We study the thermodynamics of small black holes in compactified spacetimes of the form R d−1 × S 1 . This system is analyzed with the aid of an effective field theory (EFT) formalism in which the structure of the black hole is encoded in the coefficients of operators in an effective worldline Lagrangian. In this effective theory, there is a small parameter λ that characterizes the corrections to the thermodynamics due to both the non-linear nature of the gravitational action as well as effects arising from the finite size of the black hole. Using the power counting of the EFT we show that the series expansion for the thermodynamic variables contains terms that are analytic in λ, as well as certain fractional powers that can be attributed to finite size operators. In particular our operator analysis shows that existing analytical results do not probe effects coming from horizon deformation. As an example, we work out the order λ 2 corrections to the thermodynamics of small black holes for arbitrary d, generalizing the results in the literature.

This work was mainly driven by the desire to explore, to what extent embedding some given geometry in a higher dimensional flat one is useful for understanding the causal structure of classical fields traveling in the former, in terms of that in the latter. We point out, in the 4-dimensional (4D) spatially flat Friedmann-Lemaître-RobertsonWalker universe, that the causal structure of transverse-traceless (TT) gravitational waves can be elucidated by first reducing the problem to a 2D Minkowski wave equation with a time dependent potential, where the relevant Green's function is pure tail -waves produced by a physical source propagate strictly within the null cone. By viewing this 2D world as embedded in a 4D one, the 2D Green's function can also be seen to be sourced by a cylindrically symmetric scalar field in 3D. From both the 2D wave equation as well as the 3D scalar perspective, we recover the exact solution of the 4D graviton tail, for the case where the scale factor written in conformal time is a power law. There are no TT gravitational wave tails when the universe is radiation dominated because the background Ricci scalar is zero. In a matter dominated one, we estimate the amplitude of the tail to be suppressed relative to its null counterpart by both the ratio of the duration of the (isolated) source to the age of the universe η 0 , and the ratio of the observer-source spatial distance (at the observer's time) to the same η 0 . In a universe driven primarily by a cosmological constant, the tail contribution to the background geometry a [η] 2 η µν after the source has ceased, is the conformal factor a 2 times a spacetime-constant symmetric matrix proportional to the spacetime volume integral of the TT part of the source's stress-energy-momentum tensor. In other words, massless spin-2 gravitational waves exhibit a tail-induced memory effect in 4D de Sitter spacetime. The geometry of our universe appears to be well described by Einstein's equations with a cosmological constant Λ = 3H 2 ,At zeroth order and at very large scales, T µ ν = T µ ν contains an isotropic and homogeneous background matter-energy distribution that drives the evolution of a 4-dimensional (4D) spatially flat FriedmannLemaître-Robertson-Walker (FLRW) geometry parametrized by conformal time η and 3 spatial coordinates x, i.e.,ḡAt first order, T to describe the finer structure present in the universe -clumping of dark matter, for instance -once cosmologists try to probe it at smaller scales and higher resolution. These perturbations will also produce inhomogeneities in the metric, so that nowIt is possible to perform a scalar-vector-tensor decomposition of both the matter δT α β and metric h αβ fluctuations, such that at linear order in these fields, eq. (1) would yield separate partial differential equations (PDEs) for each of the perturbations transforming differently under the rotation group SO 3 , an isometry group of the background 4D FLRW geometry in eq. (2). 1 This paper is specifically about understanding the causal stru...

Motivated by the desire to test modified gravity theories exhibiting the Vainshtein mechanism, we solve in various physically relevant limits, the retarded Galileon Green's function (for the cubic theory) about a background sourced by a massive spherically symmetric static body. The static limit of our result will aid us, in a forthcoming paper, in understanding the impact of Galileon fields on the problem of motion in the solar system. In this paper, we employ this retarded Green's function to investigate the emission of Galileon radiation generated by the motion of matter lying deep within the Vainshtein radius r v of the central object: acoustic waves vibrating on its surface, and the motion of compact bodies gravitationally bound to it. If λ is the typical wavelength of the emitted radiation, and r 0 is the typical distance of the source from the central mass, with r 0 r v , then, compared to its non-interacting massless scalar counterpart, we find that the Galileon radiation rate is suppressed by the ratio (r v /λ) −3/2 at the monopole and dipole orders at high frequencies r v /λ 1. However, at high enough multipole order, the radiation rate is enhanced by powers of r v /r 0 . At low frequencies r v /λ 1, and when the motion is non-relativistic, Galileon waves yield a comparable rate for the monopole and dipole terms, and are amplified by powers of the ratio r v /r 0 for the higher multipoles.

We argue that massless gravitons in all even dimensional de Sitter (dS) spacetimes higher than two admit a linear memory effect arising from their propagation inside the null cone. Assume that gravitational waves (GWs) are being generated by an isolated source, and over only a finite period of time η i ≤ η ≤ η f . Outside of this time interval, suppose the shear-stress of the GW source becomes negligible relative to its energy-momentum and its mass quadrupole moments settle to static values. We then demonstrate, the transverse-traceless (TT) GW contribution to the perturbation of any dS 4+2n written in a conformally flat form (a 2 η µν dx µ dx ν ) -after the source has ceased and the primary GW train has passed -amounts to a spacetime constant shift in the flat metric proportional to the difference between the TT parts of the source's final and initial mass quadrupole moments. As a byproduct, we present solutions to Einstein's equations linearized about de Sitter backgrounds of all dimensions greater than three. We then point out there is a similar but approximate tail induced linear GW memory effect in 4D matter dominated universes. Our work here serves to improve upon and extend the

Motivated by the desire to understand the causal structure of physical signals produced in curved spacetimesparticularly around black holes -we show how, for certain classes of geometries, one might obtain its retarded or advanced minimally coupled massless scalar Green's function by using the corresponding Green's functions in the higher dimensional Minkowski spacetime where it is embedded. Analogous statements hold for certain classes of curved Riemannian spaces, with positive definite metrics, which may be embedded in higher dimensional