Three Lorentz Transformations. Considering Two Rotations

Transcription

1 Ad. Stdies Theor. Phs., Vol. 6,, no., 9 Three orentz Transformations Considering To otations Mkl Chandra Das* Singhania Uniersit, ajasthan, India mkldas.@gmail.om ampada Misra Department of eletronis, Vidasagar Uniersit West Bengal, India Abstrat A formlation for ombination of three orentz transformations onsidering to rotations has been deeloped in this paper. This proedre phsiall means the omposition of three linear eloities in different diretions reslting in a single eloit. The same proess has been applied to a partile or bod possessing three simltaneos rotational motions haing mtal effets. This leads to the assmption that a bod ma possess three simltaneos sperimposed spins. Ke ords: sperimposed rotational motions, sperimposed spins.. Introdtion One of the basi qestions in orentz transformation is eloit addition. Aording to Einstein s relatiisti theorem, e hae the ell knon formla of o-linear eloit addition[], ( ) /( / ). Møller extended this approah to deelop the eloit addition formla for ombining to eloities in different diretions in planar motion []. Chandra Ier and G.M Probh desribe the transformation for omposition of to orentz boosts in different diretions in plane []. This means the omposition of to eloities and reslting in a single eloit. In this paper e first deelop the transformations ielding a * Commniation address : Satmile High Shool, P. O. Satmile 7, W. B., India

2 M. C. Das and. Misra single resltant orentz boost throgh a proper a. We extend this to ompose three sperimposed motions in different diretions into a resltant eloit and folloing this onstrtie method e sho that a bod or partile ma possess three simltaneos sperimposed spins.. Transformations In this setion e ant to derie the matrix for a onentional orenz transformation folloed b a planar rotation of the XY plane and then x folloed b another onentional orenz transformation folloed b a planar rotation of the XY plane ( ) onentional orenz transformation x x hih is again and then b another. For larit, e ma isalize six inertial frames S, S, S, S, S and S in three dimensional spae. Frames S and S hae both their o-ordinate axes aligned and S is moing at a eloit along X axis as obsered b S. The inertial frame S has another o-ordinate referene frame S, here X axis of, S are rotated b an angle onter lokise ith respet to S on XY plane. Frames S and S hae both their o-ordinate axes aligned and S is moing at a eloit along X axis as obsered b S. The inertial frame S has another oordinate referene frame S here X axis of S are rotated b an angle onter lokise ith respet to S on XY plane. Frames S and S hae both their o-ordinate axes aligned and S is moing at a eloit along X axis as obsered b S. Then the transformation of o-ordinates of an eent from frame S to frame S is gien b order matrix M ij hih is eqal to the matrix prodt. x xz x x( ) x.the matrix x,, x( ) x, xz( ) and x, are as speified in (), (), (), () and () respetiel

3 orentz transformations x x x os sin x sin os (), z z z z t t t t () x x x os sin x (), z z z sin os z t t t t () x x z z t t () The inerse of this operation is gien b N x x x xz x ij Matries M ij and N ij trn ot as M ij x xz x x x os sin sin os os sin sin os =

4 M. C. Das and. Misra os ( os ( sin ( os os os sin os sinos ) ) osos ) sin os sin sin ( ( sin sin sin os sin os ) ossin ) os ( os ( os ( sin sin os os sinos ) os os ) os ) (6) Similarl N = ij x x x xz x os sin sin os os sin sin os

5 orentz transformations (os os ( os (sin os os os sin sin ) os os os ) ) (sin os ( sin sin os sin sin ) sinos ) sin os sin os ( os ( os sin ( sin os os ossin ) os os ) os ) (7). esltant eloit de to omposition of three eloities For omposition of three eloities e ma isalize six inertial frames as disssed in setion- here, the o-ordinate of an eent at an point on S ith respet to S old be ritten as H N K (8) ij here, H x z t, K x z t Eqation (8) gies the relations as

6 M. C. Das and. Misra x N x N N z N t N x N N z N t z N x N N z N t t N x N N z N t (9) From spherial oordinate sstem one an also rite dx dt d dt dz dt sin(9 ) os os os sin(9 ) sin sin os () os(9 ) sin Folloing eqations (9) and () e old find ot the resltant eloit ' G ' of the eent of S as obsered b S, hose magnitde also an be ealated b onsidering the motion of the origin of S ( x,, z ) as dx d dz G dt dt dt () This ma be ritten as here, G P Q T P N os os N sin os N sin N Q N os os N sin os N sin N N os os N sin os N sin N T N os os N sin os N sin N () When 9 then the resltant eloit G old be gien b G ( ) ()

7 orentz transformations. esltant eloit de to three rotational motions From the aboe disssion it is seen that,, and are the eloities in three different diretions here, angle beteen and is, beteen and is and that beteen and is hih depends on and. In that sense for omposition or addition of three simltaneos sperimposed rotational motions e an onsider the eloit addition formla (). For larit e ma isalize six inertial frames as disssed in setion-, bt S, S and S don t moe ith retilinear motion. The moe onl ith anglar eloit, and respetiel, abot X, X and X axes as obsered b S and others (i.e. S, S and S ) be same as disssed in setion-, When origins of frames are same ith respet to S, then the resltant eloit of an eent of S ill be same as G in () as obsered b S here, r, r and r beome separatel relatiisti eloities and r is the position etor of the eent in S. It is to be mentioned that if a bod or partile be present at the origin of S then it ill possess three simltaneos sperimposed spins as obsered b S.. Conlsions The aboe deriations and disssions reeal the field of relatiisti effets on a bod de to three motions. The expression for resltant eloit shos that it is dependent pon all the linear eloities arising from the rotations (iz.eqation ()) eferenes [] obert esnik, Introdtion to Speial elatiit, 9, p.8 ISBN [] Møller C., The theor of elatiit, Oxford Uniersit press, Oxford, 9, - [] Chandr Ier, and G.M. Prabh, Composition of to orenz boosts throgh spatial and Spae-time rotations, Jornal of Phsial and Natral Sienes Vol., (7), no. eeied: September,