Impulsive Waves and Thin Sources

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Thin shells and impulsive waves concentrate curvature on a hypersurface while leaving ordinary smooth Einstein geometry on either side. They are controlled singularities only when the matching variables and distributional products are well defined. A non-null shell is governed by a jump in extrinsic curvature; a null shell needs a different transverse geometry; an impulsive wave can also appear through a distributional Brinkmann profile whose interpretation must be tied to a regularization or continuous coordinate form.

Required background. Analysis for Mathematical Relativity supplies distributions and traces; The Einstein Equation supplies the singular field equation; PDEs and Green Functions for Gravitation supplies weak solutions.

Helpful background. Causality and Global Structure helps distinguish a wavefront from a spacetime boundary or singular endpoint.

Timelike shells and the jump tensor

Let a timelike hypersurface $\Sigma$ divide spacetime into regions $M^-$ and $M^+$ . Choose a spacelike unit normal $s$ pointing from $M^-$ to $M^+$ , so $g(s,s)=+1$ , and require the induced metric $h_{ab}$ to agree from both sides. For this page define

\mathcal K^{\pm}_{ab} =h_a{}^\mu h_b{}^\nu\nabla^{\pm}_\mu s_\nu, \qquad [\mathcal K_{ab}]=\mathcal K^+_{ab}-\mathcal K^-_{ab}.

Here, $\mathcal K$ uses the displayed plus sign and is not the spacelike-slice $K_{ij}$ of the volume convention.

Continuity of $h$ keeps the metric continuous. A jump in its first normal derivative makes the connection discontinuous and produces a delta term in curvature, but no delta squared. Einstein’s equation gives the Israel junction condition

8\pi G S_{ab} =-\left( [\mathcal K_{ab}]-h_{ab}[\mathcal K] \right),

where $S_{ab}$ is the surface stress tensor and $[\mathcal K]=h^{ab}[\mathcal K_{ab}]$ . The labels and normal are linked: if the normal is reversed, interchange $M^+$ and $M^-$ as well. Then $[\mathcal K]_{\mathrm{new}}=(-\mathcal K^-)-(-\mathcal K^+) =[\mathcal K]$ , and the physical $S_{ab}$ is unchanged. Reversing only the normal while retaining the jump labels would incorrectly reverse the source.

The formula has three hypotheses that are often hidden: the hypersurface is non-null, the first fundamental form matches, and the metric is sufficiently regular on each side to possess traces of $\mathcal K^\pm$ . A discontinuous induced metric is not an ordinary thin shell.

Null shells require transverse curvature

On a null hypersurface, the normal is tangent and the induced metric is degenerate. There is no unit normal and no inverse three-metric with which to take the trace used above. One introduces a transverse null or non-null vector, a two-dimensional screen metric, and a transverse curvature. Its jump determines the surface stress and any impulsive gravitational part.

The split depends on the normalization of the generator and transverse vector, while the final distributional Einstein tensor is geometric. Barrabès and Israel construct this null-shell formalism in Barrabès–Israel 1991, §§II–IV. Substituting a null normal directly into the non-null Israel formula is therefore not a limiting derivation.

An impulsive plane-fronted wave

In Brinkmann coordinates, an idealized impulsive pp-wave can be written

ds^2=-2\,du\,dv+dx^2+dy^2 +H(x,y)\delta(u)\,du^2.

The wavefront is $u=0$ . Formally, the nonzero distributional curvature components are

R_{uiuj} =-\frac12\,\partial_i\partial_jH\,\delta(u), \qquad i,j\in\{x,y\},

and hence

R_{uu} =-\frac12\,\Delta_\perp H\,\delta(u).

If $H$ is harmonic in the transverse plane on the region considered, the Ricci distribution vanishes there while the trace-free Hessian carries an impulsive gravitational wave. If $\Delta_\perp H$ is nonzero, the wavefront also carries null matter.

The displayed metric coefficient itself contains a delta, so ordinary distribution theory cannot justify every intermediate inverse and product. One rigorous route regularizes the impulse and proves a distributional or generalized-function limit. Another uses a continuous coordinate form in which the metric is locally Lipschitz and geodesics can be matched across the wavefront. Podolský compares distributional and continuous forms and their geodesics in Podolský 2002, §§2–5.

Refraction across the wavefront

For the displayed pp-wave, the regions $u\ne0$ are flat. Choose $u$ as an affine parameter for a geodesic crossing the wavefront, and replace $\delta$ by a smooth unit-integral pulse before taking the limit. The transverse equations are

\frac{d^2x^i}{du^2}=\frac12\partial_iH\,\delta(u), \qquad \left[\frac{dx^i}{du}\right]=\frac12\partial_iH(x_0,y_0).

The transverse positions $(x,y)$ remain continuous. The Brinkmann coordinate $v$ , however, has the jump $[v]=H(x_0,y_0)/2$ ; it is not correct to say that every coordinate position remains continuous in this distributional chart. In a continuous chart the same geometry is represented by continuous geodesics with the appropriate regularity. The coordinate relation is described in Griffiths–Podolský 2009, §20.2, Eqs. (20.3)–(20.6), pp. 395–396. Their profile enters as $-2\mathcal H\delta(u)du^2$ , so our profile is $H=-2\mathcal H$ .

Multiplying the geodesic equation by distributions without regularization can create ambiguous products such as $\delta(u)\dot u$ evaluated on a nonsmooth trajectory. A valid result states the coordinate form, regularization or generalized solution notion, and which limiting quantities are invariant.

The transverse-curvature jump in continuous coordinates

For $H=A(x^2-y^2)$ , a continuous form of the metric near the impulse is

ds^2=-2\,du\,dV +(1+A u_+)^2dX^2+(1-A u_+)^2dY^2, \qquad u_+=\max(u,0).

Work where $|Au|\lt1$ to exclude the coordinate caustic. If the coordinates have length units, $A$ has units of inverse length. The null generator is $k=\partial_V$ and a transverse vector is $N=\partial_u$ , with $g(k,N)=-1$ . With screen metric $q_{AB}$ and the convention $C_{AB}=g(\nabla_{\partial_A}N,\partial_B)=\tfrac12\partial_uq_{AB}$ ,

[C_{AB}]=\begin{pmatrix}A&0\\0&-A\end{pmatrix}, \qquad q^{AB}[C_{AB}]=0.

The trace-free jump produces impulsive Weyl curvature and no null Ricci source. This agrees with $\Delta_\perp H=0$ in the distributional chart. The corresponding kicks are $[x']=Ax_0$ and $[y']=-Ay_0$ .

A constant reparametrization $u'=bu$ , $V'=V/b$ , $b\gt0$ , changes $k'=bk$ , $N'=N/b$ , $[C'_{AB}]=[C_{AB}]/b$ , and the Brinkmann profile to $H'=H/b$ . Nevertheless the singular curvature tensor is unchanged, since $\delta(u')du'\otimes du'=b\delta(u)du\otimes du$ . This explicit cancellation illustrates why a null-shell density or transverse-curvature component alone depends on normalization, while the complete tensor distribution does not.

Solved diagnostic: vacuum impulse or null source?

For each of the two transverse profiles below, compute $\Delta_\perp H$ and decide whether the impulse is Ricci-flat or carries a null surface source.

Solution

Take

H(x,y)=A(x^2-y^2).

Then $\Delta_\perp H=2A-2A=0$ , so $R_{uu}=0$ . Yet

\partial_x^2H=2A, \qquad \partial_y^2H=-2A,

and the trace-free curvature components are nonzero on $u=0$ . This is a vacuum gravitational impulse.

For $H=A(x^2+y^2)$ , $\Delta_\perp H=4A$ , so

R_{uu}=-2A\,\delta(u).

Here the scalar curvature vanishes, so Einstein’s equation requires $T_{uu}=-A\delta(u)/(4\pi G)$ . Nonnegative null energy therefore requires $A\le0$ for this particular sign convention and profile. The Laplacian, not the mere presence of a delta-profile metric term, distinguishes the Ricci source from the free gravitational impulse.

The final page asks what happens when singular or rapidly oscillating solutions are taken in sequences rather than prescribed on one fixed hypersurface: Compactness and Limits of Spacetimes.

References

Barrabès, Claude, and Werner Israel. “Thin Shells in General Relativity and Cosmology: The Lightlike Limit.” Physical Review D 43 (1991): 1129–1142. DOI.
Griffiths, Jerry B., and Jiří Podolský. Exact Space-Times in Einstein’s General Relativity. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2009. DOI.
Podolský, Jiří. “Exact Impulsive Gravitational Waves in Spacetimes of Constant Curvature.” In Gravitation: Following the Prague Inspiration, 205–246. World Scientific, 2002. arXiv:gr-qc/0201029.