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{SECT 0 {EXCHG {PARA 237 "" 0 "" {TEXT 252 20 "W. Tenten  Aug. 2006" }
}{PARA 237 "" 0 "" }{PARA 237 "" 0 "" {TEXT 252 51 "Berechnung mechani
scher, relativistischer  Metriken" }}}{EXCHG {PARA 237 "> " 0 "" 
{MPLTEXT 1 253 237 "restart:with(plots):with(linalg):with(tensor):with
(difforms):                                                           \+
                                                                      \+
                                               " }}{PARA 238 "" 1 "" 
{TEXT 254 49 "Warning, the name changecoords has been redefined" }
{TEXT 254 1 "\n" }}{PARA 238 "" 1 "" {TEXT 254 79 "Warning, the protec
ted names norm and trace have been redefined and unprotected" }{TEXT 
254 1 "\n" }}}{EXCHG {PARA 237 "" 0 "" {TEXT 252 22 "Spherische Koordi
naten" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 23 "coord:=[t,r,th
eta,phi];" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>I&coordG6\"7&I\"tGF%I\"
rGF%I&thetaGF%I$phiGF%" }{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG 
{PARA 237 "" 0 "" {TEXT 252 21 "Metrische Komponenten" }}}{EXCHG 
{PARA 237 "> " 0 "" {MPLTEXT 1 253 44 "g_compts:=array(symmetric,spars
e,1..4,1..4):" }}}{EXCHG {PARA 237 "" 0 "" {TEXT 252 37 " Komponenten \+
im Intervall nach d(t)^2" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 
253 29 "g_compts[1,1]:=exp(2*Phi(r)):" }}}{EXCHG {PARA 237 "" 0 "" 
{TEXT 252 38 "  Komponenten im Intervall nach d(r)^2" }}}{EXCHG {PARA 
237 "> " 0 "" {MPLTEXT 1 253 32 "g_compts[2,2]:=exp(2*Lambda(r)):" }}}
{EXCHG {PARA 237 "" 0 "" {TEXT 252 42 "  Komponenten im Intervall nach
 d(theta)^2" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 19 "g_compts
[3,3]:=r^2:" }}}{EXCHG {PARA 237 "" 0 "" {TEXT 252 39 " Komponenten im
 Intervall nach d(phi)^2" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 
253 32 "g_compts[4,4]:=r^2*sin(theta)^2:" }}}{EXCHG {PARA 237 "" 0 "" 
{TEXT 252 29 "Kovarianter metrischer Tensor" }}}{EXCHG {PARA 237 "> " 
0 "" {MPLTEXT 1 253 35 "g:=create([-1,-1 ],eval(g_compts));" }}{PARA 
239 "" 1 "" {XPPMATH 20 "6#>I\"gG6\"-I&TABLEGI*protectedGF(6#7$/I'comp
tsGF%-I'matrixG6$F(I(_syslibGF%6#7&7&-I$expGF/6#,$-I$PhiGF%6#I\"rGF%\"
\"#\"\"!F=F=7&F=-F56#,$-I'LambdaGF%F:F<F=F=7&F=F=*$F;F<F=7&F=F=F=*&F;F
<-I$sinGF/6#I&thetaGF%F</I+index_charGF%7$!\"\"FO" }{TEXT 255 1 " " }
{TEXT 255 1 " " }}}{EXCHG {PARA 237 "" 0 "" {TEXT 252 33 "Kontravarian
ter metrischer Tensor" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 
23 "ginv:=invert(g,'detg');" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>I%gin
vG6\"-I&TABLEGI*protectedGF(6#7$/I'comptsGF%-I'matrixG6$F(I(_syslibGF%
6#7&7&*$-I$expGF/6#,$-I$PhiGF%6#I\"rGF%\"\"#!\"\"\"\"!F?F?7&F?*$-F66#,
$-I'LambdaGF%F;F=F>F?F?7&F?F?*$F<!\"#F?7&F?F?F?*&F<FI-I$sinGF/6#I&thet
aGF%FI/I+index_charGF%7$\"\"\"FS" }{TEXT 255 1 " " }{TEXT 255 1 " " }}
}{EXCHG {PARA 237 "" 0 "" {TEXT 252 49 "Berechnung des Einstein Tensor
s  a) Zwischenwerte" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 23 "
D1g:=d1metric(g,coord):" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 
25 "D2g:=d2metric(D1g,coord):" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 
1 253 23 "Cf1:=Christoffel1(D1g):" }}}{EXCHG {PARA 237 "> " 0 "" 
{MPLTEXT 1 253 27 "RMN:=Riemann(ginv,D2g,Cf1):" }}}{EXCHG {PARA 237 ">
 " 0 "" {MPLTEXT 1 253 23 "RICCI:=Ricci(ginv,RMN):" }}}{EXCHG {PARA 
237 "> " 0 "" {MPLTEXT 1 253 28 "RS:=Ricciscalar(ginv,RICCI):" }}}
{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 27 "Estn:=Einstein(g,RICCI,R
S):" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 25 "displayGR(Einste
in,Estn);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6\"" }{TEXT 255 1 " " }
{TEXT 255 1 " " }}{PARA 239 "" 1 "" {XPPMATH 20 "6#I4The~Einstein~Tens
orG6\"" }{TEXT 255 1 " " }{TEXT 255 1 " " }}{PARA 239 "" 1 "" 
{XPPMATH 20 "6#I6non-zero~components~:G6\"" }{TEXT 255 1 " " }{TEXT 
255 1 " " }}{PARA 239 "" 1 "" {XPPMATH 20 "6#/I%~G11G6\"**-I$expG6$I*p
rotectedGF*I(_syslibGF%6#,$-I$PhiGF%6#I\"rGF%\"\"#\"\"\",(*&-I%diffGF*
6$-I'LambdaGF%F0F1F3F1F3F2-F(6#,$F9F2F3!\"\"F3F3F1!\"#F;F>" }{TEXT 
255 1 " " }{TEXT 255 1 " " }}{PARA 239 "" 1 "" {XPPMATH 20 "6#/I%~G22G
6\",$*&,(*&I\"rGF%\"\"\"-I%diffGI*protectedGF.6$-I$PhiGF%6#F*F*F+\"\"#
-I$expG6$F.I(_syslibGF%6#,$-I'LambdaGF%F2F3!\"\"F+F+F+F*!\"#F<" }
{TEXT 255 1 " " }{TEXT 255 1 " " }}{PARA 239 "" 1 "" {XPPMATH 20 "6#/I
%~G33G6\",$*(I\"rGF%\"\"\",,-I%diffGI*protectedGF-6$-I'LambdaGF%6#F(F(
!\"\"-F,6$-I$PhiGF%F1F(F)*&F(F)-F,6$F5-I\"$GF-6$F(\"\"#F)F)*&F(F)F3F=F
)*(F(F)F3F)F+F)F2F)-I$expG6$F-I(_syslibGF%6#,$F/F=F2F2" }{TEXT 255 1 "
 " }{TEXT 255 1 " " }}{PARA 239 "" 1 "" {XPPMATH 20 "6#/I%~G44G6\"*(I
\"rGF%\"\"\",6-I%diffGI*protectedGF,6$-I$PhiGF%6#F'F'!\"\"*&F*F(-I$cos
G6$F,I(_syslibGF%6#I&thetaGF%\"\"#F(-F+6$-I'LambdaGF%F0F'F(*&F:F(F3F9F
1*&F'F(-F+6$F.-I\"$GF,6$F'F9F(F1*(F'F(F@F(F3F9F(*&F'F(F*F9F1*(F'F(F*F9
F3F9F(*(F'F(F*F(F:F(F(**F'F(F*F(F:F(F3F9F1F(-I$expGF56#,$F<F9F1" }
{TEXT 255 1 " " }{TEXT 255 1 " " }}{PARA 239 "" 1 "" {XPPMATH 20 "6\""
 }{TEXT 255 1 " " }{TEXT 255 1 " " }}{PARA 239 "" 1 "" {XPPMATH 20 "6#
I5character~:~[-1,~-1]G6\"" }{TEXT 255 1 " " }{TEXT 255 1 " " }}}
{EXCHG {PARA 237 "" 0 "" {TEXT 252 53 "Energie Moment Tensor fuer eine
n Tropfen Fluessigkeit" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 
44 "T_compts:=array(symmetric,sparse,1..4,1..4):" }}}{EXCHG {PARA 237 
"> " 0 "" {MPLTEXT 1 253 36 "T_compts[1,1]:=exp(2*Phi(r))*rho(r):" }}}
{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 37 "T_compts[2,2]:=exp(2*Lam
bda(r))*p(r):" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 24 "T_comp
ts[3,3]:=p(r)*r^2:" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 37 "T
_compts[4,4]:=p(r)*r^2*sin(theta)^2:" }}}{EXCHG {PARA 237 "> " 0 "" 
{MPLTEXT 1 253 34 "T:=create([-1,-1],eval(T_compts));" }}{PARA 239 "" 
1 "" {XPPMATH 20 "6#>I\"TG6\"-I&TABLEGI*protectedGF(6#7$/I'comptsGF%-I
'matrixG6$F(I(_syslibGF%6#7&7&*&-I$expGF/6#,$-I$PhiGF%6#I\"rGF%\"\"#\"
\"\"-I$rhoGF%F;F>\"\"!FAFA7&FA*&-F66#,$-I'LambdaGF%F;F=F>-I\"pGF%F;F>F
AFA7&FAFA*&FIF>F<F=FA7&FAFAFA*(FIF>F<F=-I$sinGF/6#I&thetaGF%F=/I+index
_charGF%7$!\"\"FV" }{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG {PARA 
237 "" 0 "" {TEXT 252 85 "Folgend der Einstein Gleichungen (W.Pauli, T
heory of relativity, Pergamon Press 1958)" }}}{EXCHG {PARA 237 "> " 0 
"" {MPLTEXT 1 253 31 "Energy_momentum:=get_compts(T);" }}{PARA 237 "> \+
" 0 "" {MPLTEXT 1 253 28 "Einstein:= get_compts(Estn);" }}{PARA 239 ""
 1 "" {XPPMATH 20 "6#>I0Energy_momentumG6\"-I'matrixG6$I*protectedGF)I
(_syslibGF%6#7&7&*&-I$expGF(6#,$-I$PhiGF%6#I\"rGF%\"\"#\"\"\"-I$rhoGF%
F5F8\"\"!F;F;7&F;*&-F06#,$-I'LambdaGF%F5F7F8-I\"pGF%F5F8F;F;7&F;F;*&FC
F8F6F7F;7&F;F;F;*(FCF8F6F7-I$sinGF(6#I&thetaGF%F7" }{TEXT 255 1 " " }
{TEXT 255 1 " " }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>I)EinsteinG6\"-I'm
atrixG6$I*protectedGF)I(_syslibGF%6#7&7&**-I$expGF(6#,$-I$PhiGF%6#I\"r
GF%\"\"#\"\"\",(*&-I%diffGF)6$-I'LambdaGF%F5F6F8F6F8F7-F06#,$F>F7F8!\"
\"F8F8F6!\"#F@FC\"\"!FEFE7&FE,$*&,(*&F6F8-F<6$F3F6F8F7F@FCF8F8F8F6FDFC
FEFE7&FEFE,$*(F6F8,,F;FCFKF8*&F6F8-F<6$F3-I\"$GF)6$F6F7F8F8*&F6F8FKF7F
8*(F6F8FKF8F;F8FCF8F@FCFCFE7&FEFEFE*(F6F8,6FKFC*&FKF8-I$cosGF(6#I&thet
aGF%F7F8F;F8*&F;F8FgnF7FCFQFC*(F6F8FRF8FgnF7F8FWFC*(F6F8FKF7FgnF7F8FXF
8**F6F8FKF8F;F8FgnF7FCF8F@FC" }{TEXT 255 1 " " }{TEXT 255 1 " " }}}
{EXCHG {PARA 237 "" 0 "" {TEXT 252 36 "Erste Einstein Komponente fuer \+
(0,0)" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 47 "help1:=8*Pi*En
ergy_momentum[1,1]+Einstein[1,1]:" }}}{EXCHG {PARA 237 "> " 0 "" 
{MPLTEXT 1 253 35 "help2:=expand(help1/exp(Phi(r))^2):" }}}{EXCHG 
{PARA 237 "" 0 "" {TEXT 252 24 "Erste Einstein Gleichung" }}}{EXCHG 
{PARA 237 "> " 0 "" {MPLTEXT 1 253 23 "eq1:=simplify(help2)=0;" }}
{PARA 239 "" 1 "" {XPPMATH 20 "6#>I$eq1G6\"/*&,**(I#PiGI*protectedGF+
\"\"\"-I$rhoGF%6#I\"rGF%F,F0\"\"#\"\")*(-I$expG6$F+I(_syslibGF%6#,$-I'
LambdaGF%F/!\"#F,-I%diffGF+6$F:F0F,F0F,F1F,F,F4!\"\"F,F0F<\"\"!" }
{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG {PARA 237 "> " 0 "" 
{MPLTEXT 1 253 57 "eq1a:=-8*Pi*rho(r)*r^2+Diff(r*(1-exp(-2*Lambda(r)))
,r)=0;" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>I%eq1aG6\"/,&*(I#PiGI*prot
ectedGF*\"\"\"-I$rhoGF%6#I\"rGF%F+F/\"\"#!\")-I%DiffG6$F*I(_syslibGF%6
$*&F/F+,&F+F+-I$expGF46#,$-I'LambdaGF%F.!\"#!\"\"F+F/F+\"\"!" }{TEXT 
255 1 " " }{TEXT 255 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 
253 23 "op(1,op(2,op(1,eq1a)));" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#*&
I\"rG6\"\"\"\",&F&F&-I$expG6$I*protectedGF+I(_syslibGF%6#,$-I'LambdaGF
%6#F$!\"#!\"\"F&" }{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG {PARA 
237 "> " 0 "" {MPLTEXT 1 253 55 "eq1a_n:=subs(r*(1-exp(-2*Lambda(r)))=
2*m(r),lhs(eq1a));" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>I'eq1a_nG6\",&
*(I#PiGI*protectedGF)\"\"\"-I$rhoGF%6#I\"rGF%F*F.\"\"#!\")-I%DiffG6$F)
I(_syslibGF%6$,$-I\"mGF%F-F/F.F*" }{TEXT 255 1 " " }{TEXT 255 1 " " }}
}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 21 "dsolve(eq1a_n,m(r)); " 
}}{PARA 239 "" 1 "" {XPPMATH 20 "6#/-I\"mG6\"6#I\"rGF&,&-I$IntG6$I*pro
tectedGF-I(_syslibGF&6$,$*(I#PiGF-\"\"\"-I$rhoGF&F'F3F(\"\"#\"\"%F(F3I
$_C1GF&F3" }{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG {PARA 237 "" 0 
"" {TEXT 252 94 "m(r) ist die Masse innerhalb eines Sphaeren-Radius vo
n r. Berechnung der Integrationskonstante" }}{PARA 237 "" 0 "" {TEXT 
252 33 "durch Berechnung des lambda aus m" }}}{EXCHG {PARA 237 "> " 0 
"" {MPLTEXT 1 253 37 "help3:=op(1,op(2,op(1,eq1a)))=2*m(r);" }}{PARA 
239 "" 1 "" {XPPMATH 20 "6#>I&help3G6\"/*&I\"rGF%\"\"\",&F)F)-I$expG6$
I*protectedGF.I(_syslibGF%6#,$-I'LambdaGF%6#F(!\"#!\"\"F),$-I\"mGF%F4
\"\"#" }{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG {PARA 237 "> " 0 ""
 {MPLTEXT 1 253 42 "help4:=op(2,op(2,op(1,op(2,op(1,eq1a)))));" }}
{PARA 239 "" 1 "" {XPPMATH 20 "6#>I&help4G6\",$-I$expG6$I*protectedGF*
I(_syslibGF%6#,$-I'LambdaGF%6#I\"rGF%!\"#!\"\"" }{TEXT 255 1 " " }
{TEXT 255 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 28 "expan
d(solve(help3,-help4));" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$
*&I\"rG6\"!\"\"-I\"mGF'6#F&F$!\"#" }{TEXT 255 1 " " }{TEXT 255 1 " " }
}}{EXCHG {PARA 237 "" 0 "" {TEXT 256 1 " " }{TEXT 256 17 "Mit dem Ausd
ruck " }{TEXT 256 76 "g[mu,nu]  f\374r L ist die niederschrift der Lor
enz Metrik nur g\374ltig f\374r _CI=0" }}{PARA 237 "" 0 "" {TEXT 256 
46 "2. Einstein Gleichung f\374r doe (1,1) Komponente" }}}{EXCHG 
{PARA 237 "> " 0 "" {MPLTEXT 1 253 65 "eq2 := simplify( 8*Pi*Energy_mo
mentum[2,2] + Einstein[2,2] ) = 0;" }}{PARA 239 "" 1 "" {XPPMATH 20 "6
#>I$eq2G6\"/*&,***I#PiGI*protectedGF+\"\"\"-I$expG6$F+I(_syslibGF%6#,$
-I'LambdaGF%6#I\"rGF%\"\"#F,-I\"pGF%F5F,F6F7\"\")*&F6F,-I%diffGF+6$-I$
PhiGF%F5F6F,!\"#F-F,!\"\"F,F,F6FA\"\"!" }{TEXT 255 1 " " }{TEXT 255 1 
" " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 79 "eq2_2 := numer(lh
s(simplify(subs(exp(2*Lambda(r)) = 1/(1-2*m(r)/r),eq2)))) = 0;" }}
{PARA 239 "" 1 "" {XPPMATH 20 "6#>I&eq2_2G6\"/,**(I#PiGI*protectedGF*
\"\"\"I\"rGF%\"\"$-I\"pGF%6#F,F+\"\")*&F,\"\"#-I%diffGF*6$-I$PhiGF%F0F
,F+!\"#*(F,F+F4F+-I\"mGF%F0F+\"\"%F;F3\"\"!" }{TEXT 255 1 " " }{TEXT 
255 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 56 "eq2_3 := Di
ff(Phi(r),r) = solve(eq2_2, diff(Phi(r),r) );" }}{PARA 239 "" 1 "" 
{XPPMATH 20 "6#>I&eq2_3G6\"/-I%DiffG6$I*protectedGF*I(_syslibGF%6$-I$P
hiGF%6#I\"rGF%F0*(,&*(I#PiGF*\"\"\"F0\"\"$-I\"pGF%F/F5\"\"%-I\"mGF%F/F
5F5F0!\"\",&F0F5F:!\"#F<" }{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG 
{PARA 237 "" 0 "" {TEXT 256 1 " " }{TEXT 256 112 "Der Gradient des Gra
vitations-Potentials ist gr\366\337er als der der reinen Newton Beding
ung: diff(Phi,r) = m/(r^2) ," }}{PARA 237 "" 0 "" {TEXT 256 50 "dies i
st der Druck in GR in der Gravitationsquelle" }}{PARA 237 "" 0 "" 
{TEXT 256 98 "Zur weiteren Verfeinerung m\374ssen wir die relativistis
chen Gleichungen der Hydrodynamik definieren," }}{PARA 237 "" 0 "" 
{TEXT 256 47 "das sind die relativistischen Euler Gleichungen" }{TEXT 
256 2 "\n " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 37 "compts :=
 array([u_t,u_r,u_th,u_ph]):" }}}{EXCHG {PARA 237 "" 0 "" {TEXT 256 
29 "Die 4-er Geschwindigkeit ist:" }}}{EXCHG {PARA 237 "> " 0 "" 
{MPLTEXT 1 253 25 "u := create([1], compts):" }}}{EXCHG {PARA 237 "> "
 0 "" {MPLTEXT 1 253 34 "Cf2 := Christoffel2 ( ginv, Cf1 ):" }}}
{EXCHG {PARA 237 "" 0 "" {TEXT 256 57 "Radiale Komponenten der Euler G
leichung u_r = u_th=u_ph=0" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 
253 116 "(rho(r)+p(r))*get_compts(cov_diff( u, coord, Cf2 ))[1,2]/(u_t
) = -diff(p(r),r);# radial component of Euler equation," }}{PARA 239 "
" 1 "" {XPPMATH 20 "6#/*&,&-I$rhoG6\"6#I\"rGF(\"\"\"-I\"pGF(F)F+F+-I%d
iffGI*protectedGF06$-I$PhiGF(F)F*F+,$-F/6$F,F*!\"\"" }{TEXT 255 1 " " 
}{TEXT 255 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 50 "eq3 \+
:= Diff(Phi(r),r) = solve(%, diff(Phi(r),r) );" }}{PARA 239 "" 1 "" 
{XPPMATH 20 "6#>I$eq3G6\"/-I%DiffG6$I*protectedGF*I(_syslibGF%6$-I$Phi
GF%6#I\"rGF%F0,$*&-I%diffGF*6$-I\"pGF%F/F0\"\"\",&-I$rhoGF%F/F8F6F8!\"
\"F<" }{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG {PARA 237 "" 0 "" 
{TEXT 256 94 "Das ergibt die Openheiner-Volkhoff Gleichung f\374r das \+
hydrostatische Gleichgewicht eines Sterns" }}}{EXCHG {PARA 237 "> " 0 
"" {MPLTEXT 1 253 67 "Diff(p(r),r) = factor( solve(rhs(eq3) = rhs(eq2_
3),diff(p(r),r)) );" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#/-I%DiffG6$I*p
rotectedGF'I(_syslibG6\"6$-I\"pGF)6#I\"rGF)F.,$**,&*(I#PiGF'\"\"\"F.\"
\"$F+F4\"\"%-I\"mGF)F-F4F4,&-I$rhoGF)F-F4F+F4F4F.!\"\",&F.F4F7!\"#F<F<
" }{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG {PARA 237 "" 0 "" {TEXT 
256 148 "Der Druckgradient ist gr\366\337er als die klassische Grenze \+
 ( diff(p,r) =  - rho*m/(r^2) ) und dieser Gradient wird gr\366\337er \+
bei Druckanstieg (Z\344hlerterm)" }}{PARA 237 "" 0 "" {TEXT 256 89 "un
d der radius r wird kleiner (Nennerterm), wenn man sich dem Gravitatio
nszentrum n\344hert." }}{PARA 237 "" 0 "" {TEXT 256 69 "Das hei\337t, \+
diese Vorstellung ist wesentlich sch\344rfer als die Newtons." }}
{PARA 237 "" 0 "" {TEXT 256 27 " Au\337erhalb eines Stern ist " }
{TEXT 256 50 " m ( r ) =M , p= 0 ( M isdie Gesamtsternenmasse). " }}
{PARA 237 "" 0 "" {TEXT 256 15 "Dann erst gilt:" }}}{EXCHG {PARA 237 "
> " 0 "" {MPLTEXT 1 253 57 "help5:=diff(Phi(r),r) = subs(\{m(r)=M,p(r)
=0\},rhs(eq2_3));" }{MPLTEXT 1 253 30 "\neq4 := dsolve(help5, Phi(r));
" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>I&help5G6\"/-I%diffGI*protectedG
F)6$-I$PhiGF%6#I\"rGF%F.*(I\"MGF%\"\"\"F.!\"\",&F.F1F0!\"#F2" }{TEXT 
255 1 " " }{TEXT 255 1 " " }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>I$eq4G6
\"/-I$PhiGF%6#I\"rGF%,(-I#lnG6$I*protectedGF/I(_syslibGF%6#,&F*\"\"\"I
\"MGF%!\"##F3\"\"#-F-F)#!\"\"F7I$_C1GF%F3" }{TEXT 255 1 " " }{TEXT 
255 1 " " }}}{EXCHG {PARA 237 "" 0 "" {TEXT 256 23 "Wenn die Grenzbedi
ngung" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 29 "limit(rhs(eq4)
,r=infinity)=0;" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#/I$_C1G6\"\"\"!" }
{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG {PARA 237 "" 0 "" {TEXT 
256 59 "gilt, dann folgt f\374r die Schwarzschild Metrik eines Sterns:
" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 26 "g_matrix := get_com
pts(g);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>I)g_matrixG6\"-I'matrixG6
$I*protectedGF)I(_syslibGF%6#7&7&-I$expGF(6#,$-I$PhiGF%6#I\"rGF%\"\"#
\"\"!F7F77&F7-F/6#,$-I'LambdaGF%F4F6F7F77&F7F7*$F5F6F77&F7F7F7*&F5F6-I
$sinGF(6#I&thetaGF%F6" }{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG 
{PARA 237 "> " 0 "" {MPLTEXT 1 253 28 "coord := [t, r, theta, phi]:" }
}}{EXCHG {PARA 237 "" 0 "" {TEXT 256 18 "Metrik Komponenten" }}}
{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 48 "sch_compts := array(symm
etric,sparse,1..4,1..4):" }}}{EXCHG {PARA 237 "" 0 "" {TEXT 256 38 "Ko
effizienten von d(t)^2 im Intervall:" }}}{EXCHG {PARA 237 "> " 0 "" 
{MPLTEXT 1 253 81 "sch_compts[1,1] := expand( subs(Phi(r)=-1/2*ln(r)+1
/2*ln(r-2*M),g_matrix[1,1]) ):" }}}{EXCHG {PARA 237 "" 0 "" {TEXT 256 
26 "Koeffs d(r)^2 im Intervall" }}}{EXCHG {PARA 237 "> " 0 "" 
{MPLTEXT 1 253 74 "sch_compts[2,2] := expand( subs(Lambda(r)=-ln(1-2*M
/r)/2,g_matrix[2,2]) ):" }}}{EXCHG {PARA 237 "" 0 "" {TEXT 256 30 "Koe
ffs d(theta)^2 im Intervall" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 
253 31 "sch_compts[3,3]:=g_matrix[3,3]:" }}}{EXCHG {PARA 237 "" 0 "" 
{TEXT 256 26 "Koeffs d(phi) im Intervall" }}}{EXCHG {PARA 237 "> " 0 "
" {MPLTEXT 1 253 31 "sch_compts[4,4]:=g_matrix[4,4]:" }}}{EXCHG {PARA 
237 "" 0 "" {TEXT 256 34 "Und jetzt die Schwarzschild Metrik" }}}
{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 253 41 "sch := create([-1,-1], e
val(sch_compts));" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>I$schG6\"-I&TAB
LEGI*protectedGF(6#7$/I'comptsGF%-I'matrixG6$F(I(_syslibGF%6#7&7&,&\"
\"\"F5*&I\"rGF%!\"\"I\"MGF%F5!\"#\"\"!F;F;7&F;*$F4F8F;F;7&F;F;*$F7\"\"
#F;7&F;F;F;*&F7F@-I$sinGF/6#I&thetaGF%F@/I+index_charGF%7$F8F8" }
{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG {PARA 240 "" 0 "" {TEXT 
257 52 "1. Annahme: Stern aus inkompressibler Substanz mit  " }}{PARA 
240 "" 0 "" {TEXT 257 20 "  r  =  r0 = const  " }}{PARA 240 "" 0 "" 
{TEXT 257 44 "zu einem sp\344\366teren Zeitpunkt wird berechnet:" }}
{PARA 240 "" 0 "" {TEXT 257 83 "  p =( g -1) r, where  g =1 (dust), 4/
3 (noncoherent radiation), 2 (hard matter) )." }{TEXT 257 1 "\n" }
{TEXT 257 24 "\nDemnach wird die Masse:" }}}{PARA 241 "" 0 "" }{EXCHG 
{PARA 240 "> " 0 "" {MPLTEXT 1 258 28 "m(r) = int(4*Pi*rho0*r^2,r);" }
{MPLTEXT 1 258 22 "\nM = subs(r=R,rhs(%));" }}{PARA 239 "" 1 "" 
{XPPMATH 20 "6#/-I\"mG6\"6#I\"rGF&,$*(I#PiGI*protectedGF,\"\"\"I%rho0G
F&F-F(\"\"$#\"\"%F/" }{TEXT 255 1 " " }{TEXT 255 1 " " }}{PARA 239 "" 
1 "" {XPPMATH 20 "6#/I\"MG6\",$*(I#PiGI*protectedGF)\"\"\"I%rho0GF%F*I
\"RGF%\"\"$#\"\"%F-" }{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG 
{PARA 242 "" 0 "" {TEXT 259 29 "Oppenheimer-Volkoff Gleichung" }}}
{EXCHG {PARA 240 "> " 0 "" {MPLTEXT 1 258 57 "diff(p(r),r) = subs( \{r
ho(r)=rho0,m(r)=4/3*Pi*rho0*r^3\},\\" }{MPLTEXT 1 258 54 "\n-(4*Pi*r^3
*p(r)+m(r))*(rho(r)+p(r))/(r*(r-2*m(r))) );" }{MPLTEXT 1 258 23 "\neq5
 := dsolve(%,p(r));" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#/-I%diffGI*pro
tectedGF&6$-I\"pG6\"6#I\"rGF*F,,$**,&*(I#PiGF&\"\"\"F,\"\"$F(F2\"\"%*(
F1F2I%rho0GF*F2F,F3#F4F3F2,&F6F2F(F2F2F,!\"\",&F,F2F5#!\")F3F9F9" }
{TEXT 255 1 " " }{TEXT 255 1 " " }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>I
$eq5G6\"6$/-I\"pGF%6#I\"rGF%,&*(,(!\"*\"\"\"-I$expG6$I*protectedGF4I(_
syslibGF%6#,$*(I$_C1GF%F0I#PiGF4F0I%rho0GF%F0\"#;!\"$**F:F0F;F0F+\"\"#
F1F0\"\")!\"\",&!\"'F0*$,&F1F=F>F@#F0F?F?F0F;F0F0F;FA/F(,&*(F.FA,&FCF0
FD!\"#F0F;F0F0F;FA" }{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG {PARA 
243 "" 0 "" {TEXT 260 10 "Grenzwerte" }}}{EXCHG {PARA 243 "> " 0 "" 
{MPLTEXT 1 261 12 "rhs(eq5[1]);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#,&
*(,(!\"*\"\"\"-I$expG6$I*protectedGF+I(_syslibG6\"6#,$*(I$_C1GF-F'I#Pi
GF+F'I%rho0GF-F'\"#;!\"$**F2F'F3F'I\"rGF-\"\"#F(F'\"\")!\"\",&!\"'F'*$
,&F(F5F6F9#F'F8F8F'F3F'F'F3F:" }{TEXT 255 1 " " }{TEXT 255 1 " " }}}
{EXCHG {PARA 243 "> " 0 "" {MPLTEXT 1 261 61 "solve(simplify(subs(r=R,
rhs(eq5[1])))=0,exp(16*_C1*Pi*rho0));" }}{PARA 239 "" 1 "" {XPPMATH 
20 "6#*&,&\"\"$\"\"\"*$*&-I$expG6$I*protectedGF,I(_syslibG6\"6#*(I$_C1
GF.F&I#PiGF,F&I%rho0GF.F&\"#;,&!\"$F&*(F2F&F3F&I\"RGF.\"\"#\"\")F&#F&F
9F9F&F5!\"\"" }{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG {PARA 243 ">
 " 0 "" {MPLTEXT 1 261 61 "solve(simplify(subs(r=R,rhs(eq5[2])))=0,exp
(16*_C1*Pi*rho0));" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#,$*&,&!\"$\"\"
\"*$*&-I$expG6$I*protectedGF-I(_syslibG6\"6#*(I$_C1GF/F'I#PiGF-F'I%rho
0GF/F'\"#;,&F&F'*(F3F'F4F'I\"RGF/\"\"#\"\")F'#F'F9F9F'F6!\"\"F<" }
{TEXT 255 1 " " }{TEXT 255 1 " " }}}{EXCHG {PARA 243 "> " 0 "" }{PARA 
243 "> " 0 "" }}{PARA 244 "" 0 "" }{PARA 245 "" 0 "" }{PARA 246 "" 0 "
" }{PARA 247 "" 0 "" }}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 
0 1 2 33 1 1 }