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Escape Velocities in Single-Valued g Fields

Abstract. A relativistic mass, attached to an ideal spring and in a constant, opposing gravitational field, is modeled. Three cases are considered: (1) The gravitational force is such that the mass is captured by the spring, (2) The gravitational force is such that the mass escapes, stretching the spring to infinity, (3) The gravitational force is such that the computer calculates a speed greater than c, and a fatal runtime error occurs..

Discussion. In most of the literature, the escape velocity is calculated for a free (no opposing spring) mass in an inverse square g field. The mass may or may not be captured, depending on its initial velocity and the ambient field strength. One might at first surmise that, in a uniform field a mass on an opposing spring will always be captured. For one might reason that the spring force grows with increasing penetration of the mass into the field and, assuming the spring force magnitude eventually exceeds the gravitational force on the mass, the mass must eventually reverse direction and hence be captured.

But this conclusion is based on the assumption that the gravitational force on the mass, in the single-valued g field, is itself a constant m0g. In the case of mass and gravity, however, the gravitational force is not constant; it equals gm0g. And there is a threshold initial velocity and g field strength combination at which fgrav is always greater than fspring, and the mass escapes!

Note that If the gravitational field is too great, then the computer may register a runtime error (since v may exceed c in the running program). Analytically of course this would never happen. Analytically the system could either (a) result in capture (for g less than the threshold value), or (b) escape (for g greater than or equal to the threshold value).

It is also worth noting that a charge in a constant electric field would always be captured, since the electric force would be a speed-independent qE, and eventually the spring force would exceed the electric force.

The Python program that computes each case is provided in Appendix A.

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, in a

#constant gravitational field, attached to an ideal spring of spring constant k.

#The mass' speed, at x=0 and t=0, is relativistic.

#The goal of the modeling program is to demonstrate that the mass is captured

#in the first field, but escapes at more powerful fields.

#For g>=3e13, v actually exceeds c in the program

#and a 'sqrt of negative number' runtime error occurs.

import math

g=3e7    #Choose ambient field strength by uncommenting.

#g=3e8

#g=3e13

n=1000000     #Loop counter

c=3e8   #Speed of light

v=2.9e8 #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)

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

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

t=0.     #Initial time, when x=0

while v>=0.:     #Loop to determine capture or escape

t=t+dt

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

v=v+a*dt

gamma=1/math.sqrt(1.-v**2/c**2)

f=gamma*m0*g-k*x

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

if(v<0):

print('mass is captured')

elif(v>.99999999999*c):

print('mass escapes')

break

print('x=',x,'v/c=',v/c)

print('End of program')