Moving objects in retarded gravitational potentials of an expanding spherical shell/Java program

Table with computed results Java program The following Java program can be used to numerically compute the Schwarzschild distances for different linear mass densities. It implements a numerical integration for the effective masses as well as the numerical solvation for the Schwarzschild distances. /* Source file: SchwarzschildDistance.java Program: Computation of the Schwarzschild Distance within the Hubble sphere Autor: Markus Bautsch Location: Berlin, Germany Licence: public domain Date: 18th July 2024 Version: 1.3 Programming language: Java */ /* This Java-Programm computes Schwarzschild distances for any black shell masses. The black shell is at the outer rim of the Hubble space. */ public class SchwarzschildDistance { // Class constants final static double G = 6.67430e-11; // Gravitational constant in cube metres per kilogram and square second final static double c = 2.99792458e8; // Speed of light in metres per second final static double TropicalYear = 365.24219052 * 24 * 3600; // Tropical year in seconds final static double LightYear = c * TropicalYear; // Light-year in metres final static double R = 1.36e26; // Hubble length in metres final static double MUniverse = 2.97e53; // Mass of the visible universe in kilograms final static double lambdaMfirst = 3.36663e26; // start value for linear mass density of black shell final static double lambdaMlast = 3.36670e26; // startop value for linear mass density of black shell final static double deltaLambda = 0.00001e26; // step size for linear mass density of black shell // This method computes and returns the distance s of a mass m to a mass point dM in the black shell // distance is the closest distance of the mass m to to the mass point dM in the black shell in m // alpha is the angle beweteen the mass m and the mass point dM in the black shell as seen from the centre of the universe in rad private static double distanceToMassElementShell (double distance, double alpha) { double cosAlpha = java.lang.Math.cos (alpha); double squareDistance = distance * distance; double argument = 2*R*R - 2*R*distance + squareDistance - 2*R*(R-distance)*cosAlpha; double s; if (argument <= squareDistance) // because of possible rounding errors { s = distance; } else { s = java.lang.Math.sqrt (argument); } return s; } // This method computes and returns the angle of a half chord x in m as seen from the centre of the universe in rad private static double alpha (double x) { double alpha = java.lang.Math.asin (x / R); return alpha; } // This method computes and returns the integrand of the integral // distance is the closest distance of the mass m to to the mass point dM in the black shell in m // alpha is the angle beweteen the mass m and the mass point dM in the black shell as seen from the centre of the universe in rad private static double integrand (double distance, double alpha) { double s = distanceToMassElementShell (distance, alpha); double e = R * (1 - java.lang.Math.cos (alpha)); // the auxiliary sagitta e double cosBeta = (distance - e) / s; double integrand = cosBeta / s / s; return integrand; } // This method computes and returns the effective mass from 0 to alphaR for a given lambdaM and a given distance // lambdaM is the linear mass density of the black shell in kg/m // distance is the closest distance of the mass m to to the mass point dM in the black shell in m private static double integrateEffectiveMass (double lambdaM, double distance) { final long steps = 1000000; // number of steps for numerical integration double squareDistance = distance * distance; double xMax = java.lang.Math.sqrt (2*R*distance - squareDistance); // maximum half chord for a given maximum angle alphaR double x = xMax; // x runs from xMax to 0 in steps of deltaX double alpha = alpha (x); // alpha runs from alphaR to 0 double integrand = integrand (distance, alpha); double integral = 0; long step = 0; while (step <= steps) { x = xMax * (1 - (double) step / steps); double alphaNext = alpha (x); double integrandNext = integrand (distance, alphaNext); double deltaAlpha = alpha - alphaNext; double averageIntegrand = (integrandNext + integrand) / 2; double area = averageIntegrand * deltaAlpha; // numerical integration integral = integral + area; alpha = alphaNext; integrand = integrandNext; step++; } double effectiveMass = R * lambdaM * squareDistance * (2*integral); return effectiveMass; } // This method computes and returns the Schwarzschild distance d_S for a given effective mass mEff private static double schwarzschildDistance (double effectiveMass) { double schwarzschildDistance = 2 * G * effectiveMass / c / c; return schwarzschildDistance; } // This method computes and returns the effective mass mEff for a given Schwarzschild distance d_S private static double effectiveMass (double schwarzschildDistance) { double effectiveMass = schwarzschildDistance * c * c / 2 / G; return effectiveMass; } // This method solves and returns the Schwarzschild distance d_S for a given lambdaM // lambdaM is the linear mass density of the black shell in kg/m private static double solve (double lambdaM) { final double limit = 5e-16; // for relative precision of determination of Schwarzschild distance double lowerDistance = 1e20; // first low guess for Schwarzschild distance double upperDistance = 1e25; // first high guess for Schwarzschild distance double delta; // delta shall become smaller than limit double distance; double schwarzschildDistance; double deltaDistance; do { distance = (lowerDistance + upperDistance) / 2; double effectiveMass = integrateEffectiveMass (lambdaM, distance); schwarzschildDistance = schwarzschildDistance (effectiveMass); deltaDistance = (distance - schwarzschildDistance) / schwarzschildDistance; if (deltaDistance < 0) // Schwarzschild distance too large { lowerDistance = (upperDistance + lowerDistance) / 2; } else // Schwarzschild distance too small { upperDistance = (upperDistance + lowerDistance) / 2; } delta = (upperDistance - lowerDistance) / lowerDistance; } while (delta > limit); if (java.lang.Math.abs (deltaDistance) > 1e-6) { java.lang.System.out.println ("unstable numerical result:"); java.lang.System.out.println ("Computed distance = " + distance); java.lang.System.out.println ("Schwarzschild distance = " + schwarzschildDistance); } return schwarzschildDistance; } // This method outputs the table header private static void printHeader () { java.lang.System.out.print ("lambda_M in kg/m;"); java.lang.System.out.print (" M_S in kg ;"); java.lang.System.out.print (" M_S/M ;"); java.lang.System.out.print (" M_eff in kg ;"); java.lang.System.out.print (" d_S im m ;"); java.lang.System.out.print (" d_S/R ;"); java.lang.System.out.print (" d_S in ly ;"); java.lang.System.out.print (" (R-d)/R "); java.lang.System.out.println (); } // This method outputs a table result line // lambdaM is the linear mass density of the black shell in kg/m // distance is the closest distance of the mass m to to the mass point dM in the black shell in m // mEff is the effective mass of a sphere in the black shell private static void printResults (double lambdaM, double distance, double effectiveMass) { double mShell = lambdaM * R * 2 * java.lang.Math.PI; // mass of black shell double mRatio = mShell / MUniverse; double dRatio = distance / R; double dLightYears = distance / LightYear; double deltaRatio = (R - distance) / R; java.lang.System.out.printf ("%15.7e", lambdaM); java.lang.System.out.print (" ;"); java.lang.System.out.printf ("%15.6e", mShell); java.lang.System.out.print (" ;"); java.lang.System.out.printf ("%15.6f", mRatio); java.lang.System.out.print (" ;"); java.lang.System.out.printf ("%15.2e", effectiveMass); java.lang.System.out.print (" ;"); java.lang.System.out.printf ("%15.2e", distance); java.lang.System.out.print (" ;"); java.lang.System.out.printf ("%15.2e", dRatio); java.lang.System.out.print (" ;"); java.lang.System.out.printf ("%15.0f", dLightYears); java.lang.System.out.print (" ;"); java.lang.System.out.printf ("%15.9f", + deltaRatio); java.lang.System.out.println (); } // This method has a loop for computing a sequence of effective masses and Schwarzschild distances for different lambdaM values // lambdaM is iterated from lambdaMfirst to lambdaMlast in steps of deltaLambda private static void computeScharzschildRadii (double lambdaMfirst, double lambdaMlast, double deltaLambda) { double lambdaM = lambdaMfirst; // linear mass density of black shell at Hubble radius in kg/m lambdaMlast = lambdaMlast * 1.00000001; // for the reason of numerical rounding errors printHeader (); do { double schwarzschildDistance = solve (lambdaM); double effectiveMass = effectiveMass (schwarzschildDistance); printResults (lambdaM, schwarzschildDistance, effectiveMass); lambdaM = lambdaM + deltaLambda; } while (lambdaM <= lambdaMlast); } // Main program public static void main (java.lang.String [] arguments) { computeScharzschildRadii (lambdaMfirst, lambdaMlast, deltaLambda); } }