Coordinate Transformations with Variable Speed of

PIERS Proceedings, Cambridge, USA, July 5–8, 2010

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Coordinate Transformations with Variable Speed of Light Zi-Hua Weng School of Physics and Mechanical & Electrical Engineering, Xiamen University, Xiamen 361005, China

Abstract— The coordinate transformation with variable speed of light is proposed by means of the algebra of quaternions. The quaternions can be used to describe the property of electromagnetic field and gravitational field. In the quaternion space, some coordinate transformations can be deduced from the feature of quaternions, including Lorentz transformation and Galilean transformation. And that some coordinate transformations with variable speed of light can be obtained as well. The paper claims that the speed of light will be varied with the movements in the electromagnetic field and gravitational field. 1. INTRODUCTION

The variable speed of light concept states that the speed of light in a vacuum may not be constant in some particular cases. In 1937, P. Dirac and others began to investigate the consequences of natural constants changing with time, including the varying speed of light in time. In the quantum theory, R. Feynman believed that there is an amplitude for light to go faster or slower than the conventional speed of light. The light doesn’t go only in straight lines, and it doesn’t go only at the speed of light either. In the cosmology, the first variable speed of light model has been proposed by J.-P. Petit [1] from 1988. Later, the second model by J. W. Moffat [2] in 1992, and A. Albrecht and J. Magueijo in 1998 respectively. Some alternative models have also been proposed. Making use of the property of quaternions [3], we can obtain Galilean transformation and Lorentz transformation [4], when the speed of light is invariable. However the viewpoint about invariable speed of light is being doubted and challenged for a long time. Consequently the people question the validity of these coordinate transformations with invariable speed of light. Up to now, this suspicion remains as puzzling as ever. The paper attempts to explain why the above coordinate transformations remain unchanged in most cases. The quaternion was invented in 1843 by W. R. Hamilton. He made a great effort for theoretical analysis of quaternions, and tried to apply quaternions to describe several physical phenomena. In 1861, J. C. Maxwell applied the algebra of quaternions to depict the properties of electromagnetic field [5]. With the feature of quaternions, we deduce some coordinate transformations, including Galilean transformation, Lorentz transformation, and the coordinate transformations with variable speed of light. In the quaternion spaces, the speed of light will be varied with the movements in either electromagnetic field or gravitational field. 2. TRANSFORMATIONS IN THE QUATERNION SPACE

The electromagnetic theory can be described with the algebra of quaternions. In the treatise on electromagnetic theory, the algebra of quaternion was first used by J. C. Maxwell to represent the various properties of the electromagnetic field [6]. At present, the gravitational field can be described by the algebra of quaternions as well. 2.1. Coordinate Transformation

In the quaternion space, the basis vector is E = (1, i1 , i2 , i3 ), and the radius vector R(r0 , r1 , r2 , r3 ) is defined as, R = r0 + i1 r1 + i2 r2 + i3 r3 , (1) where r0 = v0 t; t denotes the time; v0 is the speed of gravitational intermediate boson. The physical quantity D(d0 , d1 , d2 , d3 ) in quaternion space is defined as, D = d0 + i1 d1 + i2 d2 + i3 d3 .

(2)

When we transform the quaternion coordinate system from one into the other, the physical quantity D is transformed into D0 (d00 , d01 , d02 , d03 ), D0 = K∗ ◦ D ◦ K, where K is the quaternion, and K∗ ◦ K = 1; ∗ denotes the conjugate of quaternion.

(3)

Progress In Electromagnetics Research Symposium Proceedings, Cambridge, USA, July 5–8, 2010

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On the one hand, in case the coordinate system is transforming, the quaternions in the above satisfy the relation as follows, D∗ ◦ D = (D0 )∗ ◦ D0 . (4) On the other hand, when the scalar part of quaternion physical quantity D does not take part in the coordinate transformations, the scalar part d0 remains the same, d0 = d00 .

(5)

From Eqs. (4) and (5), we can obtain some coordinate transformations in the quaternion space. 2.2. Galilean Transformation

In the quaternion space, the velocity V(v0 , v1 , v2 , v3 ) is V = v0 + i1 v1 + i2 v2 + i3 v3 .

(6)

As the coordinate system is transformed into other one, we have a radius vector R0(r00 , r10 , r20 , r30 ) and velocity V0 (v00 , v10 , v20 , v30 ) respectively from Eq. (3). From Eqs. (1), (3), (5), and (6), we have r0 = r00 , v0 = v00 ,

(7)

and then from Eqs. (1) and (4) and the above (j = 1, 2, 3) t0 = t00 , Σ(rj )2 = Σ(rj0 )2 .

(8)

The above means that emphasizing especially the important of radius vector Eq. (1) and velocity Eq. (6) will deduce Galilean transformation of coordinate system from Eqs. (1), (5), and (6). 2.3. Lorentz Transformation

In some special cases, we have to emphasize the importance of the power function of radius vector rather than the influence of radius vector. The physical quantity D(d0 , d1 , d2 , d3 ) is defined as D = R ◦ R = d0 + i1 d1 + i2 d2 + i3 d3 .

(9)

In the above equation, the scalar part remains the same during the quaternion coordinate system is transforming. From Eq. (5) and the above, we have (r0 )2 − Σ(rj )2 = (r00 )2 − Σ(rj0 )2 .

(10)

The above means the spacetime interval d0 remains unchanged, when the coordinate system rotates. From Eqs. (5), (6), and (9), we obtain the Lorentz transformation, (r0 )2 − Σ(rj )2 = (r00 )2 − Σ(rj0 )2 , v0 = v00 .

(11)

The above means that it will deduce Lorentz transformation of coordinate system, when we emphasize the velocity Eq. (6) and physical quantity Eq. (9) rather than the radius vector Eq. (1). It recovers that Galilean transformation and Lorentz transformation depend on the choosing from different combinations of the basic physical quantities. When r02 À Σ(rj )2 and (r00 )2 À Σ(rj0 )2 , we have r02 ≈ (r00 )2 . And then Eq. (11) is reduced to Eq. (7). 2.4. Variable Speed of Light

In some particular cases, we need to emphasize the importance of the power function of velocity rather than the influence of velocity. The physical quantity Q(q0 , q1 , q2 , q3 ) is defined as, Q = V ◦ V = q0 + i1 q1 + i2 q2 + i3 q3 .

(12)

When the coordinate system is transformed into other one, we have one physical quantity Q0 (q00 , q10 , q20 , q30 ) from Eq. (3). In the above equation, the scalar part remains the same during the coordinate system is transforming. From Eq. (5) and the above, we have, ¡ ¢2 ¡ ¢2 (v0 )2 − Σ (vj )2 = v00 − Σ vj0 . (13)

PIERS Proceedings, Cambridge, USA, July 5–8, 2010

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The above states that the speed of light v0 will be changed, when the coordinate system rotates in the quaternion spaces. From Eqs. (1), (5), and (12), we obtain the transformation A with variable speed of light, ¡ ¢2 ¡ ¢2 r0 = r00 , (v0 )2 − Σ (vj )2 = v00 − Σ vj0 . (14) From Eqs. (5), (9), and (12), we acquire the transformation B with variable speed of light, ¡ ¢2 (r0 )2 − Σ(rj )2 = (r00 )2 − Σ(rj0 )2 , (v0 )2 − Σ(vj )2 = (v00 )2 − Σ vj0 . (15) When v02 À Σ(vj )2 and (v00 )2 À Σ(vj0 )2 , we obtain v02 ≈ (v00 )2 . Therefore Eq. (14) is reduced to Eq. (7), while Eq. (15) to Eq. (11). In a similar way, the quantity D and Q can be defined as other kinds of power functions of radius vector R or velocity V, such as D = R ◦ R ◦ R ◦ R or Q = V ◦ V ◦ V, etc. Consequently we may have some more complicated coordinate transformations in the quaternion spaces. Table 1: Some coordinate transformations in the quaternion space. transformations Galilean Lorentz A B others

radius vector r0 = r00 (r0 )2 − Σ(rj )2 = (r00 )2 − Σ(rj0 )2 r0 = r00 (r0 )2 − Σ(rj )2 = (r00 )2 − Σ(rj0 )2 D = R ◦ R ◦ R ◦ R etc

velocity v0 = v00 v0 = v00 ¡ ¢2 2 (v0 ) − Σ(vj )2 = (v00 )2 − Σ vj0 ¡ ¢2 (v0 )2 − Σ(vj )2 = (v00 )2 − Σ vj0 Q = V ◦ V ◦ V etc

3. TRANSFORMATIONS IN THE OCTONION SPACE

The gravitational field and electromagnetic field both can be demonstrated by quaternions, but they are quite different from each other indeed. We add another quaternion space to the ordinary quaternion space to encompass the feature of the gravitational and electromagnetic fields [7]. 3.1. Coordinate Transformation

The basis vector of quaternion space for the gravitational field is Eg = (1, i1 , i2 , i3 ), and that for the electromagnetic field is Ee = (I0 , I1 , I2 , I3 ). While the Ee is independent of the Eg , with Ee = (1, i1 , i2 , i3 ) ◦ I0 . The basis vectors Eg and Ee can be combined together to become the basis vector E of octonion space, that is, E = Eg + Ee = (1, i1 , i2 , i3 , I0 , I1 , I2 , I3 ). The radius vector R(r0 , r1 , r2 , r3 , R0 , R1 , R2 , R3 ) in the octonion space is, R = Σ(ii ri ) + Σ(Ii Ri ),

(16)

and the velocity V(v0 , v1 , v2 , v3 , V0 , V1 , V2 , V3 ) is V = Σ(ii vi ) + Σ(Ii Vi ),

(17)

where R0 = V0 T ; T is one time-like quantity; V0 is the speed of electromagnetic intermediate boson. i0 = 1. i = 0, 1, 2, 3. When the coordinate system is transformed into other one, the octonion physical quantity D will be transformed into D0 (d00 , d01 , d02 , d03 , D00 , D10 , D20 , D30 ), D0 = K∗ ◦ D ◦ K,

(18)

where K is the octonion, and K∗ ◦ K = 1; ∗ denotes the conjugate of octonion. On the one hand, in case the coordinate system is transforming, the octonions in the above satisfy the relation as follows D∗ ◦ D = (D0 )∗ ◦ D0 .

(19)

On the other hand, when the scalar part of octonion does not take part in the coordinate transformation, the d0 remains the same, d0 = d00 . From the above, we can obtain some coordinate transformations in the octonion space.

(20)

Progress In Electromagnetics Research Symposium Proceedings, Cambridge, USA, July 5–8, 2010

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Table 2: The octonion multiplication table.

1 i1 i2 i3 I0 I1 I2 I3

1 1 i1 i2 i3 I0 I1 I2 I3

i1 i1 −1 −i3 i2 −I1 I0 I3 −I2

i2 i2 i3 −1 −i1 −I2 −I3 I0 I1

i3 i3 −i2 i1 −1 −I3 I2 −I1 I0

I0 I0 I1 I2 I3 −1 −i1 −i2 −i3

I1 I1 −I0 I3 −I2 i1 −1 i3 −i2

I2 I2 −I3 −I0 I1 i2 −i3 −1 i1

I3 I3 I2 −I1 −I0 i3 i2 −i1 −1

3.2. Galilean Transformation

When the coordinate system is rotated, we have one radius vector R0 (r00 , r10 , r20 , r30 , R00 , R10 , R20 , R30 ) and velocity V0 (v00 , v10 , v20 , v30 , V00 , V10 , V20 , V30 ) respectively from Eq. (18). From Eqs. (16), (17), (18), and (20), we have r0 = r00 , v0 = v00 ,

(21)

and then from Eqs. (16) and (19) and the above t0 = t00 , Σ(rj )2 + Σ(Ri )2 = Σ(rj0 )2 + Σ(Ri0 )2 .

(22)

The above means that emphasizing the important of radius vector Eq. (16) and velocity Eq. (17) will deduce Galilean transformation of coordinate system. The above implies that the r0 remains unchanged when the quaternion coordinate system rotates, while the R0 keeps changed as a vectorial component. When Ri ≈ Ri0 , Eq. (22) is reduced to Eq. (8). 3.3. Lorentz Transformation

In some special cases, we need to emphasize the power function of radius vector rather than the radius vector. The physical quantity D(d0 , d1 , d2 , d3 , D0 , D1 , D2 , D3 ) is defined as D = R ◦ R = Σ(ii di ) + Σ(Ii Di ).

(23)

By Eqs. (20) and (23), we have (r0 )2 − Σ(rj )2 − Σ(Ri )2 = (r00 )2 − Σ(rj0 )2 − Σ(Ri0 )2 .

(24)

The above represents that the spacetime interval d0 keeps unchanged when the coordinate system rotates in the octonion space. When the octonion space is reduced to the quaternion space, the above equation should be reduced to Eq. (10) in the quaternion space. By Eqs. (17), (20), and (23), we have Lorentz transformation, (r0 )2 − Σ(rj )2 − Σ(Ri )2 = (r00 )2 − Σ(rj0 )2 − Σ(Ri0 )2 , v0 = v00 .

(25)

The above means that emphasizing the important of velocity Eq. (17) and of physical quantity Eq. (23), we obtain Lorentz transformation of coordinate system. In the octonion space, when Ri ≈ Ri0 , Eq. (25) is reduced to Eq. (11) in the quaternion space. 3.4. Variable Speed of Light

In some particular cases, we have to emphasize the power function of velocity rather than the velocity. The physical quantity Q(q0 , q1 , q2 , q3 , Q0 , Q1 , Q2 , Q3 ) in octonion space is defined as Q = V ◦ V = Σ(ii qi ) + Σ(Ii Qi ).

(26)

When the coordinate system is rotated, we have one quantity Q0 (q00 , q10 , q20 , q30 , Q00 , Q01 , Q02 , Q03 ) from Eq. (18). In the above, the scalar part remains the same during the octonion coordinate system is transforming. From Eqs. (18) and (20) and the above, we have ¡ ¢2 (v0 )2 − Σ(vj )2 − Σ(Vi )2 = (v00 )2 − Σ vj0 − Σ(Vi0 )2 . (27)

PIERS Proceedings, Cambridge, USA, July 5–8, 2010

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The above equation represents the speed of light v0 will be changed when the coordinate system rotates in the octonion spaces. From Eqs. (16), (20), and (27), we obtain the transformation A with variable speed of light, ¡ ¢2 ¡ ¢2 ¡ ¢2 r0 = r00 , (v0 )2 − Σ (vj )2 − Σ (Vi )2 = v00 − Σ vj0 − Σ Vi0 . (28) From Eqs. (24) and (27), we gain the transformation B with variable speed of light, (r0 )2 − Σ(rj )2 − Σ(Ri )2 = (r00 )2 − Σ(rj0 )2 − Σ(Ri0 )2 , ¡ ¢2 (v0 )2 − Σ(vj )2 − Σ(Vi )2 = (v00 )2 − Σ vj0 − Σ(Vi0 )2 . In the octonion space, when Vi ≈ Vi0 , Eq. (28) is reduced to Eq. (14) in the quaternion space. It is easy to find that we may similarly have other kinds of coordinate transformations in the octonion space by defining different physical quantities. 4. CONCLUSIONS

In the quaternion and octonion spaces, Galilean transformation and Lorentz transformation etc can be deduced from the features of quaternions and octonions. This states that Lorentz transformation is only one of several coordinate transformations in electromagnetic and gravitational fields. There exist coordinate transformations with variable speed of light in quaternion and octonion spaces. In the electromagnetic and gravitational fields, different coordinate transformations depend on different choosing from the combination of basic physical quantities. And the speed of light will be changed with the movements in electromagnetic field and gravitational field. It should be noted that the study for the coordinate transformation has examined only some simple cases, including Galilean transformation and Lorentz transformation. Despite its preliminary character, this study can clearly indicate that there are several kinds of coordinate transformations with the variable speed of light. For the future studies, the research will concentrate on only some predictions about the complicated coordinate transformations with variable speed of light. ACKNOWLEDGMENT

This project was supported partially by the National Natural Science Foundation of China under grant number 60677039. REFERENCES

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