Arbab Fluid

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On The Analogy Between Electrodynamics And Hydrodynamics Using Quaternions A . I. Arbab1,2 † 1

Department of Physics, Faculty of Science, University of Khartoum, P.O. Box 321, Khartoum 11115, Sudan 2 Department of Physics and Applied Mathematics, Faculty of Applied Sciences and Computer, Omdurman Ahlia University, P.O. Box 786, Omdurman, Sudan (Received 20 December 2008)

We have derived energy conservation equations from the quaternionic Newton’s law that is compatible with Lorentz transformation. This Newton’s law yields directly the Euler equation and other relations governing the fluid motion. With this formalism, the pressure contribute positively to the dynamics of the system in the same way mass does. Hydrodynamics equations are derived from Maxwell equations by adopting an electromagnetohydrodynamics analogy. In this analogy the hydroelectric field is related to the local acceleration of the fluid and the Lorentz gauge is related to the incompressible fluid condition. An analogous Lorentz gauge in hydrodynamics is proposed. We have shown that the vorticity of the fluid is developed whenever the particle local acceleration of the fluid deviates from the velocity direction. We have shown that Lorentz force in electromagnetism corresponds to Euler force for fluids. Moreover, we have obtained a Faradaylike law in Hydrodynamics. The current and density in Schrodinger, Klein-Gordon and Dirac equations satisfy the generalized continuity equations.

1. Introduction We have recently formulated Maxwell equations using quaternion (Arbab and Satti, 2009 & Arbab, 2009). In this formalism, we have shown that Maxwell equations did not include the magnetic field produced by the charged particle. Consequently, we have shown that this magnetic field is given by an equation equivalent to Biot-Savart law. Moreover, we have shown that the magnetic field created by the charged particle is always perpendicular to the particle direction of motion. We have also generalize the ordinary continuity equation using quaternion. Consequently, we have found that Maxwell equations predict that the electromagnetic fields propagate in vacuum or a charged medium with the same speed of light in vacuum. In the present paper, using the quaternionic Newton’s second law in conjunction with the generalized continuity equations, we derive in a direct way the Euler equation and the relativistic energy conservation equation. We have shown that the quaternion Newton’s equation gives directly the non-relativistic limit of the energy momentum conservation equation. These equations are the Euler equation, the continuity equation, the energy momentum conservation equation. We have found an analogy between hydrodynamics and electrodynamics. This analogy guarantees that one can derive the equation of the former from the latter or vice versa. This analogy is quite impressing since it allows us to visualize the flow of the electromagnetic field that resembles the flow of a fluid. A Gauss-like equation in hydrodynamics is derived. The Schrodinger, Dirac † Email: [email protected]

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A. I. Arbab

and Klein-Gordon equations are in agrement with our generalized continuity equations (GCEs). This shows that these GCEs must be satisfied for any fluid flow. We have shown here that the generalized Newton’s second law yields directly the non-relativistic equations governing the motion of fluids. Moreover, we have shown earlier that the generalized continuity equations are Lorentz invariant. The fluid vorticity arises whenever the local acceleration of the fluid particles deviates from the velocity direction. The physical properties and the equations governing the hydrodynamics are derived from Newton’s second law. From this law Euler equation is then derived, together with the equation of motion of the vorticity of the fluid. We remark also that a Faraday-like law in hydrodynamics is obtained. Moreover, we have found that the diffusion equation is compatible with the GCEs. We therefore, emphasize that the GCEs should append any model dealing with fluid motion. According to this analogy and since the electrodynamics is written in terms ~ e and B, ~ w should write the hydrodynamics equations in terms of the hydroelecof the E ~ h and the vorticity ω tric field E ~ in the same way Maxwell equations are written. This is ~e ⇔ E ~h legitimate because of the analogy that exists between the two paradigms, viz., E v2 ~ ~ ⇔ ϕ. and ω ~ ⇔ B; v ⇔ A and 2

2. Continuity Equation We have recently derived the generalized continuity equation that governing the flow of any fluid (Arbab, 2009) using quaternions. These are defined as follows "  ! #  ~ ∂ρ ∂ J i 2 ~ · J~ + e Je = − ∇ ~ c +∇ ~ × J~ = 0, ∇ + ∇ρ (2.1) + ∂t c ∂t where  e = ∇

i ∂ ~ ,∇ c ∂t

 ,

  Je = iρc , J~ .

(2.2)

Equation (2.1) implies that ~ · J~ + ∂ρ = 0 , ∇ ∂t

(2.3)

~ ~ + 1 ∂J = 0 , ∇ρ 2 c ∂t

(2.4)

~ × J~ = 0 . ∇

(2.5)

and We call Eqs.(2.3)-(2.5) the generalized continuity equations(GCEs). In a covariant form, Eqs.(2.3)-(2.5) read ∂µ J µ = 0 ,

∂µ Jν − ∂ν Jµ = 0 .

(2.6)

Hence the GCEs are Lorentz invariant. We remark that the GCEs are applicable to any flow whether created by charged particles or neutral ones.

3. Newton’s second law of motion The motion of the mass (m) is governed by the Newton’s second law. The quaternionic force reads (Arbab and Satti, 2009) e Ve ) , Fe = −mVe (∇

(3.1)

On The Analogy Between Electrodynamics And Hydrodynamics Using Quaternions 3 where 

 i P , F~ , Ve = (ic , ~v ) . c The vector part of Eq.(3.1) yields the two equations      ∂~v ~ v 2 ~ × ~v +∇ − ~v × ∇ , F~ = m ∂t 2 Fe =

(3.2)

(3.3)

and ~ × ~v = − ~v × ∂~v . ∇ c2 ∂t The scalar part of Eq.(3.1) yields the two equations   ~v ∂~v 2 ~ P = mc ∇ · ~v + 2 · , c ∂t

(3.4)

(3.5)

and   ~ × ~v = 0 . ~v · ∇

(3.6)

For a continuous medium (fluid) containing a volume V , one can write Eq.(3.3) as   ∂~v ~ v 2 ~ + ∇( ) − ~v × (∇ × ~v ) = f~ , (3.7) ρ ∂t 2 where m = ρ V and f~ = Eq.(3.7) becomes

~ F V

~ v · ~v ) = ~v × (∇ ~ × ~v ) + (~v · ∇) ~ ~v , . Using the vector identity 21 ∇(~ 

 ∂~v ~ ρ + (~v · ∇) ~v = f~, . (3.8) ∂t This is the familiar Euler equation describing the motion of a fluid. For a fluid moving under pressure (Pr ) one can write the pressure force density as ~ r, f~P = −∇P

(3.9)

so that Eq.(3.8) becomes  ρ

∂~v ~ ~v + (~v · ∇) ∂t



~ r . = −∇P

(3.10)

Using Eq.(2.4), Eq.(3.5) can be written as ∂u ~ ~ + ∇ · S = f~ · ~v , ∂t

~ = (ρc2 ) ~v , S

u = ρ v2 ,

P = f~ · ~v .

(3.11)

~ is the energy flux and u is the Eq.(3.10) is an energy conservation equation, where S energy density of the moving fluid. With pressure term only, Eq.(3.10) yields ∂u ~ ~ · ~v . + ∇ · (ρ c2 + Pr ) ~v = Pr ∇ (3.12) ∂t The source term on the right hand side in the above equation is related to the work needed to expand the fluid. It is shown by Lima et al. (1997) that such a term has to be added to the usual equation of fluid dynamics to account for the work related to the local expansion of the fluid. It is thus remarkable we derive the fundamental hydrodynamics equations from just two simple quaternionic equations, the continuity and Newton’s equation. For ~ · ~v = 0 so that Eq.(3.11) becomes compressible fluids ∇ ∂u ~ + ∇ · (ρ c2 + Pr ) ~v = 0 , ∂t

(3.13)

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A. I. Arbab

which states that the pressure contributes positively to the energy flow. This means the total energy flow of the moving fluid is ~total = (ρ c2 + Pr ) ~v , S

(3.14)

and the total momentum density of the flow is given by ~p = (ρ +

Pr ) ~v , c2

(3.15)

which must be conserved. This is analogous to the general theory of relativity where the pressure and mass are sources of gravitation. Equation (3.13) states also that there is no loss or gain of energy. However, when viscous terms considered loss of energy into friction will arise. In standard cosmology the general trend of introducing the bulk viscosity (η)† is by replacing the pressure term Pr by the effective pressure (Weinberg, 1972) ~ · ~v . Peff. = Pr − η ∇

(3.16)

~ ×~v , we get Substituting this in Eq.(3.10) and defining the vorticity of the fluid by ω ~ =∇  ρ

∂~v ~ ~v + (~v · ∇) ∂t



~ r +η∇2~v +η ∇×~ ~ ω, = −∇P

(16a)

which reduces to the Navier-Stokes equation for irrotational flow (~ ω = 0). Equation (3.12) can be put in a covariant form as ∂µ T µν = 0 ,

Tµν = (ρ +

Pr ) vµ vν − Pr gµν . c2

(3.17)

where Tµν is the energy momentum tensor of a perfect fluid, vµ is its velocity and gµν is the metric tensor with signature (+ - - -). It is interesting to remark that we pass from quaternion Newton’s law to relativity without any offsetting. This is unlike the case of ordinary Newton’s law where relativistic effects can’t be included directly. Notice that for non-relativistic speed one has ~p → T 0i , i = 1, 2, 3 and T00 → ρc2 , as given by Eqs.(3.11) and (3.15). ~ × (f A) ~ = f (∇ ~ × A) ~ −A ~ × (∇f ~ ), with J~ = ρ ~v , Eq.(2.5) Using the vector identity, ∇ can be written as   ~ × J~ = ∇ ~ × (ρ ~v ) = ρ ∇ ~ × ~v − ~v × (∇ρ) ~ = 0, ∇ (3.18) which upon using Eq.(2.4) transforms into ~ × ~v = − ~v × ∂~v . ∇ c2 ∂t

(3.19)

Thus, Eq.(3.4) derived from Newton’s second law is equivalent to one of the continuity equations, viz., Eq(2.5). Taking the dot product of Eq.(2.4) with a constant velocity ~v , we get ~ ~ + 1 ~v · ∂ J = 0 , ~v · ∇ρ 2 c ∂t which yields dρ ∂ v2 = (1 − 2 ) ρ . dt ∂t c ~ = −A η † The viscous pressure can be seen from the viscous force, F

d~ v dr

⇒ Pv =

F A

~ ·~v = −η ∇

On The Analogy Between Electrodynamics And Hydrodynamics Using Quaternions 5 According to Lorentz transformation, if the density in the rest frame is ρ, it will be 2 ρ0 = (1− vc2 ) ρ in the moving inertial frame. Thus, taking the total derivative, is equivalent to taking the partial derivative of the density in a moving inertial frame (Lawden, 1968). In terms of the vorticity, the above equation becomes   ∂~v ~v ω ~ = 2× − . (3.20) c ∂t ~ by the relation ω ~ The vorticity is related to the angular velocity of the fluid (Ω) ~ = 2 Ω. Eq.(3.20) reveals that the vorticity is developed whenever the local acceleration of the fluid particle doesn’t lie along the particle motion. Hence, for irrotational fluid the local particle acceleration lies along the particle motion. This clearly shows that the fluid motion is governed by the continuity equation as well as the Newton’s equation. Moreover, notice that Eq.(3.6) is consistent with Eq.(3.4). In a recent paper, we have shown that the magnetic field produced by a moving charged particle due to an external electric field is given by (Arbab and Satti, 2009) ~. ~ = ~v × E (3.21) B c2 We remark here that there seems to be a resemblance between the vorticity of flow and the magnetic field produced by the charged particle. This analogy is evident from ~ = ∇ ~ ×A ~ and ω ~ × ~v . Moreover, Eq.(3.6) shows that the fluid the fact that B ~ = ∇ ~ =0 helicity, hf = ~v · ω ~ = 0, and Eq.(3.20) shows that the magnetic helicity, hm = ~v · B ~ (Kikuchi, 2007)†. Using vector identities, it is obvious from Eq.(3.4) that ∇ · ω ~ = 0. This ~ ·B ~ = 0. The former equation implies that vortex equations resembles the equation ∇ lines must form closed loops or be terminated at a boundary, and that the strength of a vortex line remains constant. Thus, a charged particle creates a magnetic field associated with the particle in the same way as the fluid creates a vortex that moves with the particle. According to De Broglie hypothesis, a wave nature is associated with all moving microparticles. It, thus, seems that vorticity and magnetic fields travel like a wave. With the same token, Eqs.(3.7) and (3.20) suggest that the hydroelectric field is given by ~ h = − ∂~v . E ∂t

(3.22)

This electric field is generated due to the fluid (mass) motion. Moreover, the Coulomb ~ ·A ~ = 0) in electrodynamics is equivalent to the incompressibility of the fluid gauge (∇ ~ v = 0). The Lorentz gauge ∇· ~ A+ ~ 12 ∂ϕ = 0 suggests a similar gauge for hydrodynamics (∇·~ c ∂t as ~ · ~v + 1 ∂χ = 0. ∇ (3.23) c2 ∂t However, from our above definition of hydroelectric field, we must impose the condition ~ = 0, i.e., χ is spatially independent but can depend on time, i.e., χ = χ(t). The that ∇χ unit of χ is J/kg, or m2 /s2 . This defines the energy required to move one kilogram of ~ h = − ∂~v − ∇ ~ v2 (where fluid. If we had defined the hydroelectric field in Eq.(3.22) as E ∂t 2 2 χ = v2 ), the Euler force in Eq.(3.3) would become   ~ h + ~v × ω F~h = −m E ~ ~ h . This is zero too, since χ = v 2 /2 † One may also define the hydroelectric helicity ha = ~v · E is time independent.

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A. I. Arbab

which is equivalent to Lorentz force,   ~ e + ~v × B ~ . F~em = q E ~ and This is a very interesting analogy, since for electron q < 0 so that −m < 0, ω ~ ⇔B ~ ~ Ee ⇔ Eh . The complete analogy is tabulated below. We would like to call this symmetry an electromagnetohydrodynamics (EMH) analogy. The following table shows the analogy between electromagnetics and hydrodynamics. As the electric field of an electron points ~ h ) points opposite to the direcopposite to the force direction, the hydroelectric field (E tion of flow motion.

Theory

Circulation Gauge fields Gauge condition

~ ~ =∇ ~ ×A ~ ϕ, A Electrodynamics B ~ × ~v χ, ~v Hydrodynamics ω ~ =∇

Helicity

Electric field

~ E ~ ~ ·A ~ + 12 ∂ϕ = 0 he = ~v · B ~ e = − ∂ A~ − ∇ϕ ∇ c ∂t ∂t ∂χ 1 ∂~ v ~ · ~v + 2 ~ h = − − ∇χ ~ ∇ v·ω ~ E c ∂t = 0 hf = ~ ∂t

It is an amazing analogy. Employing this analogy, we would like to derive the hydrodynamics laws from the electrodynamics corresponding ones. The Maxwell’s equations are ~ ~ ×B ~ = µ0 J~ + 1 ∂ E , ∇ (3.24) c2 ∂t ~ ~ ×E ~ + ∂ B = 0, ∇ ∂t ~ ·E ~ = ρe , ∇ ε0

(3.25) (3.26)

and ~ ·B ~ = 0, ∇

(3.27)

where ρe is the charge density. Taking the curl of both sides of Eq.(3.24) and employing Eq.(2.1) yields ~ ·E ~ h = ρm , ∇ (3.28) εh 2

where µh = a10 replaces µ0 for the EMH, where a0 = cγ2 κ , γ is the surface tension and κ is the bulk modulus. Notice that the unit of µ0 is H m−1 , ε0 is F m−1 and a0 is kg m−1 . The constant a0 may thus define the resistance (inertia) of the fluid to ~ e → E~2h and flow. Equation (3.28) is nothing but Gauss law in hydrodynamics, where E c the mass density (ρm ) replaces the charge density (ρe ). Besides, Eq.(3.28) implies the hydrodynamic permittivity, εh = ac20 . Hence, µh εh c2 = 1 (in comparison with ε0 µ0 c2 = 1). Notice however that one can relate this constant to Newton’s constant by the relation c2 a0 = 4πG . In this case Eq.(3.28) becomes ~ ·E ~ h = 4πGρm , ∇

(3.29)

1 which also suggests that G = 4πε , which is to be compared with Coulomb constant h 1 K = 4πε0 . Therefore, the constant a0 is a new fundamental constant. One can therefore −27 define the gravitational permeability as µh = 4πG m kg−1 so that εh = c2 = 9.31 × 10

On The Analogy Between Electrodynamics And Hydrodynamics Using Quaternions 7 ~ f = a0 ω 1.19 × 109 kg m−3 s2 . The vorticity field intensity of a moving fluid H ~ . Thus for a moving gravitational fluid one has Hf = 1026 ω kg m−1 s−1 . This means that such a vorticity field intensity is very huge for any a non-zero value of ω. The time dependence of the hydroscalar χ can be obtained from Eq.(3.28) using Eqs.(3.22) and (3.23) so that c2 1 d2 χ = ρm . (3.30) c2 dt2 a0 This equation will have direct consequences for cosmological applications where ρm varies with cosmic time. The velocity vector field (~v ) can then be obtained by solving Eq.(3.30) and applying the solution in Eq.(3.23). The solution of Eq.(3.30) depends on how ρm varies with time. Using the analogy shown in the above table and cross multiplying by ~v from right of Eq.(3.24), one gets   1 ∂ 2~v ~ ~ × ~v , (3.31) (∇ × ω ~ ) × ~v = µ0 (J × ~v ) − 2 c ∂t2 By vector identity, the first term on the right hand side of the above equation vanishes, since J~ = ρ ~v . Hence, 1 ∂ 2~v ~ ×ω (∇ ~ ) × ~v = 2 2 × ~v , (3.32) c ∂t Differentiating Eq.(3.19) partially with respect to time, one gets  2  ∂~ ω ~v ∂ ~v = 2× − 2 , (3.33) ∂t c ∂t so that Eq.(3.31) yields ∂~ ω ~ ×ω = ~v × (∇ ~ ), (3.34) ∂t which is the familiar vorticity equation in hydrodynamics. We remak here that this equations can’t be easily derived directly from the fluid equations alone, but resorting to Maxwell’s equations, viz., Eq.(3.24) makes its derivation easier. According to Maxwell’s equations the Poynting vector is given by ~ ~em = E × B , S µ0 so that in hydrodynamics using Eq.(3.22), it will become ~h = a0 E ~h × ω S ~, which yields ~h = (ρm c2 )~v . S This is the same as Eq.(3.14) with Pr = 0. The vanishing of the pressure is due to the ~ = 0. It is interesting to note that taking the curl of Eq.(3.22) and using the fact that ∇χ ~ × ~v , one gets fact that ω ~ =∇ ω ~ ×E ~ h + ∂~ ∇ = 0. (3.35) ∂t ~h ⇔ E ~ e and ω ~ This is the Faraday analogue of hydrodynamics, where E ~ ⇔ B. The diffusion equation is given by ~ , J~ = −D∇ρ

(3.36)

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where D is the diffusion constant. Applying Eq.(3.36) in Eq.(2.3), one finds ∂ρ = D∇2 ρ . ∂t

(3.37)

This shows that the density ρ satisfy the Schrodinger-like wave equation. The normalized solution of the Eq.(3.37) is ρ(x, t) = √

1 x2 exp (− ), 4Dt πD t

(3.38)

~ It is obvious that the which when applied in Eq.(3.36) yields the current density J. current in Eq.(3.36) satisfies the GCEs, viz., Eq.(2.5). Differentiation Eq.(3.36) partially ~ ∇ ~ · J) ~ = with respect to time and employing Eq.(2.3), (2.5) and the vector identity, ∇( 2~ ~ ~ ~ ∇ J + ∇(∇ × J) , one gets ∂ J~ = D∇2 J~ , (3.39) ∂t which shows that both ρ and J~ satisfy the same Schrodinger-like wave equation. Using Eq.(2.4) one finds that the current ρ and J~ are not unique. The new current (J~ 0 ) and density (ρ0 ) satisfy Eqs.(3.36), (3.37) and (3.39), provided that D ∂ J~ , J~ 0 = J~ + 2 c ∂t

ρ0 = ρ +

D ∂ρ . c2 ∂t

This implies that the current J~ and the density ρ are self-gauged. Now apply the above EMH analogy in Eq.(3.25), one gets   ∂ ~ ∂~ ω ω ~ × − ∂~v + ∂~ ∇ =0 ⇒ − (∇ × ~v ) + = 0. ∂t ∂t ∂t ∂t

(3.40)

(3.41)

~ × ~v . Equation (3.41) is satisfied since ω ~ =∇ Application of the EMH analogy to Eq.(3.27) yields ~ ·ω ∇ ~ = 0,

(3.42)

~ · (∇ ~ × ~v ) = 0. By the EMH analogy, Eq.(3.4) which is true using the vector identity ∇ yields the the familiar magnetohydrodynamics (MHD) equation (Davidson, 2001) ~ ∂B ~ × B). ~ = ~v × (∇ ∂t

(3.43)

This equation can also be obtained by differentiating Eq.(3.21) partially with respect to time and using Eq.(3.24). Hence, it is evident that there is a genuine one-to-one correspondence between electrodynamics and hydrodynamics. Therefore, the electrodynamics and hydrodynamics are intimately related. This would immediately imply that the electromagnetic fields propagates in the same way as a fluid moves. The electric field resembles the local acceleration and the magnetic fields resembles the vorticity of the moving fluid. Consequently, we could then say that the electromagnetic field can be visualized.

4. Dirac Equation We would like here to show that the generalized continuity equations, viz., Eq.(2.6) are compatible with Dirac equation. To this aim, we apply the current density 4-vector

On The Analogy Between Electrodynamics And Hydrodynamics Using Quaternions 9 according to Dirac formalism, in Eq.(2.6), i.e., Jµ = ψγµ ψ (Bjorken and Drell, 1964). This yields ∂ µ J ν − ∂ ν J µ = (∂ µ ψ)γ ν ψ + ψγ ν ∂ µ ψ − (∂ ν ψ)γ µ ψ − ψ(γ µ ∂ ν ψ) = 0.

(4.1)

The first term in the above equation can be written as (∂ µ ψ)γ ν ψ = (∂ µ ψ + )γ 0 γ ν ψ = (∂ µ ψ + )γ ν+ γ 0 ψ = (γ ν ∂ µ ψ)+ γ 0 ψ.

(4.2)

(∂ µ ψ)γ ν ψ = (ψγ ν ∂ µ ψ)+ .

(4.3)

(∂ ν ψ)γ µ ψ = (ψγ µ ∂ ν ψ)+ .

(4.4)

Hence, Similarly, Substituting these terms in Eq.(4.1) completes the proof, since for a plane wave ∂ µ ψ = ik µ ψ.

5. Klein-Gordon Equation In this case, the current density 4-vector is given by Jµ =

i~ (φ∗ ∂µ φ − (∂µ φ∗ )φ) , 2m

(5.1)

so that the generalized continuity equations (GCEs) becomes i~ [∂µ (φ∗ ∂ν φ − (∂ν φ∗ )φ) − ∂ν (φ∗ ∂µ φ − (∂µ φ∗ )φ)] = 0, 2m and hence, Klein-Gordon equations is compatible with our GCEs too. ∂µ Jν − ∂ν Jµ =

(5.2)

6. Schrodinger Equation The current density and probability density in Schrodinger formalism are given by ρ = ψ ∗ ψ,

~ J~ = (ψ ∗ ∇ψ − (∇ψ ∗ )ψ). 2mi

(6.1)

Applying Eq.(2.5), one gets i ~ h ~ ~ × (∇ψ ∗ )ψ) . (∇ × (ψ ∗ ∇ψ) − ∇ (6.2) 2 mi ~ × (f A) ~ = f (∇ ~ × A) ~ −A ~ × (∇f ~ ) and ∇ ~ × (∇f ~ ) = 0, Using the two vector identities ∇ Eq.(6.1) vanishes and hence one of the GCEs is satisfied. We now apply Eq.(2.4) in Eq.(6.1) to get ~ × J~ = ∇

∂ J~ 2 ~ ∗ ψ) + ~ ∂ [ ψ ∗ ∇ψ − (∇ψ ∗ ) ψ ] = 0. ~ = ∇(ψ ∇(ρc )+ ∂t 2 mi ∂t Upon using the Schrodinger equation, Hψ = i~

∂ψ , ∂t

ψ ∗ H = −i~

∂ψ ∗ ∂t

(6.3)

(6.4)

Eq.(6.3) vanishes for a plane wave equation, i.e., ψ (r, t) = A exp i(~k ·~r − ω t) , A = const.. Therefore, the GCEs are satisfied by Dirac, Klein-Gordon and Schrodinger equations. This implies that these GCEs are fundamental in formulating any field theoretic models

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A. I. Arbab

involving the motion of particles or fluids. These equations will have immense consequences when applied to the theory of electrodynamics we used to know. Such an application will lead to better formulations of the astrophysical laws governing the evolution of dense objects.

7. Concluding Remarks We have studied in this paper the consequences of the quaternionic Newton’s second law of motion. We have derived the energy conservation equation and Euler equation. These laws are compatible with the generalized continuity equations. The basic equations governing the fluid dynamics are also derived. We have then shown that the energy momentum equation is lorentz invariant. Ordinary Newton’s second law is known not to be compatible with Lorentz transformation. However, the generalized Newton’s second laws is compatible with special relativity. Moreover, we have shown that the pressure contribute equally to the energy density of the moving fluid as the mass does. Besides, the pressure is the source of the hydroelectric scalar field. Moreover, we have found an intimate analogy between electrodynamics and hydrodynamics. With this analogy, we derived the magnetohydrodynamics equations from Maxwell’s equations. Besides, we have obtained from Ampere’s law, a Gauss-like law applicable to gravitational system. Moreover, a Faraday-like law in hydrodynamics is obtained. In this analogy Euler force is corresponds to Lorentz force. The diffusion equation is in agreement with the GCEs. Moreover, the current and density in Dirac, Klein-Gordon and Schrodinger formalisms are compatible with our GCEs too.

Acknowledgments I would like to thank F. Amin for stimulating discussion.

REFERENCES Arbab, A. I., and Satti, Z. A., Progress in Physics, 2, 8 (2009). Arbab, A. I., to appear in Progress in Physics, 2, (2009). Bjorken, J.D. and Drell, S.D., Relativistic Quantum Mechanics, McGraw-Hill Book Company, (1964). Lima, J.A.S., Zanchin, V, and Brandenberger, R., Mon. Not. R. Astron. Soc., 291, L1, (1997). Kikuchi, H., Progress In Electromagnetics Research Symposium, Beijing, China, March 26-30, 996, (2007). Davidson, P.A., An Introduction to Magnetohydrodynamics, Cambridge University Press, (2001). Lawden, D.F., Tensor Calculus and Relativity, Methuen, London, (1968). Weinberg, S., Introduction to Gravitation and Cosmology, John Wiley and Sons, (1972).