Chaos in pp-wave spacetimes

Jiří Podolský, Karel Veselý

Department of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 180 00 Prague 8, Czech Republic. (July 31, 1998)

We demonstrate chaotic behavior of timelike, null and spacelike geodesics in non-homogeneous vacuum pp-wave solutions by analytic and fractal methods. This seems to be the first known example of a chaotic motion in exact radiative spacetime.

04.30.-w; 05.45.+b; 04.20.Jb

In the context of general relativity the first system for which a chaotic behavior of solutions to the Einstein equations has been recognized and thoroughly studied were Bianchi IX cosmological models (see for example [1], [2], and references therein). Complicated nonlinear effects also occur in systems with coupled gravitational and scalar fields [3] - [6].

Another type of problems providing nonlinear dynamical systems in general relativity are the studies of geodesic motion in given spacetimes. In particular, chaotic behavior of geodesics in the relativistic analogue of the two fixed-centres problem (modelled by extreme black holes) was examined in [7] - [10]. Chaotic geodesic motion was also found in (perturbed) Schwarzschild spacetime [11] - [14], in some static axisymmetric spacetimes [15], [16] and in topologically nontrivial Robertson-Walker universe [17], [18].

Here we investigate motion in exact gravitational waves, namely in the widely known class of vacuum ppwaves [19], the metric of which can be written in the form

ds^{2} = 2 dζdζ̄ − 2 dudv − (f + f̄) du^{2} ,  (1)

where f(u, ζ) is an arbitrary function of u and the complex coordinate ζ spanning the plane wave surfaces u = const. When f is linear in ζ, the metric (1) represents Minkowski universe. The case f = g(u)ζ^{2} describes plane gravitational waves (‘homogeneous’ ppwaves) which have thoroughly been investigated (see [19] for Refs). This simple example of an exact radiative spacetime has also been used for construction of sandwich and impulsive waves [20], [21].

However, here we wish to study geodesics in more general, non-homogeneous vacuum pp-waves and demonstrate their chaotic behavior. The geodesic equations for

(1) are

ζ̈ + \frac{1}{2} f̄_{,ζ̄} U^{2} = 0 ,  (2)
u̇ = U = const. ,  (3)
v̈ + (f_{,ζ} ζ̇+ f̄_{,ζ̄} ˙ζ̄) U + \frac{1}{2}(f + f̄)_{,u} U^{2} = 0 ,  (4)

where dot denotes d/dτ with τ being an affine parameter. Assuming also a condition normalizing the four-velocity such that U_{μ}U^{μ} = ϵ where ϵ = −1, 0, +1 for timelike, null or spacelike geodesics, we get

v̇ = \frac{1}{2}U^{−1} [2ζ̇˙ζ̄ − (f + f̄) U^{2} − ϵ] .  (5)

We consider U ≠ 0 (for U = 0 Eqs. (2)-(4) can be integrated yielding only some trivial geodesics). By differentiating Eq. (5) and using (2) we immediately obtain (4) which can thus be omitted. Hence it suffices to find solutions of Eq. (2) since v(τ) can then be obtained by integrating Eq. (5), and u(τ) = Uτ + u_{0}.

The remaining equation (2) has the same form for timelike, null and spacelike geodesics. Introducing real coordinates x and y by ζ = x+iy we get a system which, for f independent of u, follows from the Hamiltonian

H = \frac{1}{2} (p_{x}^{2} + p_{y}^{2}) + V (x, y) ,  (6)

where the potential is V (x, y) = \frac{1}{2}U^{2} Re f. For nonhomogeneous pp-wave spacetimes given by f = Cζ^{n}, C = const. > 0, n = 3, 4, ⋯, the corresponding potential

V (x, y) = \frac{1}{2}CU^{2} Re ζ^{n} ,  (7)

is called ‘n-saddle’. It can by visualized in polar coordinates ρ, ϕ where ζ = ρ exp(iϕ), in which it takes the form V (ρ, ϕ) = \frac{1}{2}CU^{2} ρ^{n} cos(nϕ).

Now, it was shown previously by Rod, Churchill and Pecelli in a series of papers [22] - [25] that motion in the Hamiltonian system (6) with the n-saddle potential (7) is chaotic.

Let us first briefly summarize their results for the simplest case n = 3. The corresponding potential (after removing an unimportant multiplicative factor by a suitable rescaling of τ)

V (x, y) = \frac{1}{3}x^{3} − xy^{2}  (8)

is called a ‘monkey saddle’. Interestingly, this is a special case of famous Hénon-Heiles Hamiltonian [26] which is known to be a ‘textbook’ example of a chaotic system (however, their quadratic terms are absent in our case). This particular case of the Hénon-Heiles Hamiltonian has been investigated by Rod [22]. He concentrated on bounded orbits in the energy manifolds H(x, y, p_{x}, p_{y}) = E > 0. The homogeneity of V guarantees that the orbit structure for any two positive values of E is isomorphic modulo a constant scale factor and adjustment of time: x → x̃ = λx, y → ỹ = λy and τ → τ̃ = τ/\sqrt{λ} results in E → Ẽ = λ^{3}E. Therefore, without loss of generality one can restrict to one particular value of E.

In order to describe the topological structure of all bounded orbits Rod first constructed three basic unstable periodic orbits (denoted by Π_{j}) which are isolated invariant sets for the flow. The region in which these bounded orbits occur can be decomposed into three disjoint cells R_{j} (see [22] or [27] for details); each contain only one orbit Π_{j} and no other bounded orbits. Hence, Π_{j} is the only invariant set in R_{j}, i.e. it is isolated.

Subsequently, Rod investigated orbits asymptotic to basic orbits Π_{j} as τ → ±∞ and showed that these asymptotic sets intersect transversely. This gives the existence of orbits that ‘connect’ the orbits Π_{j}: they are homoclinic (asymptotic to the same periodic orbit in both time directions) or heteroclinic (asymptotic to two different periodic orbits, one in each time direction). It is the existence of these orbits that illustrates complicated chaotic structure of the flow.

The topology of possible orbits in phase space can also be represented by symbolic dynamics given here by a set of biinfinite sequences, s ≡ ⋯, s_{k}, s_{k+1} , s_{k+2} , ⋯, where s_{k} ∈ {1, 2, 3}, s_{k} ≠ s_{k+1} . Using a topological version of the Smale horseshoe map, it was shown in [22] that to any biinfinite sequence of symbols {1, 2, 3} there exists an uncountable number of bounded orbits running through the blocks R_{j} in the prescribed order as τ goes from −∞ to +∞ . Also, the flow admits at least a countable number of non-degenerate homoclinic and heteroclinic orbits.

Rod remarked that these results could be refined if the unstable periodic orbits Π_{j} were known to be hyperbolic so that they would admit stable and unstable asymptotic manifolds. Consequently, to each periodic symbol sequence there would correspond a countable collection of periodic orbits. The difficulties in proving the hyperbolicity of Π_{j} were subsequently overcome in [23]. In [24], summarizing and generalizing some previous results [25], the Hamiltonian (6), (8) was presented as an example of a system for which the Smale horseshoe map can explicitly be embedded as a subsystem into the flow along the homoclinic and heteroclinic orbits. This completed the proof of chaotic behavior of the studied system.

Similar results hold for geodesic motion in arbitrary non-homogeneous pp-waves with the structural function of the form f = Cζ^{n}, where n ≥ 4, i.e. for a general n-saddle potential (7). It was shown in [22], [25] that the decomposition into isolating cells R_{j}, j = 1, 2, ⋯, n, each containing exactly one of the basic unstable periodic solutions Π_{j}, is analogous to the case n = 3. Subsequently, the orbits Π_{j} were proven to be hyperbolic [23] and the existence of homoclinic and heteroclinic orbits was established [24]. Again, given any biinfinite sequence, uncountably many orbits can be found which pass from one block containing Π_{j} to the other in the specified order. In order to independently support these arguments for chaotic behavior of geodesics in non-homogeneous ppwaves we investigate the motion also by a fractal method. Complementary to the analysis described above, we concentrate on unbounded geodesics and we do not restrict to the same energy manifold E = const.

Chaos is usually indicated by a sensitive dependence of the evolution on the choice of initial conditions. The coordinate independent fractal method (see for example [2], [7], [8], [10]) starts with a definition of different asymptotic outcomes (given here by ‘types of ends’ of all possible trajectories). A set of initial conditions is evolved numerically until one of the outcome states is reached. Chaos is uncovered if the basin boundaries which separate initial conditions leading to different outcomes are fractal. Such fractal partitions are the result of chaotic dynamics. As we shall now demonstrate, we observe exactly these structures in the studied system.

We integrate numerically the equations of motion given by (6), (7). The initial conditions are chosen such that the geodesics start at τ = 0 from a unit circle in the (x, y)-plane (due to the homogeneity of the n-saddle potential all other geodesics can simply be obtained by a suitable rescaling). It is natural to parametrize the initial positions by an angle ϕ ∈ [−π, π) such that x(0) = cosϕ, y(0) = sinϕ. In Fig. 1 we present typical trajectories of geodesics for n = 3, 4, 5, when ẋ(0) = 0 = ẏ(0). We observe that each unbounded geodesic escapes to infinity (where for n ≥ 3 the curvature singularity is located) only along one of the n channels in the potential. The axes of these outcome channels are given in polar coordinates by the condition cos(nϕ_{j}) = −1, j = 1, ⋯, n, and represent radial lines ‘of steepest descent’ since V → −∞ as ρ → ∞ most rapidly along them. (For non-zero initial velocities more geodesics prefer one of the channels but the character of motion does not change significantly [27].)

In fact, any unbounded geodesic oscillates around the radial axis ϕ_{j} = (2j − 1)π/n of the corresponding j-th outcome channel. Introducing Δϕ_{j}(τ) = ϕ(τ) − ϕ_{j} we find asymptotically that ρ ≈ [(\frac{n}{2}−1)\sqrt{CU^{2}}(τ_{s}−τ)]^{2/(2−n)} as ρ → ∞ and

Δϕ_{j}(τ) ≈ (τ_{s} − τ)^{α}(A cos[b ln(τ_{s} − τ)]
+B sin[b ln(τ_{s} − τ)]) ,  (9)

where α = \frac{1}{2}(n+2)/(n−2), b = \frac{1}{2} \sqrt{7n^{2} − 4n − 4}/(n−2) and A, B are constants. As the geodesics approach the singularity at ρ = ∞, τ → τ_{s}, frequency of their oscillations around ϕ_{j} grows to infinity while the amplitude of oscillations tends to zero.

FIG. 1. Geodesics starting from a unit circle escape to infinity only along one of the n channels.

Let us return back to our observation that all unbounded geodesics approach infinity through only n distinct channels. These represent possible outcomes of our system and we assign them symbol j which takes one of the corresponding values, j ∈ {1, 2, …, n} (thus, for example, j = 1 means that the geodesic approach infinity at ρ = ∞ through the first channel with the axis ϕ_{1} = π/n as τ → τ_{s} > 0). From Fig. 1 we observe that in certain regions the function j(ϕ) depends very sensitively on initial position given by ϕ. We calculated j(ϕ) numerically for n = 3, 4, 5 — the results are shown in Fig. 2. Also, in the same diagrams we plot the function τ_{s}(ϕ) which takes the value of the parameter τ when the singularity at ρ = ∞ is reached by a given geodesic.

FIG. 2. The functions j(ϕ) and τ_{s}(ϕ) indicate that basin boundaries separating different outcomes are fractal.

Clearly, the boundaries between the outcomes appear to be fractal which can be confirmed on the enlarged detail of the image and the enlarged detail of the detail etc. In Fig. 3 we show such zooming in of the fractal interval localized around the value ϕ ≈ 0 for n = 3 (there are two symmetric fractal intervals in this case around ϕ ≈ ±\frac{2}{3}π) up to the sixth level. At each level the structure has the same property, namely that between two larger connected sets of geodesics with outcome channels j_{1} and j_{2} ≠ j_{1} there is always a smaller connected set of geodesics with outcome channel j_{3} such that j_{3} ≠ j_{1} and j_{3} ≠ j_{2}. Similarly as in [7], [10], the structure of initial conditions resembles three mixed Cantor sets, and this fact is a manifestation of chaos.

FIG. 3. The fractal structure described by j(ϕ) and τ_{s}(ϕ) is clearly confirmed here by zooming in the interval around ϕ ≈ 0 for n = 3.

The above structure of j(ϕ) has its counterpart in the fractal structure of τ_{s}(ϕ), see Fig. 2 and Fig. 3. We observe that the value of τ_{s} goes to infinity on each discontinuity of j(ϕ), i.e., on any fractal basin boundary between the different outcomes. There is an infinite set of peaks corresponding to chaotic bounded orbits which never ‘decide’ on a particular outcome, and so never escape to infinity. The value of τ_{s} also increases in nonchaotic regions of ϕ as one zooms in the higher levels of the fractal. This is natural since these levels are given by geodesics which undergo ‘more bounces’ on the potential walls before falling into one of the outcome channels so that their values of τ_{s} are greater.

We demonstrated chaotic behavior of geodesics in nonhomogeneous pp-waves by invariant analytic and numerical methods. As far as we know, this is the first explicit demonstration of chaos in exact radiative spacetimes (chaotic interaction of particles with linearized gravitational waves on given backgrounds has already been studied in [11], [13], [28] - [30]). Since pp-wave solutions are the simplest gravitational waves it would be an interesting task to search for a chaotic motion in other radiative space-times.

We acknowledge the support of grants GACR-202/ 96/0206 and GAUK-230/1996.

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