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November 2013

The starting assumption of General Relativity is that spacetime is locally flat in the sense that at each event (i.e. point in spacetime) there are locally inertial coordinate systems in which the gravitational metric is Minkowskian, and its first derivatives vanish at the center. The failure of the second derivatives to vanish at the center is described by the Riemann curvature tensor, which represents the gravitational field.

Recent work of Blake Temple and former UC Davis student Moritz Reintjes has demonstrated that, in a perfect fluid, the gravitational metric is in fact not locally flat at points of shock wave collision. Instead, a new kind of singularity forms, which they have named a regularity singularity. At these points the first derivatives of the metric cannot be made to vanish at the center in any coordinate system. But Temple and Reintjes have shown that, even though the second derivatives of the metric contain delta function source terms in every coordinate system, these delta functions cancel out in the Riemann curvature tensor. Since it is generally assumed that all physical effects of the gravitational field arise from the curvature of spacetime, one may ask whether there are observable physical effects resulting from the failure of locally inertial frames at Regularity Singularities.

In recent work, Temple and Reintjes express the linearized Einstein equations describing gravity waves in an arbitrary coordinate system as a second order wave equation together with first order terms whose coefficients arise from nonzero first order deriva tives of the background spacetime metric. These first order terms would vanish in any locally inertial coordinate frame if one existed. These terms give rise to effects analogous to Coriolis forces, fictitious forces that could be removed by transformation to locally inertial coordinates, if this can be done. But at the regularity singularities caused by colliding shock waves there are no locally inertial frames. So these Coriolis-type terms cannot be made to vanish in any coordinate system in their neighborhood. In fact, they are the dominant terms in the high frequency scattering of gravity waves, and therefore their effects in principle will be measureable by every local observer. This would be a new physical effect of the gravitational field that is not due to its curvature alone, but rather due to the essential lack of regularity in the underlying spacetime geometry.

Since shock wave collision in a perfect fluid can evolve generically from smooth initial data describing a sufficiently strong expansion, (like an explosion), inside a sufficiently strong contraction, (like a collapsing star), authors suggest that regularity singularities are worth exploring as a possibly significant physical effect for gravitational wave detection, perhaps by the LIGO gravity wave sensors currently under development. In any event, as far as we know, this is the first demonstration that the scattering of gravity waves can be localized by the dynamics of a perfect fluid to create first order effects.