Comment on 'Memory effect for impulsive gravitational waves'

The symmetries of extended (i.e. non-impulsive) plane waves are identified in [7] to be a subgroup of the Carroll group. Using the symmetries, the geodesics (again in the extended case) can be calculated efficiently, especially in (R)-coordinates. This has been done in [7, section 3.2], see also [34, equation (2.11)] and [33, section IV B]. The constants of motion are given by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\xx}{{\mathbf x}} \displaystyle {\bf p}=a(u)\dot\xx,\quad {\bf k}=\xx(u)-H(u){\bf p}, \nonumber \end{align} \tag{ 7 }$$

where already the 5th conserved quantity $\mu=\dot u$ was used to parametrise the geodesics by the coordinate u. Here the matrix-valued function H is given by⁹

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:4.2} H(u)=\int_0^ua^{-1}(w)\,{\rm d}w. \nonumber \end{align} \tag{ 4.2 }$$

Additionally, the kinetic energy

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\xx}{{\mathbf x}} \displaystyle e=\frac{1}{2}g_{\mu\nu}\dot x^\mu\dot x^\nu =\frac{1}{2}\dot\xx\cdot a(u)\dot\xx+\dot v =\frac{1}{2}{\bf p}\cdot a^{-1}(u){\bf p}+\dot v \nonumber \end{align} \tag{ 8 }$$

is conserved and is set to $e=-1,0,1$ in the timelike, null and spacelike case respectively. This finally leads to the explicit expression for the geodesics, see [7, equation (3.11), [34, equation (2.11)]

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\kk}{{\mathbf k}} \newcommand{\pp}{{\mathbf p}} \newcommand{\xx}{{\mathbf x}} \displaystyle \xx(u)=H(u)\pp+\kk,\quad v(u)=-\frac{1}{2}\pp\cdot H(u)\pp+eu+d, \nonumber \end{align} \tag{ 9 }$$

where d is a constant of integration.

However, while in the extended case the symmetries allow one to nicely express the geodesics, it seems less obvious that this is also an efficient approach in the impulsive case. There the particle motion off the impulsive hypersurface is trivial and if one wants to derive its form in (R)-coordinates this can be done using the transformation (2.6) or by several other simple means, see section 3.6. This raises the question:

• (Q1)  
  Why use symmetries to calculate the geodesics in the impulsive case at all?

As a follow-up, a more subtle question arises from the further procedure applied in [35, section 5.1] (discussed in the next section)

• (Q2)  
  How to relate the values of the constants of motion in the before- and after-zones of the impulsive wave and how to argue for that?

We will come back to this questions later on in section 3.6 but first have a closer look on the just mentioned procedure.

In the main section 5 of [35] the geodesics are computed. In particular, in section 5.1 this is done in (R)-coordinates with a substantial use of (S)-coordinates. To start with, the form of the geodesics is derived using the symmetries, where the constants of motion are allowed to be different in the before- and the after-zones of the wave, denoted by $\newcommand{\pp}{{\mathbf p}} \pp_\pm$, $\newcommand{\kk}{{\mathbf k}} \kk_\pm$, and $e_\pm$, see [35, equations (4.12) and (4.13)]. This readily leads to the form of the geodesics in the before- and the after-zones (again using the $\pm$-notation)

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\kk}{{\mathbf k}} \newcommand{\pp}{{\mathbf p}} \newcommand{\xx}{{\mathbf x}} \displaystyle \label{eq:5.2} \xx_\pm(u)=\kk_\pm+H(u)\pp_\pm,\quad v_\pm=-\frac{1}{2}\pp_\pm\cdot H(u)\pp_\pm+e_\pm u+d_\pm, \nonumber \end{align} \tag{ 5.2 }$$

where now $d_\pm$ are constants of integration.

Then on physical grounds the geodesics are assumed to be continuous across u  =  0 such that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\kk}{{\mathbf k}} \newcommand{\xx}{{\mathbf x}} \displaystyle \label{eq:5.3} \kk_\pm=\xx(0)=:\xx_0,\quad d_\pm=v_\pm(0)=:v_0, \nonumber \end{align} \tag{ 5.3 }$$

where the choice that $H(0)=0$, see equation (4.2) was used as well as the fact that $a(0)=1$, see equation (2.16). Again using the latter equation one moreover obtains from (5.2) that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\pp}{{\mathbf p}} \newcommand{\xx}{{\mathbf x}} \displaystyle \label{eq:5.4} \pp_\pm=\dot\xx_\pm(0),\quad \dot v_\pm(0)=e_\pm-\frac{1}{2}|\pp_\pm|^2, \nonumber \end{align} \tag{ 5.4 }$$

where these remaining constants of motion are still allowed to be different in the two zones. Finally one may explicitly calculate H from equations (4.2) and (2.16) to obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\PP}{{P}} \displaystyle \label{eq:4.4} H(u)=u_-{{\mathbb I}} + u_+\PP^{-1}(u). \nonumber \end{align} \tag{ 4.4 }$$

This gives the following form of the geodesics

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\PP}{{P}} \newcommand{\xx}{{\mathbf x}} \displaystyle \xx(u)&=\xx_0+u_-\dot\xx_-(0)+u_+\PP^{-1}(u)\dot\xx_+(0), \nonumber \\ v(u)&= v_0+u_-\dot v_-(0)+u_+\dot v_+(0) +\frac{1}{2}\dot \xx_+(0)\cdot u_+({{\mathbb I}}-\PP^{-1}(u))\dot\xx_+(0).\label{eq:5.7} \nonumber \end{align} \tag{ 5.7 }$$

These geodesics are determined by the following set of 9 real constants $\newcommand{\xx}{{\mathbf x}} \xx_0$, $v_0$, $\newcommand{\xx}{{\mathbf x}} \dot\xx_\pm(0)$, $\dot v_\pm(0)$, while there should only be 6 since we have to subtract the two constants $u(0)=0$ and $\dot u(0)=1$, which we have used to write the geodesics in the above form, from the usual 8 initial positions and speeds in a 4-dimensional spacetime.

To determine at least some of the 'spurious constants' the authors of [35] now employ the (S)-coordinates and limit their considerations to the case of particles being at rest in the before-zone, a condition that can be expressed in the (S)-system of coordinates to be

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\xx}{{\mathbf x}} \displaystyle \label{eq:5.9} \hat\xx(\hat u)=\hat\xx_0,\quad \hat v(\hat u)=\hat v_0+e\hat u. \nonumber \end{align} \tag{ 5.9 }$$

Their decisive argument now is (see [35, p 10, bottom]):

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \nonumber & \textit{`Now, the after-zone being also flat and indeed Minkowskian when the hatted}\nonumber \\ & \textit{(S) coordinates are used, we argue that the latter have the same parametric} \hspace{1.55cm}{(*)}\nonumber \\ & \textit{form in the after-zone: (5.9) holds for all u.'}\nonumber \end{align*}$$

Using this argument the authors of [35] go on to rewrite the geodesics in (R)-form. To this end they apply the inverse of the transformation (2.25) for $\hat u=u>0$ with the impulsive profile (2.16) inserted explicitly, i.e.

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\xx}{{\mathbf x}} \newcommand{\J}{{{\mathbb J}}} \displaystyle \label{eq:5.8} \xx=({{\mathbb I}}+uk\J)^{-1}\hat\xx,\quad \hat u=u,\quad v=\hat v+\frac{1}{2}\hat\xx\cdot k\J({{\mathbb I}}+uk\J)^{-1}\hat\xx, \nonumber \end{align} \tag{ 5.8 }$$

which implies the relations of the constants (just set u  =  0)

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\xx}{{\mathbf x}} \newcommand{\J}{{{\mathbb J}}} \displaystyle \hat\xx_0=\xx_0,\quad \hat v_0=v_0-\frac{1}{2}\xx_0\cdot k\J\xx_0 \nonumber \end{align} \tag{ 10 }$$

and obtain the following geodesics

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\xx}{{\mathbf x}} \newcommand{\J}{{{\mathbb J}}} \displaystyle \xx(u)&= ({{\mathbb I}}+u_+k\J)^{-1}\xx_0,\nonumber \\ v(u)& = v_0+eu-\frac{1}{2}\xx_0\cdot k\J\Big({{\mathbb I}}-({{\mathbb I}}+u_+k\J)^{-1}\Big)\xx_0.\label{eq:5.10} \nonumber \end{align} \tag{ 5.10 }$$

After deriving the above result (5.10) the authors of [35] say:

'We see that the geodesic equation (5.10) is a special case of (5.7) where the after-zone initial velocity has been fixed by the initial conditions $\newcommand{\xx}{{\mathbf x}} \xx(0) =\xx_0$ and $\newcommand{\xx}{{\mathbf x}} \dot\xx(0-) [=\dot\xx^-_0]= 0$, namely

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\xx}{{\mathbf x}} \displaystyle \boxed{\ \dot\xx(0+)[=\dot\xx^+_0]=-c_0\xx_0.\ } \end{align} \tag{ 5.11 }$$

The impulsive GW induces a (sort of) 'percussion' [27], since

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\xx}{{\mathbf x}} \displaystyle \label{eq:5.12a} \bigtriangleup\dot\xx=\dot\xx(0+)-\dot\xx(0-)=-c_0\xx_0, \nonumber \end{align} \tag{ 5.12a }$$

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\xx}{{\mathbf x}} \displaystyle \label{eq:5.12b} \bigtriangleup\dot{v}=\dot{v}(0+)-\dot{v}(0-)=-\frac{1}{2}\vert{} c_0\xx_0\vert^2, \nonumber \end{align} \tag{ 5.12b }$$

First we note that the geodesics (5.10) are continuous not because '[...] this follows from (5.10).', see [35, section 5.1, last paragraph] but because this was assumed during the procedure explicitly in [35, p 10, first lines] and mentioned here prior to (5.3).

The main issue is, however, that the geodesics (5.10) are not continuously differentiable and, moreover, that they are actually not the correct geodesics in the first place, as we are going to argue in the following:

First, it was shown in [15, theorem 1] that the geodesics even in all impulsive (non-plane) pp-waves are C¹-curves. There, Carathéodory's solution concept is used, which is the most natural extension of classical ODE-theory using Lebesgue theory of integration. In fact, instead of solving the ODE with an $L^\infty$-right hand side one solves the classically equivalent integral equation, see e.g. [31, chapter 3, section 10, suppl. 2]. This is actually a special case of the fact that the geodesics¹⁰ of any locally Lipschitz continuous metric are C¹-curves, see [30].

Second¹¹, we explicitly demonstrate that the geodesics of (5.10) do not satisfy the geodesic equation. Let us demonstrate this just for the x¹  =  x-component, which reads

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:10} x(u)=\frac{x_0}{1+ku_+}. \nonumber \end{align} \tag{ 11 }$$

We find (using (5), as well as formally applying the product rule)

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{{\mathrm{d}}} \displaystyle \dot x(u)=-\frac{kx_0\theta(u)}{(1+ku_+)^2},\quad \ddot x(u)=-\frac{kx_0\delta(u)}{(1+ku_+)^2}+\frac{2k^2x_0\theta(u)}{(1+ku_+)^3}. \nonumber \end{align} \tag{ 12 }$$

Plugging this into the x-component of the geodesic equation for the metric (2.4) with the impulsive profile (2.16) (derived under the same 'rules' as above, see also (21) below), i.e.

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{{\mathrm{d}}} \displaystyle \ddot x(u)+\frac{2k\theta(u)}{1+ku_+}\,\dot x(u) \nonumber \end{align} \tag{ 13 }$$

one does not obtain a vanishing right hand side, but instead we have

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{{\mathrm{d}}} \displaystyle \label{eq:12} \ddot x(u)+\frac{2k\theta(u)}{1+ku_+}\,\dot x(u)=-\frac{kx_0\delta(u)}{(1+ku_+)^2} =-kx_0\delta(u), \nonumber \end{align} \tag{ 14 }$$

where in the last step we have used that for any function continuous in a neighbourhood of u  =  0 we have $f(u)\delta(u)=f(0)\delta(u)$.

The result (14) may be explained in rough terms also as follows: since the x-component of the geodesics (5.10) has a finite jump in its velocity at u  =  0 its second derivative will contain a term proportional to $\delta(u)$. On the other hand such a term cannot be present on the right hand side of the geodesic equation in (R)-coordinates. Indeed the metric is Lipschitz continuous and hence possesses an $L^\infty$-connection: the Christoffel symbols in the geodesic equation will be proportional to the step function $\theta(u)$. Also the velocity $\dot x$ will only involve a step function, so that the $\delta$-term arising from $\newcommand{\dd}{{\mathrm{d}}} \ddot x$ cannot be cancelled.

Third, one can resort to regularisation, which leads to different (and correct) geodesics, see item (A2) below, which again confirms that the geodesics of (5.10) are not the right ones.

But where is the error in the arguments of [35, section 5.1]? It is included in the statement (*). In fact, it simply makes no sense to relate the form of the geodesic equations in the before- and the after-zones in (S) coordinates. These are actually given on two non-overlapping patches and there is no way of relating quantities on either side but using the transformation to (R)-coordinates or alternatively to (B)-coordinates but in this case keeping the distributional terms.

In conclusion, the geodesics (5.10) cannot be seen as actually being geodesics of the spacetime (2.4) with the impulsive profile (2.16). They do not satisfy the geodesic equations in any meaningful way across the impulse, neither in the sense of an appropriate solution concept, nor via a regualrisation approach, nor formally. Finally the physical argument (*) put forward in deriving (5.10) is flawed.

Here we specialize the so-called C¹-matching procedure of [15, section 3] to the case of plane waves. This is a method to derive 'jump conditions' for the geodesics in impulsive waves as seen with respect to 'background coordinates' in the before- and after-zones. We suspect that this was also the idea underlying the (flawed) approach of [35, section 5.2].

In the particular case at hand we have a Minkowskian background in the before- and the after-zone and hence we can trivially derive the geodesics there in manifestly Minkowskian coordinates, i.e. in the (S)-coordinates¹². We denote these geodesics in the usual $\pm$-notation as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\xx}{{\mathbf x}} \displaystyle \label{eq:geo-s} \hat\xx^\pm(u),\quad {{\rm and}}\quad \hat v^\pm(u). \nonumber \end{align} \tag{ 15 }$$

They are clearly just straight lines and are entirely determined by the following set of $2\times 6$ constants

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\xx}{{\mathbf x}} \displaystyle \label{eq:const} \hat\xx^\pm_i:=\lim_{u\to 0\pm}\hat\xx^\pm(u),\quad &\hat v^\pm_i:=\lim_{u\to 0\pm}\hat v^\pm(u)\nonumber \\ \dot{\hat\xx}^\pm_i:=\lim_{u\to 0\pm}\dot{\hat\xx}^\pm(u),\quad &\dot{\hat v}^\pm_i:=\lim_{u\to 0\pm}\dot{\hat v}^\pm(u), \nonumber \end{align} \tag{ 16 }$$

where the subscript i stands for 'interaction time', i.e. for the instance when the geodesics cross the impulse. We can now relate the $\pm$-versions of these constants to one another using the fact that the geodesics in (R)-coordinates are C¹-curves. More precisely, we transform the geodesics (15) to (R)-coordinates in which we will denote them by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\xx}{{\mathbf x}} \displaystyle \label{eq:geoR} \xx(u),\quad {{\rm and}}\quad v(u) \nonumber \end{align} \tag{ 17 }$$

and 'match' the respective constants (16). To do so most explicitly we invoke the inverse transformation of (3) to relate the (S)- to the (R)-coordinates¹³, which reads

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\PP}{{P}} \newcommand{\xx}{{\mathbf x}} \newcommand{\J}{{{\mathbb J}}} \displaystyle \label{eq:invtrsf} \xx(u)=\PP^{-1}(u)\hat\xx(u),\quad v(u)=\hat v(u)+\frac{k}{2}\theta(u)\PP^{-1}(u)\hat\xx\cdot\J\hat\xx, \nonumber \end{align} \tag{ 18 }$$

where again $\newcommand{\J}{{{\mathbb J}}} \newcommand{\PP}{{P}} \PP(u)=({{\mathbb I}}+u_+k\J)$.

Now we can relate the $\pm$-versions of the constants (16) as follows

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\PP}{{P}} \newcommand{\xx}{{\mathbf x}} \displaystyle \label{eq:junctionx} \hat\xx^-_i=\lim_{u\to 0-}\hat\xx^-(u)&=\lim_{u\to 0-}\xx(u)= \lim_{u\to 0+}\xx(u)\nonumber \\ &=\lim_{u\to 0+}\PP^{-1}(u)\hat\xx^+(u)= \lim_{u\to 0+}(\PP^{-1}(u))\,\hat\xx^+_i=\hat\xx^+_i. \nonumber \end{align} \tag{ 19 }$$

Here we have used the definition of $\newcommand{\xx}{{\mathbf x}} \hat\xx^-_i$ in the first equality, the transformation (18) for u  <  0 in the second, the continuity of the geodesics (17) in the third, and then again the transformation (18), now for u  >  0. Finally, we have used the definition of $\newcommand{\xx}{{\mathbf x}} \hat\xx^+_i$ and the explicit form of $\newcommand{\PP}{{P}} \PP^{-1}$ to calculate the limit.

Similarly we may use the C¹-property of the geodesics (17) to relate the respective velocities on either side of the impulse and we obtain the following set of 'jump conditions':

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\xx}{{\mathbf x}} \newcommand{\J}{{{\mathbb J}}} \displaystyle \label{eq:jump} \hat\xx^-_i& = \hat\xx^+_i,\qquad \qquad \qquad \qquad \hat v^-_i = \hat v^+_i+\frac{k}{2}\hat\xx^+_i\cdot\J\hat\xx^+_i, \nonumber \\ \dot{\hat\xx}^-_i& = \dot{\hat\xx}^+_i-k\J\hat\xx^+_i,\quad \qquad \qquad \dot{\hat v}^-_i = \dot{\hat v}^+_i+k\hat\xx^+_i\cdot\J\dot{\hat\xx}^+_- -\frac{k^2}{2}\hat\xx^+_i\cdot\J\hat\xx^+_i. \nonumber \end{align} \tag{ 20 }$$

Observe that these relations are just a special case of the relations derived in [15, section 3] with the same identifications as explained below (2).

We conclude with a remark on the 'philosophy' of the C¹-matching, see [20, remark 4.1]. The matching presupposes the following knowledge of the geodesics on the entire spacetime: the geodesics heading towards the impulse have to cross it, have to be unique and of C¹-regularity. All these properties have been established for the situation at hand in [15]. Also the C¹-matching procedure has been generalised to the case of non-expanding impulsive waves in any constant curvature background in [20] and to expanding impulsive waves, again in all constant curvature backgrounds in [21].

Here we very briefly comment on the derivation of the geodesics in impulsive waves in (B)-coordinates. Indeed in [35, section 5.2] the $\newcommand{\XX}{{\mathbf X}} \XX$-components of the geodesics in impulsive plane waves are derived by basically integrating the geodesic equations and the use of the 'multiplication rules' (5) to yield

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\XX}{{\mathbf X}} \newcommand{\J}{{{\mathbb J}}} \displaystyle \label{eq:5.16} \XX(U)=P(U)\XX_0=({{\mathbb I}}+u_+k\J)\XX_0, \nonumber \end{align} \tag{ 5.16 }$$

where $\newcommand{\XX}{{\mathbf X}} \XX_0= \XX(0)$. Observe that (5.16) is in perfect agreement with the left equations in (20). The authors of [35] correctly remark in footnote 11 on p 12 that the derivation of the $V$-component is more involved from the distribution theoretic point of view.

However, an ad-hoc procedure has been employed in the pp-wave case to derive the geodesics in [8], which—to the author's best knowledge—is the earliest account explicitly calculating the geodesics in the distributional form of impulsive gravitational waves. A more reliable account has been put forward in [1], again in the pp-wave case. Here some nonlinear theory of distributions was applied but still an ad-hoc assumption (preservation of the geodesic's tangent across the impulse) was needed to derive the result. The full solution was finally given in [14, 28]. We remark that in these approaches, which essentially are based on regularisation of the impulsive profile by a sequence of general sandwich waves, it becomes a nontrivial task to show that the solutions of the now nonlinear geodesic equations live long enough to cross the regularised (and hence extended) wave zone, i.e. the impulse at all. This is done using a fixed point argument which has been subsequently refined to allow a generalisation of the procedure to ever wider classes of impulsive waves, see [22–26].

In this section we finally comment on [35, section 7], where the geodesics (5.7) are compared to the ones we have given in [29, section 4]¹⁴.

In fact [29] uses quite different conventions and there is a lapse in the geodesics presented in equation (14) there—in fact the X- the Y- components should be interchanged. However, these geodesics have been correctly transferred to the present setting in [35, section 7] to read (using (R)-coordinates $\newcommand{\xx}{{\mathbf x}} \xx=(x,y)$)

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:7.1} x(u)=x_0+\dot x_0\left(\frac{u_+}{1+u_+}+u_-\right),\quad y(u)=y_0+\dot y_0\left(\frac{u_+}{1-u_+}+u_-\right),\quad \nonumber \end{align} \tag{ 7.1 }$$

where for brevity we again restrict attention to the spatial components leaving aside the more complicated $v$-equation. Also we set k  =  1 and use the usual definition u₋  =  u if $u\leqslant 0$ and u₋  =  0 for $u\geqslant 0$.

Actually we are aware of three ways to directly derive the explicit form of the geodesics in (R)-form in the plane wave case, i.e. (7.1), all without the use of the symmetries of the spacetime and all leading to the same result:

• (A1)  
  Solving the geodesic equations, which are given e.g. in [35, equation (7.3)] and explicitly read

  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{{\mathrm{d}}} \displaystyle \label{eq:geoRe} \ddot x+2\frac{k\theta}{1+u_+}\dot x=0,\quad \ddot y-2\frac{k\theta}{1-u_+}\dot y=0, \nonumber \end{align} \tag{ 21 }$$

  separately in the before- and the after-zones and matching the integration constant to obtain a global C¹-curve.
• (A2)  
  Regularising the step function in the metric e.g. by setting $\newcommand{\ep}{\varepsilon} \newcommand{\eps}{\varepsilon} \newcommand{\e}{{\rm e}} \theta_\eps(u)=\int_{-\infty}^u\rho_\eps(t){\rm d}t$ (with $\newcommand{\ep}{\varepsilon} \newcommand{\eps}{\varepsilon} \newcommand{\e}{{\rm e}} \rho_\eps\to\delta$ in distributions), then integrating the regularised equations, and finally performing the limit $\newcommand{\ep}{\varepsilon} \newcommand{\eps}{\varepsilon} \newcommand{\e}{{\rm e}} \eps\to 0$.
• (A3)  
  Making an ad-hoc ansatz (essentially guessing the solutions) and checking that the equations do hold again using the 'multiplication rules' (5).

In [29, section 4] we actually only mention approaches (A1) and (A2). However, the fact that the C¹-property is used in the matching is explicitly stated above equation (14)—contrary to the claim made below [35, equation (7.1)]. Anyhow, it has meanwhile been proven that the C¹-property holds, see section 3.3, above.

Moreover, approach (A2) and hence also indirectly the C¹-property is confirmed in [35, caption of figure 7], where the authors acknowledge the fact that a regularisation by Gaussians leads to geodesics converging to (7.1).

Finally, we extend the calculations of equations (11)–(14) by formally showing that also the geodesics (5.7) (with arbitrary initial speeds) do not satisfy the geodesic equations [35, equation (7.3)], i.e. (21). Indeed the spatial components of (5.7) take the explicit form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:7.2} x(u)=x_0+\dot x_0^+\frac{u_+}{1+u_+}+\dot x_0^-u_-,\quad y(u)=y_0+\dot y_0^+\frac{u_+}{1-u_+}+\dot y^-_0 u_-, \nonumber \end{align} \tag{ 7.2 }$$

and using again the usual set of 'multiplication rules' (5) one obtains e.g. for the x-component in $-\infty<u<1$ that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{{\mathrm{d}}} \displaystyle \ddot x+2\frac{k\theta}{1+u_+}\dot x=\delta(u)\,(\dot x^+_0-\dot x^-_0). \nonumber \end{align} \tag{ 22 }$$

This equation again tells us that in order to satisfy the geodesic equation we need to have $\newcommand{\xx}{{\mathbf x}} \newcommand{\bi}{\boldsymbol} \bigtriangleup\dot\xx=\dot\xx^+_0-\dot\xx^-_0=0$, i.e. no jumps in the velocities of the geodesics in (R)-coordinates.

In the final section of this chapter we comment on [35, section 5.3] and the interrelations between the geodesics in (R)-form and in (B)-form. The authors of [35] say on this matter:

'The naive expectation might be that this [interrelation] could be achieved by using the transformation formula between the coordinates, (2.6), i.e.

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\PP}{{P}} \newcommand{\XX}{{\mathbf X}} \newcommand{\xx}{{\mathbf x}} \displaystyle \XX(U)=\PP(U)\,\xx(u), \label{naiveXx} \nonumber \end{align} \tag{ 5.20 }$$

which is indeed correct in the case of continuous wave profiles for particles initially at rest, [33, 34], for which $\newcommand{\xx}{{\mathbf x}} \xx(u)=\xx_0={{\rm const}}$ for all u. However, identifying the initial positions, $\newcommand{\xx}{{\mathbf x}} \newcommand{\XX}{{\mathbf X}} \xx_0=\XX_0$ and combining (5.16) and (5.10) yields instead,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\PP}{{P}} \newcommand{\XX}{{\mathbf X}} \newcommand{\xx}{{\mathbf x}} \displaystyle \boxed{\XX(U)= (\PP^T\!\PP)(u)\, \xx(u) = a(u)\, \xx(u).\, } \label{goodXx} \nonumber \end{align} \tag{ 5.21 }$$

Where does the extra $\newcommand{\PP}{{P}} \PP$-factor come from? The clue is that the delta-function $\delta(u)$ makes the velocity jump both in B [(B)] and BJR [(R)] coordinates—and does it in the opposite way, see in (5.15)¹⁵ and (5.12b) [that actually should read (5.12a)], respectively. The extra $\newcommand{\PP}{{P}} \PP$ factor takes precisely care of these jumps: the first $\newcommand{\PP}{{P}} \PP$ in (5.21) straightens the trajectory (5.10) to the trivial one, $\newcommand{\xx}{{\mathbf x}} \newcommand{\PP}{{P}} \PP(u)\xx(u)=\xx_0,$ which has zero initial BJR velocity as in the smooth case [33, 34]; then the second $\newcommand{\PP}{{P}} \PP(u)$ factor curls it up according to (5.20), yielding $\newcommand{\XX}{{\mathbf X}} \XX(u)$ in (5.16).'

This explanation remains dubious. While it is true that the combination of (5.16) and (5.10) yields (5.21), this just confirms that the geodesics in (R)-form (5.10) are not the correct ones. In fact, replacing the incorrect geodesics (5.10) by the correct ones, i.e. (7.1), which in the case at hand, i.e. vanishing speeds, simply read $\newcommand{\xx}{{\mathbf x}} \xx(u)=\xx_0$, we correctly obtain (5.20) (as is also acknowledged in the above quotation):

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\PP}{{P}} \newcommand{\XX}{{\mathbf X}} \newcommand{\xx}{{\mathbf x}} \displaystyle \XX(U)=\PP(U)\XX_0=\PP(u)\xx_0=\PP(u)\xx(u). \nonumber \end{align} \tag{ 23 }$$

The fact that formally transforming the geodesics in (R)-form (7.1) with the 'discontinuous change of coordinates' (3) yields exactly the geodesics in (B)-form (5.16) has already been noted in [29, section 4] just below equation (14)¹⁶. In fact, it does nothing else but transforming the (B)-geodesics with vanishing initial speeds into the (R)-geodesics. In other words, the broken and jumping (B)-geodesics become the new coordinate lines in the (R)-system. And this is the ultimate reason why the regularity of the metric improves from distributional in the (B)-coordinates to continuous in (R)-coordinates.

The formal calculation establishing these ideas has been turned into a solid piece of mathematics even for the pp-wave case in [13], using nonlinear distributional geometry. A good way to describe the situation in physical terms is given there in section 5: the 'discontinuous change' of coordinates is the distributional limit of a family of smooth transformations which can be obtained by a general regularisation procedure, which is adapted to the spacetime geometry. From this regularisation point of view, the (B) and (R)-forms of the impulsive metric arise as the distributional limits of the same sandwich wave in different coordinate systems. In such a scenario, in general, different spacetimes may result and the fact that in this case the geometries are 'physically equivalent' is reflected by the fact that the resulting transformation is merely discontinuous rather than unbounded. Nevertheless, it introduces finite jumps of the geodesics and their velocities.