Multidimensional gravitational waves. I. Purely radiative spacetimes

The concept of gravitational radiation in multidimensional Einstein gravity is studied. Due to technical difficulties the Petrov classification of the conformal tensor and the peeling off theorem cannot be generalized to spaces of dimensiond > 4. A multidimensionalpp wave is defined by analogy with that ford = 4. If the definition is supplemented by the Lichnerowicz condition for purely radiative spacetimes, the admitted solutions of the dimensionally reduced gravity theory describe a scalar (dilaton) and a gravitationalpp wave moving in the external spacetime in the same direction. The existence of the extra dimensions can be traced back by measuring the frequence of thepp wave in the external space. The condition is equally restrictive in a more general case of the internal space being a flat anisotropic manifold (i.e. an abelian group space).     

Complex singularity of a stokes wave

Two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth can be described by a conformal map of the fluid domain into the complex lower half-plane. Stokes wave is the fully nonlinear gravity wave propagating with the constant velocity. The increase in the scaled wave height H/λ from the linear limit H/λ = 0 to the critical value H _max/λ marks the transition from the limit of almost linear wave to a strongly nonlinear limiting Stokes wave. Here, H is the wave height and λ is the wavelength. We simulated fully nonlinear Euler equations, reformulated in terms of conformal variables, to find Stokes waves for different wave heights. Analyzing spectra of these solutions we found in conformal variables, at each Stokes wave height, the distance ν _c from the lowest singularity in the upper half-plane to the real line which corresponds to the fluid free surface. We also identified that this singularity is the square-root branch point. The limiting Stokes wave emerges as the singularity reaches the fluid surface. From the analysis of data for ν _c → 0 we suggest a new power law scaling ν _c ∝ (H _max ? H)^3/2 as well as new estimate H _max/λ ? 0.1410633.     

A new type of conformal dynamics

We consider the Lagrangian particle model introduced in [Ann. Phys. 260 (1997) 224] for zero mass but nonvanishing second central charge of the planar Galilei group. Extended by a magnetic vortex or a Coulomb potential the model exhibits conformal symmetry. In the former case we observe an additional SO(2,1) hidden symmetry. By either a canonical transformation with constraints or by freezing scale and special conformal transformations at t=0 we reduce the six-dimensional phase-space to the physically required four dimensions. Then we discuss bound states (bounded solutions) in quantum dynamics (classical mechanics). We show that the Schrödinger equation for the pure vortex case may be transformed into the Morse potential problem thus providing us with an explanation of the hidden SO(2,1) symmetry.     

Conformal transformation route to gravity’s rainbow

Conformal transformation as a mathematical tool has been used in many areas of gravitational physics. In this paper, we consider gravity’s rainbow, in which the metric can be treated as a conformal rescaling of the original metric. By using the conformal transformation technique, we get a specific form of a modified Newton’s constant and cosmological constant in gravity’s rainbow, which implies that the total vacuum energy is dependent on probe energy. Moreover, the result shows that Einstein gravity’s rainbow can be described by energy-dependent \(f(E,\tilde{R})\) gravity. At last, we study the f(R) gravity, when gravity’s rainbow is considered, which can also be described as energy-dependent \(\tilde{f}(E,\tilde{R})\) gravity.     

On the reaction pp → ppη near the threshold

The reaction pp → ppη near the kinematical threshold is considered in the framework of the model, which includes the pp final states interaction (FSI). The Coulomb and strong effects of the pp FSI are calculated correctly in the model “Coulomb + Yamaguchi potential”. The pp FSI is shown to provide the main contribution to the energy dependence of the total cross section of the pp → ppη reaction close to threshold. The energy behavior of the total cross section is predicted in agreement with the data in the excess energy region < 15MeV. The predictions for different spectra are also performed.     

Quantization of conformal gravity: Another approach to the renormalization of gravity

The non-renormalizability of quantum gravity poses a great problem to the construction of any unified field theory of all known interactions. Normally, we start with a unitary theory of gravity and investigate its renormalization properties. This is the first of a series of papers where we start with the opposite approach, beginning with a renormalizable theory and investigating its unitarity structure. In particular, we study non-perturbative approaches to the quantization of conformal gravity. Using ADM coordinates, we perform the canonical quantization of the Weyl action C_μναβ², which is renormalizable and is also local scale invariant. Although this theory is certainly not unitary in perturbation theory, we speculate that unitarity may be restored when we approach this theory non-perturbatively, by examining the possibility of different phase transitions.     

Strong binding and shrinkage of single and double nuclear systems (K?pp,K?ppn,K?K?p and K?K?pp) predicted by Faddeev-Yakubovsky calculations

Non-relativistic Faddeev and Faddeev-Yakubovsky calculations were made for K⁻pp, K⁻ppn, K⁻K⁻p and K⁻K⁻pp kaonic nuclear clusters, where the quasi bound states were treated as bound states by employing real separable potential models for the K⁻-K⁻ and the K⁻-nucleon interactions as well as for the nucleon-nucleon interaction. The binding energies and spatial shrinkages of these states, obtained for various values of the interaction, were found to increase rapidly with the interaction strength. Their behaviors are shown in a reference diagram, where possible changes by varying the interaction in the dense nuclear medium are given. Using the Λ(1405) ansatz with a PDG mass of 1405 MeV/c² for K⁻p, the following ground-state binding energies together with the wave functions were obtained: 51.5 MeV (K⁻pp), 69 MeV (K⁻ppn), 30.4 MeV (K⁻K⁻p) and 93 MeV (K⁻K⁻pp), which are in good agreement with previous results of variational calculation based on the Akaishi-Yamazaki coupled-channel potential. The K⁻K⁻pp state has a significantly increased density where the two nucleons are located very close to each other, in spite of the inner NN repulsion. Relativistic corrections on the calculated non-relativistic results indicate substantial lowering of the bound-state masses, especially of K⁻K⁻pp, toward the kaon condensation regime. The fact that the recently observed binding energy of K⁻pp is much larger (by a factor of 2) than the originally predicted one may infer an enhancement of the interaction in dense nuclei by about 25% possibly due to chiral symmetry restoration. In this respect some qualitative accounts are given based on “clearing QCD vacuum” model of Brown, Kubodera and Rho.     

Ambient Metrics for n-Dimensional pp-Waves

We provide an explicit formula for the Fefferman-Graham ambient metric of an n-dimensional conformal pp-wave in those cases where it exists. In even dimensions we calculate the obstruction explicitly. Furthermore, we describe all 4-dimensional pp-waves that are Bach-flat, and give a large class of Bach-flat examples which are conformally Cotton-flat, but not conformally Einstein. Finally, as an application, we use the obtained ambient metric to show that even-dimensional pp-waves have vanishing critical Q-curvature.     

Moduli integrals in liouville gravity

We consider moduli integrals appearing in four-point correlation functions of the (p, q) minimal models coupled to Liouville gravity on a sphere, which is sometimes called 2D minimal gravity or minimal string theory on a sphere. Liouville gravity on a sphere is the quantized metric of the spherical topology in the conformal gauge. Reviewing the previous results on such four-point functions (Y. Ishimoto and Sh. Yamaguchi: Phys. Lett. B607 (2005) 172), we show logarithmic correlation functions of ‘tachyons’ in the Liouville sector, and its moduli integrals of the full correlation functions, in particular in the Majorana fermion model coupled to 2D gravity. Further discussions and related results are given in the final section and in Y. Ishimoto and Al. Zamolodchikov: Theor. Math. Phys.147 (2006) 755.     

Conformal transformations and conformal invariance in gravitation

Conformal transformations are frequently used tools in order to study relations between various theories of gravity and Einstein's general relativity theory. In this paper we discuss the rules of these transformations for geometric quantities as well as for the matter energy‐momentum tensor. We show the subtlety of the matter energy‐momentum conservation law which refers to the fact that the conformal transformation “creates” an extra matter term composed of the conformal factor which enters the conservation law. In an extreme case of the flat original spacetime the matter is “created” due to work done by the conformal transformation to bend the spacetime which was originally flat. We discuss how to construct the conformally invariant gravity theories and also find the conformal transformation rules for the curvature invariants R², R_abR^ab, R_abcdR^abcd and the Gauss‐Bonnet invariant in a spacetime of an arbitrary dimension. Finally, we present the conformal transformation rules in the fashion of the duality transformations of the superstring theory. In such a case the transitions between conformal frames reduce to a simple change of the sign of a redefined conformal factor.     

Proton-proton scattering without Coulomb force renormalization

We demonstrate numerically that proton-proton (pp scattering observables can be determined directly by standard short-range methods using a screened pp Coulomb force without renormalization. In the examples the appropriate screening radii are given. We also numerically investigate solutions of the 3-dimensional Lippmann-Schwinger (LS) equation for a screened Coulomb potential alone in the limit of large screening radii and confirm analytically predicted properties for off-shell, half-shell and on-shell Coulomb t -matrices. These 3-dimensional solutions will form a basis for a novel approach to include the pp Coulomb interaction into the 3N Faddeev framework.     

Lanczos Potential for the van Stockung Space-Time

The Lanczos Potential is a theoretical useful tool to find the conformal Weyl curvature tensor C _abcd of a given relativistic metric. In this paper we find the Lanczos potential L _abc for the van Stockung vacuum gravitational field. Also, we show how the wave equation can be combined with spinor methods in order to find this important three covariant index tensor.     

Schrödinger-Newton equation with a complex Newton constant and induced gravity

In the reversible Schrödinger-Newton equation a complex Newton coupling Gexp(−iα) is proposed in place of G. The equation becomes irreversible and all initial one-body states are expected to converge to solitonic stationary states. This feature is verified numerically. For two-body solutions we point out that an effective Newtonian interaction is induced by the imaginary mean-fields as if they were real. The effective strength of such induced gravity depends on the local wave functions of the participating distant bodies.     

4D gravity on a BPS brane in 5D AdS-Minkowski space

We calculate small correction terms to gravitational potential near an asymmetric BPS brane embedded in a 5D AdS-Minkowski space in the context of supergravity. The normalizable wave functions of gravity fluctuations around the brane describe only massive modes. We compute such wave functions analytically in the thin wall limit. We estimate the correction to gravitational potential for small and long distances, and show that there is an intermediate range of distances in which we can identify 4D gravity on the brane below a crossover scale. The 4D gravity is metastable and for distances much larger than the crossover scale the 5D gravity is recovered.     

Bose-Einstein solitons in time-dependent linear potential

In this paper, Bose-Einstein soliton solutions of the nonlinear Schrödinger equation with time-dependent linear potential are considered. Based on the F-expansion method, we present a number of Jacobian elliptic function solutions. Particular cases of these solutions, where the elliptic function modulus equals 1 and 0, are various localized solutions and trigonometric functions, respectively. Specially, for V_ext = ZF(T) = Z[mg + Hcos (ω₁T)], we discussed the Bose-Einstein condensate trapped in the coupling external field with considering the effect of gravity; for F(T) = constant, it describes the wave (Langmuir or electromagnetic) in a linearly inhomogeneous plasma with cubic nonlinearly.     

Exact, zero-energy, square-integrable solutions of a model related to the Maxwell’s fish-eye problem

A model, which admits normalizable wave functions of the Schrödinger equation at the energy of E = 0, is exactly solved and the solutions are compared to the corresponding classical trajectories. The wave functions are proved to be square-integrable for discrete (quantized) values of the coupling constant of the used potential. We also show that our model is a specific version of the well-known Maxwell’s fish-eye. This is performed with the help of a suitably chosen conformal mapping.     

Analysis of the processes¯NN→3π,¯pp→ηπ+π− and¯pp→ωπ+π− as a test of a dual four-point-function without parity doublets

The five annihilation processes at rest:¯pn→π⁺π^?π^?,¯pp→π⁰π⁺π^?,¯pp→3π⁰,¯pp→ηπ⁺π^? and¯pp→ωπ⁺π^? are simultaneously analyzed by means of a dual quark-symmetric four-point-function without parity doublets. The qualitative features of the mass-spectra are reproduced without free parameters. Predictions for¯pp→3π⁰ are made and compared with the results of Lovelace's model. In connection with the calculation of the ratios of rates we discuss the problem of the mass extrapolation for the external particles. The ?—? puzzle turns out to be a key problem for the processes under consideration.     

Characteristic predictions of topological soliton models

Characteristic predictions of the chiral soliton models (the Skyrme model and its extensions) are discussed. The chiral soliton model predictions of low-lying dibaryon states qualitatively agree with recent evidence for the existence of narrow dibaryons in reactions of the inelastic proton scattering on deuterons and the double photon radiation pp→ppγγ. The connection between magnetic moment operators and inertia tensors valid for arbitrary SU(2) skyrmion configurations allows us to estimate the electromagnetic decay width of some states of interest. Predictions of a different type are multibaryons with a nontrivial flavor (strangeness, charm, or bottom), which can be found, in particular, in high-energy heavy ion collisions. It is shown that the large-B multiskyrmions given by the rational map ansatz can be described within the domain-wall approximation or as a spherical bag with the energy and the baryon number density concentrated at the boundary.     

Possibility of choosing between different types of nucleon-nucleon interaction on the basis of data on hard bremsstrahlung in the process pp → ppγ at beam energies in the range 350–500 MeV

Previously reported results on the differential cross sections for the process pp → ppγ occurring at beam-proton energies of 280, 350, and 450 MeV and involving the emission of hard photons are supplemented with the results of calculations at 400 and 500 MeV. The emerging pattern suggests that, in the energy range E ₀=450–500 MeV, an experiment detecting hard photons from this reaction, in which case outgoing protons escape at small angles on the different sides of the beam direction, will be very sensitive to the type of nucleon-nucleon potential (meson-exchange potentials versus the Moscow potential). The energy of E ₀=400 MeV is not optimal for this purpose because the sensitivity is higher even at E ₀=350 MeV. The possibility of distinguishing between the types of nucleon-nucleon potentials through examining the transverse analyzing power A _y (θ _γ ), which reflects the correctness of taking into account spin effects, is studied. This analyzing power is found to exhibit comparatively small changes in response to introducing short-range oscillations in the S and P waves instead of the repulsive-core-induced vanishing of the wave function at small distances.