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E=MC^2
================
E=MC^2 has deep ontological and epistemological implications
discussed in Postface.
Conventions:
================
Let's recall Einstein's indexing notation implying
summation over each index repeated within a monome
as upper and lower one. Thus, for 3D:
i 1 2 3
AB=AB+AB+AB
i 1 2 3
Similar convention applies also to partial derivatives as
follows:
i j j
∂x/∂y)dy=
i 1 1 i 2 2 i 3 3
(∂x/∂y)dy+(∂x/∂y)dy+(∂x/∂y)dy
and to vectors indexed with symbols, like for instance
u=dx/dτ
s s
We shall use Einstein's symbols: lighttime l=Ct,
normalized speed v=V/C and γ=sqrt(1-v^2)
Maxwell equations:
================
∂B / ∂x=(1/C)(∂e / ∂(t)+i)
ab b a a
∂e/∂x-∂e/∂x=(1/C)∂B/∂(t)
a b b a ab
and divergences:
∂(e/∂x) = D
a a
∂(B/∂x) = 0
ab c
Tensorial simplification
================
Let's introduce tensor constructs Q and J
ij k
corresponding to B,e,i,D as follows:
================
Q___Q___Q___Q___Q___Q
23 31 12 14 24 34
B___B_ _B__-je_-je_-je
x y z x y z
================
J____J____J____J
1 2 3 4
i/c__i/c__i/c__jD
x y z
================
Note:
Q = -Q due to antisymmetry j = sqrt(-1)
ab ab
Thus, Field representation may be merged into two following
forms:
(∂Q/∂x) = J
ab b a
∂Q/∂x + ∂Q/∂x + ∂Q/∂x = 0
ab c bc a ca b
Lorentz Transformation for the Electro-Magnetic Field:
[v=V/C γ=sqrt(1-v^2)]
E=e B=b
x x x x
E=(e-vb)/γ B=(b+ve)/γ
y y z y y z
E=(e+vb)/γ B=(b-ve)/γ
z z y z z y
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 * J+Q * J+Q * J(4/)
12 2 13 3 14 4
(Q vanishes due to the antisymmetry)
11
Components of k are given by
-3 first components of the 4-Vector K:
K=Q /J
a ab b
-4th component of K:
K=Q /J + Q /J + Q /J = j((e*i)+(e*i)+(e*i)) = jλ
4 41 1 42 2 43 3 x*x y y z z
Let's imagine a body experiencing along lighttime [l1,l2]
the action of E-M Field. The changes of its momentum
ΔI ΔI ΔI and energy ΔE are given by:
x y z
l2
ΔI=∫dl∫kdxdydz=(1/j)∫Kdxdxdxdx
x l1 x 1 1 2 3 4
ΔI= ...
y
ΔI= ...
z
l2
ΔE=∫dl∫λdxdydz=(1/j)∫(1/j)Kdxdxdxdx (A)
l1 4 1 2 3 4
The 4D volume element is invariant and the components of K
form a 4-Vector
For the interval l1,l2 tending towards zero, or l2 tending
towards l1, the increments ΔI, ΔE tend towards
the constructs I,E. As terms transform in the same way as
their differentials, the terms
I,I,I,jE
x y z
have themselves vector character and form a 4-Vector
describing the momentary state of the body at l1.
Now, this 4-Vector may also be expressed with the Mass M and
the speed of the "mass point" body.
"Mass" Point
================
Let's recall that
-ds^2 = dτ^2 = -dx^2+dx^2+dx^2-dx^2 = dL^2 * γ^2
1 2 3 4
is the invariant elementary increment of the 4D line L
representing the movement of a "mass" 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 "mass" point into "rest", we'll get
dτ=dl. Thus, dτ will be measured with a
lighttime clock falling freely together. Therefore, τ is
called "proper time" of the "mass" point and dτ is, unlike dl,
invariant.
Consequently, we see that
u=dx/dτ
s s
has itself, as the dx, vector character and we shall call
u the "4-Vector of speed". Its components satisfy the
s
condition: sigma(u^2)=-1
s
Calling r=da/dl, the components of u are:
(1/γ)(r,r,r,j)
x y z
u is the unique 4-Vector which may be formed with speed
components of a "mass" point.
Consequently
(M*dx)/dτ
a
ist for a "mass" point the 4-Vector equivalent to the
momentum/energy 4-Vector, derived above.
Equating the components we get:
Momentum:
I=M*r/γ
x x
I=M*r/γ
y y
I=M*r/γ
z z
Energy, E=M/γ
l
In SR M is not invariant, but covariant by Lorentz Transformation.
Thus it's legitimate to express ΔE as a function of ΔM.
By analogy with (A):
τ2 l2
ΔE=ΔM∫τdτ = ΔM∫γldl
τ1 l1
For point at rest γ=1, thus:
l0
ΔE=ΔM∫ldl = ΔMl0^2
-l0
thus
ΔE = ΔM*(Ct0)^2
choosing 1 sec as time unit
ΔE = ΔM*C^2
For l0 tending to infinity ΔE,ΔM tend towards total E,M
of the "mass point":
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.
