The Pythagorean (or Pythagoras’) theorem is of far reaching and fundamental importance in Euclidean Geometry and in science as such. In physics, the Pythagorean (or Pythagoras’) theorem serves especially as a basis for the definition of distance between two points. Historically, it is difficult to claim with a great degree of credibility that Pythagoras (~560 - ~480 B.C.) or someone else from his School was the first to discover this theorem.

There is some evidence, that the Pythagorean (or Pythagoras’) theorem was discovered on a Babylonian tablet [5] circa 1900 - 1600 B.C. Meanwhile, there are more than 100 published approaches proving this theorem, probably the most famous of all proofs of the Pythagorean proposition is the first of Euclid's two proofs (I.47), generally known as the Bride’s Chair. The Pythagorean (or Pythagoras’) theorem states that the sum of (the areas of) the two small squares equals (the area of) the big one square. In algebraic terms we obtain

where c represents the length of the hypotenuse (the longest side within a right angled triangle) and a and b represents the lengths of the triangle’s other two sides or legs (or catheti, singular: cathetus, greek: káthetos). Following Euclid (Elements Book I, Proposition 47) in right-angled triangles the sum of the squares on the sides containing the right angle equals the square on the side opposite the right angle.

The normalization of the Pythagorean theorem is defined as

where c represents the length of the hypotenuse (the longest side within a right angled triangle) and a and b represents the lengths of the triangle’s other two sides/legs.

We define the negation of x, denoted as n(x), as

We define the negation of anti x, denoted as n(x), as

In general, it is

In general, we define

where x and x denotes the segments on the hypotenuse c of a right angled triangle (c is the longest side within a right angled triangle).

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Form this follows that (C × X) + (C × X) = C². Due to our definition above, it is (C × X) = a² and b² = (C × X). The Pythagorean theorem is valid even if x = 1 and x = +∞ −1 while c = +∞. Under these assumptions, the Pythagorean theorem is of use to prove the validity of the claim that +1/+0 = +∞. In general, the normalized form of negation is denoted as (1 −).

According to Euclid’s (ca. 360 - 280 BC) so called geometric mean theorem or right triangle altitude theorem or Euclid’s theorem, published in Euclid’s Elements in a corollary to proposition 8 in Book VI, used in proposition 14 of Book II [6] to square a rectangle too, it is

where Δ denotes the altitude in a right triangle and x and x denote the segments on the hypotenuse c of a right angled triangle.

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The variance of a right angled triangle, denoted as s(x)², can be defined as

where Δ denotes the altitude in a right triangle and x and x denote the segments on the hypotenuse c of a right angled triangle.

The gradient, denoted as grad(a, b), a measure of how steep a slope or a line is, is defined by dividing the vertical height a by the horizontal distance b of a right angled triangle. In other words, we obtain

Scholium.

The following picture of a right angled triangle may illustrate the background of a gradient

where b denotes the run; a denotes the rise and c denotes the slope length. The gradient has several meanings. In mathematics, the gradient is more or less something like a generalization of a derivative of a function in one dimension to a function in several dimensions. Consider a n-dimensional manifold with coordinates ₁x, ₂x, _nx. The gradient of a function f(₁x, ₂x, _nx) is defined as

Due to our definition above it is equally c² ´ n(x) = a². In this case c² is not identical to the speed of the light but with the hypotenuse, the longest side within a right angled triangle. Equally, it is c² ´ n(x) = b². In general, it is true that. The raise can be calculated as

. In other words, it is a/b = y ´ (x/c) or

.

In general, it is

where _RE denotes the total (“relativistic”) energy of a system; _Rm denotes the “relativistic” mass and c denotes the speed of the light in vacuum.

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Einstein defined the matter/mass-energy equivalence as follows:

“Gibt ein Körper die Energie L in Form von Strahlung ab, so verkleinert sich seine Masse um L/V² … Die Masse eines Körpers ist ein Maß für dessen Energieinhalt” [7] .

In other words, due to Einstein, energy and mass are equivalent.

“Eines der wichtigsten Resultate der Relativitätstheorie ist die Erkenntnis, daß jegliche Energie E eine ihr proportionale Trägheit (E/c²) besitzt” [8] .

It was equally correct by Einstein to point out that matter/mass and energy are equivalent.

“Da Masse und Energie nach den Ergebnissen der speziellen Relativitätstheorie das Gleiche sind und die Energie formal durch den symmetrischen Energietensor (T_µv) beschrieben wird, so besagt dies, daß das G-Geld [gravitational field, author] durch den Energietensor der Materie bedingt und bestimmt ist” [9] .

The term relativistic mass _Rm was coined by Gilbert and Tolman [10] .

The Einsteinian matter/mass-energy equivalence [7] lies at the core of today physics. In general, due to Einstein’s special theory of relativity it is

or equally

or equally

where _OE denotes the “rest” energy; _Om denotes the “rest” mass; _RE denotes the “relativistic” energy; _Rm denotes the “relativistic” mass; v denotes the relative velocity between the two observers and c denotes the speed of light in vacuum.

The normalized relativistic energy momentum relation [10] , a probability theory consistent formulation of Einstein’s energy momentum relation, is determined as

while the “particle-wave-dualism” [10] is determined as

where _WE = (_Rp ´ c) denotes the energy of an electro-magnetic wave and _Rp denotes the “relativistic” momentum while c is the speed of the light in vacuum.

Following Einstein in his path of thoughts, we define the relativistic potential energy _pE [11] as

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The definition of the relativistic potential energy _pE is supported by Einstein’s publication in 1907. Einstein himself demands that there is something like a relativistic potential energy.

“Jeglicher Energie E kommt also im Gravitationsfelde eine Energie der Lage zu, die ebenso groß ist, wie die Energie der Lage einer ‘ponderablen’ Masse von der Größe E/c²” [12] .

Translated into English:

“Thus, to each energy E in the gravitational field there corresponds an energy of position that equals the potential energy of a ‘ponderable’ mass of magnitude E/c²”.

The relativistic potential energy _pE can be viewed as the energy which is determined by an observer P which is at rest relative to the relativistic potential energy. The observer which is at rest relative to the relativistic potential energy will measure its own time, the relativistic potential time _pt.

The relativistic kinetic energy _KE is defined [10] in general as

where _Rm denotes the “relativistic mass” and v denotes the relative velocity. In general, it is

where _PE denotes the relativistic potential energy; _KE denotes the relativistic kinetic energy; _PH denotes the Hamiltonian of the relativistic potential energy; _KH denotes the Hamiltonian of the relativistic kinetic energy. Multiplying this equation by the wave function _RY, we obtain relativity consistent form of Schrödinger’s equation as

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The historical background of the relativistic kinetic energy _KE is back grounded by the long lasting and very famous dispute between Leibniz (1646-1716) and Newton (1642-1726). In fact, the core of this controversy was the dispute about the question, what is preserved through changes. Leibnitz himself claimed, that “vis viva” [13] , [14] or the relativistic kinetic energy _KE = _Rm × v × v was preserved through changes. In contrast to Leibnitz, Newton was of the opinion that the momentum _Rp = _Rm × v was preserved through changes. The observer which is at rest relative to the relativistic kinetic energy will measure its own time, the relativistic kinetic time _kt.

An accurate clock in motion slows down with respect a stationary observer (observer at rest). The proper time _Ot of a clock moving at constant velocity v is related to a stationary observer’s coordinate time _Rt by Einstein’s relativistic time dilation [15] and defined as

where _Ot denotes the “proper” time; _Rt denotes the “relativistic” (i.e. stationary or coordinate) time; v denotes the relative velocity and c denotes the speed of light in vacuum.

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Coordinate systems can be chosen freely, deepening upon circumstances. In many coordinate systems, an event can be specified by one time coordinate and three spatial coordinates. The time as specified by the time coordinate is denoted as coordinate time. Coordinate time is distinguished from proper time. The concept of proper time, introduced by Hermann Minkowski in 1908 and denoted as _Ot, incorporates Einstein’s time dilation effect. In principle, Einstein is defining time exclusively for every place where a watch, measuring this time, is located.

“… Definition … der … Zeit … für den Ort, an welchem sich die Uhr … befindet…” [15] .

In general, a watch is treated as being at rest relative to the place, where the same watch is located.

“Es werde ferner mittels der imruhenden System befindlichen ruhenden Uhren die Zeit t [i.e. _Rt, author] des ruhenden Systems … bestimmt, ebenso werde die Zeit t [_Ot, author] des bewegten Systems, in welchen sich relativ zu letzterem ruhende Uhren befinden, bestimmt…” [15] .

Due to Einstein, it is necessary to distinguish between clocks as such which are qualified to mark the time _Rt when at rest relatively to the stationary system R, and the time _Ot when at rest relatively to the moving system O.

“Wir denken uns ferner eine der Uhren, welche relativ zum ruhenden System ruhend die Zeit t [_Rt, author], relativ zum bewegten System ruhend die Zeit t [_Ot, author] anzugeben befähigt sind…” [15] .

In other words, we have to take into account that both clocks i.e. observers have at least one point in common, the stationary observer R and the moving observer O are at rest, but at rest relative to what? The stationary observer R is at rest relative to a stationary co-ordinate system R, the moving observer O is at rest relative to a moving co-ordinate system O. Both co-ordinate systems can but must not be at rest relative to each other. The time _Rt of the stationary system R is determined by clocks which are at rest relatively to that stationary system R. Similarly, the time _Ot of the moving system O is determined by clocks which are at rest relatively to that the moving system O. In last consequence, due to Einstein’s theory of special relativity, a moving clock (_Ot) will measure a smaller elapsed time between two events than a non-moving (inertial) clock (_Rt) between the same two events.

As defined above, due to Einstein’s special relativity, it is

where _Ot denotes the “proper” time, _Rt denotes the “relativistic” (i.e. stationary or coordinate) time, v denotes the relative velocity and c denotes the speed of light in vacuum. Equally, it is

or

or

The normalized relativistic time dilation is defined as

In general, under conditions of the special theory of relativity, we define

and

and

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The following 2 × 2 table may illustrate the relationships before (Table 1).

The causal relationship k [16] under conditions of special theory of relativity (i.e. the particle-production apparatus) follows as

Under conditions [17] where

there is a relationship between the causal relationship k the Schrödinger equation in the form

The general Kronecker delta d_mn, named after Leopold Kronecker, is +1 if the variables m and n are equal, and +0 otherwise.

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For convenience, the restriction to positive integers is common, but not necessary. The general Kronecker delta, running from 1 to 4, denoted as d_mn can be displayed in matrix form as

The anti general Kronecker delta denoted as d_mn is defined as.

The special Kronecker delta d(i, j)_mn, named after Leopold Kronecker, is +1 if and only if m = i and if n = j and +0 otherwise.

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Example. The special Kronecker delta for and, running from 1 to 4, can be displayed in matrix form as

The anti special Kronecker delta denoted as d(i, j)_mn and defined as for and, running from 1 to 4, can be displayed as

The special Kronecker delta is not grounded on the equality that m = n but on the fact, the m equal to a certain value i and that n is equal to another certain value j. In other words, it is m = i and n = j.

In the following, let us define the following. Let

and

In Euclidean coordinates for an n-dimensional space the formula for the length ds² of an infinitesimal line segment due to the Pythagorean theorem follows as

or

In general, a coordinate system can be changed from the Euclidean X’s to some coordinate system of Y’s then

and

The Pythagorean theorem is defined as

While using Einstein’s summation convention, a (i.e. position dependent) metric tensor g(x)_µv is defined as

and a curved space compatible formulation of the Pythagorean theorem follows as

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The metric tensor generalizes the Pythagorean theorem of flat space in a manifold with curvature. The metric tensor can be decomposed in many different ways. Let g_µv = n_µv + n_µv where g_µv is the metric tensor of general relativity; n_µv is the tensor of special relativity and n_µv is the anti tensor of general relativity. In general theory of relativity, the scalar Newtonian gravitational potential is replaced by the metric tensor. “In particular, in general realtivity, the gravitational potential is replaced by the metric tensor g_ab” [18] . In last consequence, the gravitational potential is something like a feature of the metric tensor. Following Renn et al., the metric tensor is “… the mathematical representation of the gravitation alpotential…” [19] . On this account it is necessary to make a distinction between a gravitational potential and a gravitational field. Due to Einstein, “… the introduction of independent gravitational fields is considered justified even though no masses generating the field are defined” [2] . The question is, can a gravitational potential exist even though no masses generating the gravitational potential are defined?

In the following, we define the normalized metric tensor n_µv, while using Einstein’s summation convention, as

The line element follows in general as

Scholium.

The normalized metric tensor is not based on the gradient. The metric tensor passes over into the normalized metric tensor and vice versa. We obtain

or

Einstein field equations (EFE), originally [20] published [21] without the extra “cosmological” term Λ ´ g_µv [22] may be written in the form

where G_µv is the Einsteinian tensor; T_µv is the stress-energy tensor of matter (still a field devoid of any geometrical significance); R_µv denotes the Ricci tensor (the curvature of space); R denotes the Ricci scalar (the trace of the Ricci tensor); Λ denotes the cosmological “constant” and g_µv denotes the metric tensor (a 4 × 4 matrix) and where p is Archimedes’ constant (p = 3.1415926535897932384626433832795028841971693993751058209…) and g is Newton’s gravitational “constant” and the speed of light in vacuum is c = 299,792,458 [m/s] in S. I. units.

Scholium.

The stress-energy tensor T_µv, still a tensor devoid of any geometrical significance, contains all forms of energy and momentum which includes all matter present but of course any electromagnetic radiation too. Originally, Einstein’s universe was spatially closed and finite. In 1917, Albert Einstein modified his own field equations and inserted the cosmological constant Λ (denoted by the Greek capital letter lambda) into his theory of general relativity in order to force his field equations to predict a stationary universe.

“Ich komme nämlich zu der Meinung, daß die von mir bisher vertretenen Feldgleichungen der Gravitation noch einer kleinen Modifikation bedürfen…” [22] .

By the time, it became clear that the universe was expanding instead of being static and Einstein abandoned the cosmological constant Λ. “Historically the term containing the ‘cosmological constant’ l was introduced into the field equations in order to enable us to account theoretically for the existence of a finite mean density in a static universe. It now appears that in the dynamical case this end can be reached without the introduction of l” [23] . But lately, Einstein’s cosmological constant is revived by scientists to explain a mysterious force counteracting gravity called dark energy. In this context it is important to note that Newton’s gravitational “constant” big G is not [24] [25] a constant.

Independently of the tensors of the theory of general relativity, we introduce by definition the following covariant second rank tensors of yet unknown structure whose properties we leave undetermined as well. We define the following covariant second rank tensors of yet unknown structure as

Tensor can be decomposed (sometimes in many different ways). In the following of this publication we define the following relationships. It is

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The following 2 × 2 table may illustrate the relationships above (Table 2).

These tensors above may have different meanings depending upon circumstances. The unified field _RW_µv can be decomposed into several (sub-) fields A_µv, B_µv, C_µv, D_µv. In order to achieve unification between general relativity theory and quantum (field) theory the (sub-) fields A_µv, B_µv, C_µv, D_µv can denote the four basic fields of nature. The idea of quantum field theory is to describe a particle as a manifestation of an abstract field. In this context the particle a_i can be associated with the field A_µv, the particle b_i can be associated with the field B_µv, the particle c_i can be associated with the field C_µv, the particle d_i can be associated with the field D_µv. Thus far, we can define something like A_µv = a_i × _FA_µv and B_µv = b_i × _FB_µv and C_µv = c_i × _FC_µv and D_µv = d_i × _FD_µv where the subscript _F can denote an individual particle field. Under conditions of general relativity, Einstein field equation can be rewritten (using the tensors above) as

where and. From an epistemological point of view

is the tensor of the cause (in German: Ursache U) while is the tensor of the effect (in German: Wirkung W). As we will see, from the definition follows that

even if Einstein’s cosmological constant Λ cannot [26] be treated as a constant.

Planck defined in 1901 the constant of proportionality [27] as h. As long as Planck’s constant h is a constant, a tensor form of this constant is not needed. We define the co-variant second rank tensor of Planck’s constant _Rh_µv as

We define the co-variant second rank tensor of Dirac’s constant as

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In general it is known that

We define the co-variant second rank tensor of the speed of the light _Rc_µv, denoted by small letter c, as

where _Rf_µv denotes the stress energy tensor of frequency and _Rl_µv denotes the wave-length tensor.

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Following Einstein’s own position, the constancy of the speed of the light c is something relative and nothing absolute. Theoretically, circumstances are possible where the speed of the light is not constant. Einstein himself linked the constancy of the speed of the light c to a constant gravitational potential.

“Dagegen bin ich der Ansicht, daß das Prinzip der Konstanz der Lichtgeschwindigkeit sich nur insoweit aufrecht erhalten läßt, als man sich auf raum-zeitliche Gebiete von konstantem Gravitationspotential beschränkt. Hier liegt nach meiner Meinung die Grenze der Gültigkeit … des Prinzips der Konstanz der Lichtgeschwin- digkeit und damit unserer heutigen Relativitätstheorie” [8] .

Thus far a tensor of the speed of the light is of use to face these theoretical possibilities.

We define the co-variant second rank tensor of Newton’s gravitational constant _Rg_µv as

Scholium.

Newton’s gravitational constant is not for sure a constant. Therefore, we prefer to use the same in the form of a tensor.

We define the co-variant second rank tensor of Archimedes constant _Rp_µv as

Scholium.

Archimedes of Syracuse (ca. 287 BC - ca. 212 BC) himself was able to find p, the circumference of a circle with diameter 1 commonly approximated as 3.14159, to 99.9% accuracy about 2000 years ago. Archimedes constant p is an irrational number, p never settles into a permanent repeating pattern, the decimal representation of Archimedes constant p never ends.

We define the co-variant second rank tensor of the imaginary number i_µv as

We define the second rank tensor of space of yet unknown structure as

Under conditions of general relativity, we define

where R_µv denotes the Ricci tensor, the tensor of the curvature of space. Under conditions different form general relativity, _RS_µv can be determined in a different way. It is important to note that _RU_µv is not identical with U_µv.

Similar to general theory of relativity, it is at present appropriate to introduce a corresponding energy tensor, a tensor which represents the amounts of energy, momentum, pressure, stress et cetera in the space, a tensor which describes the energy/matter/momentum et cetera distribution (at each event) in space. The energy tensor expressed mathematically by a symmetrical tensor of the second rank of yet unknown structure is defined as

Ipso facto, the same tensor is determined by all matter present but of course any electromagnetic radiation too. Under conditions of general relativity, we define

To assure compatibility with quantum theory, we define

Due to the definition before we obtain

The tensor of probability of energy follows as

General relativity’s geometry of space and time is one but not the only one geometry of space and time. Especially general relativity’s stress-energy tensor as the source-term of Einstein’s field equations is still a field devoid of any geometrical significance. A geometrical tensorial representation of the stress energy tensor of energy is possible as

In general, we define the covariant second rank tensor of frequency _Rf_µv as

To assure compatibility with quantum theory, we define the inverse tensor _Rt_µv of the covariant second rank tensor of frequency _Rf_µv as

Per definition it follows that

In general, we define the covariant second rank tensor ₀w_µv as

Scholium.

The tensor of frequency _Rf_µv and the ₀w_µv tensor are related. Under circumstances of general relativity, there are conditions where

The matter tensor expressed mathematically by a symmetrical tensor of the second rank of yet unknown structure is defined as

Under conditions of general relativity, we define

Scholium.

This definition is based on the equivalence of mass/matter and energy due to Einstein’s special theory of relativity.

“Da Masse und Energie nach den Ergebnissen der speziellen Relativitätstheorie das Gleiche sind und die Energie formal durch den symmetrischen Energietensor (T_µv) beschrieben wird, so besagt dies, daß das G-Geld [gravitational field, author] durch den Energietensor der Materie bedingt und bestimmt ist” [9] .

We define the second rank tensor of ordinary energy ₀E_µv of yet unknown structure as

Scholium.

Under some well defined circumstances, ₀E_µv can denote the unity of strong interaction and weak interaction. Under conditions of general relativity, it is

The associated probability tensor can be achieved as

The tensor of ordinary mater expressed mathematically as a covariant second rank of yet unknown structure is defined as

We define the second rank anti tensor ₀E_µv of the tensor ₀E_µv as

Under conditions of general relativity, where ₀E_µv is tensor of ordinary energy/matter, the electromagnetic field is an anti tensor of ordinary energy/matter. Under conditions of general relativity, the tensor of the electromagnetic field is determined by an anti-symmetric second-order tensor known as the electromagnetic field (Faraday) tensor F. In general, under conditions of general relativity, the second rank covariant tensor of the electromagnetic field in the absence of “ordinary” matter, which is different from the electromagnetic field tensor F, is defined by

where F is the electromagnetic field tensor and g_µv is the metric tensor.

Scholium.

The associated probability tensor is determined as

The geometric formulation of the stress-energy tensor of the electromagnetic field follows as

The tensor ₀M_µv is defined as

A portion of the tensor of energy is due to the tensor of the electromagnetic field, another portion of the tensor of energy is due to the tensor of ordinary energy. Before going on to discuss this topic in more detail, we define in general

Under conditions of general relativity, we define

Scholium.

The stress-energy tensor of the electromagnetic field is equivalent to the portion of the stress-energy tensor of energy due to the electromagnetic field. In this approach, we are following Vranceanu in his position, that the energy tensor T_kl can be treated as the sum of two tensors one of which is due to the electromagnetic field.

“On peut aussi supposer que le tenseur d’énergie T_kl soit la somme de deux tenseurs dont un dû au champ électromagnétique…” [28] .

In English:

“One can also assume that the energy tensor T_kl be the sum of two tensors one of which is due to the electromagnetic field”.

Einstein himself demanded something similar.

“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] .

We define the second rank tensor of time of yet unknown structure as

Scholium.

All but energy is time, there is no third between energy and time. Under conditions of general theory of relativity, the associated probability tensor follows as

We define the second rank tensor _Rg_µv as

Scholium.

The tensor

is not identical with the metric tensor of general relativity, defined as

Still, circumstances may exist, where both tensors can be treated as being identical.

We define the second rank tensor ₀t_µv as

Scholium.

Under conditions of general theory of relativity, the associated probability tensor follows as

We define the second rank tensor ₀g_µv as

We define the second rank tensor _Wt_µv as

Scholium.

Under conditions of general theory of relativity, the associated probability tensor follows as

We define the second rank tensor _Wg_µv as

We define the covariant second rank wave function tensor as

Under conditions of general relativity, we define

We define the covariant second rank complex conjugate wave function tensor of yet unknown structure as

A portion of the tensor of space is due to the tensor of time, another portion of the tensor of space is determined by the tensor of energy. In general, we define

The field equation of the unified field theory follows in general as

where _RS_µv denotes the tensor of space; _RE_µv denotes the tensor of energy and _Rt_µv denotes the tensor of time.

Let _RY_µv denote a covariant second rank tensor of preliminary unknown structure. In general, we define

Scholium.

In general, the properties of the tensor _RY_µv are unknown. But one property of this tensor is known and this property assures the normalisation of the tensor of space as _RS_µv Ç _RY_µv = 1_µv. Under conditions of the general theory of relativity, it is true that as _RS_µv = R_µv and we do obtain R_µv Ç _RY_µv = 1_µv.

Let

denote a covariant second rank probability tensor of yet unknown structure as associated with a tensor _RX_µv. The probability tensor p(_RY_µv) of yet unknown structure as associated with the wave function tensor _RY_µv is defined as

Under the assumption of the validity of Born’s rule even under conditions of accelerated frames of reference, we define

where p(_RY_µv) denotes the probability tensor as associated i.e. with the wave function tensor _RY_µv and is the covariant second rank complex conjugate wave function tensor and Ç denotes the commutative multiplication of tensors.

In general, we define

where p(_RY_µv) denotes the probability tensor as associated i.e. with the wave function tensor _RY_µv and _RY_µv denote a covariant second rank tensor of preliminary unknown structure and Ç denotes the commutative multiplication of tensors.

Scholium.

The properties of the tensor _RY_µv, as mentioned already before, are still unknown. Still, another second property of this tensor is the special relationship with the wave function tensor _RY_µv. The interaction of the tensor _RY_µv with the wave function tensor _RY_µv yields the probability tensor p(_RY_µv) as associated with the wave function tensor _RY_µv. In general it is p(_RY_µv) = _RY_µv Ç _RY_µv.

In general, we define the tensor U_µv of yet unknown structure as

In general, we decompose the tensor U_µv as

Scholium.

By this definition we are following Einstein in his claim that something is determined by matter and the gravitational field. In other words, there is no third between matter and gravitational field, i.e. all but matter is gravitational field. To proceed further, in following Einstein, we make a strict distinction between matter and gravitational field too.

“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] .

The tensor _RU_µv is not identical with the tensor U_µv. In terms of set theory, we do obtain the following picture (Table 3).

In general, we define the tensor of curvature as ₀C_µv of yet unknown structure as

where G_µv is the Einsteinian tensor; R_µv is the Ricci tensor; R is the Ricci scalar and g_µv is the metric tensor of general relativity. Under conditions of the theory of general relativity it is ₀C_µv = G_µv.

Scholium.

Under conditions of general theory of relativity, the associated probability tensor follows as

In general, we define the tensor of anti-curvature as ₀C_µv of yet unknown structure as

where _RS_µv is the tensor of space, ₀C_µv is the tensor of curvature. Under conditions of general relativity, the tensor of anti-curvature is equivalent with

where G_µv is the Einsteinian tensor; R_µv is the Ricci tensor; R is the Ricci scalar and g_µv is the metric tensor of general relativity.

Scholium.

Under conditions of general theory of relativity, the associated probability tensor follows as

In general, we define the tensor of the unified field 1_µv, as

Scholium.

Every component of the tensor of the unified field is equal to +1. The tensor of the unified field is of order two, its components can be displayed in 4 × 4 matrix form as

In general, we define the zero tensor 0_µv as

Scholium.

Every component of a zero tensor is equal to +0. The zero tensor is of order two, its components can be displayed in 4 × 4 matrix form too as

In general, we define tensor of any number, i.e. the number 2_µv as

Scholium.

Every component of a tensor of the number +2 is equal to +2. The tensor of the number +2 can be displayed in 4 × 4 matrix form as

In general, we define the tensor of infinity ∞_µv as

Scholium.

Every component of the tensor of infinity is equal to +∞. The tensor of infinity is of order two, its components can be displayed in 4 × 4 matrix form as

Let ₀X_µv denote a second-tensor rank. The symmetric part of a tensor ₀X_µv is defined as

and denoted using the capital letter S and the tensor itself within the parentheses.

Let ₀X_µv denote a second-tensor rank. The anti-symmetric part of a tensor ₀X_µv is defined as

and denoted using the capital letter S underscore and the tensor itself within the parentheses.

Scholium.

In general, the tensor ₀X_µv can be written as a sum of symmetric and antisymmetric parts as

In general, let

We define the anti tensor ₀X_µv of the tensor ₀X_µv as

Scholium.

There is no third tensor between a tensor and its own anti tensor, a third is not given, tertium non datur (Aristotle). An anti tensor is denoted by the name of the tensor with underscore. Theoretically, the distinction between an anti-symmetrical tensor and an anti tensor is necessary. The simplest nontrivial antisymmetric rank-2 tensor, written as a sum of symmetric and antisymmetric parts, satisfies the equation

In general, the relationship between an anti symmetrical tensor and an anti tensor follows as

Only under conditions where we obtain

but not in general. In this context it is

The anti tensor d_µv of the Kronecker delta or Kronecker’s delta d_µv, named after Leopold Kronecker (1823-1891), follows as

Tensors independent of any coordinate system or frame of reference as generalizations of scalars (magnitude, no direction associated with a scalar) which have no indices and other mathematical objects (vectors (single direction), matrices) to an arbitrary number of indices may be operated on by tensor operators or by other tensors. In general, tensors can be represented by uppercase Latin letters and the notation for a tensor is similar to that of a matrix even if a tensor may be determined by an arbitrary number of indices. A distinction between covariant and contravariant indices is made. A component of a second-rank tensor is indicated by two indices. Thus far, a component of any tensor of any tensor rank which vanishes in one particular coordinate system, will vanish in all coordinate systems too. As is known, two tensors X and X which have the same rank and the same covariant and/or contravariant indices can be added. The sum of two tensors of the same rank is also a tensor of the same rank. In general, it is

or

or

The difference of two tensors of the same rank is also a tensor of the same rank. In general, it is

or

or

Let us display the individual components of a co-variant rank two tensor X_µv in matrix form as

Let us display the individual components of a co-variant rank two tensor Y_µv in matrix form as

The commutative multiplication of tensors (i.e. matrices), which is different from the non-commutative multiplication of tensors (i.e. matrices), is an operation of multiplying the corresponding elements of both tensors by each other. We define the commutative multiplication of tensors in general as

while the sign Ç denotes the commutative multiplication of tensors which is equally related to the Hadamard [29] product. The Hadamard product (also known as the Schur product or the point wise product), due to Jacques Salomon Hadamard (1865-1963), is an operation of two matrices of the same dimensions which is commutative, associative and distributive.

Let us introduce the notation of a co-variant rank two tensor X_µv raised to power n as

Each individual component of the tensor X_µv is multiplied by itself n-times.

Let us introduce the notation of a co-variant rank two tensor X_µv raised to power 1/n as

Each individual component of the tensor X_µv is raised to the power 1/n.

Let us once again display the individual components of a co-variant rank two tensor _RX_µv in matrix form as

The commutative division of tensors is defined by the division of the corresponding elements of both tensors by each other and displayed in matrix form as

while the sign: denotes the commutative division of tensors. The commutative division of tensors is displayed as

too.

Let E(X_µv) denote the expectation value of the covariant second rank tensor X_µv. Let p(X_µv) denote the probability tensor of the second rank tensor X_µv. In general, we define

while the sign Ç denotes the commutative multiplication of tensors.

Let E(²X_µv) denote the expectation value of the covariant second rank tensor X_µv raised to the power 2. Let p(X_µv) denote the probability tensor of the second rank tensor X_µv. In general, we define

while the sign Ç denotes the commutative multiplication of tensors.

Let s(X_µv)² denote the variance of the covariant second rank tensor X_µv. Let E(X_µv) denote the expectation value of the covariant second rank tensor X_µv. Let E(²X_µv) denote the expectation value of the covariant second rank tensor X_µv raised to the power 2. Let p(X_µv) denote the probability tensor of the second rank tensor X_µv. In general, we define

which can be written as

or as

or as

or as

while the sign Ç denotes the commutative multiplication of tensors and 1_µv is the tensor of the unified field.

Let s(X_µv) denote the standard deviation of the covariant second rank tensor X_µv. Let E(X_µv) denote the expectation value of the covariant second rank tensor X_µv. Let E(²X_µv) denote the expectation value of the covariant second rank tensor X_µv raised to the power 2. Let p(X_µv) denote the probability tensor of the second rank tensor X_µv. In general, we define

which can be written as

or as

while the sign Ç denotes the commutative multiplication of tensors and 1_µv is the tensor of the unified field. The covariant second rank tensor X_µv follows as

Let s(X_µv, Y_µv) denote the co-variance of the two covariant second rank tensors X_µv and Y_µv. Let E(X_µv, Y_µv) denote the expectation value of the two covariant second rank tensors X_µv and Y_µv. Let p(X_µv, Y_µv) denote the probability tensor of the two covariant second rank tensors X_µv and Y_µv. Let E(X_µv) denote the expectation value of the covariant second rank tensor X_µv. Let p(X_µv) denote the probability tensor of the second rank tensor X_µv. Let E(Y_µv) denote the expectation value of the covariant second rank tensor Y_µv. Let p(Y_µv) denote the probability tensor of the second rank tensor Y_µv. In general, we define

which can be written as

or as

while the sign Ç denotes the commutative multiplication. In general, it is

Let s(_RU_µv, ₀W_µv) denote the co-variance of the two covariant second rank tensors _RU_µv and ₀W_µv. Let s(_RU_µv) denote the standard deviation of the covariant second rank tensor of the cause. Let s(₀W_µv) denote the standard deviation of the covariant second rank tensor of the effect ₀W_µv. Let k(_RU_µv, ₀W_µv) denote the mathematical formula of the causal relationship in a general covariant form (i.e. Einstein’s Weltformel). In general, we define

Scholium.

In this context, the above equation is able to bridge the gap between classical field theory and quantum theory since the same enables the existence elementary particles i.e. with unequal mass but with opposite though otherwise equal electric charge.

The foundation of all what may follow is the following axiom:

Scholium.

From the standpoint of tensor calculus, it is

This article does not intend to give a review of the history of the identity law (principium identitatis). In the following it is useful to sketch, more or less chronologically, and by trailing the path to mathematics, the history of attempts of mathematizing the identity law. The identity law was used in Plato’s dialogue Theaetetus, in Aristotle’s Metaphysics (Book IV, Part 4) and by many other authors too. Especially, Gottfried Wilhelm Leibniz (1646-1716) expressed the law of identity as everything is that what it is. “Chaque chose est ce qu’elle est. Et dans autant d’exemples qu’on voudra A est A, B est B” [30] . In The problems of philosophy (1912) Russell himself is writing about the identity law too.

Lex identitatis or the identity law or principium identitatis can be expressed mathematically in the very simple form as +1 = +1. Consequently, +1 is only itself, simple equality with itself, it is only self-related and unrelated to another, +1 is distinct from any relation to another, +1 contains nothing other, no local hidden variable, but only itself, +1. In this way, there does not appear to be any relation to another, any relation to another is removed, any relation to another has vanished. Consequently, +1 is just itself and thus somehow the absence of any other determination. +1 is in its own self only itself and nothing else. In this sense, +1 is identical only with itself, +1 is thus just the “pure” +1. Let us consider this in more detail, +1 is not the transition into its opposite, the negative of +1, denoted as −1, is not as necessary as the +1 itself, +1 is not confronted by its other, +1 is without any opposition or contradiction, is not against another, is not opposed to another, +1 is identical only with itself and has passed over into pure equality with itself. But lastly, identity as different from difference, contains within itself the difference itself. Thus, it is the same +1 which equally negates itself, +1 in the same respect is in its self-sameness different from itself and thus self-contradictory. It is true, that +1 = +1, but it is equally true that −1 = −1. It is the same 1 which is related to a +1 and a −1. It is the +1 which excludes at the same time the other out of itself, the −1, out of itself, +1 is +1 and nothing else, it is not −1, it is not +2, it is not … Especially +1 is at the same time not −1, +1 is thus far determined as non being at least as non-being of its own other. In excluding its own other out of itself, +1 is excluding itself in its own self. By excluding its own other, +1 makes itself into the other of what it excludes from itself, or +1 makes itself into its own opposite, +1 is thus simply the transition of itself into its opposite, +1 is therefore determined only in so far as it contains such a contradiction within itself. The non-being of its other (−1) is at the end the sublation of its other. This non-being is the non-being of itself, a non-being which has its non-being in its own self and not in another; each contains thus far a reference to its other. Not +1 (i.e. −1) is the pure other of +1. But at the same time, not +1 only shows itself in order to vanish; the other of +1 is not. In this context, +1 and not +1 are distinguished and at the same time both are related to one and the same 1, each is that what it is as distinct from its own other. Identity is thus far to some extent at the same time the vanishing of otherness. +1 is itself and its other, +1 has its determinateness not in another, but in its own self. +1 is thus far self-referred and the reference to its other is only a self-reference. On closer examination +1 therefore is, only in so far as its Not +1 is, +1 has within itself a relation to its other. In other words, +1 is in its own self at the same time different from something else or +1 is something. It is widely accepted that something is different from nothing, thus while +1 = +1 it is at the same time different from nothing or from non- +1. From this it is evident, that the other side of the identity +1 = +1 is the fact, that +1 cannot at the same time be +1 and −1 or not +1. In fact, if +1 = +1 then +1 is not at the same time not +1. What emerges from this consideration is, therefore, even if +1 = +1 it is a self-contained opposition, +1 is only in so far as +1 contains this contradiction within it, +1 is inherently self-contradictory, +1 is thus only as the other of the other. In so far, +1 includes within its own self its own non-being, a relation to something else different from its own self. Thus, +1 is at the same time the unity of identity with difference. +1 is itself and at the same time its other too, +1 is thus contradiction. Difference as such it unites sides which are, only in so far as they are at the same time not the same. +1 is only in so far as the other of +1, the non +1 is. +1 is thus far that what it is only through the other, through the non +1, through the non-being of itself. From the identity +1 = +1 follows that +1 − 1 = 0. +1 and −1 are negatively related to one another and both are indifferent to one another, +1 is separated in the same relation. +1 is itself and its other, it is self-referred, its reference to its other is thus a reference to itself; its non-being is thus only a moment in it. +1 is in its own self the opposite of itself, it has within itself the relation to its other; it is a simple and self-related negativity. Each of them are determined against the other, the other is in and for itself and not as the other of another. +1 is in its own self the negativity of itself. +1 therefore is, only in so far as its non-being is and vice versa. Non +1 therefore is, only in so far as its non-being is, both are through the non-being of its other, both as opposites cancel one another in their combination, it is +1 − 1 = 0.

Scholium.

From the standpoint of tensor calculus, it is

Scholium.

From the standpoint of tensor calculus, it is

The law of non-contradiction (LNC) is still one of the foremost among the principles of science and equally a fundamental principle of scientific inquiry too. Without the principle of non-contradiction we could not be able to distinguish between something true and something false. There are arguably many versions of the principle of non-contradiction which can be found in literature. The method of reductio ad absurdum itself is grounded on the validity of the principle of non-contradiction. To be consistent, a claim/a theorem/a proposition/a statement et cetera accepted as correct, cannot lead to a logical contradiction. In general, a claim/a theorem/a proposition/a statement et cetera which leads to the conclusion that +1 = +0 is refuted.