Properties of Different Types of Lorentz Transformations
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1 merin Journl of Mthemtis nd ttistis 03 3(3: 05-3 DOI: 0593/jjms roperties of Different Types of Lorentz Trnsformtions tikur Rhmn izid * Md hh lm Deprtment of usiness dministrtion Leding niversity ylhet 300 ngldesh Deprtment of hysis hhjll niversity of iene nd Tehnology ylhet 3 ngldesh bstrt It is obvious tht Lorentz trnsformtion is the strting point of Reltivisti mehnis There re different types of Lorentz trnsformtions suh s peil Most generl Mixed number Geometri produt nd Quternion Lorentz trnsformtions To study reltivisti mehnis we must need to know the properties of different types of Lorentz trnsformtions In this pper we hve studied reiprol property ssoitive property isotropi property nd group property of the bove Lorentz trnsformtions Keywords peil Lorentz Trnsformtion Most Generl Lorentz Trnsformtion M ixed Nu mbe r Lo rentz Trnsformtion Geometri rodut Lo rentz Trnsformtion Reiprol roperty ssoitive roperty Isotropi roperty Group roperty Introdution In most tretments on speil reltivity the line of motion is ligned with the x-xis Th is is nturl hoie beuse in suh sitution the ( y z oordintes re invrint under the Lorentz trnsformtions Ho wever; it is of interest to study the se when the line of motion does not oinide with ny of the oordinte xes rtil instnes of suh sitution re n irplne during lnding or tke off The ground t the irfield hs nturl oordinte system with the x-xis prllel to the ground wheres the irplne sends or desends t n ngle with ground For this reson we need to study the properties of different types Lorentz trnsformtions where the line of tion is long x-xis s well s long ny rbitrry line e hve studied the Reiprol property ssoitive property Isotropi property Group property of different types of Lorentz trnsformtions peil Lorentz Trnsformtion onsider two inertil frmes of Referene nd where the frme is t rest nd the frme is moving long X-xis with veloity with respet to frme The spe nd time oordintes of nd re (x y z t nd (x y z t respetively The reltion between the oordintes of nd is lled peil Lorentz trnsformtion whih n be written s [] * orresponding uthor: susttik@gmilom (tikur Rhmn izid ublished online t opyright 03 ientifi & demi ublishing ll Rights Reserved x t x t x y y z z t ( nd x t x t x y y z z t ( Most Generl Lorentz hen the veloity of with respet to is not long X-xis ie the veloity hs three omponents x y nd z Then the reltion between the oordintes of nd is lled Most generl Lorentz trnsformtion whih n be written s [] X X X ( t (3 t ( t X nd X X X ( t ( t ( t X here in unit of
2 06 tikur Rhmn izid et l: roperties of Different Types of Lorentz Trnsformtions 3 Mixed Number Lorentz Trnsformtion In the se of Most Generl Lorentz trnsformtion the veloity of ' with respet to is not long X-xis; ie the veloity hs three omponents x y nd z Let in this se Z nd Z' be the spe prts in nd ' frmes respetively Then using the mixed produt i M ixed number Lorentz trnsformtions [3 6] n be written s t ( t Z Z ( Z t iz nd t ( t Z Z ( Z t iz Geometri rodut Lorentz Trnsformtion In this se the veloity of ' with respet to lso hs three omponents x y nd z s the Most generl Lorentz trnsformtion Let in this se Z (5 (6 nd be the spe prts in nd ' frmes respetively Then using geometri produt of two vetors the geometri produt Lorentz trnsformtion [7 8] n be written s t ( t Z Z ( Z t Z nd t ( t Z Z ( Z t Z 5 Quternion Lorentz Trnsformtion In this se the veloity of ' with respet to hs lso three omponents x y nd z s the Most generl Lorentz trnsformtion Let in this se Z (7 (8 nd be the spe prts in nd ' frmes respetively Then using uternion produt the Quternion Lorentz trnsformtion [9-] n be written s t ( t Z (9 Z ( Z t Z nd t ( t Z (0 Z ( Z t Z Reiprol roperty of Different Types of Lorentz Trnsformtions Z Z Reiprol roperty of peil Lorentz The veloity ddition formul for speil Lorentz trnsformtion [3] n be ritten s in unit of ( If we reple by where then will be hnge to Reiprol property demnds tht if Now then ( onseuently speil Lorentz trnsformtion stisfies the reiprol property Reiprol roperty of Most Generl Lorentz From the trnsformtion eutions of ddition of veloities of most generl Lorentz trnsformtion [] we hve ( in unit of (3 If we reple by where then will be hnge to where Now where ( Reiprol property demnds tht if then ( ( (
3 merin Journl of Mthemtis nd ttistis 03 3(3: n be written s o ( onseuently the most generl Lorentz trnsformtion does not stisfy the reiprol property 3 Reiprol roperty of Mi xe d Number Lorentz From the trnsformtion eutions of ddition of veloities of mixed number Lorentz trnsformtion [] we hve (5 If we reple by where then will be hnge to where Reiprol property demnds tht if then Now n be written s o (6 imilrly If we reple by where then will be hnge to where onseuently Mixed number Lorentz trnsformtion stisfies the reiprol property Reiprol roperty of Geometri rodut Lorentz From the trnsformtion eutions of ddition of veloities of geometri produt Lorentz trnsformtion [] we hve (7 If we reple by where then will be hnge to where Reiprol property demnds tht if then Now n be written s o (8 onseuently geometri produt Lorentz trnsformtion does not stisfy the reiprol property 5 Reiprol roperty of Quternion Lorentz From the trnsformtion eutions of ddition of veloities of Quternion Lorentz trnsformtion [] we hve (9 If we reple by where then will be hnge to where Reiprol property demnds tht if then Now i i i i i i i i Q Q
4 08 tikur Rhmn izid et l: roperties of Different Types of Lorentz Trnsformtions n be written s ( ( ( ( ( ( ( ( o (0 onseuently the Quternion Lorentz trnsformtion does not stisfy the reiprol property 3 ssoitive roperty of Different Types of Lorentz Trnsformtions onsider three inertil frmes of referene nd where the frme is t rest nd the frme is moving with veloity with respet to is moving with veloity with respet to is moving with veloity respet to then ssoitive property sys tht ( ( [Fig-] e re going to disuss the ssoitive property of different Lorentz trnsformtions in unit of Let most M g re the symbols of the Lorentz sum of peil Most generl Mixed number geometri produt nd Quternion produt Lorentz trnsformtions respetively 3 ssoitive roperty of peil Lorentz The veloity ddition formul for speil Lorentz trnsformtion [3] n be written s ( Now if moves with veloity with respet to then ording to the veloity ddition formul for speil Lorentz trnsformtion [3] n be written s ( ( gin Let moves with veloity with respet to nd moves with veloity with respet to then ording to the veloity ddition formul for the peil Lorentz trnsformtion [3] the resultnt veloity of nd n be written s (3 ( ( ( ( Fi gure ssoitive roperty of Lorentz trnsformtions
5 merin Journl of Mthemtis nd ttistis 03 3(3: Finlly let moves with veloity with respet to nd moves with veloity with respet to then the resultnt veloity [3] of nd n be written s ( ( Hene ( ( (5 onseuently peil Lorentz trnsformtion stisfies the ssoitive property 3 ssoiti ve roperty of Most Generl Lorentz The veloity ddition formul for most generl Lorentz trnsformtion [] n be written s ( ( (6 most ( Let us onsider moves with veloity with respet to nd moves with veloity respet to then ording to the veloity ddition formul for most generl Lorentz trnsformtion [] n be written s ( most most most ( ( ( ubstituting the vlue of we hve ( ( ( ( ( ( ( (7 ( ( ( ( ( ( gin Let moves with veloity with respet to nd moves with veloity with respet to then ording to the veloity ddition formul for most generl Lorentz trnsformtion [] the resultnt veloity of nd n be written s ( ( (8 most ( Finlly let moves with veloity with respet to nd moves with veloity with respet to then the resultnt veloity [] of nd n be written s most ( most ( ( ( ubstituting the vlue of be written s ( ( ( ( ( ( ( ( ( most (9 the bove expression n (30 Hene ( ( most most most most onseuently the Most generl Lorentz trnsformtion does not stisfy the ssoitive property (3
6 0 tikur Rhmn izid et l: roperties of Different Types of Lorentz Trnsformtions 33 ssoiti ve roperty of Mixed Number Lorentz The veloity ddition formul for mixed Number Lorentz trnsformtion [] n be written s i (3 Now if moves with veloity with respet to nd moves with veloity respet to then ording to the veloity ddition formul for mixed number Lorentz trnsformtion [] n be written s M M i M i i (33 i i i { } ( i ( gin Let moves with veloity with respet to nd moves with veloity with respet to then ording to the veloity ddition formul for mixed number Lorentz trnsformtion [] the resultnt veloity of nd n be written s i M (3 Finlly let moves with veloity with respet to nd moves with veloity with respet to then using (3 the resultnt veloity [] of n be written s n be written s Hene nd (35 (36 onseuently mixed number Lorentz trnsformtion stisfies the ssoitive property 3 ssoiti ve roperty of Geometri rodut Lorentz The veloity ddition formul for geometri produt Lorentz trnsformtion [] n be written s (37 g Now if moves with veloity with respet to nd moves with veloity respet to then ording to the veloity ddition formul for geometri produt Lorentz trnsformtion [] n be written s gin Let moves with veloity with respet to nd moves with veloity with respet to then ording to the veloity ddition formul for the Geometri produt Lorentz trnsformtion [] the resultnt veloity of nd n be written s M ( M M i i i i i i { } ( ( { } ( ( i ( ( ( M M M M ( g g g (38
7 merin Journl of Mthemtis nd ttistis 03 3(3: 05-3 g (39 Finlly let moves with veloity with respet to nd moves with veloity with respet to then using (39 the resultnt veloity [] of nd n be written s g ( g g n be written s Hene ( ( g g ( onseuently geometri produt Lorentz trnsformtion does not stisfy the ssoitive property 35 ssoiti ve roperty of Quternion Lorentz The veloity ddition formul for Quternion Lorentz trnsformtion [] n be written s ( Now if moves with veloity with respet to nd moves with veloity respet to then ording to the veloity ddition formul for the Quternion Lorentz trnsformtion [] n be written s ( (3 { ( } ( gin Let moves with veloity with respet to nd moves with veloity with respet to then ording to the veloity ddition formul for the Quternion Lorentz trnsformtion [] the resultnt veloity of nd n be written s { } g g (0 ( Finlly let moves with veloity with respet to nd moves with veloity with respet to then using ( the resultnt veloity [] of nd n be written s ( (5 n be written s { } ( ( Hene ( ( (6 onseuently Quternion Lorentz trnsformtion does not stisfy the ssoitive property It n be esily shown tht the bove Lorentz Trnsformtions stisfy the ssoitive property if y0 nd z 0 reduing these to peil Lorentz Trnsformtion Isotropi roperty of Lorentz Trnsformtions onsider three inertil frmes of Referene nd where moves with veloity with
8 tikur Rhmn izid et l: roperties of Different Types of Lorentz Trnsformtions respet to nd moves with veloity with respet to If be the veloity of with respet to Figure Isotropi property of Lorentz Trnsformtions If b nd re the three ngles of the tringle then ording to Lorentz sum we n write (7 ( nd their produt 0 ( 80 0 b sin 80 n sin (8 (9 Now the isotropi property demnds tht if then b Isotropi roperty of peil Lorentz n option to mke tringle o isotropi property is not pplible for speil Lorentz trnsformtion Isotropi roperty of Most Generl Lorentz The veloity ddition formul for most generl Lorentz trnsformtion [] n be written s ( (50 sing eution (7 nd (50 we n write (tking ( ( ( ( ( ( ( For speil Lorentz trnsformtion there will be no or utting in eution (5 we get sing eution (8 nd (50 we n write (tking ( ( ( ( ( 0 ( ± ( ( ( ( ( ± ( ( ( ( (5 (5
9 merin Journl of Mthemtis nd ttistis 03 3(3: (53 utting in eution (53 we get (5 sing eution (9 nd (50 we n write (tking (55 or or or or 0 ± ± or or or b b b b or b b or b b or
10 tikur Rhmn izid et l: roperties of Different Types of Lorentz Trnsformtions utting in eution (55 we get b b b b ( ( ( ( ( b b 0 ( ± b b ( ± (56 Hene from eutions (5 (5 nd (56 we hve b Hene most generl Lorentz trnsformtion stisfies the isotropi property 3 Isotropi roperty of Mixed Number Lorentz The veloity ddition formul for mixed number Lorentz trnsformtion [] n be written s i (57 sing eution (7 nd (57 we n write (tking i i (58 ( i sin n ( sin ( utting in eution ( we get ( 0 ± ± (59 sing eution (8 nd (57 we n write (tking ( ( i ( isin n isin n utting in eution (60 we get sing eution (9 nd (57 we n write (tking sin 0 (60 ( 0 0 ± ± i( (6
11 merin Journl of Mthemtis nd ttistis 03 3(3: i sin b n b i sin b n ( b b sin b ( ( ( 0 b b utting in eution (6 we get b b ( b b 0 (6 (63 Hene from eutions (59 (6 nd (63 we hve b Hene Mixed number Lorentz trnsformtion stisfies the isotropi property Isotropi roperty of Geometri rodut Lorentz The veloity ddition formul for Geometri produt Lorentz trnsformtion [] n be written s (6 sing eution (7 nd (6 we n write (tking (65 ( sin n ( sin b b b b b 0 ± b b ± utting in eution (65 we get ( ( ( ( 0 (66 sing eution (8 nd (6 we n write (tking utting in eution (67 we get (67 sing eution (9 nd (6 we n write (tking ( ( 6 ± 3 6 ± 3 ( sin n sin n ( sin ( ( 0 ( 0 ( ( ( 6 ( ± 3 ( 6 ( ± 3 ( 0 (68
12 6 tikur Rhmn izid et l: roperties of Different Types of Lorentz Trnsformtions ( sin b n b sin b n b ( b sin b ( ( 0 b b utting in eution (69 we get b b (69 (70 Hene from eutions (66 (68 nd (70 we hve b Hene geometri produt Lorentz Trnsformtion stisfies the isotropi property 5 Isotropi roperty of Quternion Lorentz The veloity ddition formul for Quternion Lorentz trnsformtion [] n be written s (7 sing eution (7 nd (7 we n write (tking ( b b b ( b b 0 ( ( b ( 6 ( ± 3 b ( 6 ( ± 3 ( b b 0 sin n ( ( sin utting in eution (7 we get (7 (73 sing eution (8 nd (7 we n write (tking utting in eution (7 we get (7 (75 sing eution (9 nd (7 we n write (tking ( 0 ± ( ( ± sin n sin n sin ( ( 0 ( 0 0 ± ±
13 merin Journl of Mthemtis nd ttistis 03 3(3: ( sin b n b sin b n b ( b sin b ( ( 0 b b utting in eution (76 we get (76 (77 ± b Hene from eutions (73 (75 nd (77 we hve b Hene Quternion Lorentz trnsformtion stisfies the isotropi property 5 Group roperty of Lorentz Trnsformtions The result of two Lorentz trnsformtions is itself Lorentz trnsformtion [3] 5 Group roperty of peil Lorentz onsider three inertil frmes of referene nd where hs reltive veloity with respet to long positive x-xis nd hs reltive veloity with respet to long the sme diretion s shown in fig ording to speil Lorentz trnsformtion we get x t x y y z z β b b b b ( b b 0 b b 0 ± b x t t where β β b (78 Figure 3 peil Lorentz Trnsformtions imilrly if the origins in nd oinide t t t 0 relted s the spe nd time o-ordintes in uppose be the resultnt veloity of nd then from the veloity ddition formul [] we get Now x n be written s x t β x t t β y y z z where β nd (79 (80 is the veloity of system reltive to To prove this result we hve to show tht β xt x y y z z β x t t where β β (8
14 8 tikur Rhmn izid et l: roperties of Different Types of Lorentz Trnsformtions Now using (5 n be written s o β β or β β β x x t β β x x t β x t β ( β ( β x t β (8 gin from eution (8 n be written s x x t β y y y z z x t t β β & (83 Thus the result of two Lorentz trnsformtions is itself Lorentz trnsformtion Hene peil Lorentz trnsformtions form group 5 Group roperty of Most Generl Lorentz onsider three inertil frmes of Referene nd where moves with veloity with respet to nd moves with veloity with respet to z x t β x t t β ( x t β If be the veloity of with respet to Y Y Z Y X X Z X Z Figure Most Generl Lorent z T rnsformt ion
15 merin Journl of Mthemtis nd ttistis 03 3(3: o the reltion between the o-ordintes X t in nd ( X t in n be expressed by most generl Lorentz trnsformtion by X X X ( t (8 t tx where imilrly the reltion between the o-ordintes ( X t in nd ( X t in n be expressed by most generl Lorentz trnsformtion by X X X t (85 t t X imilrly the reltion between the o-ordintes nd X t in n be expressed by most generl Lorentz trnsformtion by X X X ( t (86 t tx From the veloity ddition formul for Most Generl Lorentz trnsformtion [] we n write Now we hve from (85 where ( where w ( ( ( in unit of in unit of ( X t in unit of in t ( t X t t ( t X t here Now we hve to show X ( t X X ( ( t X ( ( X ( ( ( ( ( { t X } ( ( ( ( t e know ( ( ( n be written s ( Hene the time prt of the Most Generl Lorentz trnsformtion does not stisfy the group property gin from eution (85
16 0 tikur Rhmn izid et l: roperties of Different Types of Lorentz Trnsformtions X X X t X X ( t X X ( t ( ( tx hih n be written s X t X Θ X here the opertor in generl is different from the unit opertor is the veloity of the system reltive to nd is the veloity of the system reltive to Hene we hve Θ e get ( ( ( ( ( ( nd Now X X Θ {( d X} (87 d The rottion opertor thus represents n infinitesiml rottion round the diretion of the vetor [3] Hene most generl Lorentz trnsformtion does not stisfy the group property without rottion 53 Group roperty of Mi xe d Number Lorentz Figure- desribes tht three inertil fr mes of referene with respet to nd moves with veloity ΘX X Ω X Ω d nd where moves with veloity Ω with respet to If be the veloity of with respet to ( Zt then the reltion between the o-ordintes in nd in n be expressed by mixed number Lorentz trnsformtion by Z ( Z t iz t ( t Z (88 imilrly the reltion between the o-ordintes ( Z t in nd in n be expressed by Mixed Number Lorentz trnsformtion by Z ( Z t iz t ( t Z imilrly the reltion between the o-ordintes (89 in nd in n be expressed by mixed Number Lorentz trnsformtion by Z ( Z t iz t ( t Z (90 The veloity ddition formul for mixed Number Lorentz trnsformtion [] n be written s i (9 Now from eution (89 we n write here ( Z t where gin from eution (9 we n write ( Z t where ( Z t where in in in unit unit unit t ( t Z i t ( t Z t ( ( t Z i of { ( t Z ( Z t iz } of of ( Zt (9
17 merin Journl of Mthemtis nd ttistis 03 3(3: 05-3 Z ( Z t iz ( Z t iz ( t Z i ( Z t iz ( i ( i Z ( Z t iz ( ( Z t iz here Now we hve to show tht Hene Therefore mixed Number Lorentz trnsformtion stisfies the group property 5 Group roperty of Geometri rodut Lorentz Figure- desribes tht three inertil frmes of Referene with respet to nd moves with veloity with respet to If be the veloity of with respet to { } ( i ( then the reltion between the o-ordintes in nd in n be expressed by geometri produt Lorentz trnsformtion by Z ( Z t Z (93 t ( t Z where in unit of imilrly the reltion between the o-ordintes in nd in n be expressed by geometri produt Lorentz trnsformtion by Z ( Z t Z t ( t Z imilrly the reltion between the o-ordintes ( ( i ( ( ( ( ( ( nd where moves with veloity ( Zt ( Z t ( Z t ( Z t where in unit of (9 ( Zt in nd in n be expressed by geometri rodut Lorentz trnsformtion by Z ( Z t Z t ( t Z (95 The veloity ddition formul for geometri produt Lorentz trnsformtion [] n be written s (96 Now from eution (9 we n write Or here gin from eution (9 we n write Z ( Z t Z ( Z t Z ( t Z ( Z t Z Z Z( t t t Z Z ( Z ( Z Z t Z here ( Z t Now we hve to show tht where in unit of t ( t Z { ( t Z ( Z t Z } ( ( t Z t { } ( ( Therefore Geometri rodut Lorentz trnsformtion does not stisfy the isotropi property 55 Group roperty of Quternion rodut Lorentz ( ( ( ( Figure- desribes tht three inertil frmes of Referene nd where
18 tikur Rhmn izid et l: roperties of Different Types of Lorentz Trnsformtions moves with veloity with respet to nd moves with veloity with respet to If ( Zt Z t in be the veloity of with respet to then the reltion between the o-ordintes in nd n be expressed by Quternion rodut Lorentz trnsformtion by Z ( Z t Z t ( t Z where (97 imilrly the reltion between the o-ordintes ( Z t ( Z t in nd in n be expressed by Quternion rodut Lorentz trnsformtion by Z ( Z t Z t ( t Z where imilrly the reltion between the o-ordintes (98 in nd in n be expressed by Quternion rodut Lorentz trnsformtion by Z ( Z t Z t ( t Z ( Z t in unit of unit (99 The veloity ddition formul for Quternion rodut Lorentz trnsformtion [] n be written s where in of in unit of ( Zt Now from eution (98 we n write here gin from eution (98 we n write ( Z t Z ( t Z ( Z t Z n be written s here Now we hve to show tht Or Z t ( t Z ( t Z ( Z t Z Or t ( ( t Z { } Z ( Z t Z ( Z t ( ( Z ( ( { Z t Z } (00 Therefore Quternion rodut Lorentz trnsformtion does not stisfy the group property ( ( ( ( ( ( ( T ble omprison of the properties of different Lorentz trnsformtions Nmes of Lorentz trnsformtions Reiprol roperty ssoitive ropert y Isotropi property Group roperty peil Lorentz trnsformtion t isfy tisfies Not pplible t isfy Most generl Lorent z trnsformt ion Does not tisfy Does not tisfy t isfy Does not tisfy Mixed number Lorentz trnsformtion t isfy tisfies t isfy t isfy Geometri produt Lorentz trnsformtion Does not tisfy Does not tisfy t isfy Does not t isfy Quternion Lorentz trnsformtion Does not tisfy Does not tisfy t isfy Does not tisfy
19 merin Journl of Mthemtis nd ttistis 03 3(3: onlusions e hve disussed different properties of different Lorentz trnsformtions nd obtined tht speil nd mixed number Lorentz trnsformtions stisfy the reiprol property Most generl geometri produt nd Quternion Lorentz trnsformtions do not stisfy the reiprol property peil nd mixed number Lorentz trnsformtions stisfy ssoitive property but the most generl geometri produt nd Quternion Lorentz trnsformtions do not stisfy the ssoitive property Isotropi property is not pplible for speil Lorentz trnsformtion Most generl mixed number geometri produt nd Quternion Lorentz trnsformtions stisfy the isotropi property peil nd mixed number Lorentz trnsformtions stisfy the group property Most generl geometri produt nd Quternion Lorentz trnsformtions do not stisfy the group property d of i 37(:9- [] lm M 00 Mixed produt of vetors Journl of Theoretis 3( [5] lm M 003 omprtive study of mixed produt nd uternion produt Indin J hysis 77: 7-9 [6] lm M 00 Mixed produt of vetors Journl of Theoretis 3( [7] Dtt K De bbt nd Ronhetti L 998 Quntiztion of grvity in rel spe time Il Nuovo imento 3 [8] Dtt K Dtt R 998 Einstein field eutions in spinor formlism Foundtions of hysis letters [9] Kyrl 967 Theoretil hysis unders ompny hildelphi & London Toppn ompny Limited Tokyo Jpn [0] [] /uternionshtml REFERENE [] Resnik Robert Introdution to speil reltivity iley Estern Limited New ge interntionl limited - 99 [] Moller The Theory of Reltivity Oxford niversity ress-97 [3] lm M 000 tudy of Mixed Number ro kistn [] lm M 003 Different types of produt of vetors NewsulllMtho 6( & 3 - [3] ty roksh Reltivisti Mehnis rgti rkshn-000 Indi [] Md hh lm nd Khurshid egum Different Types of Lorentz Trnsformtions Jhngirngr hysis studies-009
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