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PLAN OF DB SPECIAL RELATIVITY dba axioms of special relativity dbb lorentz transformations dbc length contraction and time dilation dbd speed cumulation dbe E=MC^2 dbf paradox of langevins traveller BACK TO SITE PLAN Site Plan
DERIVATION OF LORENTZ TRANSFORMATIONS
Let SpaceTime be a 4D #Space of space coordinates u(/1),u(/2),u(/3) and time coordinate t, affine between space and time, which don't have a common measure. Spherical light wave emitted origin of coordinates u(/i),t reaches after time dt points du(/i)=Cdt where C= speed of light. Let's introduce del, the Kroeckner Symbol or the Fundamental Tensor of the Euclidian #Space: del(ij/)=1 for i=j del(ij/)=0 for i!=j or in matrix form: del(ij/)=[100,010,001]. Let's further introduce Einstein's indexing notation implying summation over each index repeated within a monome as upper and lower one. Then a radius dr of the sphere is given by: dr^2=del(ij/)du(/i)du(/j) where dr=Cdt, C being the speed of light. Thus: (Cdt)^2 = del(ij/)du(/i)du(/j), or (Cdt)^2 - del(ij/)du(/i)du(/j) = 0 [1] We find ourselves here at cross-roads. A.We may continue to consider two distinct affine subspaces: 1.time (dt), 2.space (del(ij/)du(/i)du(/j)) B.We may take advantage of Cdt and del(ij/)du(/i)du(/j) having the same measure of distance, thus [1] implying a 4D metric Minkowski #Space (MinSp). Question arises: Could a theory supporting invariance of C be constructed upon the assumption A? Possibly, but such a theory has never been constructed and it would not have been Einstein's SR, which is based upon B. Consequently we shall consider as an additional axiom of SR the choice of MinSp as Abstract Space of SR and we shall continue the LT derivation within MinSp. Let's recall some basic concepts of MinSp: Fundamental Tensor mu(ij/): mu(ij/) = -1 for i=j=1 mu(ij/) = 1 for i=j=2,3,4 mu(ij/) = 0 for i!=j or in matrix form: mu(ij/)=[-1000 0100 0010 0001] Base vectors: e1(1/)=i em(m/)(m=2,3,4)=1 [i=sqrt(-1)] and ek(l/)=0 for l!=k. In SR instance of Minkowski #space: x(/1)=Ct (light-time) x(/m)(m=2,3,4) space dimensions. NOTE: the fundamental difference between the Pre-SR "(t,x)" 4D #Space (t,x(/m)(m=2,3,4)) and SR's "(Ct,x)" 4D #Space consists in the first being affine and the second - metric. Indeed, there is no common measure between t and x in Pre-SR #Space, while all coordinates of SR #Space have the common measure of "distance" (including the light-time Ct). Consequently, SR #Space admits metric as described above and rotation-type transformation, namely pseudo-rotation in the pseudo-orthogonal complex plane Ct / x(/m). This pseudo-rotation is equivalent with Lorentz Transformation, as will be shown below. The invariant form ds^2 in SR #Space: ds^2=(dx(/1))^2-sigma((dx(/m))^2)(m=2,3,4) or ds^2=(Ct)^2-sigma((dx(/m))^2)(m=2,3,4) Pseudo-rotation transforming x,t to X,T moving along x(/2), keeping invariant ds^2: Ct = X(/1)sh(th) * CT ch(th) x(/2) = X(/2) ch(th) + CT sh(th) x(/3) = X(/3) x(/4) = X(/4) where sh, ch are hyperbolic functions. Putting th(th) = v/c: t = (T + (v/c^2)X(/2)) / sqrt(1 - v^2/c^2) x(/2) = (X(/2) + vT) / sqrt(1 - v^2/c^2) x(/3) = X(/3) x(/4) = X(/4) Which are the Lorentz Transformations.