Optimal time travel in the Gödel universe

Transcription

1 Optimal time travel in the Gödel universe José Natário (Instituto Superior Técnico, Lisbon) Based on arxiv: Porto, January 2013

2 Outline Newtonian cosmology Rotating Newtonian universe Newton-Cartan theory The Gödel universe Total integrated acceleration Rocket theory Optimal time travel 1

3 Newtonian cosmology Choose a galaxy as the center of the universe; all other galaxies are at r = a(t)x for some x R 3. Equation of motion: r = 4π ( 3 r3 ρ(t) Λ ) r 4π r 3 ä a = 4πρ 3 + Λ 3 Could have chosen any other galaxy to be the center of the universe: r = a(t)(x x 0 ) leads to the same equation. 2

4 Rotating Newtonian universe Choose an axis Re z to be the rotation axis of the universe; all galaxies move with velocity ṙ = ωe z r. Equation of motion: ω 2 e z (e z r) = 2π ( ρ Λ ) e z (e z r) 2ω 2 = 4πρ Λ 4π Could have chosen any other parallel axis to be the rotation axis of the universe: ṙ = ωe z (r r 0 ) leads to the same equation. 3

5 Newton-Cartan theory What are inertial frames in Newtonian cosmology? Newtonian gravity does not respect the principle of equivalence must introduce Coriolis field H. Equation of motion for a free-falling particle: r = G+ṙ H 4

6 Field equations: G = H t H = 0 G = 1 2 H2 4πρ+Λ H = 0 Rotating universe in co-rotating frame: G = 0, H = 2ωe z.

7 The Gödel universe ds 2 = 1 2ω 2 { [dt 2(cosh(r) 1)dϕ] 2 +dr 2 +sinh 2 (r)dϕ 2 +dz 2 } Solution of the Einstein equations Ric 1 Rg +Λg = 8πT 2 with Λ = ω 2, T = ω2 4π U U 5

8 ds 2 = 1 2ω 2 { [dt 2(cosh(r) 1)dϕ] 2 +dr 2 +sinh 2 (r)dϕ 2 +dz 2 } Solution of the Einstein equations Ric 1 Rg +Λg = 8πT 2 with Λ = ω 2, T = ω2 4π U U

9 ds 2 = 1 2ω 2 { [dt 2(cosh(r) 1)dϕ] 2 +dr 2 +sinh 2 (r)dϕ 2 +dz 2 } Solution of the Einstein equations Ric 1 Rg +Λg = 8πT 2 with Λ = 1 2, T = 1 8π U U

10 ds 2 = 1 2ω 2 { [dt 2(cosh(r) 1)dϕ] }{{} 2 +dr 2 +sinh 2 (r)dϕ 2 +dz 2 } θ } {{ } hyperbolic plane dθ = sinh(r) dr dϕ = da

11 Caustic Closed timelike curve Timelike geodesic Null geodesic

12 t = γ dt = γ 2θ + dτ 2 +dl 2 = 2A+ γ dl v So the projection γ of a closed timelike curve must be traversed clockwise and satisfy l < 2 A and so (using the isoperimetric inequality) A > 4π, l > 4π 2 and v max > 2 2

13 Total integrated acceleration Rocket equation: m(τ 1 ) = m(τ 0 )exp ( 1 v e τ1 τ 0 a(τ)dτ ) m(τ 0) m(τ 1 ) exp(ta), TA = τ1 τ 0 a(τ)dτ 6

14 Malament (1985) proved that TA ln(2+ 5) for any CTC, and conjectured that in fact TA 2π This would mean (Gödel, 1949) m(τ 0 ) m(τ 1 ) 1012 Manchak (2011) gave a non-periodic counter-example.

15 1 n 1 n

16 Rocket theory Problem: minimize TA = τ1 τ 0 a(τ)dτ Issue: L = a = U U is not differentiable along geodesics Issue: one has to deal with instantaneous accelerations, unlike for, say, L = a 2 (elastic curves) 7

17 U U µ = ap µ U P µ = q µ + au µ (Henriques and Natário, 2012) U q µ = R µαβγ U α P β U γ P µ P µ 1 with P µ P µ = 1 if a 0 and at instantaneous accelerations If the initial/final U µ is not specified then P µ = 0 at the beginning/end

18 U U µ = ap µ U P µ = q µ + au µ U q µ = R µαβγ U α P β U γ P µ P µ 1 with P µ P µ = 1 if a 0 and at instantaneous accelerations If the initial/final U µ is not specified then P µ = 0 at the beginning/end

19 U U µ = 0 U P µ = q µ U q µ = R µαβγ U α P β U γ P µ P µ 1 with P µ P µ = 1 if a 0 and at instantaneous accelerations If the initial/final U µ is not specified then P µ = 0 at the beginning/end

20 Optimal time travel P µ = 0 at the beginning/end means that we must begin/end with a geodesic arc Start with an ansatz depending on finitely many parameters, (numerically) minimize T A with respect to those parameters 8

21 r α ε R r ψ θ x y β ϕ

22 Satisfies the minimum conditions TA TA if we make it periodic T A for Malament s conjecture

23 Is TA the minimum? Does Malament s conjecture still hold for periodic CTCs?

24 Bibliography [1] K. Gödel, A remark about the relationship between relativity theory and idealistic philosophy, in Albert Einstein: Philosopher-Scientist, Open Court (1949), [2] D. Malament, Minimal acceleration requirements for time travel in Gödel space-time, J. Math. Phys. 26 (1985), [3] J. Manchak, On efficient time travel in Gödel spacetime, Gen. Rel. Grav. 43 (2011), [4] P. Henriques and J. Natário, The rocket problem in general relativity, J. Optim. Theory Appl. 154 (2012), (arxiv: ). [5] J. Natário, Optimal time travel in the Gdel universe, Gen. Rel. Grav. 44 (2012), (arxiv: ). 9