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A Time Machine Allowing Travel to the Past by Free Fall A time machine allowing travel to the past by free fall Livio Pizzocchero Dipartimento di Matematica, Universit`a degli Studi di Milano and Istituto Nazionale di Fisica Nucleare, Sezione di Milano TMF 2019, Torino, September 2019 Based on joint work with Davide Fermi (Classical and Quantum Gravity 35 (2018), 165003, 42 pp) Livio Pizzocchero Free fall into the past 1 / 24 Background and motivations • Time machines := spacetimes with closed timelike curves (CTCs). Literature on this topic, and on the (somehow related) subject of spacetimes with superluminal trajectories: Van Stockum (1938); G¨odel (1949); Kerr (1963); Misner (1963); Tipler (1974); Morris, Thorne and Yurtsever (1988); Gott (1991); Alcubierre (1994); Ori (1993,1995,2007), Ori and Soen (1994); Li (1994); Low (1995); Everett, Everett and Roman (1996-97); Krasnikov (1998); Tippet and Tsang (2017); Mallary, Khanna and Price (2017); Fermi and P (2018); Fermi (2018); Krasnikov (2018); ... Analysis of problematic aspects: Friedman, Morris, Novikov, Echeverria, Klinkhammer, Thorne, Yurtsever (1990); Deutsch (1991); Deser, Jackiw and ’t Hooft (1992); Hawking (1992); Ford and Roman (1995); Krasnikov (1998,2014,2018); Olum (1998); Visser (2003); Maeda, Ishibashi and Narita (1998), Lobo (2008); ... Livio Pizzocchero Free fall into the past 2 / 24 • Typical issues affecting spacetimes with CTCs: Infinitely extended structures (dust cylinders, cosmic strings, ... ) . Naked curvature singularities, or CTCs beyond event horizons. Huge tidal accelerations experienced along CTCs . Violations of energy conditions( ECs); large, negative energy densities. I Still, some quantum systems do violate the ECs (e.g., Casimir effect) . I Ori 2007 time machine: ECs fulfilled, perhaps singularities/horizons. • Chronology protection conjecture: quantum backreactions forbid CTCs. I Our proposal for a time machine (Fermi and P, 2018): ◦ A deformation of Minkowski spacetime, flat outside a torus. ◦ No singularities. No horizons. Time orientable. ECs are violated. ◦ Symmetries ⇒ explicit computation of geodesics by quadratures. ◦ A freely falling observer (timelike geodesic) can start from the outer Minkowski region, move across the toric region and return to its initial position in the outer region at an earlier time, arbitrarily far in the past. ◦ Quantitative analysis of the above time travel including proper duration, tidal forces, energy densities (all of them can be made acceptable on a human scale) . Livio Pizzocchero Free fall into the past 3 / 24 The spacetime model (in units with c = 1) • Spacetime manifold : it is R4 = R × R3. (t, ϕ, ρ, z) := natural coordinate on R + cylindrical coordinates on R3. • Main building blocks : two concentric toriT λ, TΛ with minor radii λ, Λ and common major radius R (0 <λ< Λ < R; R = scale factor): 2 2 2 2 2 2 Tλ := (ρ−R) + z = λ . TΛ := (ρ−R) + z =Λ ; • We use the shape function DETAILS p k X (ρ, z) := H (ρ/R −1)2 + (z/R)2 , H∈C ([0, +∞)) as below (26k 6∞). χ = 0 outside TΛ ; χ = 1 inside Tλ ; 0 < χ < 1 between Tλ and TΛ . / R / R 1 0.5 y 0.5 1 Livio Pizzocchero Free fall into the past 4 / 24 I We now postulate for R4 the line element (a, b > 0 dimensionless parameters) 2 2 ds2 := −(1−X )dt + a R X dϕ +(1−X )ρ dϕ − b X dt +dρ2 +dz2. 2 2 2 2 2 2 • Outside TΛ (X = 0) : ds = − dt + ρ dϕ + dρ + dz . Usual, flat Minkowski metric in cylindrical coordinates; t a timelike coord. 2 2 2 2 2 2 2 2 • Inside Tλ (X = 1) : ds := − a R dϕ + b dt + dρ + dz . A flat metric; ϕ a timelike coord.; CTCs with t, ρ, z = const., ϕ ∈ R/(2πZ). 2 k 4 I ds defines a Lorentzian metric g of class C on R (2 6 k 6 ∞), k−2 non flat between TΛ and Tλ. Riemg ∈C ; no curvature singularity . 4 I T := (R , g) is our spacetime . Spacetime region outside/insideT Λ will be called the outer Minkowski region/time machine. Livio Pizzocchero Free fall into the past 5 / 24 A tetrad. Time (and space) orientation • The explicit expression of ds2 suggests to consider the 1-forms e(0) := (1−X ) dt + a R X dϕ , e(1) := (1−X ) ρ dϕ − b X dt , e(2) := dρ , e(3) := dz , 2 2 2 2 s.t. ds2 = −e(0) +e(1) +e(2) +e(3) . (µ) µ k • The dual vector fields E(ν) s.t. he , E(ν)i = δ ν (µ, ν = 0, 1, 2, 3) are C and form an orthonormal tetrad: g(E(0), E(0)) = −1 , g(E(i), E(j)) = δij , g(E(0), E(i)) = 0 (i, j = 1, 2, 3) . I E(0) is everywhere timelike, and E(0) = ∂t in outer Minkowski region. We call E(0) the fundamental timelike vector field, and define future to be the time orientation containing E0. ⊥ I E(i) (i = 1, 2, 3) are everywhere spacelike, spanning E0 , and −1 (E(1), E(2), E(3)) = (ρ ∂ϕ, ∂ρ, ∂z ) in outer Minkowski region. ⊥ We can use (E(1), E(2), E(3)) to define the left-handed orientation of E(0). 4 ⊥ Remarks. (i) In principle each E(ν) is defined on R \{ρ = 0}, but E(0), E(3), E(0) k 4 have C extensions to the whole R . ⊥ (ii) The distribution E(0) is not involutive ⇒ @ spacelike hypersurfaces ⊥ E(0). Livio Pizzocchero Free fall into the past 6 / 24 Results on free fall motions (timelike geodesics) • Geodesics (with an affine parametrization) in our spacetime T are the solutions of Euler-Lagrange eq.s associated to the Lagrangian 1 L(ξ, ξ_) := g ξ,_ ξ_ (ξ ∈ T, ξ_ ∈ T T); 2 ξ ξ 1 h 2 2 i L(t, ϕ, ρ, z, t_, ϕ,_ ρ,_ z_) = −(1−X ) t_ + a R X ϕ_ + (1−X ) ρ ϕ_ − b X t_ +ρ _2 +z _ 2 . 2 (X = X (ρ, z)). This has the following constants of motion: I The canonical momenta associated to the coordinates t, ϕ, i.e., ∂L ∂L p := , p := , t ∂t_ ϕ ∂ϕ_ conserved since ∂L/∂t =0, ∂L/∂ϕ= 0( ∼ ∂t , ∂ϕ Killing vector fields). µ _µ I The energy E := (∂L/∂ξ )ξ − L; due to the purely kinetic form of L, 1 E(ξ, ξ_) = L(ξ, ξ_) = g ξ,_ ξ_ . 2 ξ • Free fall motions are future-directed, timelike geodesics τ 7→ ξ(τ), that we parametrize with proper time τ so that 1 1 E(ξ(τ), ξ_(τ)) = g ξ_(τ), ξ_(τ) = − , ξ_(τ) future directed. 2 ξ(τ) 2 Livio Pizzocchero Free fall into the past 7 / 24 • From now on we only consider free fall motions in the plane {z = 0} (that do exist) parametrized by proper time τ, for which _ ≡ d/dτ. • These have a Lagrangian L(t, ϕ, ρ, t_, ϕ,_ ρ_), as before with (z, z_) = (0, 0). • The canonical momenta pt := ∂L/∂t_, pϕ := ∂L/∂ϕ_ are constants of motion; we replace pt , pϕ with the dimensionless parameters 2 2 2 γ := −pt = (1−X ) −b X t_ + (a+b ρ/R)X (1−X ) R ϕ_ , pϕ 1 1 ω := = (ρ/R)2(1−X )2 − a2X 2R ϕ_ − (a+b ρ/R)X (1−X ) t_ . γ R γ γ (X ≡ X (ρ, 0) = H(|ρ/R − 1|)). In the outer Minkowski region (X = 0): γ = t_ ≡ dt/dτ ⇒ for future dir. motions γ is the Lorentz factor, γ > 1; ρ2 ω = γ R ϕ_, a sort of rescaled angular momentum . • Conversely: (ρ/R)2(1−X )2 − a2X 2 − (a+b ρ/R) X (1−X ) ω _ _ t = γ 2 ≡ t(ρ, γ, ω) , (ρ/R)(1−X )2 + a b X 2 (a+b ρ/R) X (1−X ) + (1−X )2 − b2 X 2 ω ϕ_ = γ 2 ≡ ϕ_(ρ, γ, ω) . R (ρ/R)(1−X )2 + a b X 2 2 Inside Tλ (X =1) : t_ = − γ/b (< 0 for future dir. motions from Mink. region), ϕ_ = − γ ω/(a2R) . Livio Pizzocchero Free fall into the past 8 / 24 • Let’s recall that another constant of motion is the energy E = L. • Substituting t_, ϕ_ in the expression for E gives 1 E = ρ_2 + V (ρ) , 2 γ,ω 2 2 γ2 a X +(1−X ) ω − (ρ/R)(1−X )−b X ω Vγ,ω(ρ) := 2 . 2 (ρ/R)(1−X )2 + a b X 2 This is the energy of an effective 1D conservative system: a particle of position ρ ∈ (0, +∞) with potential Vγ,ω(ρ) . γ2 R2ω2 In outer Minkowski region: (X =0): Vγ,ω(ρ) = 2 ρ2 − 1 . γ2 a2 2 Inside Tλ (X =1) : Vγ,ω(ρ) = const. = 2 a2 b2 − ω . Between Tλ and TΛ (0<X <1) : Vγ,ω depends sensibly on sign ω . • Recall we are considering motions parametrized by proper time τ, so that E = −1/2 . • Qualitative analysis: ρ(τ) is confined to a connected component of {ρ ∈ (0, +∞) | Vγ,ω(ρ) 6 −1/2}. Livio Pizzocchero Free fall into the past 9 / 24 Graphs of Vγ,ω for two choices of the parameters Vγ,ω Vγ,ω 1 R 1 R 1.5 1- /R 1+ /R 4 - / + / 1- /R 1+ /R 1- /R 1+ /R 1 2 0.5 0 ρ/R 0 ρ/R 1 2 1 2 -0.5 -2 -1 -1.5 -4 ω = 0.08 ω = − 0.08 (λ = 0.6 R, Λ = 0.8 R, a = 0.09, b = 10, k = 3, γ = 1.1) • Quantitative analysis: the conservation laws for E and pt , pϕ (or γ, ω) yield quadrature formulas for free fall motions τ 7→ t(τ), ϕ(τ), ρ(τ).
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