Gödel metric

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

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

Here, the first term is characteristic of a lambdavacuum solution and the second term is characteristic of a pressureless perfect fluid or dust solution. Notice that 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 the Einstein field equation. Gödel's original chart (given here) is geodesically complete and singularity free; therefore, it is a global chart, and the spacetime is homeomorphic to R⁴, and therefore, simply connected.

In any Lorentzian spacetime, the fourth-rank Riemann tensor is a multilinear operator on the four-dimensional space of tangent vectors (at some event), but a linear operator on the six-dimensional space of bivectors at that event. Accordingly, it has a characteristic polynomial, whose roots are the eigenvalues. In the Gödel spacetime, these eigenvalues are very simple:

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

This spacetime admits a five-dimensional Lie algebra of Killing vectors, which can be generated by time translation ${\displaystyle \partial _{t}}$, two spatial translations ${\displaystyle \partial _{y},\;\partial _{z}}$, plus two further Killing vector fields:

${\displaystyle \partial _{x}-z\,\partial _{z}}$

and

${\displaystyle -2\exp(-x)\,\partial _{t}+z\,\partial _{x}+\left(\exp(-2x)-z^{2}/2\right)\,\partial _{z}.}$

The isometry group acts transitively (since we can translate in ${\displaystyle t,y,z}$, and using the fourth vector we can move along ${\displaystyle x}$ as well), so the spacetime is homogeneous. However, it is not isotropic, as we shall see.

It is obvious from the generators just given that the slices ${\displaystyle x=x_{0}}$ admit a transitive abelian three-dimensional transformation group, so a quotient of the solution can be reinterpreted as a stationary cylindrically symmetric solution. Less obviously, the slices ${\displaystyle y=y_{0}}$ admit an SL(2,R) action, and the slices ${\displaystyle t=t_{0}}$ admit a Bianchi III (c.f. the fourth Killing vector field). We can restate this by saying that our symmetry group includes as three-dimensional subgroups examples of Bianchi types I, III and VIII. Four of the five Killing vectors, as well as the curvature tensor, do not depend upon the coordinate y. Indeed, the Gödel solution is the Cartesian product of a factor R with a three-dimensional Lorentzian manifold (signature −++).

It can be shown that the Gödel solution is, up to local isometry, the only perfect fluid solution of the Einstein field equation admitting a five-dimensional Lie algebra of Killing vectors.

The Weyl tensor of the Gödel solution has Petrov type D. This means that for an appropriately chosen observer, the tidal forces have Coulomb form.

To study the tidal forces in more detail, we compute the Bel decomposition of the Riemann tensor 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 find that the tidal tensor (with respect to ${\displaystyle {\vec {u}}={\vec {e}}_{0}}$, which components evaluated in our frame) has the form

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

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

The gravitomagnetic tensor vanishes identically

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

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 principal Lorentz 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 (the Kretschmann invariant) is constant reflects the homogeneity of the Gödel spacetime.

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

${\displaystyle -\omega {\vec {e}}_{2}}$

This means that the world lines of nearby dust particles are twisting about one another. Furthermore, the shear tensor of the congruence ${\displaystyle {\vec {e}}_{0}}$ vanishes, so the dust particles exhibit rigid rotation.

If we study the past light cone of a given observer, we find that null geodesics moving orthogonally to ${\displaystyle \partial _{y}}$ spiral inwards toward the observer, so that if he looks radially, he 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 himself. However, recall that while the first frame given above (the ${\displaystyle {\vec {e}}_{j}}$) appears static in our chart, the Fermi–Walker derivatives show that, in fact, it is spinning with respect to gyroscopes. The second frame (the ${\displaystyle {\vec {f}}_{j}}$) appears to be spinning in our chart, but it is gyrostabilized, and a nonspinning 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 nonspinning inertial observer looks along his axis of symmetry, he sees his coaxial nonspinning inertial peers apparently nonspinning with respect to himself, as we would expect.

According to Hawking and Ellis, another remarkable feature of this spacetime is the fact that, if we suppress the inessential y coordinate, 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 ${\displaystyle {\vec {e}}_{2}}$ direction can see only finitely far out, and also see themselves at an earlier time.

The cusp is a nongeodesic 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 have closed 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, and arguably succeeded in proving, 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 call presentism, whereas Gödel seems to have been arguing for something more like the philosophy of eternalism), much as he, conversely, succeeded with his incompleteness theorems in showing that intuitive mathematical concepts could not be completely described by formal mathematical systems of proof. See the book A World Without Time.[2]

Einstein was aware of Gödel's solution and commented in Albert 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. a Cauchy 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 not globally hyperbolic.

Gödel did not explain how he found his solution, but there are in fact 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 a cylindrical type chart, featuring two undetermined functions of the radial coordinate:

${\displaystyle {\vec {e}}_{0}=\partial _{t},\;{\vec {e}}_{1}=\partial _{z},\;{\vec {e}}_{2}=\partial _{r},\,{\vec {e}}_{3}={\frac {1}{b(r)}}\,\left(-a(r)\,\partial _{t}+\partial _{\varphi }\right)}$

Here, we think of the timelike unit vector field ${\displaystyle {\vec {e}}_{0}}$ 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

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

This gives the conditions

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

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 but constant 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

${\displaystyle {\vec {e}}_{3}={\frac {1}{r}}\,\partial _{\varphi }+O\left({\frac {1}{r^{2}}}\right)}$

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

${\displaystyle {\vec {e}}_{0}=\partial _{t},\;{\vec {e}}_{1}=\partial _{z},\;{\vec {e}}_{2}=\partial _{r},\;{\vec {e}}_{3}={\frac {{\sqrt {2}}\omega }{\sinh({\sqrt {2}}\omega r)}}\,\partial _{\varphi }-{\frac {{\sqrt {2}}\sinh({\sqrt {2}}\omega r)}{1+\cosh({\sqrt {2}}\omega r)}}\,\partial _{t}}$

From the metric tensor we find that the vector field ${\displaystyle \partial _{\varphi }}$, which is spacelike for small radii, becomes null at ${\displaystyle r=r_{c}}$ where

${\displaystyle r_{c}={\frac {\operatorname {arccosh} (3)}{{\sqrt {2}}\omega }}}$

This is because at that radius we find that ${\displaystyle {\vec {e}}_{3}={\frac {\omega }{2}}\,\partial _{\varphi }-\partial _{t},}$ so ${\displaystyle {\frac {\omega }{2}}\,\partial _{\varphi }={\vec {e}}_{3}+{\vec {e}}_{0}}$ and is therefore null. The circle ${\displaystyle r=r_{c}}$ at a given t 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 factor R 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 ${\displaystyle r=0}$:

Two light cones (with their accompanying frame vectors) in the cylindrical chart for the Gödel lambda dust solution. As we move outwards from the nominal symmetry axis, the cones tip forward and widen. Vertical coordinate lines (representing the world lines of the dust particles) are timelike.

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 ${\displaystyle \partial _{\varphi }}$ becomes null. For larger radii, it is timelike. Thus, corresponding to our symmetry axis we have a timelike congruence made up of circles and corresponding to certain observers. This congruence is however only defined outside the cylinder ${\displaystyle r=r_{c}}$.

This is not a geodesic congruence; rather, each observer in this family must maintain a constant acceleration in order to hold his course. Observers with smaller radii must accelerate harder; as ${\displaystyle r\rightarrow r_{c}}$ 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:

The null geodesics spiral counterclockwise toward an observer on the axis of symmetry. This shows them from "above".

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

The null geodesics are geometrically 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):

Hawking and Ellis picture of expansion and reconvergence of light emitted by an observer on the axis of symmetry.

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