High and Low Dimensions in The Black Hole Negative Mode

The negative mode of the Schwarzschild black hole is central to Euclidean quantum gravity around hot flat space and for the Gregory-Laflamme black string instability. We analyze the eigenvalue as a function of space-time dimension by constructing two perturbative expansions: one for large d and the other for small d-3, and determining as many coefficients as we are able to compute analytically. Joining the two expansions we obtain an interpolating rational function accurate to better than 2% through the whole range of dimensions including d=4.Comment: 17 pages, 4 figures. v2: added reference. v3: published versio

Matched Asymptotic Expansion for Caged Black Holes - Regularization of the Post-Newtonian Order

The "dialogue of multipoles" matched asymptotic expansion for small black holes in the presence of compact dimensions is extended to the Post-Newtonian order for arbitrary dimensions. Divergences are identified and are regularized through the matching constants, a method valid to all orders and known as Hadamard's partie finie. It is closely related to "subtraction of self-interaction" and shows similarities with the regularization of quantum field theories. The black hole's mass and tension (and the "black hole Archimedes effect") are obtained explicitly at this order, and a Newtonian derivation for the leading term in the tension is demonstrated. Implications for the phase diagram are analyzed, finding agreement with numerical results and extrapolation shows hints for Sorkin's critical dimension - a dimension where the transition turns second order.Comment: 28 pages, 5 figures. v2:published versio

The Final State of Black Strings and p-Branes, and the Gregory-Laflamme Instability

It is shown that the usual entropy argument for the Gregory-Laflamme (GL) instability for $some$ appropriate black strings and $p$-branes gives surprising agreement up to a few percent. This may provide a strong support to the GL's horizon fragmentation, which would produce the array of higher-dimensional Schwarzschild-type's black holes finally. On the other hand, another estimator for the size of the black hole end-state relative to the compact dimension indicates a second order (i.e., smooth) phase transition for some $other$ appropriate compactifications and total dimension of spacetime wherein the entropy argument is not appropriate. In this case, Horowitz-Maeda-type's non-uniform black strings or $p$-branes can be the final state of the GL instability.Comment: More emphasis on a second order phase transition. The computation result is unchange

Holography for N=1∗\mathcal{N}=1^* on S4S^4

We construct the five-dimensional supergravity dual of the $\mathcal{N}=1^*$ mass deformation of the $\mathcal{N} =4$ supersymmetric Yang-Mills theory on $S^4$ and use it to calculate the universal contribution to the corresponding $S^4$ free energy at large 't Hooft coupling in the planar limit. The holographic RG flow solutions are smooth and preserve four supercharges. As a novel feature compared to the holographic duals of $\mathcal{N} = 1^*$ on $\mathbb{R}^4$, in our backgrounds the five-dimensional dilaton has a non-trivial profile, and the gaugino condensate is fixed in terms of the mass-deformation parameters. Important aspects of the analysis involve characterizing the ambiguities in the partition function of non-conformal $\mathcal{N}=1$ supersymmetric theories on $S^4$ as well as the action of S-duality on the $\mathcal{N}=1^*$ theory.Comment: 40 pages + appendices, 4 figure

On Black-Brane Instability In an Arbitrary Dimension

The black-hole black-string system is known to exhibit critical dimensions and therefore it is interesting to vary the spacetime dimension $D$, treating it as a parameter of the system. We derive the large $D$ asymptotics of the critical, i.e. marginally stable, string following an earlier numerical analysis. For a background with an arbitrary compactification manifold we give an expression for the critical mass of a corresponding black brane. This expression is completely explicit for ${\bf T}^n$, the $n$ dimensional torus of an arbitrary shape. An indication is given that by employing a higher dimensional torus, rather than a single compact dimension, the total critical dimension above which the nature of the black-brane black-hole phase transition changes from sudden to smooth could be as low as $D\leq 11$.Comment: 1+14 pages, 2 eps figures. Replaced with the published versio

On non-uniform smeared black branes

We investigate charged dilatonic black $p$-branes smeared on a transverse circle. The system can be reduced to neutral vacuum black branes, and we perform static perturbations for the reduced system to construct non-uniform solutions. At each order a single master equation is derived, and the Gregory-Laflamme critical wavelength is determined. Based on the non-uniform solutions, we discuss thermodynamic properties of this system and argue that in a microcanonical ensemble the non-uniform smeared branes are entropically disfavored even near the extremality, if the spacetime dimension is $D \le 13 +p$, which is the critical dimension for the vacuum case. However, the critical dimension is not universal. In a canonical ensemble the vacuum non-uniform black branes are thermodynamically favorable at $D > 12+p$, whereas the non-uniform smeared branes are favorable at $D > 14+p$ near the extremality.Comment: 24 pages, 2 figures; v2: typos corrected, submitted to Class.Quant.Gra

Post-ISCO Ringdown Amplitudes in Extreme Mass Ratio Inspiral

An extreme mass ratio inspiral consists of two parts: adiabatic inspiral and plunge. The plunge trajectory from the innermost stable circular orbit (ISCO) is special (somewhat independent of initial conditions). We write an expression for its solution in closed-form and for the emitted waveform. In particular we extract an expression for the associated black-hole ringdown amplitudes, and evaluate them numerically.Comment: 21 pages, 5 figures. v4: added section with numerical evaluation of the ringdown amplitude

From Black Strings to Black Holes

Using recently developed numerical methods, we examine neutral compactified non-uniform black strings which connect to the Gregory-Laflamme critical point. By studying the geometry of the horizon we give evidence that this branch of solutions may connect to the black hole solutions, as conjectured by Kol. We find the geometry of the topology changing solution is likely to be nakedly singular at the point where the horizon radius is zero. We show that these solutions can all be expressed in the coordinate system discussed by Harmark and Obers.Comment: 6 pages, 5 figures, RevTe

Classical Effective Field Theory for Weak Ultra Relativistic Scattering

Inspired by the problem of Planckian scattering we describe a classical effective field theory for weak ultra relativistic scattering in which field propagation is instantaneous and transverse and the particles' equations of motion localize to the instant of passing. An analogy with the non-relativistic (post-Newtonian) approximation is stressed. The small parameter is identified and power counting rules are established. The theory is applied to reproduce the leading scattering angle for either a scalar interaction field or electro-magnetic or gravitational; to compute some subleading corrections, including the interaction duration; and to allow for non-zero masses. For the gravitational case we present an appropriate decomposition of the gravitational field onto the transverse plane together with its whole non-linear action. On the way we touch upon the relation with the eikonal approximation, some evidence for censorship of quantum gravity, and an algebraic ring structure on 2d Minkowski spacetime.Comment: 29 pages, 2 figures. v4: Duration of interaction is determined in Sec 4 and detailed in App C. Version accepted for publication in JHE

Static Axisymmetric Vacuum Solutions and Non-Uniform Black Strings

We describe new numerical methods to solve the static axisymmetric vacuum Einstein equations in more than four dimensions. As an illustration, we study the compactified non-uniform black string phase connected to the uniform strings at the Gregory-Laflamme critical point. We compute solutions with a ratio of maximum to minimum horizon radius up to nine. For a fixed compactification radius, the mass of these solutions is larger than the mass of the classically unstable uniform strings. Thus they cannot be the end state of the instability.Comment: 48 pages, 13 colour figures; v2: references correcte