.MCAD 301010002 1 74 .CMD PLOTFORMAT 0 0 1 0 0 0 0 1 0 0 0 1 6 0 Frame A 0 4 6 0 Frame B 0 0 2 0 NO-TRACE-STRING 0 4 3 0 NO-TRACE-STRING 0 1 4 0 NO-TRACE-STRING 0 2 5 0 NO-TRACE-STRING 0 3 6 0 NO-TRACE-STRING 0 4 0 0 NO-TRACE-STRING 0 1 1 0 NO-TRACE-STRING 0 2 2 0 NO-TRACE-STRING 0 3 3 0 NO-TRACE-STRING 0 4 4 0 NO-TRACE-STRING 0 1 5 0 NO-TRACE-STRING 0 2 6 0 NO-TRACE-STRING 0 3 0 0 NO-TRACE-STRING 0 4 1 0 NO-TRACE-STRING 1 0 21 15 .CMD FORMAT rd=d ct=10 im=i et=3 zt=15 pr=3 mass length time charge .CMD SET ORIGIN 0 .CMD SET TOL 0.001000000000000 .CMD SET PRNCOLWIDTH 8 .CMD SET PRNPRECISION 4 .CMD PRINT_SETUP 1.200000 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=Vectors .CMD DEFINE_FONTSTYLE fontID=0 family=Times points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=1 family=Times points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=2 family=Helvetica points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=4 family=Helvetica points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=5 family=Courier points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=6 family=Chicago points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=7 family=Times points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=8 family=Times points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=9 family=Times points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=10 family=Helvetica points=10 bold=1 italic=0 underline=0 .CMD UNITS U=1 .TXT 3 24 0 0 Cg a73.000000,73.000000,26 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {\fs24 \b Metric Tensors}} } .TXT 3 0 0 0 Cg a78.833333,78.833333,114 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {By Richard R. Shiffman\par Digital Graphics Assoc.\par 10318 Dunkirk Ave.\par L.A., Ca. 90025\par rrs@isi.edu}} } .TXT 12 -23 0 0 Cg a73.000000,73.000000,268 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {The metric is the function that relates an infinitesimal change in coordinates to the infinitesimal distance between the points of the space. In Euclidian 'N', which is flat, homogeneous and isotropic, distance between two points is given by Pythagorean theorem:}} } .EQN 7 0 0 0 ({0:ds}NAME)^(2)÷({0:i}NAME÷1,{0:n}NAME)$((({0:dx}NAME)[({0:i}NAME)))^(2) .TXT 0 20 0 0 Cg a51.000000,51.000000,19 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {or for 3 space:}} } .EQN 0 14 0 0 ({0:ds}NAME)^(2)÷({0:dx}NAME)^(2)+({0:dy}NAME)^(2)+({0:dz}NAME)^(2) .TXT 7 -34 0 0 Cg a73.000000,73.000000,383 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {In the above examples, all components of the coordinate system have units of length. Not all coordinate systems have this property. For example, in two dimensional polar coordinates\par one component of the position vector has a unit of length and the other is an angle. The metric function must ensure that the difference in position vectors always results in a unit of length.}} } .EQN 10 0 0 0 {10:k}NAME÷({2,1}ö{0:\q}NAMEö{0:r}NAME) .TXT 0 8 0 0 Cg a65.000000,65.000000,78 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {And the difference in position between }{\b k}{ and}{\b k}{+d}{\b k}{ is }} } .EQN 0 41 0 0 {10:dk}NAME÷({2,1}ö{0:d\q}NAMEö{0:dr}NAME) .TXT 5 -49 0 0 Cg a73.000000,73.000000,29 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {The metric in this case is}} } .EQN 0 19 0 0 ({0:ds}NAME)^(2)÷({0:dr}NAME)^(2)+({0:r}NAME)^(2)*({0:d\q}NAME)^(2) .TXT 4 -19 0 0 Cg a73.000000,73.000000,175 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Both components of ds^2 now have units of length squared.}{ }{Note that the length between two points is an invariant and is independent of the coordinate system chosen.}} } .TXT 6 0 0 0 Cg a73.000000,73.000000,637 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {If the N space you are working with (i.e. the surface of a sphere) is curved, then you can visualize it as a hypersurface in N*(N+1)/2 dimensional Euclidian space. In the case of a 2 dimensional curved space, you need to imbeded it in a 3 dimensional }{Euclidian space.}{ A 6 dimensional }{Euclidian space}{ is needed to imbed a 3 dimensional curved space. For a 4 dimensional curved space you need a 10 dimensional Euclidian space.}{The author used this technique in his first document on navigation(3). For example: when dealing with a spherical surface which is two dimensional, the general metric for E3 is shown below.}} } .EQN 19 9 0 0 ({0:ds}NAME)^(2)÷({0:dx}NAME)^(2)+({0:dy}NAME)^(2)+({0:dz}NAME)^(2) .EQN 0 19 0 0 {0:or}NAME .EQN 0 4 0 0 {0:ds}NAME÷\(((({0:dx}NAME)/({0:dt}NAME)))^(2)+((({0:dy}NAME)/({0:dt}NAME)))^(2)+((({0:dz}NAME)/({0:dt}NAME)))^(2))*{0:dt}NAME .TXT 5 -31 0 0 Cg a72.000000,72.000000,58 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {We add the constraint that we are on a sphere or radius}} } .EQN 0 41 0 0 {0:r.e}NAME~(21600)/(2*{0:\p}NAME) .TXT 0 9 0 0 Cg a28.000000,28.000000,5 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {or}} } .EQN 0 3 0 0 ({0:r.e}NAME)^(2)÷({0:x}NAME)^(2)+({0:y}NAME)^(2)+({0:z}NAME)^(2) .TXT 3 -53 0 0 Cg a72.000000,72.000000,77 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Now solving for x, y, and z as functions of latitude and longitude yields:}} } .EQN 3 0 0 0 1. .EQN 0 3 0 0 {0:x}NAME÷{0:r.e}NAME*{0:cos}NAME({0:Lat}NAME)*{0:cos}NAME(-{0:Lng}NAME) .EQN 0 22 0 0 2. .EQN 0 3 0 0 {0:y}NAME÷{0:r.e}NAME*{0:cos}NAME({0:Lat}NAME)*{0:sin}NAME(-{0:Lng}NAME) .EQN 0 21 0 0 3. .EQN 0 3 0 0 {0:z}NAME÷{0:r.e}NAME*{0:sin}NAME({0:Lat}NAME) .TXT 4 -52 0 0 Cg a72.000000,72.000000,183 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {From the chain rule we obtain derivatives of x, y, and z with respect to the parameter t. Note: \par Lat=Lat(t) and Lng=Lng(t) for a one dimensional curve on the surface of a sphere.}} } .EQN 6 0 0 0 ({0:dx}NAME)/({0:dt}NAME)÷-{0:r.e}NAME*{0:sin}NAME({0:Lat}NAME)*{0:cos}NAME({0:Lng}NAME)*({0:dLat}NAME)/({0:dt}NAME)-{0:r.e}NAME*{0:cos}NAME({0:Lat}NAME)*{0:sin}NAME({0:Lng}NAME)*({0:dLng}NAME)/({0:dt}NAME) .EQN 5 0 0 0 ({0:dy}NAME)/({0:dt}NAME)÷{0:r.e}NAME*{0:sin}NAME({0:Lat}NAME)*{0:sin}NAME({0:Lng}NAME)*({0:dLat}NAME)/({0:dt}NAME)-{0:r.e}NAME*{0:cos}NAME({0:Lat}NAME)*{0:cos}NAME({0:Lng}NAME)*({0:dLng}NAME)/({0:dt}NAME) .EQN 5 0 0 0 ({0:dz}NAME)/({0:dt}NAME)÷{0:r.e}NAME*{0:cos}NAME({0:Lat}NAME)*({0:dLat}NAME)/({0:dt}NAME) .TXT 5 0 0 0 Cg a72.000000,72.000000,227 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Substituting and simplifying the differential form of the above three derivatives back into the original 3 space metric yields the metric for a spherical surface in terms of differential changes in latitude and longitude.}} } .EQN 9 0 0 0 {0:ds}NAME÷{0:r.e}NAME*\(({0:cos}NAME({0:Lat}NAME))^(2)*({0:dLng}NAME)^(2)+({0:dLat}NAME)^(2)) .TXT 4 -1 0 0 C x1,1,0,0 .TXT 3 0 0 0 Cg a72.000000,72.000000,131 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {The distance or arc-length between any two points on the surface of the sphere connected by a curve parametrized by t would be:}} } .TXT 7 39 0 0 Cg a34.000000,34.000000,98 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Note: Lat=Lat(t) and Lng=Lng(t), also the derivatives of Lat and Lng with respect to t exist.}} } .EQN 3 -35 0 0 {0:s}NAME÷({0:r}NAME)[({0:e}NAME)*(({0:t}NAME)[(0)&({0:t}NAME)[(1)`\(({0:cos}NAME({0:Lat}NAME))^(2)*((({0:dLng}NAME)/({0:dt}NAME)))^(2)+((({0:dLat}NAME)/({0:dt}NAME)))^(2))&{0:t}NAME) .TXT 7 -4 0 0 Cg a73.000000,73.000000,888 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {If you can not enter the N*(N+1)/2 dimensional hyperspace to look down at it, }{i}{s there a way to tell if the N-space you are in is curved? Also is there a way to generate the metric function without using going through }{N*(N+1)/2}{ dimensional mathematics that reduce down to N dimensional result?\par The answer to both these questions is yes . \par \par To explore the first question, draw some common geometric figures, such as a triangle or a circle. If the sum of the internal angles in the triangle is equal to }{\f1 p}{ radians, then your space is flat. If its greater than }{\f1 p}{ radians the space you are in has positive Gaussian curvature. A sphere is a space that has positive Gaussian curvature. If the sum of the angles is less than }{\f1 p}{ radians the space has a negative Gaussian curvature. A saddle surface is a space that has negative Gaussian curvature. }{\par }} } .EQN 21 0 0 0 { 130 124 0 0 22 21}{61} 0 17198 0 0 1068 40 124 130 1 8 0 0 0 0 0 0 LVLV!-DFLG"\LVIq2RLV!$+tLG!!LVIqLSDF!-CqLG"\CtIq2ODF!$+ALG!!CtIqLP<6!-C>LG"\;4Iq2L<6!$*lLG!!;4IqLM4&!-BiLG"\2RIq2I4&!$*9LG!!2RIqLJ+t!-B6LG"\)pIq2F+t!$)dLG!!)pIqLG#d!-AaLG"\!0Iq2C#d!$)1LG!!!0IqLVL#!-DFCe"\LSAa2RL#!$+tCe!!LSAaLSCq!-CqCe"\CqAa2OCq!$+ACe!! CqAaLP;a!-C>Ce"\;1Aa2L;a!$*lCe!!;1AaLM3Q!-BiCe"\2OAa2I3Q!$*9Ce!!2OAaLJ+A!-B6Ce"\)mAa2F+A!$)dCe!!)mAaLG#1!-AaCe"\!-Aa2C#1!$)1Ce!!!-AaLVKN!-DF;%"\LP9Q2RKN!$+t;%!!LP9QLSC>!-Cq;%"\Cn9Q2OC>!$+A;%!!Cn9QLP;.!-C>;%"\;.9Q2L;.!$*l;%!!;.9QLM2|!-Bi;%"\2L9Q2I2|!$*9 ;%!!2L9QLJ*l!-B6;%"\)j9Q2F*l!$)d;%!!)j9QLG"\!-Aa;%"\!*9Q2C"\!$)1;%!!!*9QLVJy!-DF2C"\LM1A2RJy!$+t2C!!LM1ALSBi!-Cq2C"\Ck1A2OBi!$+A2C!!Ck1ALP:Y!-C>2C"\;+1A2L:Y!$*l2C!!;+1ALM2I!-Bi2C"\2I1A2I2I!$*92C!!2I1ALJ*9!-B62C"\)g1A2F*9!$)d2C!!)g1ALG")!-Aa2C"\!'1A2C") !$)12C!!!'1ALVJF!-DF)a"\LJ)12RJF!$+t)a!!LJ)1LSB6!-Cq)a"\Ch)12OB6!$+A)a!!Ch)1LP:&!-C>)a"\;()12L:&!$*l)a!!;()1LM1t!-Bi)a"\2F)12I1t!$*9)a!!2F)1LJ)d!-B6)a"\)d)12F)d!$)d)a!!)d)1LG!T!-Aa)a"\!$)12C!T!$)1)a!!!$)1LVIq!-DF!!"\LG!!2RIq!$+t!!!!LG!!LSAa!-Cq!!"\Ce!! 2OAa!$+A!!!!Ce!!LP9Q!-C>!!"\;%!!2L9Q!$*l!!!!;%!!LM1A!-Bi!!"\2C!!2I1A!$*9!!!!2C!!LJ)1!-B6!!"\)a!!2F)1!$)d!!!!)a!!LG!!!-Aa!!"\!!!!2C!!!$)1!!!!!/G-!!#B!!!!@o!!!+<7!!"K!!!!59!!!&.[!!!e!!!!&k!!!"#e!/G-!!#B!!!!@o!!!+<7!!"K!!!!59!!!&.[!!!e!!!!&k!!!"#e!/G-!!#B !!!!@o!!!+<7!!"K!!!!59!!!&.[!!!e!!!!&k!!!"#e!!#SI_G-FhFh!,@z@o"m>%<78787!(5@59!v/R.[,[,[!#&m&k!2#v#e!!!!!!)15`Aa!Q"<6+!$!(:o!!)15`Aa!Q"<6+!$!(:o!!)15`Aa!Q"<6+!$!(:o!!)15`Aa"3"{$vJj!!>{5Y+u!@J3!"4&,'!,!(9V!!&J*#!7LI>{#U"<"#!"LI6k!@J4 !"4&$W!4!(9X!!&J']!"LI6k!@J!!!A`"c!4!(9Y!!&J'M!"LI6k!@J+!!&J"c!4!(9Z!!&J'=!"LI6k!@J+!!&J"c!4!(9[!!&J'-!"LI6k!@J+!!&J"c!4!(9\!!&J&k!"LI9Q!@J+!!&J"c!4!(9]!!&J&[!"LI<7!@J*!!&J"c!4!(9^!!&J&K!"LI<7!@J*!!&J"c!4!(9_!!&J&+!#LI<7!@J*!!&J"c!4!(9`!!&J%y!"LI>{!@J* !!&J"c!4!(9a!!&J%i!"LIAa!@J)!!&J"c!0!(9b!!&J-9!"LH1A!@I{!!Iq5Z&K!@JZ!!&J$g!"LG<7#U"<$7!"LK4'!@J(!!&J"c!0!(9e!!&J,w!"LH.[!@I{!!Iq5Z.[!@JY!!&J$G!"LG<7"s"<$g!"LL<7!@I{!")15Z4'!@J'!!)0!1!"LK#e!@I{!"4'5Z6k!@J%!!&J!A!"LG#e!@JS!!&J"c!8!(9j!!&J"s!#LG+u!@Is!!&J ,7!"LG<7&;"<%9!"LG9Q!PIw!!&J!1!"LG#e!@J=!!&J$g!"LG<7%Y"<%I!"LG4'!PIy!!)0!1!"LI<7!@J)!!&J"c!4!(9l!!&J!Q!$LJ9Q!@J(!!&J"c!4!(9m!!.Z*s!#LG1A!@J!!!&J"c!8!(9n!!&J+5!"LG#e!@Iw!!&J#5!"LG<7%Y"<&+!"LJ9Q!@Ir!!&J!1!"LG4'!PI|!!&J"c!8!(9p!!&J*S!#LG&K!@Iy!!)0"c!"LG<7 #U"<&K!"LK1A!PIy!!&J"c!0!(9r!!&J-)!%LG+u!@I{!!>{5[&K!@J\!!4&"c!,!(9t!!&J.+!"LG<7"s"<'-!"LKAa!@I{!!>{5[.[!@J^!!&J"c!,!(9w!!&J-Y!"LG<7"s"<']!"LK9Q!@I{!!>{5[6k!@J[!!&J"c!,!(9z!!&J-)!"LG<7"s"<(/!"LK1A!@I{!!>{5[>{!@JX!!&J"c!,!(9}!!&J,W!"LG<7"s"<(_!"LK)1!@I{ !!>{5[G-!@JU!!&J"c!,!(:"!!&J,'!"LG<7"s"<)!!"LK#e!@I{!!>{5\!!!@JS!!&J"c!,!(:$!!&J+e!"LG<7"s"<)Q!"LJG-!@I{!!>{5\)1!@JP!!&J"c!,!(:'!!&J+5!"LG<7"s"<*#!"LJ>{!@I{!!>{5\1A!@JM!!&J"c!,!(:*!!&J*c!"LG<7"s"<*S!"LJ6k!@I{!!>{5\9Q!@JJ!!&J"c!,!(:-!!&J*3!"LG<7"s"<+%!" LJ.[!@I{!!>{5\Aa!@JG!!&J"c!,!(:0!!&J)a!"LG<7"s"<+U!"LJ&K!@I{!!>{5\Iq!@JD!!&J"c!,!(:3!!&J)1!"LG<7"s"<,'!"LIIq!@I{!!>{5]&K!@JA!!&J"c!,!(:6!!&J(_!"LG<7"s"<,W!"LIAa!@I{!!Iq5].[!@J*!!&J$7!"LG<7#U"<,g!"LH4'!@J&!!&J"c!4!(:9!!&J$g!"LG#e!@J$!!&J"c!4!(::!!&J$W!# LG#e!@J#!!&J"c!4!(:;!!&J$G!"LG&K!@J#!!&J"c!4!(:{5]Iq!@J4!!&J"c!0!(:C!!&J#%!#LH&K!@I{!!Iq5^#e !@I|!!&J$7!"LG<7#U"<.{!"LG9Q!@J'!!&J"c!0!(:F!!&J"3!"LH.[!@I{!!Iq5^+u!@Iv!!)0$W!"LG<7#U"{5^6k!PJ*!!&J"c!,!(:M!!&J$g!"LG<7"s"<0O!"LH.[!@I{!!>{5^>{!@J(!!&J"c!,!(:O!!&J$G!" LG<7"s"<0o!"LH)1!@I{!!>{5^G-!@J%!!&J"c!,!(:R!!&J#u!"LG<7"s"<1A!"LH!!!@I{!!>{5_#e!@J"!!&J"c!,!(:U!!&J#E!"LG<7"s"<1q!"LGDG!@I{!!>{5_+u!@I}!!&J"c!,!(:X!!&J"s!"LG<7"s"<2C!"LG<7!@I{!!>{5_4'!@Iz!!&J"c!,!(:[!!&J"C!"LG<7"s"<2s!"LG4'!@I{!!>{5_<7!@Iw!!&J"c!,!(:^ !!&J!q!"LG<7"s"<3E!"LG+u!@I{!!>{5_DG!@It!!&J"c!,!(:a!!&J!A!"LG<7"s"<3u!"LG#e!@I{!!4'5`!!!PI{!!4'5`#e!@I{!!)15`Aa!Q"<6+!$!(:o!!)15`Aa  .TXT 1 23 0 0 Cg a50.000000,50.000000,162 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {For the triangle on the left, w}{here Q is the Gaussian curvature of the space and A is the area of the triangle}{, }{the sum of the internal angles will be:}} } .EQN 6 0 0 0 {0:\a}NAME+{0:\b}NAME+{0:\g}NAME÷{0:\p}NAME+{0:Q}NAME*{0:A}NAME .TXT 3 0 0 0 Cg a50.000000,50.000000,285 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain { See (4) for the definition of signed curvature in general and the Gaussian curvature of a surface.\par \par Another way is to examine the measured ratio of the circumference\par of a circle to its diameter. If the ratio is not equal to –, then the space has a non-zero Gaussian curvature.}} } .TXT 13 -23 0 0 Cg a73.000000,73.000000,149 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {--------------------------------------------------------------------------------------------------------------------------------------------------}} } .TXT 3 0 0 0 Cg a73.000000,73.000000,245 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {The Metric tensor and its application to the square of the difference in position vectors is the answer to the second question. See "Cross Product, Dot Product and fun with Tensors" (5) for an introduction to tensors and their application. }} } .TXT 7 0 0 0 Cg a75.166667,75.166667,158 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {The d}{ifference in position vectors between }{\b k}{ and}{\b k}{+d}{\b k}{ is d}{\b k}{. The distance between the two position vectors is defined as:\par }} } .EQN 5 0 0 0 ({0:ds}NAME)^(2)÷({0:i}NAME÷0;{0:n}NAME-1)$({0:j}NAME÷0;{0:n}NAME-1)$({0:A}NAME)[({0:i}NAME,{0:j}NAME)*({0:dk}NAME)[({0:i}NAME)*({0:dk}NAME)[({0:j}NAME) .EQN 0 39 0 0 {0:ds}NAME .TXT 0 4 0 0 Cg a32.000000,32.000000,44 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {is the differential element of arc-length}} } .EQN 3 -4 0 0 ({0:A}NAME)[({0:i}NAME,{0:j}NAME) .TXT 0 4 0 0 Cg a30.000000,30.000000,39 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {is a component of the metric tensor}} } .EQN 3 -4 0 0 ({0:dk}NAME)[({0:x}NAME) .TXT 0 4 0 0 Cg a31.000000,31.000000,69 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {is a component of the differential change in the position vector}} } .EQN 4 -4 0 0 {0:n}NAME .TXT 0 4 0 0 Cg a26.500000,26.500000,37 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {is the dimensionality of the space}} } .TXT 2 -43 0 0 Cg a73.000000,73.000000,204 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {\b A}{, the metric tensor, is a second order symmetric tensor whose row or column size\par is equal to the dimension of the space you are working with. Some examples of a general metric tensors follow:}} } .TXT 5 0 0 0 C x1,1,0,0 .TXT 3 0 0 0 Cg a73.000000,73.000000,221 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {A second order tensor application can be represented by matrix multiplication. If we now consider }{E}{uclidian 3}{ space (E3) where the coordinate axis are orthogonal. This space is also homogeneous and isotropic.}} } .EQN 6 0 0 0 {0:n}NAME:3 .TXT 4 0 0 0 Cg a73.000000,73.000000,22 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {The metric is then:}} } .EQN 0 15 0 0 {10:A}NAME:({3,3}ö1ö0ö0ö0ö1ö0ö0ö0ö1) .TXT 0 14 0 0 Cg a44.000000,44.000000,7 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {and }} } .EQN 0 4 0 0 {10:dk}NAME÷({3,1}ö{0:dz}NAMEö{0:dy}NAMEö{0:dx}NAME) .TXT 0 8 0 0 Cg a32.000000,32.000000,84 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {is differential change of the postion vector between point or vector k and k+dk}} } .EQN 4 6 0 0 { 185 129 0 0 31 22}{61} 15752 25498 0 0 1068 40 129 185 1 8 0 0 0 0 0 0 LVLV!-DFLG"\LVIq2RLV!$+tLG!!LVIqLSDF!-CqLG"\CtIq2ODF!$+ALG!!CtIqLP<6!-C>LG"\;4Iq2L<6!$*lLG!!;4IqLM4&!-BiLG"\2RIq2I4&!$*9LG!!2RIqLJ+t!-B6LG"\)pIq2F+t!$)dLG!!)pIqLG#d!-AaLG"\!0Iq2C#d!$)1LG!!!0IqLVL#!-DFCe"\LSAa2RL#!$+tCe!!LSAaLSCq!-CqCe"\CqAa2OCq!$+ACe!! 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!PIs!!&J!A!#LG&K!@Ir!!&J!Q!"LG&K!@It!!&J!A!"LG&K!@It!!&J/]!\!(9Z!!)0!a!#LG.[!PIt!!&J!1!#LG&K!PIr!!&J!A!"LG&K!@Ir!!)0!Q!$LG)1!`Is!!)0!a!$LL4'"s"<'-!"LH&K!@Jw!!Iq5[+u!@Is!!&J#U!"LM+u!Q"<;U!$!(;1!!4'5\Iq!PJn!!>{5\G-!@Is!!&J0?!(!(:5!!&J0/!p!(9Y!!&J!q!"LG)1 !PIu!!&J!Q!"LG1A!PIt!!&J!a!#LG#e!@It!!&J!Q!#LG.[!PIr!!&J!Q!#LG+u!@It!!&J!A!#LG&K!@Is!!&J,'""!(9Y!!&J!q!"LG&K!@Is!!&J!Q!"LG)1!@Iv!!&J!A!"LG&K!@It!!&J!Q!"LG)1!@Is!!&J!A!"LG)1!@Is!!)0!A!"LG&K!@It!!&J!Q!"LG#e!@Is!!)0!Q!"LK#e11"<"C!"LG+u!PIr!!&J!a!"LG&K!@It !!&J!a!"LG+u!@Ir!!&J!Q!"LG)1!@It!!&J!1!"LG+u!@Ir!!&J!a!"LG#e!@Iu!!&J!A!"LG)1!@Ir!!&J!Q!"LG)1!@JT!')15Y6k!@It!!&J!1!"LG#e!@Iu!!&J!A!"LG)1!@Iu!!&J!a!"LG#e!@It!!&J!Q!"LG)1!@Ir!!&J!a!"LG#e!@Iu!!&J!1!"LG+u!@Is!!&J!Q!"LG&K!PIr!!&J!Q!"LK#e11"<"C!"LG&K!@Is!!&J !1!"LG+u!@Is!!&J!Q!"LG+u!@Iu!!&J!1!"LG)1!@It!!&J!Q!"LG#e!@Iu!!&J!1!"LG+u!@Ir!!&J!a!"LG&K!@It!!&J!a!#LG)1!@JT!')15Y6k!@Ir!!&J!Q!"LG&K!@Is!!&J!Q!#LG&K!@Iv!!&J!A!"LG&K!PIs!!&J!Q!#LG&K!@Is!!&J!A!"LG)1!@Is!!)0!A!"LG&K!@It!!)0!A!"LG#e!@It!!&J!Q!"LK#e-i"<"C!# LG+u!@It!!)0!a!"LG#e!PIx!!)0!Q!"LG#e""Ir!!&J!1!#LG+u!PIv!!)0!1!"LG)1!PIu!!&J!1!#LG)1!`Iu!!&J,'!8!(9Y!!&J!q!"LHDG!@It!!&J'}!"LK#e$w"<"C!"LG.[!@J0!!&J!Q!"LI9Q!@JT!!)15bAa!Q"<;U!$!(;1!$!Q  .TXT 5 -55 0 0 Cg a73.000000,73.000000,30 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {The metric tensor would be:}} } .EQN 0 23 0 0 {10:A}NAME÷({3,3}ö1ö{0:cos}NAME({0:\b}NAME)ö{0:cos}NAME({0:\g}NAME)ö{0:cos}NAME({0:\b}NAME)ö1ö{0:cos}NAME({0:\a}NAME)ö{0:cos}NAME({0:\g}NAME)ö{0:cos}NAME({0:\a}NAME)ö1) .EQN 10 -23 0 0 {10:dk}NAME÷({3,1}ö{0:dw}NAMEö{0:dv}NAMEö{0:du}NAME) .TXT 0 9 0 0 Cg a32.000000,32.000000,84 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {is differential change of the postion vector between point or vector k and k+dk}} } .EQN 14 -9 0 0 ({0:ds}NAME)^(2)÷{10:A}NAME*{10:dk}NAME*{10:dk}NAME÷({3,3}ö1ö{0:cos}NAME({0:\b}NAME)ö{0:cos}NAME({0:\g}NAME)ö{0:cos}NAME({0:\b}NAME)ö1ö{0:cos}NAME({0:\a}NAME)ö{0:cos}NAME({0:\g}NAME)ö{0:cos}NAME({0:\a}NAME)ö1)*({3,1}ö{0:dw}NAMEö{0:dv}NAMEö{0:du}NAME)*( {3,1}ö{0:dw}NAMEö{0:dv}NAMEö{0:du}NAME) .EQN 8 0 0 0 ({0:ds}NAME)^(2)÷{0:A}NAME*{0:dk}NAME*{0:dk}NAME÷({0:du}NAME)^(2)+2*{0:du}NAME*{0:cos}NAME({0:\a}NAME)*{0:dv}NAME+2*{0:du}NAME*{0:cos}NAME({0:\g}NAME)*{0:dw}NAME+({0:dv}NAME)^(2)+2*{0:dv}NAME*{0:cos}NAME({0:\b}NAME)*{0:dw}NAME+({0:dw}NAME)^(2) .TXT 4 0 0 0 Cg a73.000000,73.000000,149 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {--------------------------------------------------------------------------------------------------------------------------------------------------}} } .TXT 1 0 0 0 C x1,1,0,0 .TXT 4 0 0 0 Cg a73.000000,73.000000,652 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {In the earlier example of non-orthognal coordinates, whose basis vectors were U,V, and W, the author made the assumption that these vector were of unit length. When the basis vectors for the space aren't unitary, then the cosins of the angles between basis vectors in the metric tensor must be replace with there dot products. The follow two dimensional }{example has one basis vector, e2<0>, directed along the x-axis of R2 andis one unit of length . The other basis vector, e2<1>, is 2^(1/2) units long and is rotated counter-clockwise by 45 degrees with respected to the first vector.}{ We will now derive the metric tensor for this basis}} } .TXT 16 0 0 0 Cg a73.333333,73.333333,105 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {The two }{n}{on-orthonormal basis vectors}{ that span R2 defined below have lengths of 1 and 1.414.}} } .EQN 5 0 0 0 ({0:e2}NAME){52}(0):({2,1}ö0ö1) .EQN 0 14 0 0 ({0:e2}NAME){52}(1):({2,1}ö1ö1) .EQN 0 14 0 0 |(({0:e2}NAME){52}(0))=?_n_u_l_l_ .EQN 0 15 0 0 |(({0:e2}NAME){52}(1))=?_n_u_l_l_ .TXT 7 -43 0 0 Cg a95.666667,95.666667,66 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {The standard }{o}{rthonormal basis vectors for cartisian}{ are:}} } .EQN 0 44 0 0 ({0:x2}NAME){52}(0):({2,1}ö0ö1) .EQN 0 14 0 0 ({0:x2}NAME){52}(1):({2,1}ö1ö0) .TXT 6 -58 0 0 Cg a73.000000,73.000000,52 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Let's Plot the two non-orthonormal basis vectors.}} } .EQN 4 0 0 0 {0:i}NAME:0;1 .EQN 0 8 0 0 {0:j}NAME:0;1 .EQN 0 8 0 0 ({0:v1}NAME){52}({0:i}NAME):(({1,2}ö{0:i}NAMEö{0:i}NAME)){51} .EQN 0 17 0 0 ({0:v0}NAME){52}({0:i}NAME):(({1,2}ö0ö{0:i}NAME)){51} .EQN 4 -33 0 0 1.5&-.5&((({0:v0}NAME){52}({0:i}NAME)))[(1),((({0:v1}NAME){52}({0:i}NAME)))[(1)@1.5&-.5&((({0:v0}NAME){52}({0:i}NAME)))[(0),((({0:v0}NAME){52}({0:i}NAME)))[(0) 0 0 1 1 0 0 0 1 1 0 4 3 0 0 NO-TRACE-STRING 3 3 0 0 NO-TRACE-STRING 0 3 2 0 NO-TRACE-STRING 0 4 3 0 NO-TRACE-STRING 0 1 4 0 NO-TRACE-STRING 0 2 5 0 NO-TRACE-STRING 0 3 6 0 NO-TRACE-STRING 0 4 0 0 NO-TRACE-STRING 0 1 1 0 NO-TRACE-STRING 0 2 2 0 NO-TRACE-STRING 0 3 3 0 NO-TRACE-STRING 0 4 4 0 NO-TRACE-STRING 0 1 5 0 NO-TRACE-STRING 0 2 6 0 NO-TRACE-STRING 0 3 0 0 NO-TRACE-STRING 0 4 1 0 NO-TRACE-STRING 0 1 23 21 .TXT 3 43 0 0 Cg a36.333333,36.333333,118 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Using the Dot-Products of the basis vectors to }{define elements of the metric tensor }{we obtain the following:}} } .EQN 7 0 0 0 ({0:g}NAME)[({0:i}NAME,{0:j}NAME):({0:e2}NAME){52}({0:i}NAME)*({0:e2}NAME){52}({0:j}NAME) .TXT 4 0 0 0 Cg a29.833333,29.833333,30 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {In this case the metric is:}} } .EQN 0 19 0 0 {0:g}NAME=?_n_u_l_l_ .TXT 19 -62 0 0 Cg a72.000000,72.000000,116 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {An example, using the above metric to determine the distance between two points in the E2 coordinate system is:}} } .EQN 7 0 0 0 {0:\Da}NAME:(({1,2}ö1ö1)){51}-(({1,2}ö0ö0)){51} .EQN 0 22 0 0 {0:s2}NAME:{0:g}NAME*{0:\Da}NAME*{0:\Da}NAME .EQN 0 14 0 0 {0:s2}NAME=?_n_u_l_l_ .EQN 0 9 0 0 \({0:s2}NAME)=?_n_u_l_l_ .TXT 4 -45 0 0 Cg a72.166667,72.166667,115 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Notice that the distance between these two points in the E2 coordinates are not equal to the square root of 2. }} } .TXT 4 0 0 0 C x1,1,0,0 .TXT 2 0 0 0 Cg a73.000000,73.000000,238 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {In the above two examples the space is flat, it has no Gaussian curvature and the components of the metric tensor are constants. If the space is curved, the components of the metric tensor will be functions of your current position. }} } .TXT 7 0 0 0 Cg a73.000000,73.000000,596 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {In the case of a spherical surface revisited ( of which the earth is a good model ), the metric computes the distance between two points as a function of their differences in latitudes and longitudes. Since the surface of a sphere is two dimensional, latitude and longitude are the only two parameters needed to describe your position. }{\f1 f}{ stands for latitude and }{\f1 q}{ used for longitude. The differential displacements are orthogonal, so only the diagonal components of the metric tensor will be non-zero, but due to the curvature they will be functions of the coordinate.}} } .TXT 13 0 0 0 Cg a73.000000,73.000000,8 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Note:}} } .EQN 0 6 0 0 {0:r.e}NAME .TXT 0 3 0 0 Cg a70.000000,70.000000,51 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {is the constant radius of the spherical surface,}} } .EQN 0 35 0 0 (-{0:\p}NAME)/(2)ò{0:\f}NAMEò({0:\p}NAME)/(2) .EQN 0 9 0 0 {0:and}NAME .EQN 0 4 0 0 -{0:\p}NAMEò{0:\q}NAMEò{0:\p}NAME .EQN 2 -57 0 0 { 242 239 0 0 41 40}{61} 15752 59158 0 0 1068 40 239 242 1 8 0 0 0 0 0 0 LVLV!-DFLG"\LVIq2RLV!$+tLG!!LVIqLSDF!-CqLG"\CtIq2ODF!$+ALG!!CtIqLP<6!-C>LG"\;4Iq2L<6!$*lLG!!;4IqLM4&!-BiLG"\2RIq2I4&!$*9LG!!2RIqLJ+t!-B6LG"\)pIq2F+t!$)dLG!!)pIqLG#d!-AaLG"\!0Iq2C#d!$)1LG!!!0IqLVL#!-DFCe"\LSAa2RL#!$+tCe!!LSAaLSCq!-CqCe"\CqAa2OCq!$+ACe!! 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LG>{!@JI!!Iq5Y9Q!@JL!!&J"c!"LK#e$7"<"S!"LJ9Q!@I|!!&J"S!"LJ1A$7"<"S!"LJ9Q!PI|!!&J#e!"LIG-%Y"<"S!"LJ9Q!@Ir!!)0"c!"LG4'!@Iw!!&J)!!0!(9Z!!&J-9!"LGAa!PJB!!Iq5Y9Q!@J\!!&J!q!"LJ1A"s"<"S!"LK<7!@JN!")15Y&K!@Iw!!&J-i!"LG)1!@JI!")15Y)1!@Iv!!&J-y!"LG&K!@JI!"4'5Y)1 !@Iv!!&J.+!"LG#e!@Ir!!&J)q!0!(9U!!&J!a!"LKG-!pJG!*#e3e!!!!#d!!!!!0IqLG#d!0IqLG#d!0IqLG#d!0IqLG#d!0IqLG#d!0IqLG#d!0IqLG#d!0IqLG#d!0IqLG#d!0IqLG#d!0IqLG#d!0IqLG#d!0IqLG#dLVLVLVIq!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!")15Y)1 !@Ir!!&J!Q!"LKIq!@JI!")15Y)1!@Ir!!&J!Q!"LKG-!@JJ!!Iq5Y&K!@It!!&J!A!"LO1A#U"{5Y9Q!@J]!!&J+%!,!(9Z!!&J-9!#LJAa"s"<"S!"LK4'!PJP!!>{5Y9Q!@JY!!)0+U!,!(9Z!!&J,g!#LJIq"3"<"S!"LO1A"3"<"S!"LO1A"3"<"S!"LO1A"3"<"S!"LO1A "3"<"S!"LO1A"s"<"S!"LJDG!@J[!!>{5Y9Q!@JO!!&J-I!,!(9Z!!&J*s!#LK<7"s"<"S!"LJ9Q!PJ^!!>{5Y9Q!@JK!!&J.+!(!(9Z!!&J7m!(!(9Z!!&J7m!(!(9Z!!&J7m!(!(9Z!!&J7m!,!(9Z!!&J(o!#LL1A"s"<"S!"LIAa!PJk!!>{5Y9Q!@J{5Y9Q!@J.!!+t2c!,!(9Z!!&J%)!$LM>{"3"<"S!"LO1A"s"<"S!"LGAa!pK+!!4'5Y4'"BK7!!4'5Y9Q!@K;!!4'5Y9Q!@K;!!4'5Y9Q!@K;!!4'5Y9Q!@K;!!4'5Y9Q!@K;!!4'5Y9Q!@K;!!4'5Y9Q!@K;!!4'5Y9Q!@K;!!4'5Y9Q!@K;!!4'5Y9Q!@K;!!4'5Y9Q!@K;!!)15b!!!Q"<9Q!$!(;%!!)15b!!!Q"<9Q!$!(;% !!)15b!!!Q"<9Q!$!(;%!!)15b!!!Q"<9Q!$!(;%!&>{5Y9Q""Is!!1@"3!"LG6k!PIu!!&J!A!#LG&K!@Ir!!&J"S!$LG.[!PIv!!)0!Q!"LG.[!PIr!!&J!A!"LG&K!@It!!&J!A!#LG&K!@Is!!)0!A!#LG+u!`J'!(4'5Y9Q!@Iw!!&J!a!"LG1A!@Ix!!&J!A!"LG)1!@Ir!!&J!A!#LG&K!@Iy!!&J!Q!"LG)1!@Is!!&J!Q!"LG&K !@Is!!&J!a!"LG&K!PIs!!&J!A!"LG)1!@Ir!!&J!A!#LG&K!@It!!&J!A!"LG&K!@It!!&J$7"2!(9[!!&J!q!"LG.[!@Iv!!&J"#!"LG+u!@Is!!&J!1!"LG)1!@Is!!&J"3!"LG.[!@Ir!!&J!a!"LG#e!@Iu!!&J!1!"LG)1!@Iu!!&J!A!"LG&K!@It!!&J!1!"LG)1!@Is!!&J!A!"LG<7!@J&!'>{5Y>{!@Iu!!&J!q!"LG.[""Is !!&J!a!"LG&K!@Is!!)0!1!"LG&K!@Ix!!&J"3!"LG+u!@Ir!!&J!a!"LG#e!@It!!&J!a!"LG&K!@Is!!&J!Q!"LG&K!PIr!!&J!A!"LG&K"2Is!!+t$G".!(9]!!&J!Q!"LG.[!@Iv!!&J!a!"LG#e!@Iu!!&J!A!"LG+u!PIs!!&J"3!"LG4'!@Iu!!&J!1!"LG+u!@Ir!!&J!Q!"LG+u!@Is!!&J!A!"LG)1!@Iu!!)0!A!"LG&K!@Iu !!&J!1!"LH4'59"<#5!"LG&K!@Iv!!&J!q!"LG+u!@Is!!&J!A!"LG)1!@Ir!!&J!Q!"LG&K!PIw!!&J"C!"LG&K!@It!!&J!A!"LG&K!PIt!!&J!A!#LG&K!@Is!!)0!A!"LG#e!@It!!&J!A!"LG)1!@Is!!&J!A!"LG)1!@J&!'4'5YDG!@Is!!&J!q!"LG.[!@Iu!!&J!Q!#LG+u!@Is!!+t!Q!"LG#e!PIu!!&J!q!"LG)1!PIv!!)0 !Q!"LG#e!PIs!!)0!1!"LG&K!@Is!!&J!1!#LG)1!`It!!+t!A!#LG+u!`J'!#Iq5Y9Q!@It!!&J!A!"LG+u!@Iw!!&J!a!"LG9Q!@J$!!&J!Q!"LH9Q!@J%!!&J&[!D!(9[!!+t!Q!&LG4'""I{!!&J$'!$LH<7!@Is!!&J#U!"LI#e!Q"<9Q!$!(;%!!)15b!!!Q"<9Q!$!(;%!$!Q  .EQN 2 29 0 0 {0:f.0}NAME÷{0:x}NAME÷{0:r}NAME*{0:cos}NAME({0:\q}NAME) .EQN 0 16 0 0 {0:c.0}NAME÷{0:r}NAME .EQN 4 -16 0 0 {0:f.1}NAME÷{0:y}NAME÷{0:r}NAME*{0:sin}NAME({0:\q}NAME) .EQN 0 16 0 0 {0:c.1}NAME÷{0:\q}NAME .EQN 5 -16 0 0 {0:D}NAME({0:f.0}NAME,{0:c.0}NAME)÷{0:r}NAME"{0:r}NAME*{0:cos}NAME({0:\q}NAME)÷{0:cos}NAME({0:\q}NAME) .EQN 0 29 0 0 {0:D}NAME({0:f.1}NAME,{0:c.o}NAME)÷{0:r}NAME"{0:r}NAME*{0:sin}NAME({0:\q}NAME)÷{0:sin}NAME({0:\q}NAME) .EQN 6 -29 0 0 {0:D}NAME({0:f.0}NAME,{0:c.1}NAME)÷{0:\q}NAME"{0:r}NAME*{0:cos}NAME({0:\q}NAME)÷-{0:r}NAME*{0:sin}NAME({0:\q}NAME) .EQN 0 29 0 0 {0:D}NAME({0:f.1}NAME,{0:c.1}NAME)÷{0:\q}NAME"{0:r}NAME*{0:sin}NAME({0:\q}NAME)÷{0:r}NAME*{0:cos}NAME({0:\q}NAME) .TXT 8 -29 0 0 Cg a44.000000,44.000000,54 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Using the definition of the general metric, we get:}} } .EQN 5 -27 0 0 ({0:A}NAME)[(0,0)÷{0:D}NAME(({0:f}NAME)[(0),({0:c}NAME)[(0))*{0:D}NAME(({0:f}NAME)[(0),({0:c}NAME)[(0))+{0:D}NAME({0:f.1}NAME,{0:c.0}NAME)*{0:D}NAME({0:f.1}NAME,{0:c.0}NAME)÷({0:cos}NAME({0:\q}NAME))^(2)+({0:sin}NAME({0:\q}NAME))^(2)÷1 .EQN 5 0 0 0 ({0:A}NAME)[(0,1)÷({0:A}NAME)[(1,0)÷{0:D}NAME({0:f.0}NAME,{0:c.0}NAME)*{0:D}NAME({0:f.0}NAME,{0:c.1}NAME)+{0:D}NAME({0:f.1}NAME,{0:c.o}NAME)*{0:D}NAME({0:f.1}NAME,{0:c.1}NAME)÷{0:cos}NAME({0:\q}NAME)*(-{0:r}NAME*{0:sin}NAME({0:\q}NAME))+{0:sin}NAME( {0:\q}NAME)*{0:r}NAME*{0:cos}NAME({0:\q}NAME)÷0 .EQN 4 0 0 0 ({0:A}NAME)[(1,1)÷{0:D}NAME({0:f.0}NAME,{0:c.1}NAME)*{0:D}NAME({0:f.0}NAME,{0:c.1}NAME)+{0:D}NAME({0:f.1}NAME,{0:c.1}NAME)*{0:D}NAME({0:f.1}NAME,{0:c.1}NAME)÷((-{0:r}NAME*{0:sin}NAME({0:\q}NAME)))^(2)+(({0:r}NAME*{0:cos}NAME({0:\q}NAME)))^(2)÷({0:r}NAME)^ (2) .EQN 8 0 0 0 ({0:ds}NAME)^(2)÷{0:i}NAME${0:j}NAME$({0:A}NAME)[({0:i}NAME,{0:j}NAME)*({0:dk}NAME)[({0:i}NAME)*({0:dk}NAME)[({0:j}NAME)÷{10:A}NAME*{10:dk}NAME*{10:dk}NAME÷({2,2}ö({0:r}NAME)^(2)ö0ö0ö1)*({2,1}ö{0:d\q}NAMEö{0:dr}NAME)*({2,1}ö{0:d\q}NAMEö{0:dr}NAME)÷( {0:dr}NAME)^(2)+({0:r}NAME)^(2)*({0:d\q}NAME)^(2) .TXT 7 -2 0 0 Cg a73.000000,73.000000,474 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Above we have successfully derived the metric function for two dimensional polar coordinates from the definition of the general metric tensor and its application. Some other interesting uses of the metric tensor, both in physics and mathematics, will now be discussed. }{We have examined the metrics for two and three dimensional space in the above examples. Let's now look at four dimensions. This is the "stuff" Special and General Relativity is made out of. }{ }} } .TXT 12 0 0 0 Cg a73.000000,73.000000,149 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {--------------------------------------------------------------------------------------------------------------------------------------------------}} } .TXT 3 0 0 0 Cg a73.000000,73.000000,655 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Special (SRT) and General (GRT) Relativity, are two of the most successful theories in modern physics.}{Before examining examples from relativity, the three basic premises of these theories will now be presented. First, for SRT we have:\par \par 1) The speed of light in a vacuum, c, is a constant for all frames of reference.\par \par 2) The laws of physics are covariant in all inertial frames of reference. That is they have the same \par form in all reference frames or coordinate systems.\par \par The only extra premise that GRT adds is:\par \par 3) The effects of a uniform gravitational field is indistinguishable from the effects of constant acceleration.\par }} } .TXT 33 0 0 0 C x1,1,0,0 .TXT 3 0 0 0 Cg a75.000000,75.000000,141 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {A point or position vector in 4 dimensional space-time is called an event. This 4D point has a }{specific}{ location at a specific time.}} } .EQN 6 13 0 0 {0:x}NAME .EQN 0 3 0 0 {0:y}NAME .EQN 0 3 0 0 {0:z}NAME .TXT 0 2 0 0 Cg a24.666667,24.666667,40 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain { ... are the spacial components }} } .TXT 1 37 0 0 Cg a21.000000,21.000000,104 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {again }{\b dk}{ is the differential change in space-time between }{\b k}{ and }{\b k}{\b0 +}{\b dk.}} } .EQN 4 -56 0 0 {10:k}NAME÷({4,1}ö{0:c}NAME*{0:t}NAMEö{0:z}NAMEö{0:y}NAMEö{0:x}NAME) .EQN 0 11 0 0 {0:c}NAME*{0:t}NAME .TXT 0 8 0 0 Cg a25.166667,25.166667,110 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {... is the temporal component, \par where c is the speed of light\par in a vacuum}} } .EQN 0 27 0 0 {10:dk}NAME÷({4,1}ö{0:c}NAME*{0:dt}NAMEö{0:dz}NAMEö{0:dy}NAMEö{0:dx}NAME) .TXT 9 -48 0 0 Cg a73.000000,73.000000,413 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {We will now define the metric tensor for Special Relativity (SRT). In Euclidian 3 space length was the invariant, and it was always positive.The E3 metric was given by the identity matrix. \par \par By contrast, the invariant of Special Relativity, ds^2, is not positive by definition. It can assume both positive, negative, or zero values. The space-time interval between two events, is calculated below.}} } .EQN 17 0 0 0 ({0:ds}NAME)^(2)÷({0:i}NAME÷0,3)$({0:j}NAME÷0,3)$({0:A}NAME)[({0:i}NAME,{0:j}NAME)*({0:dk}NAME)[({0:i}NAME)*({0:dk}NAME)[({0:j}NAME)÷{10:A}NAME*{10:dk}NAME*{10:dk}NAME÷({4,4}ö1ö0ö0ö0ö0ö-1ö0ö0ö0ö0ö-1ö0ö0ö0ö0ö-1)*({4,1}ö{0:c}NAME*{0:dt}NAMEö{0:dz}NAMEö {0:dy}NAMEö{0:dx}NAME)*({4,1}ö{0:c}NAME*{0:dt}NAMEö{0:dz}NAMEö{0:dy}NAMEö{0:dx}NAME) .EQN 9 0 0 0 ({0:ds}NAME)^(2)÷{10:A}NAME*{10:dk}NAME*{10:dk}NAME÷-({0:dx}NAME)^(2)-({0:dy}NAME)^(2)-({0:dz}NAME)^(2)+({0:c}NAME)^(2)*({0:dt}NAME)^(2) .TXT 6 24 0 0 Cg a27.166667,27.166667,252 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {To the left is one form of the metric of Special Relativity. In this version of the metric tensor, the time coordinates of the events are expressed as c*t. The representation to the right moved velocity of light, c, into the metric tensor: }} } .EQN 4 30 0 0 ({4,4}ö({0:c}NAME)^(2)ö0ö0ö0ö0ö-1ö0ö0ö0ö0ö-1ö0ö0ö0ö0ö-1)*({4,1}ö{0:dt}NAMEö{0:dz}NAMEö{0:dy}NAMEö{0:dx}NAME)*({4,1}ö{0:dt}NAMEö{0:dz}NAMEö{0:dy}NAMEö{0:dx}NAME) .EQN 1 -51 0 0 {10:A}NAME:({4,4}ö1ö0ö0ö0ö0ö-1ö0ö0ö0ö0ö-1ö0ö0ö0ö0ö-1) .TXT 12 -3 0 0 Cg a73.000000,73.000000,940 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Notice that the metric tensor components are constant, so the curvature of this space is zero. The off-diagonal components are all zero, therefore this space is orthogonal. This space is called Minkowskian, after H. Minkowski, who investigated it in 1908. }{Minkowski s}{pace has three spacial dimensions and one temporal dimension. One can move back and forth at will on any of the spacial dimensions but, is forced to move forward in the temporal dimension at the speed of light. \par \par T}{he speed of light in a vacuum,c, and ds ( the interval ) must be invariant in any frame of reference, therefore both temporal and spacial differences in space-time coordinates must vary in different frames to keep ds^2 and c constant. In Special Relativity different frames of reference move at a constant velocity with respect to each other. The Lorentz transform, shown below, is used to convert one inertial frame into another.}} } .EQN 27 0 0 0 {0:c}NAME~3*(10)^(8)*({0:m}NAME)/({0:sec}NAME) .EQN 0 12 0 0 {0:\g}NAME({0:v}NAME):(1)/(\(1-((({0:v}NAME)/({0:c}NAME)))^(2))) .EQN 1 20 0 0 {0:\L}NAME({0:v}NAME):({4,4}ö{0:\g}NAME({0:v}NAME)ö0ö0ö-{0:\g}NAME({0:v}NAME)*({0:v}NAME)/({0:c}NAME)ö0ö1ö0ö0ö0ö0ö1ö0ö-{0:\g}NAME({0:v}NAME)*({0:v}NAME)/({0:c}NAME)ö0ö0ö{0:\g}NAME({0:v}NAME)) .TXT 11 -32 0 0 C x1,1,0,0 .TXT 4 0 0 0 Cg a73.000000,73.000000,306 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain { The Lorentz transform expresses the coordinates of frame B in terms of frame A. The velocity between the two frames is v, and it is directed along the X axis of frame A. This greatly simplifies the components of the transform. One can always rotate your spacial coordinate axies to make this so.}} } .TXT 9 27 0 0 Cg a46.000000,46.000000,141 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {With symbolic mathematics, the space-time coordinates of frame B moving at velocity v along the x axis of frame A are calculated below:}} } .EQN 5 -27 0 0 {10:K.a}NAME÷({4,1}ö{0:ct.a}NAMEö{0:z.a}NAMEö{0:y.a}NAMEö{0:x.a}NAME) .EQN 0 12 0 0 {10:K.b}NAME÷{10:\L}NAME({0:v}NAME)*{10:K.a}NAME .EQN 19 -12 0 0 ({4,4}ö{0:\g}NAME({0:v}NAME)ö0ö0ö-{0:\g}NAME({0:v}NAME)*({0:v}NAME)/({0:c}NAME)ö0ö1ö0ö0ö0ö0ö1ö0ö-{0:\g}NAME({0:v}NAME)*({0:v}NAME)/({0:c}NAME)ö0ö0ö{0:\g}NAME({0:v}NAME))*({4,1}ö{0:ct.a}NAMEö{0:z.a}NAMEö{0:y.a}NAMEö{0:x.a}NAME)÷({4,1}ö-{0:\g}NAME({0:v}NAME )*({0:v}NAME)/({0:c}NAME)*{0:x.a}NAME+{0:\g}NAME({0:v}NAME)*{0:ct.a}NAMEö{0:z.a}NAMEö{0:y.a}NAMEö{0:\g}NAME({0:v}NAME)*{0:x.a}NAME-{0:\g}NAME({0:v}NAME)*({0:v}NAME)/({0:c}NAME)*{0:ct.a}NAME)÷({4,1}ö{0:ct.b}NAMEö{0:z.b}NAMEö{0:y.b}NAMEö{0:x.b}NAME) .TXT 17 30 0 0 Cg a43.333333,43.333333,256 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {To the left is the}{ inverse Lorentz transform from frame B to frame A. Notice that }{\f1 L}{\f0 (-}{\f0 v) is the inverse transform of}{\f1 L}{\f0 (v). }{This is the same behavior that the rotational transformations in Euclidian 3 space exhibit.}{ }} } .EQN 6 -30 0 0 {0:I\L}NAME({0:v}NAME):({4,4}ö{0:\g}NAME({0:v}NAME)ö0ö0ö{0:\g}NAME({0:v}NAME)*({0:v}NAME)/({0:c}NAME)ö0ö1ö0ö0ö0ö0ö1ö0ö{0:\g}NAME({0:v}NAME)*({0:v}NAME)/({0:c}NAME)ö0ö0ö{0:\g}NAME({0:v}NAME)) .TXT 11 0 0 0 C x1,1,0,0 .TXT 4 0 0 0 Cg a73.000000,73.000000,19 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {For example let:}} } .EQN 0 12 0 0 {0:v}NAME:(.5*{0:c}NAME) .TXT 2 -12 0 0 Cg a64.000000,64.000000,94 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Frame B is moving with a relative velocity of v with respect to the plus X axis of frame A }} } .EQN 4 0 0 0 {0:\g}NAME({0:v}NAME)=?_n_u_l_l_ .EQN 7 0 0 0 {0:\L}NAME({0:v}NAME)=?_n_u_l_l_ .EQN 0 27 0 0 {0:I\L}NAME({0:v}NAME)=?_n_u_l_l_ .TXT 8 -27 0 0 Cg a76.166667,76.166667,98 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Let's set up a space-time grid to plot and explore the effects of the Lorentz transform on it.}} } .EQN 8 0 0 0 {0:K.a}NAME:({4,10}ö-1ö0ö0ö0ö1ö0ö0ö0ö1ö0ö0ö1ö0ö0ö0ö1ö0ö0ö0ö-1ö-1ö0ö0ö-1ö-1ö0ö0ö1ö1ö0ö0ö1ö1ö0ö0ö-1ö-1ö0ö0ö-1) .EQN 0 40 0 0 {0:i}NAME:0;{0:cols}NAME({0:K.a}NAME)-1 .TXT 9 -40 0 0 Cg a73.000000,73.000000,387 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Shown below is the space-time grid from frame A, the observer's frame and the space-time grid from frame B }{mapped by the inverse }{Lorentz transform back into the observer's frame. The solid line}{s are the space-time grid of frame A, the observer. The dashed lines are the space-time grid of frame B, moving at .5*c, mapped by the inverse transform into frame A's coordinates.}} } .EQN 10 9 0 0 ({0:K.b}NAME){52}({0:i}NAME):{0:I\L}NAME({0:v}NAME)*({0:K.a}NAME){52}({0:i}NAME) .EQN 0 22 0 0 {0:K.c}NAME:{0:I\L}NAME({0:v}NAME)*(({1,4}ö1ö0ö0ö1)){51} .EQN 0 23 0 0 ({0:K.c}NAME){51}=?_n_u_l_l_ .EQN 2 -54 0 0 &&((({0:K.a}NAME){52}({0:i}NAME)))[(3),((({0:K.b}NAME){52}({0:i}NAME)))[(3),(({0:K.c}NAME))[(3)@&&((({0:K.a}NAME){52}({0:i}NAME)))[(0),((({0:K.b}NAME){52}({0:i}NAME)))[(0),(({0:K.c}NAME))[(0) 0 0 1 0 0 0 0 1 0 0 0 1 6 0 Frame A 0 4 6 0 Frame B 2 0 5 0 NO-TRACE-STRING 0 4 3 0 NO-TRACE-STRING 0 1 4 0 NO-TRACE-STRING 0 2 5 0 NO-TRACE-STRING 0 3 6 0 NO-TRACE-STRING 0 4 0 0 NO-TRACE-STRING 0 1 1 0 NO-TRACE-STRING 0 2 2 0 NO-TRACE-STRING 0 3 3 0 NO-TRACE-STRING 0 4 4 0 NO-TRACE-STRING 0 1 5 0 NO-TRACE-STRING 0 2 6 0 NO-TRACE-STRING 0 3 0 0 NO-TRACE-STRING 0 4 1 0 NO-TRACE-STRING 0 1 46 39 .TXT 3 64 0 0 Cg a18.333333,18.333333,269 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {In the space-time plot to the left, the positive ct axis of frame A , the observer's frame, is directed upward. The positive x axis of frame A is directed to the right.\par \par The grid of frame}{B is}{ transformed into}{the}{ tilted parallelogram, in frame A. }} } .TXT 28 0 0 0 Cg a9.000000,9.000000,8 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Event}} } .EQN 0 5 0 0 (({1,4}ö1ö0ö0ö1)){51} .TXT 2 -5 0 0 Cg a21.333333,21.333333,29 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {In frame B transforms into}} } .EQN 3 0 0 0 (({1,4}ö1.732ö0ö0ö1.732)){51} .TXT 3 0 0 0 Cg a19.833333,19.833333,103 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {in frame A. Therefore, what took 1 second in frame B, takes 1.7 seconds in the observer's frame.}} } .TXT 12 -64 0 0 Cg a73.000000,73.000000,743 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain { From B's point of view frame A is moving in the -x direction at .5*c, and through the Lorentz transform frame A's time appears to slow down by the same amount to B. }{T}{his is the twin paradox}{ of special relastivity. The answer to this apparent contradiction comes from general relativity. In flat space-time if both twins continue their unaccelerated motion they will never meet again and it doesn’t matter that each twin observes the other twin's time is running slower. If one of the twins undergoes acceleration ( curving space-time) to return to his brother, he will be the younger one. Note: from ds^2=-dx^2-dy^2-dz^2+(c*dt)^2, the total space-time interval for the stationary twin is greater than the accelerated one. }} } .TXT 17 0 0 0 C x1,1,0,0 .TXT 5 0 0 0 Cg a73.000000,73.000000,153 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Now we will verify the tensor nature of the metric of Special Relativity under the Lorentz transformation. First we define a space-time interval as:}} } .EQN 6 0 0 0 {0:xa}NAME:(({1,4}ö0ö0ö0ö1)){51}-(({1,4}ö0ö0ö0ö0)){51} .TXT 3 0 0 0 Cg a73.000000,73.000000,50 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {The velocity of frame B relative to frame A is:}} } .EQN 0 33 0 0 {0:v}NAME=?{0:c}NAME .TXT 3 -33 0 0 Cg a73.000000,73.000000,42 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Given a space time event in frame A as }} } .EQN 0 31 0 0 {0:ka}NAME:(({1,4}ö0ö0ö0ö1)){51} .TXT 0 16 0 0 Cg a26.000000,26.000000,5 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {, }} } .TXT 3 -47 0 0 Cg a73.000000,73.000000,72 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {the event ka in frame A will transform with the Lorentz transform to:}} } .EQN 4 0 0 0 {0:kb}NAME:{0:\L}NAME({0:v}NAME)*{0:ka}NAME .TXT 0 12 0 0 Cg a61.000000,61.000000,7 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {and }} } .EQN 0 4 0 0 ({0:kb}NAME){51}=?_n_u_l_l_ .TXT 0 24 0 0 Cg a32.000000,32.000000,14 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {in frame B.}} } .EQN 9 -40 0 0 {10:A}NAME=?_n_u_l_l_ .TXT 0 19 0 0 Cg a54.000000,54.000000,138 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {To the left is the metric tensor of Special Relativeity defined for frame A. The calculated space-time interval for xa is shown below:}} } .EQN 5 0 0 0 {10:A}NAME*{0:xa}NAME*{0:xa}NAME=?_n_u_l_l_ .TXT 6 -19 0 0 Cg a73.000000,73.000000,168 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Next the space-time interval xa will be transformed into xb of frame B and the interval is still equal to -1. This is the invariant of }{Minkowskian}{ space-time. }} } .EQN 5 0 0 0 {0:note}NAME .EQN 0 4 0 0 {0:xb}NAME:{0:\L}NAME({0:v}NAME)*{0:xa}NAME .EQN 0 12 0 0 {0:and}NAME .EQN 0 4 0 0 ({0:xb}NAME){51}=?_n_u_l_l_ .EQN 3 -20 0 0 {10:A}NAME*({0:\L}NAME({0:v}NAME)*{0:xa}NAME)*({0:\L}NAME({0:v}NAME)*{0:xa}NAME)=?_n_u_l_l_ .TXT 3 0 0 0 Cg a73.000000,73.000000,54 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {A second order tensor is defined to transform as: }} } .EQN 0 38 0 0 ({0:t}NAME)[({0:i1}NAME,{0:j1}NAME)÷({0:j}NAME÷0,3)$({0:i}NAME÷0,3)$({0:\L}NAME)[({0:i1}NAME,{0:i}NAME)*({0:\L}NAME)[({0:j1}NAME,{0:j}NAME)*({0:t}NAME)[({0:i}NAME,{0:j}NAME) .TXT 6 -38 0 0 Cg a73.000000,73.000000,164 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {The above must be true if t is a real tensor. Using matrix multiplication to transform the metric from the first frame of reference to the second frame we have:}} } .EQN 5 0 0 0 {10:B}NAME:({0:\L}NAME({0:v}NAME)){51}*{10:A}NAME*{0:\L}NAME({0:v}NAME) .TXT 5 19 0 0 Cg a54.000000,54.000000,261 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {The metric tensor has the same components in frame B as it had in frame A. Also, the space time interval is invariant, as expected. This proves that A and B are both valid representations of the metric tensor for 4 dimensional }{Minkowski}{ space-time. }} } .EQN 5 -19 0 0 {10:B}NAME=?_n_u_l_l_ .EQN 4 18 0 0 {10:B}NAME*{0:xb}NAME*{0:xb}NAME=?_n_u_l_l_ .TXT 4 -18 0 0 Cg a73.000000,73.000000,149 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {--------------------------------------------------------------------------------------------------------------------------------------------------}} } .TXT 2 0 0 0 Cg a75.500000,75.500000,221 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Now we will examine some of the applications of the metric tensor in General Relativity. These applications deal with accelerated frames of reference and gravitational fields. In special relativity the metric was: }} } .EQN 7 0 0 0 ({0:ds}NAME)^(2)÷{10:A}NAME*{10:dk}NAME*{10:dk}NAME÷-({0:dx}NAME)^(2)-({0:dy}NAME)^(2)-({0:dz}NAME)^(2)+({0:c}NAME)^(2)*({0:dt}NAME)^(2) .TXT 0 33 0 0 Cg a40.000000,40.000000,32 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {( a flat orthagonal space ) }} } .TXT 4 -33 0 0 Cg a73.000000,73.000000,78 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {The metric needed for four dimensional space-time of general relativity is:}} } .EQN 3 0 0 0 {0:i}NAME:0;3 .EQN 0 9 0 0 {0:j}NAME:0;3 .EQN 4 -9 0 0 ({0:ds}NAME)^(2)÷{0:i}NAME${0:j}NAME$({0:A}NAME)[({0:i}NAME,{0:j}NAME)*({0:dx}NAME)[({0:i}NAME)*({0:dx}NAME)[({0:j}NAME) .TXT 6 0 0 0 Cg a35.833333,35.833333,92 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Where geometrically }{the components of the metric tensor in 4' space would be given by:}} } .EQN 0 34 0 0 ({0:A}NAME)[({0:i}NAME,{0:j}NAME)÷({0:k}NAME÷0;3)${0:D}NAME(({0:f}NAME)[({0:k}NAME),({0:c}NAME)[({0:i}NAME))*{0:D}NAME(({0:f}NAME)[({0:k}NAME),({0:c}NAME)[({0:j}NAME)) .TXT 6 -34 0 0 Cg a73.000000,73.000000,268 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Again D is the partial differential operator that differentiates f}{\dn6 k}{ with respect to c}{\dn6 j}{. F}{\dn6 k}{ is the locally flat coordinate in the neighborhood point }{\b p}{ and c}{\dn6 j}{ is one of the curvilinear coordinate of the space we are in.}{ }} } .TXT 8 0 0 0 C x1,1,0,0 .TXT 3 0 0 0 Cg a73.000000,73.000000,522 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {When using the above definition of a metric tensor to generate the metric of General Relativity, one must be careful of the sign of }{\b A}{\dn6 3,3}{. since this metric must reduce to the metric of Special Relativity in flat space-time far away from any mass-energy.}{ We must change the sign of}{\b A}{\dn6 3,3}{ to be the opposite of the signs of }{\b A}{\dn6 0,0}{, }{\b A}{\dn6 1,1}{, and }{\b A}{\dn6 2,2}{ }{. The reason for this is space-time has three }{spaciall dimensions and one temporal dimension.}} } .TXT 15 0 0 0 Cg a73.000000,73.000000,1579 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Before going into the details of Einstein's equations and General Relativity, we will introduce an important concept. This is the geodesic line between two space-time events ( points in 4-space). The simple definition of the geodesic is the shortest path ( straightest line) connecting two fixed points. It would be more general to say that the geodesic is the extremal line connecting the two points. If you are in a space with a positive Gaussian curvature, like a spherical surface, then the geodesic will be a great circle. If you travel the great circle in one direction it will be the shortest distance between point A and point B. If you travel in the opposite direction, from point A on the great circle it will be the longest possible distance between A and B. The physical meaning of the geodesic line is "the world line of a free particle", which means the path in space-time taken by a particle on which no forces are acting except for the geometrical constraints from the curvature of that space-time. An object in free fall follows such a path through space-time. It can be used to physically map the curvature of space-time (i.e. gravitational fields).}{\par \par Before investigating the geodesics of space-time in a gravitational field, "proper time" will now be defined. "Proper time" is the time measured by an observer at rest relative to what he's observing. In other words, when the astronaut in a space capsule looks at his watch, it shows him his proper time in his frame of reference. From special relativity the interval between events is:}} } .EQN 33 0 0 0 ({0:ds}NAME)^(2)÷{10:A}NAME*{10:dk}NAME*{10:dk}NAME÷-({0:dx}NAME)^(2)-({0:dy}NAME)^(2)-({0:dz}NAME)^(2)+({0:c}NAME)^(2)*({0:dt}NAME)^(2) .TXT 3 0 0 0 Cg a73.000000,73.000000,93 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {In a rest frame, the interval , ds along the object's trajectory in space-time reduces to:}} } .EQN 0 63 0 0 ({0:ds}NAME)^(2)÷({0:c}NAME)^(2)*({0:dt}NAME)^(2) .TXT 2 -63 0 0 Cg a73.000000,73.000000,283 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {All the space-like changes are zero. Therefore, the interval along an object's trajectory in its rest system is proportional to its proper time. Proper time is a useful way to parametrize the trajectory of the object. Proper time is usually assigned the symbol}{ "}{\f1 t}{". }} } .TXT 7 0 0 0 Cg a73.000000,73.000000,10 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain { Hence:}} } .EQN 0 8 0 0 {0:ds}NAME÷{0:c}NAME*{0:dt.0}NAME÷{0:c}NAME*{0:d\t}NAME .TXT 0 14 0 0 Cg a45.666667,45.666667,55 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Note: rest system coordinates are usually shown as: }} } .EQN 0 40 0 0 ({1,4}ö{0:t.0}NAMEö{0:z.0}NAMEö{0:y.0}NAMEö{0:x.0}NAME) .TXT 6 -62 0 0 Cg a73.000000,73.000000,65 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {In General Relativity, the space-time interval is defined as: }} } .EQN 0 43 0 0 ({0:ds}NAME)^(2)÷{0:i}NAME${0:j}NAME$({0:A}NAME)[({0:i}NAME,{0:j}NAME)*({0:dx}NAME)[({0:i}NAME)*({0:dx}NAME)[({0:j}NAME) .TXT 7 -43 0 0 Cg a73.000000,73.000000,66 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {The finite interval between two given events is calcutated by:}} } .EQN 6 0 0 0 ({0:s}NAME)^(2)÷({0:\t.0}NAME&{0:\t.1}NAME`((({0:ds}NAME)/({0:dt}NAME)))^(2)&{0:\t}NAME)÷({0:\t.0}NAME&{0:\t.1}NAME`{0:i}NAME${0:j}NAME$({0:A}NAME)[({0:i}NAME,{0:j}NAME)*(({0:dx}NAME)[({0:i}NAME))/({0:d\t}NAME)*(({0:dx}NAME)[({0:j}NAME))/({0:d\t}NAME)& {0:\t}NAME) .TXT 9 0 0 0 Cg a73.000000,73.000000,378 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {To find the geodesics of the local space-time, we must evaluate the above integral over all possible paths while keeping}{\f1 t}{\dn6 1}{ and}{\f1 t}{\dn6 0}{ constant, and then pick the path ( trajectory ) that makes the value of the integral extremal. This is a classic problem of Calculus of Variations, see (7) and (8) for more information on Calculus of Variations.}} } .TXT 11 29 0 0 Cg a44.000000,44.000000,276 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Much like the way you find a minimum or maximum of a function where you set the first derivative to zero, the variation (of path)of the integral must be equal zero for a geodesic path. The methods of Calculus of Variation lead to the equation shown on the next page. }} } .EQN 4 -29 0 0 {0:\d}NAME(({0:s}NAME)^(2))÷{0:\d}NAME(({0:\t.0}NAME&{0:\t.1}NAME`({0:A}NAME)[({0:i}NAME,{0:j}NAME)*(({0:dx}NAME)[({0:i}NAME))/({0:d\t}NAME)*(({0:dx}NAME)[({0:j}NAME))/({0:d\t}NAME)&{0:\t}NAME))÷0 .TXT 11 0 0 0 C x1,1,0,0 .TXT 3 0 0 0 Cg a75.333333,75.333333,658 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {From the use of Calculus of variation we obtain the following equation(s) for the geodesics in four dimensional space-time. What does this tell us? The first term in this equation is just the acceleration of a particle along the i'th coordinate of space-time. The second term containing the Christoffel symbol is made up out of a combination of components of the metric tensor and it's derivatives with respect to the coordinates. }{ If you are in a flat space ( no gravitational fields ) the value of the Christoffel symbol will be zero, since the derivatives of the components of the metric tensor with respect to their coordinates become zero.}} } .TXT 15 34 0 0 Cg a37.000000,37.000000,108 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {In this case the equations of the geodesic reduce to Newton's laws of motion for a free particle,where:}} } .EQN 1 -34 0 0 {0:\t}NAME"{0:\t}NAME"({0:x}NAME)[({0:i}NAME)+{0:j}NAME${0:k}NAME${0:\G}NAME({0:i}NAME,{0:j}NAME,{0:k}NAME)*{0:\t}NAME"({0:x}NAME)[({0:j}NAME)*{0:\t}NAME"({0:x}NAME)[({0:k}NAME)÷0 .EQN 7 34 0 0 {0:\t}NAME"{0:\t}NAME"({0:x}NAME)[({0:i}NAME)÷0 .TXT 0 10 0 0 Cg a29.000000,29.000000,101 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {When the acceleration vanishes the object will move at constant velocity along a straight line.}} } .TXT 6 -44 0 0 Cg a73.000000,73.000000,749 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {If the space-time we are exploring is curved ( i.e. has a gravitational field ), the components of the metric tensor will be functions of the coordinates, and their derivatives with respect to the coordinates will not vanish. Hence the second term in the geodesic equation is non-zero, and produces an acceleration which causes the test object to follow the curvature of the space. The second term can be interpreted as the local acceleration from gravity. Since the mass of the test object doesn't enter into the geodesic equations, the acceleration due to gravity is independent of the object's mass. Below is the definition of the Christoffel symbol of the second type that is used in the geodesic equations of General Relativity.}{ }} } .EQN 22 0 0 0 {0:\G}NAME({0:i}NAME,{0:j}NAME,{0:k}NAME)÷(1)/(2)*({0:m}NAME÷0;3)$((({0:A}NAME)^(-1)))[({0:i}NAME,{0:m}NAME)*(({0:x}NAME)[({0:k}NAME)"({0:A}NAME)[({0:j}NAME,{0:m}NAME)+({0:x}NAME)[({0:j}NAME)"({0:A}NAME)[({0:m}NAME,{0:k}NAME)-({0:x}NAME)[({0:m}NAME)"( {0:A}NAME)[({0:j}NAME,{0:k}NAME)) .TXT 8 0 0 0 Cg a73.000000,73.000000,416 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Now the question is how do we find the metric tensor for a region of space-time? Einstein's field equations are used for this purpose. These equations describe the connection between the distribution of matter and energy in space and the curvature of that space, i.e., how the matter / energy distribution generates a space's properties. These equations can be expressed in one tensor equation shown below. }} } .EQN 11 0 0 0 ({10:G}NAME)[({0:i}NAME,{0:j}NAME)÷{0:J}NAME*({10:T}NAME)[({0:i}NAME,{0:j}NAME) .TXT 0 11 0 0 Cg a62.000000,62.000000,154 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Where G is Einstein's tensor, whose components are composed of the metric tensor and its first and second derivatives with respect to the coordinates.}} } .TXT 5 0 0 0 Cg a67.166667,67.166667,98 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {J is a constant where g is the constant of gravitation and c is the speed of light in a vacuum.}} } .EQN 1 -11 0 0 {0:J}NAME÷(-4*{0:\pg}NAME)/(({0:c}NAME)^(2)) .TXT 6 0 0 0 Cg a73.000000,73.000000,960 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {\b T}{ is the energy-momentum tensor. The components of }{\b T}{ include the mass-energy density, the momentum density, and the components of the shear tensor. \par \par This tensor equation relates the curvature of space-time to the distribution of matter and energy in it. In other words, mass-energy tells space-time how to bend and space-time tells mass-energy how to move. Both the G and T are second order tensors in four dimensional space-time, therefore they have 16 components and the same symmetry as the metric tensor. If there wasn't this symmetry , Einstein's equation would form a system of 16 equations. Due to the symmetry we only have 10 separate equations to consider. Unfortunately, this is a non-linear system of differential equations of which there is no known general solution. Einstein's equation has only been solved in simple systems, with high symmetry, further simplifying the equation and reducing the independent unknowns.}} } .TXT 34 0 0 0 C x1,1,0,0 .TXT 3 0 0 0 Cg a72.833333,72.833333,1015 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {In closing the author would like to clarify the distinction between flat and curved spaces and rectilinear and curvilinear coordinate systems. A flat space is described by Euclidian geometry\par and is the only space that rectilinear coordinates will exactly fit on a global scale. The prime example of this is the Cartesian coordinates we were taught when we first were learning geometry is school. Curvilinear coordinates will also fit a flat space exactly, such as, two dimensional polar coordinates on a plain. In a curved space such as the surface of the Earth, we must use a curvilinear coordinate system, like latitude and longitude to fit the space globally. Also, the parallel postulate of Euclidian geometry is not satisfied in a curved space. Note: if you are not at a space-time singularity (in a black hole, which would be bad) or at a cusp in a geometric curved space, you can always use a rectilinear coordinate system in the neighborhood of the point of interest to fit the space locally.}} } .TXT 20 0 0 0 Cg a73.000000,73.000000,273 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Use was made of Mathcad's order of operations when using matrix multiplication to apply the metric tensor,}{\b A}{, twice to the differential position vector to calculate the square of the differential distance between the position vectors }{\b x}{ and}{\b x+dx}{. }} } .EQN 9 0 0 0 {10:A}NAME*{10:dx}NAME*{10:dx}NAME÷({10:A}NAME*{10:dx}NAME)*{10:dx}NAME÷{0:i}NAME${0:j}NAME$({0:A}NAME)[({0:i}NAME,{0:j}NAME)*({0:dx}NAME)[({0:i}NAME)*({0:dx}NAME)[({0:j}NAME)÷({0:ds}NAME)^(2) .TXT 0 39 0 0 Cg a34.000000,34.000000,16 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {not equal to:}} } .EQN 0 11 0 0 {10:A}NAME*({10:dx}NAME*{10:dx}NAME) .TXT 7 -50 0 0 Cg a73.000000,73.000000,14 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {Note that: }} } .EQN 0 8 0 0 ({0:ds}NAME)^(2)÷({10:dx}NAME){51}*{10:A}NAME*{10:dx}NAME .TXT 0 13 0 0 Cg a52.000000,52.000000,20 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {is always true}{.}} } .TXT 5 -21 0 0 Cg a73.000000,73.000000,149 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {--------------------------------------------------------------------------------------------------------------------------------------------------}} } .TXT 4 0 0 0 Cg a80.333333,80.333333,16 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {\b References}} } .TXT 4 0 0 0 Cg a86.666667,86.666667,898 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Helvetica;} {\f1\fnil Symbol;} } {\plain {\par 1. M.}{A. Akivis & V.V. Goldberg, "An Introduction to linear Algebra & Tensors", Dover Pubications,\par Mineola, N.Y., 1977\par 2. A.I.Borisenko & I.E. Tarapov, "Vector and Tensor Analysis with Applications", }{Dover Pubications,\par Mineola, N.Y., 1979\par 3}{. Richard Shiffman, LatLong to Distance 11, Mathcad Document, 1994,\par (URL:= http://www.mathsoft.com/pub/apps/navigate.mcd )\par 4. Amos Harpaz, " Relativity Theory: Concepts and Basic Principles", A.K. Peters, Ltd., 1993\par 5. }{Richard Shiffman, "The Cross Product, Dot Product and fun with Tensors", 1995,\par (URL:= http://www.mathsoft.com/pub/apps/crossdot.mcd )\par 6. }{Alfred Gray, Modern Differential Geometry of Curves and Surfaces, CRC Press, 1993}{\par 7. C. Fox, "An Introduction to the Calculus of Variation",Dover Pubications, 1987\par 8. R. Feynman, "Lectureson Physics", Addison-Wesley, Vol. II, Chapter 19, 1963}{\par }} }