Issue4(October) PROGRESSINPHYSICS Volume10(2014)

Godel’s¨ Universe Revisited

Patrick Marquet 11 rue des Chapelles, 94350 Villiers/Marne, Paris, France E-mail: [email protected]

This paper investigates the G¨odel’s exact solution of the Einstein equations which describes a stationary homogeneous cosmological Universe inducing closed timelike curves CTCs). This model is generally dismissed because it exhibits a rotational sym- metry and it requires a non zero cosmological constant in contradiction with the current astronomical observations. If the cosmological term is assumed to be slightly variable, we show that this metric can be compatible with the Hubble expansion, which makes the G¨odel model a viable representation of our Universe.

Introduction ric is an exact spherically symmetric solution. This prop- erty would imply that the Universe admits a six-parameter In his original paper [1], Kurt G¨odel has derived an exact so- group of isometries whose surfaces of transitivity are space- lution to Einstein’s field equations in which the matter takes like three-surfaces of constant curvatures. (An action of a the form of a pressure-free perfect fluid (dust solution). This group is transitive on the manifold M, if it can map any point R4 manifold is homogeneous but non-isotropic and it exhibits of M into any other point of M.) The spatial metric is ex- a specific rotational symmetry which allows for the existence pressed by of closed time like curves since the light cone opens up and tips over as the G¨odel radial coordinate increases. In addition, dr2 dl2 = + r2 sin2 θ dϕ2 + dθ2 . (1.1) it implies a non zero cosmological term and a constant scalar 1 + r2/F2   curvature, therefore it doesnot admit a Hubble expansion in In the full RW model F(t) is called the cosmic scale factor the whole, which tends to contradict all current observations. which varies with the (cosmic) proper time t of the whole We suggest here to stick to the G¨odel model which we space. consider as the true Universe, and we state that the Hubble For an open (infinite) Universe, with negative curvature expansion can yet be maintained in a particular location with k specific coordinates transformations, where the G¨odel rota- K(t) = , where k = 1. (1.2) tion is unobservable. F2 − In this distinguished location, our derivations lead to an and the three-spaces are diffeomorphic to R3. open Universe without cosmological term and as a result, no The standard formulation is given by future singularity will ever appear in this local World. 2 2 2 2 2 2 2 2 (ds )RW = F dη –dχ sinh χ sin θ dϕ + dθ (1.3) Our model however, is bound to a main restriction: for − 
 physical reasons, it provides a solution which holds only for with the usual parametrizations the existence of the cosmic scale factor, within the G¨odel dt = Fdη and r = F sinh χ. (1.4) metric. This improved G¨odel Universe which we present here, In the RW Universe, the matter with mean density ρ is non has nevertheless the advantage of agreeably coping with the interacting (dust) and the energy-momentum tensor is that of observational facts. a pressure free perfect fluid:

Tab = ρ ua ub . (1.5) Some notations From the corresponding field equations we arrive at the Space-time indices: 0, 1, 2, 3. temporal coordinate [2] Newton’s gravitation constant: G. dF The velocity of light is c = 1. η = , (1.6) ± Space-time signature: 2. Z F 8πG ρ F2 + 1 − 3 qh i F = F (cosh η 1) , (1.7) 1 Homogeneous space-times 0 − with 1.1 Roberston-Walker space 4πGρF3 F = , (1.8) Our actual observed Universe is spatially homogeneous: if 0 3 we can see these observations identically in different direc- Where the sign depends on the light emitted either from the tions, the model is said isotropic. The Robertson-Walker met- coordinates± origin or reaching this origin.

Patrick Marquet. G¨odel’s Universe Revisited 259 Volume10(2014) PROGRESSINPHYSICS Issue4(October)

1.2 TheGodel¨ metric and the covariant components The G¨odel line element is generically given by B, 0, Bex1 , 0 e2x1 (ds2) = B2 dx2 dx2 + dx2 – dx2 + G 0 − 1 2 2 3 so we obtain  1 2x1 Rab = ua ub . (1.14) + 2e (dx0 + dx2) , (1.9) B2  Since the curvature scalar is a constant, the G¨odel field where B > 0 is a constant in the original formulation. equations read This space-time has a five dimensional group of isome- tries which is transitive. It admits a five dimensional Lie al- 1 gebra of Killing vector fields generated by a time translation (Gab) = Rab gabR = 8πGρ ua ub +Λgab , (1.15) G − 2 ∂x0 , two spatial translations ∂x1 , ∂x2 plus two further Killing vector fields: where Λ is the cosmological term which is here inferred as 4πGρ, i.e.: x2 − ∂ –x ∂ and 2ex1 ∂ + x ∂ + e2x1 2 ∂ . 1 x3 2 x3 x0 2 x3 x2 = πGρ,  − 2  2 8 (1.16)   B   In all current papers, the G¨odel metric is always described R 1 as the direct sum of the metric Λ= = . (1.17) − 2 −2B2 e2x1 2 2 2 2 2 We next define new coordinates (t,w,φ) on M1 by (ds )G1 = B dx0 dx1 + dx2 +  − 2 x1 x1 E = cosh2w + cos φ sinh 2w, (1.18) + 2e (dx0 + dx2) (1.10) 

x1 on the manifold M1 = R3 and x2 e = √2 sin φ sinh 2w, (1.19)

2 2 2 (ds )G2 = B ( dx3) (1.11) 1 x0 2t 2w φ − tan φ + − = e− tan . (1.20) 2 √2 ! 2 on the manifold M2 = R1. This means that in the usual treatments, in order to ana- This leads to the new line element lyze the properties of the G¨odel solution it is always sufficient 2 2 2 2 2 4 2 2 to consider only M . The coordinate dx is deemed irrelevant (ds )G = 4B dt –dw –dy + sinh w sinh w dφ + 1 3 − and is thus simply suppressed in the classical representation,   + 2 √2 sinh2 w dφ dt (1.21) which in our opinion reveals a certain lack of completeness.  In what follows, we consider the complete solution, where we which exhibits the rotational symmetry of the solution about assign a specific meaning to dx . 3 the axis w = 0, since we clearly see that the g do not depend Let us remark that the G¨odel space is homogeneous but ab on θ. G¨odel inferred that matter everywhere rotates with the not isotropic. angular velocity 2 4πGρ. 1.3 Classical features of Godel’s¨ metric Let us considerp the reduced G¨odel metric c Computing the connection coefficients Γab from the gab given 2 2 2 2 4 2 2 (ds )G1 = 4B dt dw + sinh w sinh w dφ + in (1.9) eventually yield  − −  + 2 √2 sinh2 w dφ dt . 2x1 x1 R00 = 1, R22 = e , R02 = R20 = e . (1.12)  All light rays emitted from an event on the symmetry axis All other Rab vanish. reconvergeat a later event on this axis, with the null geodesics Hence: forming a circular cusp [3]. 1 R = . (1.13) If a curve c is defined by sinh4 w = 1, that is B2 The unit vector (world velocity) following the x0-lines is c = ln(1 + √2), (1.22) shown to have the following contravariant components

1 hence, any circle w> ln(1+ √2) in the plane t = 0,isa closed , 0, 0, 0 B timelike curve.

260 Patrick Marquet. G¨odel’s Universe Revisited Issue4(October) PROGRESSINPHYSICS Volume10(2014)

2 The modified Godel¨ metric and we retrieve the Roberston-Walker metric for an open Uni- r 2.1 Conformal transformation verse with the sole radial coordinate : 2 2 2 2 Now we will assume that the Λ-term is slightly varying with (ds )RW = F (η) dη – dχ . (2.9) the time t, so B is also variable through the dust density. See h i (1.16) for detail. Remark: The Weyl tensor of the G¨odel solution By setting R y = r cosh w, (2.1) Cab = Rab + δa δ b + 2δ[ a R b ] (2.10) cd cd 3 [ c d ] [ c d ] where r is another radial parameter, we choose: which has Petrov type D, vanishes for (2.9). Indeed, the 2 equivalent metric (2.4) implies that Cab = 0 for this con- 1 L cd B = 1– 0 (2.2) formally flat space-time. 2  2 2   2 t –y  Our observed Universe would then be devoid of the Weyl    p  curvature which explains why it is purely described in terms where L is a constant whose meaning will become apparent 0 of the Ricci tensor alone. In this view, Einstein was perhaps in the next sub-section. B is now identified with a conformal an even more exceptional visionary mind than is yet currently factor. admitted. Note: one of the Kretschmann scalar is no longer an in- variant 6 2.3 Hubble expansion R Rabcd = (2.3) abcd B4 In our local world, the null geodesics are obviously given by 2 which reflects the fact that the G¨odel space-time may be not (ds )RW = 0, that is fully homogeneous. dη = dχ (2.11) ± Anticipating on our postulate, we will state that the vari- and integrating ation of B is only localized in a certain region of the G¨odel χ = η + const. (2.12) model. The Λ-term remains constant throughoutthe complete ± metric as initially derived, thus preserving its homogeneity. Let us place ourselves at t(η), where we observe a light ray emitted at χ where its frequency is ν0. In virtue of (2.12), 2.2 The postulate the emission time will be t(η χ), and we observean apparent − Our fundamental assumption will now consist of considering frequency given by: our observed Universe as being local. By local we mean that ν F (η χ) the rotation φ is unobservable since we assume that our world ν = 0 − . (2.13) is situated at F (η) w = 0. As F (η) increases monotonically, we have ν < ν0 which Our (local) Universe is now becoming isotropic. is the expression of a red shifted light. Most observed red In this case, the G¨odel metric reduces to a standard con- shifts are rather small, so that F (η χ) can be expanded as a Taylor series about t(η χ) = t(η) and− we finally get, limiting formal solution where the light cone is centered about the t- − axis: to the first two terms 4 2 L0 2 2 (ds )G = 1 dt –dr . (2.4) F (η χ) = F (η) + t(η χ) t(η) F′(η) (2.14) " − 2 √t2 r2 # − − −     − = F (η) 1 + H t(η χ) t(η) (2.15) We now make the following transformations 0 − − n  o where F′ denotes differentiation with respect to η L0 = F0 (2.5)

F′(η) with F0 defined in (1.8) H = (2.16) 0 F (η) F0 F0 r = eη sinh χ, t = eη cosh χ, (2.6) is the present numerical value of Hubble constant. 2 2 The G¨odel solution has a non-zero cosmological term, but F not the local RW metric. 0 eη = √t2–r2 , (2.7) 2 This agrees with the fact that our open local Universe has r a singularity in the past and no singularity in the future [4], in tanh χ = , (2.8) t accordance with astronomical observations.

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Concluding remarks References Closed timelike curves turn out to exist in many other exact 1. G¨odel K. An example of a new type of cosmological solutions of Ein- solutions to Einstein’s field equations. stein’s field equations of gravitation. Reviews of Modern Physics, 1949, v. 21, no. 3, 447–450. It would seem that the first model exhibiting this property 2. Landau L. and Lifshitz E. The Classical Theory of Fields. Addison- was pioneered by C. Lanczos in 1924 [5], and later rediscov- Wesley, Reading (Massachusetts), 1962, p. 402 (French translation). ered under another form by W. J. Van Stockum in 1937 [6]. 3. Hawking S. W. The Large Scale Structure of Space-Time. Cambridge However, unlike the G¨odel solution, the dust particles University Press, Cambridge, 1987. of these Universes are rotating about a geometrically distin- 4. Fang Li Zhi and Ruffini R. Basic Concepts in Relativistic Astrophysics. guished axis. World Scientific (Singapore), 1983. Even worse, the matter density is shown to increase with 5. Lanczos C. Uber¨ eine Station¨are Kosmologie im Sinne der Einsteinis- radius w, a feature which seriously contradicts all current ob- cher Gravitationstheorie. Zeitschrift f¨ur Physik, 1924, Bd. 21, 73. servations. 6. Van Stockum W. J. The gravitational field of a distribution of particles rotating around an axis of symmetry. Proc. Roy. Soc. Edinburgh, 1937, In this sense, the G¨odel metric appears as a more plausible v. A57, 135. model characterizing a broaden Universe which is compatible with our astronomicaldata, providedone is preparedto accept the fact that our observed world is purely local.

Submitted on September 7, 2014 / Accepted on October 1, 2014

262 Patrick Marquet. G¨odel’s Universe Revisited