{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 10 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "Helvetica" 1 18 0 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Helvetica" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Hel vetica" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Helvetica" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "Helvetica" 1 18 0 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "Helvetica" 0 18 0 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "Helvetica" 0 18 0 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 277 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "War ning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Lucida Sans Typewriter" 1 10 255 0 0 1 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 52 "restart: with (linalg):with(liesymm):with(difforms) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "setup(x,y,z,t):defform(x=0,y= 0,z=0,t=0,Vx=0,Vy=0,Vz=0,D1=0,D2=0,D3=0,Ax=0,Ay=0,Az=0,C=0,Phi=0,a=con st,b=const,c=const,Lx=0,Ly=0,Lz=0,E=0,g=const);" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for close" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for `&^`" }}{PARA 7 "" 1 "" {TEXT -1 29 "Warning, new definition for d" }}{PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definition for mixpar" }}{PARA 7 "" 1 "" {TEXT -1 35 "Warning, new definition for wdegree" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#VzG*&%\"gG\"\"\"%\"tGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 38 "Using the Lo rentz map as a FRAME FIELD" }}{PARA 265 "" 0 "" {TEXT 280 3 "or " }} {PARA 266 "" 0 "" {TEXT 281 31 "Space Time as a Minkowski Fluid" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT 273 47 "R. M. \+ Kiehn (last update 08/09/99 - 08/10/99)" }}{PARA 259 "" 0 "" {TEXT 274 18 "rkiehn2352@aol.com" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 474 "Ca rtan's methods will be used to compute torsion and curvature coefficie nts induced on subspaces of R4 by assuming the Lorentz map, L, to be a Frame Field. The parameters (Vx,Vy,Vz, Lx, Ly, Lz, C,E) will NOT be \+ presumed to be global constants, a priori. For example, it may be tru e that C = C(x,y,z,t) as it does in real media or in the presence of a gravitational field. The conformal expansion factor E allows for exp ansion of the manifold and also can be E(x,y,z,t). " }}{PARA 0 "" 0 " " {TEXT -1 170 "If E is a global constant, then all torsion 2-forms of the WF type are zero, as the field \"Abnormality\" (big Omega) vanish es (See http://www22.pair.com/pdf/projfram.pdf) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 1 " " }{TEXT 270 24 "Fundamental A ssumption: " }}{PARA 0 "" 0 "" {TEXT -1 16 "The Lorentz map," }}{PARA 263 "" 0 "" {TEXT 271 26 "L: \{x,y,z,t\} ==> \{X,Y,Z,T\}" }}{PARA 0 " " 0 "" {TEXT -1 25 "must satisfy the equation" }}{PARA 262 "" 0 "" {TEXT 275 59 "x^2+y^2+z^2-(C*t)^2=0 = => X^2+Y^2+Z^2-(C*T)^ 2=0" }}{PARA 0 "" 0 "" {TEXT -1 218 "When it is recognized that the Ei konal equation (non-linear first order quadratic form of partial deriv atives) is deduced from the Maxwell PDE's as the necessary criteria f or the existence of a singular solution set, " }}{PARA 0 "" 0 "" {TEXT -1 100 "(and therefore is the fundamental equation describing th e existance of propagating discontinuities)," }}{PARA 0 "" 0 "" {TEXT -1 57 "then the key feature of the Lorentz map is that a signal " }} {PARA 0 "" 0 "" {TEXT -1 60 "( a propagating discontinuity - Not an in finite wave train) " }}{PARA 0 "" 0 "" {TEXT -1 67 "is mapped into a s ignal to all obervers related by the Lorentz map." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 187 "The fact that the Eikona l equation is equal to zero, implies that it is possible to scale all \+ quadratic forms by a non zero factor E^2. In other words the map is d efined by the equations" }}{PARA 261 "" 0 "" {TEXT 276 55 "x^2+y^2+z^2 -(C*t)^2 = E(x,y,z,t)^2\{X^2+Y^2+Z^2-(C*T)^2\}" }}{PARA 0 "" 0 "" {TEXT -1 46 " where if the RHS is zero, then so is the LHS." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 154 "The significa nce of the Lorentz transformations, L, is that they are the only Linea r transformations that preserve the Eikonal quadratic form. -- V. Foc k" }}{PARA 267 "" 0 "" {TEXT -1 39 "(AND THEREFOR MAP SIGNALS INTO SIG NALS)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 201 "The class of Lorentz maps does not require that C be a domain constan t. In fact, when C= C(x,y,z,t), the space-time variety \{x,y,z,t\} c onstrained by Lorentz maps can be viewed as a \"Minkowski Fluid\"." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 199 "In that \+ which follows, the Lorentz transformations considered will be those th at can be constructed from six matrix generators, three rotations (LRx , LRy, LRz) and three translations (LTx, LTy,LTz).." }}{PARA 0 "" 0 " " {TEXT -1 96 "The inverses of these matrices will be designated as (L Rxn, LRyn, LRzn) and (LTxn, LTyn,LTzn). " }}{PARA 0 "" 0 "" {TEXT -1 460 "The generators may be multiplied by arbitrary non-zero Expansion \+ functions, E, (which merely changes the determinant value of the map) and by 1/E for the inverse generators. The matrix generators consist of parameters( functions of x,y,z,t) and these parameters are not res tricted to be domain constants. The only requirement is that the matr ix be an element of the Lorentz group that preserves the quadratic for m defined as the Eikonal, to within a factor." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 158 "Products of these gener ators will generate new Lorentz transformations. However, the order o f application of the generators is significant to the outcome. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "LGUN stan ds for the Minkowski metric" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "LGUN:=evalm(array([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,-1]]));r:= [x,y,z,C*t]:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 357 "In that which \+ follows, six generating matrices will be constructed from maps that ta ke r = \{x,y,z,Ct) into R = \{XX,YY,ZZ,TT\}. The initial rQF (a quad ratic form defined as rQF) constructed from will be com pared to the final RQF defined as . The generators will be constructed at first for those that have positive determinant." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 290 "LRz:=subs(evalm(E*array([[(1-Lz^2) ^(1/2),Lz,0,0],[-Lz,(1-Lz^2)^(1/2),0,0],[0,0,1,0],[0,0,0,1]])));RPRIME :=simplify(innerprod(LRz,r)):XX:=RPRIME[1];YY:=RPRIME[2];ZZ:=(RPRIME[3 ]);TT:=(simplify(RPRIME[4]));DETL:=det(LRz);R:=innerprod(LRz,r):initia l_rQF:=innerprod(r,LGUN,r);LRzn:=inverse(LRz);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LGUNG-%'matrixG6#7&7&\"\"\"\"\"!F+F+7&F+F*F+F+7&F+F+ F*F+7&F+F+F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LRzG-%'matrixG 6#7&7&*&%\"EG\"\"\"-%%sqrtG6#,&F,F,*$)%#LzG\"\"#\"\"\"!\"\"F5*&F+F5F3F ,\"\"!F87&,$F7F6F*F8F87&F8F8F+F87&F8F8F8F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#XXG,&*(%\"EG\"\"\"-%%sqrtG6#,&F(F(*$)%#LzG\"\"#\"\" \"!\"\"F1%\"xGF(F(*(F'F1F/F(%\"yGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YYG,&*(%\"EG\"\"\"%#LzGF(%\"xGF(!\"\"*(F'\"\"\"-%%sqrtG6#,&F(F(* $)F)\"\"#F-F+F-%\"yGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ZZG*&% \"EG\"\"\"%\"zGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TTG*(%\"EG\"\" \"%\"CGF'%\"tGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DETLG*$)%\"EG\" \"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,initial_rQFG,**$)%\"xG \"\"#\"\"\"\"\"\"*$)%\"yGF)F*F+*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F*!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LRznG-%'matrixG6#7&7&*&*$-%%s qrtG6#,&\"\"\"F0*$)%#LzG\"\"#\"\"\"!\"\"F5F5%\"EG!\"\",$*&F3F5F7F8F6\" \"!F;7&F:F*F;F;7&F;F;*&F5F5F7F8F;7&F;F;F;F>" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 57 "Compute the final quadratic form form the Output vector R" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Final_RQF:=(factor(in nerprod(R,LGUN,R)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Final_RQFG, $*&)%\"EG\"\"#\"\"\",**$)%\"xGF)F*!\"\"*$)%\"yGF)F*F/*$)%\"zGF)F*F/*&) %\"CGF)F*)%\"tGF)F*\"\"\"F;F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 " So the map LRz is indeed a Lorentz transformation (LRz = \"Lorentz Rot ation about z axis\")." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Similar ly for rotations about x and y:" }}{PARA 0 "" 0 "" {TEXT -1 58 "An ang ular formulation for non-tachyons can be written as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 354 "LRphiz:=subs(evalm(E*array([[cos(phi),si n(phi),0,0],[-sin(phi),cos(phi),0,0],[0,0,1,0],[0,0,0,1]])));RPRIME:=s implify(innerprod(LRphiz,r)):XX:=RPRIME[1];YY:=RPRIME[2];ZZ:=(RPRIME[3 ]);TT:=(simplify(RPRIME[4]));DETL:=det(LRz);R:=innerprod(LRphiz,r);ini tial_rQF:=innerprod(r,LGUN,r);Final_RQF:=factor(simplify(innerprod(R,L GUN,R)));LRphizn:=inverse(LRphiz);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%'LRphizG-%'matrixG6#7&7&*&%\"EG\"\"\"-%$cosG6#%$phiGF,*&F+\"\"\"-%$s inGF/F,\"\"!F57&,$F1!\"\"F*F5F57&F5F5F+F57&F5F5F5F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#XXG,&*(%\"EG\"\"\"-%$cosG6#%$phiGF(%\"xGF(F(*(F'\" \"\"-%$sinGF+F(%\"yGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YYG,&*( %\"EG\"\"\"-%$sinG6#%$phiGF(%\"xGF(!\"\"*(F'\"\"\"-%$cosGF+F(%\"yGF(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ZZG*&%\"EG\"\"\"%\"zGF'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TTG*(%\"EG\"\"\"%\"CGF'%\"tGF'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DETLG*$)%\"EG\"\"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG-%'vectorG6#7&,&*(%\"EG\"\"\"-%$cosG6# %$phiGF,%\"xGF,F,*(F+\"\"\"-%$sinGF/F,%\"yGF,F,,&*(F+F3F4F3F1F3!\"\"*( F+F3F-F3F6F3F,*&F+F3%\"zGF,*(F+F3%\"CGF,%\"tGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,initial_rQFG,**$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF)F*F +*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Final_RQFG,$*&)%\"EG\"\"#\"\"\",**$)%\"xGF)F*!\"\"*$ )%\"yGF)F*F/*$)%\"zGF)F*F/*&)%\"CGF)F*)%\"tGF)F*\"\"\"F;F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(LRphiznG-%'matrixG6#7&7&*&-%$cosG6#%$phiG \"\"\"*&%\"EG\"\"\",&*$)F+\"\"#F/\"\"\"*$)-%$sinGF-F6F/F7\"\"\"!\"\",$ *&F:F/*&F1\"\"\"F3\"\"\"F=!\"\"\"\"!FD7&F?F*FDFD7&FDFD*&F/F/F1F=FD7&FD FDFDFG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 321 "LRx:=subs(evalm( E*array([[1,0,0,0],[0,(1-Lx^2)^(1/2),Lx,0],[0,-Lx,(1-Lx^2)^(1/2),0],[0 ,0,0,1]])));RPRIME:=simplify(innerprod(LRx,r)):XX:=RPRIME[1];YY:=RPRIM E[2];ZZ:=(RPRIME[3]);TT:=(simplify(RPRIME[4]));DETL:=det(LRz);R:=inner prod(LRx,r);initial_rQF:=innerprod(r,LGUN,r);Final_RQF:=innerprod(R,LG UN,R);LRxn:=inverse(LRx);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LRxG-%'matrixG6#7&7&%\"EG\"\"!F+F+7 &F+*&F*\"\"\"-%%sqrtG6#,&F.F.*$)%#LxG\"\"#\"\"\"!\"\"F7*&F*F7F5F.F+7&F +,$F9F8F-F+7&F+F+F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#XXG*&%\"EG \"\"\"%\"xGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YYG,&*(%\"EG\"\"\" -%%sqrtG6#,&F(F(*$)%#LxG\"\"#\"\"\"!\"\"F1%\"yGF(F(*(F'F1F/F(%\"zGF(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ZZG,&*(%\"EG\"\"\"%#LxGF(%\"yGF (!\"\"*(F'\"\"\"-%%sqrtG6#,&F(F(*$)F)\"\"#F-F+F-%\"zGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TTG*(%\"EG\"\"\"%\"CGF'%\"tGF'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%DETLG*$)%\"EG\"\"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG-%'vectorG6#7&*&%\"EG\"\"\"%\"xGF+,&*(F*\"\"\"-%% sqrtG6#,&F+F+*$)%#LxG\"\"#F/!\"\"F/%\"yGF+F+*(F*F/F6F+%\"zGF+F+,&*(F*F /F6F/F9F/F8*(F*F/F0F/F;F/F+*(F*F/%\"CGF+%\"tGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,initial_rQFG,**$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF)F*F +*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Final_RQFG,**&)%\"EG\"\"#\"\"\")%\"yGF)F*\"\"\"*&F'F *)%\"xGF)F*F-*&F'F*)%\"zGF)F*F-*(F'F*)%\"CGF)F*)%\"tGF)F*!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LRxnG-%'matrixG6#7&7&*&\"\"\"F+%\"E G!\"\"\"\"!F.F.7&F.*&*$-%%sqrtG6#,&\"\"\"F6*$)%#LxG\"\"#F+!\"\"F+F+F,F -,$*&F9F+F,F-F;F.7&F.F=F0F.7&F.F.F.F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Similary an angular representation is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 372 "LR thetax:=subs(evalm(E*array([[1,0,0,0],[0,cos(theta),sin(theta),0],[0,- sin(theta),cos(theta),0],[0,0,0,1]])));RPRIME:=simplify(innerprod(LRth etax,r)):XX:=RPRIME[1];YY:=RPRIME[2];ZZ:=(RPRIME[3]);TT:=(simplify(RPR IME[4]));DETL:=det(LRz);R:=innerprod(LRthetax,r);initial_rQF:=innerpro d(r,LGUN,r);Final_RQF:=factor(simplify(innerprod(R,LGUN,R)));LRthetaxn :=inverse(LRthetax);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)LRthetaxG-% 'matrixG6#7&7&%\"EG\"\"!F+F+7&F+*&F*\"\"\"-%$cosG6#%&thetaGF.*&F*\"\" \"-%$sinGF1F.F+7&F+,$F3!\"\"F-F+7&F+F+F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#XXG*&%\"EG\"\"\"%\"xGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YYG,&*(%\"EG\"\"\"-%$cosG6#%&thetaGF(%\"yGF(F(*(F'\" \"\"-%$sinGF+F(%\"zGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ZZG,&*( %\"EG\"\"\"-%$sinG6#%&thetaGF(%\"yGF(!\"\"*(F'\"\"\"-%$cosGF+F(%\"zGF( F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TTG*(%\"EG\"\"\"%\"CGF'%\"tGF '" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DETLG*$)%\"EG\"\"%\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG-%'vectorG6#7&*&%\"EG\"\"\"%\"xG F+,&*(F*\"\"\"-%$cosG6#%&thetaGF+%\"yGF+F+*(F*F/-%$sinGF2F+%\"zGF+F+,& *(F*F/F6F/F4F/!\"\"*(F*F/F0F/F8F/F+*(F*F/%\"CGF+%\"tGF+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%,initial_rQFG,**$)%\"xG\"\"#\"\"\"\"\"\"*$)%\" yGF)F*F+*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Final_RQFG,$*&)%\"EG\"\"#\"\"\",**$)%\"xGF)F*!\"\"*$ )%\"yGF)F*F/*$)%\"zGF)F*F/*&)%\"CGF)F*)%\"tGF)F*\"\"\"F;F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*LRthetaxnG-%'matrixG6#7&7&*&\"\"\"F+%\"EG !\"\"\"\"!F.F.7&F.*&-%$cosG6#%&thetaGF+*&F,\"\"\",&*$)F1\"\"#F+\"\"\"* $)-%$sinGF3F:F+F;\"\"\"F-,$*&F>F+*&F,\"\"\"F7\"\"\"F-!\"\"F.7&F.FBF0F. 7&F.F.F.F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 321 "LRy:=subs(evalm(E*array([[(1-Ly^2)^(1/2),0,- Ly,0],[0,1,0,0],[Ly,0,(1-Ly^2)^(1/2),0],[0,0,0,1]])));RPRIME:=simplify (innerprod(LRy,r)):XX:=RPRIME[1];YY:=RPRIME[2];ZZ:=(RPRIME[3]);TT:=(si mplify(RPRIME[4]));DETL:=det(LRz);R:=innerprod(LRy,r);initial_rQF:=inn erprod(r,LGUN,r);Final_RQF:=innerprod(R,LGUN,R);LRyn:=inverse(LRy);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LRyG-%'matrixG6#7&7&*&%\"EG\"\"\"- %%sqrtG6#,&F,F,*$)%#LyG\"\"#\"\"\"!\"\"F5\"\"!,$*&F+F5F3F,F6F77&F7F+F7 F77&F9F7F*F77&F7F7F7F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#XXG,&*(% \"EG\"\"\"-%%sqrtG6#,&F(F(*$)%#LyG\"\"#\"\"\"!\"\"F1%\"xGF(F(*(F'F1F/F (%\"zGF(F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YYG*&%\"EG\"\"\"%\"yG F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ZZG,&*(%\"EG\"\"\"%#LyGF(%\"x GF(F(*(F'\"\"\"-%%sqrtG6#,&F(F(*$)F)\"\"#F,!\"\"F,%\"zGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TTG*(%\"EG\"\"\"%\"CGF'%\"tGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DETLG*$)%\"EG\"\"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG-%'vectorG6#7&,&*(%\"EG\"\"\"-%%sqrtG6#,&F,F ,*$)%#LyG\"\"#\"\"\"!\"\"F5%\"xGF,F,*(F+F5F3F,%\"zGF,F6*&F+F5%\"yGF,,& *(F+F5F3F5F7F5F,*(F+F5F-F5F9F5F,*(F+F5%\"CGF,%\"tGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,initial_rQFG,**$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF) F*F+*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Final_RQFG,**&)%\"EG\"\"#\"\"\")%\"xGF)F*\"\"\"*&F'F *)%\"yGF)F*F-*&F'F*)%\"zGF)F*F-*(F'F*)%\"CGF)F*)%\"tGF)F*!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LRynG-%'matrixG6#7&7&*&*$-%%sqrtG6# ,&\"\"\"F0*$)%#LyG\"\"#\"\"\"!\"\"F5F5%\"EG!\"\"\"\"!*&F3F5F7F8F97&F9* &F5F5F7F8F9F97&,$F:F6F9F*F97&F9F9F9F<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The next exercise is to c onstruct the generators for the Lorentz translations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 359 "Delta:= 1/(1-(Vz/C)^2)^(1/2):LTz:=subs(E=1,evalm(E*array([[1,0,0,0],[0,1,0,0], [0,0,Delta,Delta*Vz/C],[0,0,Delta*Vz/C,Delta]])));RPRIME:=simplify(inn erprod(LTz,r)):XX:=RPRIME[1];YY:=RPRIME[2];ZZ:=(RPRIME[3]);TT:=(simpli fy(RPRIME[4]));DETL:=det(LTz);R:=innerprod(LTz,r):initial_rQF:=innerpr od(r,LGUN,r);Final_RQF:=factor(innerprod(R,LGUN,R));LTzn:=inverse(LTz) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LTzG-%'matrixG6#7&7&\"\"\"\" \"!F+F+7&F+F*F+F+7&F+F+*&\"\"\"F/*$-%%sqrtG6#,&F*F**&*$)%#VzG\"\"#F/F/ *$)%\"CG\"\"#F/!\"\"!\"\"F/F>*&F8F/*&-F26#F4F/F<\"\"\"F>7&F+F+F@F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#XXG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YYG%\"yG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ZZG*& ,&%\"zG\"\"\"*&%#VzGF(%\"tGF(F(\"\"\"*$-%%sqrtG6#*&,&*$)%\"CG\"\"#F,F( *$)F*F6F,!\"\"F,*$)F5\"\"#F,!\"\"F,F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TTG*&,&*&%#VzG\"\"\"%\"zGF)F)*&)%\"CG\"\"#\"\"\"%\"tGF)F)F/*&-%% sqrtG6#*&,&*$F,F/F)*$)F(F.F/!\"\"F/*$)F-\"\"#F/!\"\"F/F-\"\"\"F>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DETLG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,initial_rQFG,**$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF)F*F +*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Final_RQFG,**$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF)F*F+* $)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LTznG-%'matrixG6#7&7&\"\"\"\"\"!F+F+7&F+F*F+F+7&F+F+*&\"\"\"F /*$-%%sqrtG6#*&,&*$)%\"CG\"\"#F/F**$)%#VzGF9F/!\"\"F/*$)F8\"\"#F/!\"\" F/FA,$*&F " 0 "" {MPLTEXT 1 0 387 "D elta:=1/cos(theta):LTcosz:=subs(evalm(E*array([[1,0,0,0],[0,1,0,0],[0, 0,Delta,Delta*sin(theta)],[0,0,Delta*sin(theta),Delta]])));RPRIME:=sim plify(innerprod(LTcosz,r)):XX:=RPRIME[1];YY:=RPRIME[2];ZZ:=(RPRIME[3]) ;TT:=(simplify(RPRIME[4]));DETL:=det(LTcosz);R:=innerprod(LTcosz,r):in itial_rQF:=innerprod(r,LGUN,r);Final_RQF:=simplify(factor(innerprod(R, LGUN,R)));LTcoszn:=inverse(LTcosz); " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%'LTcoszG-%'matrixG6#7&7&%\"EG\"\"!F+F+7&F+F*F+F+7&F+F+*&F*\"\"\"-% $cosG6#%&thetaG!\"\"*&*&F*\"\"\"-%$sinGF2F7F/F0F47&F+F+F5F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#XXG*&%\"EG\"\"\"%\"xGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YYG*&%\"EG\"\"\"%\"yGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ZZG*&*&%\"EG\"\"\",&%\"zGF(*(-%$sinG6#%&thetaGF(%\"C GF(%\"tGF(F(F(\"\"\"-%$cosGF.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#TTG*&*&%\"EG\"\"\",&*&-%$sinG6#%&thetaGF(%\"zGF(F(*&%\"CGF(%\"tGF(F (F(\"\"\"-%$cosGF-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DETLG,$* &*&)%\"EG\"\"%\"\"\",&!\"\"\"\"\"*$)-%$sinG6#%&thetaG\"\"#F+F.F.F+*$)- %$cosGF3\"\"#F+!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,initial_r QFG,**$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF)F*F+*$)%\"zGF)F*F+*&)%\"CGF)F* )%\"tGF)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Final_RQFG,$*&)% \"EG\"\"#\"\"\",**$)%\"xGF)F*!\"\"*$)%\"yGF)F*F/*$)%\"zGF)F*F/*&)%\"CG F)F*)%\"tGF)F*\"\"\"F;F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(LTcoszn G-%'matrixG6#7&7&*&\"\"\"F+%\"EG!\"\"\"\"!F.F.7&F.F*F.F.7&F.F.,$*&-%$c osG6#%&thetaGF+*&F,\"\"\",&!\"\"\"\"\"*$)-%$sinGF5\"\"#F+F;\"\"\"F-F:* &*&F3F;F>F;F+*&F,\"\"\"F9\"\"\"F-7&F.F.FBF1" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 76 "NOTE If the expansion factor is E = 1/cos(Vz/C) then \+ the coefficients are " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "su bs(E=1/(cos(theta)),evalm(LTcosz));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%'matrixG6#7&7&*&\"\"\"F)-%$cosG6#%&thetaG!\"\"\"\"!F/F/7&F/F(F/F/7& F/F/*&F)F)*$)F*\"\"#F)F.*&-%$sinGF,F)*$)F*\"\"#F)F.7&F/F/F6F2" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "and related to the generators of \+ " }}{PARA 268 "" 0 "" {TEXT -1 66 "the Raliegh - Taylor the terms: \{sin(V/C)/cos^2(V/C)\} and the " }}{PARA 269 "" 0 "" {TEXT -1 49 "Ke lvin - Helmholtz the terms \{1/cos^2(V/C)\}" }}{PARA 272 "" 0 " " {TEXT -1 36 "shear or wake discontinuties.!!!!! " }}{PARA 270 "" 0 "" {TEXT -1 91 "Extraordinary that the Lorentz map generates something that can be related to a fluid!!!!. " }}{PARA 271 "" 0 "" {TEXT -1 48 " See http://www22.pair.com/csdc/car/carfre15.htm" }}{PARA 0 "" 0 " " {TEXT -1 83 "(In the Kelvin Helmoltz instability, we are suggesting \+ a reason for spiral arms!!!)" }}{PARA 0 "" 0 "" {TEXT -1 50 "(Wakes ar e related to the eikonal condition again)" }}{PARA 0 "" 0 "" {TEXT -1 176 "NOte that the Det = 1 situation corresponds to a Lorentz translat ion where the off diagonal terms have the same sign. The Lorentz tr anslation in trig form is NOT a rotation." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 346 "Delta:=1/(1- (Vx/C)^2)^(1/2):LTx:=subs(E*evalm(array([[Delta,0,0,Delta*Vx/C],[0,1,0 ,0],[0,0,1,0],[Delta*Vx/C,0,0,Delta]])));RPRIME:=simplify(innerprod(LT x,r)):XX:=RPRIME[1];YY:=RPRIME[2];ZZ:=(RPRIME[3]);TT:=(simplify(RPRIME [4]));DETL:=det(LTz);R:=innerprod(LTx,r):initial_rQF:=innerprod(r,LGUN ,r);Final_RQF:=innerprod(R,LGUN,R);LTxn:=inverse(LTx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LTxG*&%\"EG\"\"\"-%'matrixG6#7&7&*&\"\"\"F.*$-% %sqrtG6#,&F'F'*&*$)%#VxG\"\"#F.F.*$)%\"CG\"\"#F.!\"\"!\"\"F.F=\"\"!F?* &F7F.*&-F16#F3F.F;\"\"\"F=7&F?F'F?F?7&F?F?F'F?7&F@F?F?F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#XXG*&*&%\"EG\"\"\",&%\"xGF(*&%#VxGF(%\"tG F(F(F(\"\"\"*$-%%sqrtG6#*&,&*$)%\"CG\"\"#F.F(*$)F,F8F.!\"\"F.*$)F7\"\" #F.!\"\"F.F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YYG*&%\"EG\"\"\"%\" yGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ZZG*&%\"EG\"\"\"%\"zGF'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TTG*&*&%\"EG\"\"\",&*&%#VxGF(%\"xGF (F(*&)%\"CG\"\"#\"\"\"%\"tGF(F(F(F1*&-%%sqrtG6#*&,&*$F.F1F(*$)F+F0F1! \"\"F1*$)F/\"\"#F1!\"\"F1F/\"\"\"F@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%%DETLG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,initial_rQFG,**$ )%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF)F*F+*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF )F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Final_RQFG,$*&)%\"EG\" \"#\"\"\",**$)%\"xGF)F*!\"\"*$)%\"yGF)F*F/*$)%\"zGF)F*F/*&)%\"CGF)F*)% \"tGF)F*\"\"\"F;F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LTxnG-%'matri xG6#7&7&*&\"\"\"F+*&%\"EG\"\"\"-%%sqrtG6#*&,&*$)%\"CG\"\"#F+\"\"\"*$)% #VxGF7F+!\"\"F+*$)F6\"\"#F+!\"\"F+F@\"\"!FA,$*&F;F+*(F-\"\"\"-F06#F2F+ F6\"\"\"F@F<7&FA*&F+F+F-F@FAFA7&FAFAFJFA7&FBFAFAF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 346 "Delta:=1/(1-(Vy/C)^2)^(1/2):LTy:=subs(eval m(E*array([[1,0,0,0],[0,Delta,0,Delta*Vy/C],[0,0,1,0],[0,Delta*Vy/C,0, Delta]])));RPRIME:=simplify(innerprod(LTy,r)):XX:=RPRIME[1];YY:=RPRIME [2];ZZ:=(RPRIME[3]);TT:=(simplify(RPRIME[4]));DETL:=det(LTy);R:=innerp rod(LTy,r):initial_rQF:=innerprod(r,LGUN,r);Final_RQF:=innerprod(R,LGU N,R);LTyn:=inverse(LTy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LTyG-%' matrixG6#7&7&%\"EG\"\"!F+F+7&F+*&F*\"\"\"*$-%%sqrtG6#,&\"\"\"F4*&*$)%# VyG\"\"#F.F.*$)%\"CG\"\"#F.!\"\"!\"\"F.F>F+*&*&F*F4F8F4F.*&-F16#F3F.F< \"\"\"F>7&F+F+F*F+7&F+F@F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#XXG *&%\"EG\"\"\"%\"xGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YYG*&*&%\"E G\"\"\",&%\"yGF(*&%#VyGF(%\"tGF(F(F(\"\"\"*$-%%sqrtG6#*&,&*$)%\"CG\"\" #F.F(*$)F,F8F.!\"\"F.*$)F7\"\"#F.!\"\"F.F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ZZG*&%\"EG\"\"\"%\"zGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TTG*&*&%\"EG\"\"\",&*&%#VyGF(%\"yGF(F(*&)%\"CG\"\"# \"\"\"%\"tGF(F(F(F1*&-%%sqrtG6#*&,&*$F.F1F(*$)F+F0F1!\"\"F1*$)F/\"\"#F 1!\"\"F1F/\"\"\"F@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DETLG*$)%\"EG \"\"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,initial_rQFG,**$)%\" xG\"\"#\"\"\"\"\"\"*$)%\"yGF)F*F+*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F*! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Final_RQFG,$*&)%\"EG\"\"#\" \"\",**$)%\"xGF)F*!\"\"*$)%\"yGF)F*F/*$)%\"zGF)F*F/*&)%\"CGF)F*)%\"tGF )F*\"\"\"F;F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LTynG-%'matrixG6#7 &7&*&\"\"\"F+%\"EG!\"\"\"\"!F.F.7&F.*&F+F+*&F,\"\"\"-%%sqrtG6#*&,&*$)% \"CG\"\"#F+\"\"\"*$)%#VyGF;F+!\"\"F+*$)F:\"\"#F+F-F+F-F.,$*&F?F+*(F,\" \"\"-F46#F6F+F:\"\"\"F-F@7&F.F.F*F.7&F.FDF.F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Note that combinations of products of the six generato rs produce other Lorentz transformations." }}{PARA 0 "" 0 "" {TEXT -1 204 "The \"group\" is multiplicative but not additive, and is non-abel ian. The inverse of each generator is the matrix where the rotation o r translation parameter is the negative value of the original matrix. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 375 "Now it is interesting that translation by Vx followed by a translation Vz , then followed by translation of -Vx (the inverse to Vx) and then fol lowed by translation of -Vz (the inverse of Vz), does not close. The \+ product of all four translations is still a Lorentz map, but the map i s not the identity: Hence the space of Lorentz generators has curvatu re or torsion or both." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 447 "To show this, compute the \"Round Trip\" RT product of two translations followed by two inverse translations for a produc t of 4 generators. The result is still a Lorentz transformation, but \+ it is not the identity. The same is true for a Round Trip of rotation s. AS will be shown below, the RT defect of translations can produce \+ curvature and torsion 2-forms. However, the RT defect of rotations do es not produce curvature or torsion 2-forms!!!!!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "RT:=innerprod(LTzn,LTxn,LTz,LTx);RPRIME:=sim plify(innerprod(RT,r)):XX:=simplify(RPRIME[1]);YY:=RPRIME[2];ZZ:=(RPRI ME[3]);TT:=(simplify(RPRIME[4]));DETL:=simplify(det(RT));R:=innerprod( RT,r):initial_rQF:=innerprod(r,LGUN,r);Final_RQF:=subs(Lx=1,Ly=1,facto r(innerprod(R,LGUN,R)));innerprod(transpose(RT),RT);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#RTG-%'matrixG6#7&7&*&,&*&)%\"CG\"\"#\"\"\"-%%sqrt G6#*&,&*$F-F0\"\"\"*$)%#VzGF/F0!\"\"F0*$)F.\"\"#F0!\"\"F0F7*$)%#VxGF/F 0F;F0*&-F26#F4F0,&F6F7F@F;\"\"\"F?\"\"!,$*&*&F:F7FBF7F0*(-F26#F4F0)F. \"\"#F0-F26#*&FFF0*$)F.\"\"#F0F?F0F?F;*&*(F.F7FBF0,&*$F1F0F7F;F7F7F0*& -F26#F4F0FF\"\"\"F?7&FHF7FHFH7&*&*(F:F0FBF0,*F6F7F8F;*(F1F0F-F0-F26#FS F0F7F,F;F7F0*(-F26#F4F0FF\"\"\"F5\"\"\"F?FH*&,&*&F-F0F_oF0F7F8F;F0*&F5 \"\"\"-F26#FSF0F?*&*&F:F0,**&FAF0F-F0F7*&FAF0F9F0F;*()F.\"\"%F0F1F0F_o F0F7*&FcpF0F1F0F;F7F0**-F26#F4F0F.\"\"\"FF\"\"\"F5\"\"\"F?7&,$*&*(F.F0 FBF0,*F6F7F8F;*(F1F0F9F0F_oF0F7F,F;F7F0*(-F26#F4F0FF\"\"\"F5\"\"\"F?F; FH,$*&*(F.F0F:F0,&*$F_oF0F7F;F7F7F0*&F5\"\"\"-F26#FSF0F?F;,$*&,*F`pF7F apF;**F-F0F1F0F9F0F_oF0F7FepF;F0*(-F26#F4F0FF\"\"\"F5\"\"\"F?F;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%#XXG*&,.**%\"xG\"\"\")%\"CG\"\"%\"\" \"-%%sqrtG6#*&,&*$)F+\"\"#F-F)*$)%#VxGF5F-!\"\"F-*$)F+\"\"#F-!\"\"F--F /6#*&,&F3F)*$)%#VzGF5F-F9F-*$)F+\"\"#F-F=F-F)**F(F-F4F-F.F-F7F-F9**FDF )F8F)%\"zGF)F4F-F9*(FDF-)F8\"\"$F-FJF-F)*,F*F-F8F-%\"tGF)F.F-F>F-F)**F *F-F8F-FOF-F.F-F9F-**-F/6#F@F-F2\"\"\")F+\"\"#F--F/6#F1F-F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YYG%\"yG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#ZZG,$*&,>*,%#VzG\"\"\"%#VxGF*%\"xGF*-%%sqrtG6#*&,&*$)%\"CG\"\"# \"\"\"F**$)F+F5F6!\"\"F6*$)F4\"\"#F6!\"\"F6F3F6F9**)F)\"\"$F6F+F6F,F6F -F6F**,F)F6F+F6F,F6-F.6#*&,&F2F**$)F)F5F6F9F6*$)F4\"\"#F6F=F6F3F6F9**F )F6)F+F@F6F,F6FBF6F**.F)F6F+F6F,F6F-F6F3F6FBF6F***%\"zGF*FBF6)F4\"\"%F 6F-F6F9*,FOF6FBF6F3F6F-F6F8F6F***FOF6FBF6FGF6F3F6F***FOF6FBF6F8F6FGF6F 9*,F)F6%\"tGF*F-F6F8F6F3F6F9**F?F6FVF6F-F6F8F6F***F)F6FVF6FPF6FBF6F9*, F)F6FVF6F3F6FBF6F8F6F**,F)F6FVF6F-F6FPF6FBF6F*F6**-F.6#FDF6F1\"\"\"FE \"\"\"-F.6#F0F6F=F9" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#TTG*&,>**%#V xG\"\"\"%\"xGF)-%%sqrtG6#*&,&*$)%\"CG\"\"#\"\"\"F)*$)F(F3F4!\"\"F4*$)F 2\"\"#F4!\"\"F4)F2\"\"%F4F7*,F(F4F*F4F+F4)%#VzGF3F4F1F4F)*,F(F4F*F4-F, 6#*&,&F0F)*$F?F4F7F4*$)F2\"\"#F4F;F4F?F4F1F4F7**)F(\"\"$F4F*F4FBF4F?F4 F)*,F(F4F*F4F+F4F%%DETLG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,initial_rQFG,**$)%\"xG\"\"#\"\"\"\"\"\"*$ )%\"yGF)F*F+*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Final_RQFG,**$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF) F*F+*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F*!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&*&,8*$)%\"CG\"\")\"\"\"\"\"\"*&)F,\"\"' F.)%#VzG\"\"#F.!\"#*&F1F.)%#VxGF5F.F5*(F1F.F8F.-%%sqrtG6#*&,&*$)F,F5F. F/*$F3F.!\"\"F.*$)F,\"\"#F.!\"\"F.!\"%*,F3F.F;F.)F,\"\"%F.-F<6#*&,&F@F /*$F8F.FCF.*$)F,\"\"#F.FGF.F8F.FK*&FJF.)F4FKF.F/**F3F.FJF.FLF.F8F.FH*& )F9FKF.FJF.F/*(FJF.F8F.F3F.F5*(FXF.FAF.F3F.F6*&FXF.FUF.FCF.*&)FO\"\"#F .)F?\"\"#F.FG\"\"!,$*&*(F4F/F9F/,4*$FJF.F/*&FAF.F3F.FC*(F;F.F8F.F3F.F/ *&FJF.FLF.FC*(FAF.FLF.F3F.F/*&FJF.F;F.FC**FAF.F;F.F3F.FLF.F/*(FJF.F;F. FLF.F/*(FAF.F;F.F3F.FCF/F.**)F?\"\"#F.-F<6#FNF.-F<6#F>F.FO\"\"\"FGF6,$ *&*(,6*&F1F.F;F.F/*$F1F.FC*&F8F.FJF.FC*(F8F.F;F.FJF.F/**F3F.FJF.F;F.FL F.FC*(F3F.FJF.FLF.F5*,F;F.FAF.FLF.F8F.F3F.FC*(F8F.FAF.F3F.F/*&FAF.FUF. FC*&F8F.FUF.F/F/F,F/F9F.F.*&)FO\"\"#F.)F?\"\"#F.FGF67&F[oF/F[oF[o7&F\\ oF[o*&,.F^qFCFfpF/FgpFCFjpFHF]qF/*&FJF.F3F.F5F.*&)F?\"\"#F.FO\"\"\"FG, $*&*(F4F.F,F.,4*&F8F.FAF.F/*&F8F.F3F.FCFboF/*(FLF.F8F.FAF.FC*(F8F.F3F. FLF.F/FeoFCFgoF/FfoF/FhoFCF/F.**)F?\"\"#F.-F<6#FNF.-F<6#F>F.FO\"\"\"FG F67&FapF[oF]r*&,6F*F/F0F5*(F3F.F1F.FLF.FHF7F5F:FHFYFHFTF/FIFKFWF/FenFC F.*&)FO\"\"#F.)F?\"\"#F.FG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 201 "Th e same is true for a Round Trip RT product of two rotations followed b y two inverse rotations for a product of 4 generators. The result is \+ still a Lorentz transformation, but it is not the identity." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "RT:=evalm(innerprod(LRphizn ,LRthetaxn,LRphiz,LRthetax));RPRIME:=simplify(innerprod(RT,r)):XX:=sim plify(RPRIME[1]);YY:=RPRIME[2];ZZ:=(RPRIME[3]);TT:=(simplify(RPRIME[4] ));DETL:=simplify(det(RT));R:=innerprod(RT,r):initial_rQF:=innerprod(r ,LGUN,r);Final_RQF:=simplify(factor(innerprod(R,LGUN,R)));GUN:=innerpr od(transpose(RT),RT);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#RTG-%'matr ixG6#7&7&*&,(*&)-%$cosG6#%$phiG\"\"#\"\"\")-F/6#%&thetaGF2F3\"\"\"*&F- F3)-%$sinGF6F2F3F8*&)-FF3F8\"\"\",&*$F4 F3F8*$F:F3F8\"\"\"!\"\"*&*&F?F8,**&F.F8)F5\"\"$F3F8*(F.F3F5F3F:F3F8*&F .F3F4F3!\"\"FGFRF8F3*&FA\"\"\"FE\"\"\"FI*&*(F?F3F;F8,*FQF8*&F.F3F:F3F8 *&F.F3F5F3FRF5F8F8F3*&FA\"\"\"FE\"\"\"FI\"\"!7&*&*(F.F3F?F3,(FFF8FGF8F 5FRF8F3*&FA\"\"\"FE\"\"\"FI*&,**&F>F3FNF3F8*(F5F3F>F3F:F3F8F,F8FYF8F3* &FA\"\"\"FE\"\"\"FI*&*&F;F3,**&F>F3F4F3F8*&F>F3F:F3F8*&F-F3F5F3F8FZFRF 8F3*&FA\"\"\"FE\"\"\"FIFhn7&,$*&*&F;F3F?F3F3FEFIFR*&*(F5F3F;F3,&F.F8FR F8F8F3FEFI*&,&FYF8FFF8F3FEFIFhn7&FhnFhnFhnF8" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#XXG,6*&%\"xG\"\"\")-%$cosG6#%$phiG\"\"#\"\"\"F(*&F'F /-F+6#%&thetaGF(F(*(F'F/F1F/F)F/!\"\"**-%$sinGF,F(%\"yGF(F*F(F1F/F(**F 7F/F9F/F*F/)F1F.F/F5*&F7F/F9F/F5*(F;F/F7F/F9F/F(**F7F/%\"zGF(F*F/-F8F2 F(F(*,F7F/F@F/F?F/F*F/F1F/F5**F7F/F@F/F?F/F1F/F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#YYG,8*(-%$cosG6#%$phiG\"\"\"-%$sinGF)F+%\"xGF+F+**F' \"\"\"F,F0F.F0-F(6#%&thetaGF+!\"\"*&%\"yGF+F1F0F+*(F6F0F1F0)F'\"\"#F0F 4*(F6F0F8F0)F1F9F0F+*&F6F0F'F0F+*(F6F0F'F0F;F0F4*&%\"zGF+-F-F2F+F+*(F? F0F@F0F8F0F4**F@F0F?F0F8F0F1F0F+**F@F0F?F0F'F0F1F0F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ZZG,.*(-%$sinG6#%&thetaG\"\"\"-F(6#%$phiGF+%\"xGF +!\"\"**-%$cosGF)F+F'\"\"\"%\"yGF+-F3F-F+F+*(F2F4F'F4F5F4F0*&%\"zGF+F6 F4F+*(F9F4F6F4)F2\"\"#F4F0*&F9F4F;F4F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TTG*&%\"CG\"\"\"%\"tGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% %DETLG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,initial_rQFG,**$)% \"xG\"\"#\"\"\"\"\"\"*$)%\"yGF)F*F+*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F *!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Final_RQFG,**$)%\"xG\"\"# \"\"\"\"\"\"*$)%\"yGF)F*F+*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F*!\"\"" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$GUNG-%'matrixG6#7&7&*&,.*&)-%$cosG 6#%$phiG\"\"#\"\"\")-%$sinG6#%&thetaG\"\"%F3\"\"\"*()-F/F7F2F3)F5F2F3F -F3F2*&F-F3)F=F9F3F:*(F-F3)-F6F0F2F3F>F3F:*&)FCF9F3F>F3F:*&FBF3FF3F:\"\"#F3!\"\",$*&*(,4*&FB F3F>F3!\"\"*(FBF3F.F:F>F3F:*&F-F3F>F3FW*(F.F3F>F3FF3F:*&F.F3F@F3FW*&F.F3FF3F2**F-F3FBF3F>F3FF3Fen*&FBF3)F=\"\"'F3 F:F?F:*(F>F3FBF3FF3Fen*()F.F9F3F>F3FF3F:*&F.F3F>F3F:F[oFWFgnFW*&FhnF3FF3F " 0 "" {MPLTEXT 1 0 409 "RT1:=evalm(innerprod(LRzn,LRxn,LRz,LRx));RT:=in nerprod(transpose( RT1),RT1);RPRIME:=simplify(innerprod(RT,r)):XX:=sim plify(RPRIME[1]);YY:=RPRIME[2];ZZ:=(RPRIME[3]);TT:=(simplify(RPRIME[4] ));DETL:=simplify(det(RT));R:=innerprod(RT,r):initial_rQF:=innerprod(r ,LGUN,r);Final_RQF:=subs(Lx=1,Ly=1,factor(innerprod(R,LGUN,R)));innerp rod(transpose(RT),RT);innerprod(transpose(LRz),LRz);innerprod(transpos e(LTz),LTz);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$RT1G-%'matrixG6#7&7 &,(\"\"\"F+*$)%#LzG\"\"#\"\"\"!\"\"*&-%%sqrtG6#,&F+F+*$)%#LxGF/F0F1F0F -F0F+,**(F.F+F3F0-F46#,&F+F+F,F1F0F+*&F%#RTG-%'matrixG6#7&7&\"\"\"\"\"!F+F+7&F+F*F+F+7&F+F+F* F+7&F+F+F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#XXG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YYG%\"yG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ZZG%\"zG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TTG*&%\"CG\"\"\"% \"tGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DETLG\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%,initial_rQFG,**$)%\"xG\"\"#\"\"\"\"\"\"*$)%\" yGF)F*F+*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Final_RQFG,**$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF)F*F+* $)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\"\"\"\"!F)F)7&F)F(F)F)7&F)F)F(F)7&F)F)F)F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&*$)%\"EG\"\"#\"\"\"\" \"!F-F-7&F-F(F-F-7&F-F-F(F-7&F-F-F-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\"\"\"\"!F)F)7&F)F(F)F)7&F)F)*&,&*$)%\"CG\"\"#\" \"\"F(*$)%#VzGF1F2F(F2,&F.F(F3!\"\"!\"\",$*&*&F0F(F5F(F2F6F8F17&F)F)F9 F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 197 "On the other hand a translation along an axis, followed by a rotation about the same axis, followed by the inverse translatio n, followed by the inverse rotation about the same axis is the Identit y." }{TEXT -1 21 " The process closes." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 278 "RT:=innerprod(LRzn,L Tzn,LRz,LTz);RPRIME:=simplify(innerprod(RT,r)):XX:=simplify(RPRIME[1]) ;YY:=RPRIME[2];ZZ:=(RPRIME[3]);TT:=(simplify(RPRIME[4]));DETL:=simplif y(det(RT));R:=innerprod(RT,r):initial_rQF:=innerprod(r,LGUN,r);Final_R QF:=subs(Lx=1,Ly=1,factor(innerprod(R,LGUN,R)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#RTG-%'matrixG6#7&7&\"\"\"\"\"!F+F+7&F+F*F+F+7&F+F+F* F+7&F+F+F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#XXG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YYG%\"yG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ZZG%\"zG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TTG*&%\"CG\"\"\"% \"tGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DETLG\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%,initial_rQFG,**$)%\"xG\"\"#\"\"\"\"\"\"*$)%\" yGF)F*F+*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Final_RQFG,**$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF)F*F+* $)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)F*!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 299 "However this is not the case if the translation and rota tion are not about the same axis. THen the RT matrix is not the ident ity and the process of translate - rotate - inverse translate - invers e rotate is not the identity map. The RT product is still a Lorentz m ap even though it does not close." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 278 "RT:=innerprod(LRxn,LTzn,LRx,LTz);RPRIME:=simplify(in nerprod(RT,r)):XX:=simplify(RPRIME[1]);YY:=RPRIME[2];ZZ:=(RPRIME[3]);T T:=(simplify(RPRIME[4]));DETL:=simplify(det(RT));R:=innerprod(RT,r):in itial_rQF:=innerprod(r,LGUN,r);Final_RQF:=subs(Lx=1,Ly=1,factor(innerp rod(R,LGUN,R)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#RTG-%'matrixG6# 7&7&\"\"\"\"\"!F+F+7&F+,$*&,(*$-%%sqrtG6#*&,&*$)%\"CG\"\"#\"\"\"F**$)% #VzGF9F:!\"\"F:*$)F8\"\"#F:!\"\"F:F>*&)%#LxGF9F:F1F:F**$FDF:F>F:*$-F26 #F4F:FBF>*&*&FEF*,(*(-F26#,&F*F*FFF>F:F7F:F1F:F**&FNF:F7F:F>F;F*F*F:F5 FB*&**FEF:F8F*F=F*,(*&FNF:F1F:F**$FNF:F>F*F*F*F:F5FB7&F+*&*(FNF:FEF:,& F0F*F>F*F*F:*$-F26#F4F:FB*&,**(F7F:FDF:F1F:F*F6F**&F7F:FDF:F>*&FNF:FF:F5FB*&*(F8F:F=F:,*FCF*F*F*FFF>FVF>F*F:F5FB7&F+*&*&FEF:F=F:F:*&-F2 6#F4F:F8\"\"\"FB,$*&*(F8F:F=F:,&F>F*FVF*F*F:F5FBF>*&,&F\\oF>F6F*F:F5FB " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#XXG%\"xG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#YYG,$*&,>*(%\"yG\"\"\"-%%sqrtG6#*&,&*$)%\"CG\"\"#\" \"\"F**$)%#VzGF3F4!\"\"F4*$)F2\"\"#F4!\"\"F4F1F4F8*(F)F4F+F4F6F4F***F) F4F1F4)%#LxGF3F4F+F4F***F)F4F?F4F+F4F6F4F8*(F)F4F1F4F?F4F8*(F)F4F?F4F6 F4F***F@F*%\"zGF*-F,6#,&F*F**$F?F4F8F4F1F4F8**F@F4FEF4FFF4F6F4F**,F@F4 FEF4F+F4FFF4F1F4F***F@F4FEF4F+F4F6F4F8*,F@F4F1F4F7F*%\"tGF*FFF4F8**F@F 4)F7\"\"$F4FNF4FFF4F**.F@F4F1F4F7F4FNF4F+F4FFF4F**,F@F4F1F4F7F4FNF4F+F 4F8F4*&-F,6#F.F4F/\"\"\"F%#ZZG ,$*&,>*,-%%sqrtG6#,&\"\"\"F-*$)%#LxG\"\"#\"\"\"!\"\"F2F0F-%\"yGF--F*6# *&,&*$)%\"CGF1F2F-*$)%#VzGF1F2F3F2*$)F;\"\"#F2!\"\"F2F:F2F3*,F)F2F0F2F 4F2F5F2F=F2F-**F)F2F0F2F4F2F:F2F-**F)F2F0F2F4F2F=F2F3*(%\"zGF-F:F2F/F2 F3*(FGF2F/F2F=F2F-*(FGF2F5F2F:F2F3**FGF2F5F2F:F2F/F2F-**FGF2F5F2F)F2F= F2F-**F:F2F>F-%\"tGF-F/F2F3*()F>\"\"$F2FMF2F/F2F-**F:F2F>F2FMF2F5F2F3* ,F:F2F>F2FMF2F5F2F/F2F-*,F:F2F>F2FMF2F5F2F)F2F-F2*&-F*6#F7F2F8\"\"\"FB F3" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#TTG*&,.**%#LxG\"\"\"%#VzGF)% \"yGF))%\"CG\"\"#\"\"\"F)*(F(F/)F*\"\"$F/F+F/!\"\"*,F,F/F*F/%\"zGF)-%% sqrtG6#*&,&*$F,F/F)*$)F*F.F/F3F/*$)F-\"\"#F/!\"\"F/-F76#,&F)F)*$)F(F.F /F3F/F3**F,F/F*F/F5F/F6F/F)*,F,F/%\"tGF)F6F/FBF/F=F/F3*()F-\"\"%F/FIF/ F6F/F)F/*(-F76#F9F/F-\"\"\"F:\"\"\"FA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DETLG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,initial_rQFG,* *$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF)F*F+*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"t GF)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Final_RQFG,**$)%\"xG \"\"#\"\"\"\"\"\"*$)%\"yGF)F*F+*$)%\"zGF)F*F+*&)%\"CGF)F*)%\"tGF)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 -1 284 "Now for any product combinat ion of the Lorentz map generators it is possible to compute the Struct ural equations of Cartan based on the Frame field constructed from the given ( or presumed ) Lorentz map. The Cartan connection linearly co nnects the given Frame Field to its neighbors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The algebra can be quite \+ messy, but Maple with do the work." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 277 10 "Example 1:" }}{PARA 0 "" 0 "" {TEXT 278 50 "THE TORSION and CURVATURE of a LORENTZ translation" }}{PARA 0 "" 0 "" {TEXT -1 16 "along the z axis" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "FFINV:=LTz: Vz:=g*t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#VzG*& %\"gG\"\"\"%\"tGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "Z:=innerprod(FFINV,[d(x),d(y),d(z) ,d(C*t)]):sigma1:=Z[1];sigma2:=Z[2];sigma3:=Z[3];omega:=simplify((Z[4] ));" }}{PARA 256 "" 0 "" {TEXT 257 21 "The Vierbein 1-forms." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma1G-%\"dG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma2G-%\"dG6#%\"yG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma3G*&,(*&-%\"dG6#%\"zG\"\"\"%\"CGF,F,*(%\"gGF,)%\"tG\"\"# \"\"\"-F)6#F-F,F,**F/F3F1F,F-F3-F)6#F1F,F,F3*&-%%sqrtG6#,$*&,&*$)F-F2F 3!\"\"*&)F/F2F3F0F3F,F3*$)F-\"\"#F3!\"\"FBF3F-\"\"\"FH" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG*&,(*(%\"gG\"\"\"%\"tGF)-%\"dG6#%\"zGF)F) *(%\"CGF)F*\"\"\"-F,6#F0F)F)*&)F0\"\"#F1-F,6#F*F)F)F1*&-%%sqrtG6#,$*&, &*$F5F1!\"\"*&)F(F6F1)F*F6F1F)F1*$)F0\"\"#F1!\"\"FAF1F0\"\"\"FH" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "#Vol4:=wcollect(simplify(sig ma1&^sigma2&^sigma3&^Z[4]));rho:=subs(getcoeff(Vol4));" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT 258 25 "The density (determinant)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 185 "The determinant cannot go to zero for th e projective domain. The zero sets of the density function determine \+ a hypersurface. IF the hypersurface is harmonic then it can be a boun dary." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 " There is an induced metric on R4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "FF:=inverse(FFINV):Gun:=subs(innerprod(transpose(FF), FF));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GunG-%'matrixG6#7&7&\"\"\" \"\"!F+F+7&F+F*F+F+7&F+F+,$*&,&*$)%\"CG\"\"#\"\"\"F**&)%\"gGF4F5)%\"tG F4F5F*F5,&F1!\"\"F6F*!\"\"F<,$*&*(F3F*F8F*F:F*F5F;F=F47&F+F+F>F." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "From the Frame Field use the stand ard methods to compute the " }}{PARA 0 "" 0 "" {TEXT 262 36 "Cartan Ma trix of connection 1-forms." }}{PARA 0 "" 0 "" {TEXT -1 46 "See http:/ /www.uh.edu/~rkiehn/pdf/projfram.pdf" }}{PARA 0 "" 0 "" {TEXT -1 64 "f or details of the Cartan method for an arbitrary Repere Mobile." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "dFF:=array([[d(FF[1,1]),d(F F[1,2]),d(FF[1,3]),d(FF[1,4])],[d(FF[2,1]),d(FF[2,2]),d(FF[2,3]),d(FF[ 2,4])],[d(FF[3,1]),d(FF[3,2]),d(FF[3,3]),d(FF[3,4])],[d(FF[4,1]),d(FF[ 4,2]),d(FF[4,3]),d(FF[4,4])]]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "cartan:=(evalm(FFINV&*dFF)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 45 "The Interior (space-space) Connection 1 forms" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Gamma11:=factor(wcollect(cartan[1, 1]));Gamma21:=factor(wcollect(cartan[2,1]));Gamma31:=factor(wcollect(c artan[3,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma11G\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma21G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma31G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Gamma12:=factor(wcollect(cartan[1,2]));Gamma22:=factor(wcolle ct(cartan[2,2]));Gamma32:=factor(wcollect(cartan[3,2]));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%(Gamma12G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma22G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma32G \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "Gamma13:=wcollect (factor(wcollect(cartan[1,3])));Gamma23:=factor(wcollect(cartan[2,3])) ;Gamma33:=simplify(wcollect(cartan[3,3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma13G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%( Gamma23G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma33G\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 40 "The \"space-time\" connection 1-forms are:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "hh1:=simplify(wcollect(cartan[4,1] ));hh2:=factor(wcollect(cartan[4,2]));hh3:=wcollect(factor(wcollect(ca rtan[4,3])));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$hh1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$hh 2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$hh3G,&*&*(%\"gG\"\"\"%\" tGF)-%\"dG6#%\"CG\"\"\"F)*&,&F.!\"\"*&F(F/F*F/F/\"\"\",&F.F/F3F/\"\"\" !\"\"F2*&*(F(F)F.F/-F,6#F*F/F)*&F1\"\"\"F5\"\"\"F7F/" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 40 "The \"time-space connection\" 1-forms are " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "gg1:=factor(wcollect(fact or(wcollect(cartan[1,4]))));gg2:=factor(wcollect(cartan[2,4]));gg3:=wc ollect(factor(wcollect(cartan[3,4])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg2G\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg3G,&*&*(%\"gG\"\"\"%\"tGF)-%\"dG6 #%\"CG\"\"\"F)*&,&F.!\"\"*&F(F/F*F/F/\"\"\",&F.F/F3F/\"\"\"!\"\"F2*&*( F(F)F.F/-F,6#F*F/F)*&F1\"\"\"F5\"\"\"F7F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 46 "The abnormality (time-time) connection 1-form " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Note that the abnormality (Big Ome ga vanishes if expansion is a global constant. (no red shift?)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Omega:=wcollect(subs(simplif y(wcollect(cartan[4,4]))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&Omeg aG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "L:=factor(wcolle ct(hh1&^sigma1+hh2&^sigma2+hh3&^sigma3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,$*&*&%\"gG\"\"\",(*&%\"tG\"\"\"-%#&^G6$-%\"dG6#% \"CG-F26#%\"zGF-F-*(F(F-)F,\"\"#F)-F/6$F1-F26#F,F-F:*&F4F--F/6$F=F5F-! \"\"F-F)*(-%%sqrtG6#,$*&*&,&F4FB*&F(F)F,F)F-F-,&F4F-FKF-F-F)*$)F4\"\"# F)!\"\"FBF)FJ\"\"\"FL\"\"\"FPFB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "S:=(wcollect(factor(hh1&^gg1+hh2&^gg2+hh3&^gg3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "There are in general two sets of torsion two forms. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "1. Parti cle AFFINE (" }{TEXT 266 2 "PA" }{TEXT -1 158 ") torsion 2-forms whic h depend upon the product of little omega (the timelike part of the Vi erbein) and the (time-space) connection components, little gamma." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "2. WaveA FFINE (" }{TEXT 267 2 "WA" }{TEXT -1 162 ") torsion 2-forms which depe nd upon Big Omega (the time-time connection component or abnormality) \+ and again the (time-space) connection components, little gamma." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "If the ti me-space connection 1-forms vanish, small gamma, neither form of torsi on exists." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "See http://www.uh.edu/~rkiehn/pdf/projfram.pdf" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 19 "PA TORSION 2-forms " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "Sigma1:=w collect(simplify((wcollect(factor(omega&^gg1))))); Sigma2:=factor(simp lify(wcollect(factor(omega&^gg2))));Sigma3:=wcollect(simplify(wcollect (factor(simplify((omega&^gg3))))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'Sigma1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Sigma2G\"\"!" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Sigma3G,(*&**%\"CG\"\"\"%\"gG\"\" \"%\"tGF+-%#&^G6$-%\"dG6#F(-F16#F,F)F+*(-%%sqrtG6#,$*&*&,&F(!\"\"*&F*F )F,F)F)F),&F(F)F>F)F)F+*$)F(\"\"#F+!\"\"F=F+F<\"\"\"F?\"\"\"FC\"\"#*&* ()F*FFF+F,F+-F.6$-F16#%\"zGF3F)F+*(-F76#F9F+F<\"\"\"F?\"\"\"FCF)*&*(FI F+)F,FFF+-F.6$FLF0F)F+**F(\"\"\"-F76#F9F+F<\"\"\"F?\"\"\"FCF=" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 260 19 "WA TORSION 2-forms " }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Phi1:=simplify((wcol lect((factor(Omega&^gg1)))));Phi2:=wcollect(factor(Omega&^gg2));Phi3:= wcollect((factor(Omega&^gg3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% Phi1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Phi2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Phi3G\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Next compute the matrix of curvature 2-forms on the x,y,z subspace" }}{PARA 0 "" 0 "" {TEXT 261 18 "Curvature 2-forms " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "Theta:=array([[gg1&^hh1,gg1 &^hh2,gg1&^hh3],[gg2&^hh1,gg2&^hh2,gg2&^hh3],[(gg3&^hh1),(gg3&^hh2),(g g3&^hh3)]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&ThetaG-%'matrixG6#7%7%\"\"!F*F*F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The curvature 2-forms on the interior spa ce all vanish." }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 10 "DISCUSSION" } }{PARA 0 "" 0 "" {TEXT -1 61 "This example is rather remarkable in tha t it points out that " }{TEXT 268 4 "THIS" }{TEXT -1 155 " Lorentz tra nsformation does not introduce curvature on the 3D subspace (translati onal motions) even if accelerations and variable speeds C are admitted . " }}{PARA 0 "" 0 "" {TEXT -1 70 "The WA Torsion coefficients vanish if the Expansion is constant d(E)=0" }}{PARA 0 "" 0 "" {TEXT -1 72 "T he PA torsion coefficients are dependent on the Expansion as a factor ." }}{PARA 0 "" 0 "" {TEXT -1 97 "The Torsion coefficients vanish if b oth C and V are uniform and time independent, d(C)=0, d(V)=0." }} {PARA 0 "" 0 "" {TEXT -1 123 "The PA torsion is not zero (even though \+ d(C)=0) if the differential of the Velocity field does not vanish. (a ccelerations)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 268 10 "Example 2:" }}{PARA 0 "" 0 "" {TEXT 269 48 "THE TORSION and CURVATURE of a LORENTZ rotation " }} {PARA 0 "" 0 "" {TEXT -1 16 "about the z axis" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "FFINV:=LRz:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "Z:=innerprod(FFINV, [d(x),d(y),d(z),d(C*t)]):sigma1:=Z[1];sigma2:=Z[2];sigma3:=Z[3];omega: =simplify((Z[4]));" }}{PARA 256 "" 0 "" {TEXT 256 21 "The Vierbein 1-f orms." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma1G,&*(%\"EG\"\"\"-%%s qrtG6#,&F(F(*$)%#LzG\"\"#\"\"\"!\"\"F1-%\"dG6#%\"xGF(F(*(F'F1F/F(-F46# %\"yGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma2G,&*(%\"EG\"\"\" %#LzGF(-%\"dG6#%\"xGF(!\"\"*(F'\"\"\"-%%sqrtG6#,&F(F(*$)F)\"\"#F0F.F0- F+6#%\"yGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma3G*&%\"EG\"\" \"-%\"dG6#%\"zGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG*&%\"EG \"\"\",&*&%\"tGF'-%\"dG6#%\"CGF'F'*&F.F'-F,6#F*F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "#Vol4:=wcollect(simplify(sigma1&^si gma2&^sigma3&^Z[4]));rho:=subs(getcoeff(Vol4));" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 257 25 "The density (determinant)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 185 "The determinant cannot go to zero for the proj ective domain. The zero sets of the density function determine a hype rsurface. IF the hypersurface is harmonic then it can be a boundary. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "There is an induced metric on R4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "FF:=inverse(FFINV):Gun:=subs(innerprod(transpose(FF),FF));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GunG-%'matrixG6#7&7&*&\"\"\"F+*$)% \"EG\"\"#F+!\"\"\"\"!F1F17&F1F*F1F17&F1F1F*F17&F1F1F1F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "From the Frame Field use the standard met hods to compute the " }}{PARA 0 "" 0 "" {TEXT 261 36 "Cartan Matrix of connection 1-forms." }}{PARA 0 "" 0 "" {TEXT -1 46 "See http://www.uh .edu/~rkiehn/pdf/projfram.pdf" }}{PARA 0 "" 0 "" {TEXT -1 64 "for deta ils of the Cartan method for an arbitrary Repere Mobile." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "dFF:=array([[d(FF[1,1]),d(FF[1,2]) ,d(FF[1,3]),d(FF[1,4])],[d(FF[2,1]),d(FF[2,2]),d(FF[2,3]),d(FF[2,4])], [d(FF[3,1]),d(FF[3,2]),d(FF[3,3]),d(FF[3,4])],[d(FF[4,1]),d(FF[4,2]),d (FF[4,3]),d(FF[4,4])]]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "cartan:=(evalm(FFINV&*dFF)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 45 "The Interior (space-space) Connection 1 forms" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 117 "Gamma11:=factor(wcollect(cartan[1,1]));Gamma2 1:=factor(wcollect(cartan[2,1]));Gamma31:=factor(wcollect(cartan[3,1]) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma11G,$*&-%\"dG6#%\"EG\"\" \"F*!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma21G*&-%\"dG6 #%#LzG\"\"\"*$-%%sqrtG6#,$*&,&F)\"\"\"!\"\"F2F2,&F)F2F2F2F2F3F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma31G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Gamma12:=factor(wcollect(cartan[1,2]));Gam ma22:=factor(wcollect(cartan[2,2]));Gamma32:=factor(wcollect(cartan[3, 2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma12G,$*&-%\"dG6#%#LzG \"\"\"*$-%%sqrtG6#,$*&,&F*\"\"\"!\"\"F3F3,&F*F3F3F3F3F4F+!\"\"F4" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma22G,$*&-%\"dG6#%\"EG\"\"\"F*! \"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma32G\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "Gamma13:=wcollect(factor(wc ollect(cartan[1,3])));Gamma23:=factor(wcollect(cartan[2,3]));Gamma33:= simplify(wcollect(cartan[3,3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% (Gamma13G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma23G\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma33G,$*&-%\"dG6#%\"EG\"\"\"F*! \"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 40 "The \"space-time\" connection 1-forms ar e:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "hh1:=simplify(wcolle ct(cartan[4,1]));hh2:=factor(wcollect(cartan[4,2]));hh3:=wcollect(fact or(wcollect(cartan[4,3])));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$hh1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$hh2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$hh3G \"\"!" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 40 "The \"time-space conne ction\" 1-forms are " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "gg 1:=factor(wcollect(factor(wcollect(cartan[1,4]))));gg2:=factor(wcollec t(cartan[2,4]));gg3:=wcollect(factor(wcollect(cartan[3,4])));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg3G \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 46 "The abnormality (time-t ime) connection 1-form " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Note t hat the abnormality (Big Omega vanishes if expansion is a global const ant. (no red shift?)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Ome ga:=wcollect(subs(simplify(wcollect(cartan[4,4]))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OmegaG,$*&-%\"dG6#%\"EG\"\"\"F*!\"\"!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "L:=factor(wcollect(hh1&^sigm a1+hh2&^sigma2+hh3&^sigma3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"L G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "S:=(wcollect(fact or(hh1&^gg1+hh2&^gg2+hh3&^gg3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"SG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "There are in genera l two sets of torsion two forms. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 21 "1. Particle AFFINE (" }{TEXT 265 2 "PA" }{TEXT -1 158 ") torsion 2-forms which depend upon the product of lit tle omega (the timelike part of the Vierbein) and the (time-space) con nection components, little gamma." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "2. WaveAFFINE (" }{TEXT 266 2 "WA" } {TEXT -1 162 ") torsion 2-forms which depend upon Big Omega (the time- time connection component or abnormality) and again the (time-space) \+ connection components, little gamma." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "If the time-space connection 1-forms va nish, small gamma, neither form of torsion exists." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "See http://www.uh.edu/~rk iehn/pdf/projfram.pdf" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 19 "PA TORSION 2-forms " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "Sigma1:=wcollect(simplify((wcollect(fact or(omega&^gg1))))); Sigma2:=factor(simplify(wcollect(factor(omega&^gg2 ))));Sigma3:=wcollect(factor(simplify((omega&^gg3))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Sigma1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'Sigma2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Sigma3G\"\"!" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 259 19 "WA TORSION 2-forms " }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Phi1:=simplify((wc ollect((factor(Omega&^gg1)))));Phi2:=wcollect(factor(Omega&^gg2));Phi3 :=wcollect((factor(Omega&^gg3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%Phi1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Phi2G\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Phi3G\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Next compute the matrix of curvature 2-forms on the x, y,z subspace" }}{PARA 0 "" 0 "" {TEXT 260 18 "Curvature 2-forms " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "Theta:=array([[gg1&^hh1,gg1 &^hh2,gg1&^hh3],[gg2&^hh1,gg2&^hh2,gg2&^hh3],[(gg3&^hh1),(gg3&^hh2),(g g3&^hh3)]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&ThetaG-%'matrixG6#7%7%\"\"!F*F*F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The curvature 2-forms on the interior spa ce all vanish." }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 10 "DISCUSSION" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 283 87 "Single axis Lorentz Rotations d o not generate torsion or curvature on the 3D subspace !" }}{PARA 0 " " 0 "" {TEXT -1 94 "(and as is shown below, multiple Lorentz rotations do not generate torsion or curvature on 3D." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 10 "Example 3:" }}{PARA 0 "" 0 "" {TEXT 269 83 "THE TORSION and CURVATURE of a LORENTZ translation combined with a Lorentz rotation" }}{PARA 0 "" 0 "" {TEXT -1 27 "both relative to the z axis" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "FFINV:=evalm(innerpr od(LRz,LTz)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "Z:=innerprod(FFINV,[d(x),d(y),d(z),d(C*t)]):sigma1:= Z[1];sigma2:=Z[2];sigma3:=Z[3];omega:=simplify((Z[4]));" }}{PARA 256 " " 0 "" {TEXT 256 21 "The Vierbein 1-forms." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma1G,&*(%\"EG\"\"\"-%%sqrtG6#,&F(F(*$)%#LzG\"\"# \"\"\"!\"\"F1-%\"dG6#%\"xGF(F(*(F'F1F/F(-F46#%\"yGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma2G,&*(%\"EG\"\"\"%#LzGF(-%\"dG6#%\"xGF(!\" \"*(F'\"\"\"-%%sqrtG6#,&F(F(*$)F)\"\"#F0F.F0-F+6#%\"yGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma3G*&*&%\"EG\"\"\",(*&-%\"dG6#%\"zGF( %\"CGF(F(*(%\"gGF()%\"tG\"\"#\"\"\"-F,6#F/F(F(**F1F5F3F(F/F5-F,6#F3F(F (F(F5*&-%%sqrtG6#,$*&,&*$)F/F4F5!\"\"*&)F1F4F5F2F5F(F5*$)F/\"\"#F5!\" \"FDF5F/\"\"\"FJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG*&*&%\"EG \"\"\",(*(%\"gGF(%\"tGF(-%\"dG6#%\"zGF(F(*(%\"CGF(F,\"\"\"-F.6#F2F(F(* &)F2\"\"#F3-F.6#F,F(F(F(F3*&-%%sqrtG6#,$*&,&*$F7F3!\"\"*&)F+F8F3)F,F8F 3F(F3*$)F2\"\"#F3!\"\"FCF3F2\"\"\"FJ" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "#Vol4:=wcollect(simplify(sigma1&^sigma2&^sigma3&^Z[4] ));rho:=subs(getcoeff(Vol4));" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 257 25 "The density (determinant)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 185 "The determinant cannot go to zero for the projective domain. The zer o sets of the density function determine a hypersurface. IF the hyper surface is harmonic then it can be a boundary." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "There is an induced metri c on R4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "FF:=inverse(FFIN V):Gun:=subs(innerprod(transpose(FF),FF));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GunG-%'matrixG6#7&7&*&\"\"\"F+*$)%\"EG\"\"#F+!\"\"\" \"!F1F17&F1F*F1F17&F1F1,$*&,&*$)%\"CG\"\"#F+\"\"\"*&)%\"gGF:F+)%\"tGF: F+F;F+*&)F.\"\"#F+,&F7!\"\"FF;F@F;F+*&)F.\"\" #F+FD\"\"\"F0F:7&F1F1FGF4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "From the Frame Field use the standard methods to compute the " }}{PARA 0 " " 0 "" {TEXT 261 36 "Cartan Matrix of connection 1-forms." }}{PARA 0 " " 0 "" {TEXT -1 46 "See http://www.uh.edu/~rkiehn/pdf/projfram.pdf" }} {PARA 0 "" 0 "" {TEXT -1 64 "for details of the Cartan method for an a rbitrary Repere Mobile." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "dFF:=array([[d(FF[1,1]),d(FF[1,2]),d(FF[1,3]),d(FF[1,4])],[d(FF[2,1]) ,d(FF[2,2]),d(FF[2,3]),d(FF[2,4])],[d(FF[3,1]),d(FF[3,2]),d(FF[3,3]),d (FF[3,4])],[d(FF[4,1]),d(FF[4,2]),d(FF[4,3]),d(FF[4,4])]]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "cartan:=(evalm(FFINV&*dFF)): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 45 "The Interior (space-space) C onnection 1 forms" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Gamma 11:=factor(wcollect(cartan[1,1]));Gamma21:=factor(wcollect(cartan[2,1] ));Gamma31:=factor(wcollect(cartan[3,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma11G,$*&-%\"dG6#%\"EG\"\"\"F*!\"\"!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma21G*&-%\"dG6#%#LzG\"\"\"*$-%%s qrtG6#,$*&,&F)\"\"\"!\"\"F2F2,&F)F2F2F2F2F3F*!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%(Gamma31G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Gamma12:=factor(wcollect(cartan[1,2]));Gamma22:=fact or(wcollect(cartan[2,2]));Gamma32:=factor(wcollect(cartan[3,2]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma12G,$*&-%\"dG6#%#LzG\"\"\"*$-% %sqrtG6#,$*&,&F*\"\"\"!\"\"F3F3,&F*F3F3F3F3F4F+!\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma22G,$*&-%\"dG6#%\"EG\"\"\"F*!\"\"!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma32G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "Gamma13:=wcollect(factor(wcollect(cartan[1,3 ])));Gamma23:=factor(wcollect(cartan[2,3]));Gamma33:=simplify(wcollect (cartan[3,3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma13G\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma23G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma33G,$*&-%\"dG6#%\"EG\"\"\"F*!\"\"!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 40 "The \"space-time\" connection 1-forms are:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "hh1:=simplify(wcollect(cartan[4,1] ));hh2:=factor(wcollect(cartan[4,2]));hh3:=wcollect(factor(wcollect(ca rtan[4,3])));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$hh1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$hh 2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$hh3G,&*&*(%\"gG\"\"\"%\" tGF)-%\"dG6#%\"CG\"\"\"F)*&,&F.!\"\"*&F(F/F*F/F/\"\"\",&F.F/F3F/\"\"\" !\"\"F2*&*(F(F)F.F/-F,6#F*F/F)*&F1\"\"\"F5\"\"\"F7F/" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 40 "The \"time-space connection\" 1-forms are " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "gg1:=factor(wcollect(fact or(wcollect(cartan[1,4]))));gg2:=factor(wcollect(cartan[2,4]));gg3:=wc ollect(factor(wcollect(cartan[3,4])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg2G\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg3G,&*&*(%\"gG\"\"\"%\"tGF)-%\"dG6 #%\"CG\"\"\"F)*&,&F.!\"\"*&F(F/F*F/F/\"\"\",&F.F/F3F/\"\"\"!\"\"F2*&*( F(F)F.F/-F,6#F*F/F)*&F1\"\"\"F5\"\"\"F7F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 46 "The abnormality (time-time) connection 1-form " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Note that the abnormality (Big Ome ga vanishes if expansion is a global constant. (no red shift?)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Omega:=wcollect(subs(simplif y(wcollect(cartan[4,4]))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&Omeg aG,$*&-%\"dG6#%\"EG\"\"\"F*!\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "L:=factor(wcollect(hh1&^sigma1+hh2&^sigma2+hh3&^sigma 3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,$*&*(%\"EG\"\"\"%\"gG\" \"\",(*&%\"tGF)-%#&^G6$-%\"dG6#%\"CG-F36#%\"zGF)F)*(F*F))F.\"\"#F+-F06 $F2-F36#F.F)F;*&F5F)-F06$F>F6F)!\"\"F)F+*(-%%sqrtG6#,$*&*&,&F5FC*&F*F+ F.F+F)F),&F5F)FLF)F)F+*$)F5\"\"#F+!\"\"FCF+FK\"\"\"FM\"\"\"FQFC" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "S:=(wcollect(factor(hh1&^gg1 +hh2&^gg2+hh3&^gg3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG\"\"! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "There are in general two sets of torsion two forms. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 21 "1. Particle AFFINE (" }{TEXT 265 2 "PA" }{TEXT -1 158 ") torsion 2-forms which depend upon the product of little ome ga (the timelike part of the Vierbein) and the (time-space) connection components, little gamma." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "2. WaveAFFINE (" }{TEXT 266 2 "WA" }{TEXT -1 162 ") torsion 2-forms which depend upon Big Omega (the time-time conn ection component or abnormality) and again the (time-space) connectio n components, little gamma." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "If the time-space connection 1-forms vanish, sm all gamma, neither form of torsion exists." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "See http://www.uh.edu/~rkiehn/p df/projfram.pdf" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 19 "PA TORSION 2-forms " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "Sigma1:=wcollect(simplify((wcollect(factor(o mega&^gg1))))); Sigma2:=factor(simplify(wcollect(factor(omega&^gg2)))) ;Sigma3:=wcollect(factor(simplify((omega&^gg3))));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%'Sigma1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% 'Sigma2G\"\"!" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Sigma3G,(*&*,%\"tG \"\"\"%\"gGF)%\"EG\"\"\"%\"CGF,-%#&^G6$-%\"dG6#F--F26#F(F,F)*(-%%sqrtG 6#,$*&*&,&F-!\"\"*&F*F,F(F,F,F,,&F-F,F?F,F,F)*$)F-\"\"#F)!\"\"F>F)F=\" \"\"F@\"\"\"FD\"\"#*&**F(F))F*FGF)F+F)-F/6$-F26#%\"zGF4F,F)*(-F86#F:F) F=\"\"\"F@\"\"\"FDF,*&**)F(FGF)FJF)F+F)-F/6$FMF1F,F)**-F86#F:F)F-\"\" \"F=\"\"\"F@\"\"\"FDF>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 19 "WA TOR SION 2-forms " }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Phi1:=simplify((wcollect((factor(Omega&^gg1)))));Phi2:=wcollect (factor(Omega&^gg2));Phi3:=wcollect((factor(Omega&^gg3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Phi1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Phi2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Phi3G,&*&*(% \"gG\"\"\"%\"tGF)-%#&^G6$-%\"dG6#%\"EG-F/6#%\"CG\"\"\"F)*(F1\"\"\",&F4 !\"\"*&F(F5F*F5F5\"\"\",&F4F5F:F5\"\"\"!\"\"F5*&*(F4F5F(F)-F,6$F.-F/6# F*F5F)*(F1\"\"\"F8\"\"\"F<\"\"\"F>F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Next compute the matrix of curvature 2-forms on the x,y,z subsp ace" }}{PARA 0 "" 0 "" {TEXT 260 18 "Curvature 2-forms " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "Theta:=array([[gg1&^hh1,gg1&^hh2,g g1&^hh3],[gg2&^hh1,gg2&^hh2,gg2&^hh3],[(gg3&^hh1),(gg3&^hh2),(gg3&^hh3 )]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ThetaG-%'matrixG6#7%7%\"\"!F*F*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The curvature 2-forms on the interior space all vanish." }}}{EXCHG {PARA 259 "" 0 " " {TEXT -1 10 "DISCUSSION" }}{PARA 0 "" 0 "" {TEXT -1 61 "This example is rather remarkable in that it points out that " }{TEXT 267 4 "THIS " }{TEXT -1 153 " Lorentz transformation does not introduce curvature \+ on the 3D subspace (Helical like motion) even if accelerations and var iable speeds C are admitted. " }}{PARA 0 "" 0 "" {TEXT -1 70 "The WA \+ Torsion coefficients vanish if the Expansion is constant d(E)=0" }} {PARA 0 "" 0 "" {TEXT -1 72 "The PA torsion coefficients are dependent on the Expansion as a factor." }}{PARA 0 "" 0 "" {TEXT -1 97 "The To rsion coefficients vanish if both C and V are uniform and time indepen dent, d(C)=0, d(V)=0." }}{PARA 0 "" 0 "" {TEXT -1 108 "The PA torsion \+ is not zero (even though d(C)=0) if the differential of the Velocity f ield does not vanish. " }}{PARA 0 "" 0 "" {TEXT -1 66 "The Lorentz ro tation parameters do not enter into the formulas !!!" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 10 "Example 4:" }} {PARA 0 "" 0 "" {TEXT 269 112 "THE TORSION and CURVATURE of a LORENTZ \+ translation along the z axis and a Lorentz translation along the x ax is." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "FFINV:=evalm(innerprod(LTx,LTz));" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%&FFINVG-%'matrixG6#7&7&*&%\"EG\"\"\"*$-%%sqrtG6#*&, &*$)%\"CG\"\"#F,\"\"\"*$)%#VxGF6F,!\"\"F,*$)F5\"\"#F,!\"\"F,F?\"\"!*&* *F+F7F:F7%\"gGF7%\"tGF7F,*(-F/6#F1F,)F5\"\"#F,-F/6#,$*&,&F3F;*&)FCF6F, )FDF6F,F7F,*$)F5\"\"#F,F?F;F,F?*&*&F+F,F:F,F,*(-F/6#F1F,F5\"\"\"-F/6#F LF,F?7&F@F+F@F@7&F@F@*&F+F,*$-F/6#FLF,F?*&*(F+F,FCF,FDF,F,*&-F/6#FLF,F 5\"\"\"F?7&*&*&F+F,F:F,F,*&-F/6#F1F,F5\"\"\"F?F@*&*(F+F,FCF,FDF,F,*(-F /6#F1F,-F/6#FLF,F5\"\"\"F?*&F+F,*&-F/6#F1F,-F/6#FLF,F?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "Z:=innerprod(FFINV,[d(x),d(y),d(z),d(C*t)]):sigma1:= Z[1];sigma2:=Z[2];sigma3:=Z[3];omega:=simplify((Z[4]));" }}{PARA 256 " " 0 "" {TEXT 256 21 "The Vierbein 1-forms." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma1G*&*&%\"EG\"\"\",**(-%\"dG6#%\"xGF()%\"CG\"\"# \"\"\"-%%sqrtG6#,$*&,&*$F/F2!\"\"*&)%\"gGF1F2)%\"tGF1F2F(F2*$)F0\"\"#F 2!\"\"F:F2F(**%#VxGF(F=F(F?F(-F,6#%\"zGF(F(**FEF2F0F(F?F2-F,6#F0F(F(*( FEF2F/F2-F,6#F?F(F(F(F2*(-F46#*&,&F9F(*$)FEF1F2F:F2*$)F0\"\"#F2FCF2)F0 \"\"#F2-F46#F6F2FC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma2G*&%\"E G\"\"\"-%\"dG6#%\"yGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma3G*& *&%\"EG\"\"\",(*&-%\"dG6#%\"zGF(%\"CGF(F(*(%\"gGF()%\"tG\"\"#\"\"\"-F, 6#F/F(F(**F1F5F3F(F/F5-F,6#F3F(F(F(F5*&-%%sqrtG6#,$*&,&*$)F/F4F5!\"\"* &)F1F4F5F2F5F(F5*$)F/\"\"#F5!\"\"FDF5F/\"\"\"FJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG*&*&%\"EG\"\"\",**(%#VxGF(-%\"dG6#%\"xGF(-%%sq rtG6#,$*&,&*$)%\"CG\"\"#\"\"\"!\"\"*&)%\"gGF9F:)%\"tGF9F:F(F:*$)F8\"\" #F:!\"\"F;F:F(*(F>F(F@F(-F-6#%\"zGF(F(*(F8F(F@F:-F-6#F8F(F(*&F7F:-F-6# F@F(F(F(F:*(-F16#*&,&F6F(*$)F+F9F:F;F:*$)F8\"\"#F:FDF:F8\"\"\"-F16#F3F :FD" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "#Vol4:=wcollect(simp lify(sigma1&^sigma2&^sigma3&^Z[4]));rho:=subs(getcoeff(Vol4));" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 257 25 "The density (determinant)" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 185 "The determinant cannot go to ze ro for the projective domain. The zero sets of the density function d etermine a hypersurface. IF the hypersurface is harmonic then it can \+ be a boundary." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "There is an induced metric on R4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "FF:=inverse(FFINV):Gun:=subs(innerprod(transpose (FF),FF));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$GunG-%'matrixG6#7&7&* &,**$)%\"CG\"\"%\"\"\"!\"\"*()F.\"\"#F0)%\"gGF4F0)%\"tGF4F0\"\"\"*()%# VxGF4F0F5F0F7F0F1*&F3F0F;F0F1F0*()%\"EG\"\"#F0,&*$F3F0F9*$F;F0F1\"\"\" ,&FCF1*&F5F0F7F0F9\"\"\"!\"\"\"\"!,$*&*(F " 0 "" {MPLTEXT 1 0 198 "dFF:=array([ [d(FF[1,1]),d(FF[1,2]),d(FF[1,3]),d(FF[1,4])],[d(FF[2,1]),d(FF[2,2]),d (FF[2,3]),d(FF[2,4])],[d(FF[3,1]),d(FF[3,2]),d(FF[3,3]),d(FF[3,4])],[d (FF[4,1]),d(FF[4,2]),d(FF[4,3]),d(FF[4,4])]]):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "cartan:=(evalm(FFINV&*dFF)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 45 "The Interior (space-space) Connection 1 forms " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Gamma11:=factor(wcolle ct(cartan[1,1]));Gamma21:=factor(wcollect(cartan[2,1]));Gamma31:=facto r(wcollect(cartan[3,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma1 1G,$*&-%\"dG6#%\"EG\"\"\"F*!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma21G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma31G*&*, ,&%\"CG\"\"\"%#VxG!\"\"\"\"\",&F(F)F*F)F,,&*&F(F)-%\"dG6#%\"tGF)F+*&F3 F)-F16#F(F)F)F)F*F)%\"gGF,F,**)*&*&F'F)F-F)F,*$)F(\"\"#F,!\"\"#\"\"$\" \"#F,)F(\"\"$F,,&F(F+*&F7F)F3F,F)\"\"\",&F(F)FFF)\"\"\"F?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Gamma12:=factor(wcollect(cartan[1, 2]));Gamma22:=factor(wcollect(cartan[2,2]));Gamma32:=factor(wcollect(c artan[3,2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma12G\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma22G,$*&-%\"dG6#%\"EG\"\"\"F*! \"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma32G\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "Gamma13:=wcollect(factor(wc ollect(cartan[1,3])));Gamma23:=factor(wcollect(cartan[2,3]));Gamma33:= simplify(wcollect(cartan[3,3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% (Gamma13G,&*&**%#VxG\"\"\"%\"gG\"\"\"%\"tGF+-%\"dG6#%\"CGF)F+**,&F0!\" \"*&F*F)F,F)F)\"\"\",&F0F)F4F)\"\"\"F0\"\"\"-%%sqrtG6#*&*&,&F0F)F(F3F) ,&F0F)F(F)F)F+*$)F0\"\"#F+!\"\"F+FCF3*&*(-F.6#F,F)F(F+F*F+F+*(F2\"\"\" F6\"\"\"-F:6#F%(Gamma23G\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma33G,$*&-%\"dG6#%\"EG\" \"\"F*!\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 263 40 "The \"space-time\" connection 1-f orms are:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "hh1:=simplify (wcollect(cartan[4,1]));hh2:=factor(wcollect(cartan[4,2]));hh3:=wcolle ct(factor(wcollect(cartan[4,3])));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$hh1G,$*&,&*&%#VxG\"\"\"-%\"d G6#%\"CGF*!\"\"*&F.F*-F,6#F)F*F*\"\"\",&*$)F.\"\"#F3F**$)F)F7F3F/!\"\" F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$hh2G\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$hh3G,&*&*(%\"gG\"\"\"%\"tGF)-%\"dG6#%\"CG\"\"\"F)* (-%%sqrtG6#*&*&,&F.F/%#VxG!\"\"F/,&F.F/F7F/F/F)*$)F.\"\"#F)!\"\"F),&F. F8*&F(F/F*F/F/\"\"\",&F.F/F?F/\"\"\"F=F8*&*(-F,6#F*F/F.F/F(F)F)*(-F26# F4F)F>\"\"\"FA\"\"\"F=F/" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 40 "The \"time-space connection\" 1-forms are " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "gg1:=factor(wcollect(factor(wcollect(cartan[1,4])))) ;gg2:=factor(wcollect(cartan[2,4]));gg3:=wcollect(factor(wcollect(cart an[3,4])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg1G,$*&,&*&%#VxG\" \"\"-%\"dG6#%\"CGF*!\"\"*&F.F*-F,6#F)F*F*\"\"\"*&,&F.F*F)F/\"\"\",&F.F *F)F*\"\"\"!\"\"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg2G\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg3G,&*&*,,&%\"CG\"\"\"%#VxG!\"\"\" \"\",&F)F*F+F*F-%\"tGF-%\"gGF--%\"dG6#F)F*F-**)*&*&F(F*F.F*F-*$)F)\"\" #F-!\"\"#\"\"$\"\"#F-)F)\"\"#F-,&F)F,*&F0F*F/F*F*\"\"\",&F)F*FBF*\"\" \"F;F,*&**F(F-F.F--F26#F/F*F0F-F-**F)\"\"\")F6#\"\"$F>F-FA\"\"\"FD\"\" \"F;F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 46 "The abnormality (time- time) connection 1-form " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Note \+ that the abnormality (Big Omega vanishes if expansion is a global cons tant. (no red shift?)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Om ega:=wcollect(subs(simplify(wcollect(cartan[4,4]))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OmegaG,$*&-%\"dG6#%\"EG\"\"\"F*!\"\"!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "L:=factor(wcollect(hh1&^sigm a1+hh2&^sigma2+hh3&^sigma3));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"L G,$*&*(%\"EG\"\"\"-%%sqrtG6#*&*&,&%\"CGF)%#VxG!\"\"F),&F0F)F1F)F)\"\" \"*$)F0\"\"#F4!\"\"F4,D**%\"gGF)%\"tGF)-%#&^G6$-%\"dG6#F0-FA6#%\"zGF)) F0\"\"%F4F)**)F;\"\"$F4)F6$F@-FA6#F6$ F@-FA6#%\"xGF)-F+6#,$*&*&,&F0F2*&F;F4F6$-FA6#F1F@F)F FF4F2*,F1F4FKF4FfoF4FTF4FUF4F)*()F0\"\"&F4F1F4-F>6$FhoFQF)F2*,)F0FJF4F 1F4F^pF4FUF4FVF4F)*(F\\pF4-F>6$FhoFfnF)FinF4F2*,FapF4FcpF4FinF4FUF4FVF 4F)*,F1F4F;F4F6$FhoFCF)FapF4F2*,F1F4FIF4FKF4FgpF4F0F)F)*(F\\pF4F ;F4-F>6$FQFCF)F2**FapF4F;F4F[qF4FLF4F)F)F4*,-F+6#F[oF4)F/\"\"#F4)F3\" \"#F4F^o\"\"\"F`o\"\"\"F8F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "S:=(wcollect(factor(hh1&^gg1+hh2&^gg2+hh3&^gg3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "There are in general two sets of torsion two forms. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "1. Particle AFFIN E (" }{TEXT 265 2 "PA" }{TEXT -1 158 ") torsion 2-forms which depend \+ upon the product of little omega (the timelike part of the Vierbein) a nd the (time-space) connection components, little gamma." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "2. WaveAFFINE (" }{TEXT 266 2 "WA" }{TEXT -1 162 ") torsion 2-forms which depend upon B ig Omega (the time-time connection component or abnormality) and agai n the (time-space) connection components, little gamma." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "If the time-space co nnection 1-forms vanish, small gamma, neither form of torsion exists. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "See h ttp://www.uh.edu/~rkiehn/pdf/projfram.pdf" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 19 "PA TORSION 2-forms " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "Sigma1:=wcollect(simp lify((wcollect(factor(omega&^gg1))))); Sigma2:=factor(simplify(wcollec t(factor(omega&^gg2))));Sigma3:=wcollect(factor(simplify((omega&^gg3)) ));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Sigma1G,0*&**%\"EG\"\"\"%\"C GF)%#VxGF)-%#&^G6$-%\"dG6#%\"tG-F06#F*F)\"\"\"**-%%sqrtG6#*&*&,&F*F)F+ !\"\"F),&F*F)F+F)F)F5*$)F*\"\"#F5!\"\"F5-F86#,$*&*&,&F*F=*&%\"gGF)F2F) F)F),&F*F)FIF)F)F5*$)F*\"\"#F5FBF=F5F>\"\"\"F<\"\"\"FBF)*&*,F(F5F+F5FJ F5F2F5-F-6$-F06#%\"zGF3F)F5*,-F86#F:F5F*\"\"\"-F86#FEF5F>\"\"\"F<\"\" \"FBF)*&*(F(F5F+F5-F-6$-F06#%\"xG-F06#F+F)F5*(-F86#F:F5F>\"\"\"F<\"\" \"FBF=*&*(F(F5)F+\"\"#F5-F-6$F^oF3F)F5**-F86#F:F5F*\"\"\"F>\"\"\"F<\" \"\"FBF)*&**F(F5FJF5F2F5-F-6$FUFaoF)F5**-F86#F:F5-F86#FEF5F>\"\"\"F<\" \"\"FBF=*&*(F(F5)F*F[pF5-F-6$F/FaoF)F5**-F86#F:F5-F86#FEF5F>\"\"\"F<\" \"\"FBF=*&**F(F5F*F5F2F5-F-6$F3FaoF)F5**-F86#F:F5-F86#FEF5F>\"\"\"F<\" \"\"FBF=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Sigma2G\"\"!" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Sigma3G,,*&*,)%\"CG\"\"$\"\"\"%\"tGF+%\"g GF+%\"EG\"\"\"-%#&^G6$-%\"dG6#F)-F46#F,F/F+*,,&F)F/*&F-F/F,F/F/\"\"\", &F)!\"\"F:F/\"\"\"-%%sqrtG6#,$*&*&F " 0 "" {MPLTEXT 1 0 121 "Phi1:=simplify((wcollect((factor(Omega&^gg1)))));Phi2:=wcollect(f actor(Omega&^gg2));Phi3:=wcollect((factor(Omega&^gg3)));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%Phi1G*&,&*&%#VxG\"\"\"-%#&^G6$-%\"dG6#%\"EG-F .6#%\"CGF)!\"\"*&F3F)-F+6$F--F.6#F(F)F)\"\"\"*(,&F3F)F(F)\"\"\",&F3F)F (F4\"\"\"F0\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Phi2G\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Phi3G,&*&*,,&%\"CG\"\"\"%#VxG! \"\"\"\"\",&F)F*F+F*F-%\"tGF-%\"gGF--%#&^G6$-%\"dG6#%\"EG-F56#F)F*F-*, F7\"\"\")*&*&F(F*F.F*F-*$)F)\"\"#F-!\"\"#\"\"$\"\"#F-)F)\"\"#F-,&F)F,* &F0F*F/F*F*\"\"\",&F)F*FIF*\"\"\"FBF**&**F(F-F.F-F0F--F26$F4-F56#F/F*F -*,F7\"\"\"F)\"\"\")F=#\"\"$FEF-FH\"\"\"FK\"\"\"FBF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Next compute the matrix of curvature 2-forms on the x,y,z subspace" }}{PARA 0 "" 0 "" {TEXT 260 18 "Curvature 2-forms " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "Theta:=array([[gg1&^h h1,gg1&^hh2,gg1&^hh3],[gg2&^hh1,gg2&^hh2,gg2&^hh3],[(gg3&^hh1),(gg3&^h h2),(gg3&^hh3)]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%&ThetaG-%'matrixG6#7%7%\"\"!F*,(*&**%#VxG\"\" \"%\"CGF/%\"gG\"\"\"-%#&^G6$-%\"dG6#F0-F76#%\"tGF/F2*,,&F0F/F.F/\"\"\" ,&F0F/F.!\"\"\"\"\"-%%sqrtG6#*&*&F?F/F=F/F2*$)F0\"\"#F2!\"\"F2,&F0F@*& F1F/F;F/F/\"\"\",&F0F/FLF/\"\"\"FJF/*&**F0F2F1F2F;F2-F46$-F76#F.F6F/F2 *,F=\"\"\"F?\"\"\"-FC6#FEF2FK\"\"\"FN\"\"\"FJF/*&*()F0\"\"#F2F1F2-F46$ FTF9F/F2*,F=\"\"\"F?\"\"\"-FC6#FEF2FK\"\"\"FN\"\"\"FJF@7%F*F*F*7%,(*&* ,F?F2F=F2F;F2F1F2-F46$F6FTF/F2*,,&*$FinF2F/*$)F.FjnF2F@\"\"\")FE#\"\"$ FjnF2F0\"\"\"FK\"\"\"FN\"\"\"FJF/*&*,F?F2F=F2F1F2F.F2-F46$F9F6F/F2*,F \\p\"\"\"F0\"\"\")FE#\"\"$FjnF2FK\"\"\"FN\"\"\"FJF/*&**F?F2F=F2F1F2-F4 6$F9FTF/F2**F\\p\"\"\")FE#\"\"$FjnF2FK\"\"\"FN\"\"\"FJF@F*F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 284 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 10 "DISCUSSION" }}{PARA 0 "" 0 "" {TEXT -1 61 "This example i s rather remarkable in that it points out that " }{TEXT 267 4 "THIS" } {TEXT -1 24 " Lorentz transformation " }{TEXT 279 4 "DOES" }{TEXT -1 99 " introduce curvature on the 3D subspace (when accelerations and v ariable speeds C are admitted). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 147 "The curvature coefficients are not depen dent upon the Expansion, but the WA torsion coefficients require that \+ the d(E) is not zero over the domain." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 10 "Example 5:" }}{PARA 0 "" 0 "" {TEXT 269 105 "THE TORSION and CURVATURE of a LORENT Z rotation along the z axis and a Lorentz rotation along the x axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "FFINV:=evalm(innerprod(LRx,LRzn,LRx,LRz)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "Z:=innerprod(FFINV,[d(x),d(y),d(z),d(C*t)]):sigma1:=Z[1];sigma2 :=Z[2];sigma3:=Z[3];omega:=simplify((Z[4]));" }}{PARA 256 "" 0 "" {TEXT 256 21 "The Vierbein 1-forms." }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%'sigma1G,.*&-%\"dG6#%\"xG\"\"\")%\"EG\"\"#\"\"\"F+*(F'F/F,F/)%#LzGF .F/!\"\"**F'F/F1F/F,F/-%%sqrtG6#,&F+F+*$)%#LxGF.F/F3F/F+**-F(6#%\"yGF+ F,F/F2F+-F66#,&F+F+*$F1F/F3F/F+*,F=F/F2F/F,F/F5F/F@F/F3**F2F/F,F/F;F+- F(6#%\"zGF+F3" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'sigma2G,:*,-%\"dG6 #%\"xG\"\"\"%#LzGF+)%\"EG\"\"#\"\"\"-%%sqrtG6#,&F+F+*$)%#LxGF/F0!\"\"F 0-F26#,&F+F+*$)F,F/F0F8F0F+**F'F0F-F0F,F0F9F0F8*,F'F0F-F0F,F0F6F0F9F0F +**F'F0F-F0F,F0F6F0F+**-F(6#%\"yGF+F=F0F-F0F1F0F+*&FBF0F-F0F+*(FBF0F-F 0F=F0F8*(FBF0F-F0F6F0F8**FBF0F-F0F6F0F=F0F+**FBF0F9F0F-F0F6F0F8*,F7F+F -F0F1F0-F(6#%\"zGF+F9F0F+**F7F0F-F0F1F0FKF0F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'sigma3G,6*,-%\"dG6#%\"xG\"\"\"%#LzGF+)%\"EG\"\"#\"\" \"%#LxGF+-%%sqrtG6#,&F+F+*$)F,F/F0!\"\"F0F8*.F'F0F-F0F,F0-F36#,&F+F+*$ )F1F/F0F8F0F2F0F1F0F+*,F'F0F-F0F,F0F1F0F:F0F+**-F(6#%\"yGF+F7F0F-F0F1F 0F8**FAF0F-F0F:F0F1F0F8*,FAF0F-F0F:F0F1F0F7F0F+*,FAF0F2F0F-F0F1F0F:F0F 8**F-F0-F(6#%\"zGF+F>F0F2F0F8*&F-F0FHF0F+*(F-F0FHF0F>F0F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG*&)%\"EG\"\"#\"\"\",&*&%\"tG\"\"\"-%\" dG6#%\"CGF-F-*&F1F--F/6#F,F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "#Vol4:=wcollect(simplify(sigma1&^sigma2&^sigma3&^Z[4] ));rho:=subs(getcoeff(Vol4));" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 257 25 "The density (determinant)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 185 "The determinant cannot go to zero for the projective domain. The zer o sets of the density function determine a hypersurface. IF the hyper surface is harmonic then it can be a boundary." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "There is an induced metri c on R4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "FF:=inverse(FFIN V):Gun:=subs(innerprod(transpose(FF),FF));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GunG-%'matrixG6#7&7&*&\"\"\"F+*$)%\"EG\"\"%F+!\"\"\" \"!F1F17&F1F*F1F17&F1F1F*F17&F1F1F1F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "From the Frame Field use the standard methods to compute \+ the " }}{PARA 0 "" 0 "" {TEXT 261 36 "Cartan Matrix of connection 1-fo rms." }}{PARA 0 "" 0 "" {TEXT -1 46 "See http://www.uh.edu/~rkiehn/pdf /projfram.pdf" }}{PARA 0 "" 0 "" {TEXT -1 64 "for details of the Carta n method for an arbitrary Repere Mobile." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "dFF:=array([[d(FF[1,1]),d(FF[1,2]),d(FF[1,3]),d(FF[1 ,4])],[d(FF[2,1]),d(FF[2,2]),d(FF[2,3]),d(FF[2,4])],[d(FF[3,1]),d(FF[3 ,2]),d(FF[3,3]),d(FF[3,4])],[d(FF[4,1]),d(FF[4,2]),d(FF[4,3]),d(FF[4,4 ])]]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "cartan:=(evalm(FF INV&*dFF)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 45 "The Interior (spa ce-space) Connection 1 forms" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Gamma11:=factor(wcollect(cartan[1,1]));Gamma21:=factor(wcollect (cartan[2,1]));Gamma31:=factor(wcollect(cartan[3,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma11G,$*&-%\"dG6#%\"EG\"\"\"F*!\"\"!\"#" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Gamma21G*&,L*(%\"EG\"\"\"-%\"dG6#%# LzGF)-%%sqrtG6#,$*&,&%#LxGF)!\"\"F)F),&F4F)F)F)F)F5\"\"\"F)**F(F7F*F7F .F7)F-\"\"#F7F5*,)F-\"\"$F7F(F7)F4F=F7-F+6#F4F)-F/6#,$*&,&F-F)F5F)F),& F-F)F)F)F)F5F7F5*,F-F)F(F7F>F7F?F7)FC#F=F:F7F5*,F-F7F(F7F4F)F?F7FAF7F5 *,F-F7)F4F:F7-F+6#F(F)FHF7F.F7F:*,FF7F?F7FAF7F)**F-F7FZF7FMF7FHF 7F:*&F(F7F*F7F5*(F(F7F*F7FLF7F)**F-F7FZF7FMF7FAF7FQ**F-F7FLF7FMF7FAF7F :F7*(F(\"\"\"-F/6#FCF7-F/6#F1F7!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#>%(Gamma31G*&,B**%#LzG\"\"\"%\"EGF)-%\"dG6#%#LxGF)-%%sqrtG6#,$*&,&F( F)!\"\"F)F),&F(F)F)F)F)F5\"\"\"F5*,)F(\"\"$F7F*F7F+F7F/F7)F.\"\"#F7F5* ,F(F7F*F7F+F7)F2#F:FF7FCF7F <**F(F7)F.F:F7FAF7F/F7FI*,F.F7F*F7-F,6#F(F)FCF7F/F7F5**F(F7F.F7FAF7F/F 7F<*,F9F7F.F7FAF7F/F7FCF7F<*(F.F7F*F7FOF7F)**F.F7F*F7FOF7FCF7F5**F9F7F .F7FAF7F/F7FI**F9F7FMF7FAF7F/F7F<**F(F7F.F7FAF7F>F7FI**F(F7FMF7FAF7F>F 7F " 0 "" {MPLTEXT 1 0 117 "Gamma12:=factor(wcollect(cartan[1,2]));Gamma22:=factor(wcollect (cartan[2,2]));Gamma32:=factor(wcollect(cartan[3,2]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Gamma12G,$*&,V*,%#LzG\"\"\"%\"EGF*)%#LxG\"\"$\" \"\"-%\"dG6#F-F*-%%sqrtG6#,$*&,&F)F*!\"\"F*F*,&F)F*F*F*F*F9F/!\"#**)F) F.F/)F-\"\"#F/-F16#F+F*F3F/F?**F)F/F>F/F@F/)F6#F.F?F/F?**F=F/)F-\"\"%F /F@F/F3F/F;**F)F/FFF/F@F/FCF/F;**F)F/FFF/F@F/F3F/F?**F)F/F>F/F@F/F3F/F ;*,F=F/F>F/F@F/F3F/-F46#,$*&,&F-F*F9F*F*,&F-F*F*F*F*F9F/F;*,F)F/F>F/F@ F/F3F/FLF/F?*,F+F/-F16#F)F*FLF/F>F/F3F/F9*,F+F/FTF/FLF/)F)F?F/F>F/F*** F+F/FTF/FLF/F>F/F9*,F=F/F+F/F,F/F0F/F3F/F?*,F)F/F+F/F,F/F0F/FCF/F?*,F) F/F+F/F-F*F0F/F3F/F**,F)F/F>F/F@F/FCF/FLF/F;*(F+F/FTF/F>F/F**&F+F/FTF/ F9*(F+F/FTF/FLF/F**,F)F/F+F/F-F/F0F/FCF/F;*,F=F/F+F/F-F/F0F/F3F/F;*.F) F/F+F/F-F/F0F/F3F/FLF/F;*.F=F/F+F/F-F/F0F/F3F/FLF/F?*.F)F/F+F/F-F/F0F/ FCF/FLF/F?**F+F/FTF/FLF/FWF/F9**F+F/FTF/)FNFDF/FWF/F*F/*(F+\"\"\"-F46# F6F/-F46#FNF/!\"\"F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma22G,$* &-%\"dG6#%\"EG\"\"\"F*!\"\"!\"#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%( Gamma32G,$*&,fo**%\"EG\"\"\"-%\"dG6#%#LxGF*-%%sqrtG6#,$*&,&%#LzGF*!\" \"F*F*,&F5F*F*F*F*F6\"\"\")F.\"\"#F8F***F)F8F+F8)F2#\"\"$F:F8F9F8F6**F .F*)F5F:F8-F,6#F)F*F/F8\"\"%**)F.\"\"&F8F@F8FAF8FF8F@F8FAF8F/F8!\"'**F.F8F@F8FAF8F< F8!\"%**F.F8FJF8FAF8F/F8FO**FLF8F@F8FAF8F*.F)F8F+F8F/F8F@F8F9F8FgnF8F^o*,F)F8F+F8F*,F.F8F@F8FAF8F " 0 "" {MPLTEXT 1 0 129 "Gamma13:=wcollect(factor(wcollect(cartan[1,3])));Gamma23:=factor( wcollect(cartan[2,3]));Gamma33:=simplify(wcollect(cartan[3,3]));" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Gamma13G,(*&*&,2*&%\"EG\"\"\"%#LxGF +!\"\"*(F,\"\"\"F*F/-%%sqrtG6#,$*&,&%#LzGF+F-F+F+,&F6F+F+F+F+F-F/F-*() F,\"\"$F/F*F/F0F/F+*&F9F/F*F/F+**F9F/F*F/-F16#,$*&,&F,F+F-F+F+,&F,F+F+ F+F+F-F/)F6\"\"#F/F+*(F,F/F*F/F=F/F+**F,F/F*F/F=F/FCF/F-**F,F/F*F/)F?# F:FDF/FCF/F+F+-%\"dG6#F6F+F/*(F*\"\"\"-F16#F3F/-F16#F?F/!\"\"F-*&*&,B* ()F6F:F/F,F/FHF/!\"#*(F6F+F9F/F=F/FD**F6F/F9F/F=F/F0F/FD*(F6F/F,F/F=F/ FY*(FXF/F9F/F0F/FD*(F6F/F,F/)F3FIF/FY**F6F/F,F/FinF/FHF/FD**FXF/F,F/F0 F/FHF/FD*(F6F/F9F/F0F/FY**F6F/F,F/F=F/F0F/FY*(F6F/F,F/F0F/FD*(FXF/F,F/ F0F/FY*(FXF/F9F/F=F/FY*(F6F/F,F/FHF/FD*(F6F/F9F/FinF/FD*(FXF/F,F/F=F/F DF+-FK6#F*F+F/*(F*\"\"\"-F16#F3F/-F16#F?F/FSF-*&*&,8**F6F/F*F/FinF/F=F /F-*,F6F/F*F/FinF/)F,FDF/F=F/FD*(FXF/F*F/F0F/F+*,FXF/F*F/F0F/FapF/F=F/ FD**F6F/F*F/F0F/FapF/FD**FXF/F*F/F0F/FapF/FY**FXF/F*F/F0F/F=F/F-*,F6F/ F*F/F0F/FapF/F=F/FY*(F6F/F*F/FinF/F+**F6F/F*F/FinF/FapF/FY*(F6F/F*F/F0 F/F-F+-FK6#F,F+F/*(F*\"\"\"-F16#F3F/-F16#F?F/FSF-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Gamma23G,$*&,bp**%\"EG\"\"\"-%\"dG6#%#LxGF*-%%sqrtG6 #,$*&,&%#LzGF*!\"\"F*F*,&F5F*F*F*F*F6\"\"\")F.\"\"#F8F6**F)F8F+F8)F2# \"\"$F:F8F9F8F***F.F*)F5F:F8-F,6#F)F*F/F8\"\"%**)F.\"\"&F8F@F8FAF8FF8F@F8FAF8F/F8! \"'**F.F8F@F8FAF8F*.F)F8F+F8F/F8F@F8F9F8FgnF8!\"$* ,F)F8F+F8F*,F.F8F@F8FAF8F%(Gamma33G,$*&-%\"dG6#%\"EG\"\"\"F*!\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 40 "The \+ \"space-time\" connection 1-forms are:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "hh1:=simplify(wcollect(cartan[4,1]));hh2:=factor(wco llect(cartan[4,2]));hh3:=wcollect(factor(wcollect(cartan[4,3])));" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$hh1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$hh2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$hh3G\"\"!" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 40 "The \"time-space connection\" 1-forms are " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "gg1:=factor(wcollect(factor(wcolle ct(cartan[1,4]))));gg2:=factor(wcollect(cartan[2,4]));gg3:=wcollect(fa ctor(wcollect(cartan[3,4])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg 1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg2G\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$gg3G\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 46 "The abnormality (time-time) connection 1-form " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Note that the abnormality (Big Omega vani shes if expansion is a global constant. (no red shift?)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Omega:=wcollect(subs(simplify(wcoll ect(cartan[4,4]))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OmegaG,$*&- %\"dG6#%\"EG\"\"\"F*!\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "L:=factor(wcollect(hh1&^sigma1+hh2&^sigma2+hh3&^sigma3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "S:=(wcollect(factor(hh1&^gg1+hh2&^gg2+hh3&^gg3))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "There are in general two sets of torsion two forms. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "1. \+ Particle AFFINE (" }{TEXT 265 2 "PA" }{TEXT -1 158 ") torsion 2-forms \+ which depend upon the product of little omega (the timelike part of t he Vierbein) and the (time-space) connection components, little gamma. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "2. W aveAFFINE (" }{TEXT 266 2 "WA" }{TEXT -1 162 ") torsion 2-forms which \+ depend upon Big Omega (the time-time connection component or abnormali ty) and again the (time-space) connection components, little gamma." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "If the \+ time-space connection 1-forms vanish, small gamma, neither form of tor sion exists." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "See http://www.uh.edu/~rkiehn/pdf/projfram.pdf" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 19 "PA TORSION 2-forms " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "Sigma1:=w collect(simplify((wcollect(factor(omega&^gg1))))); Sigma2:=factor(simp lify(wcollect(factor(omega&^gg2))));Sigma3:=wcollect(factor(simplify(( omega&^gg3))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Sigma1G\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Sigma2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Sigma3G\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 19 "WA TORSION 2-forms " }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Phi1:=simplify((wcollect((factor(Omega&^gg1)))));Phi 2:=wcollect(factor(Omega&^gg2));Phi3:=wcollect((factor(Omega&^gg3))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Phi1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Phi2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Phi 3G\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Next compute the matri x of curvature 2-forms on the x,y,z subspace" }}{PARA 0 "" 0 "" {TEXT 260 18 "Curvature 2-forms " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "Theta:=array([[gg1&^hh1,gg1&^hh2,gg1&^hh3],[gg2&^hh1,gg2&^hh2,gg2 &^hh3],[(gg3&^hh1),(gg3&^hh2),(gg3&^hh3)]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ThetaG-%'matrix G6#7%7%\"\"!F*F*F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The curva ture 2-forms on the interior space all vanish." }}}{EXCHG {PARA 259 " " 0 "" {TEXT -1 10 "DISCUSSION" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 246 "Lorentz rotations do not seem to generate torsion or curvature on the 3D supspace, which must be due to the fact (somehow) that the tra nspose of a Lorentz rotation is proportional to its inverse. This fa ct is not true for Lorentz translations. " }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 316 "It is remarkable that the Lorentz Round Trip Rota tions do not seem to effect curvature or torsion, yet the represent a \+ defect in that the Identity is not the result of the round trip. I wo uld have suspected that this would induce a curvature or torsion defec t of some kind. But that does not seem to be the case. " }{TEXT 286 8 "Strange." }}{PARA 0 "" 0 "" {TEXT -1 189 "There is a round trip def ect must be the type of Torsion coefficient that Shipov is talking abo ut, where in this case the Frame is a Lorentz map, without Cartan curv ature and Cartan torsion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 84 "(I need to comput the curl of the vierbein and see if this is a Shipov coefficient.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "208 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }