Just as a spacetime 1-vector "point" p decomposes into "time" p⁴ and "place" P =p¹e₁+p²e₂+p³e₃ when percieved by observer E, a timelike 1-vector four-momentum m=m¹e₁+m²e₂+m³e₃+m⁴e₄ decomposes into "energy" m⁴ and "momentum" M=m¹e₁+m²e₂+m³e₃. We can measure energy and mass in m^-1 and in the geometric algebra we can regard nonnull four-moentum 1-vectors as inverted nonnull spacetime displacements.
    With regard to a particular e₄ it is natural to express timelike 1-vector m with m²=-m₀² as
    m = m₀(1-V²)^-½(e₄+V) = m_E(e₄ + V) where:

• Scalar m₀ = |m| = |m²|^½ is known the rest mass ( aka. proper mass or proper energy ) of m . We will here use the term menergy for the signed scalar squared restmass m² with (M+Ec^-1e₄)²=m₀² providing shell condition E² = M²c² + m₀²c⁴ .  
• Spacial nonunit 1-vector V = ^_e4(m) |¯_e4(m)|^-1 = (m¹e₁+m²e₂+m³e₃)m^4-1 is known as the relative velocity of m.
• Spacial nonunit 1-vector m_EV = ^_e4(m) is known as the relative momentum of m.
• Positive scalar m_E   =   m₀g_V   =   m₀(1-V²)^-½   =   m₀ + ½m₀V² + (3/8) m₀V⁴ + (5/8)m₀V⁶ + .... + 2^-2k (2k)! (k!)^-2 m₀V^2k + .. is known as the relative mass or relative energy of m. When V=0 then m_E=m₀ but as |V| increases towards 1 than, assuming m₀ is unchanging, m_E increases towards +¥.

    Relative mass-energy m_E is comprised of rest mass m₀ plus an infinite series of positive kinetic energy terms in  m₀V^2k which for |V| << 1 are approximated by the first classical kinetic energy term ½m₀V² .

    A null 1-vector has zero proper mass but nonzero relative mass equal to the magnitude of its relative momentum. It is thus percieved as a relative mass m⁴ having unit relative velocity and consequent relative momentum m⁴V = ¯_e4^*(m) .  

    Let path P = { p(t) : t Î Â } be a proper worldline for a "particle" of constant proper mass m₀ > 0 .
    Define x_E : [t₀,t₁]   ® Â^3,1 by x_E(t) º ^(p(t),e₄).
    Define t_E : [t₀,t₁] ® Â by t_E(t) º -e₄¿p(t) so that ¯(p(t),e₄) = t_E(t)e₄ .
    t_E(t) is a monontonic increasing function of t so has an inverse function t_E^-1 : [t_E(t₀),t_E(t₁)] ® [t₀,t₁] such that t_E^-1(t_E(t)) = t " t Î [t₀,t₁].
    The worldine of the particle can then be parameterised as P = { x_E(t_E^-1(t)) + te₄ : t Î [t_E(t₀),t_E(t₁)] } representing its perception by E.

    g_E(t) º dt_E(t) / dt   = (1 - v_E(t)²)^-½ .
[ Proof :

(dp(t)/dt)2 = -1	Þ	(dx(tE(t))/dt)2 - gE(t)2 = -1	Þ	(dx(tE(t)/dt)2 = gE(t)2 - 1
	Þ	((dx(tE(t))/dtE(t) gE(t))2 = gE(t)2 - 1	Þ	gE(t)2 = (1 - (dx(tE(t))/dtE(t))2)-1 .
	Þ	gE(t) = ± (1 - vE(t)2)-½		where vE(t) = dxE(t)/dtE(t) is the relative velocity.

We take the + Ö  .]

    The e₄-relative (forced Euclidean) modulus vv^† being the sum of "kinetic energy" (v¹)² +(v²)² +(v³)² and "mass energy" (v⁴)² . is known as the Hamiltonian .
    v²   = (v¹)² + (v²)² + (v³)² - (v⁴)²   provides a measure refered to by physiscists as a Lagrangian .
    Some authors represent this split as e₄m(t) = e₄¿m(t) + e₄Ùm(t) and consider the relative momentum to be a bivector but we do not take this approach here.
    E_E(t) = -m_Ee₄²     (here = m_E, elsewhere = m_Ec²).
    m_E(t) = m_Ev_E(t) .     [ Proof :  m₀g_E(t)v_E(t) = m₀ dt_E(t)/dt dx_E(t)/dt_E(t) = m₀ dx_E(t)/dt  .]
    F_E(t) º ^_e4(p") is known as the relative force assumed acting on the particle.

    We can immediately derive two standard laws of Newtonian mechanics.

• F_E(t) = dm_E(t) /dt_E     ("force = rate of change of momentum")
  [ Proof :  (dm_E(t) /dt) (dt /dt_E) = m₀ ^_e4(d² p(t)/dt² ) (g_E(t)^-1)  .]
• dE_E(t)/dt_E = F_E(t)¿v_E(t) = m₀ d² t_E/dt² (g_E(t))^-1     ("power = work")
  [ Proof : (dp(t)/dt)² = - 1 Þ (dx_E(t)/dt)² - (dt_E(t)/dt)² = - 1 Þ (dx_E(t)/dt)¿(d²x_E(t)/dt²) = (dt_E(t)/dt)(d²t_E(t)/dt²) = g_E(t)(d²t_E(t)/dt²)
      dE_E(t)/dt_E = m₀ dg_E(t)/dt_E = m₀ dg_E(t)/dt (dt/dt_E) = m₀ d² t_E/dt² (g_E(t))^-1 = m₀(dx_E(t)/dt)¿(d²x_E(t)/dt²)  .]

    Since g_E(t) = (1 - v_E(t)²)^-½ = 1 + ½v_E² + o(v_E⁴) we have E_E(t) » m₀(1 + ½v_E(t)²) for small |v_E| , corresponding to   rest and kinetic energies.
    Note that though we use here the subscript _E to denote relativity to an "observer" E, only e₄ is actually relevant and coordinate independance within E's 3D spatial universe is retained.

Justifying E=mc²
    Einsteins most famous equation has emerged (as E=m) almost tautologically here from our identification of "rest mass-energy" with the magnitude of a four-momentum, so it is reassuring to follow Pearson and demonstrate it as independantly plausible other than implicitly from a Minkowski paradigm. In atomic decay we observe that on fragmenting into rapidly seperating elements, the total (inertial) mass of the assemblage is reduced while the total kinetic energy apparently increases. It is reasonable therefore to consider that some of the (inertial) mass of the particles has been transformed into kinetic energy. If we allow two seperate atoms to decay in a similar way the composite assemblage typically gains twice as much kinetic energy and looses twice as much mass as the single atom system so it is again plausable to assume a proportional relationship between matter and energy E = k m for some constant k.
    Consider accelerating a body of mass m from rest using a continuously applied 1-D "driver" force F we have
    F = (d/dt)(mv) = (d/dt)(kEv) = k(Edv/dt + vdE/dt)   =    k v dt/dx (Edv/dt + vdE/dt)   =    k v (Edv/dx + vdE/dx)
     Now, F dx is the ammount of work done and so should equal the gain in (kinetic) energy dE so we have
    dE   =   k v (Edv + vdE) Þ dE   =   k v E (1-kv²)^-1 and so ò_E0^EE^-1 dE   =   ò₀^v dv(1-k²)^-1 Þ ln(EE₀^-1) = -½ ln(1-kv²) Þ E = E₀(1-kv²)^-½ and we see that energy E increases to infinity as v approaches upper limit k^-½ . Denoting this infinite energy requiring speed by c gives E = E₀(1-(vc^-1)²)^-½ and F = (d/dt)mv = c^-2 Ev which integrates to mv = Evc^-2 Þ E = mc² .

    Alternatively we might follow Dmitriyev and consider a particle of mass M as a bubble of volume V_v of vapour in a turbulent fluid "either" of density r_l and pressure P_l. The velocity of the particles in the volume V_l of fluid evapourating into the bubble has then form v= u₀ + u where  u₀ is the average drift over a short time interval and u is thermal jitter with <(u¹)²> = <(u²)²> = <(u³)²> = c² where <a> denotes averaging a over a small timeperiod, and so the net kinetic thermal energy inherited by the vapour is
    ½r_lV_l( <(u¹)²> + <(u²)²> + <(u³)²> ) = (3/2)r_lV_lc² = (3/2)Mc² where M=M_v=M_l is the mass of the vapourised fluid in the bubble.
    If the vapour is acting within the bubble as an ideal gas than the thermal energy of the vapour in the bubble is given by (3/2)P_vV_v so we have Mc² = P_vV_v and since P_v=P_l for equilibrium we have Mc² = P_lV_v = E, the work done needed to create the bubble against the fluid pressure P_l.

DeBroglie Waves

    Electrodynamic wave theory considers solutions of the Klein Gordan equation Ñ_p²y_p = ± l²y_p   , notably the normalised periodic solution  
    y_p   =   y₀e^ilp¿k º y₀ (ilp¿k)^↑     for p-independant null or unit timelike timelike 1-vector k and positive scalar l .
    Such waves are usually constucted over a Â_1,3 timespace as scalar-valued y(X,t) e^{-i(wt - K.X))} so that (d/dt² - Ñ_X²)y = (w² -K²)y
    y_p   =   y₀e^ilp¿k satisfies both y_p^»y_p   =   y₀^»y₀ and Ñ_p²y_p = (-l²k²y₀e^ilp¿k = l²y_p for timelike unit k  or -l²y_p for spacelike unit k. Because k²<0 the exponential power series for y_p is "bounded trigonometric" rather than "divergent hyperbolic" .

    l is frequently expressed by physicists as 2pm₀c/h with  m₀= hl(2pc)^-1 called the "mass" of y .
    In natural units c=1, h=(2p), h=1.
    Taking l = (2p)m₀/h = ih^-1m₀ gives the traditional formulation (in our unorthodox notations) of the De Broglie Wave
    y_p   =   y₀(h^-1m₀(p¿k))^↑   =   y₀(h^-1(p¿m))^↑     for positive scalar m₀ and unit timelike k with m = m₀k .

    Relativisitically, an e₄-observer percieves such a De Broglie wave vector h^-1 m=m₀k as splitting into positive scalar temporal wave frequency w=m₀k⁴h^-1 (and corresponding wave period 2p h (m₀k⁴)^-1 ) and spacial wave-vector h^-1 M_E = l^-1¯_e123(k)^~ for wavelength l = h|M_E|^-1 .

    Energy E = m₀k⁴ = hw   = m₀(1+V_E²)^½ = m₀(1 + ½V_E² - 1/8V_E² + ... + (-1)^k4k(k-1)(2k(2k-1)(2k-2))^-1 V_E^2k + ...

    Note the distinction between wave and particle energy "splits" here. Physicists traditionally "split" a particle four-momentum m=m₀k as m_El(e₄+V) but a wave four-momentum as hwe₄ + hl^-1 V^~ for wave frequency w [  Often denoted v ] and wavelength l [  Often denoted l . ]
     The two V are distinct, one being spacial momentum divided by relative mass, the other the unscaled e₁₂₃ projection. The kinetic energy series differ beyond the first, classical, term.  

    For a zero mass "electromagnetic" wave in a vacuum solving Ñ_p²y_p = 0 ( ie. l=0) we have y₀ (iap¿k)^↑     for any nonzero a and nullvector k . The e₄-relative frequency (energy) is then equal to the magnitude of the relative three-momentum hl^-1 .

    Since h » 6.625×10^-34 J s » 2^-110 J s » 2^-166 kg s , a visible wavelength photon of frequnecy 2⁴⁸ Hz has relative energy of order 2^-62 J corresponding to relative mass 2^-118 kg and we can multiply these values by factors ranging from 2²² for gamma rays down to 2^-17 for radio waves and remain within the recognised "electromagnetic spectrum".

    Since y_p is is everywhere locally normalised,  we can interpret De Broglie wave y as a "particle" only if we regard it to be equiprobably distributed everywhere, ie. having a maximally ambiguous position which in a sense means not having a position at all. Wherever it "is", however, the "particle" has a predictable definite (wholly unambiguous) four-momentum m - at least until such time as we might attempt to observe the ambiguous position.

    Fixing t=e⁴¿p we note that y_p is spacially periodic with directed wavelength h |¯_e4^*(m)|^-1 ¯_e4^*(m)^~ = h M_E^-2  ¯_e4^*(m)^~ .

    Ð_d y_p = il(d¿k)y_p   =   i2pm_Eh^-1(d¿k)y_p   =   -h^-1m_E(d¿k)y_p
which is the Schrodinger wave equation with scalar multiplier Hamiltionian operator h_p,d = -m_E(d¿k) .

    With d=e₄ we obtain Hamiltionian H_p = -e₄¿m = m_Ek⁴ = m_E(1 + v_E²)^½ [  k⁴ denotes coordinate e⁴¿k rather than |k|⁴ ] which for small v_E² we can approximate as m_E(1 + ½ v_E² + _O(v_E⁴) + ...) corresponding to "rest" , "kinetic", and "higher order"  energies respectively.
    For "slow" De Brogle waves we thus have the non-relativistic approximation y_p = y₀(h^-1m_E(p¿v_E - (1+½v_E² + _O(v_E⁴))t )^↑ y_p » (-h^-1m_E)^↑ y₀(h^-1m_E(p¿v_E - ½v_E²t)^↑
    We can incorporate the p-independant phase factor e^-h-1m_E into y₀ to obtain y_p » y₀ (h^-1m_E(p¿v_E - ½v_E²t))^↑ .

Relativistic Fluid Mechanics


Classical Flows
    Classical fluid mechanics typically represnts a flow of matter by a Â^N₀ nonunit 1-field M_P representing at a given time t the momentum per unit volume passing through a small test volume at spacial position PÎÂ^N₀ and scalar 0-field m_P representing the density (mass per unit volume) at P (as measured over a small test volume) is constant for an incompressible fluid. We have M_p = m_pV_p where V_p is the (nonunit) spacial velocity vector of the matter at p= P+te₄ is an N=(N₀+1) dimensional event point.
    The flow satisfies the conservation of matter law (aka. a continuity equation) if
    Ñ_P¿M_p = - ¶ m_P/¶ t = 0     which states that the ammount of matter leaving a small region at P per unit time as measured over a small test interval equals the fall in the ammount of matter m_P at P during that interval.
    We can express the matter conservation law as
     ¶ m_P/¶ t + Ñ_P¿(m_pV_p)   =   ¶ m_P/¶ t + (V_p¿Ñ_P)m_pÑ + m_p(Ñ_P¿V_p)   =   ((¶ /¶ t) + (V_p¿Ñ_P))m_p + m_p(Ñ_P¿V_p)   =   0     where (¶ /¶ t) + (V_p¿Ñ_P) is known as the substantial derivative corresponding to the rate of change when "following" the flow. Note that this represents the derivative in the direction of the flow scaled by the speed of the flow.
    If we define Â^N 1-vector v_p=V_p±e₄ where N=N₀+1 and e₄ is a new axis perspendicular to Â^N₀ with e₄²=±1 then the sunstantial direction becomes (v_p¿Ñ_p) with the differentiating scope of the Ñ extending only rightwards rather than encompassing the v_p.

    For a steady flow M_p = M_P with vanishing Ð_e4 derivatives the matter conservation law becomes   =   (V_p¿Ñ_P))m_p + m_p(Ñ_P¿V_p) and for in incompressible flow with m_p=m>0  this yields the incompresability condition Ñ_P¿V_p = 0, ie. the spacial divergence of the N₀-D velocity 1-vector is zero.
    For v_p=V_p±e₄ , the incompresability condition implies Ñ_p¿v_p=0.

    For an incompressible flow "momentum" and "velocity" fields become effectively equivalent with M_p=mV_p.

Spacetime 1-flows
    Consider a nonunit timelike 1-vector valued field
    m_p   =   m_pu_p   =   M_p + m_pe₄   =   m_p(1-V_p²)^-½(V_p+e⁴)   =   m_p(V_p+e⁴)   =   m_pv_p
at p as four-momentum due to matter of rest mass m_p and unit 4D timelike "four-velocity" u_p.
    An e₄-observer percieves m_p as matter of relative mass m_p º e⁴¿m_p = m_p(1-V_p²)^-½     travelling at nonunit spacial velocity 1-vector V_p = ^_e4(u_p) with relative momentum M_p = ^_e4(m_p) = m_pV_p . The matter conservation law becomes Ñ_p¿m_p = 0 .
[ Proof :  Ñ_p¿m_p = Ñ_[e123]p¿m_p + e⁴¿Ð_e4m_p = Ñ_P¿M_p + Ð_e4(e⁴¿m_p) = Ñ_P¿M_p + Ð_e4m_p = Ñ_P¿M_p + ¶m_p/¶t = 0  .]

    The e₄-substantial derivative (v_p¿Ñ_p)   =   Ð^p_vp   =   ¶/¶x⁴ + (V_p¿Ñ_[e123]) corresponds to that rate of change with respect to e₄-time when following the e₄-spacial flow V_p .
    For a static system (¶/¶x⁴ = 0) we have (v_p¿Ñ_p)   =   (V_p¿Ñ_P)   =   Ð^P_Vp .

Current

    We likewise interpret a nonunit timelike 1-vector "current" field j_p   =   j_pu_p   =   r_pu_p   =   J_p + j⁴_pe₄ as a 3D-current J_p and relative e₄-charge j⁴  with charge conservation law
    Ñ_p¿j_p   =   Ñ_[e123]p¿J_p + ¶ j⁴_p/¶ x⁴   =   0 . If j_p falls away more slowly, eg. by a spacial r^-1 factor then e₄-charge may "radiate away". [ e₄-dependant component j⁴_p is often denoted r in the literature but we here retain r for the frame-independant magnitude r_p = |j_p| = j_p. ]
    Ñ_p¿j_p=0 ensures that j⁴ (sometimes refered to as the " probable charge" at p) is "globally conserved" over (e₄-percieved) time in that its integration over a suitably large e₄-cotemporal spacial volume remains constant over time, provided r_p ® 0 faster than |p¿e₁₂₃|^-2 does as |p¿e₁₂₃| ® ¥.
[ Proof : Fix e⁴¿p = t₀ and consider a spacial 2-sphere S_t0 Ì  t₀e₄ + e₄^* of radius R . We postulate that r_p ® 0 at any point on the surface of this sphere faster than R^-2 as R becomes large (we assume the sphere still remains within B_ase) . This means that the Â₃ boundary (spherical surface) integral ò_{d St0} dp² j_p ® 0 as R ® ¥ and the geometric form of fundamental theorem of calculus (applied to Euclidean manifold S_t0) then provides that the scalar part of the volume integral ( ò_St0 dp³ Ñ_p[St0] j_p )_<0> ® 0 as R ® ¥ . Other than at the boundary dS_t0 , whose contribution to the volume integral we can neglect, Ñ_p[St0] = Ñ_p[e4^*] and so the contribution to the volume integral of a small 3-simplex d³p at p is
(d³p Ñ_p[e4^*]j_p) = -(d³p Ñ_p[e4]j_p) = -(e³e₁₂₃  e⁴Ð_e4 j_p) = (e³e₁₂₃₄Ð_e4 j_p)
    Considering just the spacial (e₁₂₃) component we obtain ò_St0 dp³ Ð_e4j⁴_p » 0 Þ (d/dt) ò_St dp³ j⁴_p » 0 where the » indicates that we can get as close to zero as we wish by making R large enough. Hence the "total probable charge" within a large enough sphere is (effectively) constant   .  .]

    So if Ñ_p¿j_p = 0 and |j_p|® 0 sufficiently fast for large |d¿p| then - ò _Wd,a d³p d¿j_p is independant of a for any given unit timelike d and the percieved total relative charge is seen as constant by all inertial observers, though differing observers see differing constants.

    If Ñ_p¿j_p =0 then we have the streamline identity
    (j_p¿Ñ_p)j_pÑ =  j_p¿w_p + Ñ_p(½j_p²)     where w_p = Ñ_pÙj_p and j_p=|j_p| .
[ Proof :  ½Ñ_pj_p² = ½(Ñ_pj_pÑj_p + Ñ_pj_pj_pÑ)_<1> = ½((Ñ_pj_p)j_p + Ñ_p(j_pÑj_p - 2j_pÑÙj_p))_<1> = (Ñ_pj_p).j_p  - Ñ_p¿(j_pÑÙj_p) = w_p.j_p   - Ñ_p¿(j_pÑÙj_p)
    = -j_p¿w_p + (Ñ_p¿j_p)j_pÑ - (Ñ_p¿j_pÑ)j_p) = -j_p¿w_p + (j_p¿Ñ_p)j_pÑ  .]

    For a static 3D flow with Ñ_P¿J_P=0 we have classical Â₃ form (J_P¿Ñ_P)J_P = (Ñ_P×J_P)×J_P + Ñ_P(½J_P²) .

Probability Flows
    We sometimes interpret r_x=|j_x| as the probability of there being a particle of unit-charge or mass at x , the particle having, if present, four-velocity j_x^~  and e₄-relative charge  j⁴|j_x|^-1 .

    For t=x⁴>R , the intersection of 3-tsphere O⁺_-R,0 and the e₄-cotemporal 2-plane W_e4,te4 = { p : e⁴¿p=x⁴=t} is a spacial 2-sphere of centre x⁴e₄ and radius
    |X| = (t²-R²)^½ = ((e⁴¿p)² + p²)^½ = |^_e4(p)| , where p = Rp^~ is any point in the intersection.
    r_x is constant throughout W_e4,te4 Ç O⁺_R,0 ,  since |x|=R=t cosh(c)^-1 ; |X|=R sinh(c) = (t²-R²)^½ ; x⁴=t ; x^~4=tR^-1 ; and c= cosh^-1(tR^-1) all are.

Classical Pressure

    The scalar static pressure s_p at p is independant of the macroscopic "net flow" m_p at p , and stems instead from microscopic "thermal" agitatation of the particles "at" p . The use of the term "static" here is traditional, and relates to "pressure not due to flow" rather than a restriction on the variability of s_p with p.
    Consider a small 3-simplex or 2-sphere V in e₁₂₃ at p in a fluid at rest from within which the fluid has been removed. Random microscopic thermal motion of the fluid will tend to cause molecular impacts on the surface of the simplex which we can regard as imparting locally normal "inward" momentum uniformly across the boundary surface. Integrating this inward component across the surface and dividing by the bondary content leads us to scalar pressure which we can regard as the magnitude of the e₄-force exterted on a small test 2-simplex parallel to an aribitary 2-blade b₂ Î e₁₂₃, divided by the simplex area.
    In an ideal gas s_pV is proportional to nQ_p where V=|V| is the volume of a test 3-simplex V at p containing n molecules of average temperature Q_p. n is proportional to total mass m_pV so s_p is proportionate to m_pQ_p .

    Now consider the body of fluid inside V. The total forces acting on this fluid are the external macroscopic forces ò_V |d³p| f_p where f_p=m_pg_p + h_p is e₄-spacial forces exerted   (typically we might have g_p=-ge₃ for uniform vertical gravity) , and the static pressure integrated over the boundary ò_dV|d²p|(-n_p)s_p   =   -ò_dVd²p e₁₂₃^-1s_p   =   -e₁₂₃^-1 ò_Vd³p Ñ_Ps_p     where n_p = I_2p^-1 e₁₂₃ = I_2p e₁₂₃^-1 is the outward normal at p Î dV  .
    Classicists typically asssume g_p=-Ñ_pG_p and h_p=-Ñ_pH_p derive from scalar 0-potentials.
    The total change in momentum incured by the mass m_p|V| of fluid as it follows the flow lines is ò_V d³p (Ñ_P¿v_p)m_p so we have
    (Ñ_P¿v_p)m_p = m_pg_p + h_p - Ñ_ps_p     which for constant m_p=m simplifies to the momentum conservation law (v_p¿Ñ_P)v_p = g_p + m_p^-1h_p - m_p^-1(Ñ_Ps_MP) aka. Euler's equation when combined with Ñ_P¿V_P=0 .
    For P in ³ we have (V_p¿Ñ_P)V_pÑ   =   ½Ñ_P(V_p²) - V_p×(Ñ_P×V_pÑ) so we can express the momentum conservation law as
    ½Ñ_P(V_p²) - V_p×(Ñ_P×V_pÑ) + m_p^-1Ñ_Ps_p - F_P = 0.
[ Proof : a×(b×c) = (a¿c)b - (a¿b)c gives V_p×(Ñ_P×V_p) = (V_p¿Ñ_P)V_pÑ - (V_p¿V_pÑ)Ñ_P = (V_p¿Ñ_P)V_pÑ - Ñ_P(V_pÑV_p)
= (V_p¿Ñ_P)V_pÑ - ½Ñ_P(V_p²)  .]

Navier-Stokes
    Suppose we combine the static pressure force -s_pn_p with a viscous force u((n_p¿Ñ_p)m_pÑ + Ñ_p(n_p¿m_pÑ)) = um((n_p¿Ñ_p)v_pÑ + Ñ_p(n_p¿v_pÑ)) for an incompressible flow with m_p=m.
    If we regard n_p as varying negligibly with p compared to v_p's variation (eg. when over a flat surface) we can view this as the n_p directed flow derivative plus the gradient of the n_p-directed flow-speed, equally wieghted by scalar u.  
    This leads to the Navier-Stokes equation
    (v_p¿Ñ_p)m_p   =   -m_p(Ñ_pG_p) - (Ñ_ps_p) - (Ñ_pH_p) + (v_p¿Ñ_p)m_pÑv_p + uÑ_p²m_p which for m_p=m becomes
    (v_p¿Ñ_p)v_p   =   -(Ñ_pG_p) - m^-1((Ñ_ps_p) + (Ñ_pH_p)) + uÑ_p²v_p where scalar u is the kinematic viscosity. Water has kinematic viscosity 10^-6 m²s^-1 at 15 ^oC while air is fifteen times that, olive oil a hundred. Treacle has viscosity of roghly 1.2×10⁷ m²s^-1 at 15 ^oC, falling rapidly with temeprature.

Vorticity
    For an incompressible (m_p=m) flow under gravity, Navier-Stokes becomes
(Ñ_p¿v_p)v_p = -Ñ_pG_p  - m^-1((Ñ_ps_p) + uÑ_p²v_p and since (v_p¿Ñ_p)v_p = v_p¿(Ñ_pÙv_p) + ½Ñ_p(v_p²) we have v_p¿w_p = -Ñ_pE_p + uÑ_p²v_p where E_p = ½v_p² + G_p + m^-1s_p where w_p º Ñ_pÙv_p has Ñ_p¿w_p=0. Hence Ñ_pÙ(v_p¿w_p) = uÑ_pÙ(Ñ_p²v_p) = uÑ_p³.v_p .

    But Ñ_pÙ(v_p¿w_p) = (Ñ_p¿(v_pÙw_p))i where (N-2)-vector kinematic vorticity w_p º w_pi^-1 = (Ñ_pÙv_p)^* so we have Ñ_p¿(v_pÙw_p) = 0
[ Proof : Ñ_pÙ(v_p.w_p) = (Ñ_p¿(v_pÙ(w_pi^-1)))i = (Ñ_p¿(v_pÙw_p))i  .]
    For incompressible flow, Ñ_p¿(v_pÙw_p) = (v_p¿Ñ_p)w_pÑ - v_pÑÙ(w_p.Ñ_p)v_pÑ so we have geometric vorticity equation (v_p¿Ñ_p)w_p = (w_p.Ñ_p)Ùv_p + u Ñ_p¿(Ñ_p²v_pi^-1) with the differentiating scope taken rightwards only. For N=3 and u=0 this reduces to (v_p.Ñ_p)w_p = (w_p.Ñ_p)v_p with 1-vector vorticity w_p.
[ Proof :  (v_p¿Ñ_p)w_pÑ + (Ñ¿v_pÑ)w_p - v_pÙ(Ñ_p¿w_pÑ) - v_pÑÙ(Ñ_p¿w_p) reduces to result since Ñ_p¿w_p = Ñ_pÙw_p = Ñ_pÙÑ_pÙv_p vanishes as does Ñ_p¿v_p  . Also u(Ñ_pÙ(Ñ_p²v_p))i^-1 = u(Ñ_p¿((Ñ_p²v_p)i^-1)  .]
     Thus the flow-directed derivative of the vorticity is the vorticity-directed derivative of the flow.
    Classically, we have (v_p¿Ñ_p)w_p = (w_p¿Ñ_p)v_p for 1-vector vorticity w_p = (Ñ_PÙV_P)^* = Ñ_P×V_P which for 2D flow becomes (Ñ_P¿V_P)w_p=0 with scalar vorticity preserved along streamlines.

    Suppose there is a streamline C₁ through p which reaches back to a source p₀ and along which the vorticity and viscosity vanish and m_p is constant.
    When f_p=0, integrating along a streamline gives Bernoulli's equation s_p + ½mv_p² = s₀ , where scalar s₀ is known as the stagnation pressure. Bernoulli's equation applies when m_p is constant and the flow is static. More generally E_p = s_p + mG_p + H_p + ½mv_p² is constant along the streamline if m_p=m and w_p=0 along it.
[ Proof :
    s_p1   =   s_p0 + ò_t0^t₁ dt v_p¿(Ñ_ps_p)   =   s_p0 + ò_t0^t₁ dt  v_p¿ (-m_p(Ñ_pG_p) - (Ñ_pH_p) - (Ñ_p¿v_p)m_pÑv_p - m_p(Ñ_p¿v_p)v_p )
      =   s_p0 + m(G₀-G₁) (H₀-H₁) - m ò_t0^t₁ dt  v_p¿((Ñ_p¿v_p)v_p )   =   s_p0 + m(G₀-G₁) (H₀-H₁) - m ò_t0^t₁ dt  v_p¿(v_p¿w_p + Ñ_pv_p² )
      =   s_p0 + m(G₀-G₁) (H₀-H₁) + ½m (v₀² - v₁²) .
    Hence s_p + mG_p + H_p + ½mv_p² is constant along the streamline.  .]

Reynolds Number
    It is frequently the case in Navier-Stokes dynamics that one of (v_p¿Ñ_p)v_p and uÑ_p²v_p will dominate to the extent that the other may be neglected and far simpler equations solved. The Reynolds number u^-1ua where u is a typical flow speed and a a characteristic length for the problem provides a rough indicator of the likely ratio of the magntiudes of the inertial (v_p¿Ñ_p)v_p term to the viscous uÑ_p²v_p term; a high Reynold's number favouring the inviscid approximation forcing u=0.
    This can fail when the large second derivatives present in thin boundary layers "seperate" from the boundary and signifcantly effect the flow far from the boundary.

Complex Potential
    The complex potential is used to emulate incompressabile, usually steady, ² planar flows V_p = V¹e₁+V²e₂ with associated complex flow v(x+i(y) = V¹+iV².

Stream Function

    Setting scalar stream function y_p º ò_C dV² V¹ - ò_C dV¹ V² for any ² path C from 0 to p yields V¹ = Ð_e2y_p ; V² = - Ð_e1y_p , ie.
    V_p = Ñ_p×(y_pe₃) = (Ñ_PÙ(y_pe₃))e₁₂₃^-1 .
    If y_p is any analytic real scalar field over ² satisfying this then its analyticity provides incompressability condition Ð_e1V¹ + Ð_e2V² = 0, ie. Ñ_PV_p=0 and also Ñ_p²y_p = 0. Hence V_p = (Ñ_P(y_pe₃))e₁₂₃^-1 = -Ñ_Py_pe₁₂ .
    Also note that Ð_Vp y_p = (V_p¿Ñ_p)y_p = 0 so y_p is constant when following the flow. Such 1-curves along which y_p is constant are known as streamlines.

Velocity Potential

    For a steady irrotational incompressible flow with Ñ_pÙV_p = 0 we can constuct a 0-potential f_p = ò_C d¹p¿V_p for any path C from 0 to p so that V_p=Ñ_pf_p , ie. Vⁱ = e_i²Ð_eif_p . If the flow is conserved we also have Ñ_p¿V_p=0 so Ñ_pV_p = Ñ_p²f_p = 0.

Complex Potential

    The Complex 0-potential f_p º f_p + iy_p satisfies Ñ_P²f_p = 0 and it is natural to here associate i with e₁₂₃. It is defined for x=P where P Î Â^N₀=² but by defining z=x¹+ix² where p=x¹e₁+x²e₂ we can regard f_p = f_z,t as a t-dependant regular function mapping C®C.
    By its construction, the complex potential satisfies the Cauchy-Riemann equations and so any regular complex function provides a complex potential with a direction-independant derivative f'(z,t) º (d/dz)f(z,t) = df/dx¹ + idy/dx¹ = V¹ - iV² = v^{^}(z,t) and |f'(z)|₊ = |v(z,t)|₊ = |V_p| .

    A regular complex 0-potential f_p with Ñ_p²f_p=0 thus fully embodies a steady incompressible irrotational 2D flow with derivative f' equal to the complex conjugate of the flow, and streamlines indicated by constant Imag(f).

Example: Uniform Flow
    The uniform flow v(z,t) = b = B(ib)^↑ has f_p = b^{^} z = B(-ib)^↑ z where p=(x,y)=[r,q] and z=x+iy.

Example: Line Vortex
    Consider the circular flow u_p = ar^-1e_q    , ie. v(z,t) = a|z|₊^-2iz. This has y_p=-a r^↓ while the velocity potential is f_p=aq yielding
    f_p =  a(q-i(r)^↓) = -ai((r)^↓+iq) = -ai(z)^↓ .

² Flow Past a Circular Boundary
    If f(z) is the complex potential of a flow having no singularities for |z|<a then f(z) + f(a²z^-1)^{^} has the same singularities over |z|>a and is purely real when |z|=a (ie. f_p=0 when r=a) so r=a is a streamline.
    From this we deduce that flow due to complex potential f(z)=b(z+a²z^-1) - ai(z)^↓ is approximately uniform for large r but has zero e_q component when var(r)=a .

    Once we have a complex potential f(z) for a flow having zero radial component along _ringa = { z : |z|₊=a } we can obtain flows around a more general closed 1-curve boundary ¦(R₀a]) where ¦: C®C is invertible simply by forming f(¦^-1(z))). The flows will be the same for large r if ¦'(z)®1 as r®¥ and the Joukowski transform ¦(z) = z+c²z^-1  is useful here.

Problems with 1-flows
    Imagine a medium consiting of two types of particle. "Green" particles of small unit "mass" and unit positive "charge", and "red" particles of unit mass and negative charge which with regard to a partcular observer e₄ , are confined to a ring of radius R centred on the origin,. Assume that there are equal large quantities of green and red particles distributed with uniform density r particles per (small) unit volume around the ring and that particles contrive to avoid impacting eachother, or do so only rarely.  Assume that the green partciles are travelling "anticlockwise" around the ring with orbital period T while the red particles travel clockwise with the same orbital period.
    The net spacial flow of mass through a given cross sectional ring element is then M_p=0, although there is kinetic energy 2r½(2pR/T)² = r(2pR/T)² . The net spacial current flow is 2 r(2pR/T) anticlockwise . If the red particles were all to reverse direction, and flow counterclockwise, we would have zero current but nonzero momentum.
    This simple example demonstrates the complications in "adding up" flows. Allowing negative and postive charges means that the sum of two charge-weighted timelike vectors may not remain timelike. Generally speaking, when adding four-momentum 1-vectors, relative masses combine additively but relative three-momentum magnitudes incurr a triangle inequality so that proper magnitude is not conseerved and tends to increase .
    When physicists speak of the flow "at" a particular point they usually mean the averaged flow over a small volume at that point, and by "small" they usually actually mean large enough to contain multiple particles. They hope by summing a large number of fundamentally discontinous point-dependant "path localised" functions to produce a continuos field. Thus all the matter at a given event p is considered to be effectively flowing with the same direction and speed. We will refer to such a flow as single flow because all the matter at p is effectively doing the same thing.

    More generally we represent a superposition of 1-flows with a 2-tensor v_p such that v_p(l,d)_aÙb provides the number of particles of the matter at p of kind classified by a matter-type indicator  l flowing across spacial 2-plane aÙb in e₄-time d. By which we mean that on fixing any timelike e₄ perpendicular to a and b the integer number of particles crossing between e₄-times t and  t+d divided by d.
    In the sense of v_p representing the documentation of an actual multiparticle flow, we regard v_p(l,d)_aÙb to be valued in integer multipes of d^-1 rather than a full real type.
    As d®0 the integer multipliers get smaller discontinuously.

    If the flow is steady in the sense that at a suffiicently small d the second derivatives become swamped by the first and we can conceptually replace individual particles with shoals of identical smaller particles having parallel trajectory then we expect v_p(l,d)_aÙb to approach a limit .
    Flow is thus more naturally regarded as a bivector which we will calla 2-flow. For N=3, the 2-flow is dual to the 1-flow .  

Multiflows
    A more general approach is to define a 1-multiflow via a point-dependant directional 0-field F_p(a) such that scalar F_p(a) = F_p(a^~) gives the ammount of matter at p flowing in direction a. For a timelike single flow m_p we have F_p(a)=a^-1¿m_p = - a¿m_p for timelike a and 0 for null or spacelike a.

    1-field flows are geometrically inadequate in that they encode only a four-direction and fail to embody the "internal freedoms" aka. the "spin" of constituent particles. We will require a multivector field y_p to properly represent the "state of the matter" at a given p.
    An obvious generalisation of a 1-vector flow m_p is a Â_1,3 multivector-valued field m_p defined over a 1-vector pointspace .   A "mass shell" normalisation condition m_p² = k_p for some scalar field k_p is satisfied by multivectors of the form m_p = m_p + m_pe₁₂₃₄ provided m_p² = m_p² + k_p.

Mobiles
    The Rigid body kinematics of U_N carry readily into R_3,1 as follows:   Let P = { p(t) : t Î [t₀,t₁] } be the proper time formulation of a worldline. Taking a₄(t) = p'(t)~ = p'(t), we associate three (spacelike) orthonormal vectors a₁(t),a²(t),a₃(t) Î a₄(t)^* with each "particle event" p(t) corresponding to the spatial orientation of the particle.
    A = { A(t) = (a₁(t),a²(t),a₃(t),a₄(t)) : t Î [t₀,t₁] } is thus an orthonormal 4D frame on a timelike curve and we call such a mobile.
    Because the mobile remains orthonormal, we can define a unique R_3,1 unit rotor R_t satisfying R_t§(a_i(t₀)) = a_i(t)     i=1,2,3,4 .
    Henceforth we will omitt writing (t) except occasionally for emphasis..
    We write A = R_§(A_t0) º RA_t0R^§ .
    Given p(t₀) and A_t0, these R_t completely specify the spacetime history of the orientated particle since p(t) = p(t₀) + ò_t0^ta₄(s) ds .

    If A = R(A_t0) then Adt = W.A where W º 2RdtR^§ is a pure bivector known as the proper angular velocity.
[ Proof : RR^§=1 Þ R(R^§)dt = -RdtR^§ = -(RdtR^§)^§ so RdtR^§ (and thus W) is a pure bivector.
    Adt = (RA_t0R^§)dt = RdtA_t0R^§ + RA_t0(R^§)dt = RdtR^§A + AR(R^§)dt = RdtR^§A - A(RdtR^§) = (RdtR^§)×A = (2RdtR^§).A = W.A  .]
      In particular a₄dt = W.a₄ Þ p" = W.pdt . We also have p" = (p"Ùpdt).pdt .
[ Proof : (pdt)² = -1 Þ p"¿pdt = 0 Þ p"pdt = p"Ùpdt. Hence p" = -(pdt²)p" = -(pdt)(pdtÙp") = (pdt)(p"Ùpdt) = (p"Ùpdt).pdt  .]

    So W = (p"Ùpdt) + b where b Î pdt^* is a pure bivector in a spacelike 3D subspace (and thus a 2-blade) rotationally accelerating the mobile around axis pdt.

    It is natural to factor R as R=LR_f4®a4 where R_f4®a4ºa₄(f₄+a₄)~ and L is a "spatial" rotation satisfying L(a₄) = a₄.

Next : Spacetime Potential Theory

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Copyright (c) Ian C G Bell 1999
Web Source: www.iancgbell.clara.net/maths or www.bigfoot.com/~iancgbell/maths
Latest Edit: 20 Jun 2007.