The geometry of small causal diamonds is systematically studied, based on three distinct constructions that are common in the literature, namely the geodesic ball, the Alexandrov interval and the lightcone cut. The causal diamond geometry is calculated perturbatively using Riemann normal coordinate expansion up to the leading order in both vacuum and non-vacuum. We provide a collection of results including the area of the codimension-two edge, the maximal hypersurface volume and their isoperimetric ratio for each construction, which will be useful for any applications involving the quantitative properties of causal diamonds. In particular, by solving the dynamical equations of the expansion and the shear on the lightcone, we find that intriguingly only the light-cone cut construction yields an area deficit proportional to the Bel-Robinson superenergy density W in four dimensional spacetime, but such a direct connection fails to hold in any other dimension. We also compute the volume of the Alexandrov interval causal diamond in vacuum, which we believe is important but missing from the literatures. Our work complements and extends the earlier works on the causal diamond geometry by Gibbons and Solodukhin, Jacobson, Senovilla and Speranza and others. Some potential applications of our results in mathematical general relativity and quantum gravity are discussed.
The Fermionic Entanglement Entropy of Causal Diamonds in Two-Dimensional Minkowski Space
Fakultät für Mathematik Universität Regensburg（雷根斯堡大学数学系）；Institut für Theoretische Physik Universität Leipzig（莱比锡大学理论物理研究所）；Dipartimento di Matematica, Università di Genova（热那亚大学数学系）
We study the covariant phase space of vacuum general relativity at the null boundary of causal diamonds. The past and future components of such a null boundary each have an infinite-dimensional symmetry algebra consisting of diffeomorphisms of the $2$-sphere and boost supertranslations corresponding to angle-dependent rescalings of affine parameter along the null generators. Associated to these symmetries are charges and fluxes obtained from the covariant phase space formalism using the prescription of Wald and Zoupas. By analyzing the behavior of the spacetime metric near the corners of the c…
We study causal diamonds in Minkowski, Schwarzschild, (anti) de Sitter, and Schwarzschild-de Sitter spacetimes using Euclidean methods. The null boundaries of causal diamonds are shown to map to isolated punctures in the Euclidean continuation of the parent manifold. Boundary terms around these punctures decrease the Euclidean action by $A_\diamond/4$, where $A_\diamond$ is the area of the holographic screen around the diamond. We identify these boundary contributions with the maximal entropy of gravitational degrees of freedom associated with the diamond.
A static observer with a finite lifetime has causal access to only a limited region of spacetime known as the causal diamond. The presence of an apparent horizon in the causal diamond, due to the observer's finite lifetime, is the origin of an Unruh-like thermal effect. Thus, even though the observer is static and the background is flat, the finite-lifetime observer experiences a thermal bath in the Minkowski vacuum. In this article, we provide an open quantum systems approach that yields a complete thermal characterization via the observer's steady-state density matrix, which is shown to be t…
It is shown that a general radial conformal Killing vector in Minkowski space-time can be associated to a generator of time evolution in conformal quantum mechanics. Among these conformal Killing vectors one finds a class which maps causal diamonds in Minkowski space-time into themselves. The flow of such Killing vectors describes worldlines of accelerated observers with a finite lifetime within the causal diamond. Time evolution of static diamond observers is equivalent to time evolution in conformal quantum mechanics governed by a hyperbolic Hamiltonian and covering only a segment of the tim…
We apply a recent proposal for a distinguished ground state of a quantum field in a globally hyperbolic spacetime to the free massless scalar field in a causal diamond in two-dimensional Minkowski space. We investigate the two limits in which the Wightman function is evaluated (i) for pairs of points that lie in the centre of the diamond (i.e. far from the boundaries), and (ii) for pairs of points that are close to the left or right corner. We find that in the centre, the Minkowski vacuum state is recovered, with a definite value of the infrared cutoff. Interestingly, the ground state is not t…
The static patch of de Sitter spacetime and the Rindler wedge of Minkowski spacetime are causal diamonds admitting a true Killing field, and they behave as thermodynamic equilibrium states under gravitational perturbations. We explore the extension of this gravitational thermodynamics to all causal diamonds in maximally symmetric spacetimes. Although such diamonds generally admit only a conformal Killing vector, that seems in all respects to be sufficient. We establish a Smarr formula for such diamonds and a "first law" for variations to nearby solutions. The latter relates the variations of t…
We study the massive scalar field Sorkin-Johnston (SJ) Wightman function restricted to a flat 2D causal diamond of linear dimension L. Our approach is two-pronged. In the first, we solve the central SJ eigenvalue problem explicitly in the small mass regime, upto order (mL)^4. This allows us to formally construct the SJ Wightman function up to this order. Using a combination of analytic and numerical methods, we obtain expressions for the SJ Wightman function both in the center and the corner of the diamond, to leading order. We find that in the center, it is more like the massless Minkowski Wi…
We study the discrete causal set geometry of a small causal diamond in a curved spacetime using the average abundance of k-element chains or total orders in the underlying causal set C. We begin by obtaining the first order curvature corrections to the flat spacetime expression for the abundance using Riemann normal coordinates. For fixed spacetime dimension this allows us to find a new expression for the discrete scalar curvature of C as well as the time-time component of its Ricci tensor in terms of the abundances of k-chains. We also find a new dimension estimator for C which replaces the f…
A more complete understanding of entanglement entropy in a covariant manner could inform the search for quantum gravity. We build on work in this direction by extending previous results to disjoint regions in $1+1$D. We investigate the entanglement entropy of a scalar field in disjoint intervals within the causal set framework, using the spacetime commutator and correlator, $i\mathbfΔ$ and $\mathbf{W}$ (or the Pauli-Jordan and Wightman functions), respectively. A new truncation scheme for disjoint causal diamonds is presented, which follows from the single diamond truncation scheme. We investi…
We argue that the degrees of freedom in a d-dimensional CFT can be re-organized in an insightful way by studying observables on the moduli space of causal diamonds (or equivalently, the space of pairs of timelike separated points). This 2d-dimensional space naturally captures some of the fundamental nonlocality and causal structure inherent in the entanglement of CFT states. For any primary CFT operator, we construct an observable on this space, which is defined by smearing the associated one-point function over causal diamonds. Known examples of such quantities are the entanglement entropy of…
The geometry of causal diamonds or Alexandrov open sets whose initial and final events $p$ and $q$ respectively have a proper-time separation $τ$ small compared with the curvature scale is a universal. The corrections from flat space are given as a power series in $τ$ whose coefficients involve the curvature at the centre of the diamond. We give formulae for the total 4-volume $V$ of the diamond, the area $A$ of the intersection the future light cone of $p$ with the past light cone of $q$ and the 3-volume of the hyper-surface of largest 3-volume bounded by this intersection valid to ${\cal O} …
In this note we prove that the volume of a causal diamond associated with an inertial observer in asymptotically de Sitter 4-dimensional space-time is monotonically increasing function of cosmological time. The asymptotic value of the volume is that of in maximally symmetric de Sitter space-time. The monotonic property of the volume is checked in two cases: in vacuum and in the presence of a massless scalar field. In vacuum, the volume flow (with respect to cosmological time) asymptotically vanishes if and only if future space-like infinity is 3-manifold of constant curvature. The volume flow …
This dissertation investigates thermodynamic, emergent and holographic aspects of gravity in the context of causal diamonds. We obtain a gravitational first law for causal diamonds in maximally symmetric spacetimes and argue that these diamonds are in thermodynamic equilibrium at negative temperature. Further, gravitational field equations, including higher curvature corrections, are derived from an equilibrium condition on the generalized entropy of small maximally symmetric diamonds. Finally, we assign three holographic microscopic quantities to causal diamonds in spherically symmetric space…
In this work we introduce the taxicab and uniform products for Lorentzian pre-length spaces. We further use these concepts to endow the space $D(R\times_T X)$ of causal diamonds with a Lorentzian length space structure, closely relating its causal properties with its geometry as a metric space furnished with its associated Hausdorff distance. Among the general results, we show that this space is geodesic and globally hyperbolic for complete $X$.
We formulate certain inequalities for the geometric quantities characterizing causal diamonds in curved and Minkowski spacetimes. These inequalities involve the red-shift factor which, as we show explicitly in the spherically symmetric case, is monotonic in the radial direction and it takes its maximal value at the centre. As a byproduct of our discussion we re-derive Bishop's inequality without assuming the positivity of the spatial Ricci tensor. We then generalize our considerations to arbitrary, static and not necessarily spherically symmetric, asymptotically flat spacetimes. In the case of…
In AdS/CFT, the non-uniqueness of the reconstructed bulk from boundary subregions has motivated the notion of code subspaces. We present some closely related structures that arise in flat space. A useful organizing idea is that of an {\em asymptotic} causal diamond (ACD): a causal diamond attached to the conformal boundary of Minkowski space. The space of ACDs is defined by pairs of points, one each on the future and past null boundaries, ${\cal I}^{\pm}$. We observe that for flat space with an IR cut-off, this space (a) encodes a preferred class of boundary ``subregions'', (b) is a plausible …
In a previous paper we obtained formulae for the volume of a causal diamond or Alexandrov open set $I^+(p) \cap I^-(q)$ whose duration $τ(p,q) $ is short compared with the curvature scale. In the present paper we obtain asymptotic formulae valid when the point $q$ recedes to the future boundary ${\cal I}^+$ of an asymptotically de-Sitter spacetime. The volume (at fixed $τ$) remains finite in this limit and is given by the universal formula $V(τ) = {4\over 3}π(2\ln \cosh{τ\over 2}-\tanh^2{τ\over 2})$ plus corrections (given by a series in $e^{-t_q}$) which begin at order $e^{-4t_q}$. The coeffi…
We investigate the implications of dark matter annihilations for cosmological parameter constraints using the causal entropic principle. In this approach cosmologies are weighted by the total entropy production within a causally connected region of spacetime. We calculate the expected entropy from dark matter annihilations within the causal diamond and investigate the preferred values of the cosmological constant and the mass and annihilation cross section of the annihilating dark matter and their dependence on the assumptions in the models. For realistic values of the cross section we typical…
In his calculation of the spacetime volume of a small Alexandrov interval in 4 dimensions Myrheim introduced a term which he referred to as a surface integral [1]. The evaluation of this term has remained opaque and led subsequent authors to evaluate the volume using other techniques [2]. It is the purpose of this work to demystify this integral. We point out that it arises from the difference in the flat spacetime volumes of the curved and flat spacetime intervals. An explicit evaluation using first order degenerate perturbation theory shows that it adds a dimension independent factor to the …
We argue that the many-worlds of quantum mechanics and the many worlds of the multiverse are the same thing, and that the multiverse is necessary to give exact operational meaning to probabilistic predictions from quantum mechanics. Decoherence - the modern version of wave-function collapse - is subjective in that it depends on the choice of a set of unmonitored degrees of freedom, the "environment". In fact decoherence is absent in the complete description of any region larger than the future light-cone of a measurement event. However, if one restricts to the causal diamond - the largest regi…
We present a theory of accelerated observers in the formalism of holographic space time, and show how to define the analog of the Unruh effect for a one parameter set of accelerated observers in a causal diamond in Minkowski space. The key fact is that the formalism splits the degrees of freedom in a large causal diamond into particles and excitations on the horizon. The latter form a large heat bath for the particles, and different Hamiltonians, describing a one parameter family of accelerated trajectories, have different couplings to the bath. We argue that for a large but finite causal diam…
The causal entropic principle aims to predict the unexpectedly small value of the cosmological constant Lambda using a weighting by entropy increase on causal diamonds. The original work assumed a purely isotropic and homogeneous cosmology. But even the level of inhomogeneity observed in our universe forces reconsideration of certain arguments about entropy production. In particular, we must consider an ensemble of causal diamonds associated with one cosmology, and we can no longer immediately discard entropy production in the far future of the universe. Depending on our choices for a probabil…
We derive the Einstein equation from the condition that every small causal diamond is a variation of a flat empty diamond with the same free conformal energy, as would be expected for a near-equilibrium state. The attractiveness of gravity hinges on the negativity of the absolute temperature of these diamonds, a property we infer from the generalized entropy.
We complete an old argument that causal diamonds in the crunching region of the Lorentzian continuation of a Coleman-Deluccia instanton for transitions out of de Sitter space have finite area, and provide quantum models consistent with the principle of detailed balance, which can mimic the instanton transition probabilities for the cases where this diamond is larger or smaller than the causal patch of de Sitter space. We review arguments that potentials which do not have a positive energy theorem when the lowest de Sitter minimum is shifted to zero, may not correspond to real models of quantum…
We show that the Feynman propagator (or the d'Alembertian) of a causal set contains the complete information about the causal set. Intuitively, this is because the Feynman propagator, being a correlator that decays with distance, provides a measure for the invariant distance between pairs of events. Further, we show that even the spectra alone (of the self-adjoint and anti-self-adjoint parts) of the propagator(s) and d'Alembertian already carry large amounts of geometric information about their causal set. This geometric information is basis independent and also gauge invariant in the sense th…
We discuss issues relating to the topology of Euclidean de Sitter space. We show that in (2+1) dimensions, the Euclidean continuation of the`causal diamond', i.e the region of spacetime accessible to a timelike observer, is a three-hemisphere. However, when de Sitter entropy is computed in a `stretched horizon' picture, then we argue that the correct Euclidean topology is a solid torus. The solid torus shrinks and degenerates into a three-hemisphere as one goes from the `stretched horizon' to the horizon, giving the Euclidean continuation of the causal diamond. We finally comment on the genera…
We show that the simplicity constraints, which define the dynamics of spin foam models, imply, and are implied by, the first law of thermodynamics, when the latter is applied to causal diamonds in the quantum spacetime. This result reveals an intimate connection between the holographic nature of gravity, as reflected by the Bekenstein entropy, and the fact that general relativity and other gravitational theories can be understood as constrained topological field theories. To state and derive this correspondence we describe causal diamonds in the causal structure of spin foam histories and gene…
We note that the observable part of universe at a certain time t_P is necessarily limited, when there is a beginning of universe. We argue that an appropriate spacetime region associated with an observer from tI to t_P is the causal diamond which is the overlap of the past/future of the observer at t_P/t_I respectively. We also note that the overlap surface \partial D of the future and the past lightcones bisects the spatial section including \partial D into two regions D and \bar D where D is the region inside the causal diamond and \bar D the remaining part of the spatial section. We propose…
Motivated by recent work suggesting observably large spacetime fluctuations in the causal development of an empty region of flat space, we conjecture that these metric fluctuations can be quantitatively described in terms of a conformal field theory of near-horizon vacuum states. One consequence of this conjecture is that fluctuations in the modular Hamiltonian $ΔK$ of a causal diamond are equal to the entanglement entropy: $\langle ΔK^2 \rangle = \langle K \rangle = \frac{A(Σ_{d-2})}{4 G_d}$, where $A(Σ_{d-2})$ is the area of the entangling surface in $d$ dimensions. Our conjecture applies to…
One of the central difficulties of settling the $L^2$-bounded curvature conjecture for the Einstein -Vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity. In this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of Enstein-Vacuum spacetimes can be controlled in terms of initial data and the total curvature flux through the hypersurface.
