*Proc. R. Soc. A* **468**, 2962–2980 (8 October 2012; published online 16 May 2012). (doi:10.1098/rspa.2011.0360)

With apologies, we must inform the reader that the result announced in our RSPA publication [1] is *false*. Although the issues and motivations regarding the problem as to whether regularity singularities exist at points of shock wave interaction in GR set out in Reintjes & Temple [1] are valid, there is a flaw in the final chapter of our proof that cannot be fixed. In trying to fix the flaw, we have discovered that the strategy can be used to prove the opposite statement: a *C*^{0,1} metric of a shock wave solution of the Einstein equations *can* always be smoothed to *C*^{1,1} by a coordinate transformation in a neighbourhood of a point of shock wave interaction between shocks from different characteristic families, in spherical symmetry. The precise statement of the result with detailed proofs is recorded in Reintjes [2]. This new result contradicts our incorrect claim in Reintjes & Temple [1] that regularity singularities are created by spherically symmetric shock wave interactions. However, the larger question as to whether regularity singularities exist in more complicated shock wave solutions of the Einstein–Euler equations remains an open problem. Thus, the larger issues laid out in Reintjes & Temple [1] remain valid.

We now identify the mistake in Reintjes & Temple [1]. To start, all of the mathematics in section 2–6 of Reintjes & Temple [1] remains correct without change. The mistake is in the last section, and this propagates into a mistake in the proof of the main theorem. Specifically, our proof in Reintjes & Temple [1] is based on the violation of the identity (7.9), which according to the argument there, is required for the smoothing coordinates to exist. But, this violation was due to the error of neglected terms that should have been present in our ansatz for the Jacobians in (7.1) of lemma 7.2 in Reintjes & Temple [1], the ansatz that meets what we refer to as the *smoothing condition* (c.f. (5.3) in Reintjes & Temple [1]). The missing terms in the expression for the Jacobians in (7.1) are the source of the error in Reintjes & Temple [1], because this Jacobian ansatz fails to meet the smoothing conditions.

To construct Jacobians that satisfy the smoothing condition, we correct the expression for the Jacobians *j*≠*i* whenever *j* and *i* appear in the same equation, and where we define *A*_{i}=*A*°*γ*_{i}, *B*_{i}=*B*°*γ*_{i} and *γ*_{i}(*t*)=(*t*,*x*_{i}(*t*)), and
*A*,*B* are the SSC metric components, the absolute value functions isolate the Lipschitz continuity and loss of derivatives in *J* on the shock surfaces *x*_{i}(*t*), *i*=1,2, and *C*^{1} functions.

The error in Reintjes & Temple [1] is the failure to include the terms in (2) containing *curl equation*,

To summarize, the larger issues regarding regularity singularities laid out in Reintjes & Temple [1] remain valid, even in the light of the corrections reported here, including the *possibility* that regularity singularities exist in more complicated shock interactions. We conclude that, at this stage, we *cannot* claim regularity singularities exist at points of shock wave interaction.

Moritz Reintjes

IMPA–Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, Brasil

Blake Temple

Department of Mathematics, University of California, Davis, CA 95616, USA

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.