Spacetime Kinematics and Mechanics

Energy and Momentum

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)(e4+V) = mE(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)m4-1 is known as the relative velocity of m.
• Spacial nonunit 1-vector mEV = ^e4(m) is known as the relative momentum of m.
• Positive scalar mE   =   m0gV   =   m0(1-V2)   =   m0 + ½m0V2 + (3/8) m0V4 + (5/8)m0V6 + .... + 2-2k (2k)! (k!)-2 m0V2k + .. is known as the relative mass or relative energy of m. When V=0 then mE=m0 but as |V| increases towards 1 than, assuming m0 is unchanging, mE increases towards +¥.

Relative mass-energy mE is comprised of rest mass m0 plus an infinite series of positive kinetic energy terms in  m0V2k 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 xE : [t0,t1]   ® Â3,1 by xE(t) º ^(p(t),e4).
Define tE : [t0,t1] ® Â by tE(t) º -e4¿p(t) so that ¯(p(t),e4) = tE(t)e4 .
tE(t) is a monontonic increasing function of t so has an inverse function tE-1 : [tE(t0),tE(t1)] ® [t0,t1] such that tE-1(tE(t)) = t " t Î [t0,t1].
The worldine of the particle can then be parameterised as P = { xE(tE-1(t)) + te4 : t Î [tE(t0),tE(t1)] } representing its perception by E.

gE(t) º dtE(t) / dt   = (1 - vE(t)2) .
[ 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.
EE(t) = -mEe42     (here = mE, elsewhere = mEc2).
mE(t) = mEvE(t) .     [ Proof :  m0gE(t)vE(t) = m0 dtE(t)/dt dxE(t)/dtE(t) = m0 dxE(t)/dt  .]
FE(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.

• FE(t) = dmE(t) /dtE     ("force = rate of change of momentum")
[ Proof :  (dmE(t) /dt) (dt /dtE) = m0 ^e4(d2 p(t)/dt2 ) (gE(t)-1)  .]
• dEE(t)/dtE = FE(t)¿vE(t) = m0 d2 tE/dt2 (gE(t))-1     ("power = work")
[ Proof : (dp(t)/dt)2 = - 1 Þ (dxE(t)/dt)2 - (dtE(t)/dt)2 = - 1 Þ (dxE(t)/dt)¿(d2xE(t)/dt2) = (dtE(t)/dt)(d2tE(t)/dt2) = gE(t)(d2tE(t)/dt2)
dEE(t)/dtE = m0 dgE(t)/dtE = m0 dgE(t)/dt (dt/dtE) = m0 d2 tE/dt2 (gE(t))-1 = m0(dxE(t)/dt)¿(d2xE(t)/dt2)  .]

Since gE(t) = (1 - vE(t)2) = 1 + ½vE2 + o(vE4) we have EE(t) » m0(1 + ½vE(t)2) for small |vE| , 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 òE0EE-1 dE   =   ò0v dv(1-k2)-1 Þ ln(EE0-1) = -½ ln(1-kv2) Þ E = E0(1-kv2) and we see that energy E increases to infinity as v approaches upper limit k . Denoting this infinite energy requiring speed by c gives E = E0(1-(vc-1)2) 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 Vv of vapour in a turbulent fluid "either" of density rl and pressure Pl. The velocity of the particles in the volume Vl 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
½rlVl( <(u1)2> + <(u2)2> + <(u3)2> ) = (3/2)rlVlc2 = (3/2)Mc2 where M=Mv=Ml 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)PvVv so we have Mc2 = PvVv and since Pv=Pl for equilibrium we have Mc2 = PlVv = E, the work done needed to create the bubble against the fluid pressure Pl.

DeBroglie Waves

Electrodynamic wave theory considers solutions of the Klein Gordan equation Ñp2yp = ± l2yp   , notably the normalised periodic solution
yp   =   y0eilp¿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 Â1,3 timespace as scalar-valued y(X,t) e-i(wt - K.X)) so that (d/dt2 - ÑX2)y = (w2 -K2)y
yp   =   y0eilp¿k satisfies both yp»yp   =   y0»y0 and Ñp2yp = (-l2k2y0eilp¿k = l2yp for timelike unit k  or -l2yp for spacelike unit k. Because k2<0 the exponential power series for yp 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
yp   =   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 ME = l-1¯e123(k)~ for wavelength l = h|ME|-1 .

Energy E = m0k4 = hw   = m0(1+VE2)½ = m0(1 + ½VE2 - 1/8VE2 + ... + (-1)k4k(k-1)(2k(2k-1)(2k-2))-1 VE2k + ...

Note the distinction between wave and particle energy "splits" here. Physicists traditionally "split" a particle four-momentum m=m0k as mEl(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 Ñp2yp = 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 yp 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 yp is spacially periodic with directed wavelength h |¯e4*(m)|-1 ¯e4*(m)~ = h ME-2  ¯e4*(m)~ .

Ðd yp = il(d¿k)yp   =   i2pmEh-1(d¿k)yp   =   -h-1mE(d¿k)yp
which is the Schrodinger wave equation with scalar multiplier Hamiltionian operator hp,d = -mE(d¿k) .

With d=e4 we obtain Hamiltionian Hp = -e4¿m = mEk4 = mE(1 + vE2)½ [  k4 denotes coordinate e4¿k rather than |k|4 ] which for small vE2 we can approximate as mE(1 + ½ vE2 + O(vE4) + ...) corresponding to "rest" , "kinetic", and "higher order"  energies respectively.
For "slow" De Brogle waves we thus have the non-relativistic approximation yp = y0(h-1mE(p¿vE - (1+½vE2 + O(vE4))t ) yp » (-h-1mE) y0(h-1mE(p¿vE - ½vE2t)
We can incorporate the p-independant phase factor e-h-1mE into y0 to obtain yp » y0 (h-1mE(p¿vE - ½vE2t)) .

Relativistic Fluid Mechanics

Classical Flows
Classical fluid mechanics typically represnts a flow of matter by a ÂN0 nonunit 1-field MP representing at a given time t the momentum per unit volume passing through a small test volume at spacial position PÎÂN0 and scalar 0-field mP representing the density (mass per unit volume) at P (as measured over a small test volume) is constant for an incompressible fluid. We have Mp = mpVp where Vp 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
ÑP¿Mp = - mP/ 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 mP at P during that interval.
We can express the matter conservation law as
mP/ t + ÑP¿(mpVp)   =   mP/ t + (Vp¿ÑP)mpÑ + mp(ÑP¿Vp)   =   (( / t) + (Vp¿ÑP))mp + mp(ÑP¿Vp)   =   0     where ( / t) + (Vp¿ÑP) 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 ÂN 1-vector vp=Vp±e4 where N=N0+1 and e4 is a new axis perspendicular to ÂN0 with e42=±1 then the sunstantial direction becomes (vp¿Ñp) with the differentiating scope of the Ñ extending only rightwards rather than encompassing the vp.

For a steady flow Mp = MP with vanishing Ðe4 derivatives the matter conservation law becomes   =   (Vp¿ÑP))mp + mp(ÑP¿Vp) and for in incompressible flow with mp=m>0  this yields the incompresability condition ÑP¿Vp = 0, ie. the spacial divergence of the N0-D velocity 1-vector is zero.
For vp=Vp±e4 , the incompresability condition implies Ñp¿vp=0.

For an incompressible flow "momentum" and "velocity" fields become effectively equivalent with Mp=mVp.

Spacetime 1-flows
Consider a nonunit timelike 1-vector valued field
mp   =   mpup   =   Mp + mpe4   =   mp(1-Vp2)(Vp+e4)   =   mp(Vp+e4)   =   mpvp
at p as four-momentum due to matter of rest mass mp and unit 4D timelike "four-velocity" up.
An e4-observer percieves mp as matter of relative mass mp º e4¿mp = mp(1-Vp2)     travelling at nonunit spacial velocity 1-vector Vp = ^e4(up) with relative momentum Mp = ^e4(mp) = mpVp . The matter conservation law becomes Ñp¿mp = 0 .
[ Proof :  Ñp¿mp = Ñ[e123]p¿mp + e4¿Ðe4mp = ÑP¿Mp + Ðe4(e4¿mp) = ÑP¿Mp + Ðe4mp = ÑP¿Mp + mp/t = 0  .]

The e4-substantial derivative (vp¿Ñp)   =   Ðpvp   =   /x4 + (Vp¿Ñ[e123]) corresponds to that rate of change with respect to e4-time when following the e4-spacial flow Vp .
For a static system (/x4 = 0) we have (vp¿Ñp)   =   (Vp¿ÑP)   =   ÐPVp .

Current

We likewise interpret a nonunit timelike 1-vector "current" field jp   =   jpup   =   rpup   =   Jp + j4pe4 as a 3D-current Jp and relative e4-charge j4  with charge conservation law
Ñp¿jp   =   Ñ[e123]p¿Jp + j4p/ x4   =   0 . If jp falls away more slowly, eg. by a spacial r-1 factor then e4-charge may "radiate away". [ e4-dependant component j4p is often denoted r in the literature but we here retain r for the frame-independant magnitude rp = |jp| = jp. ]
Ñp¿jp=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 rp ® 0 faster than |p¿e123|-2 does as |p¿e123| ® ¥.
[ Proof : Fix e4¿p = t0 and consider a spacial 2-sphere St0 Ì  t0e4 + e4* of radius R . We postulate that rp ® 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 Base) . This means that the Â3 boundary (spherical surface) integral òd St0 dp2 jp ® 0 as R ® ¥ and the geometric form of fundamental theorem of calculus (applied to Euclidean manifold St0) then provides that the scalar part of the volume integral ( òSt0 dp3 Ñp[St0] jp )<0> ® 0 as R ® ¥ . Other than at the boundary dSt0 , whose contribution to the volume integral we can neglect, Ñp[St0] = Ñp[e4*] and so the contribution to the volume integral of a small 3-simplex d3p at p is
(d3p Ñp[e4*]jp) = -(d3p Ñp[e4]jp) = -(e3e123  e4Ðe4 jp) = (e3e1234Ðe4 jp)
Considering just the spacial (e123) component we obtain òSt0 dp3 Ðe4j4p » 0 Þ (d/dt) òSt dp3 j4p » 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 Ñp¿jp = 0 and |jp|® 0 sufficiently fast for large |d¿p| then - ò Wd,a d3p d¿jp 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 Ñp¿jp =0 then we have the streamline identity
(jp¿Ñp)jpÑ =  jp¿wp + Ñpjp2)     where wp = ÑpÙjp and jp=|jp| .
[ Proof :  ½Ñpjp2 = ½(ÑpjpÑjp + ÑpjpjpÑ)<1> = ½((Ñpjp)jp + Ñp(jpÑjp - 2jpÑÙjp))<1> = (Ñpjp).jp  - Ñp¿(jpÑÙjp) = wp.jp   - Ñp¿(jpÑÙjp)
= -jp¿wp + (Ñp¿jp)jpÑ - (Ñp¿jpÑ)jp) = -jp¿wp + (jp¿Ñp)jpÑ  .]

For a static 3D flow with ÑP¿JP=0 we have classical Â3 form (JP¿ÑP)JP = (ÑP×JPJP + ÑPJP2) .

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

For t=x4>R , the intersection of 3-tsphere O+-R,0 and the e4-cotemporal 2-plane We4,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.
rx is constant throughout We4,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 ccosh-1(tR-1) all are.

Classical Pressure

The scalar static pressure sp at p is independant of the macroscopic "net flow" mp 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 sp 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 spV is proportional to nQp where V=|V| is the volume of a test 3-simplex V at p containing n molecules of average temperature Qp. n is proportional to total mass mpV so sp is proportionate to mpQp .

Now consider the body of fluid inside V. The total forces acting on this fluid are the external macroscopic forces òV |d3p| fp where fp=mpgp + hp is e4-spacial forces exerted   (typically we might have gp=-ge3 for uniform vertical gravity) , and the static pressure integrated over the boundary òdV|d2p|(-np)sp   =   -òdVd2p e123-1sp   =   -e123-1 òVd3p ÑPsp     where np = I2p-1 e123 = I2p e123-1 is the outward normal at p Î dV  .
Classicists typically asssume gp=-ÑpGp and hp=-ÑpHp derive from scalar 0-potentials.
The total change in momentum incured by the mass mp|V| of fluid as it follows the flow lines is òV d3p (ÑP¿vp)mp so we have
(ÑP¿vp)mp = mpgp + hp - Ñpsp     which for constant mp=m simplifies to the momentum conservation law (vp¿ÑP)vp = gp + mp-1hp - mp-1(ÑPsMP) aka. Euler's equation when combined with ÑP¿VP=0 .
For P in Â3 we have (Vp¿ÑP)VpÑ   =   ½ÑP(Vp2) - Vp×(ÑP×VpÑ) so we can express the momentum conservation law as
½ÑP(Vp2) - Vp×(ÑP×VpÑ) + mp-1ÑPsp - FP = 0.
[ Proof : a×(b×c) = (a¿c)b - (a¿b)c gives Vp×(ÑP×Vp) = (Vp¿ÑP)VpÑ - (Vp¿VpÑ)ÑP = (Vp¿ÑP)VpÑ - ÑP(VpÑVp)
= (Vp¿ÑP)VpÑ - ½ÑP(Vp2)  .]

Navier-Stokes
Suppose we combine the static pressure force -spnp with a viscous force u((np¿Ñp)mpÑ + Ñp(np¿mpÑ)) = um((np¿Ñp)vpÑ + Ñp(np¿vpÑ)) for an incompressible flow with mp=m.
If we regard np as varying negligibly with p compared to vp's variation (eg. when over a flat surface) we can view this as the np directed flow derivative plus the gradient of the np-directed flow-speed, equally wieghted by scalar u.
This leads to the Navier-Stokes equation
(vp¿Ñp)mp   =   -mp(ÑpGp) - (Ñpsp) - (ÑpHp) + (vp¿Ñp)mpÑvp + uÑp2mp which for mp=m becomes
(vp¿Ñp)vp   =   -(ÑpGp) - m-1((Ñpsp) + (ÑpHp)) + uÑp2vp 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 (mp=m) flow under gravity, Navier-Stokes becomes
(Ñp¿vp)vp = -ÑpGp  - m-1((Ñpsp) + uÑp2vp and since (vp¿Ñp)vp = vp¿(ÑpÙvp) + ½Ñp(vp2) we have vp¿wp = -ÑpEp + uÑp2vp where Ep = ½vp2 + Gp + m-1sp where wp º ÑpÙvp has Ñp¿wp=0. Hence ÑpÙ(vp¿wp) = uÑpÙ(Ñp2vp) = uÑp3.vp .

But ÑpÙ(vp¿wp) = (Ñp¿(vpÙwp))i where (N-2)-vector kinematic vorticity wp º wpi-1 = (ÑpÙvp)* so we have Ñp¿(vpÙwp) = 0
[ Proof : ÑpÙ(vp.wp) = (Ñp¿(vpÙ(wpi-1)))i = (Ñp¿(vpÙwp))i  .]
For incompressible flow, Ñp¿(vpÙwp) = (vp¿Ñp)wpÑ - vpÑÙ(wp.Ñp)vpÑ so we have geometric vorticity equation (vp¿Ñp)wp = (wp.Ñp)Ùvp + u Ñp¿(Ñp2vpi-1) with the differentiating scope taken rightwards only. For N=3 and u=0 this reduces to (vp.Ñp)wp = (wp.Ñp)vp with 1-vector vorticity wp.
[ Proof :  (vp¿Ñp)wpÑ + (Ñ¿vpÑ)wp - vpÙ(Ñp¿wpÑ) - vpÑÙ(Ñp¿wp) reduces to result since Ñp¿wp = ÑpÙwp = ÑpÙÑpÙvp vanishes as does Ñp¿vp  . Also u(ÑpÙ(Ñp2vp))i-1 = u(Ñp¿((Ñp2vp)i-1)  .]
Thus the flow-directed derivative of the vorticity is the vorticity-directed derivative of the flow.
Classically, we have (vp¿Ñp)wp = (wp¿Ñp)vp for 1-vector vorticity wp = (ÑPÙVP)* = ÑP×VP which for 2D flow becomes (ÑP¿VP)wp=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 mp is constant.
When fp=0, integrating along a streamline gives Bernoulli's equation sp + ½mvp2 = s0 , where scalar s0 is known as the stagnation pressure. Bernoulli's equation applies when mp is constant and the flow is static. More generally Ep = sp + mGp + Hp + ½mvp2 is constant along the streamline if mp=m and wp=0 along it.
[ Proof :
sp1   =   sp0 + òt0t1 dt vp¿(Ñpsp)   =   sp0 + òt0t1 dt  vp¿ (-mp(ÑpGp) - (ÑpHp) - (Ñp¿vp)mpÑvp - mp(Ñp¿vp)vp )
=   sp0 + m(G0-G1) (H0-H1) - m òt0t1 dt  vp¿((Ñp¿vp)vp )   =   sp0 + m(G0-G1) (H0-H1) - m òt0t1 dt  vp¿(vp¿wp + Ñpvp2 )
=   sp0 + m(G0-G1) (H0-H1) + ½m (v02 - v12) .
Hence sp + mGp + Hp + ½mvp2 is constant along the streamline.  .]

Reynolds Number
It is frequently the case in Navier-Stokes dynamics that one of (vp¿Ñp)vp and uÑp2vp 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 (vp¿Ñp)vp term to the viscous uÑp2vp 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, Â2 planar flows Vp = V1e1+V2e2 with associated complex flow v(x+i(y) = V1+iV2.

Stream Function

Setting scalar stream function yp º òC dV2 V1 - òC dV1 V2 for any Â2 path C from 0 to p yields V1 = Ðe2yp ; V2 = - Ðe1yp , ie.
Vp = Ñp×(ype3) = (ÑPÙ(ype3))e123-1 .
If yp is any analytic real scalar field over Â2 satisfying this then its analyticity provides incompressability condition Ðe1V1 + Ðe2V2 = 0, ie. ÑPVp=0 and also Ñp2yp = 0. Hence Vp = (ÑP(ype3))e123-1 = -ÑPype12 .
Also note that ÐVp yp = (Vp¿Ñp)yp = 0 so yp is constant when following the flow. Such 1-curves along which yp is constant are known as streamlines.

Velocity Potential

For a steady irrotational incompressible flow with ÑpÙVp = 0 we can constuct a 0-potential fp = òC d1p¿Vp for any path C from 0 to p so that Vp=Ñpfp , ie. Vi = ei2Ðeifp . If the flow is conserved we also have Ñp¿Vp=0 so ÑpVp = Ñp2fp = 0.

Complex Potential

The Complex 0-potential fp º fp + iyp satisfies ÑP2fp = 0 and it is natural to here associate i with e123. It is defined for x=P where P Î ÂN0=Â2 but by defining z=x1+ix2 where p=x1e1+x2e2 we can regard fp = fz,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)|+ = |Vp| .

A regular complex 0-potential fp with Ñp2fp=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 fp = b^ z = B(-ib) z where p=(x,y)=[r,q] and z=x+iy.

Example: Line Vortex
Consider the circular flow up = ar-1eq    , ie. v(z,t) = a|z|+-2iz. This has yp=-a r while the velocity potential is fp=aq yielding
fp =  a(q-i(r)) = -ai((r)+iq) = -ai(z) .

Â2 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. fp=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 eq 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 Mp=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 vp such that vp(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 vp representing the documentation of an actual multiparticle flow, we regard vp(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 vp(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 Fp(a) such that scalar Fp(a) = Fp(a~) gives the ammount of matter at p flowing in direction a. For a timelike single flow mp we have Fp(a)=a-1¿mp = - a¿mp 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 yp to properly represent the "state of the matter" at a given p.
An obvious generalisation of a 1-vector flow mp is a Â1,3 multivector-valued field mp defined over a 1-vector pointspace .   A "mass shell" normalisation condition mp2 = kp for some scalar field kp is satisfied by multivectors of the form mp = mp + mpe1234 provided mp2 = mp2 + kp.

Mobiles
The Rigid body kinematics of UN carry readily into R3,1 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 R3,1 unit rotor Rt satisfying Rt§(ai(t0)) = ai(t)     i=1,2,3,4 .
Henceforth we will omitt writing (t) except occasionally for emphasis..
We write A = R§(At0) º RAt0R§ .
Given p(t0) and At0, these Rt completely specify the spacetime history of the orientated particle since p(t) = p(t0) + òt0ta4(s) ds .

If A = R(At0) 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 = (RAt0R§)dt = RdtAt0R§ + RAt0(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=LRf4®a4 where Rf4®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.