We derive the asymptotic solutions for vacuum spacetimes with non-zero cosmological constant Λ, using the Newman-Penrose formalism. Our approach is based exclusively on the physical spacetime, i.e. we do not explicitly deal with conformal rescaling nor the conformal spacetime. By investigating the Schwarzschild-de Sitter spacetime in spherical coordinates, we subsequently stipulate the fall-offs of the null tetrad and spin coefficients for asymptotically de Sitter spacetimes such that the terms which would give rise to the Bondi mass-loss due to energy carried by gravitational radiation (i.e. involving σ o ) must be non-zero. After solving the vacuum Newman-Penrose equations asymptotically, we propose a generalisation to the Bondi mass involving Λ and obtain a positive-definite mass-loss formula by integrating the Bianchi identity involving D Ψ 2 over a compact 2-surface on I.Whilst our original intention was to study asymptotically de Sitter spacetimes, the use of spherical coordinates implies that this readily applies for Λ < 0, and yields exactly the known asymptotically flat spacetimes when Λ = 0. In other words, our asymptotic vacuum solutions with Λ = 0 reduce smoothly to those where Λ = 0, in spite of the distinct characters of I being spacelike, timelike and null for de Sitter, anti-de Sitter and Minkowski, respectively. Unlike for Λ = 0 where no incoming radiation corresponds to setting Ψ o 0 = 0 on some initial null hypersurface, for Λ = 0, no incoming radiation requires Ψ o 0 = 0 everywhere. * VeeLiem@maths.otago.ac.nz arXiv:1605.05151v3 [gr-qc]

Bus bunching is a perennial phenomenon that not only diminishes the efficiency of a bus system, but also prevents transit authorities from keeping buses on schedule. We present a physical theory of buses serving a loop of bus stops as a ring of coupled self-oscillators, analogous to the Kuramoto model. Sustained bunching is a repercussion of the process of phase synchronisation whereby the phases of the oscillators are locked to each other. This emerges when demand exceeds a critical threshold. Buses also bunch at low demand, albeit temporarily, due to frequency detuning arising from different human drivers’ distinct natural speeds. We calculate the critical transition when complete phase locking (full synchronisation) occurs for the bus system, and posit the critical transition to completely no phase locking (zero synchronisation). The intermediate regime is the phase where clusters of partially phase locked buses exist. Intriguingly, these theoretical results are in close correspondence to real buses in a university’s shuttle bus system.

We investigate a no-boarding policy in a system of N buses serving M bus stops in a loop, which is an entrainment mechanism to keep buses synchronised in a reasonably staggered configuration.Buses always allow alighting, but would disallow boarding if certain criteria are met. We let buses move with the same natural speed (applicable to programmable self-driving buses) and analytically calculate the average waiting time experienced by passengers waiting at the bus stop for a bus to arrive. Our analytical results show that a no-boarding policy can dramatically reduce the average waiting time, as compared to the usual situation without the no-boarding policy. Subsequently, we carry out simulations to verify these theoretical analyses, also extending the simulations to typical human-driven buses with different natural speeds based on real data. Finally, we describe a simple general adaptive algorithm to dynamically determine when to implement no-boarding.

We present the asymptotic solutions for spacetimes with non-zero cosmological constant Λ coupled to Maxwell fields, using the Newman-Penrose formalism. This extends a recent work that dealt with the vacuum Einstein (Newman-Penrose) equations with Λ = 0. The results are given in two different null tetrads: the Newman-Unti and Szabados-Tod null tetrads, where the peeling property is exhibited in the former but not the latter. Using these asymptotic solutions, we discuss the mass-loss of an isolated electro-gravitating system with cosmological constant. In a universe with Λ > 0, the physics of electromagnetic (EM) radiation is relatively straightforward compared to those of gravitational radiation: 1) It is clear that outgoing EM radiation results in a decrease to the Bondi mass of the isolated system. 2) It is also perspicuous that if any incoming EM radiation from elsewhere is present, those beyond the isolated system's cosmological horizon would eventually arrive at the spacelike I and increase the Bondi mass of the isolated system. Hence, the (outgoing and incoming) EM radiation fields do not couple with Λ in the Bondi mass-loss formula in an unusual manner, unlike the gravitational counterpart where outgoing gravitational radiation induces nonconformal flatness of I. These asymptotic solutions to the Einstein-Maxwell-de Sitter equations presented here may be used to extend a raft of existing results based on Newman-Unti's asymptotic solutions to the Einstein-Maxwell equations where Λ = 0, to now incorporate the cosmological constant Λ. * VeeLiem@maths.otago.ac.nz

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