The relativistic gravity train

Abstract

The gravity train that takes 42.2 min from any point A to any other point B that is connected by a straight-line tunnel through Earth has captured the imagination more than most other applications in calculus or introductory physics courses. Brachystochron and, most recently, nonlinear density solutions have been discussed. Here relativistic corrections are presented. It is discussed how the corrections affect the time to fall through Earth, the Sun, a white dwarf, a neutron star, and—the ultimate limit—the difference in time measured by a moving, a stationary and the fiducial observer at infinity if the density of the sphere approaches the density of a black hole. The relativistic gravity train can serve as a problem with approximate and exact analytic solutions and as numerical exercise in any introductory course on relativity.

Cosmic Analogues of Classic Variational Problems

“…The problem of the terrestrial brachistochrone, which has seen renewed attention recently [70,[72][73][74][75][76][77][78][79][80][81][83][84][85][86][87][88][89]], provides an explicit example of a universe dominated by a phantom fluid with non-linear equation of state, which can be solved explicitly and exhibits a finite future singularity at a finite value of the scale factor, where the Hubble function, Ricci scalar, energy density, and pressure all diverge. Finite time singularities have been the subject of much literature in the past decade [91,[98][99][100][101][102][103][104][105][106] hence, in this problem, the mechanical side of the analogy helps the cosmology side in the sense that the known exact solution for the terrestrial brachistochrone problem can be immediately translated into an analytical solution of the corresponding cosmology with complicated (non-linear) equation of state.…”

“…In the terrestrial brachistochrone problem [65,67], tunnels of various curved shape described by functions r(θ) in polar coordinates, are dug out in the uniform Earth of radius R, and one looks for the shape that minimizes the transit time between two given points. This problem has seen much renewed attention recently, mostly in the pedagogical literature [72][73][74][75][76][77][78][79][80][81][82][83][84][85][86] but not only [87][88][89].…”

Cosmic analogues of classic variational problems

“…In the terrestrial brachistochrone problem [65,67], tunnels of various curved shape described by functions r(θ) in polar coordinates, are dug out in the uniform Earth of radius R, and one looks for the shape that minimizes the transit time between two given points. This problem has seen much renewed attention recently, mostly in the pedagogical literature [72,73,74,75,76,77,78,79,80,81,83,84,85,86] but not only [87,88,89]. The corresponding Lagrangian is [67] L r, r…”

“…The problem of the terrestrial brachistochrone, which has seen renewed attention recently [70,72,73,74,75,76,77,78,79,80,81,83,84,85,86,87,88,89], provides an explicit example of a universe dominated by a phantom fluid with non-linear equation of state, which can be solved explicitly and exhibits a finite future singularity at a finite value of the scale factor, where the Hubble function, Ricci scalar, energy density, and pressure all diverge. Finite time singularities have been the subject of much literature in the past decade [98,99,100,101,102,103,104,91,105,106] hence, in this problem, the mechanical side of the analogy helps the cosmology side in the sense that the known exact solution for the terrestrial brachistochrone problem can be immediately translated into an analytical solution of the corresponding cosmology with complicated (non-linear) equation of state.…”

Radiation of relativistic ‘harmonic’ oscillator

“…The experimental implementation of the relativistic harmonic oscillator is described in [12]. Interestingly, the problem of the relativistic harmonic oscillator has been actively discussed in connection with the projects of the 'gravity train'-a train moving under the force of gravity from one point on the globe to another along a tunnel connecting these points [13,14].…”