N-th index D dimensional Einstein gravitational field equations

Abstract

Aim: The possibility of the geometrization of the 2-index, 4-index and the n-th index Einstein field equations under conditions of D space-time dimensions has been investigated. Methods: The usual tensor calculus rules were used. New rules of tensor calculus were developed too. Results: The stress-energy-tensor 8 × π × γ c 4 × T µν has been geometrized under conditions of D dimensions. The Ricci tensor Rµν has been expressed completely in terms of the metric tensor gµν under conditions of D dimensions too. Based on the geometrized Einstein field equations under conditions of D dimensions, a procedure how to calculate the value of the cosmological constant Λ depending on the space-time dimensions D has been developed. Conclusion: The Einstein field equations under conditions of D dimensions open the door to a logically consistent unified field theory.
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Fermat's last theorem refuted.

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Full text

N-th index D dimensional Einstein gravitational field
equations
Ilija Barukˇci´c
Horandstrasse, DE-26441 Jever, Germany
E-mail: Barukcic@t-online.de
https://doi.org/10.5281/zenodo.4052842
Received: 27th September 2020- accepted: 27th September 2020- published: 27th September 2020 - version 3
Abstract. Aim: The possibility of the geometrization of the 2-index, 4-index and the n-
th index Einstein field equations under conditions of D space-time dimensions has been
investigated.
Methods: The usual tensor calculus rules were used. New rules of tensor calculus were developed
too.
Results: The stress-energy-tensor 8×π×γ
c4×Tµν has been geometrized under conditions of
D dimensions. The Ricci tensor Rµν has been expressed completely in terms of the metric tensor
gµν under conditions of D dimensions too. Based on the geometrized Einstein field equations
under conditions of D dimensions, a procedure how to calculate the value of the cosmological
constant Λ depending on the space-time dimensions D has been developed.
Conclusion: The Einstein field equations under conditions of D dimensions open the door to a
logically consistent unified field theory.

1. Introduction
In point of fact, Einstein’s geometrization of gravity [see 42], the gravitational field [see 17,
10] and the various proposals for a unified field theory [see 55, 18], “a generalization of the
theory of the gravitational field”[see 24], were not yet able to produce the desired scientific
success. In particular, the stress-energy momentum tensor of the electromagnetic field and
Einstein’s stress-energy momentum tensor of matter of the general [see 22, 17, 23, 21] theory
of relativity (GTR) are the weak spots of this theory because these fields are thus far devoid
[see 29] of any geometrical significance. However, in order to complete the geometrization of
the general theory of relativity [see 20] as started but not completed by Einstein himself, it is
necessary to geometrize the electromagnetic field and the stress energy tensor of matter too. It
is necessary to expressly clarify that the dominance of geometry in physics, of the metric tensor
gµν and today’s understanding of the gravitational field as something like the manifestation of
space - time curvature has closed our scientific horizon a little bit and prevents us to much
from the possibility of handing over to future (scientific) generations the description of the
gravitational field [see 10] by other mathematical tools [see 12, 11] than geometry. Generally
speaking, although both, ‘geometrization’ and ‘unification’ are not incompatible as such, both
need not to be (mathematically) conceptually identical either. The deep hope and believe that
a complete geometrization of the Einstein’s gravitational field equations even if a delicate and
fragile plant might end up at a unified field theory in the sense of Einstein or Weyl’s and
Eddington’s classical field theory in which all fundamental interactions are described by objects
of space-time geometry might comfort and console our tortured scientific soul.
2. Material and methods
The first confrontation between GTR and experiment occurred on May 29, 1919 by a momentous
expedition at Sobral in northern Brazil (lead by Crommelin), and on the island of Pr´ıncipe off
the coast of West Africa (lead by Eddington). Astronomical observations made by this special
2

British team (Crommelin and Eddington) during the total solar eclipse occurred on May 29,
1919 provided the first empirical test of the validity of Einstein’s general theory of relativity
as discussed by the Royal Society of London and the Royal Astronomical Society announced
at their joint meeting on the sixth of November 1919 [see 52, 15]. A deeper knowledge of the
foundations of nature and physics as such has potential to reduce and diminish our shadows of
doubts lengthened by time [11].
2.1. Definitions
Definition 2.1 (Anti tensor).Let aµν denote a co-variant (lower index) second-rank tensor.
Let bµν denote another co-variant second-rank et cetera. Let Eµν denote the sum of these co-
variant second-rank tensors. Let the relationship aµν +bµν +... ≡Eµν be given. A co-variant
second-rank anti tensor [8] of a tensor aµν denoted in general as aµν is defined
aµν ≡Eµν −aµν
≡bµν +...
(1)
Let aµν denote a contra-variant (upper index) second-rank tensor. Let bµν denote another
contra-variant (upper index) second-rank et cetera. Let Eµν denote the sum of these contra-
variant (upper index) second-rank tensors. Let the relationship aµν +bµν +... ≡Eµν be given.
A co-variant second-rank anti tensor of a tensor aµν denoted in general as aµν is defined
aµν ≡Eµν −aµν
≡bµν +...
(2)
Let aµνdenote a mixed second-rank tensor. Let bµνdenote another mixed second-rank et
cetera. Let Eµνdenote the sum of these mixed second-rank tensors. Let the relationship aµν+
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bµν+... ≡Eµνbe given. A mixed second-rank anti tensor of a tensor aµνdenoted in general
as aµνis defined
aµν≡Eµν−aµν
≡bµν+...
(3)
Symmetric tensors of rank 2 may represent many physical properties objective reality. A
co-variant second-rank tensor aµν is symmetric if
aµν ≡aνµ (4)
However, there are circumstances, where a tensor is anti-symmetric. A co-variant second-rank
tensor aµν is anti-symmetric if
aµν ≡ −aνµ (5)
Thus far, there are circumstances were an anti-tensor is identical with an anti-symmetrical
tensor.
aµν ≡Eµν −bµν +... ≡Eµν −aµν ≡ −aνµ (6)
Under conditions where Eµν = 0, an anti-tensor is identical with an anti-symmetrical tensor or
it is
−aµν ≡ −aνµ (7)
However, an anti-tensor is not identical with an anti-symmetrical tensor as such.
Definition 2.2 (The n-dimensional Pythagorean theorem).The Pythagorean theorem
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of Euclidean geometry is arguably the most famous statement in mathematics. Besides of the
rarity of original sources, the Pythagorean Theorem is attributed to the Greek mathematician and
thinker Pythagoras of Samos (6th century, B.C.), the first mathematician. Meanwhile
the Pythagorean Theorem is characterised by more than 371 proofs known. The Pythagorean
Theorem is defined by one of the most beautiful equations of mathematics as
oat2+obt2≡RCt2(8)
where omay denote the point of view of a co-moving observer while Rmay denote the point of
view of a stationary observer at a certain point in space-time t. In general, it is
oat2≡oxt×RCt(9)
or
oat2n ≡oxtn×RCtn(10)
Equally, it is
obt2≡oxt×RCt(11)
or
obt2n ≡oxtn×RCtn(12)
where n denotes the number of dimensions. The Pythagorean theorem can be extended to higher
dimensions [see 60] too. In general, the Pythagorean theorem is based on the fundamental
relationship [see 12]
oxt+oxt≡RCt(13)
5

In the n dimensional case, this relationship becomes
(oxt+oxt)n≡RCtn(14)
and the n dimensional Pythagorean theorem becomes something like
(oxt+oxt)n×RCtn≡RCtn×RCtn(15)
Remark 2.1. General relativity is a theory of the geometrical properties of space-time too while
the metric tensor gµν itself is of fundamental importance for general relativity. The metric
tensor gµν is something like the generalisation of the Pythagorean theorem. Thus far, it does
not appear to be necessary to restrict the validity of the Pythagorean theorem only to certain
situations. The question is justified why the Riemannian geometry should be oppressed by the
quadratic restriction. In this context, Finsler geometry, named after Paul Finsler (1894 -
1970) who studied it in his doctoral thesis [see 28] in 1918, appears to be a kind of a metric gen-
eralization of Riemannian geometry without the quadratic restriction and justifies the attempt
to systematize and to extend the possibilities of general relativity.
Definition 2.3 (Multiplication of tensors).The distance between any two points in a given
space can be measured by a kind of a generalized Pythagorean theorem, the metric tensor gµν .
Under conditions of general relativity, the metric tensor measures lengths in various directions,
and also angles between various directions and is a symmetric rank two tensor and equally more
than just a matrix.
Let gkl or gµν denote a 2-index metric tensors. Let gklµν denote a 4-index metric tensors.
Let gklµν . . . denote a n-th index metric tensor. The n-index metric tensor gklµν . . . itself is a
6

covariant symmetric tensor and equally an example of a tensor field. If we pause for a moment
today and rely on Einstein’s “Die Grundlage der allgemeinen Relativit¨
atstheorie ”[see 17, p.
784], it is
gklµν ≡gklgµν (16)
and in the case of n-th rank order
gklµν . . . ≡gklgµν. . . (17)
The mixed and contra-variant cases are similar. Riemann defined the distance between two
neighbouring points more or less by a quadratic differential form. The geometry based on the
positive definite Riemannian metric tensor is called the Riemannian geometry. However, tensor
calculus as a generalisation of classical linear algebra should assure that formulae are invariant
under coordinate transformations and that the same are independent of any kind of the rank
order of the metric tensor chosen.
Figure 1: Einstein. Multiplication of tensors [see 17, p. 784]
Definition 2.4 (Kronecker delta).The Kronecker delta [see 61], a notation invented by
Leopold Kronecker (1823-1891) in 1868 [see 36] appears in many areas of physics, mathematics,
7

and engineering and is defined as
gµρ ×gνρ ≡gµν≡δµν(18)
Technically, the Kronecker delta itself is a mixed second-rank tensor.
Figure 2: Einstein. The Kronecker delta (named after Leopold Kronecker) [see 17, p. 787]
The quantity
δii≡δ11+δ22+... +δNN≡N(19)
is an invariant.
Definition 2.5 (The metric tensor gµν and the inverse metric tensor gµν).Einstein
described (local) stress-energy and momentum by a tensor 8×π×γ
c4×Tµν . In the same
respect, the (local) space-time curvature has been described by Einstein as Rµν −R
2×gµν +
(Λ ×gµν ). Finally, Einstein has been of the opinion that there are conditions where (local) stress-
energy and momentum and (local) space-time curvature are related to each other. In general,
Einstein field equations relate (local) space-time curvature with (local) energy and momentum
by the equation
Rµν −R
2×gµν + (Λ ×gµν )
| {z }
(local)space−time curvatur e
≡4×2×π×γ
c4×Tµν
| {z }
(local)energy and momentum
(20)
The expression on the left side of Einstein field equations represents the curvature of space-time
as determined by the metric while the expression on the right side of Einstein field equations
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represents the matter–energy content of space-time. Mathematically, it is necessary to consider
circumstances that it is possible to take the trace with respect to the metric of both sides of the
Einstein field equations. Therefore, we define in general
gµν ×gµν ≡D(21)
where D might denote the number of space-time dimensions. Vectors and scalars are invariant
under coordinate transformations. In point of fact, Einstein field equations [22, 17, 23, 21,
19] were initially formulated by Einstein himself in the context of a four-dimensional theory
even though Einstein field equations need not to break down under conditions of D space-time
dimensions [see 51]. Nonetheless, based on Einstein’s statement [17, p. 796], one gets
gµν ×gµν ≡D≡+4 (22)
or
1
gµν ×gµν ≡1
4(23)
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 and
are used to lower and raise indices. Einstein’s point of view is that
“... in the general theory of relativity ... must be ... the tensor gµν of the gravitational
potential”
[25, p. 88]
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The inverse metric tensor gµν is of the same size as the metric tensor gµν. Thus far, whatever
gµν does, gµν undoes and their product is the identity.
Definition 2.6 (The metric tensor of special relativity ηµν ).There is a fundamental
difference between Special and General Relativity regarding the metric tensor. Let ηµν denote the
metric tensor of Einstein’s special theory of relativity. In general, depending upon circumstances,
it is ηµν =



















−1 0 0 0
0−1 0 0
0 0 −1 0
0 0 0 +1



















[see 17, p. 778]. Let ηµν denote the anti-metric tensor
of Einstein’s special theory of relativity. Let gµν denote the metric tensor of Einstein’s
general theory of relativity. In general, it is (see definition 2.1, equation 1)
gµν ≡ηµν +ηµν (24)
while the n-th index relationship follows (see definition 2.1, equation 1) as
gklµν. . . ≡ηklµν. . . +ηklµν . . . (25)
Remark 2.2. Einstein field equations changes according to theorem 3.3, equation 123
S≡R
D and equation 24 to
R
D−R
2+ (Λ)×ηµν +ηµν ≡8×π×γ×T
c4×D×ηµν +ηµν (26)
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From this follows that
R
D−R
2+ (Λ)×ηµν +R
D−R
2+ (Λ)×ηµν
≡8×π×γ×T
c4×D×ηµν +8×π×γ×T
c4×D×ηµν (27)
and equally
R
D×ηµν −R
2×ηµν + (Λ ×ηµν )+R
D×ηµν −R
2×ηµν +Λ×ηµν 
≡8×π×γ×T
c4×D×ηµν +8×π×γ×T
c4×D×ηµν (28)
However, even these equations can be adapted or simplified further . . . .
Definition 2.7 (Index raising).According to Einstein [see 17, p. 790] himself or Kay et al.
[35], an order-2 tensor, twice multiplied by the contra-variant metric tensor and contracted in
different indices raises each index. In simple words, it is
F(1 3
µ c )≡g(1 2
µ ν )×g(3 4
c d )×F(ν d
2 4 )(29)
or more professionally
Fµc≡gµν ×gcd ×Fνd(30)
Definition 2.8 (The Ricci tensor Rµν ).Let Rµν denote the Ricci tensor [45] of ‘Einstein’s
general theory of relativity’[17], 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
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Figure 3: Einstein. Index lowering and raising [see 17, p. 790]
were aµν is the stress-energy tensor of ordinary matter, bµν is the stress-energy tensor of the
electromagnetic field.
Rµν ≡4×2×π×γ
c4×Tµν 
| {z }
aµν +bµν
+R
2×gµν −(Λ ×gµν )
| {z }
cµν +dµν
≡(aµν +bµν )+(cµν +dµν)
≡(aµν +cµν )+(bµν +dµν)
≡(aµν ) + (+bµν +cµν +dµν )
≡(bµν ) + (+aµν +cµν +dµν )
≡(cµν ) + (+aµν +bµν +dµν )
≡(dµν ) + (+aµν +bµν +cµν )
≡aµν +bµν +cµν +dµν
≡R
D×gµν
(31)
Remark 2.3. In general relativity, it is common to present the Riemann and Ricci tensors by
the Christoffel symbols. However, Christoffel symbols are given through the metric tensor itself.
Therefore, giving the Ricci tensor while using the metric tensor explicitly, is theoretically possi-
ble. Equation (31) provide us with one way to present the Ricci tensor in terms of the metric
12

tensor directly as Rµν ≡R
D×gµν .
Definition 2.9 (The Ricci scalar R).Under conditions of Einstein’s general [22, 17, 23, 21,
19] 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 Λand anti-lamda
[3] Λas
R≡gµν ×Rµν ≡(Λ) + (Λ) ≡D×X(32)
where D is the number of space-time dimension and X≡R
D(see theorem 3.3, equation 123).
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×gµν ≡(Λ ×gµν ) + (Λ ×gµν ) (33)
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 2.10 (The stress-energy and momentum tensor Eµν).The tensor of stress-
energy-momentum denoted as Eµν is determined in detail as follows.
13

Eµν ≡4×2×π×γ
c4×Tµν
≡Rµν −R
2×gµν + (Λ ×gµν )
≡Gµν + (Λ ×gµν )
≡Rµν −Eµν
≡aµν +bµν
≡H×gµν ≡Hµν
≡E×gµν
(34)
where E is a scalar. 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 energy tensor of matter”[see
25, p. 88]. Importantly, also, and a bit more formally put, Einstein himself elaborates on
the fundamental relationship between matter in the narrower sense and the electromagnetic
field. “Considered phenomenologically, this energy tensor is composed of that of
the electromagnetic field and of matter in the narrower sense. ”[see 25, p. 93]. The
following graphic may illustrate this relationship.
Electromagnetic field bµν
Ordinary matter aµν
Figure 4. Energy tensor as identity of ordinary matter and electromagnetic field.
However, it is necessary and tricky to explicate the purported tie between classical logic
14

and relativity theory. In Einstein’s own 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. Aristotle’s law of
excluded middle (or the principle of excluded middle), in Latin principium tertii exclusi, is a
logical foundation of Einstein’s general theory of relativity. Vranceanu [see 53] is elaborating on
the same issue too. In point of fact, 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’´energie Tkl soit la somme de deux tenseurs dont
un dˆu au champ ´electromagn´etique . . . ”[see 53]
Figure 5. Vranceanu [see 53] .
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. Matter and ordinary matter are not completely the same.
Definition 2.11 (Laue’s scalar T).Max von Laue (1879-1960) proposed the meanwhile so
called Laue scalar [37] (criticised by Einstein [26] ) as the contraction of the the stress–energy
momentum tensor Tµν denoted as T and written without subscripts or arguments. Under
conditions of Einstein’s general [22, 17, 23, 21, 19] theory of relativity, it is
T≡gµν ×Tµν (35)
Taken Einstein seriously, Tµν “denotes the co-variant energy tensor of matter”[see 25, p.
15

88]. In other words, “Considered phenomenologically, this energy tensor is composed of that of
the electromagnetic field and of matter in the narrower sense.”[see 25, p. 93]
Definition 2.12 (The scalar E or H).In general, we define the scalar E or H as
E≡H≡8×π×γ
c4×D×T
≡8×π×γ×T
c4×D(36)
where D is the space-time dimension, where c denote the speed of the light in vacuum, γdenote
Newton’s gravitational “constant”[6, 3, 9, 12], πis the number pi and T denote Laue’s scalar.
The scalar E = H is corresponding to the total energy of a (relativistic or quantum) system and
has the potential as such to bridge the gap between relativity theory and quantum mechanics
under circumstances where the same is identical with the Hamiltonian operator.
Definition 2.13 (The 4-index D dimensional stress-energy and momentum tensor
Eklµν ).The 4-index D dimensional stress-energy-momentum tenosr denoted as Eklµν is
determined in detail as
Eklµν ≡8×π×γ×T
c4×D×gklµν
≡Rklµν −R
2×gklµν + (Λ ×gklµν )
≡Gklµν + (Λ ×gklµν )
≡Rklµν −Eklµν
≡aklµν +bklµν
≡H×gklµν ≡Hklµν
≡E×gklµν
(37)
Definition 2.14 (The n-index D dimensional stress-energy and momentum tensor
16

Eklµν ...).The n-index D dimensional stress-energy-momentum tenosr denoted as Eklµν . . . is
determined in detail as
Eklµν . . . ≡8×π×γ×T
c4×D×gklµν . . .
≡Rklµν . . . −R
2×gklµν . . . + (Λ ×gklµν . . . )
≡Gklµν . . . + (Λ ×gklµν . . . )
≡Rklµν . . . −Eklµν . . .
≡aklµν . . . +bklµν . . .
≡H×gklµν . . . ≡Hklµν . . .
≡E×gklµν . . .
(38)
Definition 2.15 (The tensor of non-energy Eµν ).Under conditions of Einstein’s general
[22, 17, 23, 21, 19] theory of relativity, the tensor of non-energy or the anti tensor of the stress
energy tensor is defined/derived/determined as follows:
Eµν ≡Rµν −4×2×π×γ
c4×Tµν
≡R
2×gµν −(Λ ×gµν )
≡R
2−Λ×gµν 
≡cµν +dµν
≡Ψ×gµν ≡Ψµν
≡E×gµν
(39)
Definition 2.16 (The scalar E or t or Ψ).In general, we define the scalar E or t or Ψas
17

[see 8]
E≡t≡Ψ≡R
D−E
≡R
2−Λ(40)
Remark 2.4. In the following of research it is appropriate to prove the relationship between
(1/X) and the complex conjugate of the wave function Ψ*or the identity (1/X)≡Ψ*.
Definition 2.17 (The 4-index D dimensional tensor of non-energy Eklµν ).The 4-index
D dimensional tensor [22, 17, 23, 21, 19] of non-energy Eklµν is defined as follows:
Eklµν ≡R
D×gklµν −8×π×γ×T
c4×D×gklµν 
≡R
2×gklµν −(Λ ×gklµν )
≡R
2−Λ×gklµν 
≡cklµν +dklµν
≡Ψ×gklµν ≡Ψklµν
≡E×gklµν
(41)
Definition 2.18 (The n-th index D dimensional tensor of non-energy Eklµν ...).The
18

n-th index D dimensional tensor [22, 17, 23, 21, 19] of non-energy Eklµν . . . is defined as follows:
Eklµν . . . ≡R
D×gklµν . . . −8×π×γ×T
c4×D×gklµν . . . 
≡R
2×gklµν . . . −(Λ ×gklµν . . . )
≡R
2−Λ×gklµν . . . 
≡cklµν . . . +dklµν . . .
≡Ψ×gklµν . . . ≡Ψklµν . . .
≡E×gklµν . . .
(42)
Definition 2.19 (The Einstein’s curvature tensor Gµν).Under conditions of Einstein’s
general [22, 17, 23, 21, 19] theory of relativity, the tensor of curvature denoted by Gµν is
defined/derived/determined [see 8] as follows:
Gµν ≡Rµν −R
2×gµν 
≡R
D×gµν −R
2×gµν 
≡R
D−R
2×gµν
≡aµν +cµν
≡G×gµν
(43)
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Definition 2.20 (The scalar G).In general, we define the scalar G [8] as
G≡R
D−R
2
≡E+t−R
2
≡E+R
2−Λ−R
2
≡E−Λ
(44)
Definition 2.21 (The 4-index D dimensional Einstein’s curvature tensor Gklµν).
The Riemann tensor Rklµν does not appear explicitly in Einstein’s gravitational field equations.
Therefore, the question is justified whether Einstein’s equation of gravitation are really the most
general equations. Fr˙ed˙eric Moulin proposed in the year 2017 a kind of a generalized 4-index
gravitational field equation which contains the Riemann curvature tensor linearly [41]. Moulin
himself ascribed an energy-momentum to the gravitational field itself [41, p. 5/8] which is not
without problems. Besides of all, it is known that the Riemann curvature tensor of general
relativity Rklµν can be split into different ways, including the Weyl conformal tensor Cklµν and
the anti-Weyl conformal tensor Cklµν or in other words the parts which involve only the Ricci
tensor Rµν the curvature scalar R. Because of these properties (Rklµν ≡Cklµν +Cklµν )it is
possible to reformulate the famous Einstein equation. The 4-index D dimensional Einstein’s
curvature tensor [22, 17, 23, 21, 19] denoted by Gklµν is defined [see 8] as follows:
20

Gklµν ≡Rklµν −R
2×gklµν 
≡R
D×gklµν −R
2×gklµν 
≡R
D−R
2×gklµν
≡aklµν +cklµν
≡G×gklµν
(45)
Definition 2.22 (The n-index D dimensional Einstein’s curvature tensor Gklµν ...).
The n-index D dimensional Einstein’s curvature tensor [22, 17, 23, 21, 19] denoted by Gklµν . . .
is defined [see 8] as follows:
Gklµν . . . ≡Rklµν . . . −R
2×gklµν . . . 
≡R
D×gklµν . . . −R
2×gklµν . . . 
≡R
D−R
2×gklµν . . .
≡aklµν . . . +cklµν . . .
≡G×gklµν . . .
(46)
Definition 2.23 (The anti Einstein’s curvature tensor or the tensor or non-curvature
Gµν ).Under conditions of Einstein’s general [22, 17, 23, 21, 19] theory of relativity, the tensor
of non-curvature is defined/derived/determined [8] as follows:
21

Gµν ≡Rµν −Gµν
≡Rµν −Rµν −R
2×gµν 
≡R
2×gµν
≡bµν +dµν
≡G×gµν
(47)
Definition 2.24 (The scalar G ).In general, we define the scalar G [see 8] as
G≡R
D−G
≡R
2(48)
Definition 2.25 (The 4-index D dimensional anti Einstein’s curvature tensor
or the tensor or non-curvature G klµν ).The 4-index D dimensional anti Einstein’s
curvature tensor [22, 17, 23, 21, 19] or the tensor of non-curvature denoted as G klµν is
defined/derived/determined [8] as follows:
Gklµν ≡Rklµν −Gklµν
≡Rklµν −Rklµν −R
2×gklµν 
≡R
2×gklµν
≡bklµν +dklµν
≡G×gklµν
(49)
Definition 2.26 (The n-index D dimensional anti Einstein’s curvature tensor or the
22

tensor of non-curvature G klµν ... ).The n-index D dimensional anti Einstein’s curvature
tensor or the tensor of non-curvature denoted as G klµν . . . is defined/derived/determined [8] as
follows:
Gklµν . . . ≡Rklµν . . . −Gklµν . . .
≡Rklµν . . . −Rklµν . . . −R
2×gklµν . . . 
≡R
2×gklµν . . .
≡bklµν . . . +dklµν . . .
≡G×gklµν . . .
(50)
Definition 2.27 (The stress-energy tensor of ordinary matter aµν ).Under conditions of
Einstein’s general [22, 17, 23, 21, 19] theory of relativity, the stress-energy tensor of ordinary
matter aµν is defined/derived/determined as follows:
aµν ≡4×2×π×γ
c4×Tµν −bµν
≡Gµν + (Λ ×gµν )−bµν
≡Rµν −(R×gµν ) + (Λ ×gµν ) + dµν
≡(E−b)×gµν
≡(G−c)×gµν
≡a×gµν
(51)
23

or
aµν ≡Rµν −R
2×gµν + (Λ ×gµν )−
1
4×π×Fµc×Fνd×gcd+1
4×gµν ×Fde ×Fde (52)
Definition 2.28 (The 4-index D dimensional a klµν).The 4-index D dimensional a klµν is
defined as:
aklµν ≡(E−b)×gklµν
≡(G−c)×gklµν
≡a×gklµν
(53)
Definition 2.29 (The n-index D dimensional a klµν ...).The n-index D dimensional
aklµν . . . is defined as:
aklµν . . . ≡(E−b)×gklµν . . .
≡(G−c)×gklµν . . .
≡a×gklµν . . .
(54)
Definition 2.30 (The first quadratic Lorentz invariant F1).The inner product of
Faraday’s electromagnetic field strength tensor yields a Lorentz invariant. The Lorentz invariant
does not change from one frame of reference to another. The first quadratic Lorentz invariant,
denoted as F1is determined as
F1≡Fkl ×Fkl (55)
The electromagnetic field tensor Fkl has two Lorentz invariant quantities. One of the
two fundamental Lorentz invariant quantities of the electromagnetic field [27] is known be
24

Fkl ×Fkl = 2 ×B2−E2where E denotes the electric E and B the magnetic field in the
taken frame of reference.
Definition 2.31 (The second quadratic Lorentz invariant F2).The second quadratic
Lorentz invariant, denoted as F2is determined as
F2≡klmn ×Fkl ×Fmn (56)
Definition 2.32 (The tensor bµν ).The co-variant Minkowski’s 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. The trace of energy-momentum tensor of the electromagnetic
field is known to be null. Under conditions of Einstein’s general theory of relativity [22, 17,
23, 21, 19], the tensor bµν denotes the trace-less, symmetric stress-energy tensor for source-free
electromagnetic field is defined in cgs-Gaussian units (depending upon metric signature) as
bµν ≡1
4×π×(Fµc×Fνc) + 1
4×gµν ×Fde ×Fde (57)
[see 38, p. 13] and equally as
bµν ≡1
4×π×Fµc×Fνd×gcd−1
4×gµν ×Fde ×Fde (58)
[see 33, p. 38]. The co-variant Minkowski’s stress-energy tensor of the electromagnetic field
is expressed under conditions of D = 4 space-time dimensions more compactly in a coordinate-
25

independent [theorem 3.1, equation 80 8, p. 157] form as
bµν ≡1
4×π×Fµc×Fνd×gcd+1
4×gµν ×Fde ×Fde
≡1
4×π×(Fµc×Fµc) + F1
4×gµν
≡R
D−a−c−d×gµν
≡(E−a)×gµν
≡b×gµν
(59)
where Fde is called the (traceless) Faraday/electromagnetic/field strength tensor.
Definition 2.33 (The 4-index D dimensional stress-energy tensor of electromagnetic
field bklµν ).The 4-index D dimensional stress-energy tensor of electromagnetic field bklµν is
defined as:
bklµν ≡R
D−a−c−d×gklµν
≡(E−a)×gklµν
≡b×gklµν
(60)
Definition 2.34 (The n-index D dimensional stress-energy tensor of electromagnetic
field bklµν ...).The n-index D dimensional stress-energy tensor of electromagnetic field bklµν . . .
is defined as:
bklµν . . . ≡R
D−a−c−d×gklµν . . .
≡(E−a)×gklµν . . .
≡b×gklµν . . .
(61)
Definition 2.35 (The tensor cµν ).Under conditions of Einstein’s general [22, 17, 23, 21, 19]
theory of relativity, the tensor of non-momentum and curvature is defined/derived/determined
26

[8] as follows:
cµν ≡bµν −(Λ ×gµν )
≡(G−a)×gµν
≡R
2−Λ−d×gµν
≡(b−Λ) ×gµν
≡c×gµν
(62)
Definition 2.36 (The 4-index D dimensional tensor c klµν).The 4-index D dimensional
cklµν is defined as:
cklµν ≡(G−a)×gklµν
≡R
2−Λ−d×gklµν
≡(b−Λ) ×gklµν
≡c×gklµν
(63)
Definition 2.37 (The n-index D dimensional tensor c klµν ... ).The n-index D dimensional
cklµν . . . is defined as:
cklµν . . . ≡(G−a)×gklµν . . .
≡R
2−Λ−d×gklµν . . .
≡(b−Λ) ×gklµν . . .
≡c×gklµν . . .
(64)
Definition 2.38 (The tensor of neither curvature nor momentum dµν ).Under conditions
of Einstein’s general [22, 17, 23, 21, 19] theory of relativity, the tensor of neither curvature nor
momentum is defined/derived/determined [8] as follows:
27

dµν ≡R
2×gµν −bµν
≡R
2×gµν −(Λ ×gµν )−cµν
≡



R
D×D
2−b


×gµν
≡



R
D×D
2−Λ−c


×gµν
≡d×gµν
(65)
There may exist circumstances where this tensor indicates pure vacuum, the space devoid of any
matter.
Definition 2.39 (The 4-index D dimensional d klµν).The 4-index D dimensional d klµν is
defined as:
dklµν ≡



R
D×D
2−b


×gklµν
≡



R
D×D
2−Λ−c


×gklµν
≡d×gklµν
(66)
Definition 2.40 (The n-index D dimensional d klµν ...).The n-index D dimensional
28

dklµν . . . is defined as:
dklµν . . . ≡



R
D×D
2−b


×gklµν . . .
≡



R
D×D
2−Λ−c


×gklµν . . .
≡d×gklµν . . .
(67)
Table 1 provides an overview of the general definition of the relationships between the four
basic [12, 11] fields of nature under conditions of the general theory of relativity.
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
Definition 2.41 (The Einstein field equations).Let Rµν denote the Ricci tensor [45] of
‘Einstein’s general theory of relativity’[17], 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 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 Λdenote the Einstein’s cosmological constant.
Let Λdenote the “anti cosmological constant”[3]. Let gµν metric tensor of Einstein’s general
theory of relativity. Let Gµν denote Einstein’s curvature tensor. Let Gµν denote the “anti
tensor”[12] 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
29

ordinary matter, bµν is the stress-energy tensor of the electromagnetic field. Let c denote the
speed of the light in vacuum, let γdenote Newton’s gravitational “constant”[6, 3, 9, 12]. Let
πdenote the number pi. Einstein’s field equation, published by Albert Einstein [22] for the first
time in 1915, and finally 1916 [17] but later with the “cosmological constant”[23, 21, 19]
term are determined as
Rµν −R
2×gµν + (Λ ×gµν )≡4×2×π×γ
c4×Tµν
≡Eµν
(68)
However, the above left-hand side of the Einstein field equations represents only one part
(Ricci curvature) of the geometric structure (Weyl curvature).
Table 2 provides a more detailed overview of the definitions of the four basic [12, 11] fields
of nature.
Curvature
YES NO
Momentum YES aµν bµν
8×π×γ
c4×Tµν
NO (bµν - Λ×gµν ) (R
2×gµν - bµν ) (R
2×gµν - Λ×gµν )
Gµν
R
2×gµν Rµν
Table 2: Einstein field equations and the four basic fields of nature
Remark 2.5. There are manifolds where the stress–energy tensor arise entirely from an
electromagnetic field with the consequence that the only source for the gravitational field is the
field energy (and momentum) of the electromagnetic field. Such manifolds are characterized by
the condition that aµν = 0 (electrovacuum solutions). However, among the many well-known
exact solutions in general relativity is the lambdavacuum solution too, which is distinct from
30

electrovacuum and vacuum solutions. In general, Tµν contains all forms of energy, momentum
et cetera which includes any matter present but if there is some electromagnetic radiation then
it to must be included in Tµν . However, regions of space-time devoid of any matter but also of
radiative energy and momentum were Tµν = 0 (i. e. in a vacuum region) might exist. Under
conditions were8×π×γ
c4×Tµν ≡0the Einstein field equations becomes
Rµν −R
2×gµν + (Λ ×gµν )≡8×π×γ
c4×Tµν ≡0 (69)
or
Rµν −R
2×gµν + (Λ ×gµν )≡0 (70)
Considering Ricci-flat manifolds, manifolds with a vanishing Ricci tensor, Rµν = 0, an equivalent
formulation of the relationship above follows as
0−R
2×gµν + (Λ ×gµν )≡0 (71)
Equation 71 simplifies as
R
2×gµν ≡(Λ ×gµν ) (72)
Manipulating equation 72 it is
R
2≡(Λ) (73)
or
R≡2×(Λ) ≡(Λ) + (Λ) (74)
Rearranging equation 74, it is
R−(Λ) ≡(Λ) (75)
31

Under conditions were Tµν = 0 (i. e. in a vacuum region) and Rµν = 0 (Ricci-flat manifolds),
equation 75 simplifies (see theorem 3.18, equation 258) as
Λ≡Λ (76)
Under conditions where the Ricci scalar itself is equal to R = 0, equation 75 changes to
−Λ≡+Λ (77)
and the state of pure symmetry, a possible state nature before the beginning of this world, is
given. In other words, a nonzero cosmological constant can be positive (as in de Sitter space,
named after Willem de Sitter (1872–1934) [see 50, 49]) or negative (as in anti-de Sitter space).
Thus far, has this world developed out of the state of pure symmetry (equation 77) where
(anti-de Sitter space) = (de Sitter space)
is one among the many far-reaching questions which might follow from equation 77.
Nonetheless, equation 75 changes under conditions of manifolds where Tµν= 0 and Rµν = 0.
In general, manifolds where Tµν = 0 and Rµν = 0 are determined according to the definition 2.9,
equation 32, by the equation
Λ≡Λ (78)
Rearranging equation 70 it is
Gµν + (Λ ×gµν )≡0 (79)
or
Gµν +≡ − (Λ ×gµν ) (80)
32

An equivalent formulation of the exact lambdavacuum solutions in general relativity in terms of
the Ricci tensor is given by
Rµν ≡R
2×gµν −(Λ ×gµν ) (81)
Table 3 provides an overview of Einstein field equations and the lambdavacuum solution [12, 11].
Curvature
YES NO
Momentum YES 0 0 0
NO (-Λ×gµν ) (R
2×gµν ) (R
2×gµν - Λ×gµν )
Gµν
R
2×gµν Rµν
Table 3: Einstein field equations and the lambdavacuum solution
2.2. Axioms
2.2.1. Axioms in general Axioms [32] and rules which are chosen carefully can be of use to
avoid logical inconsistency and equally preventing science from supporting particular ideologies.
Rightly or wrongly, long lasting advances in our knowledge of nature are enabled by suitable
axioms [16] too.
2.2.2. Axiom I. Lex identitatis To say that +1 is identical to +1 is to say that both are the
same.
Axiom 1. Lex identitatis.
+ 1 ≡+1 (82)
2.2.3. Axiom II. Lex contradictionis Axiom 2. Lex contradictionis.
33

+ 0 ≡+1 (83)
2.2.4. Axiom III. Lex negationis Axiom 3. Lex negationis.
¬(0) ×(+0) ≡(+1) (84)
where ¬denotes the (natural/logical) process of negation.
3. Results
3.1. The n-dimensional Pythagorean theorem
Theorem 3.1 (The n-dimensional Pythagorean theorem).The Pythagorean theorem as
attributed to the Greek thinker Pythagoras of Samos (6th century, B.C.) is defined as
oat2+obt2≡RCt2(85)
where omay denote the point of view of a co-moving observer while Rmay denote the point of
view of a stationary observer at a certain point in space-time t.
In general, the n-dimensional Pythagorean theorem is given by the equation
oat2+obt2n≡RCt2n (86)
34

Proof by modus ponens. If the premise
+1 = +1
| {z }
(P remise)
(87)
is true, then the conclusion
oat2+obt2n≡RCt2n (88)
is also true, the absence of any technical errors presupposed. The premise
(+1) = (+1) (89)
is true. Multiplying this premise by xtit is
xt≡xt(90)
Adding oxtto equation 90 it is
xt+oxt≡xt+oxt(91)
Equation 91 changes ( see definition 2.2, equation 13 ) to
xt+oxt≡RCt(92)
which is equally the general foundation of the Pythagorean theorem [see 12]. However, the
Pythagorean theorem can be extended to higher dimensions [see 60] too. In the n-dimensional
35

case, equation 92 becomes
(oxt+oxt)n≡
(oxt+oxt)×(oxt+oxt)×(oxt+oxt)×...
| {z }
ntimes

≡RCtn(93)
Multiplying equation 93 by the term RCtnwe obtain
oxt+oxtn×RCt
n≡


oxt+oxt×oxt+oxt×oxt+oxt×...
| {z }
ntimes


×RCt
n≡RCt
n×RCt
n(94)
Equation 94 simplifies as
oxt+oxtn×RCt
n≡oxt+oxt×RCtn≡oxt×RCt+oxt×RCtn≡RCt
2n (95)
In general, it is oat2≡oxt×RCt( see definition 2.2, equation 9 ). Equation 95 simplifies as
oat2+ (oxt×RCt)n≡RCt2n (96)
Furthermore, it is obt2≡oxt×RCt( see definition 2.2, equation 11 ). The n-dimensional
Pythagorean theorem follows as
oat2+obt2n≡RCt2n (97)
In other words, our conclusion is true.
Quod erat demonstrandum.
Remark 3.1. The make a long story short, the Pythagorean theorem is defined as
oat2+obt2≡RCt2(98)
36

Raising to the power n, the n dimensional Pythagorean theorem is given as
oat2+obt2n≡RCt2n (99)
It follows from equation 99 the normalised n dimensional Pythagorean theorem as
oat2n
RCt2n +RCt2n −oat2n
RCt2n ≡RCt2n
RCt2n ≡+1 (100)
Fermat’s Last Theorem states that (oatn) + (obtn)≡RCtnwhile no three positive integers a, b,
and c satisfy the equation for any integer value of n greater than 2. Rearranging Fermat’s Last
Theorem, we obtain
((oatn)+(obtn)) 2≡(RCtn)2(101)
while a lot of positive integers oat,obt, and RCtsatisfy equation 101 for any integer value of n
(and even greater than 2). Simplifying equation 101 leads to a more general form of the equation
before as
oat2n+ (2 ×oatn×obtn) + obt2n ≡RCt2n(102)
and Fermat’s Last Theorem appears to pass over into the n dimensional Pythagorean theorem.
The metric tensor gµν of general relativity on a space is more or less a generalisation of
Pythagoras’ theorem for the distance for a certain distance between two points separated by
different distances and reproduces the usual form of the Pythagorean Theorem. In general it is
gµν dxµdxν≡ds2(103)
while reproducing the usual form of the Pythagorean Theorem. The n dimensional form follows
37

as
(gµν dxµdxν)n≡ds2n≡RCt2n(104)
3.2. Refutation of Fermat’s Last Theorem
Theorem 3.2 (Refutation of Fermat’s Last Theorem).Fermat’s last theorem known as
(oatn) + (obtn)≡RCtnwhile no three positive integers a, b, and c satisfy the equation for any
integer value of n greater than 2, is refuted.
Proof by modus ponens. If the premise
+1 = +1
| {z }
(P remise)
(105)
is true, then the conclusion
(obtn)≡RCtn(106)
is also true, the absence of any technical errors presupposed, and Fermat’s last theorem is
refuted. The premise
+ 1 ≡+1 (107)
is true. Multiplying this premise (i. e. axiom or equation 107 ) by RCtn, it is
+ 1 ×RCtn≡+1 ×RCtn(108)
or
RCtn≡RCtn(109)
Pierre de Fermat’s (1607 - 1665) Last Theorem (i. e. Observatio Domini Petri de Fermat)
38

published 1670 in the book Diophantus’s Arithmetica by Fermat’s son, often considered simply
as one of the most notable unsolved problems of mathematics, states that (oatn)+(obtn)≡RCtn
while no three positive integers a, b, and c satisfy the equation for any integer value of n greater
than 2. Equation 109 changes to
(oatn)+(obtn)≡RCtn(110)
Investigating the behaviour of Fermat’s Last Theorem under conditions where a = +0 , we
obtain
((at≡+0) n)+(obtn)≡RCtn(111)
or
(obtn)≡RCtn(112)
In other words, at least one integer, the positive zero, is in compliance with Fermat’s Last
Theorem. Consequently, Fermat’s Last Theorem is refuted. Quod erat demonstrandum.
Remark 3.2. Andrew Wiles’s 1995 corrected proof of Fermat’s Last Theorem [see 57] appears
to be none or of a limited value. Three distinct positive integers (a = +0), b, and c can satisfy
Fermat’s equation
(oatn)+(obtn)≡RCtn(113)
has a solutions in positive integers for n ≥3. However, there is justified reason to believe that
it will be disputed that the positive zero (+0) is an integer. Well, in this case, a clear and
convincing answer should be given to the question why a positive zero is not an integer. What
than is a positive zero, a non-integer, an anti-integer or .. . ?
Theorem 3.3 (The relationship between the Ricci scalar R and the number of dimensions of
39

space-time D).Einstein Field Equations are defined in space-time dimensions [see 40, p. 31]
other than 3+1 too.
In general, the scalar S is given by
X≡R
D(114)
Proof by modus ponens. If the premise
+1 = +1
| {z }
(P remise)
(115)
is true, then the conclusion
S≡R
D(116)
is also true, the absence of any technical errors presupposed. The premise
(+1) = (+1) (117)
is true. Multiplying this premise by Ricci tensor Rµν it is
Rµν ≡Rµν (118)
Theoretically, it is possible to express the Ricci tensor Rµν mathematically completely through
the metric tensor gµν in an easy and straightforward way. In general, we define the Ricci tensor
Rµν as being completely determined by a scalar X and the metric tensor gµν as
Rµν ≡X×gµν (119)
40

However, at this stage, we just don’t know the exact value of the scalar X. Rearranging it is,
Rµν ×gµν ≡X×gµν ×gµν (120)
or in accordance to definition 2.9
R≡X×gµν ×gµν (121)
According to definition 2.5 (definition 2.5, equation 21) it is
R≡X×D(122)
In general, the scalar X is depending on the number of space-time dimensions D and it is
X≡R
D(123)
In other words, our conclusion is true.
Quod erat demonstrandum.
Remark 3.3. The complete geometrization of Einstein field equations as provided by Ilija
Barukˇci´c [see 8] has been derived under conditions where the number of space-time dimensions
D is equal to D = 4.
Theorem 3.4 (The Ricci tensor Rµν given through the metric tensor gµν under conditions
of D space-time dimensions.).In general relativity, it is common to present the Riemann and
Ricci tensors using the Christoffel symbols while Christoffel symbols are given through the metric.
Mathematical formulas giving the Ricci tensor Rµν under conditions of D space-time dimensions
[see 40, p. 31] while using explicitly the metric tensor gµν are missing.
41

In general, the Ricci tensor Rµν is given by the metric tensor gµν as
Rµν ≡R
D×gµν (124)
where D is the number of space-time dimensions.
Proof by modus ponens. If the premise
+1 = +1
| {z }
(P remise)
(125)
is true, then the conclusion
Rµν ≡R
D×gµν (126)
is also true, the absence of any technical errors presupposed. The premise
(+1) = (+1) (127)
is true. Multiplying this premise by the Ricci scalar R it is
R≡R(128)
According to theorem 3.3, equation 122, the equation before (equation 128) is equivalent with
X×D≡R(129)
42

Rearranging, we obtain
X≡R
D(130)
Multiplying by the metric tensor gµν, it is
X×gµν ≡R
D×gµν (131)
According to theorem 3.3, equation 119 it is Rµν ≡X×gµν The Ricci tensor Rµν expressed
directly in terms of the metric tensor gµν under conditions of D space-time dimensions follows
as
Rµν ≡R
D×gµν (132)
In other words, our conclusion is true.
Quod erat demonstrandum.
Remark 3.4. Einstein’s general theory of relativity does not in any way privilege a particular
space-time geometry. In this context, the string theory is a theoretical framework in which
particles of particle physics are replaced by strings, a kind of one-dimensional objects and is
treated more or less as not manifestly background independent. Under conditions of D = 1
space-time dimensions (see theorem 3.4, equation 132) it is Rµν ≡R
1×gµν ≡R×gµν while
the background independent Einstein field equations changes to R
1×gµν −R
2×gµν +
(Λ ×gµν )≡8×π×γ×T
c4×1×gµν or to +R
2×gµν +(Λ ×gµν )≡8×π×γ×T
c4×gµν .
Under conditions of D = 2 space-time dimensions (see theorem 3.4, equation 132) it is necessary
to consider the possibility that Rµν ≡R
2×gµν while the Einstein field equations changes
to R
2×gµν −R
2×gµν + (Λ ×gµν )≡8×π×γ×T
c4×2×gµν or to + (Λ ×gµν )≡
4×π×γ×T
c4×gµν . In particular, under conditions of D = 3 space-time dimensions (see
theorem 3.4, equation 132) we obtain Rµν ≡R
3×gµν while the Einstein field equations changes
43

to R
3×gµν −R
2×gµν +(Λ ×gµν )≡8×π×γ×T
c4×3×gµν . Under conditions of original
general relativty (D = 4 space-time dimensions), it is (see theorem 3.4, equation 132) Rµν ≡
R
4×gµν while the Einstein field equations changes to R
4×gµν −R
2×gµν + (Λ ×gµν )≡
8×π×γ×T
c4×4×gµν . The Kaluza–Klein theory [34] is a kind of a historical precursor of
string theory and equally a classical unified field theory of gravitation and electromagnetism built
around fifth dimension which used a similar idea of Gunnar Nordstr¨
om. Nordstr¨
om suggested:
“Es wird gezeigt, daß eine einheitliche Behandlung des elektromagnetischen Feldes und des
Gravitationsfeldes m¨
oglich ist, wenn man die vierdimensionale Raumzeitwelt als eine durch eine
f¨
unfdiminsionale Welt gelegte Fl¨
ache auffaßt. “[43]. Under conditions of D = 5 space-time
dimensions (Kaluza–Klein theory), it is (see theorem 3.4, equation 132) Rµν ≡R
5×gµν while the
Einstein field equations changes to R
5×gµν −R
2×gµν +(Λ ×gµν )≡8×π×γ×T
c4×5×
gµν . In string theory, space-time is ten-dimensional (nine spatial dimensions, and one
time dimension) and it is Rµν ≡R
10 ×gµν while the Einstein field equations changes to
R
10 ×gµν −R
2×gµν + (Λ ×gµν )≡8×π×γ×T
c4×10 ×gµν . In the year 1995, Edward
Witten [see 59] suggested that the five consistent versions of superstring theory (type I, type
IIA, type IIB, and two versions of heterotic string theory) were just special limiting cases of an
eleven-dimensional theory called M-theory. In M-theory space-time is eleven-dimensional (ten
spatial dimensions, and one time dimension). Under conditions of M-theory (D = 11 space-
time dimensions), it is (see theorem 3.4, equation 132) Rµν ≡R
11 ×gµν while the Einstein field
equations changes to R
11 ×gµν −R
2×gµν +(Λ ×gµν )≡8×π×γ×T
c4×11 ×gµν . For now
there is no end in sight on the number of space-time dimensions D and the theories associated
with the same. Under conditions of a 4-index or n-th index metric tensor, the formulas need to be
adopted like R
11 ×gklµν . . . −R
2×gklµν . . . + (Λ ×gklµν . . . )≡8×π×γ×T
c4×11 ×gklµν . . .
Theorem 3.5 (The Riemann curvature tensor or Riemann–Christoffel tensor Rklµν given
explicitly through the metric tensor gµν under conditions of D space-time dimensions).The
44

Riemann curvature tensor (named after Georg Friedrich Bernhard Riemann (1826 - 1866) and
Elwin Bruno Christoffel (1829 -1900)) is more or less a central mathematical tool in the theory of
general relativity [46, 47]. The Ricci tensor Rµν itself is a contraction of the Riemann curvature
tensor but can be viewed as something like a 2-index Riemann curvature tensor and vice versa.
The Riemann curvature tensor can be viewed as an 4-index Ricci tensor.
In general, the Riemann curvature tensor Rklµν is given by the metric tensor gklµν as
Rklµν ≡R
D×gkl ×gµν ≡X×gkl ×gµν ≡R
D×gklµν (133)
where D is the number of space-time dimensions.
Proof by modus ponens. If the premise
+1 = +1
| {z }
(P remise)
(134)
is true, then the conclusion
Rklµν ≡R
D×gkl ×gµν ≡X×gkl ×gµν ≡R
D×gklµν (135)
(see definition 2.3, equation 16) is also true, the absence of any technical errors presupposed.
The premise
(+1) = (+1) (136)
is true. Multiplying this premise by the Ricci scalar R it is
R≡R(137)
45

According to theorem 3.3, equation 122, the equation before (equation 128) is equivalent with
X×D≡R(138)
Rearranging, we obtain
X≡R
D(139)
Multiplying equation 139 by the metric tensor gkl and the metric tensor gµν , it is
X×gkl ×gµν ≡R
D×gkl ×gµν (140)
According to definition 2.3, equation 16 it is gklµν ≡gkl ×gµν . We obtain
X×gklµν ≡R
D×gklµν (141)
The Riemann curvature tensor or Riemann–Christoffel tensor Rklµν expressed directly in
terms of the metric tensor gklµν under conditions of D space-time dimensions follows as
Rklµν ≡X×gklµν ≡R
D×gklµν (142)
In other words, our conclusion is true.
Quod erat demonstrandum.
Theorem 3.6 (The n-th index Riemann curvature tensor or Riemann–Christoffel tensor
Rklµν ... given explicitly through the metric tensor gklµν ... under conditions of D space-time
dimensions).The n-th index Riemann curvature tensor or Riemann–Christoffel tensor Rklµν . . .
given explicitly through the n-th index metric tensor gklµν .. . under conditions of D space-time
46

dimensions is determined as
Rklµν . . . ≡X×gklµν . . . ≡R
D×gklµν . . . (143)
Proof. It is (see theorem 3.3, equation 123)
X≡R
D(144)
Multiplying by the metric tensor gklµν ... we obtain
Rklµν ... ≡X×gklµν ... ≡R
D×gklµν ... (145)
Quod erat demonstrandum.
Remark 3.5. In general, it is Rklµν . . . ≡X×gklµν . . . ≡R
D×gklµν . . . . Taking the
trace, it is Rklµν . . . ×gklµν . . . ≡X×gklµν . . . ×gklµν . . . ≡R
D×gklµν . . . ×gklµν . . . . Since
D≡gklµν . . . ×gklµν . . . we obtain X≡R
Dand our conclusion is true. The situation is not
quite different if analysed from another point of view. As found, it is Rklµν ≡X×gkl ×gµν ≡
R
D×gkl ×gµν Rearranging, we obtain Rklµν ×gkl ≡X×gkl ×gkl ×gµν ≡R
D×gkl ×gkl ×gµν
or Rµν ≡X×D1×gµν ≡R
D×D1×gµν Rearranging again, we obtain Rµν ×gµν ≡
X×D1×gµν ×gµν ≡R
D×D1×gµν ×gµν or R≡X×D1×D2≡R
D×D1×D2
Since R = R, it is necessary to accept that D≡D1×D2or in general D≡D1×D2×. . ..
Theorem 3.7 (The relationship between the scalar E and the number of dimensions of
space-time D).Einstein field Equations in other space-time dimensions [see 40, p. 31] than
3+1 need not lead to insurmountable contradictions.
47

In general, the scalar E is determined as
E≡8×π×γ×T
c4×D(146)
Proof by modus ponens. If the premise
+1 = +1
| {z }
(P remise)
(147)
is true, then the conclusion
E≡4×2×π×γ×T
c4×D(148)
is also true, the absence of any technical errors presupposed. The premise (i. e axiom)
(+1) = (+1) (149)
is true. Multiplying this premise by the stress-energy momentum tensor it is
4×2×π×γ
c4×Tµν ≡4×2×π×γ
c4×Tµν (150)
We do expect that the stress-energy momentum tensor can be geometrized completely as
4×2×π×γ
c4×Tµν ≡E×gµν (151)
Rearranging it is,
4×2×π×γ
c4×Tµν ×gµν ≡E×gµν ×gµν (152)
48

According to definition of Laue’s scalar (definition 2.11) it is
4×2×π×γ
c4×T≡E×gµν ×gµν (153)
According to definition 2.5, equation 21 it is
4×2×π×γ×T
c4≡E×D(154)
The scalar E is depending on the number of space-time dimensions D and is given in general as
E≡8×π×γ×T
c4×D(155)
In other words, our conclusion is true.
Quod erat demonstrandum.
Theorem 3.8 (The 2-index stress-energy-momentum tensor of matter under conditions of D
space-time dimensions).The starting point of Einstein’s theory of general relativity is that gravity
as such is a property of space-time geometry. Consequently, Einstein published a geometric
theory of gravitation [17] while Einstein’s initial hope to construct a purely geometric theory of
gravitation in which even the sources of gravitation themselves would be of geometric origin has
still not been fulfilled. Einstein’s field equations have a source term, the stress-energy tensor of
matter, radiation and vacuum et cetera, which is of order two and is still devoid of any geometry
and free of any geometrical significance. In general, the completely geometrical form of the 2-
index stress-energy momentum tensor of Einstein’s theory of general relativity under conditions
49

of D space-time dimensions is given by
8×π×γ
c4×Tµν ≡8×π×γ
c4×T
D×gµν (156)
Proof by modus ponens. If the premise
+1 = +1
| {z }
(P remise)
(157)
is true, then the conclusion
8×π×γ
c4×Tµν ≡8×π×γ
c4×T
D×gµν (158)
is also true, the absence of any technical errors presupposed. The premise
(+1) = (+1) (159)
is true. Multiplying this premise by Einstein’s stress-energy-momentum tensor of matter
8×π×γ
c4×Tµν it is
8×π×γ
c4×Tµν ≡8×π×γ
c4×Tµν (160)
Again, it is possible to express the stress-energy-momentum tensor of matter completely in terms
50

of the metric tensor gµν . We obtain
Eµν ≡8×π×γ
c4×Tµν ≡E×gµν (161)
According to theorem 3.7, equation 155, equation 161 can be simplified. The stress-energy-
momentum tensor of matter under conditions of D space-time dimensions is determined by the
equation
Eµν ≡8×π×γ
c4×Tµν ≡E×gµν ≡8×π×γ
c4×T
D×gµν (162)
where D is the number of space-time dimensions. Quod erat demonstrandum.
Remark 3.6. Theorem 3.8, equation 162 demands that
8×π×γ
c4×Tµν ≡8×π×γ
c4×T
D×gµν (163)
This equation leads straightforward to the need that
Tµν ≡T
D×gµν (164)
Lemma 3.1. It is
8×π×γ
c4×Tµν ≡8×π×γ
c4×T
D×gµν (165)
Simplifying equation 165, the most simple geometrical form of the pure 2-index stress–energy
51

momentum tensor Tµν under conditions of D dimensions is determined by the equation
Tµν ≡T
D×gµν (166)
Quod erat demonstrandum.
Remark 3.7. In more detail, under conditions of D = 4 dimensions the pure 2-index
stress–energy momentum tensor Tµν is determined by the metric, enriched only by view constants
and a scalar T as
2×π×γ×T
c4×gµν (167)
However, describing the fundamental stress–energy momentum tensor Tµν , the source term of the
gravitational field in Einstein’s general theory of relativity, as an inherent geometrical structure,
as being determined and dependent on the metric field gµν is associated with several and far
reaching consequences. Theoretically, it is possible that the properties of energy, momentum,
mass, stress et cetera need no longer to be treated or understood as intrinsic properties of
matter as such. Following equation 167, the properties which material systems posses could
be determined in virtue of their relation to space-time structures too. In last consequence, the
question might arise whether the energy tensor Tµν at the end could be in different aspects less
fundamental than the metric field gµν itself. Is and why is matter more fundamental [38, 39]
than space-time? In contrast to such a position, is the assumption justified that without the
space-time structure encoded in the metric no energy (tensor)? To bring it to the point, can
space-time (and its geometric structure) exist without matter and if yes, what kind of existence
could this be? Einstein’s starting point was to derive space-time structure from the properties of
material systems. In contrast such a position, theorem 3.8 allow us to consider that the energy
tensor might depend on the metric field or in an extreme case is completely determined by the
metric field. In extreme circumstances, the matter fields themselves are potentially derivable
52

from the structure of space-time or the very definition of an energy tensor is determined by
space-time structures too. Thus far, the question is not answered definitely, which came first,
either space-time structure or energy (tensor). So it is reasonable to ask, whether the energy-
momentum tensor of matter is dependent on the structures of space-time or only mathematically
described by the structures of space-time or both or none? In other words, granddaddies either
the hen and the egg dilemma is asking for a new, innovative and comprehensive solution and
may end up in an Anti-Machian theory. However, this leads us at this point too far afield.
Theorem 3.9 (The 4-index stress-energy-momentum tensor of matter Tklµν under conditions of
D space-time dimensions completely geometrized).The very detailed discussion of the various
proposals for a unified field theory can be found in secondary literature [29] too. Nevertheless, it
seems to us that the theory of general relativity taken as the point of departure for unified field
theory has at least one weak spot. Einstein’s stress-energy tensor Tµν is more or less a field
devoid of any geometrical significance. In view of the missing geometrization of the source of
gravitation in the Einstein field equations any logically consistent development of a unified field
theory more geometrico is endangered. The 4-index D dimensional stress-energy and momentum
tensor Eklµν follows as (see definition 2.13, equation 37)
Eklµν ≡8×π×γ×T
c4×D×gklµν (168)
Proof by modus ponens. If the premise of modus ponens
+1 = +1
| {z }
(P remise)
(169)
53

is true, then the following conclusion
Eklµν ≡E×gklµν
≡8×π×γ×T
c4×D×gklµν
(170)
is also true. The premise (+1 = +1) is true. A further manipulation of the premise (+1 = +1)
yields the result (see theorem 3.7, equation 155)
E≡4×2×π×γ×T
c4×D(171)
Multiplying equation 171 by the 4-index metric tensor gklµν, it is
E×gklµν ≡4×2×π×γ×T
c4×D×gklµν (172)
The 4-index D dimensional stress-energy and momentum tensor Eklµν follows (see definition
2.13, equation 37) as
Eklµν ≡E×gklµν
≡8×π×γ×T
c4×D×gklµν
≡Rklµν −R
2×gklµν + (Λ ×gklµν )
(173)
Quod erat demonstrandum.
Theorem 3.10 (The n-index stress-energy-momentum tensor of matter Tklµν ... under
conditions of D space-time dimensions completely geometrized).The n-th index stress-energy-
momentum tensor of matter Tklµν has already been discussed in literature [41]. However, it
is possible and necessary to go even beyond 4-index stress-energy-momentum tensor of matter
Tklµν . The n-th index stress-energy-momentum tensor of matter Tklµν . . . under conditions of D
54

space-time dimensions completely geometrized follows (see definition 2.13, equation 37) as
Eklµν . . . ≡8×π×γ×T
c4×D×gklµν . . . (174)
Proof by modus ponens. If the premise of modus ponens
+1 = +1
| {z }
(P remise)
(175)
is true, then the following conclusion
Eklµν ... ≡E×gklµν ...
≡8×π×γ×T
c4×D×gklµν ...
(176)
is also true. The premise (+1 = +1) is true. A further manipulation of the premise (+1 = +1)
yields the result (see theorem 3.7, equation 155)
E≡4×2×π×γ×T
c4×D(177)
Multiplying equation 177 by the n-index metric tensor gklµν ..., it is
E×gklµν ... ≡4×2×π×γ×T
c4×D×gklµν ... (178)
The n-index D dimensional stress-energy and momentum tensor Eklµν ... follows (see definition
55

2.13, equation 37) as
Eklµν ... ≡E×gklµν ...
≡8×π×γ×T
c4×D×gklµν ...
≡Rklµν ... −R
2×gklµν ...+ (Λ ×gklµν ... )
(179)
Quod erat demonstrandum.
Theorem 3.11 (The geometrical form of the stress-energy tensor of the
electromagnetic field bµν ).The geometrization of the stress-energy tensor of the
electromagnetic fields has been left behind by Einstein [17] himself as an unsolved problem.
Besides of the many trials to extend the geometry of general relativity even to the electromagnetic
field, the conceptual differences between the geometrized gravitational field and the classical
Maxwellian theory of the electromagnetic filed were so far insurmountable.
Claim.
In general, the completely geometrical form of the stress-energy momentum tensor of the
electromagnetic field bµν is given by
bµν ≡1
4×π×4×D×((4 ×(Fµc×Fµc)) + (D×(F1))) ×gµν
≡b×gµν
(180)
Proof by modus ponens. If the premise
+1 = +1
| {z }
(P remise)
(181)
56

is true, then the conclusion
bµν ≡1
4×π×4×D×((4 ×(Fµc×Fµc)) + (D×(F1))) ×gµν
≡b×gµν
(182)
is also true, the absence of any technical errors presupposed. The premise
(+1) = (+1) (183)
is true. Multiplying this premise by the stress-energy momentum tensor of the electromagnetic
field bµν , we obtain
(+1) ×bµν ≡(+1) ×bµν (184)
or
bµν ≡bµν (185)
The tensor bµν denotes the trace-less, symmetric stress-energy tensor of the (source-free)
electromagnetic field and is defined in cgs-Gaussian units (depending upon metric
signature) as
bµν ≡1
4×π×(Fµc×Fνc) + 1
4×gµν ×Fde ×Fde (186)
[see 38, p. 13] or as
bµν ≡1
4×π×Fµc×Fνd×gcd−1
4×gµν ×Fde ×Fde (187)
[see 33, p. 38]. A completely geometrized, co-variant stress-energy tensor of the electromagnetic
field expressed under conditions of D = 4 space-time dimensions has already been published
57

[theorem 3.1, equation 80 8, p. 157]. Rearranging equation 185 in connection with equation 186
and according to the definition 2.32 it is
bµν ≡1
4×π×Fµc×Fνd×gcd+1
4×gµν ×Fde ×Fde (188)
Rearranging equation before again it is
bµν ≡1
4×π×4×D
4×D×Fµc×Fνd×gcd+ D
4×D×Fde ×Fde×gµν  (189)
where D denotes the number of space-time dimensions (see definition 2.5, equation 21).
Rearranging the equation 189 further, we obtain
bµν ≡1
4×π×4×D×4×D×Fµc×Fνd×gcd+D×Fde ×Fde×gµν  (190)
Under conditions where gµν ×gµν ≡D(see definition 2.5, equation 21) equation 190 simplifies
as
bµν ≡1
4×π×4×D×4×(gµν ×gµν )×Fµc×Fνd×gcd+D×Fde ×Fde ×gµν  (191)
or as
bµν ≡1
4×π×4×D×4×(gµν )×Fµc×Fνd×gcd×gµν +D×Fde ×Fde ×gµν  (192)
A further simplification of the relationship before (equation 192) yields the stress-energy
58

momentum tensor of the electromagnetic field bµν determined only by the metric tensor of
general relativity gµν as
bµν ≡1
4×π×4×D×4×Fµc×gµν ×gcd ×Fνd+D×Fde ×Fde×gµν (193)
However, the term Fµc×gµν ×gcd ×Fνd+Fde ×Fde of the equation 193 can be
simplified further. For the first, it is F1≡Fde ×Fde (see definition 2.30, equation 55).
Furthermore, for an order-2 tensor, twice multiplying by the contra-variant metric tensor and
contracting [see 17, p. 790] in different indices [see 35] raises each index. In other words,
according to Einstein [see 17, p. 790], it is in general F(1 3
µ c )≡g(1 2
µ ν )×g(3 4
c d )×F(ν d
2 4 )or
more professionally Fµc≡gµν ×gcd ×Fνd(see definition 2.7, equation 30) which simplifies
equation 193 as
bµν ≡1
4×π×4×D×((4 ×(Fµc×Fµc)) −(D×(F1))) ×gµν (194)
Equation 194 can be simplified further. Under conditions where
(Fµc×Fµc)≡Fde ×Fde≡F1(195)
and equation 194 simplifies under conditions of D space-time dimensions as
bµν ≡1
4×π×4×F1
4×D+D×F1
4×D×gµν (196)
or as
bµν ≡(4 + D)×F1
4×π×4×D×gµν (197)
59

In general, we define the scalar or invariant b as
b≡(4 + D)×F1
4×π×4×D(198)
The 2-index stress-energy momentum tensor of the electromagnetic field geometrized completely,
is given by
bµν ≡b×gµν (199)
In other words, our conclusion is true.
Quod erat demonstrandum.
Remark 3.8. Under the circumstances above F1≡Fde ×Fde (see definition 2.30, equation
55), the stress energy tensor of ordinary matter aµν follows as
aµν ≡Eµν −bµν
≡8×π×γ×T
c4×D−(4 + D)×F1
4×π×4×D×gµν
(200)
Theorem 3.12 (The general foundation of the Einstein field equations).Above all, it is
necessary to extend the geometrization of gravitational force to non-gravitational interactions,
in particular, to electromagnetism, in order to achieve something like a geometrical unified field
theory. Ultimately, not all are comfortable with the geometrization of physics. Besides of all
in order to describe all fundamental interactions by appropriate objects of space-time geometry,
it is necessary to work out the foundations of the Einstein field equations. The D dimensional
foundation of the Einstein field equation is given by
R
D−R
2+ (Λ) ≡8×π×γ×T
c4×D(201)
60

Proof by modus ponens. If the premise of modus ponens
+1 = +1
| {z }
(P remise)
(202)
is true, then the following conclusion
R
D−R
2+ (Λ) ≡8×π×γ×T
c4×D(203)
is also true. The premise (+1 = +1) is true. Multiplying the premise (+1 = +1) by Einstein’s
stress-energy tensor of general relativity Tµν, we obtain
(+1) ×4×2×π×γ
c4×Tµν ≡(+1) ×4×2×π×γ
c4×Tµν (204)
or
4×2×π×γ
c4×Tµν ≡2×π×γ
c4×4×Tµν (205)
Einstein offered the principle of general covariance as the foundation of the theory of general
relativity and published the relationship between curvature and momentum in the form of his
field equations as Rµν −R
2×gµν + (Λ ×gµν )≡8×π×γ
c4×Tµν (see definition 2.41,
equation 68). Equation 205 changes too
Rµν −R
2×gµν + (Λ ×gµν )≡8×π×γ
c4×Tµν (206)
Taking the trace with respect to the metric of both sides of the Einstein field equations one gets
Rµν ×gµν −R
2×gµν ×gµν + (Λ ×gµν ×gµν )≡8×π×γ
c4×Tµν ×gµν (207)
61

Equation 207 simplifies as
R−R
2×D+ (Λ ×D)≡8×π×γ
c4×T(208)
where D is the number of space time dimensions (see definition 2.5, equation 21).Dividing
equation 208 by D, the number of space-time dimensions, simplifies equation 208 further. Thus
far, from the epistemological standpoint, the generally valid D dimensional foundation of the
Einstein field equations is given by
R
D−R
2+ (Λ) ≡8×π×γ×T
c4×D(209)
Quod erat demonstrandum.
Theorem 3.13 (The 2-index D dimensional Einstein field equations completely geometrized).
Both admirers and sceptics agree that Einstein field equations are not completely geometrized.
However, a complete geometrization of Einstein field equations may go hand-in-hand with a
unification of all known interactions. Therefore, we derive now a completely geometrical form of
the 2-index Einstein field equations [17, 23, 21], the mathematical foundation of the generalised
theory of gravitation[24] under conditions of D dimensions [41]. The completely geometrized
2-index D dimensional Einstein field equations are given by the equation
R
D×gµν −R
2×gµν + (Λ ×gµν )≡8×π×γ×T
c4×D×gµν (210)
Proof by modus ponens. If the premise of modus ponens
+1 = +1
| {z }
(P remise)
(211)
62

is true, then the following conclusion
R
D×gµν −R
2×gµν + (Λ ×gµν )≡8×π×γ×T
c4×D×gµν (212)
is also true. The premise (+1 = +1) is true. A further manipulation of the premise (+1 = +1)
yields the result (see theorem 3.12, equation 209)
R
D−R
2+ (Λ) ≡8×π×γ×T
c4×D(213)
Multiplying equation 213 by the 2-index metric tensor gµν yields the 2-index D dimensional
Einstein field equations as
R
D×gµν −R
2×gµν + (Λ ×gµν )≡8×π×γ×T
c4×D×gµν (214)
Quod erat demonstrandum.
Remark 3.9. In point of fact, due to theorem 3.3, equation 123, it is X≡S≡R
D.
Equation 213 has been derived as
R
D−R
2+ (Λ) ≡8×π×γ×T
c4×D(215)
According to theorem 3.7, equation 155, it is E≡4×2×π×γ×T
c4×Dand equation 215
changes to
R
D−R
2+ (Λ) ≡E(216)
The 2-index general geometrical form of Einstein field equation under conditions of D dimensions
63

is obtained by multiplying equation 216 by the metric tensor gµν as
R
D×gµν −R
2×gµν + (Λ ×gµν )≡E×gµν (217)
Einstein’s general theory of relativity gave a very big boost to physics. However, there are
still many and deepest problems in modern physics that remain to be solved. How does space-
time develop as such? Does space-time develop from lower and more simple to higher and
more complex space-time dimensions? How will space-time develop in the future? Meanwhile,
various other theories entered the ‘scientific market’. It is known that in M-theory [see 58, p.
1129] space-time is 11-dimensional, while in in super-string theory [30] it is 10-dimensional,
and in bosonic string theory [30], it is 26-dimensional. The general geometrical form of
Einstein field equation under conditions of D dimensions is obtained as R
D×gµν −
R
2×gµν +(Λ ×gµν )≡E×gµν and should be able to cope with these theoretical challenges.
However, one starting point of string theory is the idea that the point-like particles of particle
physics can also be described as one-dimensional objects called strings. Under conditions of
one-dimension, Einsteins field equation becomes (R×gµν )−R
2×gµν + (Λ ×gµν )≡
8×π×γ
c4×T×gµν . String theory as an area of current research in theoretical physics seeks to
unite the current theory of very small objects (quantum mechanics) with the theory of very large
objects (general relativity). Under conditions of one-dimension, it should be possible to study the
nature of strings completely by the equation R
2×gµν +(Λ ×gµν )≡8×π×γ
c4×T×gµν .
Theorem 3.14 (Einstein’s field equations under conditions of D=2 space-time dimensions).
The Einstein field equations [17, 23, 21, 24] simplifies under conditions of two space-time di-
mensions as
(Λ) ×gµν ≡4×π×γ×T
c4×gµν (218)
64

Proof by modus ponens. If the premise of modus ponens
+1 = +1
| {z }
(P remise)
(219)
is true, then the following conclusion
(Λ) ×gµν ≡4×π×γ×T
c4×gµν (220)
is also true, again the absence of any technical errors presupposed. The premise
+ 1 ≡+1 (221)
is true. Multiplying this premise by Einstein’s stress-energy tensor of general relativity, we
obtain
+ 1 ×8×π×γ
c4×T
D×gµν ≡+1 ×8×π×γ
c4×T
D×gµν (222)
or
8×π×γ
c4×T
D×gµν ≡8×π×γ
c4×T
D×gµν (223)
According to Einstein it is Rµν −R
2×gµν +(Λ ×gµν )≡8×π×γ
c4×Tµν (see definition
2.41). In connection with theorem 3.12, equation 209 (see theorem 3.12, equation 209), equation
223 changes to
R
D×gµν −R
2×gµν + (Λ ×gµν )≡8×π×γ
c4×T
D×gµν (224)
which is the general form of the 2-index Einstein’s field equations under conditions of D
dimensions. Under conditions of D=2 space-time conditions, the 2-index Einstein’s field
65

equations becomes
R
2×gµν −R
2×gµν + (Λ ×gµν )≡8×π×γ
c4×T
2×gµν (225)
or
0 + (Λ ×gµν )≡8×π×γ
c4×T
2×gµν (226)
To bring it again to the point. under conditions of D dimensions it is
(Λ) ×gµν ≡4×π×γ×T
c4×gµν (227)
In other words, our conclusion is true.
Quod erat demonstrandum.
Remark 3.10. Under conditions of D=2 space-time dimensions equation 227 determines
Einstein’s cosmological constant as Λ≡4×π×γ
c4×Tor something as Λ≈T.
Theorem 3.15 (The 4-index D dimensional Einstein field equations completely geometrized).
The 4-index D dimensional Einstein field equations completely geometrized are given by
R
D×gklµν −R
2×gklµν + (Λ ×gklµν )≡8×π×γ×T
c4×D×gklµν (228)
Proof by modus ponens. If the premise of modus ponens
+1 = +1
| {z }
(P remise)
(229)
66

is true, then the following conclusion
R
D×gklµν −R
2×gklµν + (Λ ×gklµν )≡8×π×γ×T
c4×D×gklµν (230)
is also true. The premise (+1 = +1) is true. A further manipulation of the premise (+1 = +1)
yields the result (see theorem 3.12, equation 209)
R
D−R
2+ (Λ) ≡8×π×γ×T
c4×D(231)
Multiplying equation 231 by the 4-index metric tensor gklµν yields the 4-index D dimensional
Einstein field equations completely geometrized as
R
D×gklµν −R
2×gklµν + (Λ ×gklµν )≡8×π×γ×T
c4×D×gklµν (232)
Quod erat demonstrandum.
Remark 3.11. The 4-index D dimensional Einstein field equations can be simplified more or
less as Gklµν + (Λ ×gklµν )≡8×π×γ×T
c4×D×gklµν . As already pointed out, it is possible
to split the Riemannian tensor Rklµν into different ways, including the Weyl conformal tensor
[56], denoted by Cklµν , and the anti-Weyl conformal tensor Cklµν. The anti-Weyl conformal
tensor denoted as Cklµν is the part of the Riemannian tensor Rklµν which involves the Ricci
tensor Rµν and the curvature scalar R. In general, it is (Rklµν ≡Cklµν +Cklµν )(see definition
2.21). Under these circumstances, the 4-index D dimensional Einstein field equations becomes
Cklµν −R
2×gklµν + (Λ ×gklµν ) + Cklµν ≡8×π×γ×T
c4×D×gklµν .
Theorem 3.16 (The n-index D dimensional Einstein field equations completely geometrized).
67

The n-index D dimensional Einstein field equations completely geometrized are given by
R
D×gklµν . . . −R
2×gklµν . . . + (Λ ×gklµν . . . )≡8×π×γ×T
c4×D×gklµν . . . (233)
Proof by modus ponens. If the premise of modus ponens
+1 = +1
| {z }
(P remise)
(234)
is true, then the following conclusion
R
D×gklµν ...−R
2×gklµν ...+ (Λ ×gklµν ... )≡8×π×γ×T
c4×D×gklµν ... (235)
is also true. The premise (+1 = +1) is true. A further manipulation of the premise (+1 = +1)
yields the result (see theorem 3.12, equation 209)
R
D−R
2+ (Λ) ≡8×π×γ×T
c4×D(236)
Multiplying equation 236 by the n-index metric tensor gklµν ... yields the n-index D dimensional
Einstein field equations completely geometrized as
R
D×gklµν ...−R
2×gklµν ...+ (Λ ×gklµν ... )≡8×π×γ×T
c4×D×gklµν ... (237)
Quod erat demonstrandum.
Theorem 3.17 (Einstein’s cosmological constant Λ).On the whole, the level of uncertainty
[31, 14] among the scientist and especially physicist is high despite the growing evidence [4, 3,
7, 2, 13, 5, 1] to the contrary already available. An even more severe violation of our trust
68

into physics is created by the cosmological constant Λtoo, which is taken to specify the overall
vacuum energy density. Depending on the specific assumptions made, the physical value [54]
of the cosmological constant Λis found to be very contradictory. The value of the cosmological
constant Λdepends on the number of space-time dimensions D too. In general, the value of the
cosmological constant Λis given by
Λ≡8×π×γ
c4×T
D+R
2−R
D(238)
Proof by modus ponens. If the premise of modus ponens
+1 = +1
| {z }
(P remise)
(239)
is true, then the following conclusion
Λ≡8×π×γ
c4×T
D+R
2−R
D(240)
is also true, again the absence of any technical errors presupposed. The premise
+ 1 ≡+1 (241)
is true. Multiplying this premise by Einstein’s stress-energy tensor of general relativity, we
obtain
(+1) ×4×2×π×γ
c4×Tµν ≡(+1) ×4×2×π×γ
c4×Tµν (242)
or
4×2×π×γ
c4×Tµν ≡2×π×γ
c4×4×Tµν (243)
69

According to Einstein (definition 2.41), equation 243 changes to
Rµν −R
2×gµν + (Λ ×gµν )≡8×π×γ
c4×Tµν (244)
Taking the trace with respect to the metric of both sides of the Einstein field equations one gets
Rµν ×gµν −R
2×gµν ×gµν + (Λ ×gµν ×gµν )≡8×π×γ
c4×Tµν ×gµν (245)
According to definition 2.5 (definition 2.5, equation 21), equation 245 simplifies as
R−R
2×D+ (Λ ×D)≡8×π×γ
c4×T(246)
Dividing equation 246 by D, the number of space-time dimensions, it is
R
D−R
2+ (Λ) ≡8×π×γ×T
c4×D(247)
Rearranging equation 247 yields the exact value of Einstein’s cosmological constant Λ
in relation to space-time dimensions D as
Λ≡8×π×γ
c4×T
D+R
2−R
D(248)
In other words, our conclusion is true. Quod erat demonstrandum.
Remark 3.12. One important outcome of theorem 3.17, equation 248 is the discovery that the
exact value of Einstein’s cosmological constant Λdepends on D, the number of space-
time dimensions and probably vice versa. It is evident to all of us that theorem 3.17,
equation 248 provides crystal clear and objectively provable facts that Einstein’s cosmological
70

constant Λcannot be treated as a constant. Additionally, especially appropriate measurements
and experiments may proof as true that theorem 3.17 induces more than only reasonable doubts
with respect to the constancy of Einstein’s cosmological constant Λ. Furthermore, one striking
consequence of theorem 3.17, equation 248 is the fact that as soon as Λis known, it is possible
to calculate the number of space-time dimensions D of a manifold (theorem 3.17, equation 248).
Theorem 3.18 (Anti cosmological constant Λ).The value of the anti-cosmological constant Λ
can be calculated very precisely. In general, the value of the anti-cosmological constant Λ[17,
23, 21] is given by
Λ≡R
D+R
2−8×π×γ
c4×T
D(249)
Proof by modus ponens. If the premise of modus ponens
+1 = +1
|{z }
(P remise)
(250)
is true, then the following conclusion
Λ≡R
D+R
2−8×π×γ
c4×T
D(251)
is also true, again the absence of any technical errors presupposed. The premise
+ 1 ≡+1 (252)
is true. Multiplying this premise by Ricci scalar (see definition 2.8), we obtain
(+1) ×(R)≡(+1) ×(R) (253)
71

or
R≡R(254)
Adding Λ and subtracting Λ, the cosmological constant, it is
R−Λ+Λ≡R−Λ + Λ (255)
or
R−Λ+Λ≡R+ 0 (256)
According to our definition 2.8 it is
Λ + Λ ≡R(257)
and therefore
Λ≡R−Λ (258)
The exact value of the cosmological constant Λ under conditions of D space-time dimensions
was calculated by theorem 3.17, equation 248 as Λ ≡8×π×γ
c4×T
D+R
2−R
D.
The exact value of the anti cosmological constant Λ can be calculated as
Λ≡R−8×π×γ
c4×T
D+R
2−R
D (259)
or as
Λ≡R
D+2×R
2−R
2−8×π×γ×T
c4×D
≡R
D+R
2−8×π×γ
c4×T
D(260)
72

with the consequence that our conclusion is true.
Quod erat demonstrandum.
4. Discussion
Having overthrown the Newtonian gravitational theory, the general theory of relativity did not
enable general relativity’s geometrization of gravitation to non-gravitational interactions, in par-
ticular, to electromagnetism. Among Weyl and Eddington and other, Einstein was one of the
first to use explicitly the term “unified field theory”in the title [18] of a publication in 1925.
In the following, Einstein himself published more than thirty technical papers on unification of
all physical interactions. However, Einstein’s unified field theory program was in vain [12, 11].
Further lack of clarity to geometrize all fundamental interactions and to provide a completely
geometrized [24] theory of relativity stemmed from the cosmological constant Λ, the energy den-
sity of space, or vacuum energy, and the uncertainties associated with the same. Although some
physicists including Einstein himself initially opposed the cosmological constant Λ, today, there
is some experimental evidence (Perlmutter et al. [44] Supernova Cosmology Project and Riess
et al. [48] High-Z Supernova Search Team) that the expansion of the universe is accelerating,
implying the possibility of a positive nonzero value for the cosmological constant Λ. Considered
Einstein’s insight [see 17, p. 796] that gµν ×gµν ≡D= +4 (definition 2.5) it was possible to
geometrize the Einstein field equations under conditions of D dimensions. The 2-index Einstein
field equations under conditions of D dimensions (see theorem 3.13, equation 214) might be seen
as fully compliant with the relationship
R
D×gµν −R
2×gµν + (Λ ×gµν )≡8×π×γ
c4×T
D×gµν
Encouraged by this result, it was possible to calculate the exact value of the cosmological
constant Λunder conditions where gµν ×gµν ≡D(definition 2.5). However, an answer
73

to the question whether the condition gµν ×gµν ≡Dis generally given may predominantly be
found elsewhere. Under conditions where gµν ×gµν ≡Dwhere D is the number of space-time
dimensions, we are able to calculate the exact value of the cosmological constant Λ very precisely
as
Λ≡8×π×γ×T
c4×D+R
2−R
D
and much more than this. For the reasons set out above, the inevitable conclusion is that
even the value of the anti cosmological constant Λ follows as
Λ≡R
D+R
2−8×π×γ×T
c4×D.
5. Conclusion
In combination with other already published [12, 11] papers, Einstein’s general theory of
relativity is completely geometrized. The theoretical value of the cosmological constant Λ and
the value of the anti cosmological constant Λ was calculated very precisely.
Appendix
None.
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