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PaneDivider 399;
SignificantDigits 14;
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BackgroundColor 255 255 255;
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Text "Schwarzschild geometry in Eddington-Finkelstein coordinates.  Light rays.

For a very tidy treatment, see J.B. Hartle's <I>General Relativity</I>, section 12.1.

Version 0.2  3-2-11
To do:
(i) Add collapsing matter and shade non-physical region.


Mass";
Color 3;
Expr M=1;
Text "T = Schwarschild t 
v [V] is Eddington-Finkelstein advanced time 
x= Schwarzschild r
y= Eddington-Finkelstein v-r   [what Hartle calls t-tilde ... Schwarzschild time far away from the horizon.]

[Also for a brief section offer the equations of the same light rays taking simply y = v.  Plotting t-tilde = v-r  on the vertical axis is more natural since it is asymptotically the Schwarzschild time.   The other is available just in case one wants a straight plot of v vs. r.   Be careful not to confuse the two representations!   Only one should be plotted at a time.]
";
Color 4;
Expr function(W,R)=R+2*M*ln(abs(R/(2*M)-1));
Color 2;
Expr function(V,R,T)=T+function(W,R);
Color 17;
Expr ln(abs(x/(2*M)-1));
Text "Adjustable light rays:";
MathPaneSlider 25;
Expr a=slider([0,20]);
Color 3;
MathPaneSlider 47;
Expr b=slider([-20,20]);
Text "
WITH y= v-r:

Ingoing radial null rays (inner null cone) [y=v-r]";
Color 17;
Expr y=a-x,x>0;
Color 17;
Expr y+x=c,x>0,y>0,in(c,set(1,2,3,4,5,6,7,8));
Color 2;
Grain 0.7333333333333333;
Expr d*y/(d*x)=-1;
Text "Outer null cone  [y=v-r]:";
Color 17;
Expr y+x-(2*function(W,x))=b,x>0;
Color 17;
Expr y+x-(2*function(W,x))=d,x>0,y>0,in(d,set(2,3,4,5,6,7,8,9));
Color 2;
Grain 0.15;
Expr d*y/(d*x)=2/[1-2*M/x]-1;
Text "WITH y = v [given for reference only; don't switch these on while y=v-r versions above are displayed]

Ingoing radial null rays (inner null cone) [y=v]";
Color 17;
Expr y=a,x>0;
Color 17;
Expr d*y/(d*x)=0;
Text "Outer null cone  [y=v]";
Color 17;
Expr y-(2*function(W,x))=b,x>0;
Color 17;
Grain 0.5833333333333334;
Expr d*y/(d*x)=2/[1-2*M/x];
Text "END y = v section.


Horizon";
Color 8;
Expr x=2*M;
Text "Singularity";
Color 2;
Expr x=0;
Color 2;
Expr x<0;
Text "First cut at light cones,  with y = v-r: ";
Color 5;
Expr P=12.5625+4.96875*i;
Color 6;
Expr X_0=Re([P]),Y_0=Im([P]),K=0.2;
Text "Slopes of ingoing and outgoing null rays:";
Color 7;
Expr function(m_1,X,Y)=-1;
Expr function(m_2,X,Y)=2/[1-2*M/X]-1;
Text "Light cones:";
Color 17;
Expr y-Y_0=function(m_1,X_0,Y_0)*[x-X_0],[y-Y_0]^2+[x-X_0]^2<K^2;
Color 17;
Expr y-Y_0=function(m_2,X_0,Y_0)*[x-X_0],[y-Y_0]^2+[x-X_0]^2<K^2;
Text "Shading the light cones.   Future light cone only.";
Color 5;
Expr 1/function(m_1,X_0,Y_0)<(x-X_0)/(y-Y_0)<1/function(m_2,X_0,Y_0),[y-Y_0]^2+[x-X_0]^2<K^2;
Color 17;
Expr 1/function(m_1,X_0,Y_0)<(x-X_0)/(y-Y_0)<1/function(m_2,X_0,Y_0),[y-Y_0]^2+[x-X_0]^2<K^2,y>Y_0;
Text "

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