The Gödel metric, also known as the Gödel solution or Gödel universe, is an
exact solution of the
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
in which the
stress–energy tensor
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...
contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles (
dust solution
In general relativity, a dust solution is a fluid solution, a type of exact solution of the Einstein field equation, in which the gravitational field is produced entirely by the mass, momentum, and stress density of a perfect fluid that has ' ...
), and the second associated with a negative
cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is the constant coefficient of a term that Albert Einstein temporarily added to his field equ ...
(see
Lambdavacuum solution).
This solution has many unusual properties—in particular, the existence of
closed timelike curve
In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van Sto ...
s that would allow
time travel
Time travel is the concept of movement between certain points in time, analogous to movement between different points in space by an object or a person, typically with the use of a hypothetical device known as a time machine. Time travel is a w ...
in a universe described by the solution. Its definition is somewhat artificial in that the value of the cosmological constant must be carefully chosen to match the density of the dust grains, but this
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
is an important pedagogical example.
This solution was found in 1949 by
Kurt Gödel
Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...
.
Definition
Like any other
Lorentzian spacetime, the Gödel solution presents the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
in terms of some local
coordinate chart In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
. It may be easiest to understand the Gödel universe using the cylindrical coordinate system (presented below), but this article uses the chart that Gödel originally used. In this chart, the metric (or equivalently the
line element
In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc l ...
) is
:
where
is a nonzero real constant, which is the angular velocity of the surrounding dust grains around the ''y'' axis, as measured by a "non-spinning" observer riding one of the dust grains. "Non-spinning" means that the observer does not feel centrifugal forces, but in this coordinate frame it would be turning on an axis parallel to the ''y'' axis. As can be seen, the dust grains stay at constant values of ''x'', ''y'', and ''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, the
frame field can be adopted (dual to the coframe read off the metric as given above),
:
:
:
:
This frame defines a family of inertial observers who are 'comoving with the dust grains'. However, computing the
Fermi–Walker derivatives with respect to
shows that the spatial frames are ''spinning'' about
with angular velocity
. It follows that the 'non spinning inertial frame' comoving with the dust particles is
:
:
:
:
Einstein tensor
The components of the
Einstein tensor
In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field eq ...
(with respect to either frame above) are
:
Here, the first term is characteristic of a
Lambdavacuum solution and the second term is characteristic of a pressureless
perfect fluid
In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame mass density \rho_m and ''isotropic'' pressure ''p''. Real fluids are "sticky" and contain (and conduct) heat. Perfect fluids are idealized models in whi ...
or dust solution. 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 (singularity-free) solution of the
Einstein field equation
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
. Gödel's original chart is
geodesically complete In mathematics, a complete manifold (or geodesically complete manifold) is a (pseudo-) Riemannian manifold for which, starting at any point , you can follow a "straight" line indefinitely along any direction. More formally, the exponential map a ...
and singularity free; therefore, it is a global chart, and the spacetime is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to R
4, and therefore, simply connected.
Curvature invariants
In any Lorentzian spacetime, the fourth-rank
Riemann tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. I ...
is a multilinear operator on the four-dimensional space of
tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
s (at some event), but a
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
on the six-dimensional space of
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalar (mathematics), scalars and Euclidean vector, vectors. If a scalar is considered a degree-zero quantity, and a vector is a d ...
s at that event. Accordingly, it has a
characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...
, whose roots are the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s. In the Gödel spacetime, these eigenvalues are very simple:
* triple eigenvalue zero,
* double eigenvalue
,
* single eigenvalue
.
Killing vectors
This spacetime admits a five-dimensional
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
of
Killing vector
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal gene ...
s, which can be generated by '
time translation
Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the law that the laws of physics are unchanged (i ...
'
, two 'spatial translations'
, plus two further Killing vector fields:
:
and
:
The isometry group acts 'transitively' (since we can translate in
, and using the fourth vector we can move along
as well), so the spacetime is 'homogeneous'. However, it is not 'isotropic', as can be seen.
The demonstrators given demonstrate that the slices
admit a
transitive abelian three-dimensional
transformation group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
, so a quotient of the solution can be reinterpreted as a stationary cylindrical symmetric solution. The slices
admit an
SL(2,R) action, and the slices
admit a Bianchi III (c.f. the fourth Killing vector field). This can be restated by saying that the symmetry group includes 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. The Gödel solution is the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
of a factor R with a three-dimensional Lorentzian manifold (
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
−++).
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.
Petrov type and Bel decomposition
The
Weyl tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal f ...
of the Gödel solution has
Petrov type D. This means that for an appropriately chosen observer, the tidal forces are very close to those that would be felt from a point mass in
Newtonian gravity.
To study the tidal forces in more detail, the
Bel decomposition
In semi-Riemannian geometry, the Bel decomposition, taken with respect to a specific timelike congruence, is a way of breaking up the Riemann tensor of a pseudo-Riemannian manifold into lower order tensors with properties similar to the electric ...
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
, which components evaluated in our frame) has the form
:
That is, they measure isotropic tidal tension orthogonal to the distinguished direction
.
The gravitomagnetic tensor vanishes identically
:
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 invariant
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of v ...
s of the Riemann tensor are
:
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 In the theory of pseudo-Riemannian manifold, Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic curvature invariant (general relativity), scalar invariant. It was introduce ...
) is constant reflects the homogeneity of the Gödel spacetime.
Rigid rotation
The frame fields given above are both inertial,
, but the ''vorticity vector'' of the timelike geodesic congruence defined by the timelike unit vectors is
:
This means that the world lines of nearby dust particles are twisting about one another. Furthermore, the shear tensor of the congruence
vanishes, so the dust particles exhibit rigid rotation.
Optical effects
If the past
light cone
In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take thro ...
of a given observer is studied, it can be found that null geodesics moving orthogonally to
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
) appears static in the chart, the Fermi–Walker derivatives show that, in fact, it is spinning with respect to gyroscopes. The second frame (the
) 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
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.
Shape of absolute future
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
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.)
Closed timelike curves
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 curve
In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van Sto ...
s (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 call
presentism, whereas Gödel seems to have been arguing for something more like the philosophy of
eternalism).
Einstein was aware of Gödel's solution and commented in ''Albert Einstein: Philosopher-Scientist'' 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.
Globally nonhyperbolic
If the Gödel spacetime admitted any boundary-less temporal hyperslices (e.g. a
Cauchy surface In the mathematical field of Lorentzian geometry, a Cauchy surface is a certain kind of submanifold of a Lorentzian manifold. In the application of Lorentzian geometry to the physics of general relativity, a Cauchy surface is usually interpreted as ...
), 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.
A cylindrical chart
In this section, we introduce another coordinate chart for the Gödel solution, in which some of the features mentioned above are easier to see.
Derivation
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:
:
Here, we think of the timelike unit vector field
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
:
This gives the conditions
:
Plugging these into the Einstein tensor, we see that in fact we now have
. 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
. This gives
:
Finally, let us demand that this frame satisfy
:
This gives
, and our frame becomes
:
Appearance of the light cones
From the metric tensor we find that the vector field which is ''spacelike'' for small radii, becomes ''null'' at
where
:
This is because at that radius we find that
so
and is therefore null. The circle
at a given ''t'' is a closed null curve, but not a null geodesic.
Examining the frame above, we can see that the coordinate
is inessential; our spacetime is the direct product of a factor R with a signature −++ three-manifold. Suppressing
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
![Goedel lambdadust samplecones](https://upload.wikimedia.org/wikipedia/commons/c/c7/Goedel_lambdadust_samplecones.gif)
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
, the vector field
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''
.
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
the magnitude of acceleration diverges, which is just what is expected, given that
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:
![Goedel lambdadust centralcone inspiral](https://upload.wikimedia.org/wikipedia/commons/6/65/Goedel_lambdadust_centralcone_inspiral.gif)
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, when looking ''radially 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 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.
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):
![Goedel lambdadust centralcone](https://upload.wikimedia.org/wikipedia/commons/4/48/Goedel_lambdadust_centralcone.gif)
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
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving ...
, 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 to
Stephen 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.
[S. W. Hawking, ''Introductory note to 1949 and 1952'' in Kurt Gödel, ''Collected works'', Volume II (S. Feferman et al., eds).] 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".
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's principle
In theoretical physics, particularly in discussions of gravitation theories, Mach's principle (or Mach's conjecture) is the name given by Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach. The hypothe ...
,
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 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 model ''does'' accord with Mach's ideas.
Many other exact solutions that can be interpreted as cosmological models of rotating universes are known.
See also
*
van Stockum dust
In general relativity, the van Stockum dust is an exact solution of the Einstein field equations in which the gravitational field is generated by dust rotating about an axis of cylindrical symmetry. Since the density of the dust is ''increasing' ...
, for another rotating dust solution with (true) cylindrical symmetry,
*
Dust solution
In general relativity, a dust solution is a fluid solution, a type of exact solution of the Einstein field equation, in which the gravitational field is produced entirely by the mass, momentum, and stress density of a perfect fluid that has ' ...
, an article about dust solutions in general relativity.
References
Notes
*
* See ''section 12.4'' for the uniqueness theorem.
* See ''section 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 universe on arxiv.org
Vukovic R. (2014): ''Tensor Model of the Rotating Universe'', Exercise in Special Relativity
{{DEFAULTSORT:Godel Metric
Exact solutions in general relativity
Metric tensors
Metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...