__Escape
Velocities for a Relativistic Oscillator in Various g Fields
__

**Abstract.
**A**
r**elativistic 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) .

__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

#Type
"copyright", "credits" or "license()" for more
information.

#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')