GraphingCalculator 3.5;
Window 44 7 806 1018;
PaneDivider 430;
SignificantDigits 14;
FontSizes 18 14 10;
BackgroundType 0;
SliderControlValue 0;
2D.BottomLeft -1.71875 -2.9375;
2D.Axes 0;
2D.GraphPaper 0;
Text "Length Contraction – light clock.

Version 0.3, 2-7-09
To do:
(i) ?


Speed of light";
Color 7;
MathPaneSlider 4;
Expr c=slider([0,10,40]);
Text "Separation of light clock mirrors; length of mirrors; thickness of mirrors";
Color 2;
MathPaneSlider 200;
Expr D=slider([0,1]);
Color 3;
MathPaneSlider 107;
Expr L=slider([0,1]);
Color 2;
MathPaneSlider 64;
Expr d=slider([0,0.1]);
Text "Fraction of the speed of light at which the ship (frame P) travels relative to stationary observers (frame O).";
Color 5;
MathPaneSlider 37;
Expr f=slider([-1,1,40]);
Text "Speed of ship (frame P) relative to frame O; Lorentz factor.";
Color 6;
Expr V=c*f,g=1/sqrt(1-V^2/c^2);
Text "Horizontal and vertical dimensions of ship; exponent";
Color 4;
MathPaneSlider 96;
Expr a=slider([0,1]);
Color 4;
MathPaneSlider 200;
Expr b=slider([0,1]);
Color 5;
Expr m=6;
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 17;
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; ship in motion.";
Color 17;
Expr P;
Color 17;
Expr P,P+I;
Color 17;
Expr P,P+J;
Color 17;
Expr [(x-(n*V))/(D_O+a)]^m+[y/b]^m<1;
Text "Light clock mirrors in rest frame of ship";
Color 8;
Expr (D-d)/2<abs(x)<(D+d)/2,-L/2<y<L/2;
Text "Time between ticks in rest frame of clock (proper time):";
Color 2;
Expr T=2*D/c;
Text "Bouncing light signal and its path.   (Note 2-D points won't accept conditions [:-(].)";
Color 2;
Expr [x-[c*mod([n,T])-D/2]]^2+[y+d/2]^2<d^2,0<mod([n,T])<T/2;
Color 2;
Expr [x-[-(c*mod([n,T]))+3*D/2]]^2+[y-d/2]^2<d^2,T/2<mod([n,T])<T;
Color 17;
Expr y=-d/2,-D/2<x<min([c*n-D/2,D/2]);
Color 17;
Expr y=d/2,D/2>x>max([3*D/2-(c*n),-D/2]);
Text "Mirrors in (length-contracted) moving ship";
Color 17;
Expr (D_O-d)/2<abs(x-(n*V))<(D_O+d)/2,-L/2<y<L/2;
Text "Time between ticks in frame O watching ship (clock) fly by at speed V (this is time dilation); travel times for right- and left-moving clock light signals.   Note TR+TL=T.    (Contracted) length of ship moving relative to frame O. ";
Color 7;
Expr T_O=g*T,T_R=D_O/[c-V],T_L=D_O/[c+V],D_O=D/g;
Text "Light signal in moving ship and its path.   ";
Color 17;
Expr [x-(V*T_O*floor(n/T_O))-[c*mod([n,T_O])-D_O/2]]^2+[y]^2<d^2,0<mod([n,T_O])<T_R;
Color 17;
Expr [x-(V*[T_O*floor(n/T_O)+T_R])-[-(c*[mod([n,T_O])-T_R])+D_O/2]]^2+[y]^2<d^2,T_R<mod([n,T_O])<T_O;
Color 17;
Expr -d/2<y<0,0<x+D_O/2<min([n*c,D_O+V*T_R]);
Color 17;
Expr 0<y<d/2,0>x-[D_O/2+V*T_R]>max([-(c*[n-T_R]),V*T_L-D_O]);
Text "Length-contracted rod: ";
Color 17;
Expr abs(y)<d/2,abs(x-(n*V))<D_O/2;
Text "Rod at rest:";
Color 17;
Expr abs(y)<d/2,abs(x-(5*V))<D/2;
Text "

Lengths perpendicular to the motion must remain the same for all observers.   Assume for the sake of demonstration that moving perpendicular lengths contract relative to the rest length (f=0.8 or so helps):";
Color 8;
Expr L_P=L/g;
Text "Rod at rest at 5V; paint marks on rod as brushes from ship paint it:";
Color 17;
Expr -d/2<abs(x-(5*V))<d/2,-L/2<y<L/2;
Color 17;
Expr -d<abs(x-(5*V))<d,L_P/2-d<abs(y)<L_P/2+d,n>5;
Text "Rod at rest in moving ship; paint brushes at ends of rod:";
Color 17;
Expr -d/2<abs(x-(n*V))<d/2,-L_P/2<y<L_P/2;
Color 17;
Expr P+J*L_P/2;
Color 17;
Expr P-J*L_P/2;
Text "Now view from the point of view of the rod at rest in the ship.  If perpendicular moving lengths contract, the rod outside the ship moving past will contract and pass inside the paint brushes on the rod in the ship.  So is there paint on the rod outside the ship, or not?   Since the answer to this very physical question depends upon the observer (which of course it can't), it must be that lengths perpendicular to the motion are agreed upon by all observers.

Rod at rest in ship at rest:";
Color 17;
Expr -d/2<abs(x)<d/2,-L/2<y<L/2;
Color 17;
Expr J*L/2;
Color 17;
Expr [-J]*L/2;
Text "Rod moving past ship at rest:";
Color 17;
Expr -d/2<abs(x+n*V-(5*V))<d/2,-L_P/2<y<L_P/2;
Text "


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