Horizon Theorem Research Articles

Insights and guidelines on the Cauchy horizon theorems

Insights and guidelines on the Cauchy horizon theorems

No scalar-haired Cauchy horizon theorem in charged Gauss–Bonnet black holes

Recently, a “no inner (Cauchy) horizon theorem” for static black holes with non-trivial scalar hairs has been proved in Einstein–Maxwell–scalar theories and also in Einstein–Maxwell–Horndeski theories with the non-minimal coupling of a charged (complex) scalar field to Einstein tensor. In this paper, we study an extension of the theorem to the static black holes in Einstein–Maxwell–Gauss–Bonnet-scalar theories, or simply, charged Gauss–Bonnet (GB) black holes. We find that no inner horizon with charged scalar hairs is allowed for the planar (k=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=0$$\\end{document}) black holes, as in the case without GB term. On the other hand, for the non-planar (k=±1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=\\pm 1$$\\end{document}) black holes, we find that the haired inner horizon can not be excluded due to GB effect generally, though we can not find a simple condition for its existence. As some explicit examples of the theorem, we study numerical GB black hole solutions with charged scalar hairs and Cauchy horizons in asymptotically anti-de Sitter space, and find good agreements with the theorem. Additionally, in an Appendix, we prove a “no-go theorem” for charged de Sitter black holes (with or without GB terms) with charged scalar hairs in arbitrary dimensions.

No scalar-haired Cauchy horizon theorem in Einstein-Maxwell-Horndeski theories

Recently, a no inner (Cauchy) horizon theorem for static black holes with non-trivial scalar hairs has been proved in Einstein-Maxwell-scalar theories. In this paper, we study an extension of the theorem to the static black holes in Einstein-Maxwell-Horndeski theories. We study the black hole interior geometry for some exact solutions and find that the spacetime has a (space-like) curvature singularity where the black hole mass gets an extremum and the Hawking temperature vanishes. We discuss further extensions of the theorem, including general Horndeski theories from disformal transformations.

No Cauchy horizon theorem for nonlinear electrodynamics black holes with charged scalar hairs

We prove a no Cauchy horizon theorem for general nonlinear electrodynamics black holes with charged scalar hairs. By constructing a radially conserved charged, we show that there is no inner Cauchy horizon for both spherical and planar symmetric cases, independent of the form of scalar potential and nonlinear electrodynamics. After imposing the null energy condition, we are also able to rule out the existence of the Cauchy horizon for the hyperbolic black holes. We take the Born-Infeld black hole as a concrete example to study the interior dynamics beyond the event horizon. When the contribution from the scalar potential can be neglected, the asymptotic near-singularity takes a universal Kasner form. We also confirm that the intricate interior dynamics is closely associated with the instability of the inner Cauchy horizon triggered by scalar hairs.

Dynamical black holes in scalar-tensor theories

As the simplest generalization of Einstein's equations of gravitation, scalar-tensor theories have turned up a number of surprising results. Such results are the remarkable black hole properties discovered by Scheel, Shapiro and Teukolsky. In scalar-tensor theories there is a dynamical epoch where the surface area of the event horizon decreases over time and the apparent horizon which in general relativity always is confined to the interior of the black hole passes the event horizon and becomes timelike. No such processes occur in general relativity. We investigate these intriguing results further in a scalar-tensor theory with two interacting scalar fields. The scalar fields are massless and they are endowed in a space spanned by scalar fields which is described by a sigma metric. The scalar fields are coupled to collisionless matter with the problem having spherical symmetry and it is treated fully numerical. The spacetime location of the black hole horizons is calculated. The problem of gravitational collapse of matter to a black hole in a scalar-tensor theory with one scalar field and a two parameter scalar dependent coupling function (linear in the scalar field) is discussed. We show that this theory admits values for the coupling parameters where, in addition to the well-known violation of the apparent horizon theorem and the area theorem of black holes in general relativity, a further violation of these theorems occurs.

“Faster than light” photons in dilaton black hole spacetimes

We investigate the phenomenon of "faster than light" photons in a family of dilaton black hole spacetimes. For radially directed photons, we find that their light-cone condition is modified even though the spacetimes are spherically symmetric. They also satisfy the "horizon theorem" and the "polarization sum rule" of Shore. For orbital photons, the dilatonic effect on the modification of the light-cone condition can become more dominant than the electromagnetic and the gravitational ones as the orbit gets closer to the event horizon in the extremal or near-extremal cases.

‘Faster than light’ photons in gravitational fields — Causality, anomalies and horizons

A number of general issues relating to superluminal photon propagation in gravitational fields are explored. The possibility of superluminal, yet causal, photon propagation arises because of Equivalence Principle violating interactions induced by vacuum polarisation in QED in curved spacetime. Two general theorems are presented: first, a polarisation sum rule which relates the polarisation averaged velocity shift to the matter energy-momentum tensor and second, a ‘horizon theorem’ which ensures that the geometric event horizon for black hole spacetimes remains a true horizon for real photon propagation. These results are consequences of an effective action which in QED is valid only for low frequency photons. Their relevance to signal propagation and causality, which are controlled by high frequency propagation, is dependent on the dispersive properties of the modified propagation. This will be discussed elsewhere. A comparison is made with the equivalent results for electromagnetic birefringence and possible connections between superluminal photon propagation, causality and the conformal anomaly are exposed.

“Faster than light” photons and rotating black holes

The effective action for QED in curved spacetime includes equivalence principle violating interactions between the electromagnetic field and the spacetime curvature. These interactions admit the possibility of superluminal yet causal photon propagation in gravitational fields. In this paper, we extend our analysis of photon propagation in gravitational backgrounds to the Kerr spacetime describing a rotating black hole. The results support two general theorems — a polarisation sum rule and a “horizon theorem”. The implications for the stationary limit surface bounding the ergosphere are also discussed.

MARKOV DECISION PROCESSES WITH RANDOM HORIZON

In this paper we formulate Markov Decision Processes with Random Horizon (MDPRH). We show the optimality equation for the MDPRH, however there may not exist optimal stationary strategies, or ε-optimal stationary strategies for the processes. When the MDPRH has the probability distribution for the planning horizon with infinite support, we show Turnpike Planning Horizon Theorem. Then we evaluate rolling strategies and develop an algorithm obtaining an optimal first stage decision. Finally, some numerical experiments on a simple inventory model are done to understand the phenomena.

Error bound for the dynamic lot size model allowing speculative motive

This paper studies the dynamic lot size model with speculative motive for holding inventory being allowed, where speculative motive means that producing a unit in a period and holding it to a future period may be cheaper than producing it at the later time. In a rolling horizon environment, one usually makes decisions for the first few periods based on the forecast data for some fixed time horizon. This fixed time horizon may not be long enough, in the sense that information beyond that time horizon may affect decisions in the first few periods. Hence it is desirable to find a worst-case bound on the error induced by imposing a finite time horizon on the model. This paper shows how to adapt recently developed O(TlogT) algorithms for the model to find an attainable bound. Furthermore, a planning horizon theorem that is more general than that in the literature is also provided.

Using geometric techniques to improve dynamic programming algorithms for the economic lot-sizing problem and extensions

In this paper we discuss two basic geometric techniques that can be used to speed up certain types of dynamic programs. We first present the algorithms in a general form, and then we show how these techniques can be applied to the economic lot-sizing problem and extensions. Furthermore, it is illustrated that the geometric techniques can be used to give elegant and insightful proofs of structural results, like Wagner and Whitin's planning horizon theorem. Finally, we present results of computational experiments in which new algorithms for the economic lot-sizing problem are compared with each other, as well as with other algorithms from the literature.

ROLLING SCHEDULES FOR A DYNAMIC LOT-SIZING PROBLEM WITH START-UP COST

This paper considers a dynamic lot size model with start-up cost where backlogging is not allowed. The model incorporates inventory holding cost and two kinds of fixed costs—a start-up cost incurred whenever the machine (production facility) is switched from “off” to “on”, and a reservation cost incurred in each production period. The optimal (cost-minimizing) solution over a finite horizon is found in a forward dynamic programming approach, upon which a planning horizon theorem is derived to treat the model in a rolling horizon environment. Accordingly, two procedures for selecting the run length of the first production block and the amount to meet the associated production requirement are developed. The cost effectivenesses of the procedures are demonstrated by a set of simulation experiments.

A data-dependent efficient implementaton of the Wagner-Whitin algorithm for lot-sizing

The Wagner-Whitin algorithm is a well-known optimal procedure for determining production quantities X t , over the planning horizon ( N periods) when demands D t , setup cost S t , production costs C t and holding costs H t are known. Evans [1] presented a computer code and showed that W-W algorithm takes only a few seconds on the computer even for a large problem. We modify Evans code by incorporating Wagner's setup cost horizon theorem and show empirically that the new code is faster by a factor of N/4 in the best case, but is slower only by 1–2% in the worst case.

Approximating an Infinite Stage Search Problem with a Finite Horizon Model

A target is moving randomly on a lattice. A searcher, starting at the origin, wants to follow a search plan that has minimum expected cost. Very general cost functions are allowed. The cost function depends on the time to capture and the search procedure followed. At each stage, the searcher can move only to adjacent positions or remain at his current position. The target's motion may depend on the search procedure, thus allowing for the possibility that the target may take evasive action. It is shown that only a finite number of plans need to be considered in order to approximate the optimal cost. It is also shown that the leading elements of optimal finite horizon plans eventually become the leading elements of optimal infinite horizon plans. The case of search on the real line, where the cost is O(τk), where k ≥ 1 and τ is the time to capture, is considered in detail. The results of this paper also generalise the important planning horizon theorems of Bean and Smith (Bean, J. C., Smith, R. L. 1984. Conditions for the existence of planning horizons. Math. Oper. Res. 9 391–401.).

The Assortment Problem with Nonlinear Cost Functions

The early work done on the assortment problem assumed a linear production cost and a substitution cost based only on the difference in value of the item used and the item required. The research reported in this paper extends the analysis by considering concave production cost functions and substitution cost functions that add either a unit fixed cost or a unit-independent fixed charge to the previously considered “scrap” cost. The form of an optimal stocking policy is developed for each case and horizon theorems are proved to aid in the computations. Algorithms for finding optimal policies are described.

Note—Planning and Forecast Horizons for the Bond Refunding Problem

This paper discusses the planning horizon issues of the dynamic bond refunding problem, when interest, floatation and call costs are changing arbitrarily over time and are deterministic. A planning horizon theorem is proved, enabling us to identify planning and forecasting horizons and thus reducing information requirements and increasing computational efficiency.

Note—A Dominance Relation in Dynamic Lot Sizing

A computationally efficient forward algorithm for the Economic Lot Size Model is developed. It uses a Dominance Relation and the Ordering Cost to limit the solution space. The Dominance Relation developed includes the standard planning horizon theorem. Extensions for price breaks and monotonically nonincreasing production costs are presented.

An Extension to Modigliani and Hohn's Planning Horizons Results

This paper deals with the planning horizon issues of the problem of finding the optimal production schedule over a [0, T] period, for a product having: (1) deterministic demand, (2) strictly convex production costs, and (3) linear holding and back-logging costs. A set of solutions defined as extrapolations is defined, and a property of nonintersection of extrapolations is proved. Using this property uniqueness and planning horizon theorems are proved. It is shown that the reason for the planning horizons is the jump (discontinuity) in the derivative of the inventory cost function when inventory is zero. In addition, the paper deals with the following question: What kinds of changes (variations) in demand may occur such that the optimal solution (production schedule) on some time interval will remain unchanged? This question is associated with the following one: What information is needed about demand in order that the optimal solution on some time interval may be found?

Planning Horizons for Production Smoothing with Deterministic Demands

In Part I of this paper, we develop an algorithm for finding planning horizons for the deterministic production smoothing problem when all demand must be met from regular production, under rather general assumptions for the production, production smoothing, and holding cost functions. (In Part II, planning horizons will be developed when the model is extended to include backlogging and overtime.) The techniques developed here are essentially forward-looking and marginal-cost-balancing in nature, rather than total-cost-minimizing and backward-looking such as dynamic programming, or total-cost-minimizing and omni-looking in nature such as linear programming. The fact that the planning horizon theorems do not depend on having discount rates less than one illustrates that the approach developed here is fundamentally different from ordinary infinite horizon dynamic programming techniques. The algorithm is “user oriented” in the sense that only a small amount of forecasting work and computation ordinarily must be done to determine the horizon; the nature of the algorithm also makes the exact dependence of the horizon on the forecast clear for sensitivity analysis. Firms facing a seasonal demand pattern will often find that the horizon occurs within the first several periods after the peak period if there is a sufficient enough drop in demand afterwards. This result complements the findings of Modigliani and Hohn and provides insight into the nature of the optimal policy for stochastic planning problems.

Infinite Horizon Programs

Several difficulties arise when multistage optimization problems with linear constraints are extended to an infinite horizon. For linear objectives the weak duality theorem fails, duality gaps exist, complementary slackness is not a sufficient condition for optimality, and the total reward may diverge. This paper presents verifiable conditions on the problem that guarantee the existence of optimal solutions and equilibrium prices for linear and nonlinear objectives. In addition, it is ascertained that the Kuhn-Tucker conditions are sufficient for optimality if additional restrictions are imposed. If the constraints are Leontief and the objective is convex (in a maximization problem), then the existence of an extreme point optimal solution can be verified. The decision taken in each period is an extreme point of the set of solutions that are feasible in that period. The Leontief case also yields a horizon theorem. All sufficiently long finite horizon problems produce a first decision that is optimal for the infinite horizon problem.