|
|||||||
|
|
|
|||||
|
|
|||||||
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
DBE.E=MC^2
E=MC^2 has deep ontological and epistemological implication: It's an impressing illustration of P(henomenal)-Equivalence principle. P-Equivalence: A Phenomenon, say "Light" is given exclusively by its observable Aspects. Continuous Field wave and Discrete photons are P-Equivalent aspects of the Phenomenon "Light". Similarly, E=MC^2 illustrates the P-Equivalence of Mass and Energy. This simple statement has surprisingly deep implications. Indeed, Mass and Energy have no phenomenal sense and are just coefficients in formulas representing phenomenal, observable construct "Field". Mass is, for instance, an uncognizable as such singularity of Field. E=MC^2 implies P-Equivalence of Electoro-Magnetism, Inertia and Gravity, which, seen from Special Relativity appears as an anticipation of the "Equivalence Postulate" and of the General Relativity. P-Equivalence has a fundamental ontological and epistemological implications for Relativistic Dialectic (RD) of the Second Enlightenment. It supports the Dialectic view of apparently contradictory but in fact complementary Aspects. Of its implications for RD's Logic (CCA.COGNITIVE NETWORK) we shall mention here the ORR operator (exclusive or, either-or) replaced in Cognitive Network of RD: -deductively by "and" (light is wave AND corpuscle), -inductively by "or" (observed wave OR corpuscle verify the phenomenal Axiom of Light).
Notations:
Unless they are elementary displacements noted dx,dy..., vectors are usually noted with upper case letters (A,B...) and their components with lower case letters designating indexes (i,j,k...) written as upper or lower, following vector's name: A .i .j B In ASCII context we shall write them with help of brackets and slash, as follows: A(i/), B(/j) Upper indexes designate contravariant components and lower indexes covariant ones. Thus A(i/) designates the i-th covariant component of vector A and B(/j) the j-th contravariant component of vector B. In cases when the variance is not yet defined we shall skip the slash and C(k) will mean the k-th component of C of unknown variance. We shall write derivative of y with respect to x as: d(y)/d(x) and partial derivative of u with respect to v: p(u)/p(v). Let's further introduce Einstein's indexing notation implying summation over each index repeated within a monome as upper and lower one. Thus, for 3D: A(i/)B(/i)=A(1/)B(/1)+A(2/)B(/2)+A(3/)B(/3) This convention applies also to partial derivatives as follows: (px(/i)/py(/j))dy(/j) = (px(/i)/py(/1))dy(/1)+(px(/i)/py(/2))dy(/2) +(px(/i)/py(/3))dy(/3) NOTE: We shall follow Einstein's conventions: lighttime l=ct, normalized speed v=V/c and lambda=sqrt(1-v^2)
Pre-relativistic Maxwell equations:
p(B(ab/)/p(x(b/))=(1/C)(pe(c/)/p(t)+i(c/)) p(e(a/)/p(x(b/)-p(e(b/)/p(x(a/)=(1/C)p(B(ab/)/p(t) and divergences: p(e(a/))/p(x(a/) = D p(B(ab/)/p(x(c/)) = 0
Tensorial unification
Let's introduce tensor constructs Q(ij/) and J(k/) corresponding to B,e,i,D as follows: Q(23/)__Q(31/)__Q(12/)__Q(14/)__Q(24/)__Q(34/) B(x/)___B(y/)___B(z/)___-je(x/)_-je(y/)_-je(z/) J(1/)____J(2/)____J(3/)____J(4/) i(x/)/c__i(y/)/c__i(z/)/c__jD Note: Q(ab/) = -Q(ab/) due to antisymmetry j = sqrt(-1) Thus, Field representation may be merged into two following forms: p(Q(ab/))/p(x(b/)) = J(a/) p(Q(ab/)/p(x(c/) + p(Q(bc/)/p(x(a/) + p(Q(ca/)/p(x(b/) = 0 Lorentz Transformation for the Electro-Magnetic Field: [v=V/c lambda=sqrt(1-v^2)] E(x/)=e(x/) B(x/)=b(x/) E(y/)=(e(y/)-vb(z/))/lambda B(y/)=(b(y/)+ve(z/))/lambda E(z/)=(e(z/)+vb(y/))/lambda B(z/)=(b(z/)-ve(y/))/lambda Let's consider the force k acting at electricity per volume unit: k=qe + [i,B] where i: speed of electricity with unit as c [i,B]: crossproduct The first component of k is: Q(12/)J(2/)+Q(13/)J(3/)+Q(14/)J(4/) (Q(11/) vanishes due to the antisymmetry) Components of k are given by -3 first components of the 4-Vector K: K(a/)=Q(ab/)J(b/) -4th component of K: K(4/)=Q(41/)J(1/)+Q(42/)J(2/)+Q(43/)J(3/)= j(e(x/)i(x/)+(e(y/)i(y/)+e(z/)i(z/)=jL Let's imagine a Body experiencing along lighttime [l1,l2] the action of E-M Field. The changes of its momentum DelI(x/),DelI(y/),DelI(z/) and energy DelE are given by: DelI(x/)=int[l1,l2]dl(int(k(x/)dxdydz))= =(1/j)intK(1/)dx(1/)dx(2/)dx(3/)dx(4/) DelI(y/)= ... DelI(z/)= ... DelE=int[l1,l2]dl(int(Ldxdydz))= =(1/j)int((1/j)K(4/)dx(1/)dx(2/)dx(3/)dx(4/) As the 4D volume element is invariant, the components of K form a 4-Vector Terms transform in the same way as their differentials so that the terms I(x/),I(y/),I(z/),jE form a 4-Vector describing the momentary state of the Body. Now, this 4-Vector may also be expressed with the Mass m and the speed of the "material point" Body.
"Material" Point
Let's recall that -ds^2 = d(tau) = -(dx(1/)^2+dx(2/)^2+dx(3/)^2 - dx(4/)^2 = = dl^2*lambda^2 (01) is an invariant which desribes elementary increment of the 4D line representing the movement of a "material" point. If we chose the l (lighttime) axis so that its direction is that of the concerned line differentials or, as one says, we transform the "material" point into "rest", we'll get d(tau)=dl. Thus, d(tau) will be measured with a solidary light-second clock falling freely together. Therefore, tau is called proper time of the "material" point and d(tau) is invariant. Consequently, we see that u(s/)=dx(s/)/d(tau) has itself, as the dx(s/), vector character and we shall call u(s/) the "4-Vector of speed". According to (01) its components satisfy the condition: sigma(u(s/)^2)=-1 Calling r(a/)=da/dl, the components of u(s/) in the traditional notation are: (1/lambda)(r(x/),r(y/),r(z/),j) (u(s/) is the unique 4-Vector which may be formed with speed components of a "material" point. Consequently (m*dx(a/)/d(tau) ist for a "material" point the 4-Vector equivalent to the momentum/energy 4-Vector, which we have derived above. Equating the components we get: Momentum, I(x/)=m*r(x/)/lambda ... ... Energy, E(x/)=m/lambda Momentum and energy tend to infinity for V approaching c. Calling energy of immobile "material" point E(x/)=Eo and noting that for an immobile "material" point lambda=1 we get: Eo=m, energy at rest equals Mass. Choosing second as unit of time we get Eo=mc^2