expression for (32) ! ~ ✓ (2◆⇡c)4 !2 Eqn.(19) whichI(!)= afterd putting⌦ everything~ together gives ✓ ◆ (32) emm. 4 Z 2 and then0 n correspondingshould be integrated graviton over number gravitons flux with (number a wave of gravitons vector in aper given unit direction area per with unit a Then the total absorption! cross(2⇡ sectionc) is ! and the✓ corresponding graviton◆ 1 number flux2 (number3 of gravitons per unit area per unit narrow• We angular get: spread d⌦. Thus we2 replace ~ d⌦ ! d!/(2⇡c) , which then gives time) 4⇡ ~G V 16k ! 2time)⇡c 2⇡ G 4 = 2 = sin ✓ n 2m3 N gz1 n 6 (34) (35) and the corresponding graviton number flux!2 !2 (numberdc!4 2⇡|hc2N0 |c of=2⇡ gravitons(G|↵nid|!PI(↵!)n.) perR1 unit area per unit(33) emm. ~ ~ k ! 0 4 = d!2I(!). (33) n n = d⌦ ~ sin ✓ n0 2mN gz n (En0 En ~!). (31) 0! (2⇡c)3 ! ~ c4Z |h | ~! | i| ✓ = ⇡/2 ! = E EZ ✓ ◆ Z time) with andThen~ the totaln0 absorptionn from the cross delta section function. is The matrix element is given in interestingly with no reference Then to the the total neutron absorption mass cross (this section must is be due to the weak equivalence Eqn.(19) which afterWe putting identify everything the1 energy together flux per unit gives area per unit frequency as 2⇡c 2⇡ G 4 2 principle playing a role).= Thend the! I= di(!ff2)erential. 4 sin ✓ equation4 2n⇡0 2cm2⇡N Ggz2 for n4 the amplification2 of(33)(34) a beam of 2 ! c !|=h |2⇡~!c N|sink i|✓ n 2mN gz n (34) ! 4⇡ ~G I(!16)=~ d⌦ 2 4 0 ~ = 4 ! ~ c2 |h | (35) | i| (32) Z 3 6 (2⇡c) ! c (↵n0 ↵n) ✓ ◆ with ✓ = ⇡/2 and ~! = En0 En from the delta function. The matrix element is given in with ✓ = ⇡/ 2 and ! =8 En En from the delta function. The matrix element is given in Then the total absorption cross sectionand the corresponding is graviton number~ flux0 (number of gravitons per unit area per unit interestingly withEqn.( no19 reference) which to after the putting neutron everything mass (this musttogether be due gives to the weak equivalence time) Eqn.(19) which after putting everything together gives principle playing a role). Then the differential equation4⇡2 forG the amplification16 of a beam of 2⇡c 2⇡ G 4 ~ 1 2 2 = =3 d!I(!)6.4⇡ ~G 16 (35)(33) = sin ✓ n0 2mN gzc ~!n(↵n0 ↵=n) (34) (35) 2 4 8 Z c3 (↵ ↵ )6 ! ~ c |h | | i| n0 n interestinglyThen the with total no absorption reference cross to the section neutron is mass (this must be due to the weak equivalence interestingly with no reference to the neutron mass (this must be due to the weak equivalence principle playing a role). Then the2⇡ dic 2ff⇡erentialG equation for the amplification of a beam of with ✓ = ⇡/2 and ! = En En from the delta function. = sin The4 ✓ matrix2m gz element2 is given in ~ 0 principle playing a2 role).4 Then then0 diffNerentialn equation for the amplification(34) of a beam of ! ~ c |h | | i| 8 Eqn.(19) which after putting everythingwith ✓ = ⇡/ together2 and ~! = E givesn En from the delta function.8 The matrix element is given in 0 Eqn.(19) which after putting everything together gives 4⇡2~G 16 = 4⇡2~G 16 (35) 3 6= 3 6 (35) c (↵n ↵n) c (↵n ↵n) 0 0 interestingly with no reference to the neutron mass (this must be due to the weak equivalence interestingly with no reference to the neutron mass (this must be due to the weak equivalence principle playing a role). Then the differential equation for the amplification of a beam of principle playing a role). Then the differential equation for the8 amplification of a beam of
8 states per unit energy. Then integrating Eqn.(19)overtheenergyoftheemittedgraviton we get G (E E )d⌦ 4m gz 2 states per unit energy. Then integrating Eqn.(d emm19)overtheenergyoftheemittedgraviton. = n0 n (N +1) N 0 n0 n 4 2 k 2 (21) ! c 2⇡~ c (↵n ↵n) ✓ 0 ◆ we get 2 E = m gz ↵ z3 = ~ 2 Since Gn(E N E0)dn⌦where 0 2gm4m2 andgz integrating over directions, we get d emm. = n0 n (N +1) NN 0 n0 n 4 2 k 2 (21) ! c 2⇡~ c (↵n ↵n) 3 emm✓. 20 G ◆ mN gz0 2 = (N +1) 3 ~ n0 n 2 5 k Since En = mN gz0↵n where z0 = 2gm2 and integrating! over~ directions,c we↵ getn ↵n N ✓ 0 ◆ G m g2 2 2G m gz 3 N ~ emm. N =0 2 5 (Nk +1) 3 n n = (Nk +1) ~ c (↵n ↵n) 0! 2c5 ↵ ↵ 0 ~ n0 n ✓ 2 ◆ ✓ 2 G2◆ mN g G mN g=~ (Nk +1) (22) = (Nk +1) 5 3 2 5 c 3 (↵n0 ↵n) ~ c (↵n0 ↵n) ✓ ◆ ✓ 2 ◆ G mN g = (Nk +1) (22) POPULATION5 INVERSION 3 c (↵n0 ↵n) the earth. This yields an✓ additional factor◆ of 1024. Coupled with a pair of excited levels, it is
POPULATION INVERSIONnotWe impossible can obtain that a there population could be inversion a lasing amplification by mechanically of spontaneously vibrating the emitted base at gravitons. the resonant frequencyNeutron stars corresponding are supposed to to the have transition an atmosphere between of carbon, two levels only [23].cm Resonantthick and pumping of mass of a We can obtain a populationdensity inversion around 3530 bykg/m mechanically3, the density vibrating of diamond. the base This at the corresponds resonant to a number density system has the property that the probability of inducing a transition from level n to level n0 27 29 frequency correspondingof tocarbon the transition=3530/(12 between1.67 two10 levels)=1. [752]. Resonant10 carbon pumping atoms per of cubic a meter, which for is equalN to the probability⇥ of⇥ inducing a transition⇥ in the opposite direction. The long time system has the propertysimplicity that the probability we will assume of inducing all are in a an transition excited state. from level We assumen to level then carbon0 atoms replace behaviour of such a system is governed by the simple equations, with n(t) the occupation N is equal to the probabilitythe of ultra inducing cold neutrons a transition of the in terrestrial the opposite considerations, direction. The the long carbon time atoms dropping down number of the neutrons in the nth level, behaviour of such a systemfrom is excited governed states by wouldthe simple spontaneously equations, emit with gravitons.n(t) the The occupation amplification factor per unit N ˙ n (t)= n (t)+ n(t) emm. length thenn is carbon.Tocalculate0 (for details0 see Baym [5])we must divide n n by number of the neutrons in the th level,N N N N 0! the flux ˙ n(t)= n(t)+ n (t) (23) ˙ N N N 0 n0 (t)= n0 (t)+ n(t) emm. N N N = n n/ (30) where is˙ the effective rate of transitions per0! unit time. The solution of this system is n(t)= n(t)+ n0 (t) (23) where Nis the graviton N flux (gravitonsN per unit2 t area per unit time).2 t The expression for ( n0 (0)(1 + e )+ n(0)(1 e )) n (t)= N N (24) where is the effective rate emm of. transitions perN unit0 time. The solution of this2 system is n0 n should be integrated over gravitons with a wave vector in a given direction with a ! 2 t 2 t 2 t 2 t ( (0)(1 + e )+ (0)(1 e )) ( n0 (0)(1 + e )+ n(0)(1n e 1 )) n0 2 3 nnarrow(t)= angularN spread d⌦n.(t Thus)=N weN replace V ~k dN⌦ ! d! /(2(24)⇡c) , which then gives (25) N 0 N 2 Population inversion!2 2 2 t 2 2 t R ( n(0)(1 +! ed ! )+2⇡ cnN(0)(1k 2⇡ Ge ))P 1 Henceemm in. the long• If time we limit,can ~induce the0 populations the neutrons4 goto excited to n ( tstates,)= 2 inn( tthe) long( n(0) + n (0)). n(t )=n n N= d⌦ N sin ✓ n0 2mN0gz n (25)(En0 E2 n ~!). (31)0 N 0! (2time⇡c)3 limit,2 !the occupation~ c4 numbers|h | asymptoteN | iN| to: ! N N If we pumpZ the system✓ between◆ levels n and n+m, with m>1, there will be intermediate (t)= (t) 1 ( (0) + (0)) Hence in the long time limit,We identify the populations the energy go flux to pern unit0 arean per unit2 frequencyn asn0 . levels in between the two withN respectN to which! thereN willN be a population inversion. Then If we pump the system between levels•nThusand n if+ wem, pump with m>between1,4 there levels will one be2 and intermediate three, after a long ! 2⇡~!c Nk one could imaginetime, spontaneous we willI(! have)= emissiond a⌦ population of a graviton inversion by between a transition levels from the upper(32) level levels in between the two with respect to which there will be a population4 inversion.2 Then three and two. (2gravitons⇡c) is! given by to one of the intermediate levels. ✓ ◆ gravitons is given by one could imagine spontaneousand the emission corresponding• ofThe a gravitonneutrons graviton byinnumber level a transition three flux (numbercan from only the ofdecay gravitons upper by emitting level per unit a dN areak(x) per unit = carbonNk(x) (36) The amplification factor of a lasing medium is given by dx N to one of the intermediatetime) levels. graviton. The amplification factor is givendNk(x )by: = carbonNk(x) (36) dx The amplification factor of a lasing medium is given by 1with evident solution N = = ( dn!0I(!).n) (33) (26) ~! N N carbonx with evident solutionZ Nk(x)=Nk(0)e N . (37)
Then the total absorption= ( n0 crossn) section is (26) carbonx • This is the gain per meter. 6It turns outN tok( xbe)= ridiculouslyNk(0)68e N . (37) N N Numerically =5.3 10 thus on the surface of a neutron star we have carbon small. 2⇡c 2⇡ G ⇥ N ⇡ 4 3968 2 6 Numerically = =5sin9..34✓ 10 10n 0 2mperthusN gz meter. onn the Thus surface we will of a not neutron have(34) significant star we have amplification carbon except after distances !2 ~ c4 ⇥|h | | i| N ⇡ 39 39 9.4 10 per meter.of the Thus order we of will10 notmeters, have significant which is much amplification larger than except the size after of distances the observable universe. For ⇥ with ✓ = ⇡/2 and ~! = En0 En from the delta function. The matrix element is given in of the order of 1039 ameters, generic which neutron is much star larger of radius than ten the kilometers size of the with observable a carbon universe. atmosphere For of thickness 3cm, Eqn.(19) which after putting everything together gives a generic neutron starwe on of get radius a chordal ten kilometers distance with of only a carbon about 50 atmosphere meters, see of Fig. thickness (1). This 3cm, kind of amplification 4⇡2~G 16 we on get a chordal= will distance only of work only if about the quantum 50 meters, states see Fig. exist (1 in).(35) regions This kind of of much amplification stronger gravitational fields, c3 (↵ ↵ )6 for examplen0 n near the surface of some black holes will only work if the quantum states exist in regions of much stronger gravitational fields, interestingly with no reference to the neutron mass (this must be due to the weak equivalence for example near the surface of some black holes principle playing a role). Then the differential equation for the amplification of a beam of
8
FIG. 1. The chordal trajectory of the graviton inside the atmosphere of a neutron star.
Furthermore, speculatively, if mN is replaced by a considerably heavier, as yet undiscov- FIG. 1. The chordal trajectory of the graviton inside the atmosphere of a neutron star. ered, supersymmetric or dark matter particle, again there could be substantial increase in
Furthermore, speculatively,the rate of spontaneousif mN is replaced emission. by a considerably Finally, one heavier, can imagine, as yet undiscov-in an astrophysical scenario,
ered, supersymmetric or dark matter particle, again there could9 be substantial increase in the rate of spontaneous emission. Finally, one can imagine, in an astrophysical scenario,
9 the earth. This yields an additional factorgravitons of 1024. Coupled is given with by a pair of excited levels, it is gravitons is given by not impossible that there could be a lasing amplification of spontaneously emitted gravitons. dNk(x) = carbonNk(x) (36) Neutron stars are supposed to have an atmosphere of carbon, only 3cm thick and of mass dNk(x) • This rate can be enhanced by imagining the lasing takes place on the dx N = carbonNk(x) 3 (36) dx N density aroundsurface3530 of akg/m neutron, the star. density Here ofthewith diamond. atmosphere evident This solution is corresponds Carbon, and to the a number density 27 29 of carbondensity=3530 /is(12 of the1. 67order10 of: )=1.75 10 carbon atoms per cubic meter, which for carbonx with evident solution N ⇥ ⇥ ⇥ Nk(x)=Nk(0)e N . (37)
carbonx simplicity we will assume all are in an excited state. We assume the carbon atoms replace Nk(x)=Nk(0)e N . (37) 68 Numerically =5.3 10 thus on the surface of a neutron star we have carbon the ultra• But cold this neutrons still makes of the the terrestrial gain: considerations, the carbon⇥ atoms dropping down N ⇡ 68 39 Numerically =5.3 10 thus on the surface of a neutronfrom excited star states we have would spontaneouslycarbon 9 emit.4 gravitons.10 per The meter. amplification Thus we will factor not per have unit significant amplification except after distances ⇥ N ⇡ ⇥ 39 1039 emm. 9.4 10 per meter. Thus we will not have significant amplificationlength then is exceptcarbon after.Tocalculate distances (forof the details order see of Baymmeters, [5])we must which divide is muchn0 n largerby than the size of the observable universe. For ⇥ N ! of the order of 1039 meters, which is much larger than the sizethe of flux the• Which observable is still universe. absurdly small. For a generic neutron star of radius ten kilometers with a carbon atmosphere of thickness 3cm, emm. = n n/ (30) a generic neutron star of radius ten kilometers with a carbon atmosphere of thickness 3cm, we on0! get a chordal distance of only about 50 meters, see Fig. (1). This kind of amplification • What can we do? The density cannot be appreciably increased nor where is the graviton flux (gravitons perwill unit only area work per if unit the time).quantum The states expression exist for in regions of much stronger gravitational fields, we on get a chordal distance of only about 50 meters, see Fig. (1). Thiscan the kind distance of amplification through which the graviton can pass. emm. n n should be integrated over gravitons with a wave vector in a given direction with a 0! for example near the surface of some black holes will only work if the quantum states exist in regions of much stronger• But this gravitational is not true. fields, 1 2 3 narrow angular spread d⌦. Thus we replace V ~k d⌦ ! d!/(2⇡c) , which then gives for example near the surface of some black holes • Black holes admit unstable, circular, null,! geodesics (orbits). 2 2 P R emm. ! d! 2⇡~c Nk 2⇡ G 4 2 n n = d⌦ sin ✓ n0 2mN gz n (En0 En ~!). (31) 0! (2⇡c)3 ! c4 |h | | i| Z ✓ ◆ ~ We identify the energy flux per unit area per unit frequency as
4 2 ! 2⇡~!c Nk I(!)=d⌦ (32) (2⇡c)4 !2 ✓ ◆ and the corresponding graviton number flux (number of gravitons per unit area per unit time) 1 = d!I(!). (33) ! ~ Z Then the total absorption cross section is
2⇡c 2⇡ G 4 2 = sin ✓ n 2mN gz n (34) !2 ~ c4 |h 0 | | i| with ✓ = ⇡/2 and ~! = En En from the delta function. The matrix element is given in 0 Eqn.(19) which after putting everything together gives
4⇡2~G 16 = (35) c3 (↵ ↵ )6 n0 n interestingly with no reference to the neutron mass (this must be due to the weak equivalence FIG. 1. The chordal trajectory of the graviton inside the atmosphere of a neutron star. principle playing a role). Then the differential equation for the amplification of a beam of
8Furthermore, speculatively, if mN is replaced by a considerably heavier, as yet undiscov- FIG. 1. The chordal trajectory of the graviton inside the atmosphere of a neutron star. ered, supersymmetric or dark matter particle, again there could be substantial increase in the rate of spontaneous emission. Finally, one can imagine, in an astrophysical scenario, Furthermore, speculatively, if mN is replaced by a considerably heavier, as yet undiscov- ered, supersymmetric or dark matter particle, again there could be substantial increase in 9 the rate of spontaneous emission. Finally, one can imagine, in an astrophysical scenario,
9 Unstable photon orbits • A photon can scatter from infinity in almost any direction from a black hole. Only those photons heading directly towards the black hole are absorbed. • From a finite radius, the set of directions from which the photon is absorbed is a cone of finite opening angle, pointing towards the black hole. • As we descend down to the event horizon, the cone now completely flips over and closes in the outward direction. Only a photon pointing radially away from the black hole can escape. • Therefore, in between, there exists a radius at which the cone has opening angle 180 degrees. Here there must exist an unstable circular orbit. The radius of this orbit is 3GM. • Thus a photon can orbit many times before finally spiralling out from the black hole. D. BATIC, N. G. KELKAR, AND M. NOWAKOWSKI PHYSICAL REVIEW D 86, 104060 (2012) to the unstable circular orbit. We also find that the normal- where in principle ! !n, but we shall drop the subscript ized R 2 always passes through a value of 1=2 at the for convenience in what follows. Moreover, V r j lj ð Þ¼ critical value lC. f r U r (with the form of f r depending on the metric The present work differs from others [1] in view of the underð Þ ð consideration)Þ and ð Þ points mentioned above. The reflection coefficient is eval- uated exactly using the variable amplitude method as l l 1 f0 r U r ð þ2 Þ ð Þ : (3) D. BATIC, N. G. KELKAR, AND M. NOWAKOWSKIcompared to approximate PHYSICALcalculations REVIEW (third reference D 86, 104060 in (2012) ð Þ¼ r þ r Ref. [1]) [4]. We also find a potential proportional to 2 Here, a prime denotes differentiation with respect to r to the unstable circular orbit. We also find that the normal-coshÀ wherex in principlewhich gives! remarkably!n, but we good shall drop results the subscript 2 ð À Þ whereas r is a tortoise coordinate defined through ized Rl always passes through a value of 1=2 atwhen the comparedfor convenience with the numerical in what ones. follows. For this potential, Moreover, V r j j ð Þ¼ à critical value lC. the transmissionf r U r coefficients(with the form can of bef foundr depending analytically. on the metric ð Þ ð Þ ð Þ dr 1 The present work differs from others [1] in view of the under consideration)2 and à f r À : Parametrizing the coshÀ potential to fit the Regge- dr ¼ ð Þ points mentioned above. The reflection coefficient is eval-Wheeler potential, we find semi-analytical results for black hole scattering. Even though the scatteringl l 1 offf black0 r holes uated exactly using the variable amplitude method as U r ð þ Þ ð Þ : (3) compared to approximate calculations (third referenceis in a widely explored area, notð muchÞ¼ attentionr2 D.þ BATIC, hasr been N. G. KELKAR, paid AND M.A. NOWAKOWSKI Scattering from a Schwarzschild PHYSICAL black hole REVIEW D 86, 104060 (2012) to the orbiting phenomenon (mostly the gloryto the and unstable rainbow circular orbit. We also find that the normal- where in principle ! !n, but we shall drop the subscript Ref. [1]) [4]. We also find a potential proportional to 2 Here, a prime denotes differentiationized Rl always with passes respect throughIn to case ar value of of the1=2 Schwarzschildat the for convenience metric in which what we follows. shall Moreover, V r cosh 2 x which gives remarkably good resultseffects have been discussed). Here we supplementj j the ð Þ¼ À critical value lC. consider in the present work,f r Uf rr (with1 the form2M=r of fwherer dependingM on the metric ð À Þ existingwhereas literaturer withis a a tortoise detailed coordinate study of how defined the exis- through ð Þ ð Þ ð Þ when compared with the numerical ones. For this potential, à The present work differsis from the others mass [1 of] in the view black of the holeunder andð consideration) theÞ¼ tortoiseÀ and coordinate is tence of a classical orbit gets reflectedpoints in the mentioned quantum above. The reflection coefficient is eval- the transmission coefficients can be found analytically. dr given by l l 1 f0 r mechanical expressions of the scatteringuated of1 exactly a massless using the variable amplitude method as U r : (3) 2 à f r À : ð þ2 Þ ð Þ Parametrizing the coshÀ potential to fit the Regge- dr ¼ ðcomparedÞ to approximate calculations (third reference in ð Þ¼ r þ r particle from a black hole. This leadsRef. to a[1]) conjecture [4]. We also find a potential proportional to r Wheeler potential, we find semi-analytical results for black 2 r r 2M ln Here, a1 prime;r> denotes2 differentiationM: with respect to r regarding quantum corrections to the classicalcoshÀ x effective which gives remarkablyà ¼ goodþ results 2M À hole scattering. Even though the scattering off black holes ð À Þ whereas r is a tortoise coordinate defined through potential for massless particles. when compared with the numerical ones. For this potential, à is a widely explored area, not much attention has been paid the transmission coefficientsEquation can be found (2) can analytically. also be obtained for other spins. The In the nextA. section, Scattering we provide from briefly a Schwarzschild the formalism black for hole dr 1 2 à f r À : to the orbiting phenomenon (mostly the glory and rainbow Parametrizing the coshÀ correspondingpotential to fit the Regge-Wheeler Regge- potential fordr spins¼ ð sÞ 0, black hole scattering in general and go onWheeler to discuss potential, the we find semi-analytical results for black ¼ effects have been discussed). Here we supplement the In case of the Schwarzschild metric which we1, shall 2 is given as [6] D. BATIC, N. G. KELKAR, AND M. NOWAKOWSKIcritical parametersconsider PHYSICAL in relevant the present REVIEW to orbiting D work,86, and104060fhole gloryr (2012) scattering. scattering.1 2 EvenM=r thoughwhere the scatteringM off black holes existing literature with a detailed study of how the exis-In Sec. III, we discuss the calculationis ofð aÞ¼ widely the reflection exploredÀ area, not much attention has been paid A. Scattering from a2 Schwarzschild black hole to the unstable circular orbit. We also find that the normal- where inis principle the mass! of!n, the but weblack shall hole drop the andto subscript the the orbiting tortoise phenomenon coordinate (mostly is the glory and rainbow2M l l 1 2M 1 s tence of a classical2 orbit getsD. reflected BATIC, N. in G. theKELKAR, quantumcoefficient AND M. NOWAKOWSKI and present results regarding its connection PHYSICAL to V REVIEWr r D 86,1 104060 (2012)ðInþ caseÞ of theð SchwarzschildÀ Þ ; metric(4) which we shall ized Rl always passes through a value of 1=2 at the for convenience in what follows. Moreover,effectsV r have been discussed). Here we supplement the 2 3 j j given by ð Þ¼ ð ð ÃÞÞ ¼ À r r þ r mechanicalcritical value expressionsl . of the scattering of a masslessorbitingf r U r (with scattering. the form We of f alsor depending present a on conjecture theexisting metric literature related with a detailed study of how the exis- consider in the present work,f r 1 2M=r where M C to the unstable circular orbit.ð Þ Weð Þ also find that the normal-ð Þ where in principle ! !n, but we shall drop the subscriptis the mass of the black hole andð theÞ¼ tortoiseÀ coordinate is particleThe from present a black work differs hole. from This others leads [12] in to view a of conjecture the tounder these consideration) phenomena. and In Sec. IV we discussr tence how of the a classical Regge- orbit getswhere reflectedl s in. Note the quantum that as we move from r r0 2M at ized Rl always passes through a value of 1=2 at the for convenience in what follows. Moreover, Vgivenr by ¼ ¼ regardingpoints quantum mentioned above. corrections The reflection toj the coefficientj classical is eval- effectiveWheeler potentialr canr be2 reinterpretedM ln mechanical as1 ;r> an effective expressions2M: ofthe the event scattering horizon of a massless to r ,ð theÞ¼ tortoise coordinate varies critical value lC. à ¼l l þ1 f0 r 2Mf rÀUparticler (with from a the black form hole. of Thisf r leadsdepending to a conjecture on¼ the1 metric r uated exactly using the variable amplitude method as potential plusU quantumr ð þ correctionsÞ ð Þ: ð proportionalÞ ðÞ (3) ℏ. In fromð Þ to . Since the scatteringr r problem2M ln with1 ;r> the 2M: potentialcompared for massless to approximate particles. calculationsThe present (third reference work differs in from others [1ð]Þ¼ in viewr2 ofþ ther underregarding consideration) quantum corrections and toÀ the1 classical1 effective à ¼ þ 2M À Sec. V weEquation summarize (2) our can findings. also be obtainedpotential for for massless other spins. particles.radial The coordinate r in the three-dimensional (3D) case In theRef. next [1]) section, [4]. We also we providefind apoints potential briefly mentioned proportional the formalism above. to The for reflection coefficient is eval- Equation (2) can also be obtained for other spins. The 2 Here, a primecorresponding denotes differentiation Regge-Wheeler with respect potentialIn the to nextr section, for spins we providegetss mappedbriefly0, the formalism into a one-dimensional for (1D) one with the r blackcosh holeÀ scattering x which in general givesuated remarkably and exactly go on good to using results discuss the the variable amplitude method as l l 1 f0 r corresponding Regge-Wheeler potential for spins s 0, ð À Þ whereas r is a tortoise coordinate defined throughblack hole scatteringU r in generalcoordinate,ð ¼þ andÞ go on theð toÞ discuss Schro: ¨dinger-like the (3) equation (2) in black holeà ¼ criticalwhen parameters compared with relevant the numerical to orbitingcompared ones. Forand to this glory approximate potential, scattering. calculations1,à 2 is (third given reference as [6] in critical parameters relevantð Þ¼ to orbitingr2 andþ gloryr scattering. 1, 2 is given as [6] the transmission coefficients can be found analytically. II. BLACK HOLE SCATTERING scattering can be solved using standard techniques for 1D dr 1 In Sec. III, we discuss the calculation of the reflection 2M l l 1 2M 1 s2 In Sec. III, we discuss the2 calculationRef. [1]) [4 of]. Wethe reflection also find a potential proportionalà f r À to: 2 Parametrizing the coshÀ potential to fit the Regge- 2M l l coefficient1 2M and1 presents results regarding its connection to V r r 1 ð þ2 Þ ð 3À Þ ; (4) 2 We start by presentingdr ¼ someð Þ generalitiesHere, a in prime black denotes hole differentiationtunneling in quantum with respect mechanics. toð rð TheÃÞÞ ¼ asymptoticÀ r solutionsr þ r coefficientWheeler and potential, present we find results semi-analyticalcosh regardingÀ x results its connection forwhich black gives to remarkablyV r r good1 results ð þ2orbitingÞ scattering.ð 3À WeÞ also; present(4) a conjecture related ð À Þ scattering. Considerð ð ÃÞÞ the ¼ propagationÀ r whereas ofr a masslessþr is a scalar tortoiser coordinateof the Schro defined¨dinger through equation (2) are to these phenomena. In Sec. IV we discuss how the Regge- where l s. Note that as we move from r r0 2M at orbitinghole scattering. scattering. Even We though also the presentwhen scattering compared a conjectureoff black with holes the related numerical ones. For this potential, à ¼ ¼ is a widely explored area, not much attention has been paid field t; r; #; ’ governed by theWheeler wave potential equation can be reinterpreted as an effectivei!r the eventi!r horizon to r , the tortoise coordinate varies the transmission coefficientsA. can Scatteringwhere be foundl froms analytically.. a Note Schwarzschild that as we black move hole from r r 2Mc atr A ! eþ à B ! eÀ à ;r¼ 1 ; to these phenomena. In Sec. IV we discuss how the Regge- ¼ ð Þ potential plus quantum0dr corrections proportional1 ℏ. In from to . Since the scattering problem with the to the orbiting phenomenon (mostly the glory and rainbow g2 0where g denotes a static, spherically¼ ¼Ã fð rÃÞ¼À : ð Þ þ ð ÞÀ1 1 à ! À1 Wheeler potential can be reinterpretedParametrizing as the ancosh effectiveÀ Inpotentialr caserthe of event the¼ to Schwarzschild fit horizon the Regge- to metricr which, theSec. we tortoiseV shallwe summarize coordinate our findings. varies i!r radial coordinatei!r r in the three-dimensional (3D) case effects have been discussed). Here we supplement the symmetric black hole metric whose line element is dr ¼c ðr Þ C ! eþ à D ! eÀ à ;r : Wheeler potential, we findconsider semi-analytical in the present results work, forf blackr ¼1 12M=r where M gets mapped into a one-dimensional (1D) one with the r potentialexisting plus literature quantum with a corrections detailed study of proportional how the exis- ℏ. In from to . Since the scattering problem with theð ÃÞ¼ ð Þ þ ð Þ Ã ! þ1 à hole scattering. Even thoughis the the mass scattering of theÀ black1 off hole black1 andð holes theÞ¼ tortoiseÀ coordinate is coordinate, the Schro¨dinger-like equation (2) in black hole Sec. Vtencewe of summarize a classical orbit our findings. gets reflected in the quantum radial coordinate2 r in the three-dimensionalII. BLACK (3D) HOLEFor case SCATTERING waves incident on thescattering black can hole be solved from using the standard right techniques for 1D given by 2 2 dr 2 2 2 2 mechanical expressions of theis scattering aD. widely BATIC, of explored N. a G. massless KELKAR, area, not AND much M.dsgets NOWAKOWSKI attentionf mappedr dt has into been a paid one-dimensionalr d# sinA.We Scattering#d’ start (1D) PHYSICALby: presenting one from with REVIEW some(i.e., a the Schwarzschild generalitiesr D 86,), we104060 in black have black (2012) holeB ! holetunneling0, the in quantum reflection mechanics. amplitude The asymptotic solutions particle from a black hole. This leads to a conjecture ¼ ð Þ À rf r À ð þ Þ þà 1 ð Þ¼of the Schro¨dinger equation (2) are to the orbiting phenomenon (mostlyrcoordinate, ther glory2M ln and the Schrorainbowð Þ 1 ¨;r>dinger-like2M:scattering. equation Consider (2) in the black propagationR ! hole ofD a! massless=C ! scalarand the transmission amplitude regarding quantum corrections toto the the classical unstable effective circular orbit. We also find that the normal- whereIn infield principle case of ! thet; r; #;! Schwarzschild ’n, butgoverned weð shallÞ¼ by the drop metricð waveÞ the subscriptðequation whichÞ we shall II. BLACK HOLEeffects SCATTERING have been discussed). Hereà ¼ weþ supplement2M À the c r A ! e i!r B ! e i!r ;r ; 2 scattering can be solved using standard ¼ techniquesð Þ T for! 1D A ! =C ! , so that þ à À à potential for massless particles. ized Rl always passesUsing through the• Quantum a following value of mechanically1 ansatz:=2 at the suchfor considerstates convenienceg for inthe the photon in0 present whatwhere alsog follows. work, exist.ðdenotesÞ¼ f Moreover,r a static,ð Þ1 sphericallyð2VM=rÞ r where ðMÃÞ¼ ð Þ þ ð Þ Ã ! À1 existingj literaturej with a detailedEquation (study2) can of also how be obtained the exis- for other spins.r Ther ¼ ð Þ¼ À ð Þ¼ c r C ! e i!r D ! e i!r ;r : We startIn the by next presenting section, we provide some brieflycritical generalities the value formalisml inC. black for hole tunneling in quantum mechanics.f risU thersymmetric(with mass The asymptotic the of black the form hole black of metricf solutions holer whosedepending and line the element tortoise on the is metric coordinatei!r is þ à À à tence of a classical orbitcorresponding gets reflected• The effective Regge-Wheeler in the potential quantum potential for theð forÞ radial spinsð Þ smotion0, in the ð Þ c r T ! e à ;rð ÃÞ¼ ð Þ ; þ ð Þ Ã ! þ1 black hole scattering in general andThe go on present to discuss work the differs from others [1] ini!t view1 of the under consideration) and ð ÃÞ¼ ð Þ Ã ! À1 scattering. Consider the propagation of a massless scalar t; r; #;of ’ the Schroe ¨dingerc r equationY #;’given (2;) are by¼Re ! > 0; dr2 For waves incident on the black hole from the right critical parameters relevant to orbitingmechanical and glory expressions scattering. of1, 2 the is given scatteringSchwarzschild as [6] of a metric masslessn‘! is:‘m 2 2 2 2 2 i!r2 i!r points mentioned above. Theð reflectionÞ¼ coefficientr is eval-ð Þ ð Þ ds ð fÞ r dt r d#c rsin #d’e : à R !(i.e.,eÀ Ã),;r we have B ! 0:, the reflection amplitude field t; r; #; ’ governed by the wave equation i!r 2 i!r ¼ ð Þ À f r À ð þð ÃÞ¼ Þ þ ð Þ þ1 à !ð þÞ¼1 In Sec. III, we discuss the calculationparticleuated fromexactly of the a reflection using black the hole. variable This amplitude leadsc r to2M a methodA conjecturel!l e1 as 2M 1B !s e ;rl l ð Þ1 r; f0 r R ! D ! =C ! and the transmission amplitude ¼ ð Þ þ à À à U r (1)ð þ Þ ð Þ : (3) ð Þ¼ ð Þ ð Þ g coefficient and0 where present resultsg denotes regarding a its static, connection spherically to V r r ð1ÃÞ¼ ð ð Þþ2 Þ þð 3Àð ÞÞ ; r(4) r 2à M!ln2À1 1 ;r>2M: T ! A ! =C ! , so that regardingcompared quantum to approximate corrections calculationsð ð toÃÞÞ the ¼ (third classicalÀ referencer effectiver in þ r Using theà ¼ followingð Þ¼þ ansatz:r 2Mþ Àr rorbitingr scattering.¼ We also present a conjecture related i!r i!r ð Þ¼ ð Þ ð Þ symmetric black hole metric whosepotentialRef. line [1 for]) element [4 massless]. We isalso particles. find a potentialc r proportionalC ! eþ to à D ! eÀ à ;r : i!r itwhere is standardl s. Note [ that5] to as we reduce move from ther abover equation2M at to a B. Critical parameters forc blackr T hole! e orbitingà ;r ; to these phenomena. In Sec. IV we discuss2 how the Regge- ð ÃÞ¼ ð Þ þHere,Equationð a0Þ prime (2 denotes) cani!t alsoà differentiation1 ! beþ1 obtained with for respect other to spins.r Theð ÃÞ¼ ð Þ Ã ! À1 Incosh theÀ next x section, which we provide givesSchrothe event¨dinger-like remarkably• briefly horizonWith a the toSchrodinger equation goodr formalism, results the for tortoiselike for the equation radial coordinate¼ part ¼t; varies r; #; ’ e c n‘! r Y‘m #;’ ; Re ! > 0; i!r i!r Wheeler potential can be2 reinterpretedð as anÀ effectiveÞ whereas rð is a tortoiseÞ¼ coordinater ð Þ An definedð anomalousÞ throughð largeÞ angle scattering,c r e calledà R ALAS,! eÀ à ;r was : dr when compared with the numericalFor ones. waves For this¼ incident potential,1 on thecorresponding black hole Regge-Wheeler from the right potential for spins s 0,ð ÃÞ¼ þ ð Þ Ã ! þ1 potential2 plus quantum2 correctionsblack2 hole proportional2 scattering2 ℏ.2 In in generalfrom andto go. on Since to discuss the scattering the problem withà the (1) ¼ ds f r dt r thed# transmissionsin #d’ coefficients: canÀ1(i.e., be found1 2), analytically. we have B ! 1, 20 is, the given reflection as [6] amplitudeobserved often in nuclear reactions between -like nuclei Sec. V¼we summarizeð Þ À f ourr findings.Àcriticalð parametersþ relevantÞ radial to orbiting coordinate andr gloryd in the scattering. three-dimensional (3D) case dr 1 12 16 16 28 Parametrizing the cosh 2 potential toþ fit1 theV Regge-r c ð Þ¼!2 cit is; standard [5](2)à to reducefsuchr À the: as aboveC- equationO, O to- aSi, etc.,B. [7 Critical] and has parameters been attributed for black hole orbiting ð Þ In Sec. III, we discuss theÀgets calculation mappedR ! into ofaD one-dimensional2 the! =C reflection! n‘! (1D)and one the withn‘! transmission the r dr ¼ amplitudeð Þ 2 ð Þ¼À dr ðþÞ ð ðÞ Þ ¼ Schro¨dinger-likeà equation2M fortol the thel radial orbiting1 part2M mechanism1 s in scattering. The origin of this Wheeler potential, we findcoordinate, semi-analyticalT the! Schro resultsA¨dinger-likeà ! for=C black! equation, so that (2) in black hole An anomalous large angle scattering, called ALAS, was Using the followingII. BLACK ansatz: HOLEcoefficient SCATTERING and present results regarding its connection to V r r 1 ð þ2 Þ ð 3À Þ ; (4) hole scattering. Even thoughscattering the scattering canð Þ¼ be solved offð black usingÞ holesð standardÞ techniquesð forð à 1DÞÞ ¼ 2À r r þ r observed often in nuclear reactions between -like nuclei d 12 16 16 28 orbiting scattering. We alsotunneling present in quantum a conjecture mechanics. related Thei!r asymptotic solutions V r c !2 c ; (2) such as C- O, O- Si, etc., [7] and has been attributed We start by presenting some generalitiesis a widely in explored black hole area, not much attentionc hasr beenT paid! e à ;rA. Scattering fromÀ; dr2 þ a Schwarzschildð Þ n‘! ¼ blackn‘! hole 1 to these phenomena. In Sec. IV we discuss howð ÃÞ¼ the Regge-ð Þ whereÃl! Às.1 Note that as we move from r r0 to2M the orbitingat mechanism in scattering. The origin of this scattering. Consideri!t the propagationto the of orbiting a massless phenomenon scalar of (mostly the Schro the¨dinger glory equation and rainbow (2) are à ¼ ¼ t; r; #; ’ e c n‘! r Y‘m #;’ ; Re ! > 0; i!r Inthe case eventi!r of horizon the Schwarzschild to104060-2r , metric the tortoise which coordinate we shall varies ð field Þ¼ t; r; #;r ’ governedð ÞWheelereffects byð the havepotentialÞ wave been equation canð discussed).Þ be reinterpreted Here wec supplement asi!rr an effectivee theÃi!r R ! eÀ à ;r : ¼ ð Þ c r A ! eþð ÃÃÞ¼B ! eÀ þà ;rconsiderð Þ in the; presentà ! work,þ1¼f1r 1 2M=r where M g 0 where g denotespotentialexisting a static, plus literature spherically quantum with a corrections detailed(1) ð ÃÞ¼ studyð proportional ofÞ howþ theð exis-Þℏ. In fromà ! À1 to . Since the scattering problem with the r r ¼ i!r i!r is the massÀ of1 the black1 hole andð theÞ¼ tortoiseÀ coordinate104060-2 is symmetric black hole metric whoseSec.tence lineV we of element summarize a classical is orbit our findings. getsc r reflectedC ! ineþ theà quantumD ! eÀ à ;rradial coordinate: r in the three-dimensional (3D) case ð ÃÞ¼ ð Þ þ ð Þ givenà by! þ1 it is standard [5] to reduce2 themechanical above expressions equation to ofFor a the waves scattering incidentB. Critical of on a massless the parameters black hole fromgets for themapped black right hole into a orbiting one-dimensional (1D) one with the r dr à ds2 f r dt2 r2 d#particle2 sin from2#d’2 a: black hole.(i.e., This), leads we have to aB conjecture! 0, the reflectioncoordinate, amplitude the Schro¨dinger-liker equation (2) in black hole Schro¨dinger-like¼ ð equationÞ À f r forÀ theð radialþ II. part BLACKÞ HOLEþ SCATTERING1 An anomalousð Þ¼ large angle scattering,r r called2M ln ALAS,1 was;r>2M: ð Þ regarding quantum correctionsR ! toD the! =C classical! and effective the transmissionscattering amplitudeà ¼ canþ be solved2M À using standard techniques for 1D ð Þ¼ observedð Þ ð Þ often in nuclear reactions between -like nuclei Using the following2 ansatz: Wepotential start for by massless presenting particles. someT ! generalitiesA ! =C ! , so in that black hole tunneling in quantum mechanics. The asymptotic solutions d 2 ð Þ¼ ð Þ ð 12Þ 16 16 28 Equation (2) can also be obtained for other spins. The V r c In the! nextc section,; we provide(2) brieflysuch as the formalismCi!r- O, forO- Si, etc.,of the [7 Schro] and¨dinger has been equation attributed (2) are 2 scattering.n‘! Considern‘! the propagationc r ofT a! masslesse à ;r scalar corresponding; Regge-Wheeler potential for spins s 0, À dr i!tþ1 ð Þ black¼ hole scattering in general andðtoÃÞ¼ thego on orbitingð toÞ discuss mechanism theà ! À1 in scattering. The origin of this ¼ t; r; #; ’ eà c n‘!r Yfield‘m #;’ ; Ret; r;! #;> ’0; governed by thei!r wave equationi!r 1, 2 is given as [6] i!r i!r ð Þ¼ r ð Þ criticalð Þ parametersð Þ relevant to orbitingc r ande gloryà R scattering.! eÀ à ;rc r: A ! e B ! e ;r ; ¼ ð Þ ð ÃÞ¼ þ ð Þ Ã ! þ1 þ à À à g In Sec. III, we0 where discuss(1) g the calculationdenotes a of static, the reflection spherically ð ÃÞ¼ ð Þ þ ð Þ 2 à ! À1 r r ¼ 2M i!rl l 1 2M i!r1 s symmetriccoefficient black and present hole metric results whose regarding line its element connection is to V rcr r 1C ! eþ ðà þ2 DÞ ! eÀð 3Àà ;rÞ ; (4) : it is standard [5] to reduce the above equation to a B. Critical parameters for black hole orbitingð ð ðÃÞÞÃÞ¼ ¼ Àð Þr rþ ðþ Þ r à ! þ1 orbiting scattering. We also104060-2 present a conjecture related Schro¨dinger-like equation for the radial part 2 An anomalous large angle scattering,where calledFor ALAS,l wavess. was Note incident that as we on move the from blackr holer from2M at the right to these2 phenomena.2 Indr Sec. IV we2 discuss2 how2 the Regge-2 0 ds f r dt observedr d# often insin nuclear#d’ reactions: betweenthe(i.e., event -like horizon nuclei), we to r have B, the! tortoise0, the coordinate¼ reflection¼ varies amplitude d2 Wheeler¼ potentialð Þ canÀ f ber À reinterpretedð þ as an effectiveÞ 2 such as 12C-16O, 16O-28Si, etc., [7] and has beenR ! attributedþ1D ! =C ¼! 1 andð Þ¼ the transmission amplitude 2 V r c n‘!potential! c n‘! plus; quantum(2) ð correctionsÞ proportional ℏ. In from to . Since the scattering problem with the À dr þ ð Þ ¼ to the orbiting mechanism in scattering.radial The originðÀ coordinate1Þ¼ of this1ð Þr inð theÞ three-dimensional (3D) case à UsingSec. theV we following summarize ansatz: our findings. T ! A ! =C ! , so that gets mappedð Þ¼ intoð Þ a one-dimensionalð Þ (1D) one with the r i!r à coordinate,c ther Schro¨Tdinger-like! e à ;r equation (2) in black; hole II. BLACKi!t 1 HOLE SCATTERING ð ÃÞ¼ ð Þ Ã ! À1 t; r; #; ’ e 104060-2c n‘! r Y‘m #;’ ; Re ! > 0; scattering can be solvedi!r using standardi!r techniques for 1D ð Þ¼ r ð Þ ð Þ ð Þ c r e à R ! eÀ à ;r : We start by presenting some generalities in black hole tunneling in quantumð ÃÞ¼ mechanics.þ ð TheÞ asymptotic solutionsà ! þ1 scattering. Consider the propagation of a massless scalar(1) of the Schro¨dinger equation (2) are field t; r; #; ’ governed by the wave equation i!r i!r it is standard¼ ð [5] toÞ reduce the above equation to a c r B. CriticalA ! eþ parametersà B ! eÀ forà ;r black hole orbiting; g 0 where g denotes a static, spherically ð ÃÞ¼ ð Þ þ ð Þ Ã ! À1 Schro¨dinger-liker r ¼ equation for the radial part i!r i!r symmetric black hole metric whose line element is c rAn anomalousC ! eþ largeà D angle! eÀ scattering,à ;r called: ALAS, was ð ÃÞ¼ ð Þ þ ð Þ Ã ! þ1 2 observed often in nuclear reactions between -like nuclei d 2 For waves12 incident16 on16 the28 black hole from the right 2 2 dr 2 2 2 2 2 such as C- O, O- Si, etc., [7] and has been attributed ds f r dt2 V r c n‘!r d# ! sinc n‘!#d’; : (2) (i.e., ), we have B ! 0, the reflection amplitude ¼ÀðdrÞ þÀ fð rÞ À ð ¼ þ Þ toþ the1 orbiting mechanismð Þ¼ in scattering. The origin of this à ð Þ R ! D ! =C ! and the transmission amplitude Tð!Þ¼A ð! Þ=C ð! Þ, so that Using the following ansatz: ð Þ¼ ð Þ ð Þ i!r c r T ! e à ;r ; i!t 1 ð ÃÞ¼ ð Þ Ã ! À1 t; r; #; ’ e c n‘! r Y‘m #;’ ; Re ! > 0; 104060-2 i!r i!r ð Þ¼ r ð Þ ð Þ ð Þ c r e à R ! eÀ à ;r : ð ÃÞ¼ þ ð Þ Ã ! þ1 (1) it is standard [5] to reduce the above equation to a B. Critical parameters for black hole orbiting Schro¨dinger-like equation for the radial part An anomalous large angle scattering, called ALAS, was observed often in nuclear reactions between -like nuclei d2 2 such as 12C-16O, 16O-28Si, etc., [7] and has been attributed V r c n‘! ! c n‘!; (2) À dr2 þ ð Þ ¼ to the orbiting mechanism in scattering. The origin of this Ã
104060-2 ORBITING PHENOMENA IN BLACK HOLE SCATTERING PHYSICAL REVIEW D 86, 104060 (2012) mechanism can be traced back to classical dynamics, Note that part of V [i.e., the first term of V r in (4)] is where a particle approaching the potential center can get proportional to V and in the case of s 1, theð Þ entire V r eff ¼ ð Þ trapped in a circular orbit of radius r0 if its energy equals is proportional to Veff. We shall come back to this point the maximum of the effective potential at r0. Ford and later. Under such a condition, the deflection function was Wheeler [2] found the connection of this phenomenon with shown to vary logarithmically: the classical deflection function which becomes singular at l l a critical value of the angular momentum for which the  l b ln À C ;l>l; ð Þ¼ 1 þ l C circular orbit can exist and leads to divergent cross sec- C (9) tions. The analogous effect in quantum mechanical scat- lC l tering corresponds to the appearance of a diffraction  l 2 2b ln À ; 0 l
in ther form r_ =2 V const. For masslessr /r particles, quantum mechanically. In Table I we list the two sets of lC smoothly through 0 or , it leads to the phenomenon4 l=5 þ eff ¼ * 0 modes of black holes, either as an approximation [10] or Æ the same condition with V2.eff Radiusfrom the of geodesic the unstable equation orbits of and criticalinfor obtaining differentl exact values results of !r in0 the. As Nariai expected, spacetime the [difference11] for named glory. Though classically it correspondsmotion to3 a is singu- s=1 In black hole scattering Regge−Wheeler with s 1, Eq.which (4) is propor-the scalar field equation reduces to the radial equa- larity in the cross section, quantum mechanically2 one −2 tional to the classicalParametrizedV from cosh general¼ relativity.tionTABLE with Here I. the one Critical Po¨schl-Teller values of potential.l obtained using V r 3M . The expects only a prominent peak in the cross section.dV Fordeff ‘C eff 2 WKBð ¼ Þ 1 ð l=3Þ expects0;V an unstableeff rC photon;‘C orbit! ; at r (7) 3Mnumbers. Considering outside parentheses correspond to lC and those and Wheeler related the deflection function to the real partdr C QM rC ¼ ð Þ¼2 ¼ inside to l . of the quantum mechanical scattering phase shift. Detailed the potential V 3M ! with the semiclassical prescrip-B. TheC rectangular barrier approximation −10 −5 0 5 10 15 20 ð Þ¼ 2 discussions on the topic can be found in Refs.with [2,8]. Here rtion/r of l l 1 l 1=2 , it is easy to!r seeIn Ref. that the[1]s Handler0 and Matzners 1 used a rectangulars 2 critical* 0 valueð þ ofÞ ! theð þ angularÞ momentum l is0 given by ¼ ¼ ¼ we directly state their conclusion, namely, 2 2 barrier as an approximate solution to obtain the transmis- l 3p3‘=2 !rM‘ 1=2, where r 2M.sion0.5 If coefficientsone uses corresponding0.799 to the (0.892) black hole1.4203 scattering (1.484) R FIG. 2. The Regge-WheelerVC r potential0 ; compared with0 (8) the ÁÁÁ d 2 eff¼ð 2 Þ 3À ¼ 1 1.966 (2.017) 2.098 (2.145) 2.458 (2.5)  l 2 l coshÀ potential(5) fromlC ðlC Eq.Þ¼ (1142r)instead, forÀ differentr one of spins course and angularends up withproblem a quadratic of spin 1 particles. Their choice of the height of ð þ ffiffiffiÞ 2 4.632 (4.656) 4.696 (4.720) 4.885 (4.908) ð Þ¼ dl momentum. The discrepancyequation for betweenlC. The the two cases values is of morelC shouldthe barrier however is V0 with the same definition as explained whereprominent‘ has at the smaller dimension energies where of angular the Regge-Wheeler momentum poten- per 2.5 5.944 (5.963) 5.995 (6.014) 6.147 (6.166) connecting the deflection function with the real part of the be quite close for large values of2l. One couldabove. try The to find width b is energy and l dependent, b l=!. masstial which displays makes an asymmetricVeff dimensionless. tail. Notice also Replacing that increasing‘ withl 3 7.251 (7.267) 7.294 (7.310) 7.422¼ (7.437) the critical lC for the occurrence of circularThe orbits standard in the analytical results for the rectangular barrier phase shift. There exists a critical value lg correspondingl l results1 we in an return to increasing back to height. the quantum mechanical picture. 2 2 ð þ Þ scattering of spin 0 and 2 particles in the sameread for way!r too.0
2 dRl !; r V r i!r i!r 2 2 sinh a!r0 ð ÃÞ ð ÃÞ e à Rl !; r eÀ à : (20) Tl ð Þ ; dr ¼À 2i! ½ þ ð ÃÞ j j ¼ 2 2 sinh a!r0 cos =2 1 4V a2r2 à ð Þþ 0 0 The absence of reflection behind the potential at r ð À Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (16) imposes the boundary condition R !; 0à !onÀ the1 lð À1Þ¼
104060-5 EXPLORING THE STRING AXIVERSE WITH PRECISION ... PHYSICAL REVIEW D 83, 044026 (2011) We can also estimate the size of the axion cloud as the solution takes the same form as the radial wave func- tion of the Schroedinger equation with an 1=r potential, n2 rc 2 rg; (11) 2 l kr Rfar r 2kr eÀ U l 1 ; 2 l 1 ; 2kr ; (15) which is always significantly larger than the black hole ð Þ¼ð Þ ð þ À rgk ð þ Þ Þ gravitational radius, as a consequence of (3). where U is the confluent hypergeometric function of the These estimates provide the following physical picture second kind, and k is the axion momentum, ASIMINA ARVANITAKI AND SERGEI DUBOVSKY PHYSICAL REVIEWof D the83, superradiant044026 (2011) axion cloud. The cloud is composed of a wave packet of the axion field rotating on a Keplerian k2 2 !2: (16) from axion-to-photon conversion in the near-horizon where T and are the energy-momentum tensor and the ¼ a À L orbit around the black hole. This axion wave packet always In the ordinary Schroedinger equation with an 1=r poten- magnetic field. Lagrangian density for , respectively.has a For tail that a space-time goes into the near-horizon ergosphere region tial the spectrum of frequencies ! is determined by requir- Section V elaborates on an issue which is aside from the region between two constant time slices,and gets conservation amplified there of leading to the exponential growth ing the regularity of R at the origin. Instead, in the black main line of the paper, but still a very important part of the current P implies that the time-averagedof the number energy of axions flux in the packet. hole case one has to impose the regularity of the field at the Given the complexity of the Kerr metric it is not surpris- the theoretical motivations for string axions in the mass at the infinity is equal to the time-averaged energy flux horizon—the incoming wave boundary condition (13). ing that a precise analytical expression for superradiant range probed by the black hole superradiance. Namely, all through the black hole horizon. The latter is equal to One way to do this is to solve the radial Eq. (12) in the EXPLORINGrates is unavailable THE STRING (partial numerical AXIVERSE results WITH can be found PRECISION ... PHYSICAL REVIEW D 83, 044026 (2011) these axions are ‘‘anthropic’’—their initial misalignment near-horizon regime. After dropping terms of order =l in [28]). However, the2 above physical picture gives rise to angle needs to be tuned to an atypically small value in the PG @t ! @ @t ! ! m! f ; the solution in the near-horizon region, 0 044026-7 PHYSICAL REVIEW D 91, 084011 (2015) Discovering the QCD axion with black holes and gravitational waves Asimina Arvanitaki* Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada Masha Baryakhtar† and Xinlu Huang‡ Stanford Institute for Theoretical Physics, Department of Physics, Stanford University, Stanford, California 94305, USA (Received 16 December 2014; published 7 April 2015) Advanced LIGO may be the first experiment to detect gravitational waves. Through superradiance of stellar black holes, it may also be the first experiment to discover the QCD axion with decay constant above the grand unification scale. When an axion’s Compton wavelength is comparable to the size of a black hole, the axion binds to the black hole, forming a “gravitational atom.” Through the superradiance process, the PHYSICAL REVIEW D 91, 084011 (2015) number of axions occupying the boundDiscovering levels grows the exponentially, QCD axion with extracting black holes energy and gravitational and angular waves momentum from the black hole. Axions transitioning between levels of the gravitational atom and axions Asimina Arvanitaki* annihilating to gravitons can produce observablePerimeter gravitational Institute for wave Theoretical signals. Physics, The Waterloo, signals Ontario are N2L long 2Y5, Canada lasting, monochromatic, and can be distinguished from ordinary astrophysical sources. We estimate up to O 1 Masha Baryakhtar† and Xinlu Huang‡ 11 10 4 ð Þ transition events at aLIGO for an axion betweenStanford10− Instituteand for10 Theoretical− eV and Physics, up Department to 10 annihilation of Physics, Stanford events University, for an axion between 10−13 and 10−11 eV. In the event of a null search,Stanford, aLIGO California can constrain 94305, USA the axion mass for ASIMINA(Received ARVANITAKI, 16 December 2014; MASHA published BARYAKHTAR, 7 April 2015) AND XINLU HUANG PHYSICAL REVIEW D 91, 084011 (2015) a range of rapidly spinning black hole formationAdvanced rates. LIGO may Axion be the first annihilations experiment to detect are gravitational also promising waves. Through for superradiance much of lighter masses at future lower-frequency gravitationalstellar black holes,angular it wave may also velocity observatories; be the first of experiment the black the to discover hole rates the at have QCD the horizon.axion large with uncertain- decay The constant only aboveof astrophysical black holes, superradiance will operate, the grand unificationdifference scale. When is an in axion the’s Compton (dissipative) wavelength interaction is comparable to required the size of a black for hole,regardless of the model or the abundance of the boson. We ties, dominated by supermassive black holethe spin axion distributions. binds to the black hole, Our forming projections a “gravitational for atom. aLIGO” Through are the robust superradiance against process, the number of axionssuperradiance occupying the to bound occur: levels in grows the case exponentially, of a conducting extracting energy cylinder, and angularmostly refer to the QCD axion, but our result is directly 2 perturbations from the black hole environmentmomentum and from accountit the is black electromagnetism, for hole. our Axions updated transitioning while exclusion between for blacklevels on of holes, the the gravitational QCD it is axion gravity. atom and of axionsapplicable to general scalars via λ ↔ μa=fa for a scalar −13 −11 annihilating to gravitons can produce observable gravitational wave signals. The signals are long lasting, ð 4Þ 6 × 10 eV < μ < 2 × 10 eV suggested by stellarAlthough black the hole kinematic spin measurements. condition is easy to satisfy, the with mass μa and quartic interaction L ⊃ λϕ =4!. The same a monochromatic, and can be distinguished from ordinary astrophysical sources. We estimate up to O 1 amplification rate is typically small, and rotational super- ð Þ transition events at aLIGO for an axion between 10−11 and 10−10 eV and up to 104 annihilation events forresults can also be approximately applied to light vector DOI: 10.1103/PhysRevD.91.084011 an axion betweenradiance10PACS−13 and 10 in numbers:−11 particulareV. In the 04.30.Db, event is of very a null hard 04.30.-w, search, to aLIGO observe. 97.60.Lf, can constrain The 14.80. the ampli- axion massj forbosons. a range of rapidlyfication spinning rate black is hole determined formation rates. by Axion the annihilations overlap ofare also the promising scattered− for much When the superradiance effect is maximized, a macro- lighter masses atwave future with lower-frequency the rotating gravitational object; wave for observatories; nonrelativistic the rates have rotation, large uncertain-scopic “cloud” of particles forms around the black hole, ties, dominated by supermassive black hole spin distributions. Our projections2m for aLIGO are robust against perturbations fromthis the overlap black hole is environment proportional and account to forωγ ourR updatedwhere exclusion R ison the the QCD size axion ofgiving dramatic experimental signatures [4]. The signals −13 −11 ð Þ I. WHAT IS SUPERRADIANCE? 6 × 10 eV 084011-2 Lasing gain • The wave functions of the axions are essentially hydrogenic states, with nothing particularly specific. • They can drop down in energy and emit a graviton. The probability that such a graviton is emitted is given by the total amplitude for spontaneous emission of a graviton. It will not behave any differently than our previous calculation. • From this amplitude, the absorption cross section can also be computed, in the particle picture for the graviton. • In the quantization of the gravitons we must use: gµ⌫ = µ⌫ + hµ⌫ • With µ ⌫ corresponding to the Schwarzschild metric. ~G 68 • However we will still find = o(1) 10 c3 ⇥ ⇠ DISCOVERING THE QCD AXION WITH BLACK HOLES … PHYSICAL REVIEW D 91, 084011 (2015) α rgμa;rg GNM; 2 grow exponentially. If the SR condition is satisfied, the ¼ ≡ ð Þ growth will occur as long as the rate is faster than the where rg is the gravitational radius of the BH, M its mass, evolution time scales of the BH, the most relevant of which 8 and μa the boson’s mass. Throughout this paper, we use is the Eddington accretion time, τEddington 4 × 10 years. units where c ℏ 1. Like the hydrogen atom, the The growth stops when enough angular¼ momentum has ¼ ¼ • Thus the gain depends on the density of the axions. This is orbitals around the black hole are indexed by the principal, completelybeen extracted dependent so that on the the model superradiance for the axions condition and is no orbital, and magnetic quantum numbers n; l;m with dependentlonger satisfied. on the super At radiance. that point the number of bosons f g energies: • Itoccupying is calculated the that level the population is of the axionic states quickly rise to 2 2 2 α GNM 76 Δa M N a 10 à ; 708 ω μa 1 − : 3 max Δ ∼ 10 ≃ 2n2 ð Þ ≃ m à 0.1 10M ð Þ • This gives rise to a density of approximately ⊙⇠ where Δa O 0.1 is the70 difference3 between the initial The orbital velocity is approximately v ∼ α=l, and the à ¼ ð Þ 10 /Rs and final BH spin. ⇠ axions form a cloud with average distance • ThenThe the superradiance graviton would rates amplify (or with dumping gain per rates meter for of the levels that are not superradiating)~G are given by the small imagi- n2 1070/R3 = 1070/R3 o(1) nary part of the energys of3 a free-fields solution in the Kerr rc ∼ 2 rg 4 ⇥ c ⇥ α ð Þ background. Unless otherwise specified, we use the semi- analytic approach for massive2 spin-03 fields presented in [4], from the black hole. 10 /Rs o(1) which agrees well with⇠ analytical formulas⇥ for α=l 1 [2] A level with energy ω and magnetic quantum number m ≪ and the WKB approximation for α=l O 1=2 [2,4], as can extract energy and angular momentum from the black ∼ well as with partial numerical results in [13]ð .Þ Rates for hole if it satisfies the superradiance condition analogous to massive spin-1 fields are expected to be larger, and some Eq. (1): numerical progress has been made toward calculating them [14]; we choose to focus on the spin-0 case (including the ω 1 a −1 < ωþ; ωþ à r ; 5 QCD axion) for the remainder of this paper, but further 2 g m ≡ 2 1 1 − a ð Þ studies with spin-1 fields are very worthwhile. þ à pffiffiffiffiffiffiffiffiffiffiffiffiffi In Fig. 1 we show representative values of the super- where ωþ can be thought of as the angular velocity of radiance rates, Γsr, and we list sample values in Table I the black hole and 0 ≤ a < 1 is the black hole spin j Ãj along with typical BH evolution time scales. The rate varies (a a=rg in Boyer-Lindquist coordinates). The SR con- with the relevant parameters of the system as follows: ditionà ≡ requires (i) rg.—The dimensionless quantity Γsrrg depends only on the coupling α, BH spin a and the quantum α=l ≤ 1=2; 6 à ð Þ numbers of the state; the physical SR time can be as short as 100 s for stellar black holes and is longer for with the upper bound saturated for m l and extremal heavier black holes. black holes (a 1), so superradiating¼ bound states are indeed well approximatedà ¼ by solutions to a 1=r gravita- tional potential (rc ≫ rg) with subleading relativistic cor- 0.5 1. 1.5 2. 2 10 6 rections (v ≪ 1). 1 The occupation number1 N of levels that satisfy the SR a 0.99 4 2 10 a 0.9 10 8 condition grows exponentially with a rate Γsr, 3 g 2 r years 10 10 dN 10 sr 1 4 ΓsrN; dt sr ¼ sr 100 12 nlm −7 −14 −1 10 Γsr a ; α;r g O 10 –10 rg : 7 ð à ޼ ð Þ ð Þ 2 10 10 14 The boson is not required to be dark matter or be physically 0.5 1 1.5 2 2.5 present in the vicinity of the black hole: just like sponta- 10 11 eV neous emission, superradiance can start by a quantum a mechanical fluctuation from the vacuum, and proceed to FIG. 1 (color online). Superradiance times of levels l 1 to 4 (left to right) for spins a 0.99 and 0.90, fixing m ¼ l and à ¼ ¼ 1 n l 1. Time in years is shown for a 10M black hole as a The axion cloud surrounding the BH is described by a ¼ þ ⊙ classical field, and therefore does not have a well-defined function of boson mass μa; on the right axis, we show the occupation number N. In this paper we define the occupation dimensionless superradiance rate Γsrrg as a function of the number as the average value of bosons in the cloud. gravitational coupling α (top axis). 084011-3 • For a solar mass black hole the radius is about 3000 meters, giving the gain per meter to be, for 1 to 10 solar masses 8 11 10 to 10 ⇠ • Thus the graviton is amplified once every second to one thousand seconds. Eventually this would be an intense, coherent graviton beam. • For a super massive black hole of 10 7 solar masses, the gain 19 per meter would be 10 . • If the density of the lasing medium is not anywhere near the projected maximum density, it can also drastically lower the gain. • It is clear observationally, we have not seen the existence of any intense, coherent beams of gravitational radiation, essentially death rays, thus the gains have to be smaller. TIME BOMB • Thus in principle, since the beginning of the universe, about 13 billion years or 13 10 16 seconds, gravitons have been lasing in the vicinity⇥ of black holes with such super radiant axion clouds. • They are continually emitting coherent, amplified beams of gravitons. • There has not been enough time for the beams to be sufficiently amplified, to be a threat. The gain per meter has not been sufficiently high. • But there is a ticking time bomb in the process of being created. 25 • For gains per meter of the order of 10 the amplification is order 1 in the time the universe has existed. • If the axionic super radiant cloud prediction is valid, there is bad news on the horizon.