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We give a constructive proof that coordinate transformations exist whichraise the regularity of the gravitational metric tensor from $C^{0,1}$ to$C^{1,1}$ in a neighborhood of points of shock wave collision in GeneralRelativity. The proof applies to collisions between shock waves coming fromdifferent characteristic families, in spherically symmetric spacetimes. Ourresult here implies that spacetime is locally inertial and corrects an error inour earlier RSPA-publication, which led us to the false conclusion that suchcoordinate transformations, which smooth the metric to $C^{1,1}$, cannot exist.Thus, our result implies that regularity singularities, (a type of mildsingularity introduced in our RSPA-paper), do not exist at points ofinteracting shock waves from different families in spherically symmetricspacetimes. Our result generalizes Israel's celebrated 1966 paper to the caseof such shock wave interactions but our proof strategy differs fundamentallyfrom that used by Israel and is an extension of the strategy outlined in ouroriginal RSPA-publication. Whether regularity singularities exist in morecomplicated shock wave solutions of the Einstein Euler equations remains open.

No Regularity Singularities Exist at Points of General Relativistic Shock Wave Interaction between Shocks from Different Characteristic Families