General relativity

G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}




Equations


Formalisms


Advanced theory






The Gödel metric is an exact solution of the Einstein field equations in which the stress–energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles, and the second associated with a nonzero cosmological constant (see lambdavacuum solution). It is also known as the Gödel solution.
This solution has many strange properties, discussed below, in particular the existence of closed timelike curves which would allow for a form of time travel in the type of universe described by the solution. Its definition is somewhat artificial (the value of the cosmological constant must be carefully chosen to match the density of the dust grains), but this spacetime is regarded as an important pedagogical example.
The solution was found in 1949 by Kurt Gödel.
Definition
Like any other Lorentzian spacetime, the Gödel solution is defined by giving the metric tensor in terms of some local coordinate chart. It may be easiest to understand the Gödel universe using the cylindrical coordinate system presented lower down, but here we will give the chart that Gödel originally used. In this chart the line element is

ds^2= \frac{1}{2\omega^2} [ (dt + e^x dz)^2 + dx^2 + dy^2 + \tfrac{1}{2} e^{2x} dz^2], \qquad\qquad \infty < t,x,y,z < \infty,
where \omega is a nonzero real constant, which turns out to be the angular velocity around the y axis, as measured by a "nonspinning" observer riding any one of the dust grains, of the surrounding dust grains. ("Nonspinning" means that he doesn't feel centrifugal forces on his members, but in this coordinate frame he would actually be turning on an axis parallel to the y axis.) As we shall see, the dust grains stay at constant values of x,y,z. Their density in this coordinate chart increases with x, but their density in their own frames of reference is the same everywhere.
Properties
To study the properties of the Gödel solution, we can adopt the frame field (dual to the coframe read off the metric as given above)

\vec{e}_0 = \sqrt{2} \omega \, \partial_t

\vec{e}_1 = \sqrt{2} \omega \, \partial_x

\vec{e}_2 = \sqrt{2} \omega \, \partial_y

\vec{e}_3 = 2 \omega \, \left( \exp(x) \, \partial_z  \, \partial_t \right).
This frame defines a family of inertial observers who are comoving with the dust grains. However, computing the Fermi–Walker derivatives with respect to \vec{e}_0 shows that the spatial frames are spinning about \vec{e}_2 with angular velocity \omega. It follows that the nonspinning inertial frame comoving with the dust particles is

\vec{f}_0 = \vec{e}_0

\vec{f}_1 = \cos(\omega t) \, \vec{e}_1  \sin(\omega t) \, \vec{e}_3

\vec{f}_2 = \vec{e}_2

\vec{f}_3 = \sin(\omega t) \, \vec{e}_1 + \cos(\omega t) \, \vec{e}_3.
Einstein tensor
The components of the Einstein tensor (with respect to either frame above) are

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.
Topology
The Gödel spacetime is a rare example of a regular (singularityfree) solution of the Einstein field equation. The chart given here (the original chart of Gödel) is geodesically complete and singularity free; therefore, it is a global chart, and the spacetime is homeomorphic to R^{4}, and therefore simply connected.
Invariants
The curvature invariants of the Gödel spacetime are remarkable. We'll mention just one feature.
In any Lorentzian spacetime, the fourthrank Riemann tensor is a multilinear operator on the fourdimensional space of tangent vectors (at some event), but a linear operator on the sixdimensional space of bivectors at that event. Accordingly it has a characteristic polynomial, whose roots are the eigenvalues. In the Gödel spacetime, these eigenvalues are extremely simple:

triple eigenvalue zero,

double eigenvalue \omega^2,

simple eigenvalue \omega^2.
Killing vectors
This spacetime admits a remarkable fivedimensional Lie algebra of Killing vectors, which can be generated by time translation \partial_t , two spatial translations \partial_y, \; \partial_z, plus two further Killing vector fields:

\partial_x  z \, \partial_z
and

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 t,y,z, and using the fourth vector we can move along 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 x=x_0 admit a transitive abelian threedimensional transformation group, so a quotient of the solution can be reinterpreted as a stationary cylindrically symmetric solution. Less obviously, the slices y=y_0 admit an SL(2,R) action, and the slices 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 threedimensional 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 threedimensional 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 fivedimensional Lie algebra of Killing vectors.
Petrov type and Bel decomposition
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 spinspin forces on spinning test particles and other gravitational effects analogous to magnetism), and the topogravitic tensor (which represents the spatial sectional curvatures).
Interestingly enough, observers comoving with the dust particles find that the tidal tensor (with respect to \vec{u} = \vec{e}_0, which components evaluated in our frame) has the form

{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 \partial_y.
The gravitomagnetic tensor vanishes identically

{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

R_{abcd} \, R^{abcd} = 12 \omega^4, \; R_{abcd} = \mu \, \operatorname{diag}(1,0,0,0) + p \, \operatorname{diag}(0,1,1,1)
This gives the conditions

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 \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 \mu = \omega^2. This gives

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

\vec{e}_3 = \frac{1}{r} \, \partial_\phi + O \left( \frac{1}{r^2} \right)
This gives c=1/\omega, and our frame becomes

\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_\phi  \frac{\sqrt{2}\sinh(\sqrt{2} \omega r)}{1+\cosh(\sqrt{2} \omega r)} \, \partial_t
Appearance of the light cones
From the metric tensor we find that the vector field \partial_\phi, which is of course spacelike for small radii, becomes null at r=r_c where

r_c = \frac{\operatorname{arccosh}(3)}{\sqrt{2} \omega}
This is because at that radius we find that \vec{e}_3 = \frac{ \omega }2 \, \partial_\phi  \partial_t, so \frac{ \omega }2 \, \partial_\phi=\vec{e}_3+\vec{e}_0 and is therefore null. The circle 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 z is inessential; our spacetime is the direct product of a factor R with a signature ++ threemanifold. Suppressing z in order to focus our attention on this threemanifold, let us examine how the appearance of the light cones changes as we travel out from the axis of symmetry 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. Note that vertical coordinate lines (representing the world lines of the dust particles) are always timelike.
When we get to the critical radius, the cones become tangent to the closed null curve.
A congruence of closed timelike curves
At the critical radius r = r_c, the vector field \partial_\phi 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 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 r \rightarrow r_c the magnitude of acceleration diverges, which is of course just what we should expect, given that r=r_c is a null curve.
Null geodesics
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.
Note that the null geodesics are of course 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.
The absolute future
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.
Cosmological interpretation
Following Gödel, we can interpret the dust particles as galaxies, so that the Gödel solution becomes a cosmological model of a rotating universe. Besides rotating, this model exhibits no Hubble 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 nonzero cosmological constant). A less well known solution of Gödel's exhibits both rotation and Hubble expansion, and has the other qualities of his first model, so Gödel's model is really killed by the inconvenient observations that the universe is not rotating. The quality of these observations improved continually up until his death, and he would always ask "is the universe rotating yet?" and be told "no, it isn't." ^{[2]}
We have seen that observers lying on the y axis (in the original chart) see the rest of the universe rotating clockwise about that axis. However, the homogeneity of the spacetime shows that the direction but not the position 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 of Mach principle, 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 take Mach principle to mean some physical law tying the definition of nonspinning inertial frames at each event to the global distribution and motion of matter everywhere in the universe, and say that because the nonspinning inertial frames are precisely tied to the rotation of the dust in just the way such a Mach principle would suggest, this model does accord with Mach's ideas.
Many other exact solutions which can be interpreted as cosmological models of rotating universes are known. See the book by Ryan and Shepley for some of these generalizations.
See also

van Stockum dust, for another rotating dust solution with (true) cylindrical symmetry,

dust solution, an article about dust solutions in general relativity.
References

^ Einstein, Albert (1949). "Einstein's Reply to Criticisms". Albert Einstein: PhilosopherScientist. Cambridge University Press. Retrieved 29 November 2012.

^ Reflections on Kurt Gödel, by Hao Wang, MIT Press, (1987), p.183.

G.Dautcourt and M. AbdelMegied (2006). "Revisiting the Light Cone of the Goedel Universe". Classical and Quantum Gravity 23 (4): 1269–1288.

Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius; Herlt, Eduard (2003). Exact Solutions to Einstein's Field Equations (2nd ed.). Cambridge: Cambridge University Press. See section 12.4 for the uniqueness theorem.

Ryan, M. P.; and Shepley, L. C. (1975). Homogeneous Relativistic Cosmologies. Princeton: Princeton University Press.

Hawking, Stephen; and Ellis, G. F. R. (1973). The Large Scale Structure of SpaceTime. Cambridge: Cambridge University Press. See section 5.7 for a classic discussion of CTC's 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, K. (1949). "An example of a new type of cosmological solution of Einstein's field equations of gravitation". Rev. Mod. Phys. 21 (3): 447–450.

Gödel universe on arxiv.org

, Exercise in Special RelativityTensor Model of the Rotating UniverseVukovic R. (2014):
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