This paper aims to explore the physical quantities of several invariants, including the basic postulates of some types of crucial coordinate transformations, conservation laws, and continuity equations, in the electromagnetic and gravitational fields. Maxwell first utilized quaternions to describe the electromagnetic theory. Subsequent scholars make use of octonions to study the physical properties of electromagnetic and gravitational fields simultaneously, including the octonion field strength, field source, angular momentum, torque, and force. When an octonion coordinate system transforms rotationally, the scalar part of one octonion will remain unchanged, although the vector part of the octonion may alter. In the octonion space O⁠, some invariants can be derived from this octonion property. A part of these invariants can be selected as the basic postulates of Galilean transformation or Lorentz transformation. Similarly, several invariants can be derived from the octonion property in the transformed octonion space Ou⁠, and the invariants can be chosen as the basic postulates of a few new types of coordinate transformations. Furthermore, the combination of invariants in the octonion spaces can be applied as the basic postulates of some new coordinate transformations, relevant to the norm of physical quantities. Through the analysis, it is easy to find that each conserved quantity has its preconditions from the perspective of octonion spaces. This is helpful to deepen the further understanding of the physical properties of conservation laws and other invariants.

Topics
Energy conservation, Electromagnetism, Fundamental constants, Newtonian mechanics, Special relativity, Optical materials, Algebraic structures, Numerical linear algebra, Coordinate transformations, Equations of fluid dynamics

Is there any new coordinate transformation besides the Galilean transformation and Lorentz transformation? Why do you choose the Galilean transformation first in most cases? Can one conservation law (or continuity equation) be established unconditionally? For a long time, these simple and important issues have puzzled and attracted scholars. It was not until the emergence of the octonion field theory (short for the field theory described with octonions) that these questions were answered partially. When the octonion coordinate system transforms rotationally, the scalar part of one octonion will remain unchanged. By means of this octonion property, a few invariants could be achieved. The combination of these invariants can be selected as the basic postulates for the Galilean transformation, Lorentz transformation, and several new coordinate transformations in the electromagnetic and gravitational fields. It expands our understanding of the simultaneity of invariants, conservation laws, and continuity equations.

In 1756, Lomonosov first proposed the law of mass conservation. In 1777, de Lavoisier verified the law of mass conservation again. Since then, the law of mass conservation1 has been accepted. On the other hand, the further study of mass led scholars to introduce the concepts of gravitational mass and inertial mass. These extensions should be taken into account by the law of mass conservation. The coverage of the mass concept should be further expanded.

In 1842, Mayer proposed the law of energy conservation. Joule studied the law of energy conservation and measured the heat equivalent of work. In 1847, Helmholtz described the law of energy conservation strictly. The number of energy terms has been increasing over time. These variations should be taken into account by the law of energy conservation. The scope of the energy concept should be further expanded.2 

The scholars owe Franklin for the creation of the law of charge conservation. In 1747, Franklin first mentioned the law of charge conservation. After that, the scholars assume that the law of charge conservation is true on both macro- and micro-scales.3,4 In vacuum, electric charges are quantized. In one material medium,5,6 the material may possess a fractional electric charge.7,8 The influence of material media should be considered in the law of charge conservation.9 

The above-mentioned analysis shows that the existing field theories have a few defects in the exploration of some invariants, conservation laws, and continuity equations, relevant to the electromagnetic and gravitational fields.

  1. Coordinate transformations. The existing coordinate transformations mainly deal with the Galilean transformation and Lorentz transformation. The Galilean transformation involves the speed of light and radius vector. It is believed that time and mass are both invariable, and this is fit for the low-speed movement cases. The Lorentz transformation concerns the speed of light and norm of the radius vector. It deems that time and mass are both not invariable, and this is applied to the explanation of the high-speed movement cases. However, these two coordinate transformations are unable to explore some invariants related to electric charges.

  2. Unconstrained establishment. The existing field theory reckons that the electromagnetic field and gravitational field are both in the same space. It presupposes that all the conservation laws can be established simultaneously while all continuity equations can be effective simultaneously. The conservation laws and continuity equations can also be valid simultaneously. In addition, even the law of charge conservation and the law of mass conservation can be available simultaneously. However, the viewpoint leads to its inability in explaining why the charge-to-mass ratio changes.

  3. Unvarying speed of light. According to the existing experience and imagination, the classical field theory takes it for granted that the speed of light is constant in vacuum. Furthermore, this view has merely been verified in a quite limited number of experiments. Nevertheless, the speed of light is variable in a large number of physical experiments. The existing studies have not fully considered the influence of various external factors on the speed of light. Apparently, this point of view limits the scope of application of coordinate transformations.

In a stark contrast to what is mentioned above, the octonion field theory is able to solve the puzzlement of some invariants, such as the simultaneous establishment of a few invariants, in the electromagnetic and gravitational fields. In addition, it can figure out some difficult problems derived from the existing field theories. By means of the octonion field theory, it is capable of exploring the physical properties of invariants, including the basic postulates of coordinate transformations, conservation laws, and continuity equations.

Maxwell first applied not only the algebra of quaternions but also the vector analysis to explore the physical properties of electromagnetic fields. Subsequent scholars10 utilized quaternions11 and octonions12 to study the electromagnetic fields,13,14 gravitational fields,15 invariants,16,17 conservation laws, continuity equations, quantum mechanics,18,19 relativity,20 astrophysical jets, weak nuclear fields,21,22 strong nuclear fields,23,24 black holes,25 dark matters,26 and so forth.

In this paper, making use of the octonion field theory, some invariants, conservation laws, and continuity equations relevant to the electromagnetic and gravitational fields could be explored. In addition, the studies described with the octonion field theory possess the following few advantages:

  1. Norm of physical quantity. The transformed octonion space Ou is independent from the octonion space O (in Sec. III). The Galilean transformation and Lorentz transformation in the octonion space O are the same as those in the existing field theory. The Lorentz transformation deals with the norm of the octonion radius vector. Meanwhile, there are two new types of coordinate transformations, which are similar to the Galilean transformation or Lorentz transformation, respectively, in the transformed octonion space Ou⁠. Furthermore, the norms of other physical quantities can also participate in the formation of a few new coordinate transformations, in particular the norm of octonion velocity.

  2. Restrictive establishment. The invariants in the transformed octonion space Ou cannot be established simultaneously with those in the octonion space O⁠. As a result, the conservation laws are divided into two groups, and the two groups of conservation laws are unable to be effective simultaneously. Next, the continuity equations will be separated into two different sets, while the two sets of continuity equations cannot be established simultaneously. In particular, the law of mass conservation and the law of charge conservation cannot be effective simultaneously. One of its direct inferences is that the charge-to-mass ratio must be varied. Moreover, there are many factors affecting the charge-to-mass ratio.

  3. Varying speed of light. The invariable speed of light is merely a choice. This assumption is only applicable to a few physical phenomena. Meanwhile, the speed of light and even the optical refractive index are variable in a large number of physical phenomena, that is, the speed of light is not an invariant in general. In this paper, only the Galilean transformation or Lorentz transformation can choose that the speed of light is an invariant. However, the other three major coordinate transformations in this paper have to select that the speed of light is not a constant. Apparently, various types of combinations of some invariants expand the scope of application of major coordinate transformations.

In this paper, when an octonion coordinate system transforms rotationally, the scalar part of one octonion is the same, although the vector part of the octonion may vary. As a result, a few invariants can be derived from this octonion property. By means of these invariants in the octonion space O⁠, the basic postulates of Galilean transformation and Lorentz transformation could be achieved. Similarly, it is capable of inferring the basic postulates of several new coordinate transformations in the transformed octonion space Ou⁠. Furthermore, the combination of invariants may be considered as the basic postulates of some coordinate transformations, relevant to the norm of physical quantities. Through the above-mentioned analysis and comparison, it is found that the Galilean transformation must be preferred in most cases in order to obtain as many details of theoretical explanation as possible. This helps us to further understand the invariants, conservation laws, and continuity equations.

Descartes believed that space is the extension of substance. Nowadays, the Cartesian thought is improved to that the fundamental space is the extension of the fundamental field.27 The fundamental fields include the gravitational field and electromagnetic field. Each fundamental field possesses one fundamental space. Each fundamental space is one quaternion space.

Hamilton invented the algebra of quaternions in 1843. Subsequently, Graves and Cayley discovered the octonion independently. The latter is called the classical octonion. Besides, scholars have proposed some other types of octonions. In this paper, classical octonions are utilized to explore the physical properties of electromagnetic and gravitational fields.

Maxwell first applied quaternion spaces to describe the physical properties of electromagnetic fields. Subsequent scholars used quaternion spaces to study the electromagnetic theory or gravitational theory. Furthermore, two independent quaternion spaces can be combined together to become one octonion space. The present scholars study the physical properties of electromagnetic field and gravitational field simultaneously by means of octonion spaces.

In the octonion space O for the electromagnetic and gravitational fields, ij and Ij are the basis vectors, while rj and Rj are the coordinate values. The octonion radius vector is R=Rg+kegRe⁠. The octonion velocity is V=Vg+kegVe⁠. Herein, keg is one coefficient to meet the demand for dimensional homogeneity. Rg=ii0r0+Σikrk⁠, Re=iI0R0+ΣIkRk⁠, Vg=ii0v0+Σikvk⁠, and Ve=iI0V0+ΣIkVk,⁠, where rj, vj, Rj, and Vj are all real. r0 = v0t, where v0 is the speed of light, and t is the time. i0 = 1, ik2=−1⁠, Ij2=−1⁠, and Ik = ik◦I0. ◦ denotes the octonion multiplication. i is the imaginary unit. j = 0, 1, 2, 3, and k = 1, 2, 3.

In the octonion space O⁠, the octonion field strength is F⁠, the octonion field source is S⁠, and the octonion linear momentum is P⁠. From these octonion physical quantities, the octonion angular momentum L⁠, torque W⁠, and force N can be defined. Furthermore, the octonion field strength F and angular momentum L can be combined together to become the octonion composite field strength, F+=F+kflL⁠, that is, F+ is the octonion field strength of electromagnetic and gravitational fields within the material media. Herein, kfl = −μg, which is the coefficient that satisfies the needs of dimensional homogeneity.28 μg is the gravitational constant.

In the octonion space O⁠, considering the contribution of material media, within the material media, the octonion field source can be defined as μS+=−(iF+/v0+◊)*◦F+⁠, the octonion linear momentum is defined as P+=μS+/μg⁠, the gravitational strength is Fg+=i0f0++Σikfk+⁠, the electromagnetic strength is Fe+=I0F0++ΣIkFk+⁠, and the gauge condition is chosen as f0+=0 and F0+=0⁠. Within the material media, the octonion field source can be rewritten as μS+=μgSg++μeSe+−(iF+/v0)*◦F+⁠, the gravitational source is Sg+=ii0s0++Σiksk+⁠, and the electromagnetic source is Se+=iI0S0++ΣIkSk+⁠. Herein, f0+⁠, sj+⁠, F0+⁠, and Sj+ are all real, fk+ and Fk+ are both complex numbers, μ is one coefficient, μe is the electromagnetic constant, and keg2=μg/μe⁠. The quaternion operator is ◊ = ii0∂0+Σik∂k, with ∂j = ∂/∂rj. * denotes the octonion conjugate.

In the octonion composite property equation, there exists one relationship between the octonion composite linear momentum, P+⁠, with the octonion composite field strength, (F++Fext+)⁠. The term Fext+ is the octonion composite field strength from the external of the material media. In addition, kpf is one coefficient to meet the demand for dimensional homogeneity (Table I).

TABLE I.

Some equations relevant to the electromagnetic and gravitational fields in the octonion space. These octonion composite physical quantities consider the contributions of material media.

Composite physical quantityOctonion definition
Composite field source μS+=−(iF+/v0+◊)*◦F+ 
Composite linear momentum P+=μS+/μg 
Composite property equation P+=kpf(F++Fext+) 
Composite angular momentum L+=(R+krxX)×◦P+ 
Composite torque W+=−v0(iF+/v0+◊)◦{(iV×/v0)◦L+} 
Composite force N+=−(iF+/v0+◊)◦{(iV×/v0)◦W+} 
Composite physical quantityOctonion definition
Composite field source μS+=−(iF+/v0+◊)*◦F+ 
Composite linear momentum P+=μS+/μg 
Composite property equation P+=kpf(F++Fext+) 
Composite angular momentum L+=(R+krxX)×◦P+ 
Composite torque W+=−v0(iF+/v0+◊)◦{(iV×/v0)◦L+} 
Composite force N+=−(iF+/v0+◊)◦{(iV×/v0)◦W+} 
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The octonion angular momentum within the material media is defined as L+=(R+krxX)×◦P+⁠, with X being the octonion integrating function of the field potential. The octonion composite angular momentum can be rewritten as L+=Lg++kegLe+⁠, within the material media. Herein, Lg+=L10++iL1i++L1+⁠, and Le+=L20++iL2i++L2+⁠. The coefficient krx = 1/v0 is able to meet the demand for dimensional homogeneity. L1+ is the angular momentum within the material media. L1i+ is called the mass moment temporarily. L2i+ is the electric moment within the material media, while L2+ is the magnetic moment within the material media. L1+=ΣL1k+ik⁠, L1i+=ΣL1ki+ik⁠, L2+=ΣL2k+Ik⁠, and L2i+=ΣL2ki+Ik⁠. L20+=I0L20+⁠. L1j+⁠, L2j+⁠, L1ki+⁠, and L2ki+ are all real. × represents the complex conjugate.

The octonion torque is W+=−v0(iF+/v0+◊)◦{(iV×/v0)◦L+}⁠, within the material media. It can be rewritten as W+=Wg++kegWe+⁠. Herein, Wg+=iW10i++W10++iW1i++W1+⁠, and We+=iW20i++W20++iW2i++W2+⁠. W10i+ is the energy within the material media. W1i+ is the torque within the material media, including the gyroscopic torque. W20i+ is called the second-energy within the material media temporarily. W2i+ is called the second-torque within the material media temporarily. W1+=ΣW1k+ik⁠, W1i+=ΣW1ki+ik⁠, W2+=ΣW2k+Ik⁠, W2i+=ΣW2ki+Ik⁠, W20i+=W20i+I0⁠, and W20+=W20+I0⁠. W1j+⁠, W2j+⁠, W1ji+⁠, and W2ji+ are all real.

The octonion force is N+=−(iF+/v0+◊)◦{(iV×/v0)◦W+}⁠, within the material media. It will be rewritten as N+=Ng++kegNe+⁠. Herein, Ng+=iN10i++N10++iN1i++N1+⁠, and Ne+=iN20i++N20++iN2i++N2+⁠. N10+ is the power within the material media. N1i+ is the force within the material media, including the Magnus force. N20i+ is called the second-power within the material media temporarily. N2i+ is called the second-force within the material media temporarily. N1+=ΣN1k+ik⁠, N1i+=ΣN1ki+ik⁠, N2+=ΣN2k+Ik⁠, N2i+=ΣN2ki+Ik⁠, N20i+=N20i+I0⁠, and N20+=N20+I0⁠. N1j+⁠, N2j+⁠, N1ji+⁠, and N2ji+ are all real.

Through the analysis and comparison, it can be found that there are some invariants in the octonion space O⁠. These invariants mainly relate to the physical quantities in the gravitational fields. On the other hand, there are some other invariants in the transformed octonion space Ou⁠. These invariants mainly involve physical quantities in the electromagnetic fields. These two octonion spaces, O and Ou⁠, are independent of each other. The invariants in the octonion spaces O are incompatible with those in the transformed octonion space Ou⁠.

In the octonion space O⁠, when any material medium does not make a contribution to the physical quantities, it is able to define some octonion physical quantities, including the octonion field strength F⁠, field source S⁠, linear momentum P⁠, angular momentum L⁠, torque W⁠, and force N⁠. Furthermore, when the material media make a contribution to the physical quantities, we can define some more practical octonion physical quantities, including the octonion composite field strength F+⁠, field source S+⁠, linear momentum P+⁠, angular momentum L+⁠, torque W+⁠, and force N+⁠.

Similarly, in the transformed octonion space Ou⁠, when any material medium does not make a contribution to the physical quantities, it is capable of defining several transformed octonion physical quantities, including the octonion field strength Fu⁠, field source Su⁠, linear momentum Pu⁠, angular momentum Lu⁠, torque Wu⁠, and force Nu⁠. Next, when the material media make a contribution to the physical quantities, we may define some more practical octonion physical quantities, including the octonion composite field strength Fu+⁠, field source Su+⁠, linear momentum Pu+⁠, angular momentum Lu+⁠, torque Wu+⁠, and force Nu+⁠.

A few coordinate transformations and invariants within the material media can be derived from the above-mentioned octonion composite physical quantities within the material media.

Although the vector part of the octonion may vary, the scalar part of one octonion will remain unchanged in the rotational transformation of octonion coordinate systems. Making use of this property of octonion physical quantities, several classical invariants, conservation laws, and continuity equations can be inferred in the electromagnetic and gravitational fields, considering the contribution of material media.

In the octonion space O⁠, the octonion radius vector R=ii0r0+Σikrk+keg(iI0R0+ΣIkRk) in the coordinate system α can be transformed into the octonion radius vector R′=ii0′r0′+Σik′rk′+keg(iI0′R0′+ΣIk′Rk′) in the coordinate system β. In the rotational transformation of the octonion coordinate system α, the scalar part of one octonion is the same. Therefore,

(1)

where i0′=1⁠.

Similarly, the octonion velocity V(vj,Vj)⁠, linear momentum P+(pj+,Pj+)⁠, angular momentum L+(L1j+,L2j+,L1ki+,L2ki+)⁠, torque W+(W1j+,W2j+,W1ji+,W2ji+)⁠, and force N+(N1j+,N2j+,N1ji+,N2ji+) in the coordinate system α can be transformed into the octonion velocity V′(vj′,Vj′)⁠, linear momentum P+′(pj+′,Pj+′)⁠, angular momentum L+′(L1j+′,L2j+′,L1ki+′,L2ki+′)⁠, torque W+′(W1j+′,W2j+′,W1ji+′,W2ji+′) and force N+′(N1j+′,N2j+′,N1ji+′,N2ji+′) in the coordinate system β, respectively (see Ref. 28). Hence, there are

(2)
(3)
(4)
(5)
(6)
(7)
(8)

The combination of the above-mentioned invariants can be chosen as the basic postulates of some types of coordinate transformations in the octonion space O⁠, including the basic postulates, Eqs. (1) and (2), of Galilean transformation (Table II). Apparently, various combinations of these invariants can be applied as the basic postulates for different coordinate transformations, picturing a few diverse overviews of the physical world. The invariants in Table II can be established simultaneously.

TABLE II.

Some conservation laws and continuity equations and the basic postulate of Galilean transformation in the octonion space O for the gravitational and electromagnetic fields. They can be derived from the invariants when the speed of light is constant and the contribution of material media is considered.

Speed of lightInvariantConservation law
v0=v0′ r0=r0′ Galilean transformation 
t = t′ Conserved time 
v0=v0′ p0+=p0+′ ? 
m+ = m+′ Law of (gravitational) mass conservation 
v0=v0′ L10+=L10+′ ? 
v0=v0′ W10+=W10+′ ? 
v0=v0′ W10i+=W10i+′ Law of energy conservation 
W10i+/v02=W10i+′/v0′2 Law of (equivalent) mass conservation 
v0=v0′ N10+=N10+′ (Conserved) fluid continuity equation 
v0=v0′ N10i+=N10i+′ (Conserved) torque continuity equation 
Speed of lightInvariantConservation law
v0=v0′ r0=r0′ Galilean transformation 
t = t′ Conserved time 
v0=v0′ p0+=p0+′ ? 
m+ = m+′ Law of (gravitational) mass conservation 
v0=v0′ L10+=L10+′ ? 
v0=v0′ W10+=W10+′ ? 
v0=v0′ W10i+=W10i+′ Law of energy conservation 
W10i+/v02=W10i+′/v0′2 Law of (equivalent) mass conservation 
v0=v0′ N10+=N10+′ (Conserved) fluid continuity equation 
v0=v0′ N10i+=N10i+′ (Conserved) torque continuity equation 
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Table II consists of some classical conserved quantities. (a) Galilean transformation. From Eqs. (1) and (2), we can select not only the speed of light but also the scalar part of the octonion radius vector to be conserved simultaneously. It implies that the time is conserved. In addition, this is the basic postulate of familiar Galilean transformation. (b) Conserved mass. From Eqs. (2) and (3), one can choose the speed of light and the scalar part of octonion linear momentum, where both are conserved simultaneously, that is, the gravitational mass, (p0+/v0)⁠, is conserved. (c) Conserved energy. From Eqs. (2) and (6), the speed of light and energy can be selected, where both are conserved simultaneously, that is, the equivalent mass, (W10i+/v02)⁠, is conserved. Furthermore, it is able to achieve other types of conserved quantities.

The above-mentioned conserved quantities are relatively simple, so they will be applied comparatively often, in particular the basic postulates of Galilean transformation, law of mass conservation, law of energy conservation, and fluid continuity equation (see Ref. 28). The rest of the conserved quantities are relatively unfamiliar, and they are comparatively less applied. Obviously, these invariants have an important impact on the theoretical analysis. The different combinations of invariants can give their respective physical scenarios, enabling the theoretical description more colorful.

It means that some different physical quantities can be utilized to describe the physical phenomena from different perspectives. (a) According to Eqs. (1) and (2), it is able to choose the coordinate system, Sr(r1, r2, r3), to describe relevant physical phenomena. (b) According to Eqs. (2) and (3), one may select a coordinate system, Sp(p1+,p2+,p3+)⁠, to explore the relevant physical phenomena in the momentum space. The utility of linear momenta replaces that of spatial coordinates in the momentum space. (c) According to Eqs. (2) and (4)–(8), we can choose some different types of physical quantities to research the physical phenomena.

It is worth noting that, when the gravitational mass is conserved according to Eq. (3), the term relevant to the electric charge is one variable vector in the octonion space O⁠. In other words, in case the law of mass conservation is effective, the electric charge is unable to be conserved, that is, the law of charge conservation is not tenable in the octonion space O⁠.

However, the law of charge conservation will be effective in the transformed octonion space Ou relevant to the octonion space O⁠.

In the octonion space, if we multiply the basis vector, iI0, with the octonion radius vector, R⁠, from the left, it is able to achieve one new octonion radius vector, Ru=iI0◦R⁠. The latter can be considered as one octonion physical quantity in the transformed octonion space Ou⁠, that is, Ru=iI0◦(kegRe+Rg)⁠. Apparently, the octonion radius vector Ru is independent of the octonion radius vector R⁠, in particular the sequence of basis vectors or coordinate values.

In the transformed octonion space Ou⁠, the octonion radius vector, Ru=keg(R0+iΣikRk)−(I0r0+iΣIkrk)⁠, in the coordinate system ζ can be transformed into the octonion radius vector, Ru′=keg(R0′+iΣik′Rk′)−(I0′r0′+iΣIk′rk′)⁠, in the coordinate system η. In the rotational transformation of the octonion coordinate system ζ, the scalar part of the octonion radius vector is the same. As a result,

(9)

In a similar way, the octonion velocity Vu(vj,Vj)⁠, linear momentum Pu+(pj+,Pj+)⁠, angular momentum Lu+(L1j+,L2j+,L1ki+,L2ki+)⁠, torque Wu+(W1j+,W2j+,W1ji+,W2ji+)⁠, and force Nu+(N1j+,N2j+,N1ji+,N2ji+) in the coordinate system ζ can be transformed into the octonion velocity Vu′(vj′,Vj′)⁠, linear momentum Pu+′(pj+′,Pj+′)⁠, angular momentum Lu+′(L1j+′,L2j+′,L1ki+′,L2ki+′)⁠, torque Wu+′(W1j+′,W2j+′,W1ji+′,W2ji+′)⁠, and force Nu+′(N1j+′,N2j+′,N1ji+′,N2ji+′) in the coordinate system η, respectively (see Ref. 28). Hence, there are

(10)
(11)
(12)
(13)
(14)
(15)
(16)

The combination of these invariants can be chosen as the basic postulates of a few coordinate transformations in the transformed octonion space Ou⁠. Obviously, various combinations of these invariants can be applied as the basic postulates for different types of coordinate transformations, describing several diverse overviews of the physical world (Table III). The invariants in Table III can be established simultaneously.

TABLE III.

Some conservation laws and continuity equations and the basic postulate of coordinate transformation in the transformed octonion space Ou for the gravitational and electromagnetic fields. They can be derived from the invariants when the second-speed of light is constant while the contribution of material media is considered.

Second-speed of lightInvariantConservation law
V0=V0′ R0=R0′ Second-Galilean transformation 
T = T′ Conserved second-time 
V0=V0′ P0+=P0+′ ? 
q+ = q+′ Law of charge conservation 
V0=V0′ L20+=L20+′ ? 
V0=V0′ W20+=W20+′ ? 
V0=V0′ W20i+=W20i+′ Law of second-energy conservation 
W20i+/V02=W20i+′/V0′2 Law of (equivalent) charge conservation 
V0=V0′ N20+=N20+′ (Conserved) current continuity equation 
V0=V0′ N20i+=N20i+′ (Conserved) second-torque continuity equation 
Second-speed of lightInvariantConservation law
V0=V0′ R0=R0′ Second-Galilean transformation 
T = T′ Conserved second-time 
V0=V0′ P0+=P0+′ ? 
q+ = q+′ Law of charge conservation 
V0=V0′ L20+=L20+′ ? 
V0=V0′ W20+=W20+′ ? 
V0=V0′ W20i+=W20i+′ Law of second-energy conservation 
W20i+/V02=W20i+′/V0′2 Law of (equivalent) charge conservation 
V0=V0′ N20+=N20+′ (Conserved) current continuity equation 
V0=V0′ N20i+=N20i+′ (Conserved) second-torque continuity equation 
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Table III covers some conserved quantities. (a) Conserved electric charge. Choosing the two equations, Eqs. (10) and (11), shows that the electric charge, (P0+/V0)⁠, is conserved. (b) Conserved equivalent charge. The selection of two equations, Eqs. (10) and (14), states that the equivalent charge, (W20i+/V02)⁠, is conserved. (c) Simultaneity. Selecting the conservation of second-speed of light, Eq. (10), means that the speed of light, v0, is not conserved, that is, Eq. (2) is unable to be effective in the transformed octonion space Ou⁠.

According to Eqs. (10), (15), and (16), different types of coordinate systems can be chosen to study the physical phenomena from multiple perspectives. These conserved quantities are relatively simple, so they can be utilized comparatively often, in particular the law of charge conservation and current continuity equation (see Ref. 28). Although the rest of the conserved quantities are relatively strange, they also have an impact on the theoretical analysis.

It is rather remarkable that the transformed octonion space Ou is distinct to the octonion space O⁠, so the invariants in Table III cannot be established with those in Table II simultaneously. In the transformed octonion space Ou⁠, when the electric charge is conserved according to Eq. (11), the term relevant to the mass is one variable vector in the transformed octonion space Ou⁠. In other words, in case the law of charge conservation is effective, the mass must not be conserved, that is, the law of mass conservation is not effective in the transformed octonion space Ou⁠.

The norm of the octonion radius vector is one scalar and remains unchanged in the rotational transformations of octonion coordinate systems. Consequently, the above-mentioned research methods can be extended to the norm of the octonion radius vector and so forth, exploring the basic postulate of Lorentz transformation and continuity equations and others.

The norm of an octonion physical quantity is the same in the rotational transformation of octonion coordinate systems. By making use of this property of octonion physical quantities, it is capable of inferring some classical invariants, including the conservation laws and continuity equations relevant to the norm of octonion physical quantities in the electromagnetic and gravitational fields, considering the contribution of material media.

In the octonion space O⁠, the octonion radius vector R in the coordinate system α can be transformed into the octonion radius vector O′ in the coordinate system β. In the rotational transformation of the octonion coordinate system α, the norm of the octonion radius vector is the same. Therefore,

(17)

or

(18)

Similarly, the octonion linear momentum P+⁠, angular momentum L+⁠, torque W+⁠, and force N+ in the coordinate system α can be transformed into the octonion linear momentum P+′⁠, angular momentum L+′⁠, torque W+′⁠, and force N+′ in the coordinate system β, respectively. In the rotational transformation of the octonion coordinate system α, each of these norms of octonion physical quantities is the same. Hence, there are

(19)
(20)
(21)
(22)

The combination of the above-mentioned invariants is capable of deducing the basic postulates of several coordinate transformations in the octonion space O⁠, including the Lorentz transformation, Eqs. (17) and (2). Apparently, other combinations of the above-mentioned invariants can be chosen as the basic postulates for different coordinate transformations, describing a few diverse overviews of the physical world (Table IV). The invariants in Table IV can be effective simultaneously.

TABLE IV.

Several conservation laws and continuity equations and the basic postulate of Lorentz transformation are dependent on the norms of physical quantities in the octonion space O for the gravitational and electromagnetic fields. They can be derived from the invariants relevant to the norms of physical quantities when the speed of light is constant while the contribution of material media is considered.

Speed of lightInvariant relevant to the normConservation law
v0=v0′ r02−Σrk2−keg2ΣRj2=r0′2−Σrk′2−keg2ΣRj′2 Lorentz transformation 
v0=v0′ p0+2−Σpk+2−keg2ΣPj+2=p0+′2−Σpk+′2−keg2ΣPj+′2 Norm of octonion linear momentum 
(p0+/v0)2−Σ(pk+/v0)2−keg2Σ(Pj+/v0)2 Norm of mass 
=(p0+′/v0′)2−Σ(pk+′/v0′)2−keg2Σ(Pj+′/v0′)2 
v0=v0′ L+◦L+*=L+′◦(L+′)* Norm of octonion angular momentum 
v0=v0′ W+◦W+*=W+′◦(W+′)* Norm of octonion torque 
 (W+/v02)◦(W+/v02)*=(W+′/v0′2)◦(W+′/v0′2)* Norm of equivalent mass 
v0=v0′ N+◦N+*=N+′◦(N+′)* Norm of octonion force 
Speed of lightInvariant relevant to the normConservation law
v0=v0′ r02−Σrk2−keg2ΣRj2=r0′2−Σrk′2−keg2ΣRj′2 Lorentz transformation 
v0=v0′ p0+2−Σpk+2−keg2ΣPj+2=p0+′2−Σpk+′2−keg2ΣPj+′2 Norm of octonion linear momentum 
(p0+/v0)2−Σ(pk+/v0)2−keg2Σ(Pj+/v0)2 Norm of mass 
=(p0+′/v0′)2−Σ(pk+′/v0′)2−keg2Σ(Pj+′/v0′)2 
v0=v0′ L+◦L+*=L+′◦(L+′)* Norm of octonion angular momentum 
v0=v0′ W+◦W+*=W+′◦(W+′)* Norm of octonion torque 
 (W+/v02)◦(W+/v02)*=(W+′/v0′2)◦(W+′/v0′2)* Norm of equivalent mass 
v0=v0′ N+◦N+*=N+′◦(N+′)* Norm of octonion force 
View Large

Table IV includes a few classical conserved quantities. (a) Lorentz transformation. From Eqs. (17) and (2), one may select not only the speed of light but also the norm of the octonion radius vector to be conserved simultaneously. It implies that the time is not conserved. In addition, this is the basic postulate of familiar Lorentz transformation. (b) Non-conserved mass. From Eqs. (2) and (19), the speed of light and the norm of octonion linear momentum can be chosen, where both are conserved simultaneously, that is, the gravitational mass, (p0+/v0)⁠, is not conserved. (c) Non-conserved energy. From Eqs. (2) and (21), the speed of light and the norm of octonion torque are capable of being selected, where both are conserved simultaneously, that is, either the energy term, W10i+⁠, or the equivalent mass, (W10i+/v02)⁠, is not conserved. (d) Non-conserved continuity equations. From Eqs. (2) and (22), we may choose not only the speed of light but also the norm of octonion force to be conserved simultaneously. It means that the continuity equations are not invariable nor conserved.

These conserved quantities are relatively simple and often used, in particular the basic postulate of Lorentz transformation. The remaining conserved quantities are relatively unfamiliar, and they are comparatively rarely used. The combinations of remaining conserved quantities relevant to the norms have an important impact on the theoretical analysis also. Obviously, choosing the combination of different invariants as the basic postulate is able to achieve different physical scenarios.

In the transformed octonion space Ou⁠, the octonion radius vector Ru in the coordinate system ζ can be transformed into the octonion radius vector Ru′ in the coordinate system η. In the rotational transformation of the octonion coordinate system ζ, the norm of the octonion radius vector Ru is the same. It is easy to find that the norm of the octonion radius vector Ru in the transformed octonion space Ou is identical to that of the octonion radius vector R in the octonion space O⁠.

Similarly, the norm of the octonion linear momentum Pu+⁠, angular momentum Lu+⁠, torque Wu+⁠, and force Nu+ in the transformed octonion space Ou will be identical to that of the octonion linear momentum P+⁠, angular momentum L+⁠, torque W+⁠, and force N+ in the octonion space O⁠, respectively.

The combination of the above-mentioned invariants can give some coordinate transformations in the transformed octonion space Ou⁠. Apparently, a few combinations of different invariants can deduce other types of coordinate transformations, depicting several different physical scenarios (Table V). In addition, the invariants in Table V can be effective simultaneously.

TABLE V.

Some conservation laws and continuity equations and the basic postulates of coordinate transformations are dependent on the norms of physical quantities in the transformed octonion space Ou for the gravitational and electromagnetic fields. They can be derived from the invariants relevant to the norms of physical quantities when the second-speed of light is constant while the contribution of material media is considered.

Second-speed of lightInvariant relevant to the normConservation law
V0=V0′ r02−Σrk2−keg2ΣRj2=r0′2−Σrk′2−keg2ΣRj′2 Second-Lorentz transformation 
V0=V0′ p0+2−Σpk+2−keg2ΣPj+2=p0+′2−Σpk+′2−keg2ΣPj+′2 Norm of octonion linear momentum 
(p0+/V0)2−Σ(pk+/V0)2−keg2Σ(Pj+/V0)2 Norm of electric charge 
=(p0+′/V0′)2−Σ(pk+′/V0′)2−keg2Σ(Pj+′/V0′)2 
V0=V0′ L+◦L+*=L+′◦(L+′)* Norm of octonion angular momentum 
V0=V0′ W+◦W+*=W+′◦(W+′)* Norm of octonion torque 
 (W+/V02)◦(W+/V02)*=(W+′/V0′2)◦(W+′/V0′2)* Norm of equivalent charge 
V0=V0′ N+◦N+*=N+′◦(N+′)* Norm of octonion force 
Second-speed of lightInvariant relevant to the normConservation law
V0=V0′ r02−Σrk2−keg2ΣRj2=r0′2−Σrk′2−keg2ΣRj′2 Second-Lorentz transformation 
V0=V0′ p0+2−Σpk+2−keg2ΣPj+2=p0+′2−Σpk+′2−keg2ΣPj+′2 Norm of octonion linear momentum 
(p0+/V0)2−Σ(pk+/V0)2−keg2Σ(Pj+/V0)2 Norm of electric charge 
=(p0+′/V0′)2−Σ(pk+′/V0′)2−keg2Σ(Pj+′/V0′)2 
V0=V0′ L+◦L+*=L+′◦(L+′)* Norm of octonion angular momentum 
V0=V0′ W+◦W+*=W+′◦(W+′)* Norm of octonion torque 
 (W+/V02)◦(W+/V02)*=(W+′/V0′2)◦(W+′/V0′2)* Norm of equivalent charge 
V0=V0′ N+◦N+*=N+′◦(N+′)* Norm of octonion force 
View Large

Table V includes some conserved quantities. (a) Non-conserved electric charge. From Eqs. (10) and (19), the second-speed of light and the norm of octonion linear momentum can be chosen, where both are conserved simultaneously, that is, the electric charge, (P0+/V0)⁠, is not conserved. (b) Non-conserved equivalent charge. From Eqs. (10) and (21), the second-speed of light and the norm of octonion torque can be selected, where both are conserved simultaneously, that is, the second-energy term, W20i+⁠, or the equivalent charge, (W20i+/V02)⁠, is not conserved. (c) Non-conserved continuity equations. From Eqs. (10) and (22), we may choose not only the second-speed of light but also the norm of octonion force to be conserved simultaneously. It means that the continuity equations are not invariable nor conserved. The different combinations of the remaining conserved quantities relevant to the norms have an important influence on the theoretical analysis.

The norm of octonion velocity is one scalar too and is the same under the rotational transformations of octonion coordinate systems. Hence, the above-mentioned research methods will be extended to the norm of octonion velocity and others, researching the basic postulates of several coordinate transformations and continuity equations and others.

The norm of octonion velocity remains unchanged in the rotational transformation of octonion coordinate systems. According to the octonion property, a few invariants relevant to the norm of octonion velocity could be achieved, including the conservation laws and continuity equations, in the electromagnetic and gravitational fields, considering the contribution of material media.

In the octonion space O⁠, the octonion velocity V in the coordinate system α can be transformed into the octonion velocity V′ in the coordinate system β. In the rotational transformation of the octonion coordinate system α, the norm of the octonion velocity is the same. Therefore,

(23)

or

(24)

Similarly, each of the norms of the octonion radius vector R⁠, linear momentum P+⁠, angular momentum L+⁠, torque W+⁠, and force N+ is the same in the rotational transformation of the octonion coordinate system (see Tables IV and V). The selection of two invariants, Eqs. (24) and (19), will deduce the transformation of the norm of the octonion linear momentum.

In the octonion space O⁠, Eq. (24) can be simplified into

(25)

where v0(2)2=v02−Σvk2−keg2ΣVj2 and v0(2)′2=v0′2−Σvk′2−keg2ΣVj′2⁠.

This means that we may select the speed of light, v0(2), to be invariable in the octonion space O⁠. From the two equations, Eqs. (19) and (25), there is the transformation of norm of mass,

(26)

In the transformed octonion space Ou⁠, Eq. (24) is also reduced to

(27)

where −keg2V0(2)2=v02−Σvk2−keg2ΣVj2 and −keg2V0(2)′2=v0′2−Σvk′2−keg2ΣVj′2⁠.

This implies that the second-speed of light, V0(2), can be chosen to be invariable in the transformed octonion space Ou⁠. From the two equations, Eqs. (19) and (27), there is the transformation of the norm of electric charge,

(28)

In terms of the norm of octonion torque, a few similar inferences can be achieved (Table VI). The selection of the two equations, Eqs. (25) and (21), will deduce the transformation of the norm of equivalent mass when the speed of light, v0(2), is invariable in the octonion space O⁠. Next, choosing the two equations, Eqs. (27) and (21), will infer the transformation of the norm of equivalent charge when the second-speed of light, V0(2), is invariable in the transformed octonion space Ou⁠. It is a remarkable fact that some preceding invariants in Tables II and III will become the variable quantities after Eq. (24) replaces Eq. (2) or Eq. (10). In other words, these physical quantities are no longer conserved quantities. From the perspective of octonion spaces, each conserved quantity has several preconditions, in particular the time and speed of light that are often used.

TABLE VI.

Some conservation laws and continuity equations and the basic postulates of coordinate transformations are dependent on the norms of physical quantities in the octonion space for the gravitational and electromagnetic fields. They can be derived from the invariants relevant to the norms of physical quantities when the speed of light is not constant while the contribution of material media is considered.

First invariantSecond invariantConservation law
V◦V*=V′◦V′* r02−Σrk2−keg2ΣRj2=r0′2−Σrk′2−keg2ΣRj′2 Norm of octonion radius vector 
V◦V*=V′◦V′* p0+2−Σpk+2−keg2ΣPj+2=p0+′2−Σpk+′2−keg2ΣPj+′2 Norm of octonion linear momentum 
v0(2)=v0(2)′ (p0+/v0(2))2−Σ(pk+/v0(2))2−keg2Σ(Pj+/v0(2))2 Norm of mass 
=(p0+′/v0(2)′)2−Σ(pk+′/v0(2)′)2−keg2Σ(Pj+′/v0(2)′)2 
V0(2)=V0(2)′ (p0+/V0(2))2−Σ(pk+/V0(2))2−keg2Σ(Pj+/V0(2))2 Norm of charge 
=(p0+′/V0(2)′)2−Σ(pk+′/V0(2)′)2−keg2Σ(Pj+′/V0(2)′)2 
V◦V*=V′◦V′* L+◦L+*=L+′◦(L+′)* Norm of octonion angular momentum 
V◦V*=V′◦V′* W+◦W+*=W+′◦(W+′)* Norm of octonion torque 
v0(2)=v0(2)′ (W+/v0(2)2)◦(W+/v0(2)2)* Norm of equivalent mass 
=(W+′/v0(2)′2)◦(W+′/v0(2)′2)* 
V0(2)=V0(2)′ (W+/V0(2)2)◦(W+/V0(2)2)* Norm of equivalent charge 
=(W+′/V0(2)′2)◦(W+′/V0(2)′2)* 
V◦V*=V′◦V′* N+◦N+*=N+′◦(N+′)* Norm of octonion force 
First invariantSecond invariantConservation law
V◦V*=V′◦V′* r02−Σrk2−keg2ΣRj2=r0′2−Σrk′2−keg2ΣRj′2 Norm of octonion radius vector 
V◦V*=V′◦V′* p0+2−Σpk+2−keg2ΣPj+2=p0+′2−Σpk+′2−keg2ΣPj+′2 Norm of octonion linear momentum 
v0(2)=v0(2)′ (p0+/v0(2))2−Σ(pk+/v0(2))2−keg2Σ(Pj+/v0(2))2 Norm of mass 
=(p0+′/v0(2)′)2−Σ(pk+′/v0(2)′)2−keg2Σ(Pj+′/v0(2)′)2 
V0(2)=V0(2)′ (p0+/V0(2))2−Σ(pk+/V0(2))2−keg2Σ(Pj+/V0(2))2 Norm of charge 
=(p0+′/V0(2)′)2−Σ(pk+′/V0(2)′)2−keg2Σ(Pj+′/V0(2)′)2 
V◦V*=V′◦V′* L+◦L+*=L+′◦(L+′)* Norm of octonion angular momentum 
V◦V*=V′◦V′* W+◦W+*=W+′◦(W+′)* Norm of octonion torque 
v0(2)=v0(2)′ (W+/v0(2)2)◦(W+/v0(2)2)* Norm of equivalent mass 
=(W+′/v0(2)′2)◦(W+′/v0(2)′2)* 
V0(2)=V0(2)′ (W+/V0(2)2)◦(W+/V0(2)2)* Norm of equivalent charge 
=(W+′/V0(2)′2)◦(W+′/V0(2)′2)* 
V◦V*=V′◦V′* N+◦N+*=N+′◦(N+′)* Norm of octonion force 
View Large

Consequently, the ratio of charge to mass must be variable, that is, it is not an invariant.29 Moreover, the reason for the variation in the charge-to-mass ratio is quite complicated. (a) According to the invariants in Tables II and III, it can be deduced that the ratio of charge to mass is not an invariant. (b) From the invariants in Tables IV and V, it can be inferred that the ratio of charge to mass is not an invariant. (c) One can derive that the ratio of charge to mass is not an invariant, from the invariants in Table VI. (d) A part of the octonion field potential will also exert an impact on the speed of light, v0, and second-speed of light, V0, resulting in the variation in the ratio of charge to mass.

The above-mentioned research shows that a few combinations of invariants can be selected as the basic postulates of some coordinate transformations from different focuses of consideration. They can propose several theoretical explanations of physics, describing various types of physical phenomena, in particular the varying speed of light and variable optical index and others within the optical materials.

The scalar part of an octonion will be the same in case the octonion coordinate system transforms rotationally. By making use of this octonion property, some invariants of Tables II–VI in the octonion space can be deduced, including the basic postulates of Galilean transformation, Lorentz transformation, transformation of the norm of octonion velocity, and so forth.

  1. Galilean transformation. The basic postulate of Galilean transformation and other invariants, in Tables II and III, plays an important role in some cases. Meanwhile, the contributions of invariants are negligible in Tables IV–VI. This is one of the viewpoints about the coordinate transformations in physics before the 20th century. It properly describes most of the physical phenomena encountered in classical physics, in particular the cases of low-speed movement or low field strength.

  2. Lorentz transformation. The basic postulate of Lorentz transformation and other invariants, in Tables IV and V, plays a major role in certain circumstances. In addition, the contribution of invariants can be neglected in Tables II, III and VI. This is one type of viewpoint about the relativity in the 20th century. It is capable of describing some special physical phenomena, in particular the high-speed motions, with the speed of light being a constant.

  3. Transformation of the norm. The basic postulates of the transformations, relevant to the norm of octonion velocity, and other invariants, in Table VI, act as an important part in some situations. Meanwhile, it allows us to neglect the contributions of invariants in Tables II–V. This is one point of view in the paper. It is capable of depicting a few special physical phenomena, including some situations of varying speed of light and variable refractive indices.

In the octonion spaces, there are several preconditions for each conserved quantity in the field theories, in particular the time and the speed of light that are often used, from the perspective of octonion spaces. As a result, the physical scenarios you see are different from each other, from the perspective of different combinations of invariants. When some multiple combinations of invariants play an important role, they are able to explore a variety of physical phenomena. If each of these multiple combinations of invariants has a certain contribution, we can achieve the superposition of multiple physical scenarios. Consequently, the physical phenomena that field theory can describe are much more colorful than ever before (Table VII).

TABLE VII.

In the octonion spaces for the electromagnetic and gravitational fields, the comparison of physical characteristics among several main coordinate transformations, including the basic postulates and mathematical difficulty as well as the number of equations.

Coordinate transformationBasic postulateMathematical difficultyNumber of equationsOctonion space
Galilean transformation Equations (1) and (2) Simple Many O 
Second-Galilean transformation Equations (9) and (10) Simple Many Ou 
Lorentz transformation Equations (2) and (18) Complicated Few O 
Second-Lorentz transformation Equations (10) and (18) Complicated Few Ou 
Transformation of norm of octonion velocity Equation (18) and (25) or (27) More complicated Lesser O or Ou 
Coordinate transformationBasic postulateMathematical difficultyNumber of equationsOctonion space
Galilean transformation Equations (1) and (2) Simple Many O 
Second-Galilean transformation Equations (9) and (10) Simple Many Ou 
Lorentz transformation Equations (2) and (18) Complicated Few O 
Second-Lorentz transformation Equations (10) and (18) Complicated Few Ou 
Transformation of norm of octonion velocity Equation (18) and (25) or (27) More complicated Lesser O or Ou 
View Large

Compared with Tables IV and V, Tables II and III are seized of more equations, enabling them to describe more detailed physical properties of physical phenomena. Consequently, these equations in Tables II and III are easier to be widely applied as a theoretical description. On the other hand, compared with Tables IV and V, Table VI possesses fewer equations, exploring several coarser physical properties of physical phenomena. Therefore, these equations in Table VI are utilized less as a theoretical statement. By comparing the three types of coordinate transformations, it can be found that the Galilean transformation may depict a larger number of invariant details. The Lorentz transformation will picture one small number of invariant details. The comparatively complicated coordinate transformations, relevant to the norm of octonion velocity, can only describe the smallest number of invariant details. Apparently, in the theoretical interpretation of physical phenomena, the equations in Tables II and III must be preferred for the theoretical analysis of physical phenomena in order to obtain as many details of theoretical narration as possible. This is an important reason why the equations in Tables II and III can be widely used. Thereafter, Tables IV–VI serve as the supplementary means for theoretical descriptions.

In the actual theoretical description of a large number of physical phenomena that occur simultaneously, the theoretical description will be based on the physical description of invariants and Galilean transformation in Tables II and III. For some cases, it is necessary to superimpose the physical description of invariants and Lorentz transformation in Tables IV and V, affording an appropriate theoretical description of some simultaneous physical phenomena. In addition, even the physical description of invariants and transformation of norms in Table VI should be superimposed, for a fewer cases. It enables the physical descriptions to describe a large number of simultaneous physical phenomena more accurately.

In other words, most physical events involve the Galilean transformation and the remaining invariants in Tables II and III for the physical events that have been involved so far. Few physical events involve the Lorentz transformation and other invariants in Tables IV–V. In addition, fewer physical events involve the transformation of the norm of octonion velocity and other invariants in Table VI.

In terms of energy conservation, in case Tables II and III can be considered as the special case of Tables IV and V, respectively, the scope of application of the norm of octonion torque, Eq. (21), in Table IV will be much larger than that of the law of energy conservation, Eq. (5), in Table II.

Through comparison and analysis, it can be found that the theoretical explanation of some physical phenomena, relevant to the varying speed of light, is independent from that in the octonion composite spaces.30 In the octonion composite spaces, the octonion composite radius vector is R+=R+krxX⁠, the quaternion composite operator is ◊+=ii0∂/∂r0++Σik∂/∂rk+⁠, and rj+=rj+krxxj⁠.

As mentioned above, the invariants can be divided into two different groups in the gravitational and electromagnetic fields. The first group of invariants is in the octonion space O⁠, including Tables II and IV. Hence, these invariants can merely hold in the octonion space O⁠. The second group of invariants is in the transformed octonion space Ou⁠, including Tables III and V. As a result, these invariants can only be established in the transformed octonion space Ou⁠. Apparently, the first group of invariants, in the octonion space O⁠, cannot be established with the second group of invariants in the transformed octonion space Ou simultaneously.

Maxwell first utilized the algebra of quaternions to describe the physical properties of electromagnetic fields. This inspired the subsequent scholars to apply quaternions to study the physical properties of gravitational fields. Nowadays, the scholars use the algebra of octonions to explore the physical properties of electromagnetic and gravitational fields simultaneously, including the octonion radius vector, velocity, field strength, field source, angular momentum, torque, force, and others.

The scalar part of an octonion will remain unchanged when the octonion coordinate system transforms rotationally. By means of this octonion property, a few invariants, including the conservation laws and continuity equations and the basic postulates of some coordinate transformations, could be deduced. The scalar parts of the octonion radius vector and velocity both remain unchanged in the octonion coordinate system of rotational transformation. In the octonion space O⁠, the two invariants can be selected as the basic postulates of Galilean transformation. In the octonion space, the Galilean transformation believes that both time and mass are invariants. Similarly, each of the scalar parts of the octonion linear momentum, torque, and so forth is an invariant as well. The law of mass conservation and fluid continuity equation and others can be derived from these invariants and unvarying speed of light. In the transformed octonion space Ou⁠, it is capable of inferring the law of charge conservation and current continuity equation and so forth. In the octonion space, the second-Galilean transformation deems that the electric charge is an invariant. Nevertheless, the transformed octonion space Ou is independent of the octonion space O⁠. Consequently, the law of charge conservation and law of mass conservation cannot be established simultaneously. The current continuity equation and fluid continuity equation are unable to be effective simultaneously.

The norm of an octonion is a scalar as well. The norm of the octonion will remain unchanged when the octonion coordinate system transforms rotationally. By making use of this octonion property, several invariants relevant to the norms, including the conservation laws and the basic postulates of some coordinate transformations, could be inferred. In terms of the octonion radius vector, its norm is the same in the octonion coordinate system of rotational transformation. In the octonion space O⁠, this invariable norm and unvarying speed of light can be chosen as the basic postulates of Lorentz transformation. In the octonion space, the Lorentz transformation reckons that neither time nor mass is an invariant. Similarly, each of the norms of the octonion linear momentum, torque, and others is not an invariant either. The law of conservation relevant to the norm of mass and others can be derived from these invariants and unvarying speed of light. In the transformed octonion space Ou⁠, the law of conservation relevant to the norm of electric charge and so forth could be concluded. In the octonion space, the second-Lorentz transformation considers that the electric charge is not an invariant. However, the law of conservation relevant to the norm of charge is unable to be established with the law of conservation relevant to the norm of mass simultaneously because the transformed octonion space Ou is independent of the octonion space O⁠.

The norm of octonion velocity is also a scalar and remains unchanged when the octonion coordinate system transforms rotationally. From this octonion property, a few more complicated invariants related to the norms, including the conservation laws and the basic postulates of some coordinate transformations, could be derived. In the octonion coordinate system of rotational transformation, the norms of the octonion radius vector and velocity both remain unchanged. If these two invariants are selected as the basic postulates, we may get a more complicated coordinate transformation related to the norm of octonion velocity. In the coordinate transformation, time, mass, electric charge, and so forth are not invariable.

In the octonion space O⁠, compared with the number of equations related to the Lorentz transformation in Tables IV and V, the Galilean transformation in Tables II and III is relevant to more equations. The latter in Tables II and III are provided with lower mathematical difficulty, describing a larger number of invariant details. Conversely, for the complicated coordinate transformation related to the norm of octonion velocity in Table VI, the number of related equations is less than that of the Lorentz transformation in Tables IV and V. The equations in Table VI have a higher mathematical difficulty, and they can merely investigate the invariant details more roughly. In other words, the equations relevant to the Galilean transformation are able to describe the maximum number of invariant details. The relevant case of Lorentz transformation involves the second largest number of invariant details. The equations of complicated coordinate transformation, relevant to the norm of octonion velocity, can merely explore the minimum number of invariant details. Furthermore, there is also a similar situation in the transformed octonion space Ou as in the octonion space O⁠. Consequently, we must first choose the equations related to the Galilean transformation, in Tables II and III, to explain the physical phenomena in the vast majority of cases.

It is noteworthy that this paper discusses only a few simple cases of invariants and physical properties of rotational transformation of coordinate systems in the octonion spaces, including the basic postulates of the Galilean transformation and Lorentz transformation. However, it has been clearly pointed out that the basic postulates of Galilean transformation and Lorentz transformation are both invariants of the rotational transformation of the octonion coordinate systems. Choosing different combinations of invariants will infer the basic postulates of other types of coordinate transformation. Therefore, it can deduce the basic postulates of more complicated coordinate transformations, in particular several predictions related to mass and electric charge. In future studies, the plan is to explore some more complicated coordinate transformations, verifying the relevant theoretical predictions in experiments.

The author thanks Mr. Heng-Lin Wang, one of his former colleagues, for his interest in this research and financial support. In addition, the author is indebted to the anonymous referees for their valuable comments on the previous manuscripts. This project was supported partially by the National Natural Science Foundation of China, under Grant No. 60677039.

The author has no conflicts to disclose.

Zi-Hua Weng: Conceptualization (lead); Writing – original draft (lead).

The data that support the finding of this study are available within this article.

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