GraphingCalculator 4;
Window 45 0 644 796;
PaneDivider 282;
SignificantDigits 14;
FontSizes 18;
BackgroundType 0;
SliderSteps 400;
SliderControlValue 0;
T 0 5;
2D.BottomLeft -2.015625 -1.953125;
2D.Axes 0;
2D.GraphPaper 0;
Text "Relativity of simultaneity.   
Version 0.5, 2-5-09
To do:
(i) Change bulb color when they flash?  (Fluff, but fun.)
(ii) Add return flash for length contraction?
(iii) Include Galilean relativity – pulses received at same time are emitted at same time in either frame.


Two flash bulbs are situated equidistant from the center of a ship.   In the rest frame of the ship (P), the bulbs flash simultaneously, and the signals from the bulbs therefore arrive simultaneously at a photodetector situated at the center of the ship.   In a frame (O) watching the ship go by at speed V, however, the simultaneously arriving signals could not possibly have been emitted at the same time.

Speed of light";
Color 7;
MathPaneSlider 4;
Expr c=slider([0,10,40]);
Text "Distance of flash bulbs from origin";
Color 2;
MathPaneSlider 100;
Expr D=slider([0,2]);
Text "Fraction of the speed of light at which the ship (frame P) travels relative to stationary observers (frame O).  [f=0.75 is a good default.]";
Color 5;
MathPaneSlider 35;
Expr f=slider([-1,1,40]);
Text "Speed of ship (frame P) relative to frame O.";
Color 6;
Expr V=c*f;
Text "Horizontal and vertical dimensions of ship; exponent";
Color 4;
MathPaneSlider 96;
Expr a=slider([0,1]);
Color 4;
MathPaneSlider 85;
Expr b=slider([0,1]);
Color 5;
Expr m=6;
Text "Velocity vector of frame P relative to O";
Expr d=slider([0,1]);
Color 6;
Expr h=slider([0,0.1]);
Color 17;
Expr P+J*d,P+J*d+U;
Color 17;
Expr P+J*[d+h],P+J*[d+h]+U;
Text "Basis vectors";
Expr I=vector(1,0),J=vector(0,1);
Text "Origin of frame O; origin of frame P";
Color 3;
Expr O=vector(0,0),P=O+n*V*I;
Text "Frame with origin O; ship at rest";
Color 8;
Expr O;
Color 17;
Expr O,I;
Color 17;
Expr O,J;
Color 7;
Expr [x/(D+a)]^m+[y/b]^m<1;
Text "Frame with origin P; moving ship";
Color 8;
Expr P;
Color 17;
Expr P,P+I;
Color 17;
Expr P,P+J;
Color 7;
Expr [(x-(n*V))/(D+a)]^m+[y/b]^m<1;
Text "Flash bulbs in rest frame of ship; flashes simultaneous in that frame (don't arrive at P simultaneously)";
Color 5;
Expr D*I;
Color 5;
Expr -(D*I);
Color 2;
Expr [x-D]^2+y^2=[c*n]^2;
Color 2;
Expr [x+D]^2+y^2=[c*n]^2;
Text "Clock synchronization procedure:";
Color 17;
Expr M*D*I,in(M,set(-5,-4,-3,-2,-1,0,1,2,3,4,5));
Color 17;
Expr M*D*J,in(M,set(-5,-4,-3,-2,-1,0,1,2,3,4,5));
Color 17;
Expr x^2+y^2=[c*n]^2;
Text "Flash bulbs in frame relative to which ship has velocity V";
Color 5;
Expr P+D*I;
Color 5;
Expr P-(D*I);
Text "According to stationary observer, if light travels at c then left bulb must flash earlier than right in order for signal from left and right to arrive at the same time (since the signal from the left must travel further): TL << TR. 

Time of emission of each flash:";
Color 8;
Expr T_L=D/(c+V),T_R=D/(c-V);
Text "Flashes which arrive simultaneously at P were not emitted at the same time:";
Color 17;
Expr [x-[D+T_R*V]]^2+y^2=[c*[n-T_R]]^2,n>T_R;
Color 17;
Expr [x-[-D+T_L*V]]^2+y^2=[c*[n-T_L]]^2,n>T_L;
Text "Essentially the same setup can be used to illustrate length contraction as well.   Length is the distance between two points measured at the same time in a frame.   In the rest frame of a rod, if you station an observer at one end and set off flashes at both ends simultaneously in that frame (as above), that observer calculates the (rest) length as L=cT, where T is the travel time in that frame.   On the other hand, according to an observer moving relative to the ship,  those flashes were not simultaneous (as above).  Instead, set off the flashes simultaneously (in that frame) just as an observer (P+DI!)  passes one end.   The flash from the other end has to travel an extra distance Vtr to get to the observer, thus calculating the length l+Vtr.    (Turn f down to 0.5.)   The return flash travels a distance l-vtl.  Combine with the fact that tr+tl = gT, and you have length contraction, l=L/g.

Galilean simultaneity.   In Galilean relativity light would pick up/lose the forward speed of the ship, c+Vcos(theta).   (Complete.)";
Color 2;
Expr T_G=D/V;
Text "


<size=14>Author: David A. Craig <<http://web.lemoyne.edu/~craigda/>";