And Physics Nobel Prize 2020

Image from The science of interstellar, Kip Thorne

End of Time

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Physics Nobel Prize 2020 Alexander Vikman 19.11.2020 CEICO Black Holes This year’s Nobel Prize in Physics focuses on black holes, which are among the most enigmatic objects in the Universe. The Prize is awarded for establishing that black holes can form within the theory of general relativity, as well as the discovery of a supermassive compact object, compatible with a black hole, at the centre of our galaxy

Image from The science of interstellar, Kip Thorne Black Holes This year’s Nobel Prize in Physics focuses on black holes, which are among the most enigmatic objects in the Universe. The Prize is awarded for establishing that black holes can form within the theory of general relativity, as well as the discovery of a supermassive compact object, compatible with a black hole, at the centre of our galaxy

Image from The science of interstellar, Kip Thorne Roger Penrose Born: 8 August 1931, Colchester, United Kingdom Afﬁliation at the time of the award: University of Oxford, Oxford, United Kingdom Prize motivation: "for the discovery that black hole formation is a robust prediction of the general theory of relativity." Roger Penrose Born: 8 August 1931, Colchester, United Kingdom Afﬁliation at the time of the award: University of Oxford, Oxford, United Kingdom Prize motivation: "for the discovery that black hole formation is a robust prediction of the general theory of relativity."

Reinhard Genzel Born: 24 March 1952, Bad Homburg vor der Höhe, Germany Afﬁliation at the time of the award: University of California, Berkeley, CA, USA, Max Planck Institute for Extraterrestrial Physics, Garching, Germany Prize motivation: "for the discovery of a supermassive compact object at the centre of our galaxy." Roger Penrose Born: 8 August 1931, Colchester, United Kingdom Afﬁliation at the time of the award: University of Oxford, Oxford, United Kingdom Prize motivation: "for the discovery that black hole formation is a robust prediction of the general theory of relativity."

Andrea Ghez Born: 16 June 1965, New York, NY, USA Afﬁliation at the time of the award: University of California, Los Angeles, CA, USA Prize motivation: "for the discovery of a supermassive compact object at the centre of our galaxy.” Roger Penrose Born: 8 August 1931, Colchester, United Kingdom Afﬁliation at the time of the award: University of Oxford, Oxford, United Kingdom Prize motivation: "for the discovery that black hole formation is a robust prediction of the general theory of relativity."

Dark Star

John Michell (1783)

If there should really exist in nature any bodies, whose density is not less than that of the sun, and whose diameters are more than 500 times the diameter of the sun, since their light could not arrive at us; or if there should exist any other bodies of a somewhat smaller size, which are not naturally luminous; of the existence of bodies under either of these circumstances, we could have no information from sight; yet, if any other luminous bodies should happen to revolve about them we might still perhaps from the motions of these revolving bodies infer the existence of the central ones with some degree of probability, as this might afford a clue to some of the apparent irregularities of the revolving bodies, which would not be easily explicable on any other hypothesis; but as the consequences of such a supposition are very obvious, and the consideration of them somewhat beside my present purpose, I shall not prosecute them any further. Black Holes Early History

Dark Star

John Michell (1783)

Dark Star

Pierre-Simon Laplace John Michell (1783) (1796) If there should really exist in nature any bodies, whose density is not less than that of the sun, and whose diameters are more than 500 times the diameter of the sun, since their light could not arrive at us; or if there should exist any other bodies of a somewhat smaller size, which are not naturally luminous; of the existence of bodies under either of these circumstances, we could have no information from sight; yet, if any other luminous bodies should happen to revolve about them we might still perhaps from the motions of these revolving bodies infer the existence of the central ones with some degree of probability, as this might afford a clue to some of the apparent irregularities of the revolving bodies, which would not be easily explicable on any other hypothesis; but as the consequences of such a supposition are very obvious, and the consideration of them somewhat beside my present purpose, I shall not prosecute them any further. Black Holes Early History

Dark Star

Pierre-Simon Laplace John Michell (1783) (1796) If there should really exist in nature any bodies, whose density is not less than that of the sun, and whose diameters are more than 500 times the diameter of the sun, since their light could not arrive at us; or if there should exist any other bodies of a somewhat smaller size, which are not naturally luminous; of the existence of bodies under either of these circumstances, we could have no information from sight; yet, if any other luminous bodies 1 2 GmM should happen to revolve about them we might still perhaps from the motions of these mv − = 0 revolving bodies infer the existence of the central ones with some degree of probability, as this might afford a clue to some of the apparent irregularities of the revolving bodies, which 2 r would not be easily explicable on any other hypothesis; but as the consequences of such a supposition are very obvious, and the consideration of them somewhat beside my present purpose, I shall not prosecute them any further. Black Holes Early History

Dark Star

Pierre-Simon Laplace John Michell (1783) (1796) If there should really exist in nature any bodies, whose density is not less than that of the sun, and whose diameters are more than 500 times the diameter of the sun, since their light could not arrive at us; or if there should exist any other bodies of a somewhat smaller size, which are not naturally luminous; of the existence of bodies under either of these circumstances, we could have no information from sight; yet, if any other luminous bodies 1 2 GmM should happen to revolve about them we might still perhaps from the motions of these mv − = 0 revolving bodies infer the existence of the central ones with some degree of probability, as this might afford a clue to some of the apparent irregularities of the revolving bodies, which 2 r would not be easily explicable on any other hypothesis; but as the consequences of such a supposition are very obvious, and the consideration of them somewhat beside my present light cannot escape from purpose, I shall not prosecute them any further. 2GM r < r = g c2 2GM M r = ≃ 2.95 km g 2 c MSun 2GM M r = ≃ 2.95 km g 2 c MSun

For our galactic centre Sgr A* observed by Genzel and Ghez

6 MSgrA* ≃ 4 × 10 MSun 2GM M r = ≃ 2.95 km g 2 c MSun

For our galactic centre Sgr A* observed by Genzel and Ghez

6 MSgrA* ≃ 4 × 10 MSun

7 rg ≃ 10 km 2GM M r = ≃ 2.95 km g 2 c MSun

For our galactic centre Sgr A* observed by Genzel and Ghez

6 MSgrA* ≃ 4 × 10 MSun

7 rg ≃ 10 km

Distance to the Sun 15 × 107 km Birth of General Relativity (1915)

From Light Front

c2dt2 − dx2 − dy2 − dz2 = 0

2 μ ν ds = gμν (x) dx dx

To General interval / curved spacetime metric Birth of General Relativity (1915)

From Light Front

+ c2dt2 − dx2 − dy2 − dz2 = 0

2 μ ν ds = gμν (x) dx dx

To General interval / curved spacetime metric Birth of General Relativity (1915)

From Light Front

+ c2dt2 − dx2 − dy2 − dz2 = 0

2 μ ν ds = gμν (x) dx dx

To General interval / curved spacetime metric 1 R − g R = 8πG T μν 2 μν μν Birth of General Relativity (1915)

From Light Front

+ c2dt2 − dx2 − dy2 − dz2 = 0

2 μ ν ds = gμν (x) dx dx

To General interval / curved spacetime metric 1 R − g R = 8πG T μν 2 μν μν Ricci tensor Birth of General Relativity (1915)

From Light Front

+ c2dt2 − dx2 − dy2 − dz2 = 0

2 μ ν ds = gμν (x) dx dx

To General interval / curved spacetime metric 1 R − g R = 8πG T μν 2 μν μν Energy-Momentum (Stress) Ricci tensor Tensor for matter ﬁelds Birth of General Relativity (1915)

From Light Front

+ c2dt2 − dx2 − dy2 − dz2 = 0

2 μ ν ds = gμν (x) dx dx

To General interval / curved spacetime metric t

1 Future R − g R = 8πG T μν 2 μν μν

Energy-Momentum (Stress) Past Ricci tensor Tensor for matter ﬁelds Schwarzschild Solution

−1 rg rg ds2 = 1 − c2dt2 − 1 − dr2 − r2 (dθ2 + sin2 θ dϕ2) ( r ) ( r )

Karl Schwarzschild (1916) Schwarzschild Solution

−1 rg rg ds2 = 1 − c2dt2 − 1 − dr2 − r2 (dθ2 + sin2 θ dϕ2) ( r ) ( r ) t time of the observer at inﬁnity

Karl Schwarzschild (1916) Schwarzschild Solution

−1 rg rg ds2 = 1 − c2dt2 − 1 − dr2 − r2 (dθ2 + sin2 θ dϕ2) ( r ) ( r ) t time of the observer at inﬁnity r radius

Karl Schwarzschild (1916) Schwarzschild Solution

−1 rg rg ds2 = 1 − c2dt2 − 1 − dr2 − r2 (dθ2 + sin2 θ dϕ2) ( r ) ( r ) t time of the observer at inﬁnity r radius spherical symmetry

Karl Schwarzschild (1916) Schwarzschild Solution

−1 rg rg ds2 = 1 − c2dt2 − 1 − dr2 − r2 (dθ2 + sin2 θ dϕ2) ( r ) ( r ) t time of the observer at inﬁnity r radius spherical symmetry 2GM Schwarzschild radius r = Karl Schwarzschild g c2 (1916) Schwarzschild Solution

−1 rg rg ds2 = 1 − c2dt2 − 1 − dr2 − r2 (dθ2 + sin2 θ dϕ2) ( r ) ( r ) t time of the observer at inﬁnity r radius spherical symmetry 2GM Schwarzschild radius r = Karl Schwarzschild g c2 (1916) “Static solution” i gμν(t, x ) Schwarzschild Solution

Empty Space except of Tμν = 0 r = 0 Schwarzschild Solution

Empty Space except of Tμν = 0 r = 0

For r → ∞ Minkowski / ﬂat spacetime in spherical coordinates Schwarzschild Solution

Empty Space except of Tμν = 0 r = 0

For r → ∞ Minkowski / ﬂat spacetime in spherical coordinates

ds2 = c2dt2 − dr2 − r2 (dθ2 + sin2 θ dϕ2) Schwarzschild Solution

Empty Space except of Tμν = 0 r = 0

For r → ∞ Minkowski / ﬂat spacetime in spherical coordinates

ds2 = c2dt2 − dr2 − r2 (dθ2 + sin2 θ dϕ2)

r 2 μναβ 3 g Curvature invariant R R = μναβ r4 ( r ) Schwarzschild Solution

Empty Space except of Tμν = 0 r = 0

For r → ∞ Minkowski / ﬂat spacetime in spherical coordinates

ds2 = c2dt2 − dr2 − r2 (dθ2 + sin2 θ dϕ2)

r 2 μναβ 3 g Curvature invariant R R = μναβ r4 ( r ) c = 1 Flip the sign −1 rg rg ds2 = 1 − dt2 − 1 − dr2 − r2 (dθ2 + sin2 θ dϕ2) ( r ) ( r )

Radial coordinate r is not spacelike inside the Schwarzschild radius! Time coordinate t is not spacelike inside the Schwarzschild radius! Flip the sign −1 rg rg ds2 = 1 − dt2 − 1 − dr2 − r2 (dθ2 + sin2 θ dϕ2) ( r ) ( r )

Radial coordinate r is not spacelike inside the Schwarzschild radius! Time coordinate t is not spacelike inside the Schwarzschild radius!

r > rg t

Future

Past Flip the sign −1 rg rg ds2 = 1 − dt2 − 1 − dr2 − r2 (dθ2 + sin2 θ dϕ2) ( r ) ( r )

Radial coordinate r is not spacelike inside the Schwarzschild radius! Time coordinate t is not spacelike inside the Schwarzschild radius!

Schwarzschild radius

rg = 2GM r > rg t

Future

Past d s 2 = (

r 1 − r r g

Future ) Radial coordinate r is not spacelike inside the Schwarzschild radius! radius! the Schwarzschild inside r is not spacelike coordinate Radial r Time radius! the Schwarzschild inside t is not spacelike coordinate d t < 2 Flip the sign the Flip − ( r g 1 Past − r r g ) − r Schwarzschild radius Schwarzschild 1 g d = r 2 − 2 G r 2 M ( d θ 2 + sin r 2 Future Past θ t > d ϕ 2 r ) g R The END… μ ν r α β R = d s μ 2 ν 0 α = β ( =

r 1 r 3 − 4 r ( r g )

r Future r Radial coordinate r is not spacelike inside the Schwarzschild radius! radius! the Schwarzschild inside r is not spacelike coordinate Radial r g Time radius! the Schwarzschild inside t is not spacelike coordinate d ) t < 2 2 Flip the sign the Flip − ( r g 1 Past − r r g ) − r Schwarzschild radius Schwarzschild 1 g d = r 2 − 2 G r 2 M ( d θ 2 + sin r 2 Future Past θ t > d ϕ 2 r ) g End of Time! R The END… μ ν r α β R = d s μ 2 ν 0 α = β ( =

r 1 r 3 − 4 r ( r g )

For r < rg −1 rg rg ds2 = 1 − dt2 − 1 − dr2 − r2 (dθ2 + sin2 θ dϕ2) ( r ) ( r )

Homogeneous, anisotropic time-dependent metric A whole (anisotropic) Universe Inside!

For r < rg −1 rg rg ds2 = 1 − dt2 − 1 − dr2 − r2 (dθ2 + sin2 θ dϕ2) ( r ) ( r ) (t, r) ⟷ (t˜, r˜)

Homogeneous, anisotropic time-dependent metric A whole (anisotropic) Universe Inside!

For r < rg −1 rg rg ds2 = 1 − dt2 − 1 − dr2 − r2 (dθ2 + sin2 θ dϕ2) ( r ) ( r ) (t, r) ⟷ (t˜, r˜) t˜ = − r r˜ = t

Homogeneous, anisotropic time-dependent metric A whole (anisotropic) Universe Inside!

For r < rg −1 rg rg ds2 = 1 − dt2 − 1 − dr2 − r2 (dθ2 + sin2 θ dϕ2) ( r ) ( r ) (t, r) ⟷ (t˜, r˜)

t˜ = − r −rg < t˜ < 0 r˜ = t −∞ < r˜ < + ∞

Homogeneous, anisotropic time-dependent metric A whole (anisotropic) Universe Inside!

For r < rg −1 rg rg ds2 = 1 − dt2 − 1 − dr2 − r2 (dθ2 + sin2 θ dϕ2) ( r ) ( r ) (t, r) ⟷ (t˜, r˜)

t˜ = − r −rg < t˜ < 0 r˜ = t −∞ < r˜ < + ∞

Homogeneous, anisotropic time-dependent metric −1 rg rg ds2 = − 1 dt˜2 − − 1 dr˜2 − t˜2 (dθ2 + sin2 θ dϕ2) t˜ t˜ A whole (anisotropic) Universe Inside!

For r < rg −1 rg rg ds2 = 1 − dt2 − 1 − dr2 − r2 (dθ2 + sin2 θ dϕ2) ( r ) ( r ) (t, r) ⟷ (t˜, r˜)

t˜ = − r −rg < t˜ < 0 r˜ = t −∞ < r˜ < + ∞

Homogeneous, anisotropic time-dependent metric −1 rg rg ds2 = − 1 dt˜2 − − 1 dr˜2 − t˜2 (dθ2 + sin2 θ dϕ2) t˜ t˜

Stretching Contraction A spatially infinite Universe Inside!

−1 rg rg ds2 = − 1 dt˜2 − − 1 dr˜2 − t˜2 (dθ2 + sin2 θ dϕ2) t˜ t˜

−rg < t˜ < 0 −∞ < r˜ < + ∞ A spatially infinite Universe Inside!

−1 rg rg ds2 = − 1 dt˜2 − − 1 dr˜2 − t˜2 (dθ2 + sin2 θ dϕ2) t˜ t˜

−rg < t˜ < 0 −∞ < r˜ < + ∞

rg dx = − 1dr˜ dy = t˜ dθ dz = t˜ sin θ dϕ t˜ A spatially infinite Universe Inside!

−1 rg rg ds2 = − 1 dt˜2 − − 1 dr˜2 − t˜2 (dθ2 + sin2 θ dϕ2) t˜ t˜

−rg < t˜ < 0 −∞ < r˜ < + ∞

rg dx = − 1dr˜ dy = t˜ dθ dz = t˜ sin θ dϕ t˜

Inﬁnite spatial volume at t˜ = const:

+∞ rg V (t˜) = dx dy dz = − 1dr˜ ( t˜ dθ) ( t˜ sin θ dϕ) ∫ ∫−∞ t˜ ∫ A spatially infinite Universe Inside!

−1 rg rg ds2 = − 1 dt˜2 − − 1 dr˜2 − t˜2 (dθ2 + sin2 θ dϕ2) t˜ t˜

−rg < t˜ < 0 −∞ < r˜ < + ∞

rg dx = − 1dr˜ dy = t˜ dθ dz = t˜ sin θ dϕ t˜

Inﬁnite spatial volume at t˜ = const:

+∞ rg V (t˜) = dx dy dz = − 1dr˜ ( t˜ dθ) ( t˜ sin θ dϕ) ∫ ∫−∞ t˜ ∫

Finite life time −1 0 0 rg π T = dt˜ g = dt˜ − 1 = r end ∫ 00 ∫ g −rg −rg t˜ 2 Smooth transition through rg

2 dsMinkowski = du dv u = t − r , v = t + r Outgoing Ingoing Smooth transition through rg

2 dsMinkowski = du dv u = t − r , v = t + r Outgoing Ingoing For Schwarzschild rg = 2GM rg ds2 = 1 − dv2 − 2drdv − r2 (dθ2 + sin2 θ dϕ2) ( r ) r v = t + r + rg log − 1 rg

Finkelstein (1959), “Eddington-Finkelstein” coordinates Penrose singularity theorem θ Expansion

dθ 1 = − θ2 − σ2 − R kμkν dλ 2 μν

Trapped surface: closed two-surface S with the property that for both ingoing and outgoing congruences of null geodesics orthogonal to S, the expansion is negative everywhere on S.

Trapped surface

G=1 Cauchy surface Penrose singularity theorem Penrose singularity theorem i) 4 is a nonsingular (+,-,-,-) Riemannian manifold for which the null half- M+ cones form too separate systems (“past” and “future”) Penrose singularity theorem i) 4 is a nonsingular (+,-,-,-) Riemannian manifold for which the null half- M+ cones form too separate systems (“past” and “future”) ii) Every null geodesic in 4 can be extended into the future to arbitrary large M+ affine parameter values (null completeness) Penrose singularity theorem i) 4 is a nonsingular (+,-,-,-) Riemannian manifold for which the null half- M+ cones form too separate systems (“past” and “future”) ii) Every null geodesic in 4 can be extended into the future to arbitrary large M+ affine parameter values (null completeness) iii) Every timelike of null geodesic in 4 can be extended into the past until it M+ meets (Cauchy hypersurface condition) C3 Penrose singularity theorem i) 4 is a nonsingular (+,-,-,-) Riemannian manifold for which the null half- M+ cones form too separate systems (“past” and “future”) ii) Every null geodesic in 4 can be extended into the future to arbitrary large M+ affine parameter values (null completeness) iii) Every timelike of null geodesic in 4 can be extended into the past until it M+ meets (Cauchy hypersurface condition) C3 iv) At every point of 4 all time like vectors satisfy M+ 1 R − g R − λg tμtν = 8πG T tμtν ≥ 0 - non-negativeness of local ( μν 2 μν μν) μν energy Penrose singularity theorem i) 4 is a nonsingular (+,-,-,-) Riemannian manifold for which the null half- M+ cones form too separate systems (“past” and “future”) ii) Every null geodesic in 4 can be extended into the future to arbitrary large M+ affine parameter values (null completeness) iii) Every timelike of null geodesic in 4 can be extended into the past until it M+ meets (Cauchy hypersurface condition) C3 iv) At every point of 4 all time like vectors satisfy M+ 1 R − g R − λg tμtν = 8πG T tμtν ≥ 0 - non-negativeness of local ( μν 2 μν μν) μν energy v) There exists a trapped surface T2 Penrose singularity theorem

i) 4 is a nonsingular (+,-,-,-) Riemannian manifold for which the null half- M+ cones form too separate systems (“past” and “future”)

ii) Every null geodesic in 4 can be extended into the future to arbitrary large M+ affine parameter values (null completeness)