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 c² is a measure of the thermal energy of the either. 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 E = P_lV_v = Mc² as the work required 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 complex-valued _cvpsi(X,t) e^{-i(wt - K.X))} so that (d/dt² - Ñ_X²)_cvpsi = (w² -K²)_cvpsi
    More generally,   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


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.

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 A' = W.A where W º 2R'R^§ is a pure bivector known as the proper angular velocity.
[ Proof : RR^§=1 Þ R(R^§)' = -R'R^§ = -(R'R^§)^§ so R'R^§ (and thus W) is a pure bivector.
    A' = (RA_t0R^§)' = R'A_t0R^§ + RA_t0(R^§)' = R'R^§A + AR(R^§)' = R'R^§A - A(R'R^§) = (R'R^§)×A = (2R'R^§).A = W.A  .]
      In particular a₄' = W.a₄ Þ p" = W.p' . We also have p" = (p"Ùp').p' .
[ Proof : (p')² = -1 Þ p"¿p' = 0 Þ p"p' = p"Ùp'. Hence p" = -(p'²)p" = -(p')(p'Ùp") = (p')(p"Ùp') = (p"Ùp').p'  .]

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

    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: 13 Jun 2014.