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Noname Pre published at ZENODO: https://doi.org/10.5281/zenodo.3928713 (will be inserted by the editor) The field equations for gravitation and electromagnetism Ilija Barukčić Received: July 2, 2020 / Accepted: July 2, 2020 / Published: July 2, 2020 Abstract Aim: The development of a unified field theory for electromag- netism and gravitation while no material source terms are taken into account in the field equations of unified field theory is still an issue to be solved. Methods: The usual tensor calculus rules were used. Results: The gravitational and the electromagnetic field are successfully joined into one single hyper-field whose equations are determined only by the metric field gµν itself. Conclusion: The theory of gravitation and electromagnetism of Einstein and Maxwell are unified and described exclusively in terms of the metric tensor gµν . Keywords Gravitation · Electromagnetism · Einstein’s field equations · Unified field theory Internist Horandstrasse, DE-26441 Jever, Germany Tel.: +49-4466-333 Fax: +49-4466-333 E-mail: barukcic@t-online.de
2 Ilija Barukčić
1 Introduction
The many short-lived attempts to unify the electromagnetic and gravitational
fields together with other fundamental interactions within one conceptual and
formal framework has been the focus of much present research but has not yet
met with ultimate success. In point of fact, the unity of nature as the foun-
dation for the unity of science allow us to hope to find the trail of the hidden
in order to unify all the fundamental interactions, including gravitation1 and
electromagnetism, in a single theoretical framework – a unified field theory.
However, so far all attempts to tie the known fundamental forces of nature
and to unify2 the general3 theory of relativity4 , which describes the gravita-
tional5 field, with quantum mechanics made their incompatibility more or less
more apparent. Even Einstein’s efforts devoted to the search for a ‘unified
field theory’6 , “a generalization of the theory of the gravitational
field” 7 , were in vain8 . In order to reach new horizons on this issue in ques-
tion, a lot can been learned even from failed attempts at unification and much
more might be learned in the future. In brief, past experience has shown that
due to the weakness of the gravitational interaction to encounter primarily
an empirical-inductive method at unification with the goal to join the grav-
itational and electromagnetic field into a new field appears not impossible
but hardly implausible. Therefore, in this attempt at unification prefers the
deductive-hypothetical method. And it’s with this goal in mind that greater
attention will be directed in particular to the development of a new and sys-
tematic approach to the problem of unification. Fully in line with this is the
fact that all tensors of the general theory of relativity, including the Ricci
1Newton, Isaac: Philosophiae naturalis principia mathematica, Londini 1687.
2Mie, Gustav: Grundlagen einer Theorie der Materie, in: Annalen der Physik 3 (1912),
511–534. (_eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.19123420306)
Reichenbächer, Ernst: Grundzüge zu einer Theorie der Elektrizität und
der Gravitation, in: Annalen der Physik 2 (1917), 134–173. (_eprint:
https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.19173570203)
3 Einstein, Albert: Die Feldgleichungen der Gravitation, in: Sitzungsberichte der
Königlich Preußischen Akademie der Wissenschaften (Berlin), Seite 844-847. (1915); Ein-
stein, A.: Die Grundlage der allgemeinen Relativitätstheorie, in: Annalen der Physik (1916),
769–822; Einstein, Albert: Kosmologische Betrachtungen zur allgemeinen Relativitätsthe-
orie, in: Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin),
Seite 142-152. (1917); Einstein, A. / de Sitter, W.: On the Relation between the Expan-
sion and the Mean Density of the Universe, in: Proceedings of the National Academy of
Sciences of the United States of America 3 (1932), 213–214.
4 Einstein, A.: Zur Elektrodynamik bewegter Körper, in: Annalen der Physik 10 (1905),
891–921.
5 Barukčić, Ilija: The Equivalence of Time and Gravitational Field, in: Physics Procedia
(2011), 56–62.
6 Einstein, A.: Einheitliche Feldtheorie von Gravitation und Elektrizität, in: Preussische
Akademie der Wissenschaften, Phys.-math. Klasse, Sitzungsberichte (1925), 414–419.
7 Einstein, Albert: On the Generalized Theory of Gravitation, in: Scientific American
(1950), 13–17.
8 Goenner, Hubert F. M.: On the History of Unified Field Theories, in: Living Reviews
in Relativity 1 (2004).The field equations for gravitation and electromagnetism 3 tensor9 Rµν can meanwhile be presented in terms of the metric tensor10 gµν directly. However, to sum11 up, what do we do as a consequence of it? In the following, it is necessary to put some focus on the development and building of a unified field theory on the basis of a metric tensor gµν . 9 Ricci, M. M. G. / Levi-Civita, T.: Méthodes de calcul différentiel absolu et leurs applications, in: Mathematische Annalen 1 (1900), 125–201. 10 Barukčić, Ilija: Einstein’s field equations and non-locality, in: Foundations of Physics yy (2020), z1–z2 (under Review). 11 Idem: The Geometrization of the Electromagnetic Field, in: Journal of Applied Mathe- matics and Physics 12 (2016), 2135–2171.
4 Ilija Barukčić
2 Material and methods
2.1 Definitions
Evidently there several features of general relativity (particles as singularities
of a field) including experimental12 confirmation which have made the major-
ity of physicists to believe that general relativity theory is correct. However,
physicists have not yet figured out how to unify general relativity with all other
interactions and the search for a quantum theory of gravity is still without a
breakthrough in this field.
Definition 1 (Anti tensor) Let aµν denote a certain tensor. Let bµν denote
another tensor. Let Eµν denote a thrid tensor. Let the relationship aµν + bµν
+ ... ≡ Eµν be given. The anti tensor of a tensor, i. e. aµν is denoted in general
as aµν and defined as
aµν ≡ E µν − aµν
(1)
≡ bµν + ...
There are circumstances were an anti-tensor is identical with an anti-symmetrical
tensor, but both are not identical as such.
Definition 2 (The Ricci tensor Rµν ) Let Rµν denote the Ricci tensor13
of ‘Einstein’s general theory of relativity’14 , a geometric object developed by
Gregorio Ricci-Curbastro (1853 – 1925) able to measure of the degree to which
a certain geometry of a given metric differs from that of ordinary Euclidean
space. Let aµν , bµν , cµν and dµν denote the four basic fields of nature were aµν
is the stress-energy tensor of ordinary matter, bµν is the stress-energy tensor
of the electromagnetic field.
4×2×π×γ R
Rµν ≡ × T µν + × g µν − (Λ × g µν )
c4 2
| {z } | {z }
aµν +bµν cµν +dµν
≡ (aµν + bµν ) + (cµν + dµν )
≡ (aµν + cµν ) + (bµν + dµν )
(2)
≡ (aµν ) + (+bµν + cµν + dµν )
≡ (bµν ) + (+aµν + cµν + dµν )
≡ (cµν ) + (+aµν + bµν + dµν )
≡ (dµν ) + (+aµν + bµν + cµν )
≡ aµν + bµν + cµν + dµν
12 Thomson, Joseph: Joint eclipse meeting of the Royal Society and the Royal Astronom-
ical Society, in: The Observatory 545 (1919), 389–398.
13 Ricci, M. M. G. / Levi-Civita, T.: Méthodes de calcul différentiel absolu et leurs
applications (1900).
14 Einstein, A.: Die Grundlage der allgemeinen Relativitätstheorie (1916).The field equations for gravitation and electromagnetism 5
Definition 3 (The Ricci scalar R) Under conditions of Einstein’s general15
theory of relativity, the Ricci scalar curvature R as the trace of the Ricci
curvature tensor Rµν with respect to the metric is determined at each point
in space-time by lamda Λ and anti-lamda16 Λ as
R ≡ g µν × Rµν ≡ (Λ) + (Λ) (3)
A Ricci scalar curvature R which is positive at a certain point indicates
that the volume of a small ball about the point has smaller volume than a
ball of the same radius in Euclidean space. In contrast to this, a Ricci scalar
curvature R which is negative at a certain point indicates that the volume of
a small ball is larger than it would be in Euclidean space. In general it is
R × g µν ≡ (Λ × g µν ) + (Λ × g µν ) (4)
The cosmological constant can also be written algebraically as part of the
stress–energy tensor, a second order tensor as the source of gravity (energy
density).
Definition 4 (The stress-energy tensor of matter / energy Eµν ) The
stress-energy tensor of matter / energy Eµν is determined in detail as follows.
4×2×π×γ
E µν ≡ × T µν
c4
2×π×γ×T
≡ × g µν
c4
2×π×γ×T
≡ ×g µν
c4
| {z }
E (5)
≡ E × g µν
≡ Gµν + (Λ × g µν )
R
≡ Rµν − × g µν + (Λ × g µν )
2
≡ Rµν − E µν
≡ aµν + bµν
In our understanding, the stress-energy tensor of the electromagnetic field
(bµν ) is equivalent to the portion of the stress-energy tensor of matter / energy
(Eµν )due to the electromagnetic field where where Tµν “denotes the co-variant
15 Einstein, A.: Die Feldgleichungen der Gravitation (1915); Einstein, A.: Die Grundlage
der allgemeinen Relativitätstheorie (1916); Einstein, A.: Kosmologische Betrachtungen zur
allgemeinen Relativitätstheorie (1917); Einstein, A. / de Sitter, W.: On the Relation
between the Expansion and the Mean Density of the Universe (1932); Einstein, A.: Ele-
mentary Derivation of the Equivalence of Mass and Energy, in: Bulletin of the American
Mathematical Society 4 (1935), 223–230.
16 Barukčić, Ilija: Anti Einstein – Refutation of Einstein’s General Theory of Relativity,
in: International Journal of Applied Physics and Mathematics 1 (2015), 18–28.6 Ilija Barukčić
energy tensor of matter” 17 . In other words, there is no third tensor between
the stress-energy tensor of the electromagnetic field (bµν ) and the tensor of
ordinary matter or matter in the narrower sense (aµν ), a third tensor is
not given, tertium non datur! “Considered phenomenologically, this energy
tensor is composed of that of the electromagnetic field and of matter in the
narrower sense. ” 18
Electromagnetic field bµν
Ordinary matter aµν
Fig. 1. Ordinary matter and electromagnetic field.
Vranceanu19 is elaborating on this issue too. In other words, the energy
tensor Tkl is treated by Vranceanu as the sum of two tensors one of which is
due to the electromagnetic field (bµν ).
“On peut aussi supposer que le tenseur d’énergie Tkl soit la somme de
deux tenseurs dont un dû au champ électromagnétique . . . ” 20
Translated into English: ‘One can also assume that the energy tensor Tkl be
the sum of two tensors one of which is due to the electromagnetic field.’In this
context, it is necessary to make a distinction between the relationship between
ordinary matter and electromagnetic field and matter and gravitational field.
Our understanding of the relationship between matter and gravitational field
is impacted by Einstein’s train of thoughts on this relationship. Einstein wrote:
“Wir unterscheiden im folgenden zwischen ‘Gravitationsfeld ’und
‘Materie ’in dem Sinne, daß alles außer dem Gravitationsfeld als ‘Materie
’bezeichnet wird, also nicht nur die ‘Materie ’im üblichen Sinne, sondern
auch das elektromagnetische Feld.” 21
Einstein’s position translated into English:
We make a distinction hereafter between ‘gravitational field ’and ‘matter’in
this way that everything but the gravitational field is termed as ‘matter’that is
17 Einstein, Albert: The meaning of relativity. Four lectures delivered at Princeton Uni-
versity, May, 1921, Princeton 1923, p. 88.
18 Ibid., p. 93.
19 Vranceanu, G.: Sur une théorie unitaire non holonome des champs physiques, in:
Journal de Physique et le Radium 12 (1936), 514–526.The field equations for gravitation and electromagnetism 7
to say not only ‘matter’in the ordinary sense, but the electromagnetic field as
well.
Following Einstein, matter and gravitational field are complementary to
each other. In other words, all but gravitational field is matter or there is
no third between matter and gravitational field, a third is not given. Matter is
the complementary of gravitational field and vice versa. The gravitational field
is the complementary of matter. Again, each of both is equally the comple-
mentary of its own other. The following figure may illustrate this relationship
in more detail.
Gravitational field
Matter
Fig. 2. Matter and gravitational field.
Definition 5 (Laue’s scalar T) Max von Laue (1879-1960) proposed the
meanwhile so called Laue scalar22 (criticised23 by Einstein) as the contraction
of the the stress–energy momentum tensor Tµν denoted as T and written with-
out subscripts or arguments. Under conditions of Einstein’s general24 theory
of relativity, it is
T ≡ g µν × T µν (6)
where Tµν “denotes the co-variant energy tensor of matter” 25 . In other words,
“Considered phenomenologically, this energy tensor is composed of that of the
electromagnetic field and of matter in the narrower sense.” 26
Definition 6 (Einstein’s field equations ) Let Rµν denote the Ricci ten-
sor27 of ‘Einstein’s general theory of relativity’28 , a geometric object developed
by Gregorio Ricci-Curbastro (1853 – 1925) able to measure of the degree to
22 Laue, M.: Zur Dynamik der Relativitätstheorie, in: Annalen der Physik 8 (1911), 524–
542.
23 Einstein, Albert / Grossmann, Marcel: Entwurf einer verallgemeinerten Relativitäts-
theorie und einer Theorie der Gravitation: Physikalischer Teil von Albert Einstein. Mathe-
matischer Teil von Marcel Grossmann, Leipzig 1913.
24 Einstein, A.: Die Feldgleichungen der Gravitation (1915); Einstein, A.: Die Grundlage
der allgemeinen Relativitätstheorie (1916); Einstein, A.: Kosmologische Betrachtungen zur
allgemeinen Relativitätstheorie (1917); Einstein, A. / de Sitter, W.: On the Relation
between the Expansion and the Mean Density of the Universe (1932); Einstein, A.: Ele-
mentary Derivation of the Equivalence of Mass and Energy (1935).
25 Einstein, A.: The meaning of relativity. Four lectures delivered at Princeton University,
May, 1921 (1923), p. 88.
26 Ibid., p. 93.
27 Ricci, M. M. G. / Levi-Civita, T.: Méthodes de calcul différentiel absolu et leurs
applications (1900).
28 Einstein, A.: Die Grundlage der allgemeinen Relativitätstheorie (1916).8 Ilija Barukčić
which a certain geometry of a given metric differs from that of ordinary Eu-
clidean space. Let R denote the Ricci scalar, the trace of the Ricci curvature
tensor with respect to the metric and equally the simplest curvature invariant
of a Riemannian manifold. Ricci scalar curvature is the contraction of the Ricci
tensor and is written as R without subscripts or arguments. Let Λ denote the
Einstein’s cosmological constant. Let Λ denote the “anti cosmological con-
stant” 29 . Let gµν metric tensor of Einstein’s general theory of relativity. Let
Gµν denote Einstein’s curvature tensor. Let Gµν denote the “anti tensor” 30
of Einstein’s curvature tensor. Let Eµν denote stress-energy tensor of energy.
Let Eµν denote tensor of non-energy, the anti-tensor of the stress-energy tensor
of energy. Let aµν , bµν , cµν and dµν denote the four basic fields of nature were
aµν is the stress-energy tensor of ordinary matter, bµν is the stress-energy ten-
sor of the electromagnetic field. Let c denote the speed of the light in vacuum,
let γ denote Newton’s gravitational “constant” 31 . Let π denote the number
pi. Einstein’s field equation, published by Albert Einstein32 for the first time
in 1915, and finally 191633 but later with the “cosmological constant” 34
term are determined as
R 4×2×π×γ
Rµν − × g µν + (Λ × g µν ) ≡ × T µν ≡ E µν (7)
2 c4
Definition 7 (The tensor of non-energy) Under conditions of Einstein’s
general35 theory of relativity, the tensor of non-energy or the anti tensor of
the stress energy tensor is defined/derived/determined as follows:
29 Barukčić, I.: Anti Einstein – Refutation of Einstein’s General Theory of Relativity
(2015).
30 Idem: Unified Field Theory, in: Journal of Applied Mathematics and Physics 08 (2016),
1379–1438.
31 Idem: Anti Newton - Refutation Of The Constancy Of Newton’s Gravitatkional Con-
stant G: List of Abstracts, in: Quantum Theory: from Problems to Advances (QTPA 2014)
: Växjö, Sweden, June 9-12, 2014 (2014), 63; idem: Anti Einstein – Refutation of Einstein’s
General Theory of Relativity (2015); idem: Newton’s Gravitational Constant Big G Is Not
a Constant, in: Journal of Modern Physics 06 (2016), 510–522; idem: Unified Field Theory
(2016).
32 Einstein, A.: Die Feldgleichungen der Gravitation (1915).
33 Einstein, A.: Die Grundlage der allgemeinen Relativitätstheorie (1916).
34 Einstein, A.: Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie (1917);
Einstein, A. / de Sitter, W.: On the Relation between the Expansion and the Mean
Density of the Universe (1932); Einstein, A.: Elementary Derivation of the Equivalence of
Mass and Energy (1935).
35 Einstein, A.: Die Feldgleichungen der Gravitation (1915); Einstein, A.: Die Grundlage
der allgemeinen Relativitätstheorie (1916); Einstein, A.: Kosmologische Betrachtungen zur
allgemeinen Relativitätstheorie (1917); Einstein, A. / de Sitter, W.: On the Relation
between the Expansion and the Mean Density of the Universe (1932); Einstein, A.: Ele-
mentary Derivation of the Equivalence of Mass and Energy (1935).The field equations for gravitation and electromagnetism 9
4×2×π×γ
E µν ≡ Rµν − × T µν
c4
R
≡ × g µν − (Λ × g µν )
2 (8)
R
≡ − Λ × g µν
2
≡ cµν + dµν
Definition 8 (Einstein’s tensor Gµν ) Under conditions of Einstein’s gen-
eral36 theory of relativity, Einstein’s tensor Gµν is defined/derived/determined
as follows:
R
Gµν ≡ Rµν − × g µν
2
(9)
R
≡ S− × g µν
2
≡ aµν + cµν
Definition 9 (The anti Einstein’s curvature tensor or the tensor or
non-curvature ) Under conditions of Einstein’s general37 theory of relativity,
the tensor of non-curvature is defined/derived/determined as follows:
Gµν ≡ Rµν − Gµν
R
≡ Rµν − Rµν − × g µν
2
(10)
R
≡ × g µν
2
≡ bµν + dµν
Definition 10 (The stress-energy tensor of ordinary matter aµν ) Un-
der conditions of Einstein’s general38 theory of relativity, the stress-energy
tensor of ordinary matter aµν is defined/derived/determined as follows:
36 Einstein, A.: Die Feldgleichungen der Gravitation (1915); Einstein, A.: Die Grundlage
der allgemeinen Relativitätstheorie (1916); Einstein, A.: Kosmologische Betrachtungen zur
allgemeinen Relativitätstheorie (1917); Einstein, A. / de Sitter, W.: On the Relation
between the Expansion and the Mean Density of the Universe (1932); Einstein, A.: Ele-
mentary Derivation of the Equivalence of Mass and Energy (1935).
37 Einstein, A.: Die Feldgleichungen der Gravitation (1915); Einstein, A.: Die Grundlage
der allgemeinen Relativitätstheorie (1916); Einstein, A.: Kosmologische Betrachtungen zur
allgemeinen Relativitätstheorie (1917); Einstein, A. / de Sitter, W.: On the Relation
between the Expansion and the Mean Density of the Universe (1932); Einstein, A.: Ele-
mentary Derivation of the Equivalence of Mass and Energy (1935).
38 Einstein, A.: Die Feldgleichungen der Gravitation (1915); Einstein, A.: Die Grundlage
der allgemeinen Relativitätstheorie (1916); Einstein, A.: Kosmologische Betrachtungen zur
allgemeinen Relativitätstheorie (1917); Einstein, A. / de Sitter, W.: On the Relation
between the Expansion and the Mean Density of the Universe (1932); Einstein, A.: Ele-
mentary Derivation of the Equivalence of Mass and Energy (1935).10 Ilija Barukčić
aµν ≡ aµν + bµν − bµν
≡ aµν + cµν − cµν
(11)
≡ aµν − dµν + dµν
≡ Rµν − bµν − cµν − dµν
or equally as
4×2×π×γ
aµν ≡ × T µν − bµν
c4
(12)
≡ Gµν + (Λ × g µν ) − bµν
≡ Rµν − (R × g µν ) + (Λ × g µν ) + dµν
or as
R
aµν ≡ Rµν − × g µν + (Λ × g µν )
2
(13)
1 1
× F µ c × F ν d × g cd − × g µν × F de × F de
−
4×π 4
Definition 11 (The tensor bµν ) The co-variant stress-energy tensor of the
electromagnetic field, in this context denoted by bµν , is of order two and its
components can be displayed by a 4 × 4 matrix too. Under conditions of
Einstein’s general39 theory of relativity, the tensor bµν denotes the stress-
energy tensor of the electromagnetic field40 expressed more compactly and in
a coordinate-independent form as
1 1
× F µ c × F ν d × g cd − × g µν × F de × F de
bµν ≡
4×π 4
1
× (F µ c × F µ c ) − F de × F de × g µν
≡ (14)
4×4×π
R
≡ × g µν − dµν
2
where Fde is called the (traceless) Faraday/electromagnetic/field strength ten-
sor.
39 Einstein, A.: Die Feldgleichungen der Gravitation (1915); Einstein, A.: Die Grundlage
der allgemeinen Relativitätstheorie (1916); Einstein, A.: Kosmologische Betrachtungen zur
allgemeinen Relativitätstheorie (1917); Einstein, A. / de Sitter, W.: On the Relation
between the Expansion and the Mean Density of the Universe (1932); Einstein, A.: Ele-
mentary Derivation of the Equivalence of Mass and Energy (1935).
40 Hughston, L. P. / Tod, K. P.: An introduction to general relativity (London Mathe-
matical Society student texts 5), Cambridge ; New York 1990, p. 38.The field equations for gravitation and electromagnetism 11
Definition 12 (The tensor cµν ) Under conditions of Einstein’s general41
theory of relativity, the tensor of non-momentum and curvature is defined /
derived / determined as follows:
cµν ≡ bµν − (Λ × g µν ) (15)
Definition 13 (The tensor dµν (neither curvature nor momentum))
Under conditions of Einstein’s general42 theory of relativity, the tensor of
neither curvature nor momentum is defined/derived/determined as follows:
R
dµν ≡ × g µν − bµν
2
(16)
R
≡ × g µν − (Λ × g µν ) − cµν
2
There may exist circumstances where this tensor indicates pure vacuum, the
space devoid of any matter.
Table 1 provides an overview of the definitions of the four basic43 fields of
nature.
Curvature
YES NO
Momentum YES aµν bµν Eµν
NO cµν dµν Eµν
Gµν Gµν Rµν
Table 1 Einstein field equations and the four basic fields of nature
41 Einstein, A.: Die Feldgleichungen der Gravitation (1915); Einstein, A.: Die Grundlage
der allgemeinen Relativitätstheorie (1916); Einstein, A.: Kosmologische Betrachtungen zur
allgemeinen Relativitätstheorie (1917); Einstein, A. / de Sitter, W.: On the Relation
between the Expansion and the Mean Density of the Universe (1932); Einstein, A.: Ele-
mentary Derivation of the Equivalence of Mass and Energy (1935).
42 Einstein, A.: Die Feldgleichungen der Gravitation (1915); Einstein, A.: Die Grundlage
der allgemeinen Relativitätstheorie (1916); Einstein, A.: Kosmologische Betrachtungen zur
allgemeinen Relativitätstheorie (1917); Einstein, A. / de Sitter, W.: On the Relation
between the Expansion and the Mean Density of the Universe (1932); Einstein, A.: Ele-
mentary Derivation of the Equivalence of Mass and Energy (1935).
43 Barukčić, I.: Unified Field Theory (2016); idem: The Geometrization of the Electro-
magnetic Field (2016).12 Ilija Barukčić
Definition 14 (The inverse metric tensor gµν and the metric tensor
gµν ) Under conditions of Einstein’s general44 theory of relativity, it is45 :
g µν × g µν ≡ +4 (17)
or
1 1
µν
≡ (18)
g µν ×g 4
where gµν is the matrix inverse of the metric tensor gµν . The inverse metric
tensor or the metric tensor, which is always symmetric, allow tensors to be
transformed into each other.
Einstein’s point of view is that “... in the general theory of relativity ...
must be ... the tensor gµν of the gravitational potential” 46
Definition 15 (Index raising) For an order-2 tensor, twice multiplying by
the contra-variant metric tensor and contracting in different indices47 raises
each index. In simple words, it is
1 3 1 2 3 4
F( µ c ) ≡ g( µ ν ) × g( c d )×F
( ν d
) (19)
2 4
or more professionally
F µ c ≡ g µν × g cd × F ν d (20)
44 Einstein, A.: Die Feldgleichungen der Gravitation (1915); Einstein, A.: Die Grundlage
der allgemeinen Relativitätstheorie (1916); Einstein, A.: Kosmologische Betrachtungen zur
allgemeinen Relativitätstheorie (1917); Einstein, A. / de Sitter, W.: On the Relation
between the Expansion and the Mean Density of the Universe (1932); Einstein, A.: Ele-
mentary Derivation of the Equivalence of Mass and Energy (1935).
45 Idem: Die Grundlage der allgemeinen Relativitätstheorie (1916), p. 796.
46 Einstein, A.: The meaning of relativity. Four lectures delivered at Princeton University,
May, 1921 (1923), p. 88.
47 Kay, David C.: Schaum’s outline of theory and problems of tensor calculus (Schaum’s
outline series. Schaum’s outline series in mathematics), New York 1988.The field equations for gravitation and electromagnetism 13
3.2 Axioms
3.3 Axioms in general
Axioms48 and rules which are chosen carefully can be of use to avoid logical
inconsistency and equally preventing science from supporting particular ide-
ologies. Rightly or wrongly, long lasting advances in our knowledge of nature
are enabled by suitable axioms49 too. Einstein himself brings it again to the
point.50
“Die wahrhaft großen Fortschritte der Naturerkenntnis sind
auf einem der Induktion fast diametral entgegengesetzten Wege
entstanden.” 51
Einstein’s previous position now been translated into English: The truly
great advances in our understanding of nature originated in a manner almost
diametrically opposed to induction. It is worth mentioning in this matter, Ein-
stein himself advocated especially basic laws (axioms) and conclusions derived
from the same as a main logical foundation of any ‘theory’.
“Grundgesetz (Axiome)
und
Folgerungen
zusammen bilden das was man
eine ‘Theorie’
nennt. ” 52
Albert Einstein’s (1879-1955) message translated into English as: Basic
law (axioms) and conclusions together form what is called a ‘theory’ has still
to get round. However, it is currently difficult to ignore completely these his-
torical and far reaching words of wisdom. The same taken more seriously and
put into practice, will yield an approach to fundamental scientific problems
which is more creative and sustainably logically consistent. Historically, Aris-
totle himself already cited the law of excluded middle and the law of
contradiction as examples of axioms. However, lex identitatis is an axiom
too, which possess the potential to serve as the most basic and equally as the
48 Hilbert, David: Axiomatisches Denken, in: Mathematische Annalen 1 (1917), 405–415.
49 Easwaran, Kenny: The Role of Axioms in Mathematics, in: Erkenntnis 3 (2008), 381–
391.
50 see Einstein, Albert: Induktion and Deduktion in der Physik, in: Berliner Tageblatt
and Handelszeitung (1919), Suppl. 4, hier p. 17.14 Ilija Barukčić
most simple axiom of science. Something which is really just itself is equally
different from everything else. In point of fact, is such an equivalence which
everything has to itself inherent or must the same be constructed by human
mind and consciousness. Following Gottfried Wilhelm Leibniz (1646-1716):
“Chaque chose est ce qu’elle est.
Et dans autant d’exemples qu’on voudra
A est A, B est B. ” 53
or A = A, B = B or +1 = +1. In this context, lex contradictionis,
the negative of lex identitatis, or +0 = +1 is of no minor importance too.
3.3.1 Axiom I. Lex identitatis
To say that +1 is identical to +1 is to say that both are the same.
Axiom 1. Lex identitatis.
(g µν × g µν ) − 3 ≡ +1 (21)
or
+1 ≡ +1 (22)
However, even such a numerical identity which seems in itself wholly unprob-
lematic, for it indicates just to a relation which something has to itself and
nothing else, is still subject to controversy. Another increasingly popular view
is that the same numerical identity implies the controversial view that we are
talking about two different numbers +1. The one +1 is on the left side on
the equation, the other +1 is on the right side of an equation. The basicness
of the relation of identity implies the contradiction too while circularity is
avoided. In other words, how can the same +1 be identical with itself and be
equally different from itself? We may usefully state that identity is an utterly
problematic notion and might be the most troublesome of all.
3.3.2 Axiom II. Lex contradictionis
Axiom 2. Lex contradictionis.
(g µν × g µν − 3) − (g µν × g µν − 3) ≡ +1 (23)
or
+0 ≡ +1 (24)
A considerable obstacle to understanding contemporary usage of the term
contradiction, however, is that contradiction does not seem to be a unitary
one. How can something be both, itself (a path is a straight line from the
standpoint of a co-moving observer at a certain point in space-time) and theThe field equations for gravitation and electromagnetism 15
other of itself, its own opposition (the same path is not a straight line, the
same path is curved, from the standpoint of a stationary observer at a certain
point in space-time)54 . We may simply deny the existence of objective or of
any other contradictions. Furthermore, even if it remains especially according
to Einstein’s special theory of relativity that it is not guaranteed that the
notion of an absolute contradiction is justified, Einstein’s special theory of
relativity insist that contradictions are objective and real. That this is so
highlights the fact that from the standpoint of a co-moving observer, under
certain circumstances, a path is a straight line and nothing else. However,
under the same circumstances of special theory of relativity where the relative
velocity v > 0, from the standpoint of a stationary observer the same path
is a not a straight line, the path is curved. The justified question is, why
should and how can an identical be a contradictory too?
3.3.3 Axiom III. Lex negationis
Axiom 3. Lex negationis.
¬(+0) × 0 = (+1) (25)
where ¬ denotes the (natural “determinatio negatio est” 55 / logical56 )
process of negation.
54 Barukčić, Ilija: Aristotle’s law of contradiction and Einstein’s special theory of relativ-
ity, in: Journal of Drug Delivery and Therapeutics 2 (2019), 125–143.
55 see Spinoza, Benedictus De: Opera quae supersunt omnia / iterum edenda curavit,
praefationes, vitam auctoris, nec non notitias, quae ad historiam scriptorum pertinent, Ienae:
In Bibliopolio Academico, Heinrich Eberhard Gottlob Paulus, 1802, p. 634.
56 Boole, George: An investigation of the laws of thought, on which are founded mathe-
matical theories of logic and probabilities 1854.16 Ilija Barukčić
4 Results
Theorem 1 (The field cµν + dµν )
Claim.
If the premise
+1 = +1 (26)
| {z }
(P remise)
is true, then the conclusion
R
cµν + dµν ≡ × g µν − (Λ × g µν ) (27)
2
is also true, the absence of any technical errors presupposed.
Proof The premise
(+1) = (+1) (28)
of this theorem is true. Multiplying the premise of this theorem by the Ricci
tensor Rµν it is
1 × Rµν ≡ 1 × Rµν (29)
or
Rµν ≡ Rµν (30)
According to definition 2, this equation is rearranged as
aµν + bµν + cµν + dµν ≡ Rµν (31)
The sum of the tensors cµν + dµν follows as
cµν + dµν ≡ Rµν − aµν − bµν (32)
or as
cµν + dµν ≡ Rµν − (aµν + bµν ) (33)
or according to definition 4 as
2×π×γ×T
cµν + dµν ≡ Rµν − × g µν (34)
c4
Based on Einstein’s field equations(definition 6), this equation is equivalent
with
R
cµν + dµν ≡ × g µν − (Λ × g µν ) (35)
2
t
uThe field equations for gravitation and electromagnetism 17
Theorem 2 (Einstein’s tensor Gµν )
Claim.
If the premise
+1 = +1 (36)
| {z }
(P remise)
is true, then the conclusion
aµν + cµν ≡ Gµν (37)
is also true, the absence of any technical errors presupposed.
Proof The premise
(+1) = (+1) (38)
of this theorem is true. Multiplying the premise of this theorem by the Ricci
tensor Rµν it is
1 × Rµν ≡ 1 × Rµν (39)
or
Rµν ≡ Rµν (40)
According to definition 2, this equation is rearranged as
aµν + bµν + cµν + dµν ≡ Rµν (41)
The sum of the tensors aµν + cµν follows as
aµν + cµν ≡ Rµν − bµν − dµν (42)
or as
aµν + cµν + 0 ≡ Rµν + 0 − bµν − dµν (43)
or as
R R
aµν + cµν + 0 ≡ Rµν − × g µν + × g µν − bµν − dµν (44)
2 2
or according to definition 8
R
aµν + cµν + 0 ≡ Gµν + × g µν − bµν − dµν (45)
2
Under conditions in nature where
R
× g µν − bµν − dµν ≡ 0 (46)
2
equation 45 can be rearranged and the tensor Gµν need to be decomposed into
the tensors aµν and cµν as
+aµν + cµν + 0 ≡ Gµν + 0 (47)
and at the end as
+aµν + cµν ≡ Gµν (48)18 Ilija Barukčić
t
u
Remark 1 At least since gravitational waves were finally captured57 , it appears
to be that general relativity theory is more and more on the way to become one
of the icons of modern science. Therefore and in connection with other aspects,
we might infer from this theorem thatEinstein’s
tensor can be decomposed
un-
R
der some well defined circumstances × g µν − bµν − dµν ≡ 0 very
2
precisely. Under theses conditions, it is (Gµν ≡ +aµν + cµν ). However, even
if no sweeping generalisation can be made that this theorem (theorem 2) is
generally valid, this theorem would be of little
or none worth if there
were
R
no circumstances where the condition × g µν − bµν − dµν ≡ 0 is
2
valid. Because of these objections, the phrase is always right that such a re-
grettable generalisation is wrong as long as it is not possible to verify the
result of this theorem (equation 47) through other theoretical or experimen-
tal
investigations.
However and in deductivist style, under conditions where
R
× g µν − bµν − dµν ≡ 0 the propositions are true and the infer-
2
R
ences are valid. It is × g µν − bµν − dµν ≡ 0. Further research
2
may provide evidence, whether this requirement is generally valid.
57 Abbott, B. P.: Observation of Gravitational Waves from a Binary Black Hole Merger,
in: Physical Review Letters 6 (2016), 061102–1 – 061102–16.The field equations for gravitation and electromagnetism 19
Theorem 3 (The unity of gravitation and electromagnetism bµν +
cµν ) What is gravitation? What is electromagnetism? Do they exist indepen-
dently of each other and the geometry of space-time? Historically, the trial to
describe all fundamental interactions in the sense of a unified field theory by
objects of space-time geometry is ascribed especially to Hermann Klaus Hugo
Weyl (1885-1955)58 and Sir Arthur Stanley Eddington (1882-1944)59 . Weyl’s
generalisation of Riemannian geometry and his classical geometrical approach
connected the gravitational and the electromagnetic fields as represented by the
metrical field and a vector field. Eddington tried to provide physics with new
foundations that covered everything in the physical world, from the universe
at large to the tiny particle. However, one of the grand failures of Eddington’s
philosophy of physics was that the laws of physics were treated as construc-
tions60 of the physicists and were not expressions of processes or regularities
in an external, objective and real world. Kaluza’s61 introduced a five dimension
in order to approach to the unification of gravitation and electromagnetism. In
point of fact and from an epistemological standpoint, neither Einstein nor the
majority of other physicist were impressed by these and similar contributions.
Claim.
If the premise
+1 = +1 (49)
| {z }
(P remise)
is true, then the conclusion
1 µ c de
bµν + cµν ≡ × (F µ c × F ) − F de × F − Λ × g µν (50)
8×π
is also true, the absence of any technical errors presupposed.
Proof The premise
(+1) = (+1) (51)
of this theorem is true. Multiplying the premise of this theorem by the Ricci
tensor Rµν it is
2×π×γ×T 2×π×γ×T
1× × g µν ≡ 1 × × g µν (52)
c4 c4
or
2×π×γ×T 2×π×γ×T
× g µν ≡ × g µν (53)
c4 c4
58 Weyl, Hermann: Reine Infinitesimalgeometrie, in: Mathematische Zeitschrift 3 (1918),
384–411; idem: Raum. Zeit. Materie. Vorlesungen über allgemeine Relativitätstheorie,
Dritte, umgearbeitete Auflage, 1919.
59 Eddington, Arthur Stanley: Space, time and gravitation : an outline of the general
relativity theory 1920; idem: The philosophy of physical science, Cambridge (England) 1939.
60 Ibid.
61 Kaluza, T: Zum Unitätsproblem der Physik, in: Sitzungsber. Königl. Preuss. Akad.
Wiss. (1921), 966–972.20 Ilija Barukčić
Equation 53 changes according to equation 5 (defintion 4)
2×π×γ×T
(Gµν + (Λ × g µν )) ≡ × g µν (54)
c4
According to equation 48 (theorem 2) it is +aµν + cµν ≡ Gµν . Equation 54 can
be rearranged as
2×π×γ×T
(+aµν + cµν ) + (Λ × g µν ) ≡ × g µν (55)
c4
2×π×γ×T
Equation 5 (definition 4) is +aµν + bµν ≡ × g µν . There-
c4
fore, equation 55 can be rearranged as
(+aµν + cµν ) + (Λ × g µν ) ≡ (+aµν + bµν ) (56)
Equation 56 simplifies as
(+cµν ) + (Λ × g µν ) ≡ (+bµν ) (57)
The tensor of the gravitational field (cµν ) as determined by ordinary matter
(aµν ) follows as
(+cµν ) ≡ (+bµν ) − (Λ × g µν ) (58)
or in more detail as
(+cµν ) ≡ (+bµν ) − (Λ × g µν ) (59)
1
It is bµν ≡ × (F µ c × F µ c ) − F de × F de × g µν as demonstrated
4×4×π
by equation 14 (definition 11). Equation 59 becomes more concrete as
1 µ c de
(+cµν ) ≡ × (F µ c × F ) − F de × F × g µν − (Λ × g µν )
4×4×π
(60)
or as
1
× (F µ c × F µ c ) − F de × F de
cµν ≡ − Λ × g µν (61)
16 × π
Equation 59 demands that cµν ≡ bµν − (Λ × g µν ) Adding the tensor bµν to
equation 59 yields the tensor of gravitation and electromagnetism as
bµν + cµν ≡ bµν + bµν − (Λ × g µν ) (62)
or in more detail
1 µ c de
bµν +cµν ≡ 2× × (F µ c × F ) − F de × F × g µν −(Λ × g µν )
4×4×π
(63)The field equations for gravitation and electromagnetism 21
or
1
× (F µ c × F µ c ) − F de × F de × g µν
bµν + cµν ≡ − (Λ × g µν )
8×π
(64)
and finally
1 µ c de
bµν + cµν ≡ × (F µ c × F ) − F de × F − Λ × g µν (65)
8×π
t
u
Remark 2 It is the successful geometrization of Einstein’s field equations which
is the foundation of “. . . joining the gravitational and the electromag-
netic field into one single hyperfield whose equations represent the
conditions imposed on the geometrical structure . . . ” 62
5 Discussion
Historically, there have been many different attempts63 to unify Maxwell’s
theory of the electromagnetic64 field with Einstein’s general theory of grav-
itation65 into a unified field theory66 . A geometrization of the stress-energy
tensor of the electromagnetic field and the stress-energy tensor of matter of
Einstein’s field equations in particular has been an indispensable foundation
for the development of a completely geometrized67 theory of relativ-
ity in order to overcome the difficulties inherent in this matter. Advantage
is taken of a previous gemetriziation68 of Einstein’s field equation. A gener-
ally covariant field equation
for gravitation and electromagnetism is devel-
1 µc de
oped as bµν + cµν ≡ × (F µ c × F ) − F de × F − Λ × g µν
8×π
by considering especially the relationship to the metric tensor gµν . In order
to formulate a logically consistent theory of quantum gravity or an unified
field theory, it is necessary to change drastically several points of view. In this
context, equation 65 justifies the question, does the gravitational field itself
possess any kind of an energy density, stress et cetera or not? Einstein himself
introduced a mathematical object in his final general theory of relativity for
describing the energy components of the gravitational field by tσ α . “Es ist
62 see Tonnelat, Marie-Antoinette: La théorie du champ unifié d’Einstein et quelques-uns
de ses développements, Paris (France) 1955, p. 5.
63 Barukčić, I.: Unified Field Theory (2016); idem: The Geometrization of the Electro-
magnetic Field (2016).
64 Maxwell, J. C.: On physical lines of force, in: Philosophical Magazine (1861), 11–23.
65 Einstein, A.: Die Feldgleichungen der Gravitation (1915); Einstein, A.: Die Grundlage
der allgemeinen Relativitätstheorie (1916); Einstein, A.: Kosmologische Betrachtungen zur
allgemeinen Relativitätstheorie (1917); Einstein, A. / de Sitter, W.: On the Relation
between the Expansion and the Mean Density of the Universe (1932).
66 Einstein, A.: Einheitliche Feldtheorie von Gravitation und Elektrizität (1925).
67 Einstein, A.: On the Generalized Theory of Gravitation (1950).
68 Barukčić, I.: Einstein’s field equations and non-locality (2020).22 Ilija Barukčić zu beachten dass tσ α kein Tensor ist; . . . Die Größen tσ α bezeichnen wir als die ‘Energiekomponenten’des Gravitationsfelds. ” 69 However, soon Schrödinger70 and Bauer71 showed that the energy components of the gravitational field are equal to zero when the same should not be, and are not equal to zero when the same should be. The problem of the energy of the gravity field in general relativity appears to be solved or is at least solvable by equation 65. It is to be noted, that per definitionem all energy, radiation, stress et cetera is included exclusively within the stress-energy tensor of mat- ter, denoted by Eµν = aµν + bµν . Thus far, there is and cannot be any further energy left, which can be treated as being part of the gravitational field cµν itself. Therefore, the energy components of the gravitational field72 of Einstein’s general theory of relativity and require a further and definite clarification. 69 see Einstein, A.: Die Grundlage der allgemeinen Relativitätstheorie (1916), p. 806. 70 Schrödinger, Erwin: Die Energiekomponenten des Gravitationsfeldes, in: Physikalische Zeitschrift (1918), 4–7. 71 Bauer, Hans Adolf: Über die Energiekomponenten des Gravitationsfeldes, in: Physikalische Zeitschrift (1918), 163–165. 72 Ibid.; Schrödinger, E.: Die Energiekomponenten des Gravitationsfeldes (1918); Ein- stein, Albert: Notiz zu E. Schrödingers Arbeit: „Die Energiekomponenten des Gravitations- feldes“, in: Physikalische Zeitschrift (1918), 115–116.
The field equations for gravitation and electromagnetism 23 6 Conclusion If was possible to couple the stress-energy tensor of electromagnetic field (bµν ) and the tensor of the gravitational field (cµν ) to the metric tensor (gµν )itself and to unify both fields into a hyper-field of gravitation and electromagnetism. Acknowledgements Zotero (https://www.zotero.org/) is a free and easy-to-use tool which was off great help for me to collect, organize, cite, and share research. Overleaf (https://de.overleaf.com/), an online LaTeX Editor and GnuPlot were of no minor help too. I am very grateful for the possibility for being able to use these tools. Author contributions statement The author is the only author of this manuscript. Conflict of Interest Statement The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. There are no conflict of interest. Financial support and sponsorship Nil. References Newton, Isaac: Philosophiae naturalis principia mathematica, Londini 1687 url: https://doi.org/10.5479/sil.52126.39088015628399 (visited on 02/12/2019). Mie, Gustav: Grundlagen einer Theorie der Materie, in: Annalen der Physik 3 (1912), 511–534 url: https : / / onlinelibrary . wiley . com / doi / abs / 10 . 1002 / andp . 19123420306 (visited on 06/30/2020). (_eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.19123420306) Reichenbächer, Ernst: Grundzüge zu einer Theorie der Elektrizität und der Gravitation, in: Annalen der Physik 2 (1917), 134–173 url: https : / / onlinelibrary.wiley.com/doi/abs/10.1002/andp.19173570203 (vis- ited on 06/30/2020). (_eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.19173570203)
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