arXiv subjects

Eric Woolgar

Publications and source records attributed to Eric Woolgar.

24 records · Page 2Linked to original sources

arXiv2017

Curvature-dimension bounds for Lorentzian splitting theorems

We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-Émery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions $N\le 1$, including all negative synthetic dimensions. The rigidity of the timelike splitting reduces to a warped product splitting when $N=1$. We also extend the null splitting theorem of Lorentzian geometry, showing that it holds under a null curvature-dimension bound on the Bakry-Émery-Ricci tensor for all $N\in (-\infty, 2]\cup (n,\infty)$ and for the $N=\infty$ case as well, with reduced rigidity if $N=2$. In consequence, the basic singularity and splitting theorems of Lorentzian Bakry-Émery theory now cover all synthetic dimensions for which such theorems are possible. The splitting theorems are found always to exhibit reduced rigidity at the critical synthetic dimension.

math.DG↗

arXiv2017

The rigid Horowitz-Myers conjecture

The "new positive energy conjecture" Horowitz and Myers (1999) probes a possible nonsupersymmetric AdS/CFT correspondence. We consider a version formulated for complete, asymptotically Poincaré-Einstein Riemannian metrics $(M,g)$ with bounded scalar curvature $R\ge -n(n-1)$. This version then asserts that any such $(M,g)$ must have mass not less than the mass $m_0$ of a metric $g_0$ induced on a time-symmetric slice of a certain AdS soliton spacetime. The conjecture remains unproved, having so far resisted standard techniques. Little is known other than that the conjecture is true for metrics which are sufficiently small perturbations of $g_0$. We pose another test for the conjecture. We assume its validity and attempt to prove as a corollary the corresponding scalar curvature rigidity statement, that $g_0$ is the unique asymptotically Poincaré-Einstein metric with mass $m=m_0$ obeying $R\ge -n(n-1)$. Were a second such metric $g_1$ not isometric to $g_0$ to exist, it then may well admit perturbations of lower mass, contradicting the assumed validity of the conjecture. We find that the minimum mass metric must be static Einstein, so the problem is reduced to that of static uniqueness. When $n=3$ the manifold is isometric to a time-symmetric slice of an AdS soliton spacetime, unless it has a non-compact horizon. En route we study the mass aspect, obtaining and generalizing known results. The mass aspect is (i) related to the holographic energy density, (ii) a weighted invariant under boundary conformal transformations when the bulk dimension is odd, and (iii) zero for negative Einstein manifolds with Einstein conformal boundary.

math.DG↗

arXiv2015

Cosmological singularities in Bakry-Émery spacetimes

We consider spacetimes consisting of a manifold with Lorentzian metric and a weight function or scalar field. These spacetimes admit a Bakry-Émery-Ricci tensor which is a natural generalization of the Ricci tensor. We impose an energy condition on the Bakry-Émery-Ricci tensor and obtain singularity theorems of a cosmological type, both for zero and for positive cosmological constant. That is, we find conditions under which every timelike geodesic is incomplete. These conditions are given by "open" inequalities, so we examine the borderline (equality) cases and show that certain singularities are avoided in these cases only if the geometry is rigid; i.e., if it splits as a Lorentzian product or, for a positive cosmological constant, a warped product, and the weight function is constant along the time direction. Then the product case is future timelike geodesically complete while, in the warped product case, worldlines of certain conformally static observers are complete. Our results answer a question posed by J Case. We then apply our results to the cosmology of scalar-tensor gravitation theories. We focus on the Brans-Dicke family of theories in 4 spacetime dimensions, where we obtain "Jordan frame" singularity theorems for big bang singularities.

math.DG↗

arXiv2015

Cosmological singularity theorems and splitting theorems for N-Bakry-Emery spacetimes

We study Lorentzian manifolds with a weight function such that the $N$-Bakry-Émery tensor is bounded below. Such spacetimes arise in the physics of scalar-tensor gravitation theories, including Brans-Dicke theory, theories with Kaluza-Klein dimensional reduction, and low-energy approximations to string theory. In the "pure Bakry-Émery" $N= \infty$ case with $f$ uniformly bounded above and initial data suitably bounded, cosmological-type singularity theorems are known, as are splitting theorems which determine the geometry of timelike geodesically complete spacetimes for which the bound on the initial data is borderline violated. We extend these results in a number of ways. We are able to extend the singularity theorems to finite $N$-values $N\in (n,\infty)$ and $N\in (-\infty,1]$. In the $N\in (n,\infty)$ case, no bound on $f$ is required, while for $N\in (-\infty,1]$ and $N= \infty$, we are able to replace the boundedness of $f$ by a weaker condition on the integral of $f$ along future-inextendible timelike geodesics. The splitting theorems extend similarly, but when $N=1$ the splitting is only that of a warped product for all cases considered. A similar limited loss of rigidity has been observed in prior work on the $N$-Bakry-Émery curvature in Riemannian signature when $N=1$, and appears to be a general feature.

math.DG↗

arXiv2014

Renormalized volume and the evolution of APEs

We study the evolution of the renormalized volume functional for asymptotically Poincare-Einstein metrics (M,g) which are evolving by normalized Ricci flow. In particular, we prove that the time derivative of the renormalized volume along the flow is the negative integral of scal(g(t)) + n(n-1) over the manifold. This implies that if scal(g(0))+n(n-1) is non-negative at t=0, then the renormalized volume decreases monotonically. We also discuss how, when n=4, our results describe the Hawking-Page phase transition. Differences in renormalized volumes give rigorous meaning to the Hawking-Page difference of actions and describe the free energy liberated in the transition.

math.DG↗

arXiv2002

Positive Mass from Holographic Causality

For n+1 dimensional asymptotically AdS spacetimes which have holographic duals on their n dimensional conformal boundaries, we show that the imposition of causality on the boundary theory is sufficient to prove positivity of mass for the spacetime when n > 2, without the assumption of any local energy condition. We make crucial use of a generalization of the time-delay formula calculated in gr-qc/9404019, which relates the time delay of a bulk null curve with respect to a boundary null geodesic to the Ashtekar-Magnon mass of the spacetime. We also discuss holographic causality for the negative mass AdS soliton and its implications for the positive energy conjecture of Horowitz and Myers.

hep-th↗