The action of a casual set

A causal set is a model for a discrete spacetime in which the “atoms of spacetime” carry a relation of ancestry. This order relation is mathematically given by a partial order, and is is taken to underly the macroscopic causal notions of before and after. The work presented in this thesis proposes a definition for the action of a causal set analogous to the continuum Einstein-Hilbert action. The path taken towards the definition of this action is somewhat indirect. We first construct a retarded wave operator on causal sets well-approximated by 4-dimensional spacetimes and prove, under certain assumptions, that this operator gives the usual continuum d’Alembertian and the scalar curvature of the approximating spacetime in the continuum limit. We use this result to define both the scalar curvature and the action of a causal set. This definition can be shown to work in any dimension, so that an explicit form of the action exists in all dimensions. We conjecture that, under certain conditions, the continuum limit of the action is given by the Einstein-Hilbert action up to boundary terms, whose explicit form we also conjecture. We provide evidence for this conjecture through analytic and numerical calculations of the expected action of various spacetime regions. The 2-dimensional action is shown to possess topological properties by calculating its expectation value for various regions of 2-dimensional spacetimes with different topologies. We find that the topological character of the 2d action breaks down for causally convex regions of the trousers spacetime that contain the singularity, and for non-causally convex rectangles. Finally, we propose a microscopic account of the entropy of causal horizons based on the action. It is a form of “spacetime mutual information” arising from the partition of spacetime by the horizon. Evidence for the proposal is provided by analytic results and numerical simulations in 2- dimensional examples. Further evidence is provided by numerical results for the Rindler and cosmic deSitter horizons in both 3 and 4-dimensions, and for a non-equilibrium horizon in a collapsing shell spacetime in 4-dimensions.Open Acces

The Scalar Curvature of a Causal Set

A one parameter family of retarded linear operators on scalar fields on causal sets is introduced. When the causal set is well-approximated by 4 dimensional Minkowski spacetime, the operators are Lorentz invariant but nonlocal, are parametrised by the scale of the nonlocality and approximate the continuum scalar D'Alembertian, $\Box$, when acting on fields that vary slowly on the nonlocality scale. The same operators can be applied to scalar fields on causal sets which are well-approximated by curved spacetimes in which case they approximate $\Box - {{1/2}}R$ where $R$ is the Ricci scalar curvature. This can used to define an approximately local action functional for causal sets.Comment: Typo in definition of equation (3) and definition of n(x,y) corrected. Note: published version still contains typ

The continuum limit of a 4-dimensional causal set scalar d'Alembertian

The continuum limit of a 4-dimensional, discrete d'Alembertian operator for scalar fields on causal sets is studied. The continuum limit of the mean of this operator in the Poisson point process in 4-dimensional Minkowski spacetime is shown to be the usual continuum scalar d'Alembertian $\Box$. It is shown that the mean is close to the limit when there exists a frame in which the scalar field is slowly varying on a scale set by the density of the Poisson process. The continuum limit of the mean of the causal set d'Alembertian in 4-dimensional curved spacetime is shown to equal $\Box - \frac{1}{2}R$, where $R$ is the Ricci scalar, under certain conditions on the spacetime and the scalar field.Comment: 31 pages, 2 figures. Slightly revised version, accepted for publication in Classical and Quantum Gravit

Lebanon: new president, old politics

Filippo Dionigi argues that despite the recent presidential election nothing much has changed in Lebanon's domestic politics or its international rediolation

Book review: the second Arab awakening and the battle for pluralism by Marwan Muasher

In this book Marwan Muasher proposes a broad analysis and assessment of the Arab Spring, calling on the West to rethink political Islam and the Arab-Israeli conflict. He also discusses steps all parties can take to encourage positive state-building in a freshly unsettled Arab world. Filippo Dionigi finds that some of the author’s arguments constitute wishful thinking, but overall his optimism may inspire successful reconciliation between parties in the future

Terrains minés

article d'introduction du n°2001/1 de la revue d'Ethnologie Français

Terrains minés

article d'introduction du n°2001/1 de la revue d'Ethnologie Français

Dynamics between Hezbollah and the Special Tribunal for Lebanon

Although the main threat to Lebanese stability over the last months was the possibility of an Israeli attack, Lebanon is now dealing with internal tensions in connection to the indictment of Hezbollah members by the Special Tribunal for Lebanon (STL).The tribunal, in charge of investigating the killing of Rafiq al-Hariri in 2005, initially focused on the involvement of the Syrian regime. Following the release of the initial suspects on the grounds of insufficient evidence, a stream of leaks indicated the possible indictment of members of Hezbollah