In this and the two following papers II and III we study the axisymmetric collision of two black holes at the speed of light, with a view to understanding the more realistic collision of two black holes with a large but finite incoming Lorentz factor y. The curved radiative region of the space-time, produced after the two incoming impulsive plane-fronted shock waves have collided, is treated using perturbation theory, following earlier work by Curtis and Chapman. The collision is viewed in a frame to which a large Lorentz boost has been applied, giving a strong shock with energy v off which a weak shock with energy A. && v scatters. This yields a singular perturbation problem, in which the Einstein field equations are solved by expanding in powers of A, /v around flat space-time. When viewed back in the center-ofmass frame, this gives a good description of the regions of the space-time in which gravitational radiation propagates at small angles L9 but a large distance from the symmetry axis, near each shock as it continues to propagate, having been 'distorted and deflected in the initial collision. The news function co(v, 8) describing the gravitational radiation is expected to have a convergent series expansion co(r, 9) = g"" Oa, "(r)sin "8, where r" is a retarded time coordinate. First-order perturbation theory gives an expression for ao(~) in agreement with that found previously by studying the finite-y collisions.Second-order perturbation theory gives a&( f) as a complicated integral expression. A new mass-loss formula is derived, which shows that if the end result of the collision is a single Schwarzschild black hole at rest, plus gravitational radiation which is (in a certain precise sense) accurately described by the above series for co(~, 8), then the final mass can be determined from knowledge only of ao(~) and a, (~). This leads to an interesting test of the cosmic censorship hypothesis. The numerical calculation of a, (~) is made practicable by analytical simplifications described in the following paper II, where the perturbative field equations are reduced to a system in only two independent variables. Results are presented in the concluding paper III, which discusses the implications for the energy emitted and the nature of the radiative space-time.PACS number(s): 04.30.+x, 97.60.Lf

This paper summarizes results following from the two preceding papers, I and 11, on the gravitational radiation emitted in the head-on collision of two black holes, each with energy p, at or near the speed of light. The radiation (in the speed-of-light case) near the forward and backward directions 8=0, a, where 8 is the angle from the symmetry axis in the center-of-mass frame, is given by the series A co( ?, 0 )=2,7=0a2n ( r^/p )sin2"8 for the news function co of retarded time ? and angle 8; successive terms can in principle be found from a perturbation treatment. Here the form of a,(r^/p) is presented. Knowledge of a, allows the new mass-loss formula of paper I to be applied, giving a calculation of the mass of the (assumed) final Schwarzschild black hole. Since the "final mass" resulting from the calculation exceeds 2p, the assumptions of the new mass-loss formula must not all hold. The most likely explanation is that there is a "second burst" of radiation present in the space-time, centered for small angles 6 on retarded times roughly 18p ln6( later than the "first burst" described above. A more realistic crude estimate of the energy emitted in gravitational waves is given by the Bondi expression, taking only the first two terms a . and a2 in co; this gives an efficiency of 16.4% for gravitational wave generation.PACS number(s): 04.30. + x, 97.60.Lf

We show that Smarr's calculation of the zero-frequency limit of the energy spectrum for the gravitational radiation produced during the scattering or collision of two particles is a linearized approximation valid only when the radiation is weak. In particular, it cannot be applied to the head-on collision of two black holes, so Smarr's conclusions concerning the isotropy of the radiation emitted during the high-speed, equal-mass, black-hole collision have no firm foundation.

Herd immunity, a process in which resistant individuals limit the spread of a pathogen among susceptible hosts has been extensively studied in eukaryotes. Even though bacteria have evolved multiple immune systems against their phage pathogens, herd immunity in bacteria remains unexplored. Here we experimentally demonstrate that herd immunity arises during phage epidemics in structured and unstructured Escherichia coli populations consisting of differing frequencies of susceptible and resistant cells harboring CRISPR immunity. In addition, we develop a mathematical model that quantifies how herd immunity is affected by spatial population structure, bacterial growth rate, and phage replication rate. Using our model we infer a general epidemiological rule describing the relative speed of an epidemic in partially resistant spatially structured populations. Our experimental and theoretical findings indicate that herd immunity may be important in bacterial communities, allowing for stable coexistence of bacteria and their phages and the maintenance of polymorphism in bacterial immunity.