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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
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DBE.E=MC**2
E=MC**2 has deep ontological and epistemological implications
discussed in Postface.
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
is an invariant which describes elementary increment of the
4D line L 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)=dx(4/)**2. Thus, d(tau) will be measured with a
lighttime 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". 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, 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". 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
I(y/)=M*r(y/)/lambda
I(z/)=M*r(z/)/lambda
Energy, E(l/)=M/lambda
Back to our original notation:
Energy, E(l/)=Mdl/lambda
Momentum and energy tend to infinity for V approaching C
and for low speeds momentum approximates that of traditional
physics.
Calling energy and mass of immobile "material" point
respectively E(l/)=Eo, Mo and noting that for an immobile
"material" point lambda=1 we get:
Eo=Modl=Mod(Ct)**2
Choosing second as unit of time and dropping "o" for
cosmetic reasons we get:
E=MC**2
POSTFACE
E=MC**2 is an impressing illustration of the
P(henomenal)-Equivalence.
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". Under the disguise
of pure abstractions, E=MC**2 implies P-Equivalence of
Electro-Magnetism and the fields constructing subatomic
particles. A decaying particle converts involved quantity of
a strong field to enormous amount of radiation weaker by
factor C**2.
Seen from Special Relativity, E=MC**2 appears as anticipation
not only of the General Relativity, but also of the Quantum
Field Theory.
