The gravitational field at the shortest scales

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The gravitational field at the shortest scales Piero Nicolini

1 The gravitational field at the shortest scales Piero Nicolini FIAS & ITP Johann Wolfgang Goethe-Universität Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität, Frankfurt a. M. - May 24, 2012 Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

2 A physical quantity Measurement of X x M X(x) = { } = X 1 (x),..., X j (x), X j+1 (x),..., X n (x) x is a point (event) of the manifold (spacetime) M. {X 1 (x),..., X n (x)} R n X(x) k < e.g. the gravitational field A lunar gravimeter - Apollo 17 (1972) Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

The gravitational field Newtonian gravity g = F m = r d2 dt 2 = GM ˆr = Φ r 2 Gauss law for gravity g = 2 Φ = 4πGρ Solution e.g. ρ in = ρ 0 and ρ out = 0 r > R Φ(r) = G M r, g = G M r 2 r < R Φ(r) = G M 2R 3 r 2, g = G M R 3 r Earth s gravitational field Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a.

3 The gravitational field Newtonian gravity g = F m = r d2 dt 2 = GM ˆr = Φ r 2 Gauss law for gravity g = 2 Φ = 4πGρ Solution e.g. ρ in = ρ 0 and ρ out = 0 r > R Φ(r) = G M r, g = G M r 2 r < R Φ(r) = G M 2R 3 r 2, g = G M R 3 r Earth s gravitational field Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

4 Poisson/Laplace equation for gravity Poisson: point-like mass g( x) = 2 Φ( x) = 4πGMδ( x) Divergence theorem (g n) ds = ( g) dv = 4πGM δ( x)dv. S V V g(r) = G M r 2 Laplace: homogeneous equation in R 3 \ {0} div g = g = 1 r 2 r (r 2 g) = 0, r 2 g = const. g(r) = const r 2, const = GM (boundary conditions) Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

the Sun 3 the gravitational redshift of light The Pound-Rebka experiment (1959) Piero Nicolini (FIAS &

5 Einstein gravity Tensor field equations R µν 1 2 g µν R + g µν Λ = 8πG c 4 T µν Curvature R, R µν, R µ νρσ. Classical tests of GR 1 the perihelion precession of Mercury s orbit 2 the deflection of light by the Sun 3 the gravitational redshift of light The Pound-Rebka experiment (1959) Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

6 Static, spherically symmetric gravitational field Perfect fluid stress-energy tensor for the star T αβ = The outer geometry (ρ + p c 2 ) u α u β + pg αβ, ρ(r) = [Schwarzschild, Sitzungsber. Preuss. Akad. Wiss. 189 (1916)] ds 2 = { ρ 0, if r R 0, if r > R ( 1 2GM ) ( c 2 c 2 dt 2 1 2GM ) 1 ( r c 2 dr 2 r 2 dθ 2 + sin 2 θ dϕ 2) r The inner geometry [Schwarzschild, Sitzungsber. Preuss. Akad. Wiss. 424 (1916)] ( ds 2 = e 2Φ c 2 dt 2 1 2Gm(r) ) 1 ( c 2 dr 2 r 2 dθ 2 + sin 2 θ dϕ 2) r r m(r) = 4π ρ(r )r 2 dr = πr 3 ρ 0 Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

The Schwarzschild black hole Laplace: vacuum solution i.e. R µν = 0 Curvature singularity The mathematically rigorous solution may not be physically rigorous - R.P.

7 The Schwarzschild black hole Laplace: vacuum solution i.e. R µν = 0 Curvature singularity The mathematically rigorous solution may not be physically rigorous - R.P. Feynman in Feynman & Hibbs, Quantum mechanics and path integrals p. 94 (McGraw-Hill, 1965) R αβγδ R αβγδ = 48G2 M 2 c 4 r 6. Poisson: [Balasin & Nachbagauer, CQG 10, 2271 (1993)] T (x) = Mδ (3) (x) Karl Schwarzschild ( ) ( dt t + dr r 1 2 dθ θ 1 ) 2 dϕ ϕ Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

The Hawking radiation The vacuum state 0 is ill defined in curved space = Black hole thermal emission d 2 N dtdω = 1 2π 1 e ω k BT ± 1 T = c3 Gk B 1 8πM how important are quantum effects for

8 The Hawking radiation The vacuum state 0 is ill defined in curved space = Black hole thermal emission d 2 N dtdω = 1 2π 1 e ω k BT ± 1 T = c3 Gk B 1 8πM how important are quantum effects for black holes r H m = M kg Hawking emission Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

9 The case of mini black holes Particle Black hole λ Mc r H GM c 2 Particle black hole λ r H c M M P G G r H L P Evaporating black holes c 3 L P m < r H < m M P 10 8 kg < M < kg Gravity self completeness Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

Quantum gravity S = 1 2πα String theory d 2

Loop quantum gravity A j [Σ] = L 2 P j i (j

= Θ ij Asymptotically safe gravity G 0 G AS

10 Quantum gravity S = 1 2πα String theory d 2 Σ (Ẋ X ) 2 (Ẋ)2 (X ) 2. Loop quantum gravity A j [Σ] = L 2 P j i (j i + 1) i Noncommutative geometry [x i, x j ] = Θ ij Asymptotically safe gravity G 0 G AS (p) = 1 + αg 0 p 2 Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

11 Effective quantum gravity equations A simple example GR: G µ ν = 8 πt µ ν T 0 0 = M δ(3) ( x ) l 0 l 0 l 0 QG: G µ ν l = 8 π T µ ν l T 0 0 Energy density M i) ((4πl ) 2 δ (3) ( x) ρ l ( x) = M ii) π 2 ii)... Stress-energy tensor l = ρ l ( x ) ( exp 3/2 l ( x 2 + l 2 ) 2 x 2 4l 2 ) T (x) = {ρ l (dt t ) + P r (dr r ) + P (dθ θ + dϕ ϕ )} Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

12 The static, spherically symmetric case Line element ds 2 = e 2Φ( r ) ( 1 2Gm ( r ) /r ) dt 2 ( 1 2Gm ( r ) /r ) 1 dr 2 r 2 dω 2 Fluid-like gravity field equations dm dr dp r dr Equation of state Gaussian profile solution = 4π r 2 ρ l (r) m(r) = 4π r 0 ρ(r )r 2 dr = G m(r) + 4π r 3 P r r(r 2Gm(r)) ( ρ l + P r ) + 2 r ( P P r ) P r = ρ l Φ(r) = 0 [PN, Smailagic & Spallucci, PLB 632, 547 (2006)], [Modesto, Moffat & PN, PLB 695, 397 (2011)] m(r) = M γ ( 3/2, r 2 /4l ) ( ) 2 M 1 r l π = exp( r 2 /4l 2 ) ( ) Γ (3/2) 1 r 3 M l 3 6 π with γ ( 3/2, r 2 /4l 2 ) r 2 /4l 2 0 dt t 1/2 e t r l r l Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

13 A regular static black hole g 1 rr vs r, for various values of M/l. M > M 0 two horizons M < M 0 no horizon M = M l/g one degenerate horizon The Ricci scalar near the origin is R ( 0 ) = 4M π l 3 The curvature is constant and positive (desitter geometry). Energy conditions If M < M 0 no BH and no naked singularity (mini-gravastar?) If M M 0 inner horizon origin outer horizon 2M Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

14 Improved thermodynamics new temperature profile ( T l = c 1 r ) + 3 e r 2 + /4l2 4πk B r + 4l 2 γ(3/2; r+/4l 2 2 ) transition to a positive heat capacity phase ( ) ( ) 1 ds dtl C(r + ) = T l, dr + dr + ds = 1 M dr + T l r + SCRAM phase (evaporation switching off) remnant formation l for the BH is like for the black body Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

15 The measure of the gravitational field at short scales deviations of Newton s law Φ = G M ) (1 + αe r/r r torsion balance experiments [Hoyle et al., PRL 86, 1418 (2001)] R < 44 µm Scale drawing of the torsion pendulum and rotating attractor. Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

16 What if... Additional spatial dimensions (ADD scenario) [Arkani-Hamed, Dimopoulos, Dvali. Phys.Lett.B429: ,1998] SM fields brane gravity bulk R must be SMALL... Φ(r) = k+1 c 1 k M M 2+k R k r as r R. R must be LARGE... c M MP 2 r M ( LP R ) k k+2 MP 1 TeV Flux lines with one extradimension = k 2 Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

17 Higher dimensional black holes Higher-dimensional Schwarzschild geometry ds(k+4) 2 = F (r)dt 2 + F 1 (r)dr 2 + r 2 dω 2 3+k ( rh ) k+1 F(r) = 1 r ( ) [ 1 8Γ( k+3 2 r H = ) ] 1 k+1 ( ) 1 M k+1 π M c (k + 2) M for M M 1 TeV r H 10 4 fermi validity of spherical symmetry r H R black hole formation r H λ black hole factory T GeV σ BH πr 2 H 1nb N BH/s = σ BH L LHC 10 Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

Data analysis at the 95 % CL they rule out BHs with M = 3.5 4.5 TeV for M up to 3 TeV. [CMS collaboration, Phys.Lett.

6396 [hep-ex] ] Results are questioned: semiclassical approximation not valid [Park, Phys.Lett.

18 Data analysis at the 95 % CL they rule out BHs with M = TeV for M up to 3 TeV. [CMS collaboration, Phys.Lett.B697:434,2011] The CMS detector Update, February 2012 M > 5.3 TeV [CMS collaboration, arxiv: [hep-ex] ] Results are questioned: semiclassical approximation not valid [Park, Phys.Lett.B701:587,2011] The CMS control room Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

19 Beyond the semiclassical approximation Higher dimensional non-singular black hole Temperature ds(4+k) 2 = F (r)dt 2 + F 1 (r)dr 2 + r 2 dω 2 3+k ( rh ) k+1 F(r) = 1 r ( ) [ 1 8γ( k+3 2 r h =, ] r 2 1 k+1 ( ) 1 ) 4l M 2 k+1 π M c (k + 2) M ds 2 = F (r) dt 2 + F 1 (r) dr 2 + r 2 dω 2 3+k ds 2 E = F (r) dτ 2 + F 1 (r) dr 2 + r 2 dω 2 3+k. T l = k + 1 4πr h 1 2 k + 1 ( rh 2l ) k+3 e r 2 h /4l2 γ ( ). k+3 2, r h 2 4l 2 Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

20 Evaporation and grey body factors Maximum temperatures k l [GeV] T [GeV] T max negligible quantum back reaction longer decaying times no Doomsday scenario t (10 26 ) s [Bleicher & PN, J.Phys.Conf.Ser. 237 (2010) ] Particle spectrum where g (s) jk dn (s) dt = j N j g (s) jn (ω) exp ( ω/k B T l ) ± 1 dω (2π) is the grey body factor and N j is the j-state multiplicity. Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

21 Scalar emission Brane emission with varying M Bulk emission with varying M Energy fluxes Energy fluxes Schwarzschild black hole (grey) with the same mass as the non-singular BH at Tl max (purple). dashed curves correspond to additional quantum gravity corrections on matter fields the units are l = 1 Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

22 Scalar emission Brane emission with varying k Bulk emission with varying k Energy fluxes Energy fluxes Non-singular BHs with maximum temperature T max l dashed curves correspond to additional quantum gravity corrections on matter fields the units are l = 1 Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

23 Bulk/brane emission Figure: Bulk/brane emission The units are l = 1. Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

24 The signature Total fluxes k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 Particles Power brane emission (normalized to 4d) k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 Particles Power bulk emission (normalized to 4d) Bulk/brane emission decreases with increasing k. e.g. for k = 7 bulk/brane 0.02% (in marked contrast to the usual result 93%) emission dominated by soft particles mostly on the brane. [PN & Winstanley JHEP 1111, 075 (2011)] Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

Extragalactic neutrino oscillations Transition probability (two-flavour model) ( m P(ν α ν β ) = sin 2 (2θ) sin 2 2 ) L 4E Modified transition probability ( m P l (ν α ν β ) = sin 2 (2θ) sin 2 2 ) L

25 Extragalactic neutrino oscillations Transition probability (two-flavour model) ( m P(ν α ν β ) = sin 2 (2θ) sin 2 2 ) L 4E Modified transition probability ( m P l (ν α ν β ) = sin 2 (2θ) sin 2 2 ) L 2 4E e l E 2 Probability difference p P(α β) P l (α β) Extragalactic high energy neutrinos would not oscillate! [Sprenger, PN & Bleicher, CQG 28, (2011)] p for l m. Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

Phys, 541, 1(2000)] 2 3 Holographic model [t Hooft, in Salamfest p.

26 Back to 4d: an astronomical test Modified dispersion relation p 2 = E 2 [1 ± a 0 (E/E P ) α ], α = Farthest quasar image degradation [Tamburini et al., A& A 533, A71 (2011)] 1 2 Random walk model [Amelino Camelia, Lect.Not.Phys, 541, 1(2000)] 2 3 Holographic model [t Hooft, in Salamfest p. 284(World Scientific 2003)] 1 Standard model [Smolin, hep th/ ] 2 Gaussian smearing model (a 0 = 2/3) Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

27 Back to 4d: detection of the Hawking radiation Schwarzschild T = c3 Gk B 1 8πM 10 7 ( M M ) K t ( M M ) 3 Gyr primordial BHs t Gyr M kg, r h m, T K EMP 1 GHz [Rees, Nature 266, 333 (1977)] [Blandford, MNRAS 181, 489 (1977)] the scenario needs revision! quantum mechanical desitter instability suppressed nucleation of Planck size regular BHs [Mann & PN, PRD 84, (2011)] new temperature profile Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

28 Spontaneous dimensional reduction at short scales? Dream look for a mechanism of dimensional reduction to 2D to have gravity a power-counting renormalizable theory [G 2 ] = (length) 0 [t Hooft, in Salamfest p. 284 (World Scientific 1993)] [Carlip, in 25th Max Born Symposium (2009)] the quantum spacetime is like a fractal D D s A dream come true [Modesto & PN, PRD 81, (2010)] D s = s s + l 2 D for the here presented regular spacetimes In a) a Cantor set In b) holed structure with self-similarity. Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

29 Phase transitions via gauge-gravity duality Witten: Hawking-Page p.t. in d + 1 dimension = deconfinement in d dimensions for N c regular BHs with AdS background Λ = 1/L 2 Hawking-Page Van der Waals l 0 l is the excluded volume b temperature parameter q 2l/L q < q 1st order p. t. small to large BHs q = q = critical point q > q analytical cross over q = N f /N c deconfinement/crossover [PN & Torrieri, JHEP 1108, 097 (2011)] Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

30 Conclusions Summary: we easily passed all checks required by quantum gravity singularity avoidance black hole thermodynamics stability spontaneous dimensional reduction at the Planck scale to 2D. distinctive signatures Theoretical developments new effective formulations of quantum gravity LHC phenomenology and quantum gravity at the terascale primordial and astrophysical black holes further developments about nuclear matter phase transition via gauge-gravity duality [Kaminski, Sprenger, Torrieri & PN, in progress] [Frassino & PN, in progress]... Observational tests Gravitational waves horizon imaging of black hole candidates black hole formation in particle detectors... Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

31 Conclusions Summary: we easily passed all checks required by quantum gravity singularity avoidance black hole thermodynamics stability spontaneous dimensional reduction at the Planck scale to 2D. distinctive signatures Theoretical developments new effective formulations of quantum gravity LHC phenomenology and quantum gravity at the terascale primordial and astrophysical black holes further developments about nuclear matter phase transition via gauge-gravity duality [Kaminski, Sprenger, Torrieri & PN, in progress] [Frassino & PN, in progress]... Observational tests Gravitational waves horizon imaging of black hole candidates black hole formation in particle detectors... Dankeschön! Piero Nicolini (FIAS & ITP, JWGU, Frankfurt a. M.) The gravitational field at the shortest scales ITP, JWGU, Franfurt a. M., / 31

κ = f (r 0 ) k µ µ k ν = κk ν (5)

κ = f (r 0 ) k µ µ k ν = κk ν (5) 1. Horizon regularity and surface gravity Consider a static, spherically symmetric metric of the form where f(r) vanishes at r = r 0 linearly, and g(r 0 ) 0. Show that near r = r 0 the metric is approximately