arXiv subjects

Klaus Kassner

Publications and source records attributed to Klaus Kassner.

At least 19 recordsLinked to original sources

arXiv2021

Large amplitude behavior of the Grinfeld instability, Part I: High-order weakly nonlinear analysis

Amplitude expansions are used to determine steady states of a semi-infinite solid subject to the Grinfeld instability in systems with a fixed (wave)length. We present two methods to obtain high-order weakly nonlinear results. Using the system size as a control parameter, we circumvent the problem that there is no instability threshold for an extended system in the absence of gravity. This way, the case without gravity becomes accessible to a weakly nonlinear treatment. The dependence of the branch structure of solution space on the level of gravity (or density difference) is exhibited. In the zero-gravity limit, we recover the solution branch obtained by Spencer and Meiron. A transition from a supercritical to a subcritical bifurcation is observed as gravity is increased or the nonhydrostatic stress is decreased at fixed gravity. At given values of the system parameters, we find a discrete, possibly infinite, set of solution branches. This is reminiscent of dendritic or eutectic growth, where similar solution sets exist, of which only a particular one is linearly stable. Despite the high order of our expansions, the approach is restricted to relatively small nondimensional amplitudes ($\lesssim 0.2$), a disadvantage we can overcome by a variational approach that will be discussed in a companion paper. At the critical point, we find that not only the first Landau coefficient is negative but all of them up to the highest amplitude order (15) we could compute so far.

cond-mat↗

arXiv2021

The Vaidya metric: expected and unexpected traits of evaporating black holes

The ingoing Vaidya metric is introduced as a model for a non-rotating uncharged black hole emitting Hawking radiation. This metric is expected to capture the physics of the spacetime for radial coordinates up to a small multiple $(>1)$ of the Schwarzschild radius. For larger radii, it will give an excellent approximation to the spacetime geometry in the case of astrophysical black holes $(M\ge M_{\astrosun})$, except at extremely large distances from the horizon (exceeding the cosmic particle horizon). In the classroom, the model may serve as a first exploration of non-stationary gravitational fields. Several interesting predictions are developed. First, particles dropped early enough before complete evaporation of the black hole cross its horizon as easily as with an eternal black hole. Second, the Schwarzschild radius takes on the properties of an apparent horizon, and the true event horizon of the black hole is \emph{inside} of it, because light can escape from the shrinking apparent horizon. Third, a particle released from rest close enough to the apparent horizon is strongly repelled and may escape to infinity. An interpretation is given, demonstrating that such a particle would be able to compete, for a short time, in a race with a photon.

gr-qc↗

arXiv2019

Reduction of a family of metric gravities

A recent proposal by Shuler regarding a postulate-based derivation of a family of metrics describing the gravitational field outside a static spherically symmetric mass distribution is reviewed. All of Shuler's gravities agree with the Schwarzschild solution in the weak-field limit, but they differ in the strong-field domain, i.e., close enough to a sufficiently compact source of the field. It is found that the evoked postulates of i) momentum conservation and ii) consistency of field strength measurement are satisfied in all metric theories of gravity compatible with the Einstein equivalence principle, no matter what the form of the metric. Therefore, they cannot be used, within any correct deduction, to derive a particular metric. Shuler's derivations are based on an inconsistent set of correspondences between local and distant quantities. Furthermore, it is shown here that out of the family of possible metrics given by Shuler only one member, the Schwarzschild metric, satisfies a standard relativistic generalization of Newton's law of gravitation, suggesting the others to be unphysical.

gr-qc↗

arXiv2018

Radially falling test particle approaching an evaporating black hole

A simple model for an evaporating non-rotating black hole is considered, employing a global time that does not become singular at the putative horizon. The dynamics of a test particle falling radially towards the center of the black hole is then investigated. Contrary to a previous approach, we find that the particle may pass the Schwarzschild radius before the black hole has gone. Backreaction effects of Hawking radiation on the space-time metric are not considered, rather a purely kinematical point of view is taken here. The importance of choosing an appropriate time coordinate when describing physical processes in the vicinity of the Schwarzschild radius is emphasized. For a shrinking black hole, the true event horizon is found to be inside the sphere delimited by that radius.

gr-qc↗

arXiv2017

A Physics-First Approach to the Schwarzschild Metric

As is well-known, the Schwarzschild metric cannot be derived based on pre-general-relativistic physics alone, which means using only special relativity, the Einstein equivalence principle and the Newtonian limit. The standard way to derive it is to employ Einstein's field equations. Yet, analogy with Newtonian gravity and electrodynamics suggests that a more constructive way towards the gravitational field of a point mass might exist. As it turns out, the additional physics needed is captured in two plausible postulates. These permit to deduce the exact Schwarzschild metric without invoking the field equations. Since they express requirements essentially designed for use with the spherically symmetric case, they are less general and powerful than the postulates from which Einstein constructed the field equations. It is shown that these imply the postulates given here but that the converse is not quite true. The approach provides a fairly fast method to calculate the Schwarzschild metric in arbitrary coordinates exhibiting stationarity and sheds new light on the behavior of waves in gravitational fields.

gr-qc↗

arXiv2016

Dust ball physics and the Schwarzschild metric

A physics-first derivation of the Schwarzschild metric is given. Gravitation is described in terms of the effects of tidal forces (or of spacetime curvature) on the volume of a small ball of test particles (a dust ball), freely falling after all particles were at rest with respect to each other initially. The possibility to express Einstein's equation this way and some of its ramifications have been enjoyably discussed by Baez and Bunn [Am. J. Phys. 73, 644 (2005)]. Since the formulation avoids the use of tensors, neither advanced tensor calculus nor sophisticated differential geometry are needed in the calculation. The derivation is not lengthy and it has visual appeal, so it may be useful in teaching.

gr-qc↗

arXiv2016

Why ghosts don't touch: a tale of two adventurers falling one after another into a black hole

The case for the utility of Kruskal-Szekeres coordinates in classroom made by Augousti et al. in this journal (Eur. J. Phys. 33:1--11, 2012) is strengthened by extending their discussion beyond the event horizon of the black hole. Observations made by two adventurers following one another into a Schwarzschild black hole are examined in terms of these nonsingular coordinates. Two scenarios are considered, the first corresponding to one observer following the other closely, the second to a significant distance between the two of them, precluding the existence of a common inertial system. In particular, the concepts of distance and temporal separation near the horizon and the redshift of the first infaller's image as seen by the second are investigated. The results show that the notion of "touching ghosts" does not correspond to the local physics of two observers falling into a black hole. The story line is interesting enough and the mathematical details are sufficiently simple to use the example in a general relativity course, even at the undergraduate level.

gr-qc↗

arXiv2015

Simulations support protocol independency of the granular temperature

A possible approach to the statistical description of granular assemblies starts from Edwards' assumption that all blocked states occupying the same volume are equally probable (S.F. Edwards, R. Oakeshott, Physica A 157, 1080 (1989)). We performed computer simulations using two-dimensional polygonal particles excited periodically according to two different protocols: excitation by pulses of "negative gravity" and excitation by "rotating gravity". The first protocol exhibits a non-monotonous dependency of the mean volume fraction on the pulse strength. The overlapping histogram method is used in order to test whether or not the volume distribution is described by a Boltzmann-like distribution, and to calculate the inverse compactivity as well as the logarithm of the partition sum. We find that the mean volume is a unique function of the measured granular temperature, independently of the protocol and of the branch in $ϕ(g)$ and all determined quantities are in agreement with Edwards' theory.

cond-mat.stat-mech↗

arXiv2015

Classroom reconstruction of the Schwarzschild metric

A promising way to introduce general relativity in the classroom is to study the physical implications of certain given metrics, such as the Schwarzschild one. This involves lower mathematical expenditure than an approach focusing on differential geometry in its full glory and permits to emphasize physical aspects before attacking the field equations. Even so, in terms of motivation, lacking justification of the metric employed may pose an obstacle. The paper discusses how to establish the weak-field limit of the Schwarzschild metric with a minimum of relatively simple physical assumptions, avoiding the field equations but admitting the determination of a single parameter from experiment. An attractive experimental candidate is the measurement of the perihelion precession of Mercury, because the result was already known before the completion of general relativity. It is shown how to determine the temporal and radial coefficients of the Schwarzschild metric to sufficiently high accuracy to obtain quantitative predictions for all the remaining classical tests of general relativity.

gr-qc↗

arXiv2013

Selection theory of free dendritic growth in a potential flow

The Kruskal-Segur approach to selection theory in diffusion-limited or Laplacian growth is extended via combination with the Zauderer decomposition scheme. This way nonlinear bulk equations become tractable. To demonstrate the method, we apply it to two-dimensional crystal growth in a potential flow. We omit the simplifying approximations used in a preliminary calculation for the same system [T. Fischaleck, K. Kassner, EPL 81, 54004 (2008)], thus exhibiting the capability of the method to extend mathematical rigor to more complex problems than hitherto accessible.

cond-mat.mtrl-sci↗

arXiv2013

Ways to resolve Selleri's paradox

Selleri's paradox, based on an analysis of rotating frames, appears to show that the speed of light in an inertial system is not normally isotropic. This in turn seems at odds with the second postulate of special relativity requiring a universal light speed in inertial systems. First, it is demonstrated how to circumvent Selleri's argument using Einstein synchronization in rotating frames. Then the nature of Selleri's result is exposed: it simply corresponds to the adoption of a synchronization procedure different from Einstein's. In this scheme, anisotropic one-way speeds of light by no means contradict special relativity.

physics.class-ph↗

arXiv2012

Spatial geometry of the rotating disk and its non-rotating counterpart

A general relativistic description of a disk rotating at constant angular velocity is given. It is argued that conceptually this direct approach poses fewer problems than the special relativistic one. For observers on the disk, the geometry of their proper space is hyperbolic. This has interesting consequences concerning their interpretation of the geometry of a non-rotating disk having the same radius. The influence of clock synchronization on spatial measurements is discussed.

physics.pop-ph↗

arXiv2007

Comparison of phase-field models for surface diffusion

The description of surface-diffusion controlled dynamics via the phase-field method is less trivial than it appears at first sight. A seemingly straightforward approach from the literature is shown to fail to produce the correct asymptotics, albeit in a subtle manner. Two models are constructed that approximate known sharp-interface equations without adding undesired constraints. Linear stability of a planar interface is investigated for the resulting phase-field equations and shown to reduce to the desired limit. Finally, numerical simulations of the standard and a more sophisticated model from the literature as well as of our two new models are performed to assess the relative merits of each approach. The results suggest superior performance of the new models in at least some situations.

physics.comp-ph↗

arXiv2007

Extending the scope of microscopic solvability: Combination of the Kruskal-Segur method with Zauderer decomposition

Successful applications of the Kruskal-Segur approach to interfacial pattern formation have remained limited due to the necessity of an integral formulation of the problem. This excludes nonlinear bulk equations, rendering convection intractable. Combining the method with Zauderer's asymptotic decomposition scheme, we are able to strongly extend its scope of applicability and solve selection problems based on free boundary formulations in terms of partial differential equations alone. To demonstrate the technique, we give the first analytic solution of the problem of velocity selection for dendritic growth in a forced potential flow.

nlin.PS↗

arXiv2006

Influence of external flows on crystal growth: numerical investigation

We use a combined phase-field/lattice-Boltzmann scheme [D. Medvedev, K. Kassner, Phys. Rev. E {\bf 72}, 056703 (2005)] to simulate non-facetted crystal growth from an undercooled melt in external flows. Selected growth parameters are determined numerically. For growth patterns at moderate to high undercooling and relatively large anisotropy, the values of the tip radius and selection parameter plotted as a function of the Peclet number fall approximately on single curves. Hence, it may be argued that a parallel flow changes the selected tip radius and growth velocity solely by modifying (increasing) the Peclet number. This has interesting implications for the availability of current selection theories as predictors of growth characteristics under flow. At smaller anisotropy, a modification of the morphology diagram in the plane undercooling versus anisotropy is observed. The transition line from dendrites to doublons is shifted in favour of dendritic patterns, which become faster than doublons as the flow speed is increased, thus rendering the basin of attraction of dendritic structures larger. For small anisotropy and Prandtl number, we find oscillations of the tip velocity in the presence of flow. On increasing the fluid viscosity or decreasing the flow velocity, we observe a reduction in the amplitude of these oscillations.

cond-mat.mtrl-sci↗

arXiv2006

How to model surface diffusion using the phase-field approach

It is demonstrated that the description of surface-diffusion controlled dynamics via the phase-field method is less trivial than it appears at first sight. A seemingly straightforward approach previously used in the literature is shown to fail to produce the correct asymptotics, albeit in a subtle manner. An apparently obvious alternative fails for a complementary reason. Finally, a model is constructed that asymptotically approaches known sharp-interface equations without adding undesired constraints. In order to provide a complete physical example, the model is exhibited for the elastically induced Grinfeld instability with material transport by surface diffusion. The whole analysis is carried out for three-dimensional systems to pave the way for simulations in 3D.

cond-mat.other↗

arXiv2005

Phase Field Modeling of Fast Crack Propagation

We present a continuum theory which predicts the steady state propagation of cracks. The theory overcomes the usual problem of a finite time cusp singularity of the Grinfeld instability by the inclusion of elastodynamic effects which restore selection of the steady state tip radius and velocity. We developed a phase field model for elastically induced phase transitions; in the limit of small or vanishing elastic coefficients in the new phase, fracture can be studied. The simulations confirm analytical predictions for fast crack propagation.

cond-mat.mtrl-sci↗

arXiv2003

Large amplitude behavior of the Grinfeld instability: a variational approach

In previous work, we have performed amplitude expansions of the continuum equations for the Grinfeld instability and carried them to high orders. Nevertheless, the approach turned out to be restricted to relatively small amplitudes. In this article, we use a variational approach in terms of multi-cycloid curves instead. Besides its higher precision at given order, the method has the advantages of giving a transparent physical meaning to the appearance of cusp singularities and of not being restricted to interfaces representable as single-valued functions. Using a single cycloid as ansatz function, the entire calculation can be performed analytically, which gives a good qualitative overview of the system. Taking into account several but few cycloid modes, we obtain remarkably good quantitative agreement with previous numerical calculations. With a few more modes taken into consideration, we improve on the accuracy of those calculations. Our approach extends them to situations involving gravity effects. Results on the shape of steady-state solutions are presented at both large stresses and amplitudes. In addition, their stability is investigated.

cond-mat↗