Intrinsic Time Geometrodynamics: Explicit Examples

Transcription

1 CHINESE JOURNAL OF PHYSICS VOL. 53, NO. 6 November 05 Intrinsic Time Geometrodynamics: Explicit Examples Huei-Chen Lin and Chopin Soo Department of Physics, National Cheng Kung University, Tainan 70, Taiwan Received July 7, 05) Intrinsic time quantum geometrodynamics resolved the problem of time and bridged the deep divide between quantum mechanics and canonical quantum gravity with a Schrödinger equation which describes evolution in an intrinsic time variable. In this formalism, Einstein s general relativity is a particular realization of a wider class of theories. In this work, explicit classical black hole and cosmological solutions and the motion of test particles are derived and analyzed in the context of constant three-curvature solutions in intrinsic time geometrodynamics, and we exemplify how this formalism yields results which agree with the predictions of Einstein s theory. DOI: 0.6/CJP PACS numbers: 04.0.Cv, 04.0.Jb, Ds I. INTRODUCTION TO INTRINSIC TIME GEOMETRODYNAMICS A framework for geometrodynamics without the paradigm of space-time covariance has been advocated in Refs. [ 3]. With a Schrödinger equation for quantum geometrodynamics which describes first-order evolution in intrinsic time, it resolved the problem of time and bridged the deep divide between quantum mechanics and conventional canonical formulations of quantum gravity. In Horava-Lifshitz gravity [4], the deep conflict between gravity as a unitary field theory and space-time 4-covariance is overcome by retaining only spatial covariance in the theory; and power-counting renormalizability is achieved by supplementing the potential of Einstein s theory with higher spatial curvature terms which explicitly break 4-covariance. In this work, classical black hole and cosmological solutions and the motion of point particles are derived and discussed from the point of view of intrinsic time geometrodynamics, and we demonstrate how this formalism can yield results which agree with the predictions of Einstein s theory. To recapitulate the theory, we start with the Arnowitt-Misner-Deser ADM) decomposition ds = N + q ij dx i + N i ) dx j + N j ). The canonical action of General Relativity GR) may be written as NH S = π ij q ij d 3 x + N i ) H i d 3 x + boundary term, ) wherein the super-hamiltonian H = κ q [ Gijkl π ij π kl + V q ij ) ], and H i = q ij k π kj = 0 is the super-momentum constraint. The DeWitt supermetric, with deformation parameter Electronic address: l89808@mail.ncku.edu.tw Electronic address: cpsoo@mail.ncku.edu.tw c 05 THE PHYSICAL SOCIETY OF THE REPULIC OF CHINA

2 00- INTRINSIC TIME GEOMETRODYNAMICS: EXPLICIT EXAMPLES VOL. 53 l, is G ijkl = q ikq jl + q il q jk ) lq ij q kl. In Einstein s GR, l = and the potential V = q R Λ κ) eff ). With β = l 3, the super-hamiltonian constraint factorizes in an interesting way as q κ H = G ijkl π ij π kl + V q ij ) = βπ H ) βπ + H ) = 0 ) wherein H = Ḡijkl π ij π kl + V q ij ) = [ q ik q jl + q il q jk ] π ij π kl + V q ij ). In the symplectic potential π ij δq ij = π ij δ q ij +πδ ln q 3, clean separation of the conjugate pair, ln q 3, π), consisting of one-third of) the logarithm of the determinant of the spatial metric and the trace of the momentum, from q ij, π ij ), the unimodular part of the spatial metric with traceless conjugate momentum, allows a deparametrization of the theory wherein ln q 3 plays the role of the intrinsic time variable for β > 0. In the quantum context, with ˆπ = h δ i in δ ln q 3 the metric representation, the Hamiltonian constraint implies a Schrödinger equation of the form βˆπ + H)Ψ = 0 with β = ± β, and H/β generates time - development in intrinsic time ln q 3. The Hodge decomposition for compact manifolds yields δ ln q 3 = δt + i δy i, wherein δt = 3 δ ln V spatial which is proportional to the logarithmic change in the spatial volume) is a three-dimensional diffeomorphism invariant 3dDI) quantity which serves as the global spatially independent) intrinsic time interval; whereas i δy i can be gauged away, since the Lie derivative L δn ln q 3 = 3 iδn i. With respect to 3dDI time-development in δt, the 3dDI physical Hamiltonian is thus H Phys := Hx) β d3 x, with the fundamental 3dDI Schrödinger equation, i h δψ δt = H PhysΨ. In the classical context, it has been demonstrated, through the Hamilton-Jacobi and Hamilton equations, in Refs. [, ] that the resultant classical spacetime that is produced by this theory has an emergent ADM lapse function which is precisely N = q t ln q /3 3 in i ) 4βκ H. 3) In the conventional canonical formulation of Einstein s GR, the EOM dq ij = { q ij, NH + N i H i }P.. = 4Nκ q G ijkl π kl +L N q ij, relates the extrinsic curvature to π ij by K ij := N dq ij L N q ij ) = κ q G ijkl π kl. Taking the trace gives 3 K = N t ln q 3 3 in i) = κβ q H, wherein the constraint βπ + H) = 0 has been used to arrive at the last step. Thus it is noteworthy that Einstein s theory yields an a posteriori lapse function N which is in complete agreement with the result 3) of the classical spacetime produced by H Phys in the formalism of intrinsic time geometrodynamics. From the perspective of intrinsic time geometrodynamics and the paradigm shift to 3dDI, Einstein s GR with its corresponding V and β ) is a particular realization of a wider class of theories. The requirement of a real physical Hamiltonian density compatible with spatial diffeomorphism symmetry suggests supplementing the kinetic term with a quadratic

3 VOL. 53 HUEI-CHEN LIN AND CHOPIN SOO 00-3 form, i.e., H = Ḡ ijkl π ij π kl + [ ] δw q δw ikq jl + q jk q il ) + γq ij q kl. 4) δq ij δq kl H is then real if γ > 3. Dependent only on 3-geometry, W is the sum of a Chern- Simons action of the spatial affine connection and the spatial Einstein-Hilbert action with the cosmological constant, i.e., [ q W = br Λ) + g ϵ ikj Γ l im j Γ m kl + )] 3 Γl imγ m jnγ n kl. 5) A slight generalization is to replace δw δq ij in the positive semidefinite quadratic form in H with qλ q ij + b Rq ij + c Rij + g C ij ), which is the most general symmetric second rank tensor density) containing up to third derivatives of the spatial metric []. Quantum considerations which prompt further improvements in the precise expression of the Hamiltonian are detailed in Ref. [3]. II. EXPLICIT SOLUTIONS In addition to the spatial Ricci scalar and cosmological constant, higher curvature terms, such as the traceless part of the Ricci tensor, Rij, and the Cotton-York tensor, C ij, are thus present in the potential in the generalized Hamiltonian density H/β. ut these vanish identically for solutions with constant spatial curvature slicings, which implies that the physical content of well-known solutions of Einstein s theory can be captured by H Phys of intrinsic time geometrodynamics in this setting. II-. Constant spatial curvature slicings We consider constant 3-curvature slicings with t-independent lapse and the following shift vector for simplicity to obtain exact solutions of the full theory. As we shall show explicitly, these will include the Robertson-Walker and Painlevé-Gullstrand form of the Schwarzschild-deSitter solutions. The metric may then be expressed rather generically as [ ) ds = Nr) + a dr t) + nr) + r dθ + sin θdϕ )], 6) kr with the constant spatial scalar curvature R = 6k, and k = 0, ± determines the topology of a the slicings. For the above constant 3-curvature slicings, the super-hamiltonian constraint simplifies to H K ij K ij + l 3l K +λ = 0, with λ := Λ eff R+ R 8Λ eff = Λ eff 6k + 9k a Λ eff. a 4 And the extrinsic curvature can thus be determined for t-independent a) as K ij = N tq ij i N j j N i ) = a r n) N kr ra n kr N ra n sin θ kr N. 7)

4 00-4 INTRINSIC TIME GEOMETRODYNAMICS: EXPLICIT EXAMPLES VOL. 53 The super-hamiltonian and super-momentum constraints reduce to the restrictions 4 r G l)6g G 3 G) + 6 l) = r, 8) with G = r rn) n, and N = { kr )[ l)n +4lrn rn)+r l) rn) ] r 3l )λ }. From the solutions Gr) of Eq. 8), the lapse and the shift can be determined. However, it is hard to solve Eq. 8) explicitly, except in the l = limit of Einstein s theory. The implication is that exact solutions of deformed GR expressed in terms of r are complicated. Remarkably, it is possible to solve for the metric in terms of the dynamical variable π for arbitrary l. We discuss how this can be carried out below. II-. Solution of the constraints The momentum π ij q p ij for the t-independent constant 3-curvature metric of the λ 0 0 form compatible with 7) is p i j = 0 λ 0 ; correspondingly 0 0 λ λ = p + 3 λ = 3 6p ij p ij p ), p ) 6p ij p ij p, 9) with the decomposition p i j = pi j + 3 δi j p. The traceless part pi j p i j = 6 l 3) p 6 λ l 3) p 6 λ can thus be expressed as l 3) p 6 λ.. 0) In Eq. 0) we have used the relation p ij = K ij + l 3l q ijk and the super-hamiltonian constraint H = 0 which relates λ to K ij. And the super-momentum constraint now reduces to the single requirement 3 rp + 6 l ) p r λ + r l ) p 6 3 λ = 0. ) 6 It is helpful to convert Eq. ) into ln r p = 3 [ 3λ 3l )p ] 3l )p + 3l )p 3λ. )

5 VOL. 53 HUEI-CHEN LIN AND CHOPIN SOO 00-5 The solution reveals the relation between r and p as r = [ 3l )p + 3l ] 3l )p 3 3λ 3l ) c [3l )p 3λ] 6 =: gp), 3) wherein c is the constant of integration. For the constant 3-curvature spacetimes of Eq. 6), there are two further equations for consistency: Ki j = p i j and K = 3l)p. In terms of r and p these are expressed as kr n r rn) 3rN = 6 l 3) p 6 λ, 4) kr n+r rn) = 3l )p. rn With Eq. ), n which is related to the shift) and the lapse function N can be determined from Eq. 4) as n = [ 3l )p + 3l ] 3l )p 3 3λ 3l [3l )p 3λ] 6 3 c [3l )p + 3l 3l )p 3λ 3l ) kgp) N =. 5) 3l )p + 3l )p 3λ, ] This completes the solution. Note that while we are unable to express p in terms of r nicely, we are able to achieve the inversion r = gp) in 3) and thus the metric of 6) which now satisfies all constraints is expressible in terms of coordinate variables t, p, θ, ϕ) and functions a, Np), np) for arbitrary l. II-3. Constant curvature slicings with t-independent scale factor a With a hence λ) being a space-time independent constant, the generic solution for arbitrary l can be written down from the formulas displayed earlier. In the special case of l = for GR, inverting Eq. 3) yields p = c r) 3 lapse N are then determined from Eq. 5) to be n = 3c ) + 3c3 r3 λ +3c 3 r 3 λ. The function n and the 9c 3 r + λ 6 r ; N = 3c kr. 6)

6 00-6 INTRINSIC TIME GEOMETRODYNAMICS: EXPLICIT EXAMPLES VOL. 53 Comparison with physics identifies the constant c as c = 3c yields the Schwarzschild-deSitter solution as [ ds = kr ) PG + a dr Gm + PG kr r = GM c R Λ eff 3 R 3k R ) 4a 4 dτ dr + Λ eff 9 Gm ) 3. Defining PG = + λ ) ] 6 r + r dθ + sin θdϕ ),7) ) + R dω ; 8) GM c R Λ eff 3 R 3k R 4a 4 Λ eff Λeff ), wherein λ = Λ eff 6k + 9k a Λ eff = 3k a 4 a Λ R := ar, and M := ma 3 c. eff Expression 7) is the solution written in Painlevé-Gullstrand PG) form, with constant adr kr GM R + λ 6 R 3-curvature slicings [5], while the identification dτ = P G yields [ GM R λ 6 R kr ] the metric in the standard form of 8), which suffers from coordinate singularities and is a priori defined only in the region between the horizons. The PG form of the metric 7) is free of coordinate singularities and extends the manifold beyond the horizons. In the case of spatially compact k = ) slicing, the range of the radial coordinate is 0 R = ra a, and to cover a region of sufficient interest of the Schwarzschild-deSitter manifold we should choose a 3 3k Λ eff, which is the radius of the desitter horizon which implies 4a 4 Λ eff Λ eff 3 ). In fact by choosing either k = 0, or a Λ eff for k =, the PG form 7) remains valid except at the physical singularity at R = 0) for the full range R > 0 of the radial coordinate. Moreover, such a choice guarantees the physical requirement that the metric is asymptotically de Sitter with Λ eff as the value of the cosmological constant. II-4. Time-dependent scale factor, vanishing shift vector, and Robertson- Walker solution The case with zero shift vector and t-dependent at) in Eq. 6) can be solved readily. For vanishing shift vector, the extrinsic curvature becomes K i j = δi j ta Na. Since K i j = p i j = 0, we obtain H/ qβ) p = λ from Eq. 0), and the lapse can then be l 3 determined from the relation K = 3l)p as N = t a This is a specialization of 3 )λ. at) l the formula 3). The explicit form of the metric is then ds = 3 ta) [ dr λa 3l ) + a kr + r dθ + sin θdϕ ) ] 9) [ dr = + at ) kr + r dθ + sin θdϕ ) ], 0) which is cast in the usual Robertson-Walker form after reparametrizing the metric by identifying := 3 t a) = λ3l ) 3d ln a wherein λ = Λ eff 6k + 9k depends on a. a λ3l ) a Λ eff a 4 This can be integrated to yield the time dependence of a in terms of t as a t ) = [ ] 3k + e t t 0 ) 3 3l )Λ eff. ) Λ eff

7 VOL. 53 HUEI-CHEN LIN AND CHOPIN SOO 00-7 At large values of t t 0 ) Λ eff the resultant metric expands exponentially regardless of k, and with l = as in Einstein s theory) it then yields the usual de Sitter expansion with at ) e Λeff 3 t t 0 ). III. MOTION OF TEST PARTICLE The motion of a test point particle of mass m 0 described canonically by x i P, P i) can be derived from the particle Hamiltonian [ ] H P = N q ij P i P j + m 0 N i P i δ x x P )d 3 x = N x P, t) q ij x P, t)p i P j + m 0 N i x P, t)p i. ) The canonical equation dxi P N i = Nq ij P j q ij P i P j +m 0 S = = {x i P, H P} P.. relates the velocity and momentum by dxi P. Inverting for P i in terms of dxi P [ dx i ] P P i H P = m 0 + N i results in the action N q ij dxi P + N i ) dxj P + N j ) + = m 0 g µν x P, t)dx µ dx ν, 3) which is just the usual proper time action on identifying the ADM metric ds = g µν dx µ dx ν = N + q ij dx i + N i )dx j + N j ). Conversely, starting with the Lagrangian in the final step of 3), the Hamiltonian of ) is obtained. It follows that the particle will obey the geodesic equation in the background ADM metric. The derivation is insensitive to the particular form of the background lapse function and holds, in particular, q t ln q /3 3 in i ) for N = 4βκ H in the intrinsic time formulation. The particle Hamiltonian of ) is motivated by the fact that in the presence of the particle, the total Hamiltonian constraint, H T = H pure GR + qe P = 0, is equivalently 0 = H T = κ βπ + H)βπ + H) + E P q = κ ) qep [ βπ + H + βπ + q κ This implies a Schrödinger equation [ˆπ + β H + )] qep H +. 4) κ qep κ )] Ψ = 0 with evolution in intrinsic

8 00-8 INTRINSIC TIME GEOMETRODYNAMICS: EXPLICIT EXAMPLES VOL. 53 time ln q /3. The Hamiltonian for evolution w.r.t. ADM coordinate time t is [, ] H ADM = d 3 x t ln q /3 3 in i ) qep H β + κ + N i H i 5) = d 3 x t ln q /3 ) ) H 3 in i qep + β 4κ H + + N i H i 6) d 3 x t ln q /3 ) H 3 in i β + NE P + N i H i, 7) with N = q t ln q /3 3 in i ) 4βκ H being the lapse function of the background geometry, and qep retaining the expansion in 6) only to first-order in κ H this ratio compares the particle s energy to the Hamiltonian density of the rest of the universe). The EOM of the particle are then dx i { ) } P = {x i P, H ADM } P.. x i P, d 3 xne P N i P i = {x i P, H P } P.., P.. dp i = {P i, H ADM } P.. {P i, H P } P.. ; 8) with E P = q ij P i P j + m 0 δ x x P) and H P as in ). III-. Geodesic equation, perihelion shift, and bending of light for Painlevé- Gullstrand metric It was already demonstrated that a test particle will obey the geodesic equation. We now analyze this motion in Schwarzschild-de Sitter spacetime expressed in constantcurvature PG form for simplicity we use the notation t := t P G in this subsection), [ dr ds = + + ) ] A + R dω ; A = GM Rc + Λ eff 6k ) 6 a + 9k Λ eff a 4 R, = k R a. With x µ P = t, R, θ, ϕ), the geodesic equation d x µ dp + Γ µ να dxν dp dx α dp = 0 is equivalent to R A θ R A sin θ ϕ Ṙ A Aṫ A ) A 3/ ṫṙa ) + ẗ RA ) θ +RA ) sin θ ϕ + A )A )ṫ + AṫṘA ) Ṙ θ R sin θ cos θ ϕ + θ + cot θ θ ϕ + ϕ Ṙ ϕ R + Ṙ A ) + R = 0;

9 VOL. 53 HUEI-CHEN LIN AND CHOPIN SOO ) wherein derivatives w.r.t. p and R are denoted by and. Choosing the initial motion to lie in the equatorial plane θ = π, θ = 0) implies θ Ṙ ϕ ) = 0 as well. Furthermore, R + ϕ = R ϕ = 0 leads to conservation of R ϕ =: L. On geodesics, there is also the R d dp constancy of g µν ẋ µ ẋ ν = A )ṫ + AṫṘ geodesics, we may use dp = cdτ and ẋ µ = dxµ cdτ A )ṫ + AṫṘ + Ṙ + R ϕ =, = Consequently ṫ = RṘ A± ) + Ṙ + R ϕ = c dτ dp =: E. For time-like, whereas E = 0 for null geodesics. In general, { 0 null geodesic time-like. R 4Ṙ L R A +L R 3 R 4 A ) R A ) 30). Substituting this into the geodesic equation for the radial coordinate results in L A ) RL + R )A ) + R = 0, R 3 so the effects are dependent only upon A = GM Λ effr Rc 3 3k R 4Λ eff ). The a 4 remaining dynamical equation is thus R + 3GL M R 4 c L GM R 3 + R c RΛ ) eff 3k R 3 4a 4 = 0. 3) Λ eff The trajectories Rϕ) with dr dp = dϕ dp dr dϕ = L R dr dϕ ; u := R, dr dp = u LLu ) d R) = L u d R)), are thus governed by L u d u dϕ dϕ ) GMu Λ c eff 3u 3k 4a 4 Λ eff = 0, or equivalently, u d u dϕ + u 3GMu c GM = L c dϕ ) L du du R dϕ = L dϕ, d R = dp + 3GML u 4 L u 3 + c Λ )) eff 3L u 3 + 9k 4a 4 Λ. 3) eff This is precisely the same equation as in Einstein s theory with cosmological constant provided, as motivated in the earlier discussion, 0. Thus the motion of a test particle k a 4 Λ eff in constant curvature PG exact solution of intrinsic time geometrodynamics is in complete agreement with the predictions of Einstein s GR. III-. Comparison with non-constant curvature solutions in Hořava Gravity There exists other solutions with zero shift in Hořava gravity [6 9]. The solution of Ref. [6] is ds = N + dr f + r dθ + sin θdϕ ), wherein f = 3 Λ effr αr + 9 l) + 6l ). Even when l =, the solution with f = 3 Λ effr α r deviates from the Schwarzschild form and it is not of constant 3-curvature. Setting the deformation parameter to l = for comparison with Einstein s theory yields ds = N + dr f + r dθ + sin θdϕ ) ; N = f = 3 Λr + α r. 33)

10 00-0 INTRINSIC TIME GEOMETRODYNAMICS: EXPLICIT EXAMPLES VOL. 53 FIG. : FIG. : FIG. 3: FIG. 4: Fig. shows the usual precession of perihelion behavior predicted by Eq. 3); whereas Figs. 4 are the starkly different predictions of Eq. 34) as the parameter α is varied over a wide range. All figures have the same L and initial conditions u0), u 0); and the orbits are shown for two revolutions, i.e., 0 ϕ 4π, with solid lines for 0 ϕ < π and dashed lines for π ϕ 4π. It can be shown that the corresponding geodesic equation takes the form d u dϕ + u + 3α u α = 4 4L u 3/ Λ ) { 0 null geodesic 3L u 3, = time-like, 34) which differs from Eq. 3), and thus from the predictions of Einstein s theory. Solving for the geodesics numerically with fixed L and initial conditions u0), u 0) = 0 yields the results in Figs. 4 which provide stark graphical comparisons of the predictions of 3) and 34). Unlike 3) in intrinsic time geometrodynamics, 34) which has a very different dependence on u fails to produce the normal precession of perihelion behavior in Einstein s theory for time-like bound geodesics even when the parameter in the solution α is varied over a wide range.

11 VOL. 53 HUEI-CHEN LIN AND CHOPIN SOO 00- Acknowledgments This work was supported in part by the Ministry of Science and Technology R.O.C.) under Grant Nos. NSC0--M MY3 and MOST04--M We would like to thank Eyo Eyo Ita III and Hoi-Lai Yu for beneficial discussions during the course of this work. References [] C. Soo and H.-L. Yu, Prog. Theor. Exp. Phys. 04, 03E0 04). [] N. Ó Murchadha, C. Soo, and H. L. Yu, Class. Quantum Grav. 30, ). [3] E. E. Ita III, C. Soo, and H.-L. Yu, Prog. Theor. Exp. Phys. 083E0 05). [4] P. Hořava, Phys. Rev. D 79, ). [5] C. Y. Lin and C. Soo, Phys. Lett. 67, ). [6] H. Lü, J. Mei, and C. N. Pope, Phys. Rev. Lett. 03, ). [7] E. O. Colgain and H. Yavartanoo, JHEP 0908, 0 009). [8] A. Ghodsi, Int. J. Mod. Phys. A 6, 95 0). [9] A. N. Aliev and C. Senturk, Phys. Rev. D 8, ).