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TheGödel metric, also known as theGödel solution orGödel universe, is anexact solution, found in 1949 byKurt Gödel,^[1] of theEinstein field equations in which thestress–energy tensor contains two terms: the first representing the matter density of a homogeneous distribution of swirling dust particles (seeDust solution), and the second associated with a negativecosmological constant (seeLambdavacuum solution).

This solution has many unusual properties—in particular, the existence ofclosed time-like curves that would allowtime travel in a universe described by the solution. Its definition is somewhat artificial, since the value of the cosmological constant must be carefully chosen to correspond to the density of the dust grains, but thisspacetime is an important pedagogical example.

Like any otherLorentzian spacetime, the Gödel solution represents themetric tensor in terms of a localcoordinate chart. It may be easiest to understand the Gödel universe using the cylindrical coordinate system (see below), but this article uses the chart originally used by Gödel. In this chart, the metric (or, equivalently, theline element) is

${\displaystyle g=\frac{1}{2{\omega }^2}\left[-(dt+e^{x}dy{)}^2+d{x}^2+\frac{1}{2}{e}^{2x}d{y}^2+d{z}^2\right]}$

where${\displaystyle \omega }$ is a non-zero real constant that gives the angular velocity of the surrounding dust grains about they-axis, measured by a "non-spinning" observer riding on one of the dust grains. "Non-spinning" means that the observer does not feel centrifugal forces, but in this coordinate system, it would rotate about an axis parallel to they-axis. In this rotating frame, the dust grains remain at constant values ofx,y, andz. Their density in this coordinate diagram increases withx, but their density in their own frames of reference is the same everywhere.

To investigate the properties of the Gödel solution, theframe field can be assumed (dual to the co-frame read from the metric as given above),

${\displaystyle {\overrightarrow{e}}_0=\sqrt{2}\omega {\mathrm{\partial }}_t}$

${\displaystyle {\overrightarrow{e}}_1=\sqrt{2}\omega {\mathrm{\partial }}_x}$

${\displaystyle {\overrightarrow{e}}_2=\sqrt{2}\omega {\mathrm{\partial }}_y}$

${\displaystyle {\overrightarrow{e}}_3=2\omega \left(\exp (-x){\mathrm{\partial }}_z-{\mathrm{\partial }}_t\right).}$

This framework defines a family of inertial observers that are 'comoving with the dust grains'. The computation of theFermi–Walker derivatives with respect to${\displaystyle {\overrightarrow{e}}_0}$ shows that the spatial frames arespinning about${\displaystyle {\overrightarrow{e}}_2}$ with the angular velocity⁠${\displaystyle -\omega }$⁠. It follows that the 'non spinning inertial frame' comoving with the dust particles is

${\displaystyle {\overrightarrow{f}}_0={\overrightarrow{e}}_0}$

${\displaystyle {\overrightarrow{f}}_1=\cos (\omega t){\overrightarrow{e}}_1-\sin (\omega t){\overrightarrow{e}}_3}$

${\displaystyle {\overrightarrow{f}}_2={\overrightarrow{e}}_2}$

${\displaystyle {\overrightarrow{f}}_3=\sin (\omega t){\overrightarrow{e}}_1+\cos (\omega t){\overrightarrow{e}}_3.}$

The components of theEinstein tensor (with respect to either frame above) are

${\displaystyle G^{\hat{a}\hat{b}}={\omega }^2\mathrm{diag}(-1,1,1,1)+2{\omega }^2\mathrm{diag}(1,0,0,0).}$

Here, the first term is characteristic of aLambdavacuum solution and the second term is characteristic of a pressurelessperfect fluid or dust solution. The cosmological constant is carefully chosen to partially cancel the matter density of the dust.

The Gödel spacetime is a rare example of a regular (singularity-free) solution of theEinstein field equations. Gödel's original chart isgeodesically complete and free of singularities. Therefore, it is a global chart, and the spacetime ishomeomorphic toR⁴, and therefore, simply connected.

In any Lorentzian spacetime, the fourth rankRiemann tensor is a multilinear operator on the four-dimensional space oftangent vectors (at some event), but alinear operator on the six-dimensional space ofbivectors at that event. Accordingly, it has acharacteristic polynomial, whose roots are theeigenvalues. In Gödelian spacetime, these eigenvalues are very simple:

• triple eigenvalue zero,
• double eigenvalue${\displaystyle -{\omega }^2}$,
• single eigenvalue${\displaystyle {\omega }^2}$.

This spacetime admits a five-dimensionalLie algebra ofKilling vectors, which can be generated by 'time translation'${\displaystyle {\mathrm{\partial }}_t}$, two 'spatial translations'${\displaystyle {\mathrm{\partial }}_y,{\mathrm{\partial }}_z}$, plus two further Killing vector fields:

${\displaystyle {\mathrm{\partial }}_x-y{\mathrm{\partial }}_y}$

and

${\displaystyle -2\exp (-x){\mathrm{\partial }}_t+y{\mathrm{\partial }}_x+\left(\exp (-2x)-y^2/2\right){\mathrm{\partial }}_y.}$

The isometry group acts 'transitively' (since we can translate into${\displaystyle t,y,z}$, and with the fourth vector we can move along${\displaystyle x}$), so spacetime is 'homogeneous'. However, it is not 'isotropic', as can be seen.

The given demonstrators show that the slices${\displaystyle x=x_0}$ admit atransitiveabelian three-dimensionaltransformation group, so that a quotient of the solution can be reinterpreted as a stationary cylindrically symmetric solution. The slices${\displaystyle y=y_0}$ allow for anSL(2,R) action, and the slices${\displaystyle t=t_0}$ admit a Bianchi III (cf. the fourth Killing vector field). This can be rewritten as the symmetry group containing three-dimensional subgroups with examples of Bianchi types I, III, and VIII. Four of the five Killing vectors, as well as the curvature tensor do not depend on the coordinate y. The Gödel solution is theCartesian product of a factorR with a three-dimensional Lorentzian manifold (signature −++).

It can be shown that, except for thelocal isometry, the Gödel solution is the only perfect fluid solution of the Einstein field equation which admits a five-dimensional Lie algebra of the Killing vectors.

TheWeyl tensor of the Gödel solution hasPetrov typeD. This means that for an appropriately chosen observer, the tidal forces are very close to those that would be felt from a point mass inNewtonian gravity.

To study the tidal forces in more detail, theBel decomposition of the Riemann tensor can be computed into three pieces, the tidal or electrogravitic tensor (which represents tidal forces), the magnetogravitic tensor (which represents spin-spin forces on spinning test particles and other gravitational effects analogous to magnetism), and the topogravitic tensor (which represents the spatial sectional curvatures).

Observers comoving with the dust particles would observe that the tidal tensor (with respect to${\displaystyle \overrightarrow{u}={\overrightarrow{e}}_0}$, which components evaluated in our frame) has the form

${\displaystyle {E\left[\overrightarrow{u}\right]}_{\hat{m}\hat{n}}={\omega }^2\mathrm{diag}(1,0,1).}$

That is, they measure isotropic tidal tension orthogonal to the distinguished direction${\displaystyle {\mathrm{\partial }}_y}$.

The gravitomagnetic tensor vanishes identically

${\displaystyle {B\left[\overrightarrow{u}\right]}_{\hat{m}\hat{n}}=0.}$

This is an artifact of the unusual symmetries of this spacetime, and implies that the putative "rotation" of the dust does not have the gravitomagnetic effects usually associated with the gravitational field produced by rotating matter.

The principalLorentz invariants of the Riemann tensor are

${\displaystyle R_{abcd}{R}^{abcd}=12{\omega }^4,R_{abcd}{{}^{\star }R}^{abcd}=0.}$

The vanishing of the second invariant means that some observers measure no gravitomagnetism, which is consistent with what was just said. The fact that the first invariant (theKretschmann invariant) is constant reflects the homogeneity of the Gödel spacetime.

The frame fields given above are both inertial,${\displaystyle {\mathrm{\nabla }}_{{\overrightarrow{e}}_0}{\overrightarrow{e}}_0=0}$, but thevorticity vector of the timelike geodesic congruence defined by the timelike unit vectors is

${\displaystyle -\omega {\overrightarrow{e}}_2}$

This means that the world lines of nearby dust particles are twisting about one another. Furthermore, the shear tensor of the congruence${\displaystyle {\overrightarrow{e}}_0}$ vanishes, so the dust particles exhibit rigid rotation.

If the pastlight cone of a given observer is studied, it can be found that null geodesics moving orthogonally to${\displaystyle {\mathrm{\partial }}_y}$ spiral inwards toward the observer, so that if one looks radially, one sees the other dust grains in progressively time-lagged positions. However, the solution is stationary, so it might seem that an observer riding on a dust grain will not see the other grains rotating about oneself. However, recall that while the first frame given above (the${\displaystyle {\overrightarrow{e}}_j}$) appears static in the chart, the Fermi–Walker derivatives show that it is spinning with respect to gyroscopes. The second frame (the${\displaystyle {\overrightarrow{f}}_j}$) appears to be spinning in the chart, but it is gyrostabilized, and a non-spinning inertial observer riding on a dust grain will indeed see the other dust grains rotating clockwise with angular velocity${\displaystyle \omega }$ about his axis of symmetry. It turns out that in addition, optical images are expanded and sheared in the direction of rotation.

If a non-spinning inertial observer looks along his axis of symmetry, one sees one's coaxial non-spinning inertial peers apparently non-spinning with respect to oneself, as would be expected.

According to Hawking and Ellis, another remarkable feature of this spacetime is the fact that, if the inessential y coordinate is suppressed, light emitted from an event on the world line of a given dust particle spirals outwards, forms a circular cusp, then spirals inward and reconverges at a subsequent event on the world line of the original dust particle. This means that observers looking orthogonally to the${\displaystyle {\overrightarrow{e}}_2}$ direction can see only finitely far out, and also see themselves at an earlier time.

The cusp is a non-geodesic closed null curve. (See the more detailed discussion below using an alternative coordinate chart.)

Because of the homogeneity of the spacetime and the mutual twisting of our family of timelike geodesics, it is more or less inevitable that the Gödel spacetime should haveclosed timelike curves (CTCs). Indeed, there are CTCs through every event in the Gödel spacetime. This causal anomaly seems to have been regarded as the whole point of the model by Gödel himself, who was apparently striving to prove that Einstein's equations of spacetime are not consistent with what we intuitively understand time to be (i. e. that it passes and the past no longer exists, the position philosophers callpresentism, whereas Gödel seems to have been arguing for something more like the philosophy ofeternalism).^[2]

Einstein was aware of Gödel's solution and commented inAlbert Einstein: Philosopher-Scientist^[3] that if there are a series of causally-connected events in which "the series is closed in itself" (in other words, a closed timelike curve), then this suggests that there is no good physical way to define whether a given event in the series happened "earlier" or "later" than another event in the series:

In that case the distinction "earlier-later" is abandoned for world-points which lie far apart in a cosmological sense, and those paradoxes, regarding the direction of the causal connection, arise, of which Mr. Gödel has spoken.

Such cosmological solutions of the gravitation-equations (with not vanishing A-constant) have been found by Mr. Gödel. It will be interesting to weigh whether these are not to be excluded on physical grounds.

If the Gödel spacetime admitted any boundaryless temporal hyperslices (e.g. aCauchy surface), any such CTC would have to intersect it an odd number of times, contradicting the fact that the spacetime is simply connected. Therefore, this spacetime is notglobally hyperbolic.

In this section, we introduce another coordinate chart for the Gödel solution, in which some of the features mentioned above are easier to see.

Gödel did not explain how he found his solution, but there are many possible derivations. We will sketch one here, and at the same time verify some of the claims made above.

Start with a simple frame in acylindrical type chart, featuring two undetermined functions of the radial coordinate:

${\displaystyle {\overrightarrow{e}}_0={\mathrm{\partial }}_t,{\overrightarrow{e}}_1={\mathrm{\partial }}_z,{\overrightarrow{e}}_2={\mathrm{\partial }}_r,{\overrightarrow{e}}_3=\frac{1}{b(r)}\left(-a(r){\mathrm{\partial }}_t+{\mathrm{\partial }}_{\phi }\right)}$

Here, we think of the timelike unit vector field${\displaystyle {\overrightarrow{e}}_0}$ as tangent to the world lines of the dust particles, and their world lines will in general exhibit nonzero vorticity but vanishing expansion and shear. Let us demand that the Einstein tensor match a dust term plus a vacuum energy term. This is equivalent to requiring that it match a perfect fluid; i.e., we require that the components of the Einstein tensor, computed with respect to our frame, take the form

${\displaystyle G^{\hat{i}\hat{j}}=\mu \mathrm{diag}(1,0,0,0)+p\mathrm{diag}(0,1,1,1)}$

This gives the conditions

${\displaystyle b^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}=\frac{b^{\mathrm{\prime }\mathrm{\prime }}{b}^{\mathrm{\prime }}}{b},{\left(a^{\mathrm{\prime }}\right)}^2=2b^{\mathrm{\prime }\mathrm{\prime }}b}$

Plugging these into the Einstein tensor, we see that in fact we now have${\displaystyle \mu =p}$. The simplest nontrivial spacetime we can construct in this way evidently would have this coefficient be some nonzero butconstant function of the radial coordinate. Specifically, with a bit of foresight, let us choose${\displaystyle \mu ={\omega }^2}$. This gives

${\displaystyle b(r)=\frac{\sinh (\sqrt{2}\omega r)}{\sqrt{2}\omega },a(r)=\frac{\cosh (\sqrt{2}\omega r)}{\omega }+c}$

Finally, let us demand that this frame satisfy

${\displaystyle {\overrightarrow{e}}_3=\frac{1}{r}{\mathrm{\partial }}_{\phi }+O\left(\frac{1}{r^2}\right)}$

This gives${\displaystyle c=-1/\omega }$, and our frame becomes

${\displaystyle {\overrightarrow{e}}_0={\mathrm{\partial }}_t,{\overrightarrow{e}}_1={\mathrm{\partial }}_z,{\overrightarrow{e}}_2={\mathrm{\partial }}_r,{\overrightarrow{e}}_3=\frac{\sqrt{2}\omega }{\sinh (\sqrt{2}\omega r)}{\mathrm{\partial }}_{\phi }-\frac{\sqrt{2}\sinh (\sqrt{2}\omega r)}{1+\cosh (\sqrt{2}\omega r)}{\mathrm{\partial }}_t}$

From the metric tensor we find that the vector field${\displaystyle {\mathrm{\partial }}_{\phi }}$, which isspacelike for small radii, becomesnull at${\displaystyle r=r_c}$ where

${\displaystyle r_c=\frac{\mathrm{arccosh}(3)}{\sqrt{2}\omega }}$

This is because at that radius we find that⁠${\displaystyle {\overrightarrow{e}}_3=\frac{\omega }{2}{\mathrm{\partial }}_{\phi }-{\mathrm{\partial }}_t}$⁠, so${\displaystyle \frac{\omega }{2}{\mathrm{\partial }}_{\phi }={\overrightarrow{e}}_3+{\overrightarrow{e}}_0}$ and is therefore null. The circle${\displaystyle r=r_c}$ at a givent is a closed null curve, but not a null geodesic.

Examining the frame above, we can see that the coordinate${\displaystyle z}$ is inessential; our spacetime is the direct product of a factorR with a signature −++ three-manifold. Suppressing${\displaystyle z}$ in order to focus our attention on this three-manifold, let us examine how the appearance of the light cones changes as we travel out from the axis of symmetry${\displaystyle r=0}$:

When we get to the critical radius, the cones become tangent to the closed null curve.

At the critical radius${\displaystyle r=r_c}$, the vector field${\displaystyle {\mathrm{\partial }}_{\phi }}$ becomes null. For larger radii, it istimelike. Thus, corresponding to our symmetry axis we have a timelikecongruence made up ofcircles and corresponding to certain observers. This congruence is howeveronly defined outside the cylinder${\displaystyle r=r_c}$.

This is not a geodesic congruence; rather, each observer in this family must maintain aconstant acceleration in order to hold his course. Observers with smaller radii must accelerate harder; as${\displaystyle r\to r_c}$ the magnitude of acceleration diverges, which is just what is expected, given that${\displaystyle r=r_c}$ is a null curve.

If we examine the past light cone of an event on the axis of symmetry, we find the following picture:

Recall that vertical coordinate lines in our chart represent the world lines of the dust particles, butdespite their straight appearance in our chart, the congruence formed by these curves has nonzero vorticity, so the world lines are actuallytwisting about each other. The fact that the null geodesics spiral inwards in the manner shown above means that when our observer, when lookingradially outwards, sees nearby dust particles not at their current locations, but at their earlier locations. This is what we would expect if the dust particles are in fact rotating about one another.

The null geodesics aregeometrically straight; in the figure, they appear to be spirals only because the coordinates are "rotating" in order to permit the dust particles to appear stationary.

According to Hawking and Ellis (see monograph cited below), all light rays emitted from an event on the symmetry axis reconverge at a later event on the axis, with the null geodesics forming a circular cusp (which is a null curve, but not a null geodesic):

This implies that in the Gödel lambda dust solution, theabsolute future of each event has a character very different from what we might naively expect.

Following Gödel, we can interpret the dust particles as galaxies, so that the Gödel solution becomes acosmological model of a rotating universe. Besides rotating, this model exhibits noHubble expansion, so it is not a realistic model of the universe in which we live, but can be taken as illustrating an alternative universe, which would in principle be allowed by general relativity (if one admits the legitimacy of a negative cosmological constant). Less well known solutions of Gödel's exhibit both rotation and Hubble expansion and have other qualities of his first model, but traveling into the past is not possible. According toStephen Hawking,these models could well be a reasonable description of the universe that we observe, however observational data are compatible only with a very low rate of rotation.^[4] The quality of these observations improved continually up until Gödel's death, and he would always ask "Is the universe rotating yet?" and be told "No, it isn't".^[5]

We have seen that observers lying on they axis (in the original chart) see the rest of the universe rotating clockwise about that axis. However, the homogeneity of the spacetime shows that thedirection but not theposition of this "axis" is distinguished.

Some have interpreted the Gödel universe as a counterexample to Einstein's hopes that general relativity should exhibit some kind ofMach's principle,^[4] citing the fact that the matter is rotating (world lines twisting about each other) in a manner sufficient to pick out a preferred direction, although with no distinguished axis of rotation.

Others^{[citation needed]} take Mach principle to mean some physical law tying the definition of non-spinning inertial frames at each event to the global distribution and motion of matter everywhere in the universe, and say that because the non-spinning inertial frames are precisely tied to the rotation of the dust in just the way such a Mach principle would suggest, this modeldoes accord with Mach's ideas.

Many other exact solutions that can be interpreted as cosmological models of rotating universes are known.^[6]

• van Stockum dust, for another rotating dust solution with (true) cylindrical symmetry,
• Dust solution, an article about dust solutions in general relativity.

• G. Dautcourt and M. Abdel-Megied (2006). "Revisiting the Light Cone of the Goedel Universe".Classical and Quantum Gravity.23 (4):1269–1288.arXiv:gr-qc/0511015.Bibcode:2006CQGra..23.1269D.doi:10.1088/0264-9381/23/4/013.S2CID 14666907.
• Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius; Herlt, Eduard (2003).Exact Solutions to Einstein's Field Equations (2nd ed.). Cambridge: Cambridge University Press.ISBN 0-521-46136-7. Seesection 12.4 for the uniqueness theorem.
• Hawking, Stephen; Ellis, G. F. R. (1973).The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press.ISBN 0-521-09906-4. Seesection 5.7 for a classic discussion of CTCs in the Gödel spacetime.Warning: in Fig. 31, the light cones do indeed tip over, but they also widen, so that vertical coordinate lines are always timelike; indeed, these represent the world lines of the dust particles, so they are timelike geodesics.
• Gödel, Kurt (1949-07-01)."An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation"(PDF).Reviews of Modern Physics.21 (3):447–450.Bibcode:1949RvMP...21..447G.doi:10.1103/RevModPhys.21.447.ISSN 0034-6861.
• Vukovic R. (2014):Tensor Model of the Rotating Universe, Exercise in Special RelativityArchived 2014-11-13 at theWayback Machine.

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