Quantum Entanglement in Pleba´nski-Demia´nski Spacetimes Hugo Garc´ıa-Compe´an Cinvestav FCFM, UANL II Taller de Gravitaci´on,F´ısica de Altas Energ´ıasy Cosmolog´ıa ICF, UNAM, Cuernavaca Morelos Agosto 2014 Work with Felipe Robledo-Padilla Class. Quantum Grav. 30 (2013) 235012 August 3, 2014 I. Introduction II. Pleba´nski-Demia´nskiSpacetime III. Frame-dragging IV. Spin precession V. EPR correlation VI. Spin precession angle in Expanding and Twisting Pleba´nski-Demia´nskiBlack Hole I. Introduction For many years quantum states of matter in a classical gravitational background have been of great interest in physical models. Examples: I Neutron interferometry in laboratories on the Earth. It captures the effects of the gravitational field into quantum phases associated to the possible trajectories of a beam of neutrons, following paths with different intensity of the gravitational field. (Colella et al 1975) I Another instance of the description of quantum states of matter in classical gravitational fields is Hawking's radiation describing the process of black hole evaporation. This process involves relativistic quantum particles and uses quantum field theory in curved spacetimes. II. Pleba´nski-Demia´nski Spacetime The metric " 1 D 2 ρ2 ds2 = − dt − (a sin2 θ + 2l(1 − cos θ))dφ + dr 2 Ω2 ρ2 D # P 2 sin2 θ + adt − (r 2 + (a + l)2)dφ + ρ2 dθ2 ; ρ2 P (1) with ρ2 = r 2 + (l + a cos θ)2; α Ω = 1 − (l + a cos θ)r; ! 2 2 P = sin θ(1 − a3 cos θ − a4 cos θ); α α2 Λ D = (κ + e2 + g 2) − 2mr + "r 2 − 2n r 3 − κ + r 4; ! !2 3 (2) and where α α2 Λ a = 2a m − 4al (κ + e2 + g 2) − 4 al; 3 ! !2 3 α2 Λ a = −a2 (κ + e2 + g 2) − a2; 4 !2 3 κ α α2 Λ " = + 4l m − (a2 + 3l 2) (κ + e2 + g 2) + ; a2 − l 2 ! !2 3 κl α α2 Λ n = − (a2 − l 2) m + (a2 − l 2)l (κ + e2 + g 2) + ; a2 − l 2 ! !2 3 1 + 2l α m − 3l 2 α2 (e2 + g 2) − l 2Λ κ = ! !2 : 1 2 α2 a2−l2 + 3l !2 (3) paremeters: α; !; m; e; g; Λ; `; a: Equation (1) can be represented by the line element 2 2 2 2 2 ds = g00dt + 2g03dtdφ + g11dr + g22dθ + g33dφ ; (4) which has a non-diagonal element that represents the axial symmetry and the metric coefficients are: −D + Pa2 g = ; 00 Ω2ρ2 1 D P g dx i = (a sin2 θ + 2l(1 − cos θ)) − a(r 2 + (a + l)2) dφ, 0i Ω2 ρ2 ρ2 ρ2 sin2 θ 1 D g dx i dx j = dr 2 + ρ2 dθ2 + − (a sin2 θ + 2l(1 − cos θ))2 ij Ω2D Ω2P Ω2 ρ2 P + (r 2 + (a + l)2)2 dφ2: ρ2 (5) In order to describe the motion of spinning particles in a curved spacetime, the vierbein chosen is: µ 1 0 p g03 e0 (x) = p (1; 0; 0; 0); e µ = −g00 1; 0; 0; ; −g00 g00 µ 1 1 p e1 (x) = p (0; 1; 0; 0); e µ = g11(0; 1; 0; 0); g11 µ 1 2 p e2 (x) = p (0; 0; 1; 0); e µ = g22(0; 0; 1; 0); g22 s s 2 µ −g00 g03 3 g03 − g00g33 e3 (x) = 2 − ; 0; 0; 1 ; e µ = (0; 0; 0; 1:); g03 − g00g33 g00 −g00 (6) µ ν ea (x)eb (x)gµν(x) = ηab; a ν ν e µ(x)ea (x) = δµ ; (7) a µ a e µ(x)eb (x) = δ b: III. Frame-dragging As seen by long distance observers, the hovering position has a four-velocity defined by µ −1=2 uh = (dt=dτ; 0; 0; 0) = ((−g00) ; 0; 0; 0): For a free-falling particle the four-velocity due the frame-dragging, as seen by the same distant observers, is described by r µ −g33 g03 ufd = 2 1; 0; 0; − : g00g33 − (g03) g33 In the present work, the frame-dragging velocity has to be measured by the hovering observer as a local inertial frame velocity and it can be obtained by µ projecting out the four-momentum mufd of the particle over the four-vector velocity uhµ of the hovering observer µ ufduhµ = −E = −γfd; (8) where E is the relativistic energy per unit mass of the particle with respect to a local (hovering) 2 −1=2 observer. Here γfd = (1 − vfd) . Moreover a ab µ uh = η eb uhµ and therefore a ufd = (cosh η; 0; 0; sinh η): (9) IV. Spin precession Now we consider two observers and one EPR source on the equator plane θ = π=2. The observers are placed at azimuthal angles φ = ±Φ and the EPR source is located at φ = 0. The observers and the EPR source are assumed to be hovering. From the perspective of a zero angular momentum observers (ZAMO), the local velocity of the entangled particles is given by a uEPR = (cosh ζ; 0; 0; sinh ζ); (10) where vEPR = tanh ζ is the speed of particles in the local inertial frame of the ZAMO. After the pair of entangled spin-1/2 particles is generated at the EPR source, they leave it and follow a circular path around a black hole. In spherical coordinates on the equatorial plane θ = π=2, the velocity of particles has two relevant components, the temporal one and the spatial one with φ-coordinate at constant radius r. Thus, for the hovering observer, the motion is measured by the proper-velocity with v = tanh ξ. That is, ua = (cosh ξ; 0; 0; sinh ξ), therefore the general contravariant four-velocity is t t t u = e0 cosh ξ + e3 sinh ξ; φ φ (11) u = e3 sinh ξ; µ such that u uµ = −1. In order the particles describe circular motion, we must apply an external force that compensates both the centrifugal force and the gravity. The acceleration due to this external force is µ ν µ a (x) = u (x)rνu (x): (12) On the equatorial plane the acceleration then becomes r t 2 1 2 a = (e0 ) Γ0 0 cosh ξ t 2 1 φ 2 1 t φ 1 2 + (e3 ) Γ0 0 + (e3 ) Γ3 3 + 2e3 e3 Γ0 3 sinh ξ t φ 1 t 1 +2e0 (e3 Γ0 3 + e3 Γ0 0) sinh ξ cosh ξ; (13) µ where Γρ σ are the usual Christoffel’s symbols. The change of the local inertial frame consists of a boost along the 1-axis and a rotation about the 2-axis calculated by a ν a χ b(x) = −u !ν b(x); (14) where the connection one-forms are defined as a ν a a ν !µ b(x) = −eb (x)rµe ν(x) = e ν(x)rµeb (x): (15) In our particular situation, the connections of interest are given by: 0 r 0 0 r 0 3 !t 1 = e1 e tΓ0 1 + e1 e φΓ0 1; 1 t 1 1 φ 1 1 !t 3 = e3 e r Γ0 0 + e3 e r Γ0 3; 0 r 0 0 r 0 3 (16) !φ 1 = e1 e tΓ1 3 + e1 e φΓ1 3; 1 t 1 1 φ 1 1 !φ 3 = e3 e r Γ0 3 + e3 e r Γ3 3: The relevant boost is described by 0 t r 0 0 0 3 χ 1 = −e0 e1 (e t Γ0 1 + e φΓ0 1) cosh ξ r φ 0 0 0 3 t 0 0 0 3 −e1 e3 (e t Γ1 3 + e φΓ1 3) + e3 (e t Γ0 1 + e φΓ0 1) sinh ξ; while the rotation about the 2-axis is given by 1 t 1 t 1 φ 1 χ 3 = −e0 e r (e3 Γ0 0 + e3 Γ0 3) cosh ξ 1 t t 1 φ 1 φ t 1 φ 1 −e r e3 (e3 Γ0 0 + e3 Γ0 3) + e3 (e3 Γ0 3 + e3 Γ3 3) sinh ξ: The infinitesimal Lorentz transformation 1 λa (x) = − [aa(x)p (x) − pa(x)a (x)] + χa (x): (17) b m b b b The boost along the 1-axis and the rotation about the 2-axis are respectively 0 1 t 2 1 2 t 2 1 φ 2 1 t φ 1 2 λ 1 = e r (e0 ) Γ0 0 cosh ξ + ((e3 ) Γ0 0 + (e3 ) Γ3 3 + 2e3 e3 Γ0 3) sinh ξ t φ 1 t 1 + 2e0 (e3 Γ0 3 + e3 Γ0 0) sinh ξ cosh ξ cosh ξ t r 0 0 0 3 −e0 e1 (e t Γ0 1 + e φΓ0 1) cosh ξ r φ 0 0 0 3 t 0 0 0 3 −e1 e3 (e t Γ1 3 + e φΓ1 3) + e3 (e t Γ0 1 + e φΓ0 1) sinh ξ; 1 1 t 2 1 2 t 2 1 φ 2 1 t φ 1 2 λ 3 = −e r (e0 ) Γ0 0 cosh ξ + ((e3 ) Γ0 0 + (e3 ) Γ3 3 + 2e3 e3 Γ0 3) sinh ξ t φ 1 t 1 + 2e0 (e3 Γ0 3 + e3 Γ0 0) sinh ξ cosh ξ sinh ξ t 1 t 1 φ 1 −e0 e r (e3 Γ0 0 + e3 Γ0 3) cosh ξ 1 t t 1 φ 1 φ t 1 φ 1 −e r e3 (e3 Γ0 0 + e3 Γ0 3) + e3 (e3 Γ0 3 + e3 Γ3 3) sinh ξ: (18) The change of the spin is obtained by computing the infinitesimal Wigner rotation λi (x)p (x) − λ (x)pi (x) #i (x) = λi (x) + 0 k k0 : (19) k k p0(x) + m In particular, the rotation about the 2-axis through a certain angle reads: 1 1 h t 2 1 2 t 2 1 φ 2 1 t φ 1 2 # 3 = −e r (e0 ) Γ0 0 cosh ξ + ((e3 ) Γ0 0 + (e3 ) Γ3 3 + 2e3 e3 Γ0 3) sinh ξ t φ 1 t 1 i + 2e0 (e3 Γ0 3 + e3 Γ0 0) sinh ξ cosh ξ sinh ξ t 1 t 1 φ 1 −e0 e r (e3 Γ0 0 + e3 Γ0 3) cosh ξ 1 h t t 1 φ 1 φ t 1 φ 1 i −e r e3 (e3 Γ0 0 + e3 Γ0 3) + e3 (e3 Γ0 3 + e3 Γ3 3) sinh ξ (20) sinh ξ n 1 h t 2 1 2 t 2 1 φ 2 1 t φ 1 2 + cosh ξ+1 e r (e0 ) Γ0 0 cosh ξ + ((e3 ) Γ0 0 + (e3 ) Γ3 3 + 2e3 e3 Γ0 3) sinh ξ t φ 1 t 1 i + 2e0 (e3 Γ0 3 + e3 Γ0 0) sinh ξ cosh ξ cosh ξ t r 0 0 0 3 −e0 e1 (e t Γ0 1 + e φΓ0 1) cosh ξ r h φ 0 0 0 3 t 0 0 0 3 i o − e1 e3 (e t Γ1 3 + e φΓ1 3) + e3 (e t Γ0 1 + e φΓ0 1) sinh ξ : Finally, from the tetrad given in Eq.