Einstein's Special Relativity Theory and Mach's Principle by Lars ...

Einstein’s Special Relativity Theory and Mach’s Principle
by
Lars Wåhlin
Colutron Research, Boulder, CO USA
Presented February 14, 2002 at the AAAS annual Meeting in Boston (General Poster Session)
Abstract: Many historical works on Einstein describe his approval of Mach’s
philosophy and his effort to incorporate Mach’s Principle into his relativity theories.
Einstein eventually abandoned Mach’s Principle but with some reservations.
However, Mach’s Principle still persists and its presumed incompatibility with
Einstein’s Relativity continues to be an obstacle for many in their attempts to
understand Einstein’s theory. This essay intends to resolve the issue by showing that
Einstein’s Special Relativity, in fact, is subject to Mach’s Principle and that the proof
can be found in the relativistic velocities of atomic orbits.


Ever since Einstein published his papers on Special Relativity [1,2]
there have been many scientists who have not been fully satisfied with
the theory. Perhaps most noteworthy is Walter Ritz, who collaborated
with Einstein in 1909, and more recently the late Professor Petr
Beckmann, who was the founder of the journal Galilean Electrodynamics,
the principal aim of which is to refute Einstein’s Special Theory of Relativity.
From time to time other scientific journals have accepted articles critical of
Special Relativity, and associations have been formed by philosophically
minded groups who do not wholly accept the concept of space-time and the
rejection of absolute space and absolute velocity, as upheld by the Special
Theory of Relativity. One such organization is the Natural Philosophy
Alliance, which boasts an impressive list of members.
However, scientists and philosophers who believe that Einstein’s
Special Theory of Relativity is one of the greatest achievements in
science and irrefutable, by far outnumber those who are not convinced of
its validity. Further, the relativistic velocity equations have been proven
repeatedly in high energy particle accelerators.
I believe that for many scientists Special Relativity is difficult to
understand except with respect to solving the equations. There is no
doubt that Einstein’s energy-velocity equation is valid and indisputable,
and one of the greatest achievements in science. Why therefore, is there
still scope for debate? It is the elimination of absolute space and absolute
velocity or, in other words, the rejection of Mach’s Principle [3,4], that

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Einstein himself was forced to discard, which creates the conflict. It was
in fact Einstein’s mathematics teacher, Herman Minkowski [5], who
introduced the purely mathematical concept of space-time, which
discarded absolute space and absolute velocity, which Einstein reluctantly
accepted [6].
It is possible to verify that Einstein was correct in believing that Mach’s
Principle should be incorporated into his relativity theory. Mach’s
Principle requires that inertia of mass and consequently potential energy
of inertial mass must be generated by the rest of the Universe. In
mathematical terms Mach’s Principle can be written as
2
φ
= GM
/ R = c where φ is the cosmic gravitational tension or the
univ
univ
univ
amount of potential energy per mass generated by the Universe; G is the
gravitational constant; c the speed of light; R the absolute distance to the
center of mass of the system and Muniv the total mass of matter within the
radius of curvature R. Technically, Mach’s Principle can be applied to the
Earth and the solar system by simply using
2
φ = GM / r = v where r is
sol
sol
the distance to the center of mass of the solar system and M the mass of
sol
the solar system within r. The gravitational tension φ at the orbital
sol
radii r of the different planets equals the square of their orbital
velocities. Mach’s Principle can be further extended to our galaxy or to
clusters of galaxies and ultimately to the Universe as a whole, at which
point
2
φ = c .
univ
This leads to a velocity effect peculiar to Mach’s Principle. For example,
should we want to sling the Earth in its orbit at r2 out to the orbit of
Mars at r , then the amount of kinetic energy that needs to be added to
3
Earth is
1
ΔE = m φ −φ , where m is the Earth’s mass and φ
φ the
2
(
)
2
3
2 and 3
gravitational tension of the solar system at r and r respectively. The
2
3
difference in orbital velocity is thus Δv = φ − φ . However, decreasing
2
3
the Earth’s orbit by the same amount of energy, ∇ = Δ to a smaller

E
E
radius r means a loss of potential energy (∇E = loss of energy) in the
1
form of friction and radiation or
1
∇E = m φ −φ , and the difference in
2
(
)
1
2
orbital velocity becomes ∇v = φ − φ . Note, that when E
Δ = ∇E then
1
2
v
Δ ≠ v
∇ , which is a consequential effect of Mach’s Principle.
The Special Theory of Relativity has so far ignored the above effect, since
it considers matter at relative rest (thus the term rest mass energy
2
E = m c ) and cannot deduct velocities from rest or zero velocity. It can only
0
0
accurately be applied to velocities that are produced by an increase in

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3
rest mass energy or E + E
Δ . Einstein’s relativistic velocity equation can
0
be written in a Mach’s format as
2
⎛
E
⎞

0
Δv = φ
−φ
.
(1)
univ
univ ⎜⎜
⎟
⎟
⎝ E + ΔE
0
⎠
In cases where energy is lost to radiation such as when electrons are
captured in high speed atomic orbits, Einstein’s relativistic equation
becomes obsolete and must be replaced by a second equation that can be
used in cases where loss of rest mass energy occurs, such as E − ∇E or
0
2
⎛ E − ∇E ⎞

0
∇v = φ
−φ
.
(2)

univ
univ ⎜⎜
⎟
⎟
⎝
E0
⎠
This becomes evident if we apply both the above velocity equations to the
inner orbits of atoms and compare the results to published measured
values that currently appear in The Handbook of Chemistry and Physics
(under Ionization energies or Ionization potentials of the Elements).
The circumference of the innermost atomic orbit as determined by Louis
de Broglie’s wave theory is
1
2 h∇v
Zq2

=
, ( 1 wavelength)
(3)
∇E
4ε ∇E
2
0
and solving for ∇E by inserting ∇v from Equation (2) we obtain
⎡
⎛ Zq
2
2
⎤
⎞
m

n
∇E = E ⎢1 − 1 −
⎥ ×
, (Joules)
(4)
e
0 ⎢
⎜
⎜ 2ε hc⎟⎟
⎝ 0
⎠ ⎥ m + m
n
e
⎣
⎦
where Z is the atomic number;; E the electron’s rest mass energy; q the
0
electron’s electric charge; ε the permittivity constant and h Max
0
Planck’s constant. The term, m /(m + m )where m
m are masses

n
n
e

n and
e
of the atomic nucleus and electron respectively, reduces the orbital
energy to that of the electron only. For example, Z=29 (Cu) yields a
∇E / q =
.
11573 3 eV

5
or 5.7 eV higher than the published data.
e
Inserting v
Δ for Z = 29 from Einstein’s Special Relativity Equation (1)
into the above de Broglie Equation (3)

⎡⎛
⎞
⎤
1
m

⎜
⎟
n
E
Δ
=
⎢
− ⎥
1 ×
e
E0
, (Joules) (5)
⎢⎜ 1 − (Zq2 /(2ε hc 2 ⎟
))
⎥
+
n
m
e
m
⎣⎝
0
⎠
⎦

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4
results in a critical error of 274 eV higher than published data which has
prompted investigators to introduce several correction factors such as the
Dirac-Fock correction [7]; self energy correction [8]; Uehling vacuum
polarization correction [9]; higher order vacuum polarization correction
[10] and nuclear size correction etc., in order to match the measured
values.



Fig. 1. Deviation in percent between measured values and results obtained
from Equations (4) and (5). Also shown are values obtained from
Newton’s non-relativistic Equation.

The curves in Fig. 1, which are constructed from Equations (4) and (5)
and Newton’s non-relativistic relation
2
1
E
Δ = mv , show a deviation
2
from measured values in percent. The results of Equation (5) seem to
indicate a very small systematic Compton red shift of
∇λ = 1
( − cosα )h /(m c) in the published measurements which accounts
e
for the 5.7 eV discrepancy at Z=29. The Compton red shift could be
caused by a recoil or deflection angle of cosα = (
+
1
k log eV )
2
k affecting
the spectrometric measurements where k = 0.0197565 and
1
k = 0.89794 are proportionality constants and the energy of the spectra in
2
electron volts, see Fig. 2.

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Fig. 2. Compton scattering angle α for the different elements
−1
α = cos [(0.0197565 log eV ) + 0.89794
1
]

My personal conclusion is that the mathematics of Einstein’s Special
Theory of Relativity is only correct for cases of relative increase in rest
mass energy, as in particle accelerators for example, and that Mach’s
Principle should be included in the theoretical interpretation of the
theory to account for the energy-velocity relationship in cases where rest
mass energy is lost, such as in atomic orbits. The close agreement
between measured values and the results from Equation (4), after
corrected for Compton red shifts, see Fig. 3, should prove this point.


Fig. 3. Difference in parts per million between measured values (corrected for
Compton red shifts) and the theoretical values from Equation (4).

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The fact that Einstein’s Special Theory of Relativity works well in
particle accelerators, but fails for atomic orbits, is quite serious since
there are by far more atoms in the world than particle accelerators.
It is remarkable that Mach’s Principle has to be invoked in order to
explain relativistic atomic orbits when Mach himself did not believe in
atoms while Einstein, on the other hand, who was first to prove that
atoms exist (Brownian movement and the photoelectric effect) chose to
abandon Mach’s Principle

References:
[1] A. Einstein, AdP 17, 891, (1905).
[2] A. Einstein, AdP 20, 627, (1906).
[3] E. Mach, History and Root of the Conservation of Energy, (1872).
[4] E. Mach, The Science of Mechanics, 6th ed., (1904).
[5] H. Minkowski, Goett. Nachr., 53, (1908).
[6] A. Pais, Subtle is the Lord, Oxford University Press, Oxford, (1982).
[7[ J.P. Desclaux, Numerical Dirac-Fock Calculations for Atoms:
Relativistic effects in Atoms, Molecules and Solids, ed. G.L. Malli,
Plenum Press, New York, p.133,(1981).
[8] P.J. Mohr, Proceedings of the Workshop on Foundations of the
Relativistic Theory of Atomic Structure, ANL-80-126, (1981).
[9] E.A. Uehling, Polarization effects in Positron Theory, Phys. Rev.,
48, 55. [1935].
[10] G. Källen et al., Fourth Order Vacuum Polarization, Kong. Dansk.
Vidensk. Selsk. Mat. Fys. Med., 29, p.17 (1972).

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WORK SHEET
Measured groundstate energies (highest ionization potential) for one-electron atoms from Handbook of
Chemistry and Physics
TABLE 1
Z Element Energy eV1
Z Element Energy eV1
Z Element Energy eV1
1 H 13.59844
11 Na 1648.702
21 Sc 6033.712
2 He 54.41778
12 Mg 1962.665
22 Ti 6625.82
3 Li 122.45429
13 Al 2304.141
23 V 7246.12
4 Be 217.71865
14 Si 2673.182
24 Cr 7894.81
5 B 340.22580
15 P 3069.842
25 Mn 8571.94
6 C 489.99334
16 S 3494.1892 26 Fe 9277.69
7 N 667.046
17 Cl 3946.296
27 Co 10 012.12
8 O 871.4101
18 Ar 4426.2296
28 Ni 10775.40
9 F 1103.1176
19 K 4934.046
29 Cu 11567.617
10 Ne 1362.1995
20 Ca 5469.864

Above values corrected for a minor Compton red-shift of ∇λ = (1 − cosα )h / (m c), where
e
cosα = ( .
0 0197565 log eV
, and by using the following equation:
1 ) +
.
0 8979399
2

∇λ q
(eV )
eV
1
=
+
.
2
eV1
ch

TABLE 2
Z Element Energy eV2
Z Element Energy eV2
Z Element Energy eV2
1 H 13.59846
11 Na 1648.906
21 Sc 6035.661
2 He 54.41817
12 Mg 1962.943
22 Ti 6628.102
3 Li 122.4560
13 Al 2304.511
23 V 7248.77
4 Be 217.72383
14 Si 2673.662
24 Cr 7897.86
5 B 340.23758
15 P 3070.453
25 Mn 8575.44
6 C 490.01632
16 S 3494.955. 26 Fe 9281.678
7 N 667.0862
17 Cl 3947.241
27 Co 10016.63
8 O 871.4754
18 Ar 4427.3808
28 Ni 10780.48
9 F 1103.217
19 K 4935.432
29 Cu 11573.322
10 Ne 1362.3452
20 Ca 5471.515


Theoretical values using Equation

E ⎡
⎛ Zq
2
2
⎤
⎞
m

0
n
eV =
⎢1− 1−
⎥ ×
3
q ⎢
⎜
⎜ 2ε hc⎟⎟ ⎥
⎝
0
⎠
(m + m
n
e )
⎣
⎦
m / m
(
+ m ) = 1821A / (1821A + )
1
n
n
e

TABLE 3
Z A Energy eV3
Z A Energy eV3
Z A Energy eV3
1 H 1.008 13.598470
11 Na 22.99 1648.9105
21 Sc 44.96 6035.6852
2 He 4.003 54.418224
12 Mg 24.32 1962.9465
22 Ti 47.9 6628.0684
3 Li 6.94 122.456266
13 Al 26.98 2304.5127
23 V 50.95 7248.7505
4 Be 9.013 217.72428
14 Si 28.09 2673.6593
24 Cr 52.01 7897.8330
5 B 10.82 340.23846
15 P 30.975 3070.4526
25 Mn 54.94 8575.4320
6 C 12.011 490.01767
16 S 32.66 3494.9520 26 Fe 55.85 9281.6569
7 N 14.008 667.08850
17 Cl 35.457 3947.2312
27 Co 58.94 10016.6635
8 O 16.0 871.47793
18 Ar 39.94 4427.3654
28 Ni 58.71 10780.485
9 F 19.0 1103.2206
19 K 39.1 4935.4223
29 Cu 63.54 11573.353
10 Ne 20.183 1362.3488
20 Ca 40.08 5471.4978


h = 6.626075540 E-34, c = 2.99792458 E+8, ε0 = 8.854187817 E-12, e
m = 9.109389754 E-31, Proton-
electron mass ratio m / m = 1836 , q=1.6021773349 E-19,
2
E = m c =8.18711121654 E-14
p
e
0
e

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