Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Some mathematical results of Stephen Hawking

Zoe Wyatt

PG Colloquium, University of Edinburgh

26 April 2018

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

1 Introduction

2 Early classical GR work (1962-73)

3 Quantum Gravity work (post 1973)

4 Voyager Mission and Conclusion

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Stephen Hawking

Born 8 January 1942, died 14 March 2018 age 76

Lucasian Professor of Mathematics at Cambridge 1979 to 2009

Fellow of the Royal Society (1974)

early-onset slow-progressing form of motor neurone disease

Obituary “Mind over Matter” by Penrose on Guardian website

A Brief History of Time (1988) sold over 25m copies worldwide.

Stephen Hawking and Jane Wilde, 1965

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Stephen Hawking on InspireHEP

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Laws of Physics

Newton’s law (1687)

~F =GMm

4πr 2r

Predictions

universal law of gravitation

Maxwell’s equations (1861)

∇ · ~E = ρ/ε0 ∇ · ~B = 0

∂~B

∂t= −∇× ~E ∂E

∂t=

1

µ0ε0∇× ~B − 1

ε0

~J

Predictions

propagation of electromagnetic field

generation of electric and magnetic fields by charges, currents, andchanges of each other

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

General Relativity

Einstein equations (1915)

Rµν [g ]− 1

2R[g ]gµν = Tµν

Unknown: Lorentzian metric gµν

Predictions

interaction of curvature of spacetime withmatter and energy

black holes (1958)

gravitational waves (LIGO 2016)

advance of the perihelion of Mercury

binary pulsars (Taylor and Hules,PSR1913+16 in 1974)

gravitational lensing (1919)

gravitational redshift (1950s)

GPS corrections

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Solutions of the Einstein Equations

Examples of the stress energy tensor

EM field Tµν = µ−10 (FµρF ν

ρ − 14 gµνFρσF ρσ)

Perfect fluid Tµν = (ρ+ p)uµuν + pgµν

vacuum outside the Sun, Earth etc.energy including gravitational energy

Geometry around a spherical object of mass M and radius R described by

gS(M) = −(

1− rSr

)dt2 +

(1− rS

r

)−1

dr 2 + r 2(dθ2 + sin2 θdφ2)

for r > R.

Examples of such objects

the Sun rS = 3× 103mSagittarius A* rS = 1.3× 1010m

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Einstein’s Static Universe

Early 1900s, stars in the Milky Way were not moving in anysystematic way. Einstein believed the universe was static.

Static: spacetime does not change over time and is irrotational

Static solutions to EE possible if one adds in Λ the cosmologicalconstant

Gµν + Λgµν = Tµν

Idea: repulsive force that balances out gravity to avoid everythingcollapsing under attractive force of gravity

Hubble observed redshift (1929) ⇒ cosmic expansion ⇒ dynamicuniverse

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Robertson-Walker Cosmological Model

Galaxy distribution appears to be homogeneous and isotropic

Homogeneous (no priviledged positions)

Isotropic (no preferred directions)

Roberston-Walker metric takes the form:

ds2 = −dt2 + a2(t)

(dr 2

1− kr 2+ r 2(dθ2 + sin2 θdφ2)

)

Parameter k|k| ∈ −1, 0, 1 is curvature of spatial

slices, unknown positive function a(t).

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Robertson-Walker Cosmological Model

Model is dynamic by using above ansatz in the Einstein equationscoupled to a perfect fluid of density ρ and pressure p:

Gµν = (ρ+ p)uµuν + pgµν + Λgµν

∇µTµν = 0

Examples of matter

gas of non-relativistic particles (nuclei, electrons), dark matter(p << ρ)

dark matter behaves in same way as above but no EM interaction

radiation like photons, neutrinos (p = ρ/3)

early universe dominated by photons −→ CMB

dark energy (p = −ρ) constant energy density

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Robertson-Walker Cosmological Model

Dynamical Friedmann equations:

3a2

a2= ρ− 3

k

a2+ Λ , 3

a

a= −(ρ+ 3p) + Λ

Take sum of contributions ρ = ρr + ρm + ρΛ. Each have different scalings⇒ single component dominates

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

The Big Bang

Vacuum universe (Λ = 0) cannot be static:

if ρ > 0, p ≥ 0 then a 6= 0 and negative hence non static

Hubble experiments show universe is expanding, a > 0

∃ time T = aa |t0 ' 1010 at which a(T ) = 0.

Also true if Λ at most small and positive

Hubble Law v = H0d and Hubble parameter H = a/a

Experiments of Type 1a supernovae 1998: accelerated expansion(a > 0) since Λ > 0

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

The Singularity Theorems

FRW spacetimes begin from a singular ‘Big Bang’ state. Could thisjust be due to the homogeneity and isotropy?No, thanks to Hawking’s PhD thesis!

Theorem (Cosmological Singularity Theorem, Hawking 1967)

Take an Einstein spacetime (M, g) satisfying the SEC. Suppose thereexists a compact (...) achronal spacelike hypersurface S with extrinsiccurvature supp∈S Kp ≤ C < 0. Then there is at least one past directedtimelike geodesic that is inextendible.

Compare to: Penrose’s 1965 result just looking at gravitational collapse.

Theorem (Singularity Theorem, Hawking and Penrose 1970)

Einstein spacetime (M, g) with the SEC, some generic conditions and notimelike curves. Assume either (1) ∃ compact achronal set, (2) ∃ trappedsurface, (3) exists point where expansion becomes negative alonggeodesics. Then (M, g) must contain at least one incomplete at least oneincomplete geodesic.

Beware: GR should fail at extremely high densities, need to includequantum effects at this point.

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Black Holes

Schwarzshild (1916) solution: static (time-independent, nonrotating)spherically symmetric object of radius R, vacuum solution to the Einsteinequations

gS(M) = −(

1− rSr

)dt2 +

(1− rS

r

)−1

dr 2 + r 2(dθ2 + sin2 θdφ2)

Take R < rS then get a black hole region: no signal or material body canescape to infinity

Oppenheimer and Snyder (1939) showedthe collapse of a spherically symmetric dustfluid leads to a Schwarzshild black hole.Singularity theorems remove this sphericalsymmetry.

Rotating black hole is not spherically symmetric, but could beaxisymmetric (symmetric under rotations about an axis) like Kerr (1963).

gK = gK (M, J)

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Topology of the Event Horizon

Move from static to less restrictive stationary (time-independent)black holes

Event horizon is boundary of the black hole region

Asymptotic flatness is a way to capture an ‘isolated system’ (noglobal inertial coord. system to define fall off rates)

Theorem (Topology of Event Horizons, Hawking 1972)

A (3 + 1)-dimensional a.f. stationary black hole with an energy conditionhas an event horizon with cross sections that are topologically an S2.

uses Gauss-Bonnet theorem. Result fails in higher dimensions (blackring S2 × S1 in 4 + 1 dimensions

energy conditions capture physically reasonable matter

W(N)EC:Tabξaξb for ξ timelike (null). ie, energy density

non-negativeDEC: speed of the energy-momentum current of the matter is lessthan the speed of light

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Uniqueness theorems

Black holes form by dynamical gravitational collapse

Isolated black hole expected to settle down to time-independentequilibrium state

AIM: classify all stationary (equilibrium) solutions to the vacuum EEs

Israel (‘67) showed static, a.f. black hole (with some regularity) issph. sym. and thus (by Birkhoff) isometric to Schwarzschild

Theorem (Uniqueness of Kerr)

Kerr is the only stationary a.f. analytic solution to the vacuum EEs.

Hawking (‘73) showed that a stationary, a.f. real analytic solution tothe Einstein-Maxwell equations is static or axisymmetric (usedtopology theorem)

Carter (‘71) and Robinson (‘75) then showed that stationary,axisymmetric, a.f. vacuum spacetimes are Kerr

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

What is Thermodynamics

Analogies between black holes and thermodynamicsphysical quantities (temperature, energy, and entropy) thatcharacterize thermodynamic systems at thermal equilibriumsteam engine, insulated piston/cylinder arrangement

Four rather vague laws:

0 thermal equilibrium is an equivalence relation1 energy is conserved and can only be transferred or converted

δE = heat across boundaries + W

2 the entropy of any isolated system always increases (towardsmaximum entropy, thermal equilibrium)

3 the entropy of a system approaches a constant value as thetemperature approaches zero.

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Black Hole Thermodynamics

Analogies between black holes and thermodynamics.

Look at evolution of the event horizon (boundary of a black hole)

Theorem (Second law of black hole mechanics, Hawking 1971)

In a ‘nice’ spacetime (M, g) with the NEC and an event horizon H. Thearea of H2 = H ∩ Σ2 (a 2-dim submanifold of Σ2) is greater than orequal to the area of H1 = H ∩ Σ1.

Put simply: δA ≥ 0.

Compare with second law of thermodynamics: δS ≥ 0

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Theorem (Killing Horizons, Hawking 1972)

Take a stationary, analytic, a.f vacuum black hole spacetime with (future)event horizon H. Then H is a null hypersurface with a Killing vector fieldξ defined in an n‘hood of H such that ξ is normal to H.

Leads to surface gravity κ:

∇a(ξbξb)|H = −2κξa

in static, a.f. spacetime can interpret κ surface gravity, ie our weightcomes from force mg where g = 9.8m/s2

Theorem (Zeroth law of black hole mechanics)

κ is constant on the horizon H, for a stationary black hole spacetime withthe DEC.

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Black Hole Thermodynamics

Mass Formula of Bardeen, Carter and Hawking (‘73)

A stationary, axisymmetric vacuum spacetime containing a blackhole has mass

M =1

4πκA + 2ΩHJ

where J is the angular momentum of the black hole.

Theorem (First law of black hole mechanics, Bardeen, Carter andHawking 1973)

A stationary, axisymmetric black hole spacetime under a stationary,axisymmetric perturbation satisfies

δM =κ

8πδA + ΩHδJ

Proof for all perturbations by Sudasky and Wald (‘92).Compare with the first law of thermodynamics δE = T δS + PδV

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Black Hole Thermodynamics

Make the identifications: T ↔ ακ, S ↔ A8πα and E ↔ M.

Suggests black holes are thermodynamic objects.

Law Thermodynamics Black HolesZeroth T constant throughout body κ constant on horizon

in thermal equilibrium of stationary black hole

First δE = TdS + PδV δM = 18πκδA + ΩHδJ

Second δS ≥ 0 δA ≥ 0

Third T 6= 0 κ 6= 0

Means a black hole would have a temperature and emit radiation!

Hawking (1974) showed effective emmision of particles (Hawkingradiation) from a black hole with a blackbody spectrum at Hawkingtemperature TH = ~κ/(2π)

needs QFT on a curved background

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Conclusion

Voyager I and II spacecrafts, launched 1977

fly by all 4 Jovians (Jupiter, Saturn, Uranus, Neptune)

alignment (once per 175 years) of Jovians meant spacecraft couldvisit all planets using gravity assists

Voyager I first man-made object in interstellar space (Sept 2013)

2017 film: The Farthest

Experiencing zero gravity in 2007

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Voyager Missions 1977

Family Portrait

60 pictures from Voyager I in 1990

6.4 billion km from Earth

Mars and Mercury lost in the Sun glare, Pluto too faint

Introduction Early classical GR work (1962-73) Quantum Gravity work (post 1973) Voyager Mission and Conclusion

Pale Blue Dot