.MCAD 304020000 1 0 282 0 .CMD FORMAT rd=d ct=10 im=i et=3 zt=15 pr=3 mass length time charge temperature tr=0 vm=0 .CMD SET ORIGIN 0 .CMD SET TOL 0.001000000000000 .CMD SET PRNCOLWIDTH 8 .CMD SET PRNPRECISION 4 .CMD PRINT_SETUP 1.200000 0.000000 1.200000 1.200000 0 .CMD HEADER_FOOTER 1 1 *empty* *empty* *empty* 0 1 *empty* *empty* *empty* .CMD HEADER_FOOTER_FONT fontID=14 family=Arial points=10 bold=0 italic=0 underline=0 colrid=-1 .CMD HEADER_FOOTER_FONT fontID=15 family=Arial points=10 bold=0 italic=0 underline=0 colrid=-1 .CMD DEFAULT_TEXT_PARPROPS 0 0 0 .CMD DEFINE_FONTSTYLE_NAME fontID=0 name=Variables .CMD DEFINE_FONTSTYLE_NAME fontID=1 name=Constants .CMD DEFINE_FONTSTYLE_NAME fontID=2 name=Text .CMD DEFINE_FONTSTYLE_NAME fontID=4 name=User^1 .CMD DEFINE_FONTSTYLE_NAME fontID=5 name=User^2 .CMD DEFINE_FONTSTYLE_NAME fontID=6 name=User^3 .CMD DEFINE_FONTSTYLE_NAME fontID=7 name=User^4 .CMD DEFINE_FONTSTYLE_NAME fontID=8 name=User^5 .CMD DEFINE_FONTSTYLE_NAME fontID=9 name=User^6 .CMD DEFINE_FONTSTYLE_NAME fontID=10 name=User^7 .CMD DEFINE_FONTSTYLE fontID=0 family=Times^New^Roman points=10 bold=0 italic=0 underline=0 colrid=-1 .CMD DEFINE_FONTSTYLE fontID=1 family=Times^New^Roman points=10 bold=0 italic=0 underline=0 colrid=-1 .CMD DEFINE_FONTSTYLE fontID=2 family=Arial points=10 bold=0 italic=0 underline=0 colrid=-1 .CMD DEFINE_FONTSTYLE fontID=4 family=Arial points=10 bold=0 italic=0 underline=0 colrid=-1 .CMD DEFINE_FONTSTYLE fontID=5 family=Courier^New points=10 bold=0 italic=0 underline=0 colrid=-1 .CMD DEFINE_FONTSTYLE fontID=6 family=System points=10 bold=0 italic=0 underline=0 colrid=-1 .CMD DEFINE_FONTSTYLE fontID=7 family=Script points=10 bold=0 italic=0 underline=0 colrid=-1 .CMD DEFINE_FONTSTYLE fontID=8 family=Roman points=10 bold=0 italic=0 underline=0 colrid=-1 .CMD DEFINE_FONTSTYLE fontID=9 family=Modern points=10 bold=0 italic=0 underline=0 colrid=-1 .CMD DEFINE_FONTSTYLE fontID=10 family=Times^New^Roman points=10 bold=0 italic=0 underline=0 colrid=-1 .CMD UNITS U=1 .CMD DIMENSIONS_ANALYSIS 0 0 .CMD COLORTAB_ENTRY 0 0 0 .CMD COLORTAB_ENTRY 128 0 0 .CMD COLORTAB_ENTRY 0 128 0 .CMD COLORTAB_ENTRY 128 128 0 .CMD COLORTAB_ENTRY 0 0 128 .CMD COLORTAB_ENTRY 128 0 128 .CMD COLORTAB_ENTRY 0 128 128 .CMD COLORTAB_ENTRY 128 128 128 .CMD COLORTAB_ENTRY 192 192 192 .CMD COLORTAB_ENTRY 255 0 0 .CMD COLORTAB_ENTRY 0 255 0 .CMD COLORTAB_ENTRY 255 255 0 .CMD COLORTAB_ENTRY 0 0 255 .CMD COLORTAB_ENTRY 255 0 255 .CMD COLORTAB_ENTRY 0 255 255 .CMD COLORTAB_ENTRY 255 255 255 .TXT 2 28 69 0 0 Cg a81.200000,81.200000,19 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}{\f1\fcharset0\fmodern Courier;}}\plain\cf1\fs20 \pard {\f1\fs24 Lorentz Transforms} } .TXT 3 -26 71 0 0 Cg a82.200000,82.200000,200 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard There are two different ways of representing the time dimention. We can also choose units so that the speed of light is 1 leading to a simplification of the terms Here I have used c = 1 and real time.} .TXT 10 0 108 0 0 Cg a82.200000,82.200000,6 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard [L] =} .EQN 0 4 107 0 0 ({4,4}ö{0:\g}NAMEö0ö0ö-{0:\g}NAME*{0:v}NAMEö0ö1ö0ö0ö0ö0ö1ö0ö-{0:\g}NAME*{0:v}NAMEö0ö0ö{0:\g}NAME) .TXT 0 13 119 0 0 Cg a65.200000,65.200000,5 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard where} .EQN 0 5 122 0 0 {0:\g}NAME .TXT 0 2 120 0 0 Cg a61.200000,61.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 0 2 121 0 0 (1)/(\(1-({0:v}NAME)^(2))) .TXT 0 6 109 0 0 Cg a60.200000,60.200000,6 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard giving} .EQN 0 7 110 0 0 ({4,1}ö{0:t_1}NAMEö{0:z_1}NAMEö{0:y_1}NAMEö{0:x_1}NAME) .TXT 0 4 111 0 0 Cg a49.200000,49.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 0 2 112 0 0 ({4,4}ö{0:\g}NAMEö0ö0ö-{0:\g}NAME*{0:v}NAMEö0ö1ö0ö0ö0ö0ö1ö0ö-{0:\g}NAME*{0:v}NAMEö0ö0ö{0:\g}NAME)*({4,1}ö{0:t_0}NAMEö{0:z_0}NAMEö{0:y_0}NAMEö{0:x_0}NAME) .TXT 0 19 113 0 0 Cg a28.200000,28.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 0 2 114 0 0 ({4,1}ö-{0:\g}NAME*{0:v}NAME*{0:x_0}NAME+{0:\g}NAME*{0:t_0}NAMEö{0:z_0}NAMEö{0:y_0}NAMEö{0:\g}NAME*{0:x_0}NAME-{0:\g}NAME*{0:v}NAME*{0:t_0}NAME) .TXT 8 -67 123 0 0 Cg a82.200000,82.200000,111 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard (Mathcad responds to the use of the hyphen key by plaxing brackets around the expression, so x' cannot be used)} .TXT 9 0 124 0 0 Cg a83.200000,83.200000,9 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard For v =} .EQN 0 7 125 0 0 ({4,1}ö0ö0ö{0:v}NAME*{0:sin}NAME({0:\q}NAME)ö{0:v}NAME*{0:cos}NAME({0:\q}NAME)) .TXT 0 9 138 0 0 Cg a66.200000,66.200000,20 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard the matrix transform} .EQN 0 15 139 0 0 ({4,4}ö1ö0ö0ö0ö0ö1ö0ö0ö0ö0ö{0:cos}NAME({0:\q}NAME)ö{0:sin}NAME({0:\q}NAME)ö0ö0ö-{0:sin}NAME({0:\q}NAME)ö{0:cos}NAME({0:\q}NAME))*({4,1}ö0ö0ö{0:v}NAME*{0:sin}NAME({0:\q}NAME)ö{0:v}NAME*{0:cos}NAME({0:\q}NAME)) .TXT 0 24 141 0 0 Cg a25.200000,25.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 0 2 142 0 0 ({4,1}ö0ö0ö0ö({0:cos}NAME({0:\q}NAME))^(2)*{0:v}NAME+({0:sin}NAME({0:\q}NAME))^(2)*{0:v}NAME) .TXT 0 16 143 0 0 Cg a25.200000,25.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 0 2 144 0 0 ({4,1}ö0ö0ö0ö{0:v}NAME) .TXT 13 -75 151 0 0 Cg a83.200000,83.200000,24 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard The transform for v is} .EQN 0 17 158 0 0 ({4,4}ö1ö0ö0ö0ö0ö1ö0ö0ö0ö0ö{0:cos}NAME({0:\q}NAME)ö-{0:sin}NAME({0:\q}NAME)ö0ö0ö{0:sin}NAME({0:\q}NAME)ö{0:cos}NAME({0:\q}NAME))*({4,4}ö{0:\g}NAMEö0ö0ö-{0:\g}NAME*{0:v}NAMEö0ö1ö0ö0ö0ö0ö1ö0ö-{0:\g}NAME*{0:v}NAMEö0ö0ö{0:\g}NAME)*({4,4}ö1ö0ö0ö0ö0ö1ö0ö0ö0ö0ö {0:cos}NAME({0:\q}NAME)ö{0:sin}NAME({0:\q}NAME)ö0ö0ö-{0:sin}NAME({0:\q}NAME)ö{0:cos}NAME({0:\q}NAME))*({4,1}ö{0:t}NAMEö{0:z}NAMEö{0:y}NAMEö{0:x}NAME) .TXT 13 -14 165 0 0 Cg a80.200000,80.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 0 2 166 0 0 ({4,4}ö{0:\g}NAMEö0ö-{0:sin}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAMEö-{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAMEö0ö1ö0ö0ö-{0:sin}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAMEö0ö({0:sin}NAME({0:\q}NAME))^(2)*{0:\g}NAME+({0:cos}NAME({0:\q}NAME))^(2)ö{0:cos}NAME( {0:\q}NAME)*{0:\g}NAME*{0:sin}NAME({0:\q}NAME)-{0:sin}NAME({0:\q}NAME)*{0:cos}NAME({0:\q}NAME)ö-{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAMEö0ö{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:sin}NAME({0:\q}NAME)-{0:sin}NAME({0:\q}NAME)*{0:cos}NAME({0:\q}NAME)ö( {0:cos}NAME({0:\q}NAME))^(2)*{0:\g}NAME+({0:sin}NAME({0:\q}NAME))^(2))*({4,1}ö{0:t}NAMEö{0:z}NAMEö{0:y}NAMEö{0:x}NAME) .EQN 13 0 197 0 0 ({4,1}ö{0:t_1}NAMEö{0:z_1}NAMEö{0:y_1}NAMEö{0:x_1}NAME) .TXT 0 6 195 0 0 Cg a80.200000,80.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 1 2 196 0 0 ({4,1}ö-{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAME*{0:x}NAME-{0:sin}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAME*{0:y}NAME+{0:\g}NAME*{0:t}NAMEö{0:z}NAMEö{0:x}NAME*{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:sin}NAME({0:\q}NAME)-{0:x}NAME*{0:sin}NAME({0:\q}NAME)* {0:cos}NAME({0:\q}NAME)+{0:y}NAME*({0:sin}NAME({0:\q}NAME))^(2)*{0:\g}NAME+{0:y}NAME*({0:cos}NAME({0:\q}NAME))^(2)-{0:sin}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAME*{0:t}NAMEö{0:x}NAME*({0:cos}NAME({0:\q}NAME))^(2)*{0:\g}NAME+{0:x}NAME*({0:sin}NAME({0:\q}NAME) )^(2)+{0:y}NAME*{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:sin}NAME({0:\q}NAME)-{0:y}NAME*{0:sin}NAME({0:\q}NAME)*{0:cos}NAME({0:\q}NAME)-{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAME*{0:t}NAME) .TXT 10 -13 199 0 0 Cg a83.200000,83.200000,69 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Those with Mathcad 6.0 can download the mathcad file and verify that } .EQN 5 8 200 0 0 ({0:x_1}NAME)^(2)+({0:y_1}NAME)^(2)+({0:z_1}NAME)^(2)-({0:t_1}NAME)^(2) .TXT 0 19 201 0 0 Cg a56.200000,56.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 0 4 202 0 0 ({0:x}NAME)^(2)+({0:y}NAME)^(2)+({0:z}NAME)^(2)-({0:t}NAME)^(2) .TXT 4 -31 203 0 0 Cg a83.200000,83.200000,62 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard So we are confident that this is indeed a Lorentz transform. } .TXT 6 0 204 0 0 Cg a83.200000,83.200000,384 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard If we now consider three observers in inertial reference frames S_0, S_1 and S_2 such that at some moment, their axes are coincident and that they zero their clocks at that momen. If the velocity of S_1 in S_0 is u along the x axis and the velocity of S_2 in S_1 is the vector v defined above, we can apply first a Lorentz from S_0 to S_1 and then a Transform from S_1 TO S_2.} .EQN 9 13 216 0 0 {0:\b}NAME .TXT 0 2 217 0 0 Cg a76.200000,76.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 0 3 218 0 0 (1)/(\(1-({0:u}NAME)^(2))) .EQN 16 -15 221 0 0 ({4,1}ö{0:t_2}NAMEö{0:z_2}NAMEö{0:y_2}NAMEö{0:x_2}NAME) .TXT 0 6 220 0 0 Cg a80.200000,80.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 0 2 219 0 0 ({4,4}ö{0:\g}NAMEö0ö-{0:sin}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAMEö-{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAMEö0ö1ö0ö0ö-{0:sin}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAMEö0ö({0:sin}NAME({0:\q}NAME))^(2)*{0:\g}NAME+({0:cos}NAME({0:\q}NAME))^(2)ö{0:cos}NAME( {0:\q}NAME)*{0:\g}NAME*{0:sin}NAME({0:\q}NAME)-{0:sin}NAME({0:\q}NAME)*{0:cos}NAME({0:\q}NAME)ö-{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAMEö0ö{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:sin}NAME({0:\q}NAME)-{0:sin}NAME({0:\q}NAME)*{0:cos}NAME({0:\q}NAME)ö( {0:cos}NAME({0:\q}NAME))^(2)*{0:\g}NAME+({0:sin}NAME({0:\q}NAME))^(2))*({4,4}ö{0:\b}NAMEö0ö0ö-{0:\b}NAME*{0:u}NAMEö0ö1ö0ö0ö0ö0ö1ö0ö-{0:\b}NAME*{0:u}NAMEö0ö0ö{0:\b}NAME)*({4,1}ö{0:t_0}NAMEö{0:z_0}NAMEö{0:y_0}NAMEö{0:x_0}NAME) .TXT 14 -11 227 0 0 Cg a83.200000,83.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 0 2 228 0 0 ({4,4}ö{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAME*{0:\b}NAME*{0:u}NAME+{0:\g}NAME*{0:\b}NAMEö0ö-{0:\b}NAME*{0:u}NAME*{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:sin}NAME({0:\q}NAME)+{0:\b}NAME*{0:u}NAME*{0:sin}NAME({0:\q}NAME)*{0:cos}NAME({0:\q}NAME)- {0:sin}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAME*{0:\b}NAMEö-{0:\b}NAME*{0:u}NAME*({0:cos}NAME({0:\q}NAME))^(2)*{0:\g}NAME-{0:\b}NAME*{0:u}NAME*({0:sin}NAME({0:\q}NAME))^(2)-{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAME*{0:\b}NAMEö0ö1ö0ö0ö-{0:sin}NAME( {0:\q}NAME)*{0:\g}NAME*{0:v}NAMEö0ö({0:sin}NAME({0:\q}NAME))^(2)*{0:\g}NAME+({0:cos}NAME({0:\q}NAME))^(2)ö{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:sin}NAME({0:\q}NAME)-{0:sin}NAME({0:\q}NAME)*{0:cos}NAME({0:\q}NAME)ö-{0:cos}NAME({0:\q}NAME)*{0:\g}NAME* {0:v}NAME*{0:\b}NAME-{0:\g}NAME*{0:\b}NAME*{0:u}NAMEö0ö{0:\b}NAME*{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:sin}NAME({0:\q}NAME)-{0:\b}NAME*{0:sin}NAME({0:\q}NAME)*{0:cos}NAME({0:\q}NAME)+{0:sin}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAME*{0:\b}NAME*{0:u}NAMEö {0:\b}NAME*({0:cos}NAME({0:\q}NAME))^(2)*{0:\g}NAME+{0:\b}NAME*({0:sin}NAME({0:\q}NAME))^(2)+{0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAME*{0:\b}NAME*{0:u}NAME)*({4,1}ö{0:t_0}NAMEö{0:z_0}NAMEö{0:y_0}NAMEö{0:x_0}NAME) .TXT 10 -2 229 0 0 Cg a83.200000,83.200000,170 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard It can be seen that the second column remains unchange by the post multiplication. If the Lorentz group exists then we should be able to equate this to a single tranform:} .EQN 10 5 233 0 0 ({4,4}ö1ö0ö0ö0ö0ö1ö0ö0ö0ö0ö{0:cos}NAME({0:\f}NAME)ö-{0:sin}NAME({0:\f}NAME)ö0ö0ö{0:sin}NAME({0:\f}NAME)ö{0:cos}NAME({0:\f}NAME))*({4,4}ö{0:\a}NAMEö0ö0ö-{0:\a}NAME*{0:w}NAMEö0ö1ö0ö0ö0ö0ö1ö0ö-{0:\a}NAME*{0:w}NAMEö0ö0ö{0:\a}NAME)*({4,4}ö1ö0ö0ö0ö0ö1ö0ö0ö0ö0ö {0:cos}NAME({0:\f}NAME)ö{0:sin}NAME({0:\f}NAME)ö0ö0ö-{0:sin}NAME({0:\f}NAME)ö{0:cos}NAME({0:\f}NAME)) .TXT 14 17 235 0 0 Cg a61.200000,61.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 0 2 234 0 0 ({4,4}ö{0:\a}NAMEö0ö-{0:sin}NAME({0:\f}NAME)*{0:\a}NAME*{0:w}NAMEö-{0:cos}NAME({0:\f}NAME)*{0:\a}NAME*{0:w}NAMEö0ö1ö0ö0ö-{0:sin}NAME({0:\f}NAME)*{0:\a}NAME*{0:w}NAMEö0ö({0:sin}NAME({0:\f}NAME))^(2)*{0:\a}NAME+({0:cos}NAME({0:\f}NAME))^(2)ö{0:cos}NAME( {0:\f}NAME)*{0:\a}NAME*{0:sin}NAME({0:\f}NAME)-{0:sin}NAME({0:\f}NAME)*{0:cos}NAME({0:\f}NAME)ö-{0:cos}NAME({0:\f}NAME)*{0:\a}NAME*{0:w}NAMEö0ö{0:cos}NAME({0:\f}NAME)*{0:\a}NAME*{0:sin}NAME({0:\f}NAME)-{0:sin}NAME({0:\f}NAME)*{0:cos}NAME({0:\f}NAME)ö( {0:cos}NAME({0:\f}NAME))^(2)*{0:\a}NAME+({0:sin}NAME({0:\f}NAME))^(2)) .TXT 11 -23 236 0 0 Cg a82.200000,82.200000,62 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard But this requires us to equate the terms of the second columns} .EQN 4 15 255 0 0 {0:cos}NAME({0:\q}NAME)*{0:\g}NAME*{0:sin}NAME({0:\q}NAME)-{0:sin}NAME({0:\q}NAME)*{0:cos}NAME({0:\q}NAME) .TXT 0 23 254 0 0 Cg a44.200000,44.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 0 3 253 0 0 {0:cos}NAME({0:\f}NAME)*{0:\a}NAME*{0:sin}NAME({0:\f}NAME)-{0:sin}NAME({0:\f}NAME)*{0:cos}NAME({0:\f}NAME) .EQN 4 -18 262 0 0 ({0:sin}NAME({0:\q}NAME))^(2)*{0:\g}NAME+({0:cos}NAME({0:\q}NAME))^(2) .TXT 0 15 257 0 0 Cg a44.200000,44.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 0 3 263 0 0 ({0:sin}NAME({0:\f}NAME))^(2)*{0:\a}NAME+({0:cos}NAME({0:\f}NAME))^(2) .EQN 4 -13 265 0 0 -{0:sin}NAME({0:\q}NAME)*{0:\g}NAME*{0:v}NAME .TXT 0 10 260 0 0 Cg a44.200000,44.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 0 3 264 0 0 -{0:sin}NAME({0:\f}NAME)*{0:\a}NAME*{0:w}NAME .TXT 5 -42 271 0 0 Cg a83.200000,83.200000,218 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard We note that these equations are independent of u and conclude that no solution is possible which will be valid for all u less than c. Therefore, we are unable to show that there is a Lorentz transform from S_0 to S_2.} .TXT 6 0 274 0 0 Cg a83.200000,83.200000,28 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard However, the condition that } .EQN 0 22 275 0 0 ({0:x_0}NAME)^(2)+({0:y_0}NAME)^(2)+({0:z_0}NAME)^(2)-({0:t_0}NAME)^(2) .TXT 0 19 276 0 0 Cg a56.200000,56.200000,1 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard =} .EQN 0 4 277 0 0 ({0:x_2}NAME)^(2)+({0:y_2}NAME)^(2)+({0:z_2}NAME)^(2)-({0:t_2}NAME)^(2) .TXT 0 18 278 0 0 Cg a20.200000,20.200000,13 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard is satisfied.} .TXT 4 -63 279 0 0 Cg a83.200000,83.200000,153 {\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0 \fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard This can be varified by doing the sums on a separate mathcad sheet. (It has a habbit of filling its memory and crashing if too much is done in one sheet)}