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Lorentz_transform_of_world_line.gif (200 × 200 pixels, file size: 166 KB, MIME type: image/gif, looped, 100 frames)

Summary

Description
English: Changing views of spacetime along the world line of a rapidly accelerating observer

In this animation, the vertical direction indicates time and the horizontal direction indicates distance, the dashed line is the spacetime trajectory ("world line") of an accelerating observer. The small dots are arbitrary events in spacetime that are stationary relative to each other. The events passing the two diagonal lines in the lower half of the picture (the past light cone of the observer) are those that are visible to the observer.

The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime changes when the observer accelerates. In particular, absolute time is a concept not applicable in Lorentzian spacetime: events move up-and-down in the figure depending on the acceleration of the observer.

Compare this to the absolute time apparent in Image:Galilean transform of world line.gif.
Source Own work
Author Cyp
Source code
InfoField
Source of program used to generate image:
//GPL
#include <stdio.h>
#include <stdlib.h>
#include <math.h>

#define PI 3.141592653589793238462

#define SX 256
#define SY 256
#define PL 100
#define DN 2000

unsigned char imgSX][SY];

double pathPL+1][2], dotsDN][2];

void dodot(int x, int y, double f) {
  if(x<0||x>=SX||y<0||y>=SY) return;
  imgy][x*=f;
}

void dospot(int x, int y) {
  dodot(x, y, .5);
  dodot(x+1, y, .75);
  dodot(x-1, y, .75);
  dodot(x, y+1, .75);
  dodot(x, y-1, .75);
}

void dobigspot(int x, int y) {
  int a, b;
  for(b=-3;b<=3;++b) for(a=-3;a<=3;++a) if(a*a+b*b<=9) dodot(x+a, y+b, (a*a+b*b)/10.);
}

void dospotd(double t, double x) {
  dospot((x+1)*(SX/2.), (-t+1)*(SY/2.));
}

void dosmallspotd(double t, double x) {
  dodot((x+1)*(SX/2.), (-t+1)*(SY/2.), .25);
}

void dobigspotd(double t, double x) {
  dobigspot((x+1)*(SX/2.), (-t+1)*(SY/2.));
}

int main() {
  char fn100];
  int n, x, y, t, i, w;
  double a, b, da, db, ta, tb;
  FILE *f;
  path0][0=path0][1=0;
  for(t=0;t<=PL;++t) patht][1=0;
  for(n=1;n<10;++n) {
    a=rand()%20000/10000.-1; a/=n*n*n*n/200.; b=rand()%20000*(PI/10000);
    for(t=0;t<=PL;++t) {
      patht][1+=a*sin((2*PI/PL)*n*t+b);
    }
  }
  for(t=PL;t>=0;--t) patht][1-=path0][1];
  path0][0=0;
  for(t=1;t<=PL;++t) {
    a=patht][1-patht-1][1];
    patht][0=patht-1][0+sqrt(1+a*a);
  }
  for(t=0;t<DN;++t) {
    a=rand()%20000/10000.-1; b=rand()%20000/10000.-1;
    dotst][0=a*pathPL][0/2; dotst][1=b*1000;
  }
  for(n=0;n<100;++n) {
    i=PL*n/100;
    a=pathi+1][0-(da=pathi][0]); b=(db=pathi][1])-pathi+1][1];
    ta=pathPL][0]; tb=pathPL][1];
    a/=50.; b/=50.;
    for(y=0;y<SY;++y) for(x=0;x<SX;++x) imgy][x=255;
    for(y=0;y<SY;++y) imgy][y*SX/SY*=.5;
    for(y=0;y<SY;++y) imgy][(SY-y-1)*SX/SY*=.5;
    for(w=-20;w<=20;++w)
      for(t=0;t<PL;++t) dospotd(a*(patht][0-da-w*ta)+b*(patht][1-db-w*tb),
                                b*(patht][0-da-w*ta)+a*(patht][1-db-w*tb));
    for(w=-20;w<=20;++w)
      for(t=0;t<PL;t+=10) dobigspotd(a*(patht][0-da-w*ta)+b*(patht][1-db-w*tb),
                                  b*(patht][0-da-w*ta)+a*(patht][1-db-w*tb));
    for(w=-20;w<=20;++w)
      for(t=0;t<DN;++t) dospotd(a*(dotst][0-da-w*ta)+b*(dotst][1-db-w*tb),
                                b*(dotst][0-da-w*ta)+a*(dotst][1-db-w*tb));
//if(n==0) printf("%lf; %lf, %lf, %lf; %lf, %lf, %lf, %lf, %lf\n", a*(path[PL][0]-da-1*ta)+b*(path[PL][1]-db-1*tb), path[PL][0], da, 1*ta, path[PL][1], db, 1*tb, path[0][0], path[0][1]);
    sprintf(fn, "lor%04d.pgm", n);
    f=fopen(fn, "wb");
    fprintf(f, "P5\n%d %d\n255\n", SX, SY);
    fwrite(img, 256*256, 1, f);
    fclose(f);
  }
}

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GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.
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Date/TimeThumbnailDimensionsUserComment
current 09:24, 21 August 2005 Thumbnail for version as of 09:24, 21 August 2005200 × 200 (166 KB) CypNew and improved - now with bigger and better dots
08:29, 21 August 2005 Thumbnail for version as of 08:29, 21 August 2005200 × 200 (151 KB) Cyp{{GFDL}}

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