Mathematics Colloquia and Seminars

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A family of global exact shock-wave solutions of the Einstein field equations are constructed in which the expansion wave behind the shock is modelled by a family of self-similar expanding waves. This two parameter family of expanding waves extends the one parameter family found by Smoller and Temple by incorporating a more general equation of state. Prior to this extension, the one parameter family of expanding waves was only known locally, and this prevented consideration of conservation across any shock surface away from the centre of expansion. The two parameter family of expanding waves solve the Einstein field equations for a perfect fluid under the assumptions of spherical symmetry and self-similarity in the variable $z = r/t$. Such assumptions reduce the Einstein field equations from a system of PDE, to a system of ODE in the single self-similar variable $z$. The two-parameter family of shock-wave spacetimes are obtained by matching this two-parameter family of expanding waves Lipschitz continuously to the one-parameter family of spherically symmetric, self-similar, perfect fluid, static spacetimes across a spherical shock surface. Both the expanding wave and static families are assumed to have isothermal equations of state of the form $p = \sigma\rho$ and $\bar{p} = \bar{\sigma}\bar{\rho}$ respectively, with the strictly positive constants $\sigma$ and $\bar{\sigma}$ representing a single parameter for each family. The remaining parameter, denoted by $a$, represents a perturbation from the one-parameter family of Friedmann-Lemaitre-Robertson-Walker spacetimes with an identical equation of state. By imposing conservation of mass and momentum across the shock surface, a constraint on the parameters is obtained and a local coordinate system can be found that raises the regularity of the shock-wave spacetimes by one full derivative. When $\sigma = \bar{\sigma}$, these shock-wave spacetimes model the general relativistic version of an explosion within a static, singular, isothermal sphere. Given that stars may be modelled as a static, singular isothermal spheres, such spacetimes find an immediate application to the modelling of supernovae. Interestingly, Smoller and Temple found that despite the fact their family of expanding waves solve the Einstein field equations in the absence of a cosmological constant, a comic acceleration was still present, with such an acceleration being parameterised by $a$. In the context of these new shock-wave spacetimes, this means that supernovae can be expected to induce local spacetime expansion and that the acceleration of such expansion is directly determined by the equation of state of the former star. In addition to the modelling of supernovae, the presence of this acceleration in the absence of a cosmological constant opens up the question of whether a vast primordial shock-wave could give rise to the cosmic acceleration observed today without the need for dark energy.