The Physics of Black Holes

Trial Lecture Nikolai Fomin University of Bergen

25/08/2020

1/43 The Physics of Black Holes Plan of the Lecture

1 Introduction

2 Formation and Classification of Black Holes

3 Black Hole Mechanics

4 Experimental Results

5 Hawking Radiation

6 Conclusion

7 Backup

2/43 The Physics of Black Holes Introduction

Enigmatic objects Wide spread in media and fiction Naive definition Black holes are objects with such strong gravitational pull that not even light can escape

Image source: European Southern Observatory 3/43 The Physics of Black Holes Historical Developments

1915 Einstien Field Equations (EFE)

Gµν + Λgµν = κTµν 1916 Schwarzschild metric – solution of EFE 2GM Schwarzschild radius rs = c2

1958 Finkelstein and Misner – identified rs with the event horizon of a black hole Images source: wikimedia

4/43 The Physics of Black Holes Historical Developments

1963 Kerr – EFE solution for rotating black holes 1965 Newman – rotating, charged black hole 1967 Israel et al. – no-hair theorem 1970s Hawking et al. – laws of black hole mechanics Image source: 1974 Hawikng – Hawking radiation The royal society No-hair theorem An isolated black hole is fully described by it’s mass, angular momentum and charge

Image source: University of Pittsburgh 5/43 The Physics of Black Holes What Is a Black Hole?

Classical Definition (Hawking, S. & Ellis, G.) A region of the spacetime that is not contained in the causal past of the infinite future.

Divide universe in two mutually exclusive regions Exterior causally connected to ”infinitely far away” Interior is a region such that anything that enters it remains there The boundary is defined to be it’s event horizon

6/43 The Physics of Black Holes What Is a Black Hole?

The definition of black holes and event horizons is global No local experiment can map out the event horizon Whether a ray can escape to infinity starting today depends on the position of the event horizon next week

What we know There are objects in the universe that approximately behave like the idealized black holes for sufficiently long time

7/43 The Physics of Black Holes Black Holes Formation

An active star with radius R and mass M Equilibrium of kinetic and gravitational energy When the star runs out of fuel – gravitational collapse Possible equilibrium states Image source: ASPIRE

8/43 The Physics of Black Holes Chandrasekhar limit

Gravitational energy Electron degeneracy pressure GM2 α Egrav ∼ − R = − R , 3 M ∼ nmpR =⇒ Kinetic energy M n = ne ∼ m R3 E ∼ nR3hEi p kin For non-relativistic electrons −1/3 ~ 2 ne ∼ hpe i hpe i hEi ∼ me 2 Pauli exclusion principle becomes ~ M 5/3 1 β Ekin ∼ m ( m ) R2 = R2 relevant e p

9/43 The Physics of Black Holes Chandrasekhar limit

α β E = Egrav + Ekin ∼ − R + R2 has a minimum for non-zero R Relativistic electrons 1/3 hEi ∼ hpeic ∼ ~cne E ∼ c( M )4/3 1 = γ kin ~ mp R R α γ E ∼ − R + R 1 ~c 3/2 =⇒ Mcritical ∼ 2 ( ) mp G Mcritical ≈ 1.4M White dwarf Image source: P.K. Townsend

10/43 The Physics of Black Holes Tolman–Oppenheimer–Volkoff limit

M > Mcritical Electron capture Collapse to nuclear matter density e− + p+ → n + ν e Neutron degeneracy pressure ∆m ∼ 3me Tolman–Oppenheimer–Volkoff limit Electrons in white dwarfs are MTOV ≈ 2.3M non-relativistic 2 Neutron star EF / mec Heavier stars collapse into black holes

11/43 The Physics of Black Holes Black Holes Classification

Intermediate-mass black holes Primordial black holes 2 5 Mass ranges 10 − 10 M Formed in the early universe Too massive to come from a Can have masses below TOV limit gravitational collapse of a star ”Normal” or stellar black holes Supermassive black holes Produced in the gravitational Mass ranges 105 − 109M collapse of a star Centers of galaxies Typical mass ranges ∈ [3, 100]M Sagittarius A∗

12/43 The Physics of Black Holes Black Hole Mechanics

Variables typically used to Surface gravity κ describe black holes Formal definition Horizon area A Magnitude of the gradient of the norm of the Mass M horizon generating Killing field χα evaluated at Energy E the horizon.

Angular velocity Ω Equivalently Angular momentum J κ is the force per unit mass that must be Electric surface potential Φ applied at infinity in order to hold a particle on Electric charge Q its path.

13/43 The Physics of Black Holes Surface Gravity

Massless inextensible string Raise a unit mass particle by the distance dl at radius r

dEr = gr dl Distant observer pulling the string

dE∞ = g∞dl Gravitational redshift Image source: wikimedia

14/43 The Physics of Black Holes Surface Gravity

gr = Er = (1 − 2GM )−1/2 g∞ E∞ r Surface gravity for Schwarzschild Schwarzschild black hole black holes g = GM (1 − 2GM )−1/2 r r 2 r κ = 1 GM 4GM =⇒ g∞ = 2 r For rotating Kerr black holes As r → rs = 2GM, 1 2 κ = 4GM − MGΩ gr → ∞, g∞ → κ

15/43 The Physics of Black Holes Mechanics of Black Holes

Zeroth law of thermodynamics If two systems are both in thermal equilibrium with a third system then 1970s – Bardeen, Carter, Hawking they are in thermal equilibrium with each other. Classical thermodynamics analogy κ local → global Black hole equivalent Surface gravity κ is constant on the κ ↔ classical temperature horizon of a stationary black hole. Dominant energy condition is assumed.

16/43 The Physics of Black Holes Mechanics of Black Holes

First law of thermodynamics Conservation of energy Angular velocity Ω dE = TdS − pdV + µdN Electric potential of the horizon Φ Black hole equivalent First term – ”entropy” If a black hole with mass M, charge Q Other terms – changes due to and angular momentum J is perturbed rotation and electromagnetism κ dM = 8π dA + ΩdJ + ΦdQ

17/43 The Physics of Black Holes Mechanics of Black Holes

Second law of thermodynamics Entropy of a closed system never decreases Hawking’s area theorem dS ≥ 0 Black hole cannot split in two

Black hole equivalent M3 → M1 + M2, M3 ≥ M1 + M2 p 2 2 The area of the future event horizon is M3 ≤ M1 + M2 ≤ M1 + M2 a non-decreasing function of time. dA ≥ 0

18/43 The Physics of Black Holes Mechanics of Black Holes

Third law of thermodynamics It is impossible for any process, no Weaker version of the Third Law than the matter how idealized, to reduce the Planck version entropy of a system to its absolute-zero 1 Simplest black hole – κ ∼ 4M value in a finite number of operations. Adding charge Q =⇒ electrostatic Black hole equivalent repulsion Surface gravity κ cannot be reduced to Similar for angular momentum J zero in a finite number of operations.

19/43 The Physics of Black Holes Gravitational Waves

Gravitation – manifestation of the curvature of the spacetime Gravitational waves Propagate with the speed of light Typical sources – binaries Hulse - Taylor pulsar 1913+16

20/43 The Physics of Black Holes Gravitational Waves

Laser interferometric detectors LIGO, VIRGO, GEO 600, TAMA 300 2015 LIGO – first direct observation Observation of Gravitational Waves from a Binary Black Hole Merger LISA, DECIGO – new generation of space-based detectors

Image source: wikimedia

21/43 The Physics of Black Holes Gravitational Waves

Images sourse: LIGO 22/43 The Physics of Black Holes Photon Orbit

Black holes invisible to radio telescopes Spacetime around black holes curved Photon orbit Lower bound for a stable orbit 2019 Event Horizon Telescope – supermassive black hole in the Messier 87 galaxy Image source: Event Horizon Telescope 23/43 The Physics of Black Holes Accretion Disks

Accumulation of matter around a central body Orbiting in inward spirals Gravitational potential energy converted into heat Black holes – hot enough to emit x-rays Hubble Space Telescope 5 Intermediate-mass black hole M ∼ 5 × 10 M Messier 87 black-hole-powered jet Images source: NASA

24/43 The Physics of Black Holes Black Hole Entropy

So far T = 0 dStot dSBH dSmatter Second law of thermodynamics dt = dt + dt ≥ 0 Throw a box of gas in the black Do we also get the TdSBH = dE? hole 1974 Hawking confirmed the What happens to the entropy? conjecture A SBH = 2 1973 Bekenstein – black hole 4Lp entropy postulate

25/43 The Physics of Black Holes Black Hole Entropy

Box with gas – linear size L, mass ρ ∼ L if δr ∼ L2/GM m and temperature T The energy is red shifted q 2GM mL Black hole – mass M, horizon ∆M ∼ m 1 − 2GM+δr ∼ GM radius rs = 2GM Assume L ∼ ~/T ∆S ∼ −m/T ∆S ∼ −m/T ∼ −mL/~ ∼ Schwarzschild black hole proper −GM∆M/ ∼ − ∆A ~ ~G distance Generalizes to Kerr black holes √ ρ ∼ GMδr and more general systems

26/43 The Physics of Black Holes Hawking Radiation

Inside the horizon components of In 1974 Hawking – black holes the metric change signs emit blackbody radiation Ingoing virtual particle with Virtual particle-antiparticle pairs negative energy relative to external Energy conservation – ±E observer can have positive energy Exist for t ∼ ~/E relative to an observer inside the Energy frame dependent horizon Black hole evaporation

27/43 The Physics of Black Holes Hawking Radiation

Black hole radiating with Proper time to reach hoziron √ characteristic temperature ∼ GMδr kT ∼ ~/GM |E| ∼ √ ~ GMδr ∆S = ∆M/T ∼ GM∆M/~ Redshifted at infinity q =⇒ ∆S ∼ ∆A 2GM ~G E∞ ∼ √ ~ 1 − GMδr 2GM+δr Precise calculations give ~ E∞ ∼ GM kTHawking = ~κ/2π

28/43 The Physics of Black Holes Hawking Radiation - Experiment

−8 THawking ∼ 6 × 10 M /M K No chance to observe In TeV-scale gravity models could be studied at LHC Analogs in condensed matter systems Acoustic black holes, Bose-Einstein condensates, slow light, superfluid quasiparticles Observation of Stimulated Hawking Radiation in an Optical Analogue

29/43 The Physics of Black Holes Entanglement

For entangled states only the Consider two separated electrons with spin state of the whole system up and spin down states makes sense |Ψi = | ↑i1 ⊗ | ↓i2 Can write down a density Entangled states cannot be written as a matrix ρ = |ΨihΨ| and trace product of electron 1 ⊗ electron 2 over it |Ψi = √1 (| ↑i ⊗ | ↓i + | ↓i ⊗ | ↑i ) 2 1 2 1 2 Statistical construct, not a quantum state

30/43 The Physics of Black Holes Information Loss

Consider Hawking radiation Information about the original The quanta escaping to infinity matter that went into the black and falling into the black hole are hole entangled Can quantum information be At some point the black hole destroyed? evaporates Information loss paradox The wave function has complete String theory, black holes as information about the system fuzzballs

31/43 The Physics of Black Holes Conclusions

Presented basic ideas about black holes and their physics Some experimental results relating to the black holes Active branch of physics, lots of experimental and theoretical developments! Thank you for your attention!

32/43 The Physics of Black Holes Backup

33/43 The Physics of Black Holes Schwarzschild Metric 2 2GM 2 1 2 2 2 2 2 ds = (1 − r )dt − 2GM dr − r (dΘ + sin Θdφ ) 1− r Singularities at r = 2GM and r = 0 Schwarzschild radius The metric describes the spacetime outside the body Ignore for a second that far away observer can’t see what is inside a black hole 2GM Inside the horizon factor (1 − r ) changes sign The ”nature of time and radial distance” have changed Light cone points towards the region the object is allowed to travel Inside the horizon you can only travel towards the Image source: singularity MTW 34/43 The Physics of Black Holes Black Hole Entropy

Consider a box with (hot) gas So far treating black holes classically Throw it in a black hole! Previously discussed four laws of Initial state clearly has some black hole mechanics have very entropy close parallel to the laws of What happens to the entropy of thermodynamics the gas when it goes into the Crucial component needed for singularity? black hole thermodynamics is Second law of thermodynamics missing says the black hole should have T = 0, no thermal radiation gained entropy? What about entropy? J. Bekenstein postulated black hole entropy in 1973 35/43 The Physics of Black Holes Hawking Radiation

We have two different vacuum states For two different frames of |0i and |0¯i annihilating ai anda ¯i reference with time coordinates t † Number operators Ni = ai ai and and t¯ we get † N¯i =a ¯ a¯i  †  i φ = Σ a u + a u? and 2 i i i i i Computing h0¯|Ni |0¯i = Σj |βji |  † ? Particle content of vacuum depends φ = Σi a¯i u¯i +a ¯ u¯ i i on the frame! Expressu ¯ as a function of (u , u?) j i i Hawking considered a mass collapsing ? u¯j = Σi (αji ui + βji ui ) into a black hole and computed Bogoliubov transformation Bogoliubov coefficients connecting initial and final vacuums

36/43 The Physics of Black Holes Hawking Radiation

The important physical feature is a

The number Np of the particles ”pile up” of vacuum modes at the that reach the observer at infinity horizon leading to the exponential is given by relationship N = Γ Initially computed by Hawking for p e~ω/TH −1 Thermal distribution of particles particular choices of vacuum with temperature TH Generalizations are possible Γ is the greybody factor correcting Alternative derivations based on for the emissivity of the black various assumptions and holes which differs from perfect approaches confirm the prediction black bodies Of course, the proper way to do this would be quantum gravity... 37/43 The Physics of Black Holes Hawking Radiation

† Interpret ak and ak and Let us look at a simple example annihilation and creation operators Free real scalar field φ quantization respectively  † ?  Vacuum state |0i that annihilates φ = Σk ak uk (t, x) + ak uk (t, x) all ak , ak |0i = 0 u = exp(ik · x − iω t) k k In a curved spacetime standard p 2 2 ωk = |k| + m Fourier modes are not available Fourier modes uk is a set of However, given time coordinate t, orthonormal functions satisfying we can find modes with the same 2 ( + m )uk (t, x) = 0 form, decompose the field and ∂tuk (t, x) = −iωk uk (t, x) obtain creation/annihilation operators

38/43 The Physics of Black Holes Rindler Coordinates

Describe observer moving with constant acceleration a = ξ−1 in Minkowski spacetime x = ξ cosh η t = ξ sinh η η = aτ where τ is proper time x ds2 = dt2 − dz2 = ξ2dη2 − dξ2 Calculate Bogoliubov coefficients in Rindler coordinates Accelerated observer sees thermal Rindler wedge bath – Unruh radiation k T = a/2π B Unruh ~ 39/43 The Physics of Black Holes Rindler Decomposition

Divide Hilbert space into HL acted on by fields with x < 0 and HR acted on by fields with x > 0 To find vacuum state of a quantum system we can act on a generic state |χi with Change variables so that instead of e−HT , T → ∞ tE ∈ [−∞, 0] we integrate over 1 −HT hφ|0i = h0|χi limT →∞hφ|e |χi the angular direction φˆ(t =0) R E ˆ −IE Boost operator Kx in Euclidean hφ|0i ∝ ˆ Dφe φ(tE =−∞) plane generates rotation

I Euclidean action, φˆ depends on −πKR E hφLφR |0i ∝ hφR |e Θ|φLiL both position and time Θ anti-unitary operator 40/43 The Physics of Black Holes Rindler Decomposition

Plug in eigenstates of KR Why is temperature dimensionless? hφLφR |0i ∝ −πωi ? During choice of Rindler Σi e hφL|i iLhφR |iiR coordinates we suppress an |i ?i = Θ†|ii L R arbitrary choice of length scale l |0i = √1 Σ e−πωi |i ?i |ii Z i L R Inverse of proper acceleration Reduced density matrix for the 1 a = l right wedge Observer with acceleration a 1 −2πωi ρR = Z Σi e |iiR hi| observes a Thermal density matrix with TUnruh = ~ 1 2πkB c temperature T = 2π

41/43 The Physics of Black Holes Trans-Planckian problem

Hawking’s calculation includes a reservoir of trans-Planckian modes near the event horizon Wavelength arbitrarily shorter than Planck length Is it safe to extrapolate QFT beyond what is typically considered the range of validity? The Hawking radiation spectrum turns out to be robust against these high frequency modes While the role of trans-Planckian excitations is still not perfectly understood various analytical and numerical calculations confirm the macroscopic thermodynamic properties of the black holes

42/43 The Physics of Black Holes More About the Third Law

Black hole with angular momentum J and charge Q κ = 4πµ/A, A = 4π[2M(M + µ) − Q2] µ = (M2 − Q2 − J2/M2)1/2 M2 < Q2 + J2/M2 – ”naked” singularity not shielded by event horizon 1969 Penrose – Cosmic censorship hypothesis

43/43 The Physics of Black Holes