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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 71 "To display the (extende
d) Schwarzschild manifold in Kruskal coordinates" }}{PARA 256 "" 0 "" 
{TEXT 257 29 "and to look at some geodesics" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 98 "The black lines are th
e singularities, r=0; the black hole \"crunch\" and the white hole \"e
xplosion\"" }}{PARA 0 "" 0 "" {TEXT -1 123 "The hyperbolae are lines o
f constant r;  the red lines are the coordinate singularity at r=2m an
d t= plus or minus infinity" }}{PARA 0 "" 0 "" {TEXT -1 78 "The point \+
in the middle, i.e., (u,v)=(0,0) is r=2m and all finite values of t." 
}}{PARA 0 "" 0 "" {TEXT -1 81 "u and v are dimensionless; quantitites \+
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,u,-u,sqrt(u^2-1),-sqrt(u^2-1),sqrt(u^2-2),-sqrt(u^2-2),sqrt(u^2-.5),-
sqrt(u^2-.5)],u=0..a,0..a,labels=[u,v],color=[black,black,blue,green,b
lue,green,red,red,plum,plum,plum,plum,plum,plum],thickness=[7,7,3,3,3,
3,5,5,3,3,3,3,3,3],scaling=constrained,title=`Kruskal coordinates`):" 
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1),u,-u,sqrt(u^2-1),-sqrt(u^2-1),sqrt(u^2-2),-sqrt(u^2-2),sqrt(u^2-.5)
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3,3,3,5,5,3,3,3,3,3,3],scaling=constrained,title=`Kruskal coordinates`
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Kruskal~coordinates|+~for~the~Schwarzschild~manifold;|+~red~lines~are~
r=2m~and~t=infinityG-%%VIEWG6$;$!#D!\"\"$\"#DF^\\sF[\\s" 1 2 0 1 10 0 
2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+
3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve \+
10" "Curve 11" "Curve 12" "Curve 13" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 90 "The turquoise-colored lines are lines of constant value for the
 Schwarzschild coordinate t" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Now le
t's find some timelike geodesics.  These are radial infall, with R the
 location where infall began:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 61 "r:=R->(R/2)*(1+cos(eta)); tau:=R->(R^(3/2)/2)*(eta+sin(eta));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"RG6\"6$%)operatorG%&arro
wGF(,$*&#\"\"\"\"\"#F/*&9$F/,&F/F/-%$cosG6#%$etaGF/F/F/F/F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tauGf*6#%\"RG6\"6$%)operatorG%&arro
wGF(,$*&#\"\"\"\"\"#F/*&)9$#\"\"$F0F/,&%$etaGF/-%$sinG6#F7F/F/F/F/F(F(
F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "t1:=R->sqrt(R-1)*(eta
+(R/2)*(eta+sin(eta)))+2*arctanh(sqrt((R/r(R)-1)/(R-1)));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#t1Gf*6#%\"RG6\"6$%)operatorG%&arrowGF(,&*&-%
%sqrtG6#,&9$\"\"\"F3!\"\"F3,&%$etaGF3*&#F3\"\"#F3*&F2F3,&F6F3-%$sinG6#
F6F3F3F3F3F3F3*&F9F3-%(arctanhG6#-F/6#*&,&*&F2F3-%\"rG6#F2F4F3F3F4F3F1
F4F3F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "t2:=R->sqrt
(R-1)*(eta+(R/2)*(eta+sin(eta)))+2*arccoth(sqrt((R/r(R)-1)/(R-1)));" }
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wGF(,&*&-%%sqrtG6#,&9$\"\"\"F3!\"\"F3,&%$etaGF3*&#F3\"\"#F3*&F2F3,&F6F
3-%$sinG6#F6F3F3F3F3F3F3*&F9F3-%(arccothG6#-F/6#*&,&*&F2F3-%\"rG6#F2F4
F3F3F4F3F1F4F3F3F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "The jum
p from t1 to t2 occurs when r reaches the value 1, i.e., r=2m;" }}
{PARA 0 "" 0 "" {TEXT -1 39 "let us denote that value of eta by eta2" 
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rrowGF(-%'arccosG6#,&*&\"\"#\"\"\"9$!\"\"F2F2F4F(F(F(" }}}{EXCHG 
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\\wzg%eHF/7$$\"$?\"F*$\"3;)pXo$RweHF/-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-
%+AXESLABELSG6$Q\"R6\"Q!Fc]q-%%VIEWG6$;F(Fd\\q%(DEFAULTG" 1 2 0 1 10 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 101 "So eta2 varies from 0 (at R=1) to Pi, at
 R=infinity, in a monotone way, but rising reasonably slowly!" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "NEXT task
 is to define the Kruskal coordinates in terms of those above" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "u1:=R->sqrt(r(R)-1)*exp(r(R)
/2)*cosh(t1(R)/2); v1:=R->sqrt(r(R)-1)*exp(r(R)/2)*sinh(t1(R)/2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u1Gf*6#%\"RG6\"6$%)operatorG%&arrow
GF(*(-%%sqrtG6#,&-%\"rG6#9$\"\"\"F5!\"\"F5-%$expG6#,$*&#F5\"\"#F5F1F5F
5F5-%%coshG6#,$*&F<F5-%#t1GF3F5F5F5F(F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#v1Gf*6#%\"RG6\"6$%)operatorG%&arrowGF(*(-%%sqrtG6#,&
-%\"rG6#9$\"\"\"F5!\"\"F5-%$expG6#,$*&#F5\"\"#F5F1F5F5F5-%%sinhG6#,$*&
F<F5-%#t1GF3F5F5F5F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 
"u2:=R->sqrt(1-r(R))*exp(r(R)/2)*sinh(t2(R)/2); v2:=R->sqrt(1-r(R))*ex
p(r(R)/2)*cosh(t2(R)/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u2Gf*6#
%\"RG6\"6$%)operatorG%&arrowGF(*(-%%sqrtG6#,&\"\"\"F1-%\"rG6#9$!\"\"F1
-%$expG6#,$*&#F1\"\"#F1F2F1F1F1-%%sinhG6#,$*&F<F1-%#t2GF4F1F1F1F(F(F(
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2Gf*6#%\"RG6\"6$%)operatorG%&a
rrowGF(*(-%%sqrtG6#,&\"\"\"F1-%\"rG6#9$!\"\"F1-%$expG6#,$*&#F1\"\"#F1F
2F1F1F1-%%coshG6#,$*&F<F1-%#t2GF4F1F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "uu
:=sqrt(rr-1)*exp(rr/2)*cosh(tt/2); vv:=sqrt(rr-1)*exp(rr/2)*sinh(tt/2)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#uuG*(,&%#rrG\"\"\"F(!\"\"#F(\"
\"#-%$expG6#,$*&F+F)F'F(F(F(-%%coshG6#,$*&F+F)%#ttGF(F(F(" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#vvG*(,&%#rrG\"\"\"F(!\"\"#F(\"\"#-%$expG6#,$
*&F+F)F'F(F(F(-%%sinhG6#,$*&F+F)%#ttGF(F(F(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 106 "The actual shapes of the curves depend somewhat on where
 on begins the trajectory, i.e., on the value of R" }}{PARA 0 "" 0 "" 
{TEXT -1 129 "Therefore, let us try to look at these curves in various
 different sets of coordinates, for several different, fixed values of
 R." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 259 84 
"We begin with R=1.5, which means that it begins very close to the bla
ck hole, at 3m!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "mk1p5a:
=plot([r(1.5),t1(1.5)],eta=0..eta2(1.5),thickness=3): mk1p5b:=plot([r(
1.5),t2(1.5)],eta=eta2(1.5)..Pi,0..8,thickness=3):" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "display(mk1p5a,mk1p5b);" }}{PARA 13 "" 1 "" 
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CP6G<k()QF,7$$\"3++++%opK#GF,$\"3QV8%Rd)e()QF,7$$\"3=+++XEzjGF,$\"3]hM
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\"3u5;V$3Hv)QF,7$$\"39+++M=B$)HF,$\"3u=QyeX_()QF,7$$\"3$)*****\\n'3@IF
,$\"3]\"HPr2Bv)QF,7$$\"35+++IyWhIF,$\"3?1sZOE_()QF,7$$\"3&)*****fgy+5$
F,$\"3mwZ5tD_()QF,7$$\"+]EfTJ!\"*$\"+oD_()QF]`oF\\^mF`[l-%+AXESLABELSG
6%Q$eta6\"Q!Fd`o-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"+aEfTJF]`o;F($\"\"
)F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" "Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
94 "The red curve is r, which simply decreases nicely from 1.5 to zero
, as eta varies from 0 to Pi" }}{PARA 0 "" 0 "" {TEXT -1 155 "The gree
n curve is t, which increases all the way to positive infinity at 1, a
nd then decreases down from there to some value about +3.9 as r reache
s zero." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "On the other hand, no
w let's display those same variables as an actual trajectory, as it wo
uld appear on a Minkowski-type diagram, " }}{PARA 0 "" 0 "" {TEXT -1 
33 "with t vertical and r horizontal." }}{PARA 0 "" 0 "" {TEXT -1 123 
"This time the colors are red for motion PRIOR to the horizon, and gre
en for motion AFTER crossing the horizon, at r/2m = 1." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "plot([[r(1.5),t1(1.5),eta=0..eta2(
1.5)],[r(1.5),t2(1.5),eta=eta2(1.5)..Pi]],labels=[r,t],thickness=4);" 
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2$\"3eA#yN@ay)QF*7$$\"3-?n2#eDR*eF2$\"3yr=ki)Gx)QF*7$$\"3#)>t?avo`ZF2$
\"3-\"p3\"G<k()QF*7$$\"3yfF_x$yyw$F2$\"3y9#QRd)e()QF*7$$\"3yp[#z-=a(GF
2$\"3Sb;Oeeb()QF*7$$\"3!yg#G\"e(H2@F2$\"3o\"*[Hix`()QF*7$$\"3rV&R\\hB_
]\"F2$\"3CA6V$3Hv)QF*7$$\"3_LIoU$oYQ*!#?$\"3GoNyeX_()QF*7$$\"3'**H6D,Z
!RaFi_m$\"35Fs8xI_()QF*7$$\"3y`\"=D-3uS#Fi_m$\"3k!>xkjAv)QF*7$$\"3_w&z
X=\")=Y'!#@$\"3MvZ5tD_()QF*7$F+$\"+rD_()Q!\"*-F^^l6&F`^lF+Fa^lF+-%+AXE
SLABELSG6$%\"rG%\"tG-%*THICKNESSG6#\"\"%-%%VIEWG6$%(DEFAULTGF^bm" 1 2 
0 1 10 4 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2
" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "Now let's look at the same \+
trajectory, BUT in Kruskal coordinates, where both u and v are well-be
haved as the horizon is crossed." }}{PARA 0 "" 0 "" {TEXT -1 47 "   Th
e ONLY change at the horizon is the color!" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "plot([[u1(1.5),v1(1.5),eta=
0..eta2(1.5)],[u2(1.5),v2(1.5),eta=eta2(1.5)..Pi]],labels=[u,v],thickn
ess=4,scaling=constrained,title=`v vs. u,\\n radial infall from R=3m`)
;" }}{PARA 13 "" 1 "" {GLPLOT2D 521 521 521 {PLOTDATA 2 "6)-%'CURVESG6
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45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 115 "Lastly, I would like to show the u,v - trajectory abov
e, but on top of the usual quartering of Kruskal coordinates;" }}
{PARA 0 "" 0 "" {TEXT -1 120 "we only need the top quadrant of those c
oords, and only out far enough to see the trajectory hit the singulari
ty at r=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "fall1p5:=plo
t([[u1(1.5),v1(1.5),eta=0..eta2(1.5)],[u2(1.5),v2(1.5),eta=eta2(1.5)..
Pi]],labels=[u,v],thickness=4,scaling=constrained,title=`v vs. u,\\n r
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y(corduv(3.5),fall1p5);" }}{PARA 13 "" 1 "" {GLPLOT2D 521 521 521 
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0 "" 0 "" {TEXT -1 93 "Interesting to see how long the t-variable take
s before it suddenly takes off toward infinity" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 160 "df3:=plot([[u1(3),v1(3),eta=0..eta2(3)],[u2(3
),v2(3),eta=eta2(3)..Pi]],labels=[u,v],thickness=4,scaling=constrained
,title=`v vs. u,\\n radial infall from R=6m`):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "display(df3,corduv(145));" }}{PARA 13 "" 1 "" 
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mg+%37F]z7$Fghr$!3N0j6HW:R7F]z7$F\\ir$!3!p;U(pNlp7F]z7$Fair$!3_A%zH4zw
H\"F]z7$Ffir$!3NhgI$[*zH8F]z7$F[jr$!37G$eQKG&e8F]z7$F`jr$!3!Hn>ECf\"*Q
\"F]z7$Fejr$!3$>!=dBpZ=9F]z7$Fa]n$!3M$==&eF)*\\9F]zF\\[sFj`o-%+AXESLAB
ELSG6%%\"uG%\"vG-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%&TITL
EG6#%Bv~vs.~u,|+~radial~infall~from~R=6mG-%%VIEWG6$;F+Fa]nFa_v" 1 2 0 
1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" 
"Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" 
"Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Cur
ve 16" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "ALSO of some interest a
re the paths of null rays." }}{PARA 0 "" 0 "" {TEXT -1 95 "Of course i
n Kruskal coord's ingoing light rays are simply straight lines at 135 \+
degree slopes:" }}{PARA 0 "" 0 "" {TEXT -1 93 "but in Schwarzschild co
ords they travel along the following paths, at least so long as r > 2m
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "nul:=R->R-r + log((R-1)
/(r-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nulGf*6#%\"RG6\"6$%)ope
ratorG%&arrowGF(,(9$\"\"\"%\"rG!\"\"-%$logG6#*&,&F-F.F.F0F.,&F/F.F.F0F
0F.F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "I create the followi
ng list of R values, where of course R is simply a point on the trajec
tory, " }}{PARA 0 "" 0 "" {TEXT -1 27 "from which I begin plotting" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "lst:=[1.1,1.278,1.5,2,3];" 
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\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot([seq(nul
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Corresponding to some starting poi
nt---at v=0 = t---in Kruskal coords are the following values" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "uv0:=sqrt((r-1)*exp(r));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$uv0G*$*&,&%\"rG\"\"\"F)!\"\"F)-%$ex
pG6#F(F)#F)\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "lstuv0:
=[evalf(seq(eval(uv0,r=n),n=lst))]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 58 "Which then generate the very straightforward curves below." }}
{PARA 0 "" 0 "" {TEXT -1 99 "One sees that it is even possible for the
se curves to come into that part of Region II where u < 0." }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "nools:=[seq(plot
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "display(nools,corduvm(4));" 
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