Gowdy Spacetimes
| Robert H. Gowdy (2014), Scholarpedia, 9(3):31673. | doi:10.4249/scholarpedia.31673 | revision #143314 [link to/cite this article] |
Gowdy Spacetimes have become useful test cases for studying the dynamics of Einstein's theory of general relativity. Because they are simple enough to analyze and admit arbitrary wavelength gravitational waves, they provide insights into the full dynamics of general relativity that cannot be provided by homogeneous cosmological models. This review is intended as a basic conceptual introduction to these spacetimes. For a review that is oriented towards proving mathematical theorems about Gowdy spacetimes, see the Living Reviews in Relativity article by Hans Ringström (http://www.livingreviews.org/lrr-2010-2, 2010) and the references cited there.
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Background information about \(G_{2}\) spacetimes
Isometry groups and group orbits
An isometry group is a group of motions that leaves the metric of a geometry unchanged. An orbit of an isometry group is the set of points that can be reached by letting all operations of the group act on a single point. \(G_{2}\) spacetimes have two-parameter Lie groups as their isometries. When the orbits are required to be closed and orientable, the \(G_{2}\) group with an effective action on the manifold is actually \(SU\left( 1\right) \times SU\left( 1\right) \) and the orbits can only be tori or circles. This limited variety makes it possible to work out all of the possible structures of these spacetimes. For example, connected, group-invariant spacelike hypersurfaces in these spacetimes must be homeomorphic to the three-torus $T^{3} = S^{1}\times S^{1}\times S^{1}$, or the three-handle $S^{1}\times S^{2}$, or a space covered by the three sphere $S^{3}$. (Chruściel 1990)
Metric on a general \( SU\left( 1\right) \times SU\left( 1\right) \) invariant spacetime
The metric tensor in a coordinate patch on a $G_{2}$ spacetime with closed orbits can always be brought into a special form by choosing the coordinates properly. The SU$\left( 1\right) \times $SU$\left( 1\right) $ group coordinates $x^{2}=\sigma $ and $x^{3}=\delta $ (each ranging from $0$ to $2\pi $) are used as coordinates on each orbit. Each orbit is labeled by a space coordinate $x^{1}=\theta $ and a time coordinate $x^{0}=t$. The metric intrinsic to the two-dimensional orbit labeled by $t,\theta $ is represented as \[ d\mu ^{2}/L^{2}=R\left[ \left( q_{22}d\sigma +q_{23}d\delta \right) ^{2}+\left( q_{32}d\sigma +q_{33}d\delta \right) ^{2}\right] \] where the matrix formed by the coefficients $q_{mn}$ has unit determinant. The constant length $L$ is included so that all of the coordinates and metric functions can be dimensionless. The geometrical significance of this representation is that the area of the group orbit labeled by the coordinates $t,\theta $ is given by $4\pi ^{2}L^{2}R\left( t,\theta \right) $. The function $R$ is usually referred to as the orbit area function. Where this function is zero there is either a lower-dimensional orbit (e.g. a rotation axis) or a null orbit at a causality violation, or possibly a spacetime singularity. With these definitions, the complete spacetime metric, in a neighborhood of a two-dimensional orbit, can be brought into the form \[\tag{1} ds^{2}/L^{2}=e^{2a}\left( d\theta ^{2}-dt^{2}\right) +d\mu ^{2}/L^{2}+2\left( N_{2}d\sigma +N_{3}d\delta \right) dt \] where $a$, $N_{m}$ are functions of the orbit label coordinates $t,\theta $.
The Gowdy family of metrics makes the additional specialization of two-surface orthogonality
\[\tag{2} N_{2}=N_{3}=0 \]
The specialization can be understood by using the Killing vectors $X=\frac{\partial }{\partial \sigma } $ and $Y=\frac{\partial }{\partial \delta } $ to construct the twist parameters $c_{1}=\epsilon ^{\alpha \beta \gamma \delta }X_{\alpha }Y_{\beta }X_{\gamma ;\delta } $ and $c_{2}=\epsilon ^{\alpha \beta \gamma \delta }X_{\alpha }Y_{\beta }Y_{\gamma ;\delta } $ where Greek letters range from $0$ to $3,$ $\epsilon ^{\alpha \beta \gamma \delta }$ is the totally antisymmetric Levi-Civita density, and the semicolon denotes covariant derivative components. For a vacuum Einstein spacetime, these parameters are constant over the whole manifold (See Geroch 1972). A straightforward computation using the assumed form of the metric shows that these parameters have expressions that are proportional to $R$ and can be solved for the derivatives $\partial N_{a}/\partial \theta .$ If these parameters are zero at a point in the manifold where $R$ is not zero, one can choose $N_{1}=N_{2}=0$ at that point and then integrate the resulting ordinary differential equations to obtain $N_{1}=N_{2}=0$ throughout the $R>0$ region.
In the $S^{3}$ and $S^{1}\times S^{2}$ cases, which have rotation axes (where $R=0 $), the twist parameters are zero. For these cases, the assumption of two-surface orthogonality does not discard any solutions. However, for the $T^{3}$ case, there are no rotation axes and the twist parameters can be non-zero. In that case, the two-surface orthogonality assumption discards a complex (and difficult to analyze) set of spacetimes.
Parameterizing the metric
The coefficients of the unimodular matrix $q_{mn}$ can be parameterized in various ways. The parametrization that is used in most of the literature on Gowdy spacetimes is
\[\tag{3} d\mu ^{2}/L^{2}=R\left[ e^{P}\left( d\sigma +Qd\delta \right) ^{2}+e^{-P}d\delta ^{2}\right] \] which corresponds to the matrix components $q_{22}=e^{P/2}, q_{23}=e^{P/2}Q, q_{33}=e^{-P/2}, $ and $q_{32}=0 $.
This parameterization does not treat the two Killing vectors, $\partial /\partial \sigma $ and $\partial /\partial \delta $ symmetrically, so another parameterization can be obtained by switching the Killing vectors. That parameterization renders the orbit metric as
\[ d\mu ^{2}/L^{2}=R\left[ e^{Y}\left( d\delta +Xd\sigma \right) ^{2}+e^{-Y}d\sigma ^{2}\right] \] Changing from one parameterization to the other corresponds to an inversion in the hyperbolic plane \[\tag{4} Y=P+\ln \left( e^{-2P}+Q^{2}\right) ,\qquad X=Q/\left( e^{-2P}+Q^{2}\right) \]
Field equations
The need for fortunate circumstances
In terms of the Ricci tensor $R_{\mu \nu }$, the curvature scalar ${}^{4}R$, and the spacetime metric tensor $g_{\mu \nu }$ the Einstein tensor is \[ G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}{}^{4}Rg_{\mu \nu } \] and Einstein's Field Equations in the presence of matter with stress-energy tensor $T_{\mu \nu }$ are \[ G_{\mu \nu }=8\pi T_{\mu \nu }. \] The metric tensor is assumed to have the G$_{2}$-symmetric, two-surface orthogonal form (see Equations (1), (2) and (3).) \[\tag{5} ds^{2}=L^{2}\left\{ e^{2a}\left( d\theta ^{2}-dt^{2}\right) +R\left[ e^{P}\left( d\sigma +Qd\delta \right) ^{2}+e^{-P}d\delta ^{2}\right] \right\} \] where $a,R,P,Q$ are assumed to depend only on the coordinates $t$ and $% \theta $. This form of the metric suggests using the orthonormal basis forms \[ \omega ^{0}=Le^{a}dt,\qquad \omega ^{1}=Le^{a}d\theta ,\qquad \omega ^{2}=L% \sqrt{R}e^{P/2}\left( d\sigma +Qd\delta \right) ,\qquad \omega ^{3}=L\sqrt{R}% e^{-P/2}d\delta \] to define an orthonormal reference frame.
The Einstein equations provide a system of partial differential equations, which must be solved for the four functions $a,R,P,Q$. The earlier discussion of G$_{2}$ spacetimes provides assurance that solutions exist, but finding them requires several fortunate circumstances:
The orbit area $R$ decouples
Two orthonormal frame components of the Einstein tensor combine to yield a simple result\[ G_{11}-G_{00}=L^{-2}e^{-2a}R^{-1}\left( R_{\theta \theta }-R_{tt}\right) \] where subscripts $\theta $ and $t $ are being used to denote partial derivatives with respect to those variables. If the stress-energy tensor obeys the condition $T_{11}-T_{00}=0$, then the orbit area function decouples from the rest and obeys the simple and familiar vibrating string equation \[\tag{6} R_{\theta \theta }-R_{tt}=0. \] By choosing a solution to this equation, one links the arbitrary coordinates $\theta ,t$ to the geometrically invariant orbit area. Different choices for $R$ lead to different global spacetime structures (See Gowdy 1974).
The red-shift potential $a$ is integrable
Define retarded and advanced time coordinates $u=t-\theta $ and $v=t+\theta $ and the corresponding derivatives $f_{-}=\frac{\partial f}{\partial u} $ and $f_{+}=\frac{\partial f}{\partial v} $ to express two more combinations of orthonormal frame Einstein tensor components: \[\tag{7} G_{00}\pm G_{01}=4L^{-2}e^{-2a}R^{-1}\left[ a_{\pm }R_{\pm }-\frac{1}{2}% R_{\pm \pm }+\frac{1}{4}R\left( R_{\pm }/R\right) ^{2}-\frac{1}{4}R\left( P_{\pm }{}^{2}+Q_{\pm }{}^{2}e^{2P}\right) +\frac{1}{2}R_{+-}\right] . \] which can be solved for the derivatives of the function $a$ wherever the gradient of $R$ is not lightlike. The function $a$ can then be reduced to an integral of the other metric functions.
Where $R$ has a lightlike gradient $% \left( R_{\pm }=0\right) $, the corresponding equation yields a constraint on the values of $P,Q,$ and their first derivatives. It can be shown that the remaining Einstein equations preserve this constraint if it is satisfied on an initial hypersurface. Such a constraint is called a matching constraint because it joins spacetime regions where the local character of the solution changes.
The dynamical equations decouple from the red-shift potential
The orbit area equation, eqn (6) and the integrable equations (7) imply that the Einstein tensor components $G_{00}, G_{01}=G_{10}, G_{11}$ all vanish. That leaves $G_{ab}$ where $a,b$ range from 2 to 3 as the remaining possible non-zero components of the Einstein tensor. If the full Einstein tensor is zero, then the Ricci tensor will also be zero. Thus, $R_{ab}=0$ is a necessary condition to have a vacuum solution of Einstein's equations. As was noted by Gowdy 1974, it is not algebraically sufficient, but, because of the contracted Bianchi identities satisfied by the Einstein tensor, it is a sufficient condition in almost all cases.
For the metric given by Eqn. (5) and the corresponding orthonormal frame, these Ricci tensor components are: \[ L^{2}e^{2a}R_{22}=-L^{2}e^{2a}R_{33}=\frac{1}{R}\left( RP_{t}\right) _{t}-% \frac{1}{R}\left( RP_{\theta }\right) _{\theta }+\left( Q_{\theta }^{2}-Q_{t}^{2}\right) e^{2P} \] \[ L^{2}e^{2a-2P}R_{23}=\frac{1}{R}\left( RQ_{t}e^{2P}\right) _{t}-\frac{1}{R}% \left( RQ_{\theta }e^{2P}\right) _{\theta } \] where subscripts are being used to denote partial derivatives with respect to $t$ and $\theta$. In the vacuum case, these components are set equal to zero, yielding a pair of dynamical equations for the potentials $P$ and $Q$. We can choose a solution for the orbit area function $R$, solve these two dynamical equations for $P,Q$ and then perform an integration to determine the red-shift potential $a$. For this procedure to work, it is critically important that the dynamical equations have decoupled from the red-shift potential $a$. If matter with $T_{ab}\neq 0$ is present, its dependence on $a $ must be appropriate to permit this same decoupling.
The dynamical equations form a wave-map or nonlinear sigma model
The dynamical equations define coupled scalar waves on a background spacetime. The background spacetime can be chosen in a variety of ways. One background choice is a four-dimensional spacetime $\mathcal{M}_{\text{ref}}$ with the metric given in Eq. (5) but with $a=0$ and the functions $% P,Q$ chosen to be some arbitrary functions $P_{\text{ref}},Q_{\text{ref}}$. The orbit-area function $R$ is equal to $\sqrt{-g}$ for this metric, so we can use the well-known coordinate-basis relation (with semicolons for covariant derivatives and commas for partial derivatives) \[ \square f=-g^{\alpha \beta }f_{;\alpha \beta }=-\frac{1}{\sqrt{-g}}g^{\alpha \beta }\left( \sqrt{-g}f_{,\alpha }\right) _{,\beta } \] to see that the dynamical equations take the form $\square P+e^{2P}\nabla Q\cdot \nabla Q=0 $ and $\square Q-2\nabla Q\cdot \nabla P=0 $ where the wave operator and the dot product use the background metric and it is assumed that $P,Q$ are independent of the group coordinates $\sigma ,\delta $.
Another possible background choice, which avoids the need to choose arbitrary functions, is a fictitious three-dimensional $SU\left( 1\right) $-invariant spacetime $\mathcal{N}$ with metric $ds^{2}=d\theta ^{2}-dt^{2}+Rd\phi ^{2} $. In this case, the dynamical equations take the same form, but with $R,P,Q$ now assumed to be independent of the $SU\left( 1\right) $ group coordinate $% \phi $.
When the function $Q$ is zero, this system reduces to a linear wave equation for $P$. That case is usually referred to as a Polarized Gowdy spacetime. Solutions for the polarized case can be constructed at will from well-known functions. When $Q$ is not zero, the system is nonlinear and difficult to solve. However, it is a very special type of nonlinear system. The system can be obtained by extremizing the action \[ I=\int \left( \nabla P\cdot \nabla P+e^{2P}\nabla Q\cdot \nabla Q\right) \mu \] where $\mu $ is the volume element on the background spacetime, (either $% \mathcal{M}_{\text{ref}}$ or $\mathcal{N}$).
One can think of the pair of functions $\left( P,Q\right) $ as a field on the background spacetime with values in a two-dimensional space whose metric tensor is $\gamma =dP\otimes dP+e^{2P}dQ\otimes dQ $. The action then takes a form similar to the "energy" functional of a harmonic map. \[ I=\int g^{\alpha \beta }\gamma _{AB}f_{;\alpha }^{A}f_{;\beta }^{B}\mu \] This type of nonlinear wave equation is called a wave-map system by mathematicians (See Tataru 2004) and a nonlinear sigma model (Ketov 2009) by physicists. The target space, with metric $\gamma $ is also familiar as the hyperbolic space. Because this system has been studied extensively, it has been possible to analyze its properties in some detail without actually solving it.
Varieties
The $T^{3}$ model
The choice $R=t$ with periodic boundary conditions at $\theta =0$ and $% \theta =2\pi $ leads to an expanding spacetime with no rotation axes. It has a possible initial singularity at $t=0$ and can be regarded as a toy model of a standard Big Bang cosmology. It is technically the simplest of the Gowdy models and is well suited to addressing issues connected with the initial singularity. In the absence of matter, the dynamical equations \[ \begin{eqnarray*} P_{tt}+\frac{1}{t}P_{t}-P_{\theta \theta }+\left( Q_{\theta }^{2}-Q_{t}^{2}\right) e^{2P} &=&0 \\ Q_{tt}+\frac{1}{t}Q_{t}-Q_{\theta \theta }+2\left( P_{t}Q_{t}-P_{\theta }Q_{\theta }\right) &=&0 \end{eqnarray*} \] may be obtained by varying the action integral \[ I=\int \int \left[ P_{\theta }^{2}-P_{t}^{2}+\left( Q_{\theta }^{2}-Q_{t}^{2}\right) e^{2P}\right] d\theta dt. \] The equations satisfied by the red-shift potential $a$ are \[ \begin{eqnarray*} a_{t} &=&-\frac{1}{4t}+\frac{1}{4}t\left( P_{t}^{2}+P_{\theta }^{2}+\left( Q_{t}^{2}+Q_{\theta }^{2}\right) e^{2P}\right) \\ a_{\theta } &=&\frac{1}{2}t\left( P_{t}P_{\theta }+Q_{t}Q_{\theta }e^{2P}\right) \end{eqnarray*} \] A slight simplification of these equations can be obtained by writing $a$ in the form $a=-\frac{1}{4}\ln t+\frac{1}{2}\lambda $ so that the spacetime metric for the model takes the form \[ ds^{2}/L^{2}=t^{-1/2}e^{\lambda /2}\left( d\theta ^{2}-dt^{2}\right) +t\left[ e^{P}\left( d\sigma +Qd\delta \right) ^{2}+e^{-P}d\delta ^{2}\right] \] and the function $\lambda $ is given by the integral \[ \lambda \left( \theta ^{\prime },t\right) =t\int_{0}^{\theta ^{\prime }}\left( P_{t}P_{\theta }+Q_{t}Q_{\theta }e^{2P}\right) d\theta \] The periodic boundary conditions on $\lambda $ then lead to the integral constraint \[ \int_{0}^{2\pi }\left( P_{t}P_{\theta }+Q_{t}Q_{\theta }e^{2P}\right) d\theta =0. \tag{8} \] This constraint is preserved by the dynamical equations, so it only needs to be imposed on the initial conditions.
The $S^{1}\times S^{2}$ model
Consider the general Gowdy spacetime metric (Eqn. (5)) with the choice $R=\sin t\sin \theta $. A spacetime with constant time sections that have $S^{1}\times S^{2}$ topology would have a term proportional to $\sin ^{2}\theta d\delta ^{2}$, near the rotation axes at $\theta =0$ and $\theta =\pi $. In this case, $\delta $ plays the role of the rotation angle at each axis. Our metric has $Re^{-P}d\delta ^{2}=\sin t\sin \theta e^{-P}d\delta ^{2}$, so we need to define a new wave potential that differs from $P$ by a term proportional to $-\ln \sin \theta $.
To avoid complicating the dynamical equations, we would also like the extra term to give zero when the dynamical wave operator acts on it. Fortunately, the term $\ln R$ has that property. Define a new wave potential $W$ by $P=W-\ln R $ and write the redshift potential in the form $a=\frac{1}{2}\lambda -\frac{1}{2}W $. The spacetime metric then takes the form \[ ds^{2}/L^{2}=e^{\lambda -W}\left( d\theta ^{2}-dt^{2}\right) +e^{W}\left( d\sigma +Qd\delta \right) ^{2}+e^{-W}\sin ^{2}t\sin ^{2}\theta d\delta ^{2}. \]
For this metric to be regular at the rotation axes, the ratio of the circumference to the radius of a circle around the axis must approach $2\pi $ so that \[ \left. e^{\lambda }\right\vert _{\sin \theta =0}=\sin ^{2}t+\lim_{\sin \theta \rightarrow 0}\frac{Q^{2}}{\sin ^{2}\theta }. \] This regularity constraint requires $Q$ to be zero at both axes. In addition, for the metric to be differentiable at the axes, $Q$ must be a differentiable function on the sphere so that its derivative with respect to $\theta $ must also vanish at the axes. Similarly, the function $W$ must be differentiable on the sphere. The boundary conditions at the axes are then \[ \left. Q\right\vert _{\theta =0}=\left. Q\right\vert _{\theta =\pi }=\left. Q_{\theta }\right\vert _{\theta =0}=\left. Q_{\theta }\right\vert _{\theta =\pi }=\left. W_{\theta }\right\vert _{\theta =0}=\left. W_{\theta }\right\vert _{\theta =\pi }=0 \tag{9} \] and \[ \left. \lambda \right\vert _{\theta =0}=\left. \lambda \right\vert _{\theta =\pi }=\ln \sin ^{2}t. \tag{10} \] The vacuum Einstein equations for this case are \[ \lambda _{\pm }R_{\pm }=R_{\pm \pm }+\frac{1}{2}\left( RW_{\pm }^{2}+R^{-1}Q_{\pm }{}^{2}e^{2W}\right) \tag{11} \] \[ \left( RW_{t}\right) _{t}-\left( RW_{\theta }\right) _{\theta }+\frac{1}{R}% \left( Q_{\theta }^{2}-Q_{t}^{2}\right) e^{2W}=0 \] \[ \left( R^{-1}e^{2W}Q_{t}\right) _{t}-\left( R^{-1}e^{2W}Q_{\theta }\right) _{\theta }=0 \] and the dynamical equations may be obtained by varying the action \[ I=\int \int_{0}^{\pi }\left[ RW_{\theta }^{2}-RW_{t}^{2}+\frac{1}{R}\left( Q_{\theta }^{2}-Q_{t}^{2}\right) e^{2W}\right] d\theta dt. \] subject to the regularity constraints (which eliminate the surface terms in the variation).
So long as the matching constraints have been satisfied where $R_{\pm }=0$, Equation (11) may be solved for $\lambda _{\theta }$ which can then be integrated to find $\lambda $ just as in the $T^{3}$ case. Here there is no periodicity requirement, but the boundary conditions on $\lambda $ given by Equation (10) yield a conserved integral constraint similar to the one (Eq. (8)) that occurs in the $% T^{3}$ case.
The $S^{3}$ Model
Again consider the general Gowdy spacetime metric (Eqn. (5)) with the choice $R=\sin t\sin \theta $. A spacetime with constant time sections with three-sphere topology would have a metric with a rotation-axis term proportional to $\theta ^{2}d\delta ^{2}$ near $\theta =0$ and another rotation axis term proportional to $\left( \pi -\theta \right) ^{2}d\sigma ^{2}$ near $\theta =\pi $. In this case, the angles $\sigma $ and $\delta $ reverse roles at the two rotation axes. Now find terms to subtract from the potential $P$ that give zero when the dynamical operator acts on them and also produce rotation axis terms of the required form. Define a new wave potential $W$ by $P=W-\ln \tan \frac{\theta }{2}-\ln \tan \frac{t}{2} $ and write the redshift potential in the form $a=\frac{1}{2}\lambda $. The space-time metric then takes the form \[ ds^{2}/L^{2}=e^{\lambda }\left( d\theta ^{2}-dt^{2}\right) +4\left[ e^{W}\cos ^{2}\frac{t}{2}\cos ^{2}\frac{\theta }{2}\left( d\sigma +Qd\delta \right) ^{2}+e^{-W}\sin ^{2}\frac{t}{2}\sin ^{2}\frac{\theta }{2}d\delta ^{2}% \right] . \] The regularity constraint at $\theta =0$ looks very much like the $% S^{1}\times S^{2}$ case and requires $Q$ to vanish there. Invert the representation using Eqn (4) and find that the regularity constraint at $\theta =\pi $ requires the function $X$ to vanish there. However, Eqn (4) implies that $X$ can vanish if and only if $Q$ vanishes. The resulting boundary conditions on the functions $W$ and $Q$ are then exactly the same as for the $S^{1}\times S^{2} $ case and are given by Eqn. (9). The regularity conditions then require boundary conditions on the function $\lambda $: \[\tag{12} \begin{eqnarray*} \left. \lambda \right\vert _{\theta =0} &=&-\left. W\right\vert _{\theta =0}+\ln \left( \frac{1-\cos t}{2}\right) \\ \left. \lambda \right\vert _{\theta =\pi } &=&+\left. W\right\vert _{\theta =0}+\ln \left( \frac{1+\cos t}{2}\right) \end{eqnarray*} \] The Einstein equations for this case are: \[ \lambda _{\pm }R_{\pm }=\pm \frac{1}{4}+\frac{1}{2}R_{\pm \pm }+\frac{1}{2}% W_{\pm }\left( \sin t\pm \sin \theta \right) +\frac{1}{2}R\left( W_{\pm }^{2}+Q_{\pm }{}^{2}e^{2W}\cot ^{2}\frac{1}{2}\theta \cot ^{2}\frac{1}{2}% t\right) \tag{13} \] \[ \left( RW_{t}\right) _{t}-\left( RW_{\theta }\right) _{\theta }+\left( Q_{t}^{2}-Q_{\theta }^{2}\right) e^{2W}R\cot ^{2}\frac{1}{2}\theta \cot ^{2}% \frac{1}{2}t=0 \] \[ \left[ e^{2W}R\left( \cot ^{2}\frac{1}{2}\theta \right) \left( \cot ^{2}% \frac{1}{2}t\right) Q_{t}\right] _{t}-\left[ e^{2W}R\left( \cot ^{2}\frac{1}{% 2}\theta \right) \left( \cot ^{2}\frac{1}{2}t\right) Q_{\theta }\right] _{\theta }=0 \] and the dynamical equations may be obtained by varying the action \[ I=\int \int_{0}^{\pi }\left[ RW_{\theta }^{2}-RW_{t}^{2}+\frac{1}{R}\left( Q_{\theta }^{2}-Q_{t}^{2}\right) \left( \cot ^{2}\frac{1}{2}\theta \right) \left( \cot ^{2}\frac{1}{2}t\right) e^{2W}\right] d\theta dt. \]
Just as for the $S^{1}\times S^{2}$ model, once the dynamical equations have been solved and the matching constraints have been satisfied, Equation (13) may be solved for $\lambda _{\theta} $, which can then be integrated for $\lambda$. The boundary conditions on $\lambda $ given by Equation (12) yield a conserved integral constraint similar to the one (Eq. (8)) that occurs in the $% T^{3}$ case.
Matching Constraints
For the $S^{1}\times S^{2}$ model and the $S^{3}$ model, it is useful to show the region in which the orbit area function is positive, as in Figure 1. In this region, the metric tensor is regular as long as the metric functions $\lambda, W, Q$ are regular. Spacetime expands from a Big Bang at $t=0$ to a final crunch at $t=\pi $.
The derivatives $R_{+}$ and $R_{-}$ are zero along the dotted diagonals of this figure. Along the line $t+\theta =\pi $ Equations (11) and (13) become quadratic constraints on $W_{+}$ and $Q_{+}$. Similarly, along the line $t-\theta =0,$ those equations become quadratic constraints on $W_{-}$ and $Q_{-}$. If the initial data on a space-like hypersurface satisfies the matching constraints, the constraints will be propagated along the dotted lines by the dynamical equations.
The matching constraints occur where the gradient of the orbit-area function $R$ changes between spacelike and timelike. The gradient is timelike in the regions marked "Expanding" and "Collapsing." There, it is possible to perform a coordinate transformation to make $R=t$ and obtain a metric that is exactly the same locally as the $% T^{3}$ metric, but with different boundary conditions. Where the gradient is spacelike, the metric can be transformed locally into an Einstein-Rosen cylindrical wave metric.
Applications
Inhomogeneous cosmology
Although these spacetimes are sometimes referred to as inhomogeneous cosmologies, they are not legitimate cosmological models because their matter content is limited to massless fields and their G$_{2}$ symmetry is quite special. Massless scalar fields and electromagnetic fields can be included, so long as they have the same G$_{2}$ symmetry as the spacetime (See Charach 1979). However, the inclusion of other types of matter destroys the fortunate circumstances that make these models tractable.
Numerical exploration of gravitational dynamics
The dynamical equations for these models have just two component fields in one space dimension and are of a very well-studied type. Simulating these equations numerically on a computer is quite straightforward (Berger 2002). One noteworthy result of such exploration (Berger 1993) was the observation that regular initial data, evolved back toward the singularity would often develop features that have come to be called spikes. Given the clue provided by the numerical work, it was then possible to provide an analytical explanation for the phenomenon (See Rendall 2001). An explicit solution for a spike has since been found (Lim 2008).
Validation of numerical computer simulations
Most astrophysical systems lack the symmetry and simplicity that would be needed for analytical methods to work. For complex situations such as the in-spiraling merger of two collapsed objects, numerical simulation methods are required to compare the predictions of Einstein's general relativity to observation. The computer codes used for these simulations are complex and the part of the code that simulates Einstein's field equations is particularly delicate. Although Gowdy spacetimes do not model realistic situations, they do provide highly complex exact solutions (the polarized models) and well-tested numerical solutions for validating these computer codes (See Alcubierre et al. 2004).
The nature of the Big Bang singularity
Although the Gowdy spacetimes are not realistic cosmological models, they do have early stages that resemble the Big Bang. The nature of the Big Bang in cosmological models is the subject of several conjectures.
The Strong Cosmic Censorship Conjecture (SCCC) says, very roughly, that one cannot (in the generic case) extend spacetime beyond the region where causality holds. In particular, there should be nothing before the Big Bang and the entire region from the Big Bang to the Big Crunch should be regular (technically, it should be globally hyperbolic). The original version of the conjecture was due to Roger Penrose (Penrose 1969 and 1969a) and its most recent version is described by Moncrief and Eardly (1981).
The Belinski-Khalatnikov-Lifshitz Scenario (BKL) says that, near the Big Bang, (in the generic case) spatial gradients become unimportant and each small spatial neighborhood evolves independently like a homogeneous anisotropic universe, oscillating infinitely many times as the Big Bang is approached. The BKL Scenario and the SCCC are both discussed in detail in the Living Reviews article by Ringström (2010).
The first part of proving a conjecture is to state it correctly and that requires searching for counterexamples to the proposed statement. Gowdy spacetimes have offered a promising source of counterexamples, particularly since one very well-known example of a spacetime that can be extended before its Big Bang is the Taub Universe, which is actually a particular Gowdy $S^3 $ spacetime. The existence of such counterexamples is the motivation for specifying that the conjecture holds only in the generic case.
The SCCC is sometimes described as forbidding naked singularities. The $S^{1}\times S^{2}$ and $S^3 $ models offer a situation where one might expect such a thing. At the rotation axes in such a spacetime, cylindrical gravitational waves come to a focus and one might expect that they could generate a singularity. In Figure 1 that would correspond to some sort of problem at the sides of the square. Piotr Chruściel (1990) showed that such a thing cannot happen.
The SCCC also asserts that the extension of a spacetime past its initial singularity is possible only in finely balanced, exceptional circumstances. For a set of initial conditions that is dense in the set of all possible initial conditions, evolving backwards in time leads to curvature invariants that diverge so that no extension is possible. As Ringström (2010) describes, that has been proven to be the case for the Gowdy $T^3 $ models. For the $S^{1}\times S^{2}$ and $S^3 $ models, it is clear that spacetime near the initial singularity can always be described in coordinates that make it locally identical to a $T^3 $ model. Only the rotation axes cannot be described in such a way and one can argue that such symmetry axes would not even occur in a generic spacetime.
The BKL scenario is not as well-stated as the SCCC, but it does seem to be correct for Gowdy Spacetimes (Isenberg 1990). In the generic case, the spatial curvature terms in Einstein's equations drop out, exactly as the BKL scenario describes.
References
Alcubierre, M et al. (2004). Towards standard testbeds for numerical relativity Class. Quantum Grav. 21: 589-613.
Berger(1993). Numerical investigation of cosmological singularities Phys. Rev. D 48: 4676-4687.
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