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Escape Velocities for a Relativistic Oscillator in Various g Fields

Abstract. A relativistic oscillator is modeled on a computer. Four cases are considered: (1) The mass oscillates in gravity-free space (g=0), (2) The mass oscillates in the earth’s gravitational field at its surface (g=9,8), (3) The mass oscillates in the Sun’s approximate gravitational field at its surface (g=3e3), (4) The mass oscillates in a black hole’s gravitational field at its surface (g=3e7) .

The oscillator in each case can be considered itself to be a clock. Successive ticks can be defined to be each time the mass passes through x=0 when traveling in the positive x-direction.

The Python program that computes the Amplitude and period of oscillation in each case is provided in Appendix A. The results are as follows.

Case 1 (g-0).

Amplitude=723294299.7603947 meters

Period=11.362235849308732 seconds

Case 2 (g=9.8).

Amplitude=723204328.5442391 meters

Period=11.362235849308732 seconds

Case 3 (g=3e3).

Amplitude=723213111.2409652 meters

Period=11.362336380273646 seconds

Case 4 (g=3e7).

Amplitude=823741919.0294896 meters

Period=12.624753972187094 seconds.

It is clear in the cases considered that in general the amplitudes and periods of the oscillator increase as the strength of the ambient gravitational field increases.

Appendix A

#Python 3.3.4 (v3.3.4:7ff62415e426, Feb 10 2014, 18:12:08) [MSC v.1600 32 bit (Intel)] on win32

#This program computes the period of a 'clock', which consists of a mass,

#attached to an ideal spring of spring constant k.

#The mass oscillates back and forth on the end of the spring.

#The mass' speed, at x=0 and t=0, is maximum and relativistic. Thus x=0 is the center of

#oscillation. The first of four legs of oscillation is assumed to end when the

#mass speed is approximately zero. The goal of the modeling program is to compute the amplitude

#and the period of oscillation in various constant gravitational fields.

import math

g=0.    #Choose ambient field strength by uncommenting.

#g=9.8

#g=3e3

#g=3e7

n=1000000     #Loop counter

c=3e8   #Speed of light

v=2.9e8 #Maximum of mass speed (at x=0 and t=0)

m0=1.     #Rest mass

k=1.     #Spring constant

PeriodNonrel=2.*math.pi*math.sqrt(m0/k)     #Non-relativistic period

dt=PeriodNonrel/n     #Time between computation loops

x=0.     #Initial position of mass

gamma=1/math.sqrt(1-v**2/c**2)  #Un-comment this for relativistic mass

f=gamma*m0*g-k*x     #Spring force on mass

a=f/(gamma**3*m0)     #Acceleration of mass

t=0.     #Time at beginning of oscillation

while v>=0.:     #Loop for first quarter of oscillation

t=t+dt

x=x+v*dt+.5*a*dt**2

v=v+a*dt

gamma=1/math.sqrt(1.-v**2/c**2)    #Uncomment this for relativistic mass.

f=gamma*m0*g-k*x

a=f/(gamma**3*m0)

Period=4*t

print('Amplitude=',x,'   Period=',Period)

print('End of program')