Formation Of Curvature Singularity Research Articles

Black hole singularity, generalized (holographic) c-theorem and entanglement negativity

In this paper we revisit the question that in what sense empty AdS5 black brane geometry can be thought of as RG-flow. We do this by first constructing a holographic c-function using causal horizon in the black brane geometry. The UV value of the c-function is aUV and then it decreases monotonically to zero at the curvature singularity. Intuitively, the behavior of the c-function can be understood if we recognize that the dual CFT is in a thermal state and thermal states are effectively massive with a gap set by the temperature. In field theory, logarithmic entanglement negativity is an entanglement measure for mixed states. For example, in two dimensional CFTs at finite temperature the renormalized entanglement negativity of an interval has UV (Low-T) value cUV and IR (High-T) value zero. So this is a potential candidate for our c-function. In four dimensions we expect the same thing to hold on physical grounds. Now since the causal horizon goes behind the black brane horizon the holographic c-function is sensitive to the physics of the interior. Correspondingly the field theory c-function should also contain information about the interior. So our results suggest that high temperature (IR) expansion of the negativity (or any candidate c-function) may be a way to probe part of the physics near the singularity. Negativity at finite temperature depends on the full operator content of the theory and so perhaps this can be done in specific cases only.The existence of this c-function in the bulk is an extreme example of the paradigm that space-time is built out of entanglement. In particular the fact that the c-function reaches zero at the curvature singularity correlates the two facts: loss of quantum entanglement in the IR field theory and the end of geometry in the bulk which in this case is the formation of curvature singularity.

Formation of curvature singularities on the interface between dielectric liquids in a strong vertical electric field

The nonlinear dynamics of the interface between two deep dielectric fluids in the presence of a vertical electric field is studied. We consider the limit of a strong external electric field where electrostatic forces dominate over gravitational and capillary forces. The nonlinear integrodifferential equations for the interface motion are derived under the assumption of small interfacial slopes. It is shown in the framework of these equations that, in the generic case, the instability development leads to the formation of root singularities at the interface in a finite time. The interfacial curvature becomes infinite at singular points, while the slope angles remain relatively small. The curvature is negative in the vicinity of singularities if the ratio of the permittivities of the fluids exceeds the inverse ratio of their densities, and it is positive in the opposite case (we consider that the lower fluid is heavier than the upper one). In the intermediate case, the interface evolution equations describe the formation and sharpening of dimples at the interface. The results obtained are applicable for the description of the instability of the interface between two magnetic fluids in a vertical magnetic field.

Singularities in the complex physical plane for deep water waves

AbstractDeep water waves in two-dimensional flow can have curvature singularities on the surface profile; for example, the limiting Stokes wave has a corner of $2\lrm{\pi} / 3$ radians and the limiting standing wave momentarily forms a corner of $\lrm{\pi} / 2$ radians. Much less is known about the possible formation of curvature singularities in general. A novel way of exploring this possibility is to consider the curvature as a complex function of the complex arclength variable and to seek the existence and nature of any singularities in the complex arclength plane. Highly accurate boundary integral methods produce a Fourier spectrum of the curvature that allows the identification of the nearest singularity to the real axis of the complex arclength plane. This singularity is in general a pole singularity that moves about the complex arclength plane. It approaches the real axis very closely when waves break and is associated with the high curvature at the tip of the breaking wave. The behaviour of these singularities is more complex for standing waves, where two singularities can be identified that may collide and separate. One of them approaches the real axis very closely when a standing wave forms a very narrow collapsing column of water almost under free fall. In studies so far, no singularity reaches the real axis in finite time. On the other hand, the surface elevation $y(x)$ has square-root singularities in the complex $x$ plane that do reach the real axis in finite time, the moment when a wave first starts to break. These singularities have a profound effect on the wave spectra.

Formation of curvature singularity along vortex line in an axisymmetric, swirling vortex sheet

We consider an axisymmetric, swirling vortex sheet in an inviscid and incompressible flow. Caflisch et al. pointed out that the vortex sheet acquired a singularity in finite time, but the property of the singularity was not revealed. In the present paper we show convincing numerical evidences of the singularity formation by applying the same numerical methods as what was used in the study of a two-dimensional (2D) vortex sheet. We find that the radial and axial components of the axisymmetric vortex sheet behave like the 2D singularity that has been observed in many vortex-sheet motions, while the azimuthal component of the sheet behaves differently. Furthermore, the singularity appears along the vortex line and the first derivative of the vortex sheet strength forms a cusp, while the known singularities are associated with the curvature along curves perpendicular to the vortex lines and the sheet strength has a cusp.

Time-independent bounds on solutions with shocks to the relativistic Euler equations on conformally flat space–times with curvature singularities

We consider the initial value problem of the relativistic Euler equations when the underlying space–time is not flat but conformally flat, and demonstrate existence of solutions with shocks, with time-independent bounds, in some cases from big bang till big crunch. Our theorem requires that the space–time metric satisfy certain bounds, but these do not constrain the curvature of space–time, and hence our theorem may guarantee uniform bounds on solutions until the formation of curvature singularities.

Quantum fluctuations and curvature singularities in Jackiw-Teitelboim gravity.

Jackiw-Teitelboim gravity with matter degrees of freedom is considered. The classical model is exactly solvable and its solutions describe nontrivial gravitational scattering of matter wave packets. For a huge amount of the solutions the scattering space-times are free of curvature singularities. However, the quantum corrections to the field equations inevitably cause the formation of curvature singularities, vanishing only in the limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$. The singularities cut the space-time and disallow propagation to the future. The model is inspired by the dimensional reduction of 4D pure Einstein gravity, restricted to space-times with two commuting spacelike Killing vectors. The matter degrees of freedom also stem from the 4D ansatz. The measures for the continual integrations are judiciously chosen and one-loop contributions (including the graviton and the dilaton ones) are evaluated. For the number of the matter fields $N=24$ we obtain even the exact effective action, applying the David-Distler-Kawai procedure. The effective action is nonlocal, but the semiclassical equations can be solved by using some theory of the Hankel transformations.