arXiv subjects

Stephen McCormick

Publications and source records attributed to Stephen McCormick.

23 records · Page 2Linked to original sources

arXiv2016

The asymptotically flat scalar-flat Yamabe problem with boundary

We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with boundary, in dimension $n\geq3$. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric $g$, there is a conformally equivalent asymptotically flat scalar-flat metric that agrees with $g$ on the boundary. We then replace the metric boundary condition with a condition on the mean curvature: Given a function $f$ on the boundary that is not too large, we show that there is an asymptotically flat scalar-flat metric, conformally equivalent to $g$ whose boundary mean curvature is given by $f$. The latter case involves solving an elliptic PDE with critical exponent using the method of sub- and supersolutions. Both results require the usual assumption that the Sobolev quotient is positive.

math.AP↗

arXiv2015

The Hilbert manifold of asymptotically flat metric extensions

In [Comm. Anal. Geom., 13(5):845-885, 2005.], Bartnik described the phase space for the Einstein equations, modelled on weighted Sobolev spaces with local regularity $(g,π)\in H^2\times H^1$. In particular, it was established that the space of solutions to the contraints form a Hilbert submanifold of this phase space. The motivation for this work was to study the quasilocal mass functional now bearing his name. However, the phase space considered there was over a manifold without boundary. Here we demonstrate that analogous results hold in the case where the manifold has an interior compact boundary, where Dirichlet boundary conditions are imposed on the metric. Then, still following Bartnik's work, we demonstrate the critical points of the mass functional over this space of extensions correspond to stationary solutions. Furthermore, if this solution is sufficiently regular then it is in fact a static black hole solution. In particular, in the vacuum case, critical points only occur at exterior Schwarzschild solutions; that is, critical points of the mass over this space do not exist generically. Finally, we briefly discuss the case when the boundary data is Bartnik's geometric data.

gr-qc↗

arXiv2015

A note on mass-minimising extensions

A conjecture related to the Bartnik quasilocal mass, is that the infimum of the ADM energy, over an appropriate space of extensions to a compact 3-manifold with boundary, is realised by a static metric. It was shown by Corvino [Comm. Math. Phys. 214(1), (2000)] that if the infimum is indeed achieved, then it is achieved by a static metric; however, the more difficult question of whether or not the infimum is achieved, is still an open problem. Bartnik [Comm. Anal. Geom. 13(5), (2005)] then proved that critical points of the ADM mass, over the space of solutions to the Einstein constraints on an asymptotically flat manifold without boundary, correspond to stationary solutions. In that article, he stated that it should be possible to use a similar construction to provide a more natural proof of Corvino's result. In the first part of this note, we discuss the required modifications to Bartnik's argument to adapt it to include a boundary. Assuming that certain results concerning a Hilbert manifold structure for the space of solutions carry over to the case considered here, we then demonstrate how Bartnik's proof can be modified to consider the simpler case of scalar-flat extensions and obtain Corvino's result. In the second part of this note, we consider a space of extensions in a fixed conformal class. Sufficient conditions are given to ensure that the infimum is realised within this class.

gr-qc↗

arXiv2014

The Phase Space for the Einstein-Yang-Mills Equations and the First Law of Black Hole Thermodynamics

We use the techniques of Bartnik (2005) to show that the space of solutions to the Einstein-Yang-Mills constraint equations on an asymptotically at manifold with one end and zero boundary components, has a Hilbert manifold structure; the Einstein-Maxwell system can be considered as a special case. This is equivalent to the property of linearisation stability, which was studied in depth throughout the 70s. This framework allows us to prove a conjecture of Sudarsky and Wald (1992), that is, the validity of the first law of black hole thermodynamics is a suitable condition for stationarity. Since we work with a single end and no boundary conditions, this is equivalent to critical points of the ADM mass subject to variations fixing the Yang-Mills charge corresponding exactly to stationary solutions. The natural extension to this work is to prove the second conjecture of Sudarsky and Wald, which is the case where an interior boundary is present; this will be addressed in future work.

gr-qc↗

arXiv2014

First Law of Black Hole Mechanics as a Condition for Stationarity

In earlier work [arXiv:1302.1237], we provided a Hilbert manifold structure for the phase space for the Einstein-Yang-Mills equations, and used this to prove a condition for initial data to be stationary. Here we use the same phase space to consider the evolution of initial data exterior to some closed 2-surface boundary, and establish a condition for stationarity in this case. It is shown that the differential relationship given in the first law of black hole mechanics is exactly the condition required for the initial data to be stationary; this was first argued non-rigorously by Sudarsky and Wald in 1992. Furthermore, we give evidence to suggest that if this differential relationship holds then the boundary surface is the bifurcation surface of a bifurcate Killing horizon.

gr-qc↗