Penrose junction conditions with $\Lambda$:
Geometric insights into low-regularity metrics
for impulsive gravitational waves

In this case the usual null coordinates of flat space are employed, namely

in which the Minkowski metric takes the form

Now this spacetime ${\cal M}$ is cut along the null plane ${\cal N}$ given by ${{\mathcal{U}}=0}$, and the half-spaces ${\cal M}^{-}$ and ${\cal M}^{+}$ are defined by ${{\mathcal{U}}<0}$ and ${{\mathcal{U}}>0}$, respectively, see Figure 1.

The warp at ${{\mathcal{U}}=0}$ is then given by a deformed shift along ${\mathcal{V}}$, specified by an arbitrary function ${\mathcal{H}}(\eta,\bar{\eta})$, while keeping $\eta$ (that is $x,y$) fixed. These Penrose junction conditions at ${{\mathcal{U}}=0}$ are explicitly given by

Such an identification of the boundary points leads to an impulsive plane wave characterized by the (arbitrary) real function ${\mathcal{H}}(\eta,\bar{\eta})$, as we will see more explicitly in Section 3.

To obtain spheres expanding at the speed of light, alternative coordinates of Minkowski space (2), namely

must be employed in which ${U=0}$ represents a null cone ${\cal N}$. In fact, these are the Robinson–Trautman coordinates

where ${V=0}$ labels the vertex of the cone, from which the spherical impulse expands (these coordinates degenerate along ${t=-z}$). The “half-spaces” ${\cal M}^{-}$ and ${\cal M}^{+}$, again given by ${U<0}$ and ${U>0}$, are now the interior and the exterior of the null cone, respectively, see Figure 2.

The warp at ${U=0}$ is performed by the Penrose junction conditions

where $h(\zeta)$ is an arbitrary holomorphic function of the complex coordinate $\zeta$ (which is actually a stereographic representation of the spherical angles $\theta,\phi$ on the expanding impulse). Such an identification of the boundary points represented by the mapping ${\zeta\mapsto h(\zeta)}$ leads to an impulsive spherical wave, whose specific character is determined by $h(\zeta)$.

More details and many references can be found in Chapter 20 of GriffithsPodolsky:2009 , in the review Podolsky:2002 , or in BH:03 . Recent summaries of nonexpanding impulsive waves are contained in PodolskySaemannSteinbauerSvarc:2015 , and of expanding (spherical) impulsive waves in PodolskySaemannSteinbauerSvarc:2016 .

Our present contribution concentrates on nonexpanding impulses propagating in Minkowski, de Sitter and anti-de Sitter spaces (maximally symmetric vacuum spacetimes with any value of the cosmological constant $\Lambda$), and is based mainly on our recent papers on this topic SaemannSteinbauerLeckePodolsky:2016 ; SaemannSteinbauer:2017 , and most of all PodolskySaemanSteinbauerSvarc:2019 .

In his work Penrose:1972 Roger Penrose not only introduced the geometrical “cut-and-paste” construction method (described in Section 2.1) but also presented both a continuous and a distributional metric form of impulsive pp-waves, and their mutual relation.

While in Penrose:1972 only a particular warping function was considered explicitly, namely the quadratic expression ${{\mathcal{H}}=\eta^{2}+\bar{\eta}^{2}}$ (which enters the metric (11) below), the procedure also works for the complete family of pp-waves parametrized by an arbitrary function ${{\mathcal{H}}(\eta,\bar{\eta})}$. Indeed, extending Penrose’s original idea we may apply to the flat metric (2), that is to ${\,{\rm{d}}s_{0}^{2}=2\,{\rm{d}}\eta\,{\rm{d}}\bar{\eta}-2\,{\rm{d}}{\mathcal{U}}\,{\rm{d}}{\mathcal{V}}\,}$, the transformation

where

is any smooth enough real-valued function of the complex variable $Z$ and its complex conjugate $\bar{Z}$. Moreover, $u_{+}$ is the (Lipschitz) continuous kink function, while $\Theta$ is the (locally bounded, i.e., $L^{\infty}_{\mbox{{\tiny loc}}}$) Heaviside step function

Therefore, ${h(Z,\bar{Z})=h(\eta,\bar{\eta})}$ at ${u=0}$, which may be identified with ${\mathcal{H}}(\eta,\bar{\eta})$.

If (3.1) is applied separately to (2) for ${u<0}$ defining ${\mathcal{M}}^{-}$ (where it is just an identity ${{{\mathcal{U}}}=u,{{\mathcal{V}}}=v,\eta=Z}$) and to ${u>0}$ defining ${\mathcal{M}}^{+}$, the metric becomes

This is the continuous Rosen form of a pp-wave AB:97 ; PodolskyVesely:1998 , which is impulsive due to the (mere) Lipschitz continuity of the coefficient $u_{+}$.³³3Rosen presented this type of the metric only for (extended) plane waves, which are a special subcase of the complete family of pp-waves.

Interestingly, applying the transformation (3.1) to (2) for any ${u\in\mathbb{R}}$ we formally get the metric with the Dirac delta $\delta({{\mathcal{U}}})$,⁴⁴4More precisely, by applying the discontinuous transformation (3.1) on the metric (11), with the distributional identities ${u_{+}^{\prime}=\Theta}$, ${\Theta^{\prime}=\delta}$, ${\Theta u_{+}=u_{+}}$ and the identification ${{\mathcal{H}}\lvert_{u=0}\,=h}$ (using ${\eta\lvert_{u=0}\,=Z}$), one obtains the continuous Rosen form (10).

This is the Brinkmann form of a pp-wave, which is impulsive due to it being explicitly distributional in $\,{\mathcal{U}}\,$. Its warping metric function is given by ${\,{\mathcal{H}}(\eta,\bar{\eta})\equiv h(Z,\bar{Z})\,}$ evaluated at ${\,u=0\,}$.

Moreover, it can be immediately observed that the transformation (3.1) is discontinuous due to the presence of the Heaviside function $\Theta(u)$ entering ${\mathcal{V}}$, which exactly represents the Penrose junction conditions (3), namely ${\big{(}{\mathcal{V}},\eta,\bar{\eta}\big{)}_{{\mathcal{M}}^{-}}\,=\,\big{(}{\mathcal{V}}-{\mathcal{H}}(\eta,\bar{\eta}),\eta,\bar{\eta}\big{)}_{{\mathcal{M}}^{+}}}$. Recall that there is no change in $\eta$ at ${{\mathcal{U}}\equiv u=0}$ because $\eta(Z,\bar{Z})$ given by expression (3.1) is continuous.

There are thus close relations between the continuous Rosen metric form (10), the distributional Brinkmann metric form (11), and the Penrose junction conditions (3) for impulsive pp-waves. However, at this stage, these relations have to be considered only formal, because they involve distributions and also their products. A more rigorous treatment of the related mathematical subtleties occurring in low regularity is thus required to clarify exact meaning of these relations.

To begin with, we discuss the regularities of the involved metrics. The continuous form (10) of the impulsive pp-wave is actually locally Lipschitz continuous, a class of metrics which is often denoted by $C^{0,1}$ or $C^{1-}$. Such metrics are well within the Geroch–Traschen (or GT) class of metrics GT:87 which possess regularity ${H^{2}_{\mbox{{\tiny loc}}}\cap L^{\infty}_{\mbox{{\tiny loc}}}}$ and are uniformly nondegenerate LM:07 ; SV:09 . In their classical paper GT:87 Robert Geroch and Jennie Trashen have shown that such metrics allow to (stably) define the Riemann tensor as a tensor distribution, and that they are well-suited to describe spacetimes which possess a distributional curvature supported on a hypersurface. This is in fact the case for the metric (10) which has the Riemann and the Ricci tensor proportional to $\delta(u)$, and hence its curvature concentrated on the impulse.

On the other hand, the distributional form of the impulsive pp-wave metric (11) clearly is outside the GT-class, and hence there is no consistent distributional framework (such as Mar:68 ) available to study its curvature. Nevertheless, using the special Brinkmann coordinates it is formally possible to compute the curvature which then is again concentrated on the impulse and proportional to $\delta({\mathcal{U}})$. As a warning it has to be remarked that we have definitely reached the “grey areas” of distribution theory since e.g. only the mixed components ${R^{i}}_{j}$ of the Ricci tensor can be computed, but not those with both upper or lower indices. Moreover, the discontinuous change of coordinates is literally non-sensical within distribution theory since it boils down to performing the distributional pullback of the metric (11) by a merely $L^{\infty}$-map.

A rigorous investigation of impulsive pp-waves was performed in 1998–1999 by Michael Kunzinger and the second author in the series of articles Steinbauer:1998 ; KunzingerSteinbauer:1999a ; KunzingerSteinbauer:1999b . First, in Steinbauer:1998 the geodesics (and the geodesic deviation) for the distributional form of impulsive pp-waves were studied using a careful regularization procedure. Thereby the Dirac $\delta$ in (11) was replaced by a general class of smooth functions, the so-called model delta nets $\delta_{\varepsilon}$ defined as follows: Choose a smooth function $\rho$ with unit integral, supported in $[-1,1]$, and set $\delta_{\varepsilon}(x)=(1/\varepsilon)\,\rho(x/\varepsilon)$.⁵⁵5All results are actually independent of the specific choice of $\rho$. Technically, the geodesic equation for the regularized metric(s) become nonlinear, and the fact that (at least for small regularization parameters $\varepsilon$) the geodesics exist long enough to cross the regularised impulse⁶⁶6By this we mean the support of the sandwich profile $\delta_{\varepsilon}$, i.e., the region $-\varepsilon\leq u\leq\varepsilon$. and hence are complete, is established using a fixed point argument. The resulting geodesics for the distributional metric (obtained by a distributional limit) are then independent of the specific regularisation used, and reproduce earlier ad-hoc results of e.g. FPV:88 ; Bal:97 .

Then in KunzingerSteinbauer:1999a this analysis was put into the framework of nonlinear distributional geometry (GKOS:01, , Ch. 3.2) providing a solution concept for the geodesic (and geodesic deviation) equation for the (generalised version of the) metric (11), which is obtained by replacing the Dirac delta by a so-called generalized delta function.⁷⁷7Technically, this is an equivalence class of smooth functions represented by a model delta net. This setting, based on Colombeau’s construction of algebras of generalized functions Col:85 ; Col:92 , allows for a consistent treatment of products of distributions in (semi-)Riemannian geometry. In particular, this allowed to clarify the meaning of the Penrose junction conditions and the equivalence of the distributional and continuous forms of the metric KunzingerSteinbauer:1999b .

The geometric key idea was to employ a privileged (natural) family of null geodesics which cross the impulse. Indeed, the existence (and uniqueness) of such geodesics was proven, enabling a study of their interaction with the impulse, see Figure 3. Their “shift” and their “refraction” can explicitly be expressed in terms of the profile ${{\mathcal{H}}}$ and its derivatives at the point the corresponding geodesic hits the impulse. Finally, this family of null geodesics gives the “comoving” Rosen coordinates, and hence allows to smoothly and explicitly transform the generalized version of (11) to the generalized version of (10).

Neglecting the mathematical details, we can summarize that:

• •

  a suitable family of null geodesics in the distributional pp-wave metric (11) is explicitly constructed,

• •

  they cross the impulse, they are complete, and define the comoving coordinates of the Rosen metric (10), hence

• •

  they allow us to “geometrically regularize” the discontinuous transformation (3.1).

Put somewhat more vividly, it turns out that within nonlinear distributional geometry the impulsive pp-wave spacetime can be equivalently described by two metrics, the generalized distributional Brinkmann form and the generalized continuous Rosen form which are related by a generalized coordinate transform. The distributional limits of the respective metrics are precisely (11) and (10), and the distributional limit of the corresponding transformation is the discontinuous transformation (3.1), which explicitly encodes the Penrose junction conditions (3), see also EG:11 ; Erl:13 and the diagram in (KunzingerSteinbauer:1999b, , p. 1261).

Finally, we mention that this way of dealing with the intricacies of low-regularity metrics in the context of impulsive plane waves has recently lead to the clarification of a lapse in the literature on the wave memory effect in these geometries Ste:19 .

Let us begin by recalling the (anti-)de Sitter geometries. They are constant-curvature spaces which are also maximally symmetric (admitting 10 Killing vector fields), and conformally flat vacuum solutions of Einstein’s equations with $\Lambda$. Their metric can be very conveniently written as

which nicely represents Minkowski, de Sitter, and anti-de Sitter space, for ${\Lambda=0}$, ${\Lambda>0}$, and ${\Lambda<0}$, respectively, and is explicitly conformally flat.

The coordinates of (12) cover the full (anti-)de Sitter hyperboloid

embedded in 5-dimensional flat space

where

and

The global coordinate parameterization of the (anti-)de Sitter manifold determined by (13) in (14) corresponding to the conformally flat metric (12) is

where the conformal factor is

Recalling ${\sqrt{2}\,\eta\equiv x+{\mathrm{i}}\,y}$, see (1), we obtain ${Z_{2}=x/\Omega}$ and ${Z_{3}=y/\Omega}$. The inverse expressions to (17) are thus simply given by

with ${\Omega=2a/(Z_{4}+a)}$.

There exists an interesting geometric method by which impulsive gravitational waves can be constructed in de Sitter and anti-de Sitter spaces. This was introduced in the work of Hotta and Tanaka in 1993 HottaTanaka:1993 (which also employed the “shift function” method by Dray and ’t Hooft DraytHooft:1985 ), and further developed in PodolskyGriffiths:1997 ; PodolskyGriffiths:1998 ; HorowitzItzhaki:1999 . It employs the 5-dimensional representation (13), (14) of the background (anti-)de Sitter manifold.

In fact, the use of the $5$-dimensional formalism initially occurred in the works HottaTanaka:1993 ; PodolskyGriffiths:1997 ; PodolskyGriffiths:1998 where special impulsive waves in de Sitter space were constructed by boosting the Schwarzschild–de Sitter black hole solution to the speed of light. This itself was a natural generalization of the ultrarelativistic boost of the Schwarzschild solution, which in 1971 has lead to the famous Aichelburg–Sexl impulsive pp-wave in Minkowski space AS:71 . In cases with any $\Lambda$, this procedure produces specific impulsive waves generated by null particles, i.e., sources moving at the speed of light.

The embedding method is based on considering impulsive pp-waves in flat $5$-dimensional semi-Riemannian space (14), but constraining them to the $4$-dimensional hyperboloid (13). Specifically, we consider the $5$-dimensional impulsive pp-wave, generalizing the Brinkmann distributional metric (11),

By applying the constraint (13), the manifold is reduced to the 4-dimensional (anti-)de Sitter space whenever ${U\neq 0}$. However, due to the presence of $\delta(U)$ in (20), the distributional curvature arises at the single wave-surface ${U=0}$, which represents the impulse in de Sitter or anti-de Sitter space. In view of (17), such an impulse is located at

The location of the impulse is indicated in Fig. 4 as a pair of null lines. Moreover, from (13) we find its geometry to be given by

This is clearly a sphere $S^{2}$ in de Sitter space (since ${\sigma=1}$ for ${\Lambda>0}$), and a hyperboloid $H^{2}$ in anti-de Sitter space (since ${\sigma=-1}$ for ${\Lambda<0}$). Moreover, their “radius” is determined by the constant ${a=\sqrt{3/(\sigma\Lambda)}}$, which means that such an impulse is indeed nonexpanding.

The impulsive waves discussed above, be they constructed by embedding the 4-dimensional (anti-)de Sitter hyperboloid (13) into the $5$-dimensional pp-wave spacetime (20), or by considering an ultrarelativistic limit of special sources, belong to the Kundt class of spacetimes.

All Kundt-type solutions of vacuum Einstein’s equations with ${\Lambda\neq 0}$ of algebraic type N (i.e., those which represent “pure” gravitational waves) were explicitly found in GarciaPlebanski:1981 ; OzsvathRobinsonRozga:1985 . In these works it was demonstrated that there are two distinct subclasses of such spacetimes when ${\Lambda=0}$, only one subclass when ${\Lambda>0}$, but (interestingly) three subclasses when ${\Lambda<0}$, including a special Siklos solution Siklos:1985 , representing gravitational waves with hyperbolic surfaces in the anti-de Sitter universe (see Podolsky:1993 ; Podolsky:1998Siklos ; BicakPodolsky:1999a , or the review GriffithsPodolsky:2009 for more details concerning the classification and mutual relations between the subclasses).

These various subclasses are geometrically distinct for a general (smooth) wave-profile. However, in the impulsive case, i.e., if the wave profile is taken to be a Dirac delta thus localizing the Kundt waves to just one impulsive wave surface, only a single (locally) unique impulsive class of solutions exists — a surprising result proved by the first author in Podolsky:1998 .

A number of questions thus naturally arose, namely: Can such nonexpanding impulsive waves with $\Lambda$, which possess a unique distributional form, also be written in some continuous metric form? What are their mutual coordinate relations? Can the Penrose “cut-and-paste” method be extended to such a cosmological setting? And, what are the corresponding Penrose junction conditions generalized to any cosmological constant $\Lambda$?

These questions were basically answered in 1999 in the work PodolskyGriffiths:1999 of the first author with Jerry Griffiths. The trick was to employ the convenient metric (12) which describes the full (anti-)de Sitter background space via the simple parametrization (19), (18) of the hyperboloid (13). Moreover, the metric (12) is conformally flat, so the only difference with respect to impulsive pp-waves is the presence of the overall conformal factor ${\Omega=1+\tfrac{1}{6}\,\Lambda(\eta\bar{\eta}-{\mathcal{U}}{\mathcal{V}})}$. This approach allowed us to generalize in PodolskyGriffiths:1999 the continuous Rosen form (10) and the distributional Brinkmann form (11) of impulsive pp-waves, to find their mutual transformation, and to obtain the relation to the embedding metric (20). The results are as follows:

Nonexpanding impulsive gravitational waves in (anti-)de Sitter space can be written in the following 3 alternative forms:

• •

  A continuous metric

  $\displaystyle{\rm{d}}s^{2}=\frac{2\,\big{\lvert}{\rm{d}}Z+u_{+}(u)(h_{,\bar{Z}Z}\,{\rm{d}}Z+{h}_{,\bar{Z}\bar{Z}}\,{\rm{d}}\bar{Z})\big{\lvert}^{2}-2\,{\rm{d}}u\,{\rm{d}}v}{\big{[}\,1+\frac{1}{6}\Lambda(Z\bar{Z}-uv+u_{+}(u)\,G)\,\big{]}^{2}}\,,$

  (23)

  where $u_{+}(u)$ is the kink function (9), ${h\equiv h(Z,\bar{Z})}$ is any real-valued function as in (8), and the function $G(Z,\bar{Z})$ is defined as ${G\equiv Z\,h_{,Z}+\bar{Z}\,h_{,\bar{Z}}-h}$. Again, the non-differentiability of $u_{+}$ gives rise to a $\delta$-function curvature located at $u=0$. For ${\Lambda=0}$ the metric (23) clearly reduces to the Rosen form of impulsive pp-waves (10).

• •

  A distributional metric

  $\displaystyle{\rm{d}}s^{2}=\frac{2\,{\rm{d}}\eta\,{\rm{d}}\bar{\eta}-2\,{\rm{d}}{\mathcal{U}}\,{\rm{d}}{\mathcal{V}}+2\,{\mathcal{H}}(\eta,\bar{\eta})\,\delta({{\mathcal{U}}})\,{\rm{d}}{{\mathcal{U}}}^{2}}{[\,1+\frac{1}{6}\Lambda(\eta\bar{\eta}-{{\mathcal{U}}}{{\mathcal{V}}})\,]^{2}}\,,$

  (24)

  where $\delta({\mathcal{U}})$ is the Dirac delta distribution. The corresponding impulse in (anti-)de Sitter space (12) is thus clearly located at ${{\mathcal{U}}=0}$, and its geometry is ${\,2\,{\rm{d}}\eta\,{\rm{d}}\bar{\eta}\,[\,1+\frac{1}{6}\Lambda\,\eta\bar{\eta}\,]^{-2}}$, that is a sphere $S^{2}$ in de Sitter space and a hyperboloid $H^{2}$ in anti-de Sitter space, in full agreement with (22). For ${\Lambda=0}$ we immediately recover planar impulsive wave in Minkowski space, namely the Brinkmann form of impulsive pp-waves (11).

• •

  An embedding metric, as introduced in Section 4.2, namely

  ${\rm{d}}s^{2}={\rm{d}}Z_{2}^{2}+{\rm{d}}Z_{3}^{2}+\sigma{\rm{d}}Z_{4}^{2}-2{\rm{d}}U{\rm{d}}V+H(Z_{2},Z_{3},Z_{4})\,\delta(U)\,{\rm{d}}U^{2}\,,$

  (25)

  with the constraint (13), that is

  ${Z_{2}}^{2}+{Z_{3}}^{2}+\sigma{Z_{4}}^{2}-2UV=3/\Lambda\,,$

  (26)

  where ${\sigma=1}$ for ${\Lambda>0}$ while ${\sigma=-1}$ for ${\Lambda<0}$. This is the 5-dimensional pp-wave metric in the distributional (Brinkmann) form constrained to the (anti-)de Sitter hyperboloid (26). The nonexpanding impulse is located at ${U=0\Leftrightarrow Z_{0}=Z_{1}}$, see (21) and Fig. 4, and its geometry is given by (22).

It is now straightforward to formally obtain the coordinate relation between the 5-dimensional embedding metric (25) and the distributional metric (24). By applying the parametrization (17) of the (anti-)de Sitter hyperboloid — which satisfies the constraint (26) — to the metric (25), the (anti-)de Sitter background part ${\rm{d}}s_{0}^{2}$ takes the form (12), and there is an additional impulsive term ${H\,\delta(U)\,{\rm{d}}U^{2}}$. Because ${{U}={\mathcal{U}}\,\Omega^{-1}}$, using the distributional identities ${\delta(U)=\Omega\,\delta({\mathcal{U}})}$ and ${{\mathcal{U}}\delta({\mathcal{U}})=0}$ it can be written as ${H\,\Omega^{-1}\,\delta({{\mathcal{U}}})\,{\rm{d}}{{\mathcal{U}}}^{2}}$. This is exactly the last term ${2\,{\mathcal{H}}\,\Omega^{-2}\,\delta({{\mathcal{U}}})\,{\rm{d}}{{\mathcal{U}}}^{2}}$ in the distributional metric (24), with the obvious identification

Similarly, we can find the transformation between the distributional metric (24) and the continuous metric (23). Since their numerators are exactly the Brinkmann and Rosen forms of impulsive pp-waves, respectively, which are related by the transformation (3.1), we only need to compare their conformal factors. Applying the relations (3.1) and the identity (of $L^{\infty}$-functions) ${u_{+}=u\,\Theta}$ we observe that actually

We thus conclude that the metric forms (24) and (23) of nonexpanding impulses in (anti-)de Sitter space are related by the same discontinuous transformation (3.1) as in flat space, namely

This fact enables us to generalize the Penrose “cut-and-paste” construction method and to formulate the corresponding Penrose junction conditions for any value of the cosmological constant $\Lambda$.

The transformation between the continuous metric (23) and the distributional metric (24) is given by the discontinuous transformation (4.4), which is independent of $\Lambda$. The Penrose “cut-and-paste” method for construction of these nonexpanding impulsive waves can thus be used similarly as in flat Minkowski space ${\cal M}$, provided the more general, conformally flat metric (12) is employed. The only difference is that the cut is performed in de Sitter manifold $d{\cal S}$ or anti-de Sitter manifold $Ad{\cal S}$ along the null surface ${\cal N}$, expressed as ${\,{\mathcal{U}}=0}$ in the background coordinates of (12). Such a cut is indicated in Fig. 4. The pasting is then performed using the Penrose junction conditions at ${{\mathcal{U}}=0}$

which are formally the same as (3) in the original case ${\Lambda=0}$. The geometric reason is that the cosmological constant enters the metric (12) only via the conformal factor.

Moreover, the Penrose junction conditions (30) are again implicitly encoded in the transformation (4.4), namely via the deformed shift along ${\mathcal{V}}$ given by the real function ${\mathcal{H}}(\eta,\bar{\eta})$ which corresponds to the step-term $\Theta(u)\,h$ therein. Indeed, it follows from (4.4) that ${{\mathcal{H}}(\eta,\bar{\eta})=h(Z,\bar{Z})}$ because ${\eta=Z}$ at ${u={\mathcal{U}}=0}$ due to the continuity of the kink function $u_{+}(u)$.

All geodesics crossing any nonexpanding impulse in (anti-)de Sitter space were investigated and (formally) found in 2001 by the first author and Marcello Ortaggio PodolskyOrtaggio:2001 . Using the embedding formalism, they are explicitly given by

in the cases ${\sigma e=0}$, ${\sigma e<0}$, and ${\sigma e>0}$, respectively. Here ${\sigma=\mathrm{sign}\,\Lambda}$, and $e$ determines the velocity normalization (${e=-1}$ for timelike, ${e=0}$ for null, and ${e=1}$ for spacelike geodesics, respectively). The affine parameter $t$ is chosen in such a way that each geodesic crosses the impulse ${U=0}$ at ${t=0}$.

Without loss of generality, $\dot{U}^{0}$ can be taken to be positive, so that (31) are increasing functions. Using $U$ as a convenient parameter and applying some distributional identities, the general solution of the remaining geodesic equations can be derived. For null geodesics with ${\dot{U}^{0}=1}$, it can be written⁸⁸8See Sec. 5 in SaemannSteinbauerLeckePodolsky:2016 , namely Proposition 5.3. Relation to the original form presented in PodolskyOrtaggio:2001 (which uses a different definition of $C$) is contained in Remark 5.4.

The constants ${Z_{p}^{0},\dot{Z}^{0}_{p},V^{0},\dot{V}^{0}}$ for ${p=2,3,4}$ are determined by the initial data, while the coefficients $A_{p},B,C$ are

${H(0)=H(Z^{0}_{p})}$, ${G(0)=G(Z_{p}^{0})\equiv\delta^{pq}\,Z_{p}^{0}\,H_{,q}(Z_{r}^{0})-H(Z_{r}^{0})}$, ${H_{,q}(0)=H_{,q}(Z^{0}_{p})}$ are functions evaluated on the impulse ${U=0}$, given by $H(Z_{2},Z_{3},Z_{4})$ of (25).

Hence we see that, analogous to the case of impulsive pp-waves, the global geodesics suffer a jump in $V$ across the impulse (determined by $B$) and, in addition, are broken in the $V$ and $Z_{p}$-directions (determined by $C$ and $A_{p}$, respectively). The coefficients specifying the magnitude of these effects are again given by the profile function $H$ and its derivatives, evaluated at the points where the respective geodesic hits the impulse.

These results were fully confirmed in 2016 SaemannSteinbauerLeckePodolsky:2016 ; SaemannSteinbauer:2017 by a rigorous investigation of the geodesic equations using a regularisation technique. Again the Dirac $\delta$ in the metric, now in (25), was replaced by a model delta nets $\delta_{\varepsilon}$. In physical terms this means that the formal distributional form of the impulsive metric is understood as a limit of a family of spacetimes with ever shorter but stronger sandwich gravitational waves with smooth profile $\delta_{\varepsilon}$.

Although the resulting geodesic equation for the regularized metric(s) forms a highly coupled system, it was possible to prove the existence and uniqueness of geodesics that (at least for small regularization parameters $\varepsilon$) exist long enough such that they cross the regularized wave impulse, i.e., the region $-\varepsilon\leq U\leq\varepsilon$. Since off that region we are dealing with the (anti-)de Sitter background this immediately implies completeness of the geodesics of the regularized spacetime(s). The proof is based on an application of Weissinger’s fixed point theorem Wei:52 . Remarkably, when taking the impulsive limit ${\varepsilon\to 0}$, the geodesics converge to the the same limit for any choice of profile $\rho$ of the sandwich gravitational waves, i.e., they are independent of the specific regularization. Moreover, they fully agree with (5.1)–(5.1), and extend these formulae to any value of the initial data ${\dot{U}^{0},Z_{p}^{0},\dot{Z}^{0}_{p},V^{0},\dot{V}^{0}}$.

Moreover, we have also investigated the geodesics in these spacetimes employing the continuous form of the metric  (23). This analysis is based on the local Lipschitz continuity of this metric and the observation made in Steinbauer:2014 that every locally Lipschitz metric has $C^{1}$-geodesics in the sense of Filipov Fil:88 . This solution concept for ordinary differential equations with discontinuous right-hand sides was also used in PodolskySaemannSteinbauerSvarc:2015 to establish the existence and uniqueness of continuously differentiable geodesics crossing the wave impulse. We also explicitly derived their form using a $C^{1}$-matching procedure.

A natural question thus arose about the mutual consistency of these two results, both obtained in a rigorous way but starting from two different forms of the metric, namely the continuous and the embedding $5$-dimensional distributional form. It can indeed formally be shown that the form of the geodesics derived in both ways are the same when appropriate coordinate transformations are applied. This result confirmed that both our approaches are consistent, and that the understanding of geodesics in the complete family of spacetimes with nonexpanding impulsive gravitational waves and any cosmological constant now rests on firm mathematical grounds, preparing us for the next step.

With these explicit geodesics at hand, we were able in PodolskySaemanSteinbauerSvarc:2019 to geometrically derive the discontinuous transformation (4.4) and the Penrose junction conditions (30), thus clarifying their nature and meaning, and laying the foundations for their rigorous treatment analogous to the one for the pp-wave case in KunzingerSteinbauer:1999b . Let us summarize the key steps and the main results:

• •

  First, in the distributional metric (24) with coordinates ${({\mathcal{U}},{\mathcal{V}},x,y)}$ we identify a special family of global null geodesics which cross the impulse located at ${{\mathcal{U}}=0}$. We get them by naturally setting their initial values ${{\mathcal{V}}_{0},x_{0},y_{0}}$ to be constants (with no velocity) in (anti-)de Sitter space ($A$)${{d{\cal S}}^{-}}$ in front of the impulse, that is for ${{\mathcal{U}}<0}$. So in front of the impulse they are the generators of the (anti-)de Sitter space, when expressed in the $5$-dimensional flat space.

• •

  Such null geodesics are explicitly given by

  $$\gamma({\lambda})=\Big{(}\frac{\alpha\lambda}{1-\beta\lambda},{\mathcal{V}}_{0},x_{0},y_{0}\Big{)}\,,$$

  (34)

  where $\lambda$ is the affine parameter and

  $$\alpha\equiv 1+\frac{\Lambda}{12}\left(x_{0}^{2}+y_{0}^{2}\right)\,,\qquad\beta\equiv-\frac{\Lambda}{6}\,{\mathcal{V}}_{0}\,.$$

  (35)

  Notice that ${{\mathcal{U}}=0}$ iff ${\lambda=0}$.

• •

  Next we express these null geodesics in the $5$-dimensional embedding form of (anti-)de Sitter space in the coordinates ${(U,V,Z_{2},Z_{3},Z_{4})}$ of (14) using (17) as

  $$\gamma(\lambda)=\frac{1-\beta\lambda}{\alpha}\,\bigg{(}\frac{\alpha\lambda}{1-\beta\lambda},{\mathcal{V}}_{0},x_{0},y_{0},a\Big{(}2-\frac{\alpha}{1-\beta\lambda}\Big{)}\bigg{)}\,.$$

  (36)

  Notice that ${\,U=\lambda}$.

• •

  Now we employ the results on global null geodesics crossing the impulse at ${U=0}$ to the region ${U>0}$ behind it, expressed in the $5$-dimensional coordinates $(U,V,Z_{2},Z_{3},Z_{4})$. As explained in Section 5.1, these were rigorously derived using a general regularisation. After a distributional limit they take the form

  $$\gamma_{5D}(\lambda)\equiv\left(\begin{array}[]{c}U(\lambda)\\ V(\lambda)\\ Z_{p}(\lambda)\end{array}\right)=\left(\begin{array}[]{c}\lambda\\ V^{0}+\dot{V}^{0}\,\lambda+B\,\Theta(\lambda)+C\,\lambda_{+}(\lambda)\\ Z_{p}^{0}+\dot{Z}^{0}_{p}\,\lambda+A_{p}\,\lambda_{+}(\lambda)\end{array}\right),$$

  (37)

  cf. (5.1), (5.1), with the initial data (for ${U<0}$)

  		$\displaystyle V^{0}=\frac{{\mathcal{V}}_{0}}{\alpha}\,,$	$\displaystyle Z_{2}^{0}=\frac{x_{0}}{\alpha}\,,$	$\displaystyle Z_{3}^{0}=\frac{y_{0}}{\alpha}\,,$	$\displaystyle Z_{4}^{0}=a\,\Big{(}\frac{2}{\alpha}-1\Big{)}\,,$
  		$\displaystyle\dot{V}^{0}=-\frac{\beta}{\alpha}\,{\mathcal{V}}_{0}\,,$	$\displaystyle\dot{Z}^{0}_{2}=-\frac{\beta}{\alpha}\,x_{0}\,,$	$\displaystyle\dot{Z}^{0}_{3}=-\frac{\beta}{\alpha}\,y_{0}\,,$	$\displaystyle\dot{Z}^{0}_{4}=-2a\,\frac{\beta}{\alpha}\,.$		(38)

  Here $\lambda_{+}(\lambda)\equiv\lambda\,\Theta(\lambda)$, while the coefficients are

  	$\displaystyle A_{j}={\mathcal{H}}^{\mathrm{i}}_{,j}+\frac{x_{0}^{j}}{2\sigma\alpha a^{2}}\,{\mathcal{G}},\quad A_{4}=\frac{1}{\sigma\alpha a}{\mathcal{G}},\quad B=\frac{1}{\alpha}{\mathcal{H}}^{\mathrm{i}}\,,$
  	$\displaystyle C=\frac{1}{2}\big{(}({\mathcal{H}}^{\mathrm{i}}_{,x})^{2}+({\mathcal{H}}^{\mathrm{i}}_{,y})^{2}\big{)}+\frac{1}{2\sigma\alpha a^{2}}\big{(}({\mathcal{H}}^{\mathrm{i}}+{\mathcal{G}}){\mathcal{V}}_{0}+{\mathcal{H}}^{\mathrm{i}}{\mathcal{G}}\big{)}\,,$		(39)

  with ${{\mathcal{G}}\equiv{\mathcal{H}}^{\mathrm{i}}-x_{0}{\mathcal{H}}^{\mathrm{i}}_{,x}-y_{0}{\mathcal{H}}^{\mathrm{i}}_{,y}}$. Here the superscript $\,{}^{\mathrm{i}}$ denotes the evaluation of the respective functions at ${U=0}$, i.e., “at the impulse”.

• •

  Next we express these null geodesics in the distributional coordinates $({\mathcal{U}},{\mathcal{V}},x,y)$ for the (anti-)de Sitter space ${{(A)d{\cal S}}^{+}}$ behind the impulse, that is for ${{\mathcal{U}}>0}$, using the relations (19),

  $${\mathcal{U}}=\Omega\,U\,,\quad{\mathcal{V}}=\Omega\,V\,,\quad x=\Omega\,Z_{2}\,,\quad y=\Omega\,Z_{3}\,,\quad\hbox{with}\ \,\Omega=\frac{2a}{Z_{4}+a}\,.$$

  (40)

  A somewhat lengthy calculation gives the explicit global null geodesics crossing the impulse in the form

  	$\displaystyle\gamma_{4D}[{\mathcal{V}}_{0},x_{0},y_{0}](\lambda)$		(41)
  	$\displaystyle\ \equiv\left(\!\!\begin{array}[]{c}{\mathcal{U}}(\lambda)\\ {\mathcal{V}}(\lambda)\\ x^{j}(\lambda)\end{array}\!\!\right)=\left(\begin{array}[]{c}{\displaystyle\frac{\alpha\lambda}{1-\beta\lambda+\frac{\Lambda}{6}{\mathcal{G}}\,\lambda_{+}}\vspace{2.0mm}}\\ \!\!{\mathcal{V}}_{0}+{\displaystyle\frac{\Theta(\lambda)}{1-\beta\lambda+\frac{\Lambda}{6}{\mathcal{G}}\,\lambda_{+}}}\,{\mathcal{H}}^{\mathrm{i}}+{\displaystyle\frac{\alpha\lambda_{+}}{2(1-\beta\lambda+\frac{\Lambda}{6}{\mathcal{G}}\,\lambda_{+})}}\,{\mathcal{F}}\!\vspace{2.0mm}\\ x_{0}^{j}+{\displaystyle\frac{\alpha\lambda_{+}}{1-\beta\lambda+\frac{\Lambda}{6}{\mathcal{G}}\,\lambda_{+}}}\,{\mathcal{H}}_{,j}^{\mathrm{i}}\\ \end{array}\right).$		(48)

  We have denoted the dependence on the initial data $[{\mathcal{V}}_{0},x_{0},y_{0}]$ explicitly, and we have used the abbreviation

  $${\mathcal{F}}\equiv({\mathcal{H}}_{,x}^{\mathrm{i}})^{2}+({\mathcal{H}}_{,y}^{\mathrm{i}})^{2}+\frac{{\mathcal{H}}^{\mathrm{i}}}{\sigma\alpha a^{2}}({\mathcal{V}}_{0}+{\mathcal{G}})\,.$$

  (49)

  Recall also that the constants $\alpha$, $\beta$ are given by (35).

• •

  Finally, using the relations ${{\mathcal{U}}(\lambda)=\alpha\lambda/(1-\beta\lambda+\frac{\Lambda}{6}{\mathcal{G}}\,\lambda_{+})}$ and ${\Theta(\lambda)=\Theta({\mathcal{U}})}$ implying ${\,{\mathcal{U}}_{+}=\alpha\lambda_{+}/(1-\beta\lambda+\frac{\Lambda}{6}{\mathcal{G}}\,\lambda_{+})}$, the result (41) can be simply rewritten as

  $\displaystyle\gamma_{4D}[{\mathcal{V}}_{0},x_{0},y_{0}]({\mathcal{U}})\equiv\left(\!\!\begin{array}[]{c}{\mathcal{U}}\\ {\mathcal{V}}({\mathcal{U}})\\ x^{j}({\mathcal{U}})\end{array}\!\!\right)\!=\!\left(\!\!\!\begin{array}[]{c}{\mathcal{U}\vspace{2.0mm}}\\ {\mathcal{V}}_{0}+\Theta({\mathcal{U}})\,{\mathcal{H}}^{\mathrm{i}}+{\mathcal{U}}_{+}\,\frac{1}{2}\big{[}({\mathcal{H}}_{,x}^{\mathrm{i}})^{2}+({\mathcal{H}}_{,y}^{\mathrm{i}})^{2}\big{]}\vspace{2.0mm}\\ x_{0}^{j}+{\mathcal{U}}_{+}\,{\mathcal{H}}_{,j}^{\mathrm{i}}\\ \end{array}\!\!\!\right)\!.$ (56)

The explicit form of global null geodesics in (anti-)de Sitter space with nonexpanding impulsive gravitational waves just obtained can be employed as a transformation from the continuous to the corresponding distributional form of the metric ${(u,v,X,Y)\mapsto({\mathcal{U}},{\mathcal{V}},x,y)}$ with ${u={\mathcal{U}}}$. The trick is:

Take the initial values ${{\mathcal{V}}_{0},x_{0},y_{0}}$ as the comoving coordinates. They define the coordinates of the continuous metric form.

$$\left(\!\!\!\begin{array}[]{c}u\\ v\\ X\\ Y\end{array}\!\!\!\right)\!\!\mapsto\gamma_{4D}[v,X,Y](u)\!=\!\!\left(\begin{array}[]{c}u\\ \!\!\!v+\Theta(u)\,{\mathcal{H}}^{\mathrm{i}}+u_{+}\,\frac{1}{2}\big{[}({\mathcal{H}}_{,X}^{\mathrm{i}})^{2}+({\mathcal{H}}_{,Y}^{\mathrm{i}})^{2}\big{]}\!\!\!\\ X+u_{+}{\mathcal{H}}_{,X}^{\mathrm{i}}\\ Y+u_{+}{\mathcal{H}}_{,Y}^{\mathrm{i}}\end{array}\right)\!\!=\!\!\left(\!\!\begin{array}[]{c}{\mathcal{U}}\\ {\mathcal{V}}\\ x\\ y\end{array}\!\!\right).$$

Moreover, by inspecting (5.3) it is seen that the coordinates $\eta,\bar{\eta}$ are continous across the impulse at ${{\mathcal{U}}=0}$, but there is a jump in ${\mathcal{V}}$ having the value $h(Z,\bar{Z})={\mathcal{H}}(\eta,\bar{\eta})$, i.e., ${{\mathcal{V}}\mapsto{\mathcal{V}}-{\mathcal{H}}}$ across the impulse ${\cal N}$. This provides the justification — and, in fact, a systematic derivation — of the Penrose junction conditions (30) in de Sitter and anti-de Sitter space.

In a nutshell, our overall procedure employed in this section was:

• •

  We have derived the discontinuous transformation (5.3) from a special family of global null geodesics in (anti-)de Sitter space with impulsive waves in the distributional form, obtained by a general regularisation. To do so, we employed the $5$-dimensional embedding formalism where these geodesics are easily seen to be the generators of the (anti-)de Sitter hyperboloid.

• •

  This transformation turns these special null geodesics into coordinate lines, hence the coordinates $(u,v,X,Y)$, equivalent to $(u,v,Z,\bar{Z})$, are comoving with the corresponding null particles.

• •

  In fact, that is the way how the distributional metric (24) is transformed to the much more regular (locally Lipschitz) continuous metric (23).

• •

  This generalises (and is in perfect agreement with) the case ${\Lambda=0}$, in which the continous Rosen coordinates (10) for pp-waves are also comoving, and are related to the distributional Brinkmann coordinates (11) by the same transformation (3.1).

• •

  Moreover, the rigorously derived discontinuous transformation (5.3) justifies and proves the uniqueness of the Penrose junction conditions (30) for construction of nonexpanding impulsive waves in de Sitter and anti-de Sitter spaces.

• •

  Finally, this approach also allows us to extend the mathematically precise analysis of the Penrose “discontinuous coordinate transformation” described in Section 3.2 to the case of non-vanishing cosmological constant $\Lambda$. Its detailed implementation will appear in a forthcoming, more technical paper.