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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
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  • Black Hole Entropy Entanglement And

    Black Hole Entropy Entanglement And

    BLACK HOLE ENTROPY ! ENTANGLEMENT! AND! HOLOGRAPHIC SPACETIME" Ted Jacobson" University of Maryland" Goddard Scientific Colloquium, Feb. 7, 2018 Holographic " Information" principle" paradox" Area 3 4~GN /c geometry" black hole " Einstein eqn " from" entropy" as vacuum " entanglement " thermodynamics" QFT " renormalization" Albert Einstein, aged 33, 1912" GRAVITY IS CURVATURE OF SPACETIME" Spatial curvature analogy:" " Spacetime" Initially parallel lines " don’t stay parallel" time" earth" apple" Time runs slower lower down!" B" Apple free-fall " apple" is the straightest path " in spacetime between A & B…" and the path of longest time." How much slower? " One billionth of a second per year per foot" A" ’ 2 earth" at the earth s surface (g/c )." Spiral of Mercury’s orbit: didn’t fit Newton’s theory, " by 43’’/century…" That’s about 9 minutes advance time for the transit per century… " General relativity nailed it." To calculate the rate of perihelion advance Einstein needed only the first approximation to the line element outside the sun: " " 2 2 2 2 2 2 2 ds = (1− rS r)dt − (1+ rS r)dr − r (dθ +sin θ dϕ ) 2 rS = 2GM c = 3km The Schwarzschild Singularity (1916) 2 2 −1 2 2 2 2 2 ds = (1− rS r)dt − (1− rS r) dr − r (dθ +sin θ dϕ ) 2 rS = 2GM c = 3km "Schwarzschild radius" Schwarzschild (1916): “in order that this discontinuity coincides with the origin” one should define the radial coordinate appropriately. Droste (1916): “a moving particle can never pass that sphere because it would take an infinite amount of time” The true, non-singular nature of the Schwarzschild “singularity” was not widely understood until 42 years later… but it was understood perfectly well by one man in 1932… Georges Lemaitre First person to understand the nature of the Schwarzschild singularity as an event horizon (1932).