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