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Godel's universe revisited.

Introduction

In his original paper [1], Kurt Godel has derived an exact solution to Einstein's field equations in which the matter takes the form of a pressure-free perfect fluid (dust solution). This R4 manifold is homogeneous but non-isotropic and it exhibits a specific rotational symmetry which allows for the existence of closed time like curves since the light cone opens up and tips over as the Godel radial coordinate increases. In addition, it implies a non zero cosmological term and a constant scalar curvature, therefore it doesnot admit a Hubble expansion in the whole, which tends to contradict all current observations.

We suggest here to stick to the Godel model which we consider as the true Universe, and we state that the Hubble expansion can yet be maintained in a particular location with specific coordinates transformations, where the Godel rotation is unobservable.

In this distinguished location, our derivations lead to an open Universe without cosmological term and as a result, no future singularity will ever appear in this local World.

Our model however, is bound to a main restriction: for physical reasons, it provides a solution which holds only for the existence of the cosmic scale factor, within the Godel metric.

This improved Godel Universe which we present here, has nevertheless the advantage of agreeably coping with the observational facts.

Some notations

Space-time indices: 0, 1, 2, 3.

Newton's gravitation constant: G.

The velocity of light is c = 1.

Space-time signature: -2.

1 Homogeneous space-times

1.1 Roberston-Walker space

Our actual observed Universe is spatially homogeneous: if we can see these observations identically in different directions, the model is said isotropic. The Robertson-Walker metric is an exact spherically symmetric solution. This property would imply that the Universe admits a six-parameter group of isometries whose surfaces of transitivity are spacelike three-surfaces of constant curvatures. (An action of a group is transitive on the manifold M, if it can map any point of M into any other point of M.) The spatial metric is expressed by

[dl.sup.2] = [[dr.sup.2]/1 + [r.sup.2]/[F.sup.2]] + [r.sup.2] ([sin.sup.2] [theta] d [[phi].sup.2] + d[[theta].sup.2]). (1.1)

In the full RW model F(t) is called the cosmic scale factor which varies with the (cosmic) proper time t of the whole space.

For an open (infinite) Universe, with negative curvature

K(t) = [k/[F.sup.2]], where k = -1. (1.2)

and the three-spaces are diffeomorphic to [[real part].sub.3].

The standard formulation is given by

[([ds.sup.2]).sub.RW] = [F.sup.2]{d[[eta].sup.2] - d[[chi].sup.2] - [sinh.sup.2] [chi] ([sin.sup.2] [theta] d[[phi].sup.2] + d[[theta].sup.2])) (1.3)

with the usual parametrizations

dt = Fd[eta] and r = F sinh [chi]. (1.4)

In the RW Universe, the matter with mean density [rho] is non interacting (dust) and the energy-momentum tensor is that of a pressure free perfect fluid:

[T.sub.ab] = [rho] [u.sub.a][u.sub.b]. (1.5)

From the corresponding field equations we arrive at the temporal coordinate [2]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.6)

F = [F.sub.0](cosh [eta] - 1), (1.7)

with

[F.sub.0] = [4[pi]G[rho][F.sup.3]/3], (1.8)

Where the [+ or -] sign depends on the light emitted either from the coordinates origin or reaching this origin.

1.2 The Godel metric

The Godel line element is generically given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.9)

where B > 0 is a constant in the original formulation.

This space-time has a five dimensional group of isometries which is transitive. It admits a five dimensional Lie algebra of Killing vector fields generated by a time translation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], two spatial translations [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] plus two further Killing vector fields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In all current papers, the Godel metric is always described as the direct sum of the metric

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)

on the manifold [M.sub.1] = [[real part].sub.3] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)

on the manifold [M.sub.2] = [[real part].sub.1].

This means that in the usual treatments, in order to analyze the properties of the Godel solution it is always sufficient to consider only [M.sub.1]. The coordinate [dx.sub.3] is deemed irrelevant and is thus simply suppressed in the classical representation, which in our opinion reveals a certain lack of completeness. In what follows, we consider the complete solution, where we assign a specific meaning to [dx.sub.3].

Let us remark that the Godel space is homogeneous but not isotropic.

1.3 Classical features of Godel's metric

Computing the connection coefficients [[GAMMA].sub.ab.sup.c] from the [g.sub.ab] given in (1.9) eventually yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.12)

All other [R.sub.ab] vanish.

Hence:

R = [1/[B.sup.2]]. (1.13)

The unit vector (world velocity) following the [x.sub.0]-lines is shown to have the following contravariant components

[1/B], 0, 0, 0

and the covariant components

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so we obtain

[R.sub.ab] = [1/[B.sup.2]] [u.sub.a][u.sub.b]. (1.14)

Since the curvature scalar is a constant, the Godel field equations read

[([G.sub.ab]).sub.G] = [R.sub.ab] - [1/2] [g.sub.ab]R = 8[pi]G[rho][u.sub.a][u.sub.b] + [LAMBDA][g.sub.ab], (1.15)

where [LAMBDA] is the cosmological term which is here inferred as -4[pi]G[epsilon], i.e.:

[1/[B.sup.2] = 8[pi]G[rho], (1.16)

[LAMBDA] = -[R/2] = -[1/2[B.sup.2]]. (1.17)

We next define new coordinates (t, w, [phi]) on [M.sub.1] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.19)

tan [1/2]([phi] + [[x.sub.0] - 2t/[square root of 2]]) = [e.sup.-2w] tan [[phi]/2] (1.20)

This leads to the new line element

[([ds.sup.2]).sub.G] = 4[B.sup.2]([dt.sup.2] - [dw.sup.2] - [dy.sup.2] + [sinh.sup.4] w - [sinh.sup.2] w) d[[phi].sup.2] + 2[square root of 2] [sinh.sup.2] w d[phi] dt) (1.21)

which exhibits the rotational symmetry of the solution about the axis w = 0, since we clearly see that the [g.sub.ab] do not depend on 9. Godel inferred that matter everywhere rotates with the angular velocity 2 [square root of 4[pi]G[rho]].

Let us consider the reduced Godel metric

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

All light rays emitted from an event on the symmetry axis reconverge at a later event on this axis, with the null geodesics forming a circular cusp [3].

If a curve c is defined by [sinh.sup.4] w = 1, that is

c = ln(l + [square root of 2]), (1.22)

hence, any circle w > ln(l + [square root of 2]) in the plane t = 0, is a closed timelike curve.

2 The modified Godel metric

2.1 Conformal transformation

Now we will assume that the A-term is slightly varying with the time t, so B is also variable through the dust density. See (1.16) for detail. By setting

y = r cosh w, (2.1)

where r is another radial parameter, we choose:

B = [1/2] [(1 - [[L.sub.0]/2[square root of [t.sup.2] - [y.sup.2]]]).sup.2] (2.2)

where [L.sub.0] is a constant whose meaning will become apparent in the next sub-section. B is now identified with a conformal factor.

Note: one of the Kretschmann scalar is no longer an invariant

[R.sub.abcd] [R.sup.abcd] = [6/[B.sup.4]] (2.3)

which reflects the fact that the Godel space-time may be not fully homogeneous.

Anticipating on our postulate, we will state that the variation of B is only localized in a certain region of the Godel model. The [LAMBDA]-term remains constant throughout the complete metric as initially derived, thus preserving its homogeneity.

2.2 The postulate

Our fundamental assumption will now consist of considering our observed Universe as being local. By local we mean that the rotation [phi] is unobservable since we assume that our world is situated at

w = 0.

Our (local) Universe is now becoming isotropic.

In this case, the Godel metric reduces to a standard conformal solution where the light cone is centered about the t-axis:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.4)

We now make the following transformations

[L.sub.0] = [F.sub.0] (2.5)

with [F.sub.0] defined in (1.8)

r = [[F.sub.0]/2] [e.sup.[eta]] sinh [chi], t = [[F.sub.0]/2] [e.sup.[eta]] cosh [chi], (2.6)

[[F.sub.0]/2] [e.sup.[eta]] = [square root of [t.sup.2] - [r.sup.2]] (2.7)

tanh [chi] = [r/t], (2.8)

and we retrieve the Roberston-Walker metric for an open Universe with the sole radial coordinate r:

[([ds.sup.2]).sub.RW] = [F.sup.2]([eta])[d[[eta].sup.2] - d[[chi].sub.2]]. (2.9)

Remark: The Weyl tensor of the Godel solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.10)

which has Petrov type D, vanishes for (2.9). Indeed, the equivalent metric (2.4) implies that [C.sup.ab.sub.cd] = 0 for this conformally flat space-time.

Our observed Universe would then be devoid of the Weyl curvature which explains why it is purely described in terms of the Ricci tensor alone. In this view, Einstein was perhaps an even more exceptional visionary mind than is yet currently admitted.

2.3 Hubble expansion

In our local world, the null geodesics are obviously given by [([ds.sup.2]).sub.RW] = 0, that is

d[eta] = [+ or -] d[chi] (2.11)

and integrating

X = [+ or -] n + const. (2.12)

Let us place ourselves at t([eta]), where we observe a light ray emitted at [chi] where its frequency is [v.sub.0]. In virtue of (2.12), the emission time will be t([eta] - [chi]), and we observe an apparent frequency given by:

v = [[v.sub.0]F([eta] - [chi])/F([eta])]. (2.13)

As F([eta]) increases monotonically, we have v < [v.sub.0] which is the expression of a red shifted light. Most observed red shifts are rather small, so that F([eta] - [chi]) can be expanded as a Taylor series about t([eta] - [chi]) = t([eta]) and we finally get, limiting to the first two terms

F([eta] - [chi]) = F([eta]) + [t ([eta] - [chi]) - t([eta])] F'([eta]) (2.14)

= F([eta]){1 + [H.sub.0][t ([eta] - [chi]) - t([eta])]) (2.15)

where F' denotes differentiation with respect to [eta]

[H.sub.0] = [F'([eta])/F([eta])] (2.16)

is the present numerical value of Hubble constant.

The Godel solution has a non-zero cosmological term, but not the local RW metric.

This agrees with the fact that our open local Universe has a singularity in the past and no singularity in the future [4], in accordance with astronomical observations.

Concluding remarks

Closed timelike curves turn out to exist in many other exact solutions to Einstein's field equations.

It would seem that the first model exhibiting this property was pioneered by C. Lanczos in 1924 [5], and later rediscovered under another form by W. J. Van Stockum in 1937 [6].

However, unlike the Godel solution, the dust particles of these Universes are rotating about a geometrically distinguished axis.

Even worse, the matter density is shown to increase with radius w, a feature which seriously contradicts all current observations.

In this sense, the Godel metric appears as a more plausible model characterizing a broaden Universe which is compatible with our astronomical data, provided one is prepared to accept the fact that our observed world is purely local.

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

References

(1.) Godel K. An example of a new type of cosmological solutions of Einstein's field equations of gravitation. Reviews of Modern Physics, 1949, v. 21, no. 3, 447-450.

(2.) Landau L. and Lifshitz E. The Classical Theory of Fields. Addison-Wesley, Reading (Massachusetts), 1962, p. 402 (French translation).

(3.) Hawking S. W. The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge, 1987.

(4.) Fang Li Zhi and Ruffini R. Basic Concepts in Relativistic Astrophysics. World Scientific (Singapore), 1983.

(5.) Lanczos C. Uber eine Stationare Kosmologie im Sinne der Einsteinischer Gravitationstheorie. Zeitschriftfur Physik, 1924, Bd. 21, 73.

(6.) Van Stockum W. J. The gravitational field of a distribution of particles rotating around an axis of symmetry. Proc. Roy. Soc. Edinburgh, 1937, v. A57, 135.

Patrick Marquet

11 rue des Chapelles, 94350 Villiers/Marne, Paris, France

E-mail: patrick.marquet6@wanadoo.fr
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Date:Oct 1, 2014
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