Narrow Results

We revisit the free-fall energy density of scalar fields semiclassically by employing the trace anomaly on a two-dimensional Schwarzschild black hole with respect to various black hole states in order to clarify whether something special at the horizon happens or not. For the Boulware state, the energy density at the horizon is always negative divergent, which is independent of initial free-fall positions. However, in the Unruh state the initial free-fall position is responsible for the energy density at the horizon and there is a critical point to determine the sign of the energy density at the horizon. In particular, a huge negative energy density appears when the freely falling observer is dropped just near the horizon. For the Hartle–Hawking state, it may also be positive or negative depending on the initial free-fall position, but it is always finite. Finally, we discuss physical consequences of these calculations.

In this paper, we take a closer and new look at the effects of tidal forces on the free fall of a quantum particle inside a spherically symmetric gravitational field. We derive the corresponding Schrödinger equation for the particle by starting from the fully relativistic Klein–Gordon equation in order (i) to briefly discuss the issue of the equivalence principle and (ii) to be able to compare the relativistic terms in the equation to the tidal-force terms. To the second order of the nonrelativistic approximation, the resulting Schrödinger equation is that of a simple harmonic oscillator in the horizontal direction and that of an inverted harmonic oscillator in the vertical direction. Two methods are used for solving the equation in the vertical direction. The first method is based on a fixed boundary condition, and yields a discrete-energy spectrum with a wavefunction that is asymptotic to that of a particle in a linear gravitational field. The second method is based on time-varying boundary conditions and yields a quantized-energy spectrum that is decaying in time. Moving on to a freely-falling reference frame, we derive the corresponding time-dependent energy spectrum. The effects of tidal forces yield an expectation value for the Hamiltonian and a relative change in time of a wavepacket’s width that are mass-independent. The equivalence principle, which we understand here as the empirical equivalence between gravitation and inertia, is discussed based on these various results. For completeness, we briefly discuss the consequences expected to be obtained for a Bose–Einstein condensate or a superfluid in free fall using the nonlinear Gross–Pitaevskii equation.

Obviously, in Galilean physics, the universality of free fall implies an inertial frame, which in turns implies that the mass $m$ of the falling body is omitted (because it is a test mass; put otherwise, the center of mass of the system coincides with the center of the main, and fixed, mass $M$; or else, we consider only a homogeneous gravitational field). Conversely, an additional (in the opposite or same direction) acceleration proportional to $m/M$ would rise either for an observer at the center of mass of the system, or for an observer at a fixed distance from the center of mass of $M$. These elementary, but overlooked, considerations fully respect the equivalence principle (EP) and the (local) identity of an inertial or a gravitational pull for an observer in the Einstein cabin. They value as fore-runners of the self-force and gauge dependency in general relativity. Because of its importance in teaching and in the history of physics, coupled to the introductory role to Einstein’s EP, the approximate nature of Galilei’s law of free fall is explored herein. When stepping into general relativity, we report how the geodesic free fall into a black hole was the subject of an intense debate again centered on coordinate choice. Later, we describe how the infalling mass and the emitted gravitational radiation affect the free fall motion of a body. The general relativistic self-force might be dealt with to perfectly fit into a geodesic conception of motion. Then, embracing quantum mechanics, real black holes are not classical static objects any longer. Free fall has to handle the Hawking radiation, and leads us to new perspectives on the varying mass of the evaporating black hole and on the varying energy of the falling mass. Along the paper, we also estimate our findings for ordinary masses being dropped from a Galilean or Einsteinian Pisa-like tower with respect to the current state of the art drawn from precise measurements in ground and space laboratories, and to the constraints posed by quantum measurements. Appendix A describes how education physics and high impact factor journals discuss the free fall. Finally, case studies conducted on undergraduate students and teachers are reviewed.

In order to improve the simulation accuracy for free-fall lifeboat in ship life-saving training system, this paper analyzes and models the motion of boat’s launching from the skid. The whole launching is divided into four phases, namely: sliding down, rotation, free fall and water entry. According to the theory of momentum and strip theory, hydrodynamic forces of the boat at water entry are calculated under the effect of waves. The method of interpolation is used for calculating the half width and added mass of cross-sections at water entry. The model is used for numerical investigation about the boat launching from skid under different conditions and applied to ship life-saving simulation training system. The following conclusions are finally obtained: (1) When the initial inclination angle is 30${}^{\circ }$, the horizontal distance between the point of water entry of the boat and the lower end of the slide is about 7.2m. The horizontal distance will be smaller, when the initial inclination angle increases. There is no obvious law between forward distance and waves. (2) When the initial inclination angle is 45${}^{\circ }$, the setback may occur after the boat entering the wave. When the initial angle is 60${}^{\circ }$, the setback occurs after the boat entering the water. (3) When the center of gravity is 1.5m in front of the midship of the boat, the boat will turn over.

In this study, we focus on the concept of weightlessness in an experiment that involves magnetic forces, and discuss the motion of objects and the movement of a system’s center of mass from the perspective of inertial observer.