time_dilation.py

"""
Defines gravitational time dilation/Lorentz transformations for particle at a relativistic speed near a Schwarzschild black hole. 
"""

import numpy as np 
from scipy import constants

def sch_rad(hole_mass):
    """
    Computes the Schwarschild radius for a black hole of mass hole_mass. 
    """
    return 2*constants.G*hole_mass/(constants.speed_of_light)**2 

def grav_dilation(s_rad, radius, far_time):
    """
    Computes graviational time dilation for a Schwarzschild black hole of radius sch_rad. This function assumes that 
    both observers are along the same radial coordinate extending from the black hole center; far_time is the time interval 
    measured by the observer at radius_far > radius, where radius is the distance of the "closer" observer to the black hole. 
    That is, the observer at radius_far measures a weaker gravitational field than the observer at radius. 

    Note that far_time is a time interval, i.e. far_time = t_1 - t_0 for some event the far observer measures as beginning at 
    t_0 and ending at t_1 
    """

    return far_time*np.sqrt(1 - s_rad/radius)

def lorentz_dilation(beta, time_interval):
    """
    Computes time dilation due to special relativistic effects: in other words, time dilation due to high velocity (beta --> c). 
    Assumes the time_interval provided is measured in the rest frame of the moving particle.
    """

    gamma = 1.0/np.sqrt(1 - beta**2)
    return gamma*time_interval

def lorentz_grav_dilation(beta, time_interval, s_rad, radius):
    """
    Combines the effects of special and general relativity for a massive particle traveling in the radial direction 
    from a Schw. black hole of mass hole_mass. This can be derived by introducing a non-zero beta into the Schw. metric. 
    As in the grav_dilation case, time_interval is the time interval measured by an observer at rest far from the black hole 
    event horizon. The variable sch_rad is the Schwarschild radius of the black hole (computed using the sch_rad function above).
    """

    return time_interval*np.sqrt( (1 - s_rad/radius)*( 1 - (beta**2)/(1 - s_rad/radius)**2 ) )