Impulsive gravitational waves generated by null particles in de Sitter and anti-de Sitter backgrounds

Jiří Podolský

Department of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 18000 Prague 8, Czech Republic.

Jerry B. Griffiths

Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, U.K. (October 8, 1997)

By boosting the metric of the Schwarzschild–de Sitter and Schwarzschild–anti-de Sitter space-times in the ultrarelativistic limit as the mass parameter vanishes, impulsive gravitational waves are obtained in backgrounds with a nonzero cosmological constant. We determine the geometry of these impulsive wave surfaces using various coordinate charts. In the de Sitter background they are spheres containing two null particles located at the poles. In the anti-de Sitter background they are hyperboloidal surfaces of constant negative curvature containing a single null particle on the axis of symmetry. The global structure and conformal diagrams for the complete manifolds are also determined

04.20.Jb, 04.30.Nk

I. INTRODUCTION

In a now classic paper, Aichelburg and Sexl [1] obtained an exact solution of Einstein’s equations for a null particle in a Minkowski background by boosting the Schwarzschild metric to the ultrarelativistic limit as the mass becomes arbitrarily small. The resulting space-time is an impulsive pp-wave. Generalisations of this solution have been obtained by a number of authors [2]– [5] by similarly boosting the Kerr or Kerr–Newman solutions.

A similar approach has been adopted by Hotta and Tanaka [6] in boosting the Schwarzschild–de Sitter metric to a similar limit. Effectively, they consider this as a perturbation of the de Sitter metric and boost the particle to the ultrarelativistic limit as its mass vanishes in an appropriate way. This approach also results in impulsive gravitational waves. However, the background in this case is not flat and the wavefronts are not simple planes in Minkowski space. Rather they are surfaces of constant curvature in the de Sitter background. The calculations can easily be adapted to describe impulsive gravitational waves in an anti-de Sitter background. However, the global properties of the solution in this case differ significantly.

It is the purpose of the present paper to clarify certain aspects of these solutions and to investigate the geometrical properties of the impulsive wave surfaces as they would be seen by different families of observers.

II. BOOSTING THE SCHWARZSCHILD–DE SITTER SOLUTION

Let us start with the Schwarzschild–de Sitter solution in the familiar form

ds^{2} = (1 − \frac{r^{2}}{a^{2}} − \frac{2m}{r})dt^{2} − (1 − \frac{r^{2}}{a^{2}} − \frac{2m}{r})^{−1} dr^{2}
−r^{2}(dθ^{2} + sin^{2} θ dϕ^{2}) (2.1)

in which the cosmological constant is given by Λ = 3/a^{2} and m is the mass parameter. In the limit when m = 0, this is simply the de Sitter solution which, as is well known, can be represented as a four-dimensional hyperboloid

Z_{0}^{2} − Z_{1}^{2} − Z_{2}^{2} − Z_{3}^{2} − Z_{4}^{2} = −a^{2}  (2.2)

embedded in a five-dimensional Minkowski space-time

ds^{2} = dZ_{0}^{2} − dZ_{1}^{2} − dZ_{2}^{2} − dZ_{3}^{2} − dZ_{4}^{2}.  (2.3)

For m = 0, this can be achieved by putting

Z_{0}= \sqrt{a^{2} − r^{2}} sinh(t/a)
Z_{4}= ±\sqrt{a^{2} − r^{2}} cosh(t/a)
Z_{1}= r cosθ  (2.4)
Z_{2}= r sinθ cosϕ
Z_{3}= r sinθ sinϕ.

From this it can immediately be observed that the origin r = 0 corresponds to the two world lines

(Z_{0}, Z_{1}, Z_{2}, Z_{3}, Z_{4}) = (a sinh(t/a), 0, 0, 0, ±a cosh(t/a))

(2.5)

along the hyperboloid, as indicated in figure 1 in which the coordinates Z_{2} and Z_{3} are suppressed. It may seem surprising that the Schwarzschild–de Sitter solution (2.1) does not just have a single source in the limit when the mass is small. However, it is known that the coordinates of (2.1) do not cover the complete space-time whose analytic extension must include both the black hole and white hole parts. This fact can be deduced from its conformal structure (see eg. Bičák and Podolský, [7]).

Fig. 1

Following the approach of Aichelburg and Sexl [1], we may now boost the singularity representing the point mass in the Schwarzschild–de Sitter solution to the limit in which its speed approaches that of light and its mass is scaled to zero in an appropriate way.

As observed by Hotta and Tanaka [6], in this case it is first appropriate to take the linear approximation of (2.1) for small m in the form

ds^{2}= (1 − \frac{r^{2}}{a^{2}} − \frac{2m}{r})dt^{2}

− (1 − \frac{r^{2}}{a^{2}} )^{−1}[1 + (1 − \frac{r^{2}}{a^{2}} )^{−1} \frac{2m}{r}] dr^{2}

−r^{2}(dθ^{2} + sin^{2} θ dϕ^{2}).  (2.6)

This can be considered as a first order perturbation of the de Sitter line element. Using the coordinates Z_{i} of (2.4), the first order perturbation of the line element (2.3) can be written as

ds_{1}^{2}= −\frac{2ma^{2}}{r} [ (\frac{Z_{4}dZ_{0} − Z_{0}dZ_{4}}{Z_{4}^{2} − Z_{0}^{2}} )^{2}

+\frac{a^{2}}{r^{2}} (\frac{Z_{0}dZ_{0} − Z_{4}dZ_{4}}{Z_{4}^{2} − Z_{0}^{2}} )^{2} ],  (2.7)

where r = (a^{2} + Z_{0}^{2} − Z_{4}^{2})^{1/2}. A boost can then be performed in any direction orthogonal to Z_{4} (a boost in the Z_{4} direction simply corresponds to a time shift). Taking the boost in the Z_{1} direction, we can put

Z_{0} → \frac{Z_{0} + vZ_{1}}{\sqrt{1 − v^{2}}} , Z_{1} → \frac{vZ_{0} + Z_{1}}{\sqrt{1 − v^{2}}} ,

Z_{2} → Z_{2},  Z_{3} → Z_{3},  Z_{4} → Z_{4},  (2.8)

where v is the boost parameter. It can then be seen that, in the limit as v → 1, the linear perturbation term vanishes everywhere except on the hypersurface Z_{0}+Z_{1} = 0. Then, by also scaling the mass according to

m = p\sqrt{1 − v^{2}},

where p is a constant, it can be seen that the only nonvanishing term in (2.7) is that which becomes a multiple of (dZ_{0} + dZ_{1})^{2}. The line element thus takes the form

ds^{2}= dZ_{0}^{2} − dZ_{1}^{2} − dZ_{2}^{2} − dZ_{3}^{2} − dZ_{4}^{2}
−2pa^{2}H(Z_{4}) δ(Z_{0} + Z_{1}) (dZ_{0} + dZ_{1})^{2},  (2.9)

where, putting

x^{2} = \frac{(Z_{0} + vZ_{1})^{2}}{1 − v^{2}} ,

H(Z_{4})δ(Z_{0} + Z_{1})  (2.10)

= lim_{v→1} \frac{1}{\sqrt{1 − v^{2}}} \frac{(a^{2} − Z_{4}^{2})Z_{4}^{2} + (a^{2} + Z_{4}^{2})x^{2}}{(a^{2} − Z_{4}^{2} + x^{2}) ^{3/2} (Z_{4}^{2} − x^{2})^{2}} .

In evaluating this limit, we follow Hotta and Tanaka [6], in making use of the identity

lim_{v→1} \frac{1}{\sqrt{1 − v^{2}}} f (\frac{(Z_{0} + vZ_{1})^{2}}{1 − v^{2}} )
= δ(Z_{0} + Z_{1}) ∫_{−∞}^{∞} f(x^{2})dx,  (2.11)

which can easily be proven in the theory of distributions. However, in this case, the integrand involved contains singularities at x = ±Z_{4}. Nevertheless, it is possible to remove the effect of these by first making a singular coordinate transformation such that

dZ_{0}−dZ_{1} → dZ_{0} − dZ_{1}

− lim_{v→1} \frac{4pa^{3}}{\sqrt{1 − v^{2}}} (a^{2} − \frac{(Z_{0} + Z_{1})^{2}}{1 − v^{2}} )^{−2} (dZ_{0} + dZ_{1}).

Then, using (2.11), it can be shown that the function H(Z_{4}) in the line element (2.9) takes the form

H(Z_{4}) = −\frac{2}{a^{2}} [2 − \frac{Z_{4}}{a} ln |\frac{a + Z_{4}}{a − Z_{4}} |].  (2.12)

It can thus be seen that the resulting space-time differs from the de Sitter space-time only by the inclusion of an impulsive wave located on the null hypersurface given by

Z_{0} + Z_{1} = 0,  Z_{2}^{2} + Z_{3}^{2} + Z_{4}^{2} = a^{2}.  (2.13)

It may be observed that the shock wave is a 2-sphere in the 5-dimensional Minkowski space at any time Z_{0}.

In addition, as shown in [6], the singularities that were originally located at r = 0 have now been boosted to become two null particles described by delta functions in the energy-momentum tensor on the wave surface. These have the worldlines

Z_{0} + Z_{1} = 0,  Z_{2} = Z_{3} = 0,  Z_{4} = ±a.  (2.14)

These are the two null straight lines on the fourdimensional hyperboloid as indicated in figure 2.

Fig. 2

III. GLOBAL COORDINATES (Λ > 0)

Let us now investigate the geometry of the shock wave in the de Sitter background. To do this, we set up some appropriate coordinate system. At each event, we introduce some “time” and “space” labels. We can then ask the question: what are the space coordinates of the shock wave at a given “synchronized” time? Of course, although the geometry is unique, the coordinates can be chosen arbitrarily, so that the coordinate shape is not unique. In practice, however, “comoving coordinates” can always be introduced, so the coordinate shape of the shock wave will correspond to its shape as seen by the associated family of observers.

In fact there are a number of privileged families of observers in a de Sitter space. These correspond to those in which the spatial sections at any synchronous time have constant curvature. We will consider the three cases of positive, zero and negative curvature. Of these the most natural choice of coordinates are those associated with spatial sections of constant positive curvature. These will be considered in the present section, with the other cases considered in section 4. Further properties of the coordinates used for the de Sitter spacetime and the relations between them have been described by Eriksen and Grøn [8], although in a slightly different notation.

The most natural set of coordinates covering the fourdimensional hyperboloid (2.2) are the “global coordinates” given by

Z_{0}= a sinh(τ/a)
Z_{1}= a cosh(τ/a) cosχ
Z_{4}= a cosh(τ/a) sinχ cosθ  (3.1)
Z_{2}= a cosh(τ/a) sinχ sinθ cosϕ
Z_{3}= a cosh(τ/a) sinχ sinθ sinϕ

in which the coordinates θ and ϕ are different from those of section 2. Indeed the same symbols are used for a number of different coordinate parametrisations below. With this, the line element becomes

ds^{2} = dτ^{2} − a^{2} cosh^{2}(τ/a)  (3.2)

×(dχ^{2} + sin^{2} χ(dθ^{2} + sin^{2} θdϕ^{2})).

In this case, the privileged family of timelike observers have worldlines on which χ, θ and ϕ are constant. For this family, the 3-spaces τ = const., which represent the universe at any time, have the geometry of 3-spheres S^{3} which appear to contract to a minimum size at τ = 0 and then re-expand.

Fig. 3

Let us now return to consider the shock wave, which is given by (2.13) with the null particles having the worldlines (2.14). Using (3.1), we see that the shock wave is located at cosχ = − tanh(τ/a). Thus, the hyperspherical coordinate χ monotonically increases from χ = 0 as τ → −∞ to χ = π/2 at τ = 0 and χ = π as τ → +∞. This has a natural physical interpretation shown in figure 3. As the synchronous (proper) time τ of comoving observers at fixed points P given by constant χ_{0}, θ_{0} and ϕ_{0} increases, the impulse propagates from the “North pole” (χ = 0) to the “South pole” (χ = π) of the closed space S^{3}. Moreover, the radius of S^{3} is given by R = a cosh(τ/a); i.e. as τ increases, for τ < 0, the space contracts to a minimum radius a (at τ = 0) and then re-expands for τ > 0.

This is visualized in figure 3 where the coordinate ϕ was suppressed so that the evolution of the universe is represented by a sequence of 2-spheres (with coordinates χ and θ) of which only hemispheres are shown. At any time, the impulse is here given by a (semi-)circle which is located exactly in the circular intersection of the corresponding sphere with a cylinder of radius a. Indeed, the distance of the impulse from the axis through the poles

is given by R sinχ = a cosh(τ/a)\sqrt{1 − tanh^{2}(τ/a)} ≡ a. The null particles which are the singular sources of the wave are located at θ = 0 and θ = π, as follows immediately from (2.14) and (3.1).

In order to investigate the global conformal structure of the space-time, it is useful to introduce the conformal time η_{p} defined by

η_{p} = 2 arctan(e^{τ/a}).  (3.3)

Then sinh(τ/a) = − cotη_{p}, cosh(τ/a) = 1/ sinη_{p} and the line element (3.2) becomes

ds^{2} = \frac{a^{2}}{sin^{2} η_{p}} (dη_{p}^{2} − dχ^{2} − sin^{2} χ(dθ^{2} + sin^{2} θdϕ^{2})).

(3.4)

In this case, the appropriate conformal diagram which is shown in figure 4 covers the coordinate ranges from η_{p} = 0 (τ = −∞) to η_{p} = π (τ = ∞), and from χ = 0 to χ = π.

Fig. 4

It may then be observed that the condition Z_{0}+Z_{1} = 0 corresponds to the single root χ = η_{p}, and hence that

δ(Z_{0} + Z_{1}) = \frac{1}{a} δ(χ − η_{p}).

The impulsive wave in the conformal diagram in figure 4 is simply located along the line χ = η_{p}. Moreover, since the function H(Z_{4}) only need be evaluated for χ = η_{p}, it can be seen that the impulsive addition to the de Sitter line element (3.4) is given by

ds_{1}^{2} = 4pa [2 − cosθ ln |\frac{1 + cosθ}{1 − cosθ} |]
×δ(η_{p} − χ)(dη_{p} − dχ)^{2}.  (3.5)

From this, it can immediately be seen that the two null particles are located at the points θ = 0 and θ = π, which correspond to the opposite poles of the spherical wave surface.

IV. OTHER COORDINATE CHARTS (Λ > 0)

Although the coordinates introduced in the previous section are the most natural in the sense that they cover the entire de Sitter space-time and provide a simple interpretation of the shock geometry, it is also appropriate to consider some alternative coordinate systems as they can be interpreted in terms of different families of observers.

Taking the case in which spatial sections at synchronous times have zero curvature, we can introduce the conformally flat coordinates

Z_{0}= \frac{1}{2η}(a^{2} − η^{2} + (x − a)^{2} + y^{2} + z^{2})
Z_{1}= \frac{a}{η}(x − a)
Z_{2}= \frac{a}{η}y  (4.1)
Z_{3}= \frac{a}{η}z
Z_{4}= \frac{1}{2η}(a^{2} + η^{2} − (x − a)^{2} − y^{2} − z^{2}).

With this, the line element becomes

ds^{2} = \frac{a^{2}}{η^{2}} (dη^{2} − dx^{2} − dy^{2} − dz^{2}).  (4.2)

When coming to describe the geometry of the impulsive wave in these coordinates, it is convenient to put

x = ρ cosθ,  y = ρ sinθ cosϕ,  z = ρ sinθ sinϕ,

where ρ ∈ [0, ∞), θ ∈ [0, π], ϕ ∈ [0, 2π), so that

ρ^{2} = x^{2} + y^{2} + z^{2}

and then we have

Z_{0} + Z_{1} = −\frac{1}{2η}(η^{2} − ρ^{2})  (4.3)

and

Z_{4} = \frac{1}{2η}(η^{2} − ρ^{2} + 2ax).

Thus, on the impulsive wave on which Z_{0} + Z_{1} = 0, we have η = ±ρ, and

Z_{4} = \frac{ax}{η} = a\frac{ρ}{η} cosθ.

It may be noted that, in order to include the maximum analytic extension of the space-time, it is necessary to permit η to cover the range (−∞, ∞). In addition, from (4.3) it follows that

δ(Z_{0} + Z_{1}) = δ(η − ρ) + δ(η + ρ)

which indicates that the impulsive wave has the two components:

when η = ρ,  Z_{4} = a cosθ
when η = −ρ,  Z_{4} = −a cosθ.

Combining these, we see that the impulsive addition to the de Sitter line element (4.2) is given by

ds_{1}^{2}= 4p [2 − cosθ ln |\frac{1 + cosθ}{1 − cosθ} |]
× [δ(η − ρ)(dη − dρ)^{2} + δ(η + ρ)(dη + dρ)^{2}] . (4.4)

This looks like two impulses. However, both components are required for the conformal picture to be geodesically complete: since ρ ≥ 0, the term δ(η − ρ) is nonvanishing only for η ≥ 0 while δ(η + ρ) applies only for η ≤ 0. Again it can be seen that the two null source particles are located at the poles θ = 0 and θ = π on the spherical wave surface.

The conformal diagram of the section y = z = 0 is shown in figure 5. In this, the whole de Sitter universe is covered by two regions in which η ≥ 0 and η ≤ 0. It is also convenient to use x = ρ cosθ where θ = 0 or π. The diagram thus illustrates the paths of the two null particles propagating in opposite directions. The impulsive waves ρ = η and ρ = −η can also be seen to represent either a contracting 2-sphere in an exponentially expanding universe or an expanding 2-sphere in a contracting universe.

Fig. 5

This point can also be clarified by introducing the proper time τ defined by

η = ±ae^{−τ/a},

so that the line element (4.2) of de Sitter space-time becomes

ds^{2} = dτ^{2} − e^{2τ/a}(dρ^{2} + ρ^{2}(dθ^{2} + sin^{2} θdϕ^{2})).  (4.5)

It can thus be seen that the region in which η > 0 corresponds to an exponentially expanding “inflationary” universe. The region included in the conformal diagram in which η < 0 corresponds to a time reversed exponentially contracting universe.

It may also be observed that the “radius” of the spherical impulsive wave is given by

ρ = |η| = ae^{−τ/a}.

Thus, in the expanding universe with the 3-dimensional spaces τ = const. being flat, the impulsive wave is an exponentially contracting 2-sphere. Conversely (changing τ → −τ), it can also be considered as an expanding spherical wave in an exponentially contracting universe.

Let us finally consider how the impulsive wave would appear to the family of observers which, at any synchronous time, are located in a space of a negative curvature. In this case, the four-dimensional hyperboloid (2.2) can be parametrised by

Z_{0}= a sinh(τ/a) coshψ
Z_{1}= a cosh(τ/a)
Z_{4}= a sinh(τ/a) sinhψ cosθ  (4.6)
Z_{2}= a sinh(τ/a) sinhψ sinθ cosϕ
Z_{3}= a sinh(τ/a) sinhψ sinθ sinϕ

and the de Sitter line element takes the form

ds^{2} = dτ^{2} − a^{2} sinh^{2}(τ/a)  (4.7)

×(dψ^{2} + sinh^{2} ψ(dθ^{2} + sin^{2} θdϕ^{2})).

However, it may be noted that these coordinates only cover part of the hyperboloid.

Again it is convenient to introduce a conformal time defined here by

η_{n} = \{ \begin{matrix}2 coth^{−1} e^{τ/a} & for τ ≥ 0 \\ 2 tanh^{−1} e^{τ/a} & for τ ≤ 0. \end{matrix}  (4.8)

It may be noted that, as τ increases from −∞ to 0 and then to ∞, η_{n} increases from 0 to ∞ and then decreases back to 0. Using this, the hyperboloid is parametrised by (4.6) where sinh(τ/a) = ±1/ sinhη_{n}, cosh(τ/a) = coshη_{n}/ sinhη_{n} and the line element (4.7) becomes

ds^{2} = \frac{a^{2}}{sinh^{2} η_{n}} (dη_{n}^{2} − dψ^{2} − sinh^{2} ψ(dθ^{2} + sin^{2} θdϕ^{2})).

(4.9)

Since

Z_{0} + Z_{1} = \frac{a}{sinhη_{n}} (coshη_{n} ± coshψ),

we have

δ(Z_{0} + Z_{1}) = \frac{1}{a}[δ(η_{n} − ψ) + δ(η_{n} + ψ)].

Thus, on the impulsive wave, ψ = ±η_{n} and it can again be shown that the impulsive addition to the de Sitter line element (4.9) is given by

ds_{1}^{2}= 4pa [2 − cosθ ln |\frac{1 + cosθ}{1 − cosθ} |]  (4.10)

×[δ(η_{n} − ψ)(dη_{n} − dψ)^{2} + δ(η_{n} + ψ)(dη_{n} + dψ)^{2}].

This again looks like two impulses. However, the first term is nonvanishing only for ψ > 0, while the second term is nonvanishing only for ψ < 0. Both components are required for the conformal picture to be geodesically complete. This is described in figure 6.

Fig. 6

The spherical pulse is now propagating in a 3-space τ = const. which has a negative curvature and contracts for τ < 0 according to a| sinh(τ/a))|. The impulse is located at |ψ| = η_{n} = 2 tanh^{−1} e^{τ/a}, i.e. it is expanding from ψ = 0 to |ψ| = ∞.

V. A NULL PARTICLE IN AN ANTI-DE SITTER BACKGROUND

For a vacuum universe with a negative cosmological constant (Λ < 0), the Schwarzschild–anti-de Sitter solution can be written in the form (2.1) but with a^{2} replaced by −a^{2}. It is then possible to proceed as in section 2 to boost the singular point r = 0 in the limit as m → 0 in an appropriate way. This again produces an impulsive wave, but this time in an anti-de Sitter background.

Before proceeding, however, it is first appropriate to recall that the anti-de Sitter solution can be represented as the four-dimensional hyperboloid

Z_{0}^{2} − Z_{1}^{2} − Z_{2}^{2} − Z_{3}^{2} + Z_{4}^{2} = a^{2}  (5.1)

embedded in a five-dimensional Minkowski space-time

ds^{2} = dZ_{0}^{2} − dZ_{1}^{2} − dZ_{2}^{2} − dZ_{3}^{2} + dZ_{4}^{2}  (5.2)

with two timelike dimensions, Z_{0} and Z_{4}. The natural parametrisation of this is given by

Z_{4}= \sqrt{a^{2} + r^{2}} cos(t/a)
Z_{0}= \sqrt{a^{2} + r^{2}} sin(t/a)
Z_{1}= r cosθ  (5.3)
Z_{2}= r sinθ cosϕ
Z_{3}= r sinθ sinϕ.

From this it can be seen that a source at the origin r = 0 is located on the line

(Z_{0}, Z_{1}, Z_{2}, Z_{3}, Z_{4}) = (a sin(t/a), 0, 0, 0, a cos(t/a))(5.4)

which is a closed timelike geodesic which returns to the same event after Δt = 2πa. This is illustrated in figure 7 in which the spacelike coordinates Z_{2} and Z_{3} are suppressed. It can thus be seen that the m → 0 limit of the Schwarzschild–anti-de Sitter solution has a single source following a closed timelike line.

Fig. 7

In this case, the first order perturbation of the antide Sitter line element, equivalent to (2.7), can be written as

ds_{1}^{2}= −\frac{2ma^{2}}{r} [ (\frac{Z_{4}dZ_{0} − Z_{0}dZ_{4}}{Z_{4}^{2} + Z_{0}^{2}} )^{2}

+\frac{a^{2}}{r^{2}} (\frac{Z_{0}dZ_{0} + Z_{4}dZ_{4}}{Z_{4}^{2} + Z_{0}^{2}} )^{2} ],  (5.5)

where r = (Z_{0}^{2} + Z_{4}^{2} − a^{2})^{1/2}. A boost can then be performed in the Z_{1} direction using (2.8). Again it can be seen that, in the limit as v → 1, this linear perturbation term vanishes everywhere except on the hypersurface Z_{0} + Z_{1} = 0. By putting m = p\sqrt{1 − v^{2}}, the line element then takes the form

ds^{2}= dZ_{0}^{2} − dZ_{1}^{2} − dZ_{2}^{2} − dZ_{3}^{2} + dZ_{4}^{2}
−2pa^{2}H(Z_{4}) δ(Z_{0} + Z_{1}) (dZ_{0} + dZ_{1})^{2},  (5.6)

where using (2.11) with x^{2} = (Z_{0} + vZ_{1})^{2}/(1 − v^{2}),

H(Z_{4}) = ∫_{−∞}^{∞} \frac{(Z_{4}^{2} − a^{2})Z_{4}^{2} + (a^{2} + Z_{4}^{2})x^{2}}{(Z_{4}^{2} − a^{2} + x^{2}) ^{3/2} (Z_{4}^{2} + x^{2}) ^{2}} dx. (5.7)

Since in this case the integrand is nonsingular we obtain immediately

H(Z_{4}) = −\frac{2}{a^{2}} [2 − \frac{Z_{4}}{a} ln |\frac{a + Z_{4}}{a − Z_{4}} |].  (5.8)

It can be seen that in this case the resulting space-time differs from the anti-de Sitter space-time by the inclusion of an impulsive wave located on the null hypersurface given by

Z_{0} + Z_{1} = 0,  Z_{4}^{2} − Z_{2}^{2} − Z_{3}^{2} = a^{2}.  (5.9)

Unlike the case in a de Sitter background, this is a now an evolving 2-hyperboloid in a 5-dimensional Minkowski space with two time dimensions.

Also, the source is now a null particle having the worldline

Z_{0} + Z_{1} = 0,  Z_{2} = Z_{3} = 0,  Z_{4} = ±a.

This is represented by the two null straight lines on the four-dimensional hyperboloid as indicated in figure 8. However, since time is here effectively wrapped around the hyperboloid, this represents a null particle which propagates to infinity and then comes back again in an endless cycle.

Fig. 8

VI. A GLOBAL COORDINATE CHART (Λ < 0)

In an anti-de Sitter space-time, it is appropriate initially to consider the natural global coordinate system obtained by putting r = a sinhR in (5.3) by which

Z_{4}= a coshR cos(t/a)
Z_{0}= a coshR sin(t/a)
Z_{1}= a sinhR cosθ  (6.1)
Z_{2}= a sinhR sinθ cosϕ
Z_{3}= a sinhR sinθ sinϕ

and the line element becomes

ds^{2} = cosh^{2}R dt^{2} − a^{2}(dR^{2} + sinh^{2}R(dθ^{2} + sin^{2} θdϕ^{2})).

(6.2)

In this case, the privileged family of timelike observers have worldlines on which R, θ and ϕ are constant, their proper times are given by τ = t coshR. The timelike slices t = const. are three dimensional spaces of constant negative curvature.

We may now consider the impulsive wave on which Z_{0} + Z_{1} = 0. It can be seen that this is given by

tanhR cosθ = − sin(t/a).  (6.3)

It may be noticed that this does not give such a convenient coordinate condition as does the equivalent case in a de Sitter background. Nevertheless, we can show that

δ(Z_{0} + Z_{1}) = \frac{δ(t + a sin^{−1}(tanhR cosθ))}{\sqrt{1 + sinh^{2} R sin^{2} θ}}

and, evaluating H(Z_{4}) using (6.1), we obtain that the impulsive addition to the anti-de Sitter line element (6.2) is given by

ds_{1}^{2}= 4p [\frac{2}{coshR cos(t/a)} − ln |\frac{1 + coshR cos(t/a)}{1 − coshR cos(t/a)}|]
×δ(t + a sin^{−1}(tanhR cosθ))  (6.4)
×(a\frac{cosθ}{coshR}dR − a sinhR sinθdθ + coshR cos(t/a)dt)^{2},

which looks somewhat complicated.

We would now like to visualize this wave as it would be observed by the natural family of observers in the 3-dimensional space (R, θ, ϕ). For this, we consider a section t = const. through the space-time. It is then convenient to introduce the coordinate Z̃_{0} = a coshR which, together with the coordinates Z_{1}, Z_{2} and Z_{3} span a 4-dimensions pseudo-Euclidean space

ds_{4}^{2} = dZ̃_{0}^{2} − dZ_{1}^{2} − dZ_{2}^{2} − dZ_{3}^{2}.  (6.5)

The hypersurface of the impulsive wave can now be visualised at any time as the intersection of the plane Z_{1} = − sin(t/a)Z̃_{0} with half of a rotational hyperboloid

Z̃_{0}^{2} − Z_{1}^{2} − (\sqrt{Z_{2}^{2} + Z_{3}^{2}} )^{2} = a^{2}  (6.6)

in the pseudo-Euclidean half space (Z̃_{0}, Z_{1}, \sqrt{Z_{2}^{2} + Z_{3}^{2}} ) representing the 3-dimensional space of constant negative curvature with the metric ds_{3}^{2} = −a^{2}(dR^{2}+sinh^{2}R(dθ^{2}+ sin^{2} θdϕ^{2})). This is illustrated in figure 9.

Fig. 9

The source of the impulsive wave is a single null particle located on the wave at the point \sqrt{Z_{2}^{2} + Z_{3}^{2}} = 0, Z_{1} = −a tan(t/a). This indicates that the null particle moves from Z_{1} = ∞ to Z_{1} = −∞ as t goes from −\frac{π}{2}a to \frac{π}{2}a.

It is then possible to visualize the wavefront at any time by projecting the intersection of the plane and the hyperboloid onto the space of Z_{1} and \sqrt{Z_{2}^{2} + Z_{3}^{2}} as illustrated in figure 9, and then rotating it about the Z_{1} axis. In this way, the Z_{2} and Z_{3} coordinates are reintroduced by putting

Z_{2} = \sqrt{Z_{2}^{2} + Z_{3}^{2}} cosϕ, Z_{3} = \sqrt{Z_{2}^{2} + Z_{3}^{2}} sinϕ.

The impulsive wave is thus represented by the sequence of hyperboloidal surfaces illustrated in figure 10. While the angle ϕ is a “polar” angle associated with the axial symmetry of the surfaces, the angle θ is measured from the (positive-oriented) axis of symmetry. It can be shown that each hyperboloidal surface representing the impulsive wave front has an asymptotic angle θ_{a} given by θ_{a} = t/a+π/2. Therefore, as t/a grows from −π/2 to π/2 (and the source null particle moves from Z_{1} = ∞ to Z_{1} = −∞) the wave surfaces open from θ_{a} = 0 to θ_{a} = π.

Fig. 10

Since the time coordinate t is periodic, it can be seen that the impulsive wave propagates to Z_{1} = −∞ and then back to Z_{1} = ∞ in an endless cycle.

In order to investigate the conformal structure of the space-time in these coordinates, we introduce the conformal time χ defined by

χ = 2 arctan(e^{R}) − π/2.  (6.7)

In this case we have coshR = secχ, sinhR = tanχ and line element becomes

ds^{2} = \frac{a^{2}}{cos^{2}χ} (\frac{1}{a^{2}} dt^{2} − dχ^{2} − sin^{2}χ(dθ^{2} + sin^{2} θdϕ^{2})) .

(6.8)

In this case, the impulsive wave on which Z_{0} + Z_{1} = 0 can be seen to occur when sinχ cosθ = − sin(t/a) in agreement with (6.3). The location of the wave in the conformal diagram of the anti-de Sitter space-time for θ = 0 is shown in figure 11.

Fig. 11

VII. OTHER COORDINATE CHARTS (Λ < 0)

Let us now consider how the impulsive wave would appear to the alternative family of observers which, at any synchronous time, are located in a space of constant negative curvature. In this case, the four-dimensional hyperboloid (5.1) can be parametrised by

Z_{0}= a sin(τ/a)
Z_{1}= a cos(τ/a) sinhψ cosθ
Z_{2}= a cos(τ/a) sinhψ sinθ cosϕ  (7.1)
Z_{3}= a cos(τ/a) sinhψ sinθ sinϕ
Z_{4}= a cos(τ/a) coshψ

and the anti-de Sitter line element takes the form

ds^{2} = dτ^{2} − a^{2} cos^{2}(τ/a)  (7.2)
×(dψ^{2} + sinh^{2} ψ(dθ^{2} + sin^{2} θdϕ^{2})).

The time-slices τ = const. are 3-dimensional spaces of constant negative curvature, but which do not have a constant “size” as occurs for the global coordinates (6.2) in the previous section. Their curvature change according to a cos(τ/a). However, it may be noted that these coordinates only cover part of the hyperboloid.

Again it is convenient to introduce a conformal time defined by

η_{n} = tanh^{−1} ( sin(τ/a)).  (7.3)

It may be noted that, as τ/a increases from −π/2 to 0 and then to π/2, η_{n} increases from −∞ to ∞. With these parameters, the hyperboloid is now given by (7.1) with sin(τ/a) = tanhη_{n}, cos(τ/a) = sechη_{n} and line element (7.2) becomes

ds^{2} = \frac{a^{2}}{cosh^{2} η_{n}} (dη_{n}^{2} − dψ^{2} − sinh^{2} ψ(dθ^{2} + sin^{2} θdϕ^{2})).

(7.4)

Proceeding as before, we can consider the impulsive wave on which Z_{0} + Z_{1} = 0. However, this again does not give such a convenient coordinate condition as does the equivalent case in a de Sitter background. Calculations can again be performed to yield an expression similar to (6.4). However, this does not provide any further insights. The conformal diagram with the impulse for θ = 0 is given in figure 12.

Fig. 12

It is finally appropriate here to consider a conformally flat coordinate system. Accordingly, we can put

Z_{0}= \frac{1}{2x}(a^{2} − η^{2} + x^{2} + y^{2} + (z − a)^{2})
Z_{1}= \frac{a}{x}(z − a)

Z_{2}= \frac{1}{2x}(a^{2} + η^{2} − x^{2} − y^{2} − (z − a)^{2})  (7.5)
Z_{3}= \frac{a}{x}y
Z_{4}= \frac{a}{x}η.

With this, the line element becomes

ds^{2} = \frac{a^{2}}{x^{2}} (dη^{2} − dx^{2} − dy^{2} − dz^{2}).  (7.6)

It is also convenient to put

x = ρ cosθ, y = ρ sinθ cosϕ, z = ρ sinθ sinϕ (7.7)

so that the line element becomes

ds^{2} = \frac{a^{2}}{ρ^{2} cos^{2} θ} (dη^{2} − dρ^{2} − ρ^{2}(dθ^{2} + sin^{2} θdϕ^{2})). (7.8)

It can also be shown that the form of the impulse is given by

δ(Z_{0} + Z_{1}) = \frac{|x|}{ρ}[δ(η − ρ) + δ(η + ρ)]  (7.9)

and the impulsive addition to the anti-de Sitter line element is given by

ds_{1}^{2}= \frac{4p}{| cosθ|} [2 − \frac{1}{cosθ} ln |\frac{1 + cosθ}{1 − cosθ} |]  (7.10)

× [δ(η − ρ)(dη − dρ)^{2} + δ(η + ρ)(dη + dρ)^{2}] .

This is clearly the simplest coordinate form of the shock geometry (5.6) in an anti–de Sitter background. Again this is only one impulse for η > 0 and η < 0. Also, the impulsive wave is given by η = ±ρ. The metric is regular for θ = π/2 so that it is singular only for θ = 0 and θ = π. The conformal diagram is given in figure 13.

Fig. 13

For any value of η, the surfaces ρ = const. according to (7.7) look like spheres. However, they are not spheres: they are surfaces that are conformal to spheres. Actually, they are hyperboloidal. Indeed, performing the transformation ξ = tanh^{−1}(sinθ) we can write the section η = ±ρ of the line element (7.8) as dσ^{2} = −a^{2}(dξ^{2} + sinh^{2} ξdϕ^{2}). This represents the hyperboloidal 2-space of constant negative curvature −a.

We may thus conclude that the conformally flat coordinate system (7.8) is well adapted for describing the impulsive wave (5.6) in anti-de Sitter space-time. Conversely, the coordinates (6.2) and (7.2) are not suitable due to the fact that their corresponding 2-surfaces are spheres having the line element dσ^{2} = −C^{2}(dθ^{2} + sin^{2} θdϕ^{2}), C = const. This naturally explains the complicated forms of the impulsive additions to the antide Sitter line element (6.4) in these coordinates. By contrast, we have previously observed that all three coordinate systems (3.2), (4.2) and (4.7) that we have introduced in the Λ > 0 case are suitable for describing spherical impulsive gravitational waves in the de Sitter universe since they also have dσ^{2} = −C^{2}(dθ^{2}+sin^{2} θdϕ^{2}).

VIII. CONCLUSIONS

Following the method of Hotta and Tanaka [6], we have boosted the Schwarzschild–de Sitter and the Schwarzschild–anti-de Sitter metrics to the ultrarelatistic limit and have thus constructed exact solutions for impulsive waves in backgrounds with a nonzero cosmological constant. In a de Sitter background, a spherical gravitational wave is generated by two null particles propagating in opposite directions. We have used three privileged families of observers in which the constant time surfaces have constant positive, negative or zero curvature. We have shown that each of these coordinate systems are suitable for describing the geometry of the impulsive wave. In the closed form of the de Sitter universe which contracts to a minimum size and then re-expands, the impulse propagates from the “North pole” to the “South pole”. In an expanding flat and open de Sitter universe it is a contracting sphere; alternatively in a contracting universe the sphere is expanding.

By contrast, the impulsive wave generated in an antide Sitter background is hyperboloidal. It is generated by a single null particle. We have also shown that it is only the conformally flat coordinates that form a suitable natural parameterisation of this solution.

It may also be noted that a family of impulsive spherical gravitational waves have been constructed by Hogan [9] which appear to propagate either in a de Sitter or an anti-de Sitter background. However, these are not related to the solutions given above.

ACKNOWLEDGMENTS

JP is grateful for hospitality at the Mathematical Sciences Department at Loughborough University and for support from the grant GACR-202/96/0206 of the Czech Republic and the grant GAUK-230/96 of Charles University while this work was undertaken.

[1] P.C. Aichelburg and R.U. Sexl, Gen. Relativ. Gravit. 2, 303 (1971).

[2] V. Ferrari and P. Pendenza, Gen. Relativ. Gravit. 22, 1105 (1990).

[3] H. Balasin and H. Nachbagauer, Class. Quantum Grav. 12, 707 (1995).

[4] H. Balasin and H. Nachbagauer, Class. Quantum Grav. 13, 731 (1996).

[5] C.O. Loustó and N. Sánchez, Nucl. Phys. B383, 377 (1992).

[6] M. Hotta and M. Tanaka, Class. Quantum Grav. 10, 307 (1993).

[7] J. Bičák and J. Podolský, Phys. Rev. D 52, 887 (1995).

[8] E. Eriksen and Ø. Grøn, Int. J. Mod. Phys. D 4, 115 (1995).

[9] P.A. Hogan, Phys. Lett. 171A, 21 (1992).

FIG. 1. The de Sitter space-time represented as a four-dimensional hyperboloid embedded in a five-dimensional Minkowski space-time. The coordinates Z_{2} and Z_{3} have been suppressed, and the timelike world lines are indicated of the two test particles which correspond to the weak limit of the source of the Schwarzschild–de Sitter space-time.

FIG. 2. The de Sitter space-time represented as a four-dimensional hyperboloid embedded in a five-dimensional Minkowski space-time with the coordinates Z_{2} and Z_{3} suppressed. The null world lines are indicated of the two particles which correspond to the source of the Schwarzschild–de Sitter space-time boosted to the limit in which its velocity is that of light and its mass vanishes.

FIG. 3. In natural coordinates the de Sitter space-time is a closed universe with the spatial geometry of a 3-sphere whose radius is given by R = a cosh(τ/a). Suppressing the coordinate ϕ, its evolution can be visualized as a sequence of 2-spheres which are indicated here by hemispheres which contract to a minimum size and then re-expand. At any time, the impulsive wave is a 2-sphere represented here by a semi-circle located at the circular intersection of a cylinder of radius a with the corresponding sphere representing the universe. This propagates from the “North pole” χ = 0 as τ → −∞ across the equator χ = π/2 at τ = 0 to the “South pole” χ = π as τ → ∞ while the whole universe contracts to a minimum size R = a at τ = 0 and then re-expands.

FIG. 4. The conformal diagram of the de Sitter space-time with natural global coordinates η_{p}, χ. Each point of the diagram represents a 2-sphere parametrized by coordinates θ, ϕ. The lines χ = 0 and χ = π are identified. In contrast to Minkowski space-time, de Sitter has a space-like future and past infinity for null and time-like lines. The impulsive wave is located along χ = η_{p} with two singular points sources at θ = 0 and θ = π.

FIG. 5. The conformal diagram of the section y = z = 0 of the de Sitter space-time with coordinates η, ρ = |x|. The wave is located along η = ±ρ and represents either an exponentially contracting 2-sphere in an exponentially expanding flat inflationary universe or an expanding 2-sphere in a contracting space.

FIG. 6. The conformal diagram of the de Sitter space-time with coordinates τ, ψ. For the corresponding comoving observers the spherical impulsive wave expands in a contracting space of negative curvature.

FIG. 7. The anti-de Sitter space-time represented as a four-dimensional hyperboloid embedded in a five-dimensional space-time with two timelike dimensions Z_{0} and Z_{4}. Here the coordinates Z_{2} and Z_{3} have been suppressed, and the closed world line of the particle which corresponds to the weak limit of the source of the Schwarzschild–anti-de Sitter space-time is indicated.

FIG. 8. The anti-de Sitter space-time represented as a four-dimensional hyperboloid embedded in a five-dimensional Minkowski space-time with two timelike dimensions Z_{0} and Z_{4}. The null world line is indicated of the particle which correspond to the source of the Schwarzschild–anti-de Sitter space-time boosted to the limit in which its velocity is that of light and its mass vanishes.

FIG. 9. The geometry of the impulsive wave in anti-de Sitter space-time as observed by natural family of observers at any time is given by the intersection of the plane Z_{1} = − sin(t/a)Z̃_{0} with half of a rotational hyperboloid in the pseudo-Euclidean space so that the wavefronts are the spaces of constant negative curvature.

FIG. 10. By projecting the intersections representing the wave fronts from figure 9 onto the plane of Z_{1}, Z_{2}^{2} + Z_{3}^{2} and rotating them about the Z_{1} axis we get the visualization of the impulsive wave in the anti-de Sitter space-time as a sequence of hyperboloidal surfaces. The source of the wave is a null particle moving along the axis of symmetry which at any time is located in the corresponding surface.

FIG. 11. The conformal diagram of the anti-de Sitter space-time with natural global coordinate chart t, χ. Each point represents a 2-sphere parametrized by θ, ϕ, antipodal points on the spheres correspond to ±χ. Null and space-like infinity form a time-like surface χ = π/2, (R = ∞). The impulsive wave for θ = 0 is also indicated.

FIG. 12. The synchronous coordinates τ, ψ, θ, ϕ cover only a part of the whole anti-de Sitter space-time. The time-like geodesics ψ, θ, ϕ constants all converge. In this diagram drawn for θ = 0 the impulsive wave propagates along the indicated null lines.

FIG. 13. The conformal diagram of the anti-de Sitter space-time and conformally flat coordinates η, ρ which cover the whole universe. Here we consider the section Z_{2} = Z_{3} = 0 so that ϕ = π/2 and sin θ = (ρ^{2} − η^{2})/(2aρ). The impulsive wave is given simply by η = ±ρ.