Biquaternion Relativity - Gravitation as an Effect of Spatial Varying Speed of Light
Andre´ Waser CH-8840 Einsiedeln, Switzerland; [email protected] The biquaternion notation - or more precisely the semi-biquaternion notation - is ap- plied to mechanics and special relativity. The Lorentz transformation as well as electro- magnetic field transformations result from a pure multiplication with the biquaternion relative velocity. By a straight forward development of the biquaternion notation, it turns out that the biquaternion velocity plays a crucial role regarding the principle of equivalence. The total time derivation of the four-velocity delivers inertial as well as gravitational acceleration simultaneously. It our model, the cause of gravitation is a spatial varying speed of light.
Issued: January 01, 2011
1 Introduction The different brackets shall indicate that the two semi-biqua- ternions are not of identical structure. The multiplication of A Hamilton quaternion q [1] is an expansion of real numbers√ by the Hamilton units i, j, k having the magnitude of −1 two biquaternions and forming a four-number as " ! # a ib A = a + b = 0 + 0 q = q0 + iq1 + jq2 + kq3 , i~a ~b whereas q ...q are real numbers. The Hamilton units consti- " ! # 0 3 x iy tute the associative but not commutative multiplications X = x + y = 0 + 0 i~x ~y i2 = j2 = k2 = i jk = −1 , is i j = k jk = i ki = j , i j = − ji jk = −k j ki = −ik . a x − b y + ~a · ~x − ~b · ~y 0 0 0 0 AX = ~ ~ + The real part of a quaternion is often called the scalar part i a0~x + x0~a + b0~y + y0b + ~a × ~y − b × ~x while the other three imaginary parts are called a three-vector. ~ i a0y0 + b0 x0 − ~a · ~y − b · ~x Both together constitute a four vector: . (1) a0~y + x0~b − b0~x − y0~a − ~a × ~x + ~b × ~y q = q0, ~q , The biquaternion multiplication is not commutative but asso- where ~q = iq1 + jq2 + kq3. Later Hamilton [2] presented the ciative. For the biquaternion scalar multiplication the Hamil- biquaternion as an extension of a quaternion by introducing ton units are not multiplied and thus complex numbers q → q +ip , where i denotes the Gaussian n n n imaginary unit. A · X = a0 x0 − ~a · ~x − b0y0 − ~b · ~y . Q = q + ip + i (q + ip ) + j (q + ip ) + k (q + ip ) . 0 0 1 1 2 2 3 3 Thereof follows the magnitude of a biquaternion Generally a biquaternion can be grouped into two four-vec- p q tors |Q| = Q · Q = a2 − ~a · ~a − b2 − ~b · ~b . (2) 0 0 Q = q0, i~q + ip0, ~p = q + p . Both terms q and p shall be called semi-biquaternions. These A special case is given when the second of the two semi- two semi-biquaternions do not have the same structure. The biquaternions is set to zero. With b = 0 and ~y = 0, the multi- first term is very suitable to cast translational physical four- plication of two biquaternions reduces to quantities whereas the second may represent a complemen- " ! # a x + ~a · ~x 0 tary rotational four-quantity, if existing. AX = 0 0 + . − × For a more convenient readability a biquaternion can be i a0~x + x0~a ~a ~x written as the sum of a ’left’ and ’right’ semi-biquaternion as The multiplication of a semi-biquaternion with an other one " ! # q ip results almost in another semi-biquaternion, except of an ad- Q = q + p = 0 + 0 . i~q ~p ditional cross product.
Andre´ Waser, Biquaternion Relativity - Gravitation as an Effect of Spatial Varying Speed of Light1 2 Biquaternion Velocity With equation (6) follows We have shown some time ago that such semi-biquaternions dX · dX V0 · V0 = (8) can be applied well to electrodynamics [3]. In this paper we dX0 · dX0 focus on mechanics, and especially on special relativity. We This directly leads to the biquaternion coordinate transforma- start with the position biquaternion representing an event in 0 space-time tion from S to S " ! # 0 0∗ ct 0 X = XV (9) X = + (3) i~x 0 as well as the backward transformation where the constant c is the speed of light. With X = X0V0 (10) " ! # cdt 0 dX = + Explicitly the forward and backward transformations are id~x 0 " ! # " ! # ct0 0 ct − ~v·~v 0 and with the proper time interval dτ X0 = + = γ c + (11) i~x0 0 i ~x − ~vt ~x×~v r c p · " ! # " 0 ~v·~v ! # 2 2 ~v ~v ct 0 ct + 0 d |X| c dt − d~x · d~x cdt − cdτ . c 0 = ( ) = 1 2 = X = + = γ 0 0 + ~x ×~v . (12) c i~x 0 i ~x + ~vt − c follows the biquaternion velocity Equation (11) and equation (12) reveal, that " ! # dX c 0 0 V = = γ + (4) ~x × ~v = 0 and ~x × ~v = 0 . dτ i~v 0 Above transformations are valid for the special case of colli- with 0 1 near vectors ~xk~v and ~x k~v only. The general case is found γ = q . by transforming this part of position vector X with a spatial − v2 1 c2 vector parallel to ~v and add the remaining part of X with a spatial vector orthogonal to ~v. With Independent of the magnitude of ~v, the magnitude of the bi- " ! # quaternion velocity is always c. Thus the unity biquaternion ct 0 velocity is Xk = ~x·~v ~v + " ! # i v v 0 0 1 0 V = γ ~v + (5) i c 0 the general forward transformation is
or 0 0∗ 0∗ dX V X = X⊥ + XkV = X − Xk 1 − V , (13) = = V0 (6) d|X| |V| or explicitly The inverse of a unity biquaternion velocity is identical to ~v·~v # its biconjugate, which in turn is again a unity biquaternion γ ct − 0 X0 = c + , velocity i ~x − (1 − γ) ~x·~v ~v − γ~vt 0 1 ∗ v v = V0 (7) V0 whereas the general backward transformation is with " ! # 0 0 0 0 0 0 ∗ 1 0 X = X + X V = X − X 1 − V , (14) V0 = γ + . ⊥ k k −i~v 0 c or explicitly 3 Biquaternion Lorentz Transformation ~v·~v # γ ct + c 0 S shall be the inertial system where an observer is at rest. X = 0 + . i ~x0 − (1 + γ) ~x ·~v ~v + γ~vt 0 Another inertial system S0 shall move with constant velocity v v ~v against S. The distance between two events in four-space is No new Lorentz transformation has been described above. invariant for both systems But it might be of some interest to see that the Lorentz trans- d|X| = d|X0| . formation can be described with a multiplication of the po- sition biquaternion with the velocity biquaternion of relative Also the square of these length elements is an invariant and movement. The component of the position biquaternion par- therefore allel to the velocity biquaternion is dX · dX = dX0 · dX0 = d2|X| = d2|X0| . X · V0 V0 .
2 Andre´ Waser, Biquaternion Relativity - Gravitation as an Effect of Spatial Varying Speed of Light Its Lorentz transformation is The electromagnetic fields follow from the differentiation of ! the potential fields. With the definition of the biquaternion ∗ ~x · ~v X · V0 V0V0 = γ ct − . differential operator c " ! # The result has no spatial position vector at all and has a pure ∂ 1 ∂ 0 ∇ ≡ = c ∂t + . (17) timely character in all cases, i.e. independent of the direction ∂X ∇~ 0 of the unprimed position biquaternion or of its spatial vector. Thus, for every observer in S, the direction of the time vec- follow the biquaternion electric and magnetic fields 0 tor (scalar part) in any moving system S coincides with the 0 1 ∂ϕ ~ ~ ! # direction V of the relative four-velocity. c 2 − ∇ · A 0 E = c∇A = c ∂t + , To find the addition law of two biquaternion velocities iE~ cB~ we denote the relative biquaternion velocity between two sys- tems S and S0 as V, and the velocity of a moving body mea- " ! 1 ∂ϕ ~ ~ 0 i( 2 − ∇ · A) 0 0 B = −i∇A = + c ∂t , sured in S shall be denoted as U . The total biquaternion iB~ − E~ velocity W measured in S is found by the backward transfor- c mation of U0 as where E = −icB. With the Lorentz gauge " 0 # " # ~u ·~v 0 0 0 c + c 1 ∂ϕ W = U V = γuγv 0 + γuγv ~u0×~v − ∇~ · A~ = 0 i ~u + ~v c c2 ∂t with the biquaternion electric and magnetic fields without a scalar 1 1 γu = q and γv = q . part are found 02 2 " ! # 1 − u 1 − v 0 0 c2 c2 E = + , (18) iE~ cB~ Some calculation directly leads to the total velocity " ! # " ! # c 0 0 0 B = + ~ . (19) W = γw + w~×~v iB~ − E iw~ − c c with With the electric field biquaternion (18), the forward transfor- 1 mation is γw = q 1 − w2 " ! # c2 0 0 0 E = 0 + 0 and the law of velocity addition known from Special Relativ- iE~ B~ 1 ity ∗ − ~v · E~ ~v · B~ ~u0 + ~v = EV0 = γ c + γ . (20) w~ = . i E~ + ~v × B~ − E~ ~u0·~v c 1 + c2 Again, the magnitude of W is equal to the speed of light c. Obviously this transformation is only applicable for
4 Biquaternion Field Transformations ~v · E~ = 0 and ~v · B~ = 0 .
The biquaternion electromagnetic potential Thus only the field components orthogonal to the spatial ve- " ! # 1 ϕ 0 locity vector are transformed, but not the collinear field com- A ≡ c + (15) A~ 0 ponents. With 0 0 is transformed exactly in the same way as the biquaternion E = ~ + k i ~v·E ~v 0 position vector. For completeness, the forward transforma- v v tion is ~ we find the general transformation for electromagnetic fields ϕ − ~v·A 0 A0 = γ c c + , ~ − ϕ A~×~v 0 0∗ 0∗ 0∗ i A c2 ~v c E = Ek + E⊥ V = EV + Ek 1 − V . (21) ~ k which again is valid for A ~v only. The general case is found 5 Biquaternion Total Time Derivative with With any biquaternion field ~ γ ϕ − A·~v 0 c c 0 " ! # A = + . (16) a0 0 ~ − − A~·~v ~v − ϕ 0 A = + i A (1 γ) v v γ c2 ~v i~a 0
Andre´ Waser, Biquaternion Relativity - Gravitation as an Effect of Spatial Varying Speed of Light3 that is a function of a biquaternion position The biquaternion force to accelerate a mass with non-relativi- stic velocities is then A = A (X) = A (X (ct, x1, x2, x3)) . " P ! 0 # F ≡ ma = m + , (26) the total time differential is iF~ cΩ~ ∂A (X) ∂A ∂X ∂A ∂X dA = dX = + d~x . or explicitly ∂X ∂X ∂t ∂X ∂~x ∂c + ~v · ∇~ c + c∇~ · ~v + ~v · ~v With F = m ∂t ∂t + ∂ ∂ i ∂~v + ~v ∇~ · ~v + c∇~ c + ~v ∂c − ~v × ∇~ × ~v = ∇ and = ∇~ ∂t c ∂t ∂X ∂~x i~v · ∇~ × ~v follows after some calculation . (27) −c∇~ × ~v − ~v × ∂~v + ∇~ c c∂t dA ∂ + ~v · ∇~ 0 = ∂t + A , ~ ~v ∂ − ◦ ∇~ The non-relativistic spatial Force F~ becomes dt i c∇ + c ◦ ∂t ~v ∂~v ~v ∂c where the multiplication symbol ◦ denotes that for the scalar F~ = m~a = + ~v ∇~ · ~v − ~v × ∇~ × ~v + c∇~ c + . (28) part the scalar product and for the vector part the cross prod- ∂t c ∂t uct must be applied. Explicitly this becomes With c = constant and replacing the second term with a vector identity follows ∂a0 ~ ~ ~a dA + ~v · ∇a0 + c∇ · ~a + ~v · = ∂t ∂t + 2 ! ∂~a ∇~ · ∇~ ~v ∂a0 − × ∇~ × ∂~v v dt i ∂t + ~v ~a + c a0 + c ∂t ~v ~a ~a = + ∇ − ~v × ∇~ × ~v , ∂t 2 ~ i~v · ∇ × ~a . (22) which corresponds to the well known acceleration field in −c∇~ × ~a − ~v × ∂~a + ∇~ a c∂t 0 fluid mechanics [4]. Interestingly this is completely equivalent to 7 Gravitation Potential dA 1 = V ∇ A In equation (28) we find two other terms that are of particular dt γ interest: ~v ∂c such that the relativistic total time derivative operator in bi- F~ ∼ c∇~ c + . quaternion form is c ∂t d With c = constant, thus independent of space and time, both = V ∇ . (23) dτ terms would be zero. But there are already theories of time varying speed of light (VSL theories), which have been intro- 6 Biquaternion Acceleration and Force duced to explain the horizon problem of cosmology and pro- With c = constant and with the relativistic spatial acceleration pose an alternative to cosmic inflation (see for example [5] and citations therein). As a straight forward approach we as- d~v ~a ≡ (24) sume that c=c(~x). To still have a four-space (x0, ~x) with four dt independent dimensions, the time interval dx0 needs to be in- the relativistic biquaternion acceleration a can be written di- dependent of spatial interval d~x and vice-versa. With c , rectly as constant, the interval dx0 shall be invariant to the local speed of light. Thus we have " ! # · # a 0 dV γ4 ~v ~a 0 ≡ 0 c a + = = 2 2 ~v·~a ~v + . dx0 = c0dt = cdtc , i~a 0 dτ iγ γ c c + ~a 0 where c is the constant speed of light in vacuum as com- Instead of using equation (24), the operator (23) can be ap- 0 monly defined. The time interval in a region with c constant plied to find for the relativistic biquaternion acceleration , changes according to dV c a ≡ = V∇V . (25) dt = 0 dt . (29) dτ c c
We will not evaluate this equation for relativistic velocities Thus the interval of time dimension x0 is independent of c. in this paper. With the non-relativistic velocity U, the non- The velocity in a region with c , constant is measured with relativistic acceleration becomes local time interval " ! # dU c 0 d~x c d~x a = = U∇U with U ≡ c0 + . ~v = = . dt i c ~v 0 dtc c0 dt
4 Andre´ Waser, Biquaternion Relativity - Gravitation as an Effect of Spatial Varying Speed of Light This satisfies the principle of causality where local velocity 8 Gravitation of two Body System ~v cannot be higher than local speed of light c. The invariant In a system with two point masses Mm, where M is the length element of four-space is now central mass, m the orbiting mass, and φ the gravity potential s " ! # 2 of M, the speed of light shall be a function of M as well as of c 0 c dX = dt + = cdt 1 − 0 v2 = cdτ (30) the distance ~r measured from M to m, i.e. c C0 4 c i c ~v 0 c c = c f(M,~r) . and the biquaternion velocity (4) changes to 0 " ! # The function f(M,~r) shall fulfill these boundary conditions: dXc c 0 Vc = = γc C0 + (31) • 0 ≤ f(M,~r) ≤ 1 for 0 ≤ r ≤ ∞, and dτc i c ~v 0 • f(M,~r) = 1 with r → ∞, and with 1 • the spatial variation of c must be below the today’s pre- γc = q . c2 cision of light speed measurements. 1 − 0 v2 c4 In plain spherical coordinates with radial symmetry, where M Also the biquaternion differential operator (17) changes to is in the origin, the inertia force Fg to mass m is " ! # ∂ 1 ∂ c ∂t 0 ~ 2 ∂ f (M, r) ∇c ≡ = + , (32) Fg = −mc∇c = mc0 f (M, r) , ∂Xc ∇~ 0 ∂~r where c is not a constant anymore. The total time derivative where r denotes the radius measured from M to m. Above operator (23) further changes to equation shall be equal to the Newton gravity force Mm d F = −G c ~r0 . = Vc ∇c . (33) g r2 dτc In non-relativistic case, the acceleration a of a non-relativistic Here the proper mass mc has been used, since - with conser- velocity U can then be written as vation of total energy-momentum - the mass is also subject to f(M,~r) as dU dVc a = = U∇U ≈ V ∇ V = = a . 2 2 2 2 2 c c c c E = γcmc ≈ mc = mc f (M, r) = mcc . (38) dt dτc 0 0 In non-relativistic case this still corresponds to equation (27). Thus we have The inertia force to a mass m just becomes Fg = −ma. Since ∂ f (M, r) Mm mc2 f (M, r) = −G f 2(M, r)~r0 . we are only interested in derivations of c, above equation can 0 ∂~r r2 be reduced to dc By integration and using above boundary conditions the ex- g = = U∇c . (34) dt plicit solution becomes − GM The inertia force Fg acting to an accelerated mass m is then c2r f (M, r) = e 0 , ~v ∂c F = −mg = −mc∇~ c − m . and thereof the spatial varying speed of light around M to g c ∂t GM r − 2 S Considering only the spatial variation, thus with c constant in c r − r c(M, r) = c0 e 0 = e 2 , (39) time, above equation reduces further to where r is the Schwarzschild radius. The same spatial vari- 2 S c ation of c has been proposed by Puthoff [6]. Newton’s law of Fg = −mc∇~ c = −m∇ = −m∇φ . (35) 2 gravity attraction between two bodies changes slightly to The scalar φ can be interpreted as a gravity potential with Mm − GM 2 2 c c2r 0 units [m /s ] of any mass M as Fg = −G e 0 ~r with ~r = ~rm − ~rM . r2 2 c M,~r 2 φ ~r = . (36) For rrS the deviation of 1/r is far beyond today’s measure- 2 ment capabilities, but below rS the gravity attraction for point In our model, the non-relativistic gravity potential energy V masses turns to zero instead to infinity. of a mass m in any gravity field φ is then Using 39 and the first terms of the corresponding Taylor 1 series, the total energy of a mass m becomes approximately V = mφ = mc2 , (37) −GM 1 GM 2 2 2 c r 2 2 E = γcmc = γcmc e 0 ≈ mc + mv − m + ..., what could be interpreted as the ’kinetic energy’ of a mass m 0 0 2 r ’moving’ with local speed of light c along the time coordinate where the last term represents the traditional gravity potential axis x0. energy of m in gravity field of M.
Andre´ Waser, Biquaternion Relativity - Gravitation as an Effect of Spatial Varying Speed of Light5 9 Planet Orbit with new Gravity Potential 10 Gravitation Time Dilation and Redshift The biquaternion momentum is Using equation (39) with (29), the time dilation in a gravity " ! # field of M is found approximatively with Taylor series as c 0 p = mV = m + . (40) c c c0 s i c ~v 0 − GM c2r GM 2GM dt = e 0 dt ≈ 1 − dt ≈ 1 − dt . (43) For a constant speed of light, the magnitude of any four- c 0 2 0 2 0 c0r c0r dimensional speed is constantly c and the four-dimensional momentum of a body with mass m is always conserved. With The gravitational redshift is defined to presence of gravitation, this momentum shall still be con- λ0 served. This is achieved by equalizing the magnitudes of z = − 1 , λc four-dimensional velocity without and with gravity field. We start with the squared magnitude of four-dimensional velocity where the wavelength λ0 is measured at a distant point of ob- servation and λc closer to the mass M that causes the gravity c2 2 2 2 0 2 potential. The closer point usually is regarded as the emit- c = γc c − v . 0 c2 ting point (of the sun or other celestial bodies) and the distant For non-relativistic velocities and with (39) this can be writ- point as the receiving point (on Earth). For simplicity we as- ten as sume, that emitter and receiver do maintain a constant radial − GM GM 2 2 c2r 2 c2r distance r. The photon energy does not change along its way mc = mc e 0 − mv e 0 . 0 0 according to (38). Thus, the frequency at emitting point must Introducing plain polar coordinates (r: radius, ϕ: rotation an- be identical to receiver point (i.e. there is no doppler effect). gle) and replacing the exponential functions by the first terms With λ=c/ν we directly find of its Taylor series, it follows c0 1 1 −1 z = − 1 = − 1 ≈ − 1 . (44) 2 2 rS 2 2 2 rS − GM q mc = mc 1 − − m r˙ + r ϕ˙ 1 − . c c2r rS 0 0 0 − r r e 1 r For radial symmetric and isotropic conditions this reduces This is the gravity redshift well known from General Relativ- further to ity. But in our model the photon frequency and thus photon 2 2 2 L L rS rS rS energy (four-dimensional momentum) is conserved, where mr˙2 + − − mc2 = mc2 1 − − mc2 , mr2 mr2 r 0 r 0 r 0 the speed of light changes instead. where the angular momentum L = mr2ϕ˙ has been used. The 11 Generalized Biquaternion Inertia and Gravitation angular momentum is conserved for central force fields. Fi- nally, by using the potential energy A generalization of inertia and gravitation can be achieved by consequently interpret the four-velocity Vc as a ’potential rS E = mc2 ≈ mc2 1 − field for inertia and gravity’. 0 r With the derivation of four-velocity V the equation of motion for m around M known from General c # Relativity is found as 1 ∂c + ∇~ · ~u 0 c ∂t ∇cVc = 1 ∂~u + . (45) 2 2 2 2 ∇~ ∇~ × 1 Mm L L M E mc i c + c ∂t ~u mr˙2 − G + − G = − 0 . (41) r 2 2 3 2 2 2mr mc0r 2mc0 2 and with the substitutions The perihelion precession can now be calculated as done in 1 ∂c g = − − ∇~ · ~u standard text books. Finally, by using the four-dimensional 0 c ∂t length element (30) together with equation (39), whereas the ∂~u ~g = −c∇~ c − first terms of its Taylor series are used again, it follows ∂t −1 ~ 2GM 2GM ω~ = ∇ × ~u dX2 = dt2c2 1 − − dx2 1 − . 0 2 2 c0r c0r the a biquaternion ’acceleration field’ can be identified to Using polar coordinates and again assuming radial symmetry " ! # cg 0 and isotropic conditions follows directly the Schwarzschild g = −c∇U = 0 + . (46) i~g cω~ metric ~ 2 2 2 − 2GM − The similarity between ~g and the electric field E as well as dX = dt c0 1 c2r 0 between ω~ and the magnetic induction B~ is obvious. The vec- −1 2 − 2GM − 2 2 − 2 2 . (42) tor field ~g is an acceleration field. The vector field ω~ might be dr 1 c2r r dϕ r sin ϕdθ 0 visualized as a spin (or rotation) field.
6 Andre´ Waser, Biquaternion Relativity - Gravitation as an Effect of Spatial Varying Speed of Light With the biquaternion d’Alembert operator where m is the particle’s rest mass. To find the electric field of a point charge, the modified biquaternion electric potential 2 2 ∗ 1 ∂ ∆ ≡ |∇ | = ∇ ∇ = − ∇~ (47) " 1 ! # c c c c 2 2 ϕ 0 c ∂t A ≡ c + with c = c f (~x) . (49) iA~ 0 0 equation (45) can be expanded further to is required. The electric field biquaternion follows directly as ∗ G G ∇ ∇cVc = P = ρVc , c 2 2 1 ∂ϕ ∇~ · ~ − ϕ ∂c # c c c ∂t + c A c2 ∂t 0 E = −c∇A = + . ~ ∂A~ ϕ ~ ∇~ × ~ where G is the gravity constant, ρ is the mass density and P i ∇ϕ + ∂t − c ∇c c A is the biquaternion momentum density Thereof follows the electric field of an electric potential ϕ as " ! # c 0 P ≡ ρV = ρ + with ρ = γ ρ . (48) ϕ c i c0 ~v 0 c 0 E~ = −∇~ ϕ + ∇~ c . c c
With the ’kinetic Lorentz condition’ g0=0 follow four equa- In particular, for a point charge, this becomes tions similar to Maxwell’s equations as − Gm e c2r 0 E~ e 0 ~r . = 2 (50) ∇~ · ω~ = 0 , 4π0r ∇~ · ~g = Gρ , The electric field of a charged point particle has the same ∂ω~ characteristics as it’s gravity field. + ∇~ × ~g = 0 , ∂t 13 Resume 1 ∂~g G ∇~ × ω~ − = ρ~u . c2 ∂t c2 Casting physical quantities with biquaternions is not essential as such, but it allows a straight-forward, ’intuitive’ approach The further analysis of these equations is beyond the scope to describe physical models with ’natural numbers’ just using of this paper. They might be interesting for fluid mechanics the basic arithmetics of addition and multiplication together as well as for cosmology, for example the analysis of large with some calculus algebra. cosmic structures. With The four-dimensional velocity plays a key role to under- G stand special relativity, field transformations, energy-momen- ∆Vc = ρVc and g0 = 0 tum conservation as well as gravitation. c2 In our proposed model, the speed of light is not a constant follow the inhomogeneous wave equations but depends on masses (or localized energy densities). As ! we have roughly shown, this model can offer an alternative 1 ∂~g G ∂ ρ~u − ∇~ 2~g = − ∇~ (ρc) + , to understand the cause of gravity. The principle of equiva- c2 ∂t c2 ∂t lence, that laid the foundation of General Relativity, has been 1 ∂ω~ G enhanced by a - let’s say - ’principle of identity’. Both, iner- − ∇~ 2ω~ = ∇~ × ρ~u . c2 ∂t c2 tia and gravitation have the same identical origin, namely the total time derivative of the velocity four-vector. Thereof follow, in absence of mass densities, the homogenous wave equations. The result is the prediction of free transverse Issued on January 01, 2011. gravitation waves with propagation velocity c. References 12 Electric Field of a Point Charge 1. Hamilton W. R. ”On a new Species of Imaginary Quantities connected with a theory of Quaternions”, Proceedings of the Royal Irish Academy Since c is not constant, also the permittivity 0 and permeabil- 2 (Nov 13, 1843) 424–434 ity µ0 of the vacuum must change together with c 2. Hamilton W. R. ”Lectures on Quaternions”, Macmillan & Co, Cornell University Library (1853) 2 2 f (~x) f (~x) 3. van Flanderen K. J., Waser A. ”Generalisation of Classical Electrody- c0 f (~x) = . 0 µ0 namics to Admit a Scalar Field and Longitudinal Waves”, Hadronic Journal 24 (2001) 609–628 Thus both vacuum properties 0 and µ0 change inversely to 4. Truckenbrodt E. ”Fluidmechanik”, Springer Verlag Berlin, 4. Auflage c, whereas the vacuum impedance remains constant [6]. For (1996) 73 example, the electric potential of a single point charge is 5. Ellis G. F. R. ”Note on Varing Speed of Light Cosmologies”, arXiv: astro-ph/0703751v1 (Feb 05, 2008) − Gm e c2r 0 6. Puthoff H. E. ”Polarizable-Vacuum (PV) Approach to General Relativ- ϕ = e 0 ~r ity”, Foundations of Physics 32/6 (Jun 2002) 927-943 4π0r
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