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File:Lorentz transform of world line.gif

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


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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.

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Source of program used to generate image:


#define PI 3.141592653589793238462

#define SX 200
#define SY 200
#define PL 100
#define DN 2000

unsigned char img[SX][SY];

double path[PL+1][2], dots[DN];

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

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 fn[100];
  int n, x, y, t, i, w;
  double a, b, da, db, ta, tb;
  FILE *f;
  for(t=0;t<=PL;++t) path[t][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) {
  for(t=PL;t>=0;--t) path[t][1]-=path[0][1];
  for(t=1;t<=PL;++t) {
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