Basic postulates of some coordinate transformations within material media

The 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. J. C. Maxwell first utilized the quaternions to describe the electromagnetic theory. Subsequent scholars make use of the 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 $\mathbb{O}$ , 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, it is able to derive several invariants from the octonion property, in the transformed octonion space $\mathbb{O}_u$ . And that the invariants can be chosen as the basic postulates of a few new types of coordinate transformations. Further, 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.


I. INTRODUCTION
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 the 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, it is able to achieve a few invariants. The combination of these invariants can be selected as the basic postulates for the Galilean transformation, The scholars owe B. Franklin the creation of the law of charge conservation. In 1747, B. 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 a vacuum, the 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 analysis shows that the existing field theories have a few defects in the ex-ploration 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 believes that the time and mass both are invariable, and this is fit for the low-speed movement cases. The Lorentz transformation concerns the speed of light and norm of radius vector. It deems that the time and mass both are not invariable, and this is applied to explaining the high-speed movement cases.
But these two coordinate transformations are unable to explore some invariants related to the electric charges.
(2) Unconstrained establishment. The existing field theory reckons that the electromagnetic field and gravitational field both are in the same space. It presupposes that all of 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. And even the law of charge conservation and the law of mass conservation can be available simultaneously. However, the viewpoint leads to its inability to explain why the charge-to-mass ratio changes. In a stark contrast to the 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. Also 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. J. C. Maxwell first applied not only the algebra of quaternions but also the vector analysis to explore the physical properties of electromagnetic fields. The subsequent scholars 10 uti-lize the quaternions 11 and octonions 12 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 , and dark matters 26 and so forth.
In the paper, making use of the octonion field theory, it is able to explore some invariants, conservation laws and continuity equations, relevant to the electromagnetic and gravitational fields. And that the studies described with the octonion field theory possess a few advantages as follows. (2) Restrictive establishment. The invariants in the transformed octonion space O u cannot be established simultaneously with those in the octonion space O . As a result, the conservation laws are divided into two different types of 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 the 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 the 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 the paper, when an octonion coordinate system transforms rotationally, the scalar part of one octonion will keep 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, it is able to achieve the basic postulates of Galilean transformation and Lorentz transformation. Similarly, it is capable of inferring the basic postulates of several new coordinate transformations in the transformed octonion space O u .
Further, 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 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.

II. OCTONION SPACES
R. Descartes believed that the space is the extension of substance. Nowadays, the Cartesian thought is improved to that the fundamental space is the extension of fundamental field 27 . The fundamental fields include the gravitational field and electromagnetic field. In the octonion space O for the electromagnetic and gravitational fields, i j and I j both are the basis vectors, while r j and R j both are the coordinate values. The octonion radius vector is, R = R g + k eg R e . The octonion velocity is, Herein k eg is one coefficient, to meet the demand for dimensional homogeneity.
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, it is able to define the octonion angular momentum L, torque W and force N . Further, the octonion field strength F and angular momentum L can be combined together to become the octonion composite field strength, F + = F+k f l L . That is, F + is the octonion field strength of electromagnetic and gravitational fields within the material media. Herein k f l = −µ g , which is the coefficient to satisfy the needs of dimensional homogeneity 28 . µ g is the gravitational constant.
In the octonion space O considering the contribution of material media, the octonion field source can be defined as, µS + = −(iF + /v 0 + ♦) * • F + , within material media. The octonion linear momentum is defined as, P + = µS + /µ g , within material media. The gravitational strength is, F + g = i 0 f + 0 + Σi k f + k , within material media. The electromagnetic strength is, F + e = I 0 F + 0 + ΣI k F + k , within material media. The gauge condition is chosen as, f + 0 = 0 and F + 0 = 0, within material media. The octonion field source can be rewritten as, µS + = µ g S + g + µ e S + e − (iF + /v 0 ) * • F + , within material media. The gravitational source is, S + g = ii 0 s + 0 + Σi k s + k , within material media. The electromagnetic source is, S + e = iI 0 S + 0 + ΣI k S + k , within material media. Herein f + 0 , s + j , F + 0 , and S + j are all real. f + k and F + k both are complex numbers. µ is one coefficient. µ e is the electromagnetic constant. k 2 eg = µ g /µ e . The quaternion operator is, ♦ = ii 0 ∂ 0 + Σi k ∂ k , with ∂ j = ∂/∂r j . * 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, . The term, F + ext , is the octonion composite field strength from the external of material media. And k pf is one coefficient, to meet the demand for dimensional homogeneity ( Table 1).
The octonion angular momentum within material media is defined as, L + = (R+k rx X) × • P + , with X being the octonion integrating function of field potential. The octonion composite angular momentum can be rewritten as, L + = L + g + k eg L + e , within material media.
The coefficient, k rx = 1/v 0 , is able to meet the demand for dimensional homogeneity. L + 1 is the angular momentum within material media. L i+ 1 is called as the mass moment temporarily. L i+ 2 is the electric moment within material media, while L + 2 is the magnetic moment within material media.
2k are all real. × represents the complex conjugate. The octonion torque is, And it can be rewritten as, is the torque within material media, including the gyroscopic torque. W i+ 20 is called as the second-energy within material media temporarily. W i+ 2 is called as the second-torque within material media temporarily.
And it will be rewritten as, is the power within material media. N i+ 1 is the force within material media, including the Magnus force. N i+ 20 is called as the second-power within material media temporarily. N i+ 2 is called as the second-force within material media temporarily. 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. Further, 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 O u , 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 F u , field source S u , linear momentum P u , angular momentum L u , torque W u , and force N u . 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 F + u , field source S + u , linear momentum P + u , angular momentum L + u , torque W + u , and force N + u . It is capable of deriving a few coordinate transformations and invariants within material media, from the above octonion composite physical quantities within material media.

III. GALILEAN TRANSFORMATION
Although the vector part of the octonion may vary, the scalar part of one octonion will remain unchanged still, in the rotational transformations of octonion coordinate systems.
Making use of this property of octonion physical quantities, it is capable of inferring several classical invariants, conservation laws and continuity equations, in the electromagnetic and gravitational fields considering the contribution of material media.

A. Law of mass conservation
In the octonion space O , the octonion radius vector, R = ii 0 r 0 + Σi k r k + k eg (iI 0 R 0 + ΣI k R k ) , in the coordinate system α can be transformed into the octonion radius vector, , in the coordinate system β. In the rotational transformation of octonion coordinate system α, the scalar part of one octonion keeps the same. Therefore, where The combination of the above 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 2). 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 the Table 2 can be established simultaneously.
The Table 2 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 octonion radius vector to be conserved simultaneously. It implies that the time is conserved.
And 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 both are conserved simultaneously. That is, the gravitational mass, (p + 0 /v 0 ), is conserved. (c) Conserved energy. From Eqs. (2) and (6), it is capable of selecting the speed of light and energy both are conserved simultaneously. That is, the equivalent mass, (W i+ 10 /v 2 0 ), is conserved. Further, it is able to achieve other types of conserved quantities.
The above 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, S r (r 1 , r 2 , r 3 ), to describe relevant physical phenomena. (b) According to Eqs. (2) and (3), one may select a coordinate system, S p (p + 1 , p + 2 , p + 3 ), 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 Eq.(2) and Eqs.(4) to (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 .
But the law of charge conservation will be effective, in the transformed octonion space

B. Law of charge conservation
In the octonion space, if we multiply the basis vector, iI 0 , by the octonion radius vector, R , from the left, it is able to achieve one new octonion radius vector, The  TABLE II. Some conservation laws, 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.
(conserved) torque continuity equation latter can be considered as one octonion physical quantity in the transformed octonion space O u , that is, R u = iI 0 •(k eg R e +R g ). Apparently, the octonion radius vector R u is independent of the octonion radius vector R , in particular the sequence of basis vectors or coordinate values.
In the transformed octonion space O u , the octonion radius vector, R u = k eg (R 0 + iΣi k R k ) − (I 0 r 0 + iΣI k r k ), in the coordinate system ζ can be transformed into the octonion , in the coordinate system η. In the rotational transformation of octonion coordinate system ζ, the scalar part of the octonion radius vector keeps the same. As a result, In a similar way, the octonion velocity V u (v j , V j ), linear momentum P + u (p + j , P + j ), angular momentum L + u (L + 1j , L + 2j , L i+ 1k , L i+ 2k ), torque W + u (W + 1j , W + 2j , W i+ 1j , W i+ 2j ) and force N + u (N + 1j , N + 2j , N i+ 1j , N i+ 2j ), in the coordinate system ζ , can be transformed into the octonion velocity and force N +′ u (N +′ 1j , N +′ 2j , N i+′ 1j , N i+′ 2j ) in the coordinate system η, respectively (see Ref. [28]). So there are, The  Table 3 can be established simultaneously.
The Table 3  According to Eq.(10) and Eqs. (15) to (16), it is able to choose different types of coordinate systems 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 O u is distinct to the octonion space O , so the invariants in the Table 3 cannot be established with those in the Table 2 simultaneously. In the transformed octonion space O u , 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 O u . 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 O u .
The norm of octonion radius vector is one scalar also, and remains unchanged in the rotational transformations of octonion coordinate systems. Consequently, the above research methods can be extended to the norm of octonion radius vector and so forth, exploring the basic postulate of Lorentz transformation, and continuity equations and others.

A. Octonion space O
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 octonion coordinate system α, the norm of octonion radius vector keeps the same. Therefore, or 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 octonion coordinate system α, each of these norms of octonion physical quantities keeps the same. So there are, The combination of the above 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 invariants can be chosen as the basic postulates for different coordinate transformations, describing a few diverse overviews of the physical world ( Table 4). The invariants in the Table 4 can be effective simultaneously.
The Table 4 includes a few classical conserved quantities. (a) Lorentz transformation.
From Eqs. (17) and (2) In the transformed octonion space O u , the octonion radius vector R u in the coordinate system ζ can be transformed into the octonion radius vector R ′ u in the coordinate system η. In the rotational transformation of octonion coordinate system ζ, the norm of octonion radius vector R u keeps the same. It is easy to find that the norm of octonion radius vector R u , in the transformed octonion space O u , is identical to that of octonion radius vector R, Similarly, the norm of octonion linear momentum P + u , angular momentum L + u , torque W + u and force N + u , in the transformed octonion space O u , will be identical to that of octonion linear momentum P + , angular momentum L + , torque W + and force N + in the octonion space O, respectively.
The combination of the above invariants can give some coordinate transformations, in the transformed octonion space O u . Apparently, a few combinations of different invariants can deduce other types of coordinate transformations, depicting several different physical scenarios ( Table 5). And that the invariants in the Table 5 can be effective simultaneously.
The Table 5

V. NORM OF OCTONION VELOCITY
The norm of octonion velocity remains unchanged, in the rotational transformations of octonion coordinate systems. According to the octonion property, it is able to achieve a few invariants relevant to the norm of octonion velocity, 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 octonion coordinate system α, the norm of octonion velocity keeps the same. Therefore, or Similarly, each of the norms of octonion radius vector R , linear momentum P + , angular momentum L + , torque W + and force N + keeps the same, in the rotational transformation of octonion coordinate system (see Tables 4 and 5). The selection of two invariants, Eqs. (24) and (19), will deduce the transformation of the norm of octonion linear momentum.
In the octonion space O , Eq.(24) can be simplified into, where The above means that we may select the speed of light, v 0(2) , to be invariable, in the octonion space O . From two equations, Eqs. (19) and (25) , there is the transformation of norm of mass, In the transformed octonion space O u , Eq.(24) is also reduced into, The above implies that it is able to choose the second-speed of light, V 0(2) , to be invariable, in the transformed octonion space O u . From two equations, Eqs. (19) and (27) , there is the transformation of norm of electric charge, In terms of the norm of octonion torque, it is able to achieve a few similar inferences ( Table 6). The selection of two equations, Eqs. (25) and (21) Consequently, the ratio of charge to mass must be variable, that is, it is not an invariant 29 .
Moreover, the reason for the variation of the charge-to-mass ratio is quite complicated. (a) According to the invariants in the Tables 2 and 3, it is able to deduce that the ratio of charge to mass is not an invariant. (b) From the invariants in the Tables 4 and 5, it is capable of inferring 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 the Table 6

VI. DISCUSSIONS
The scalar part of an octonion will keep the same, in case the octonion coordinate system transforms rotationally. Making use of this octonion property, it is able to deduce some invariants of the Tables 2 to 6    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 7).
Compared with the Tables 4 and 5, the Tables 2 and 3  Therefore, these equations in the Table 6 Table 6.
In terms of the energy conservation, in case the Tables 2 and 3 can be considered as   the special case of the Tables 4 and 5, respectively, the scope of application of the norm of octonion torque, Eq.(21), in the Table 4 will be much larger than that of the law of energy conservation, Eq.(5), in the Table 2.
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 the that in the octonion composite spaces 30 . In the octonion composite spaces, the octonion composite radius vector is, R + = R + k rx X . And the quaternion composite operator is, ♦ + = ii 0 ∂/∂r + 0 + Σi k ∂/∂r + k , and r + j = r j + k rx x j . 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 the Tables 2 and 4 Tables 4 and 5, the Galilean transformation in the Tables 2 and 3 is relevant to more equations. The latter in the Tables 2 and 3 are provided with lower mathematical difficulty, describing one larger number of invariant details. Conversely, for the complicated coordinate transformation related to the norm of octonion velocity in the Table 6, the number of related equations is less than that of the Lorentz transformation in the Tables 4 and 5. The equations in the Table 6