Just as a spacetime 1-vector "point" p decomposes into "time" p^{4} and "place" P
=p^{1}e_{1}+p^{2}e_{2}+p^{3}e_{3} when percieved by
observer E, a timelike 1-vector four-momentum
m=m^{1}e_{1}+m^{2}e_{2}+m^{3}e_{3}+m^{4}e_{4} decomposes into "energy" m^{4} and "momentum"
M=m^{1}e_{1}+m^{2}e_{2}+m^{3}e_{3}.
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 e_{4} it is natural to express timelike
1-vector m with m^{2}=-m_{0}^{2}
as
m = m_{0}(1-V^{2})^{-½}(e_{4}+V) = m_{E}(e_{4} + V) where:
Relative mass-energy m_{E} is comprised of rest mass m_{0} plus an infinite series of positive kinetic energy terms in m_{0}V^{2k} which for |V| << 1 are approximated by the first classical kinetic energy term ½m_{0}V^{2} .
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 m^{4} having unit relative velocity and consequent relative momentum m^{4}V = ¯_{e4*}(m) .
Let path P = { p(t) : t Î Â } be a proper worldline
for a "particle" of constant proper mass m_{0} > 0 .
Define x_{E} :
[t_{0},t_{1}] ® Â^{3,1} by
x_{E}(t) º ^(p(t),e_{4}).
Define t_{E} : [t_{0},t_{1}] ® Â by t_{E}(t) º -e_{4}¿p(t)
so that ¯(p(t),e_{4}) = t_{E}(t)e_{4} .
t_{E}(t) is a monontonic increasing function of t so has an inverse function
t_{E}^{-1} : [t_{E}(t_{0}),t_{E}(t_{1})] ® [t_{0},t_{1}] such that
t_{E}^{-1}(t_{E}(t)) = t " t Î [t_{0},t_{1}].
The worldine of the particle can then be parameterised as P =
{ x_{E}(t_{E}^{-1}(t)) + te_{4} : t Î [t_{E}(t_{0}),t_{E}(t_{1})] }
representing its perception by E.
g_{E}(t) º dt_{E}(t) / dt
= (1 - v_{E}(t)^{2})^{-½}
.
[ Proof :
We take the + Ö
.]
(dp(t)/dt)^{2} = -1
Þ (dx(t_{E}(t))/dt)^{2} - g_{E}(t)^{2} = -1
Þ (dx(t_{E}(t)/dt)^{2} = g_{E}(t)^{2} - 1
Þ ((dx(t_{E}(t))/dt_{E}(t) g_{E}(t))^{2} = g_{E}(t)^{2} - 1
Þ g_{E}(t)^{2} = (1 - (dx(t_{E}(t))/dt_{E}(t))^{2})^{-1} .
Þ g_{E}(t) = ± (1 - v_{E}(t)^{2})^{-½}
where v_{E}(t) = dx_{E}(t)/dt_{E}(t) is the relative velocity.
The e_{4}-relative (forced Euclidean) modulus vv^{†} being the sum of "kinetic energy"
(v^{1})^{2}
+(v^{2})^{2}
+(v^{3})^{2}
and "mass energy" (v^{4})^{2} .
is known as the Hamiltonian .
v^{2}
= (v^{1})^{2}
+ (v^{2})^{2}
+ (v^{3})^{2}
- (v^{4})^{2}
provides a measure refered to by
physiscists as a Lagrangian .
Some authors represent this split as e_{4}m(t)
=
e_{4}¿m(t)
+ e_{4}Ùm(t)
and consider the relative momentum to be a bivector but we do not take this approach here.
E_{E}(t) =
-m_{E}e_{4}^{2}
(here = m_{E}, elsewhere = m_{E}c^{2}).
m_{E}(t) = m_{E}v_{E}(t) .
[ Proof : m_{0}g_{E}(t)v_{E}(t)
= m_{0} dt_{E}(t)/dt dx_{E}(t)/dt_{E}(t)
= m_{0} 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.
Since g_{E}(t) = (1 - v_{E}(t)^{2})^{-½} = 1 + ½v_{E}^{2} + o(v_{E}^{4})
we have E_{E}(t) » m_{0}(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 e_{4} is actually relevant and coordinate independance within E's
3D spatial universe is retained.
Justifying E=mc^{2}
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-kv^{2})^{-1}
and so
ò_{E0}^{E}E^{-1} dE = ò_{0}^{v} dv(1-k^{2})^{-1}
Þ
ln(EE_{0}^{-1}) = -½ ln(1-kv^{2})
Þ E = E_{0}(1-kv^{2})^{-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 = E_{0}(1-(vc^{-1})^{2})^{-½} and
F = (d/dt)mv = c^{-2} Ev which integrates to mv = Evc^{-2} Þ
E = mc^{2} .
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= u_{0} + u where u_{0} is the average drift over a
short time interval and u is thermal jitter with
<(u^{1})^{2}> =
<(u^{2})^{2}> =
<(u^{3})^{2}> = c^{2}
where <a> denotes averaging a over a small timeperiod
and c^{2} is a measure of the thermal energy of the either.
The net kinetic thermal energy inherited by the vapour is
½r_{l}V_{l}( <(u^{1})^{2}> + <(u^{2})^{2}> + <(u^{3})^{2}> ) =
(3/2)r_{l}V_{l}c^{2} = (3/2)Mc^{2} 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_{v}V_{v} so we have Mc^{2} = P_{v}V_{v} and since P_{v}=P_{l} for equilibrium we have
E = P_{l}V_{v} = Mc^{2} 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_{p}^{2}y_{p} = ± l^{2}y_{p} ,
notably the normalised periodic solution
y_{p} = y_{0}e^{ilp¿k}
º y_{0} (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/dt^{2} - Ñ_{X}^{2})_cvpsi = (w^{2} -K^{2})_cvpsi
More generally,
y_{p} = y_{0}e^{ilp¿k}
satisfies both
y_{p}^{»}y_{p} = y_{0}^{»}y_{0}
and
Ñ_{p}^{2}y_{p} =
(-l^{2}k^{2}y_{0}e^{ilp¿k}
= l^{2}y_{p} for timelike unit k or -l^{2}y_{p} for spacelike unit k.
Because k^{2}<0 the exponential power series for y_{p} is "bounded trigonometric"
rather than "divergent hyperbolic" .
l is frequently expressed by physicists as 2pm_{0}c/h
with m_{0}= hl(2pc)^{-1} called the "mass" of y .
In natural units c=1, h=(2p), h=1.
Taking l = (2p)m_{0}/h = ih^{-1}m_{0} gives the traditional
formulation (in our unorthodox notations) of the
De Broglie Wave
y_{p}
= y_{0}(h^{-1}m_{0}(p¿k))^{↑}
= y_{0}(h^{-1}(p¿m))^{↑}
for positive scalar m_{0} and unit timelike k with m = m_{0}k .
Relativisitically, an e_{4}-observer percieves such a De Broglie wave vector h^{-1} m=m_{0}k as splitting into positive scalar temporal wave frequency w=m_{0}k^{4}h^{-1} (and corresponding wave period 2p h (m_{0}k^{4})^{-1} ) and spacial wave-vector h^{-1} M_{E} = l^{-1}¯_{e123}(k)^{~} for wavelength l = h|M_{E}|^{-1} .
Energy E = m_{0}k^{4} = hw = m_{0}(1+V_{E}^{2})^{½} = m_{0}(1 + ½V_{E}^{2} - 1/8V_{E}^{2} + ... + (-1)^{k}4k(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=m_{0}k as m_{El}(e_{4}+V)
but a wave four-momentum as
hwe_{4} +
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 e_{123} projection. The
kinetic energy series differ beyond the first, classical, term.
For a zero mass "electromagnetic" wave in a vacuum solving Ñ_{p}^{2}y_{p} = 0 ( ie. l=0) we have y_{0} (iap¿k)^{↑} for any nonzero a and nullvector k . The e_{4}-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 2^{48} 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 2^{22} 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=e^{4}¿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_{E}h^{-1}(d¿k)y_{p}
= -h^{-1}m_{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=e_{4} we obtain Hamiltionian
H_{p} = -e_{4}¿m = m_{E}k^{4}
= m_{E}(1 + v_{E}^{2})^{½}
[ k^{4} denotes coordinate e^{4}¿k rather than |k|^{4} ]
which for small v_{E}^{2}
we can approximate as
m_{E}(1 + ½ v_{E}^{2} + _{O}(v_{E}^{4}) + ...)
corresponding to "rest" , "kinetic", and "higher order" energies respectively.
For "slow" De Brogle waves we thus have the non-relativistic approximation
y_{p} = y_{0}(h^{-1}m_{E}(p¿v_{E} - (1+½v_{E}^{2} + _{O}(v_{E}^{4}))t )^{↑}
y_{p} »
(-h^{-1}m_{E})^{↑} y_{0}(h^{-1}m_{E}(p¿v_{E} - ½v_{E}^{2}t)^{↑}
We can incorporate the p-independant phase factor e^{-h-1mE} into y_{0}
to obtain
y_{p} » y_{0}
(h^{-1}m_{E}(p¿v_{E} - ½v_{E}^{2}t))^{↑} .
Relativistic Fluid Mechanics
Spacetime 1-flows
Consider a nonunit timelike 1-vector valued field
m_{p} = m_{p}u_{p} = M_{p} + m_{p}e_{4} = m_{p}(1-V_{p}^{2})^{-½}(V_{p}+e^{4})
= m_{p}(V_{p}+e^{4})
= m_{p}v_{p}
at p as four-momentum due to matter of rest mass m_{p} and unit 4D timelike "four-velocity" u_{p}.
An e_{4}-observer percieves m_{p}
as matter of relative mass
m_{p} º e^{4}¿m_{p} = m_{p}(1-V_{p}^{2})^{-½}
travelling at nonunit spacial velocity 1-vector V_{p} = ^_{e4}(u_{p})
with relative momentum M_{p} = ^_{e4}(m_{p}) = m_{p}V_{p} .
The matter conservation law becomes
Ñ_{p}¿m_{p} = 0 .
[ Proof : Ñ_{p}¿m_{p} =
Ñ_{[e123]p}¿m_{p} + e^{4}¿Ð_{e4}m_{p}
= Ñ_{P}¿M_{p} + Ð_{e4}(e^{4}¿m_{p})
= Ñ_{P}¿M_{p} + Ð_{e4}m_{p}
= Ñ_{P}¿M_{p} + ¶m_{p}/¶t
= 0 .]
The e_{4}-substantial derivative
(v_{p}¿Ñ_{p}) = Ð^{p}_{vp} =
¶/¶x^{4} + (V_{p}¿Ñ_{[e123]})
corresponds to that rate of change with respect to e_{4}-time when following the e_{4}-spacial flow V_{p} .
For a static system (¶/¶x^{4} = 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_{p}u_{p} = r_{p}u_{p}
= J_{p} + j^{4}_{p}e_{4} as a 3D-current J_{p} and relative e_{4}-charge
j^{4} with charge conservation law
Ñ_{p}¿j_{p} = Ñ_{[e123]p}¿J_{p} + ¶ j^{4}_{p}/¶ x^{4} = 0 .
If j_{p} falls away more slowly, eg. by a spacial r^{-1} factor then e_{4}-charge may
"radiate away".
[
e_{4}-dependant component j^{4}_{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 j^{4}
(sometimes refered to as the " probable charge" at p)
is "globally conserved" over (e_{4}-percieved) time in that its integration over a suitably large e_{4}-cotemporal spacial volume remains constant over time, provided r_{p} ® 0 faster than |p¿e_{123}|^{-2} does as
|p¿e_{123}| ® ¥.
[ Proof :
Fix e^{4}¿p = t_{0} and consider a spacial 2-sphere S_{t0} Ì
t_{0}e_{4} + e_{4}^{*}
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
Â_{3} boundary (spherical surface) integral
ò_{d St0} dp^{2} 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} dp^{3} 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 d^{3}p at p is
(d^{3}p Ñ_{p}_{[e4*]}j_{p})
= -(d^{3}p Ñ_{p}_{[e4]}j_{p})
= -(e^{3}e_{123} e^{4}Ð_{e4} j_{p})
= (e^{3}e_{1234}Ð_{e4} j_{p})
Considering just the spacial (e_{123}) component we obtain
ò_{St0} dp^{3} Ð_{e4}j^{4}_{p} » 0
Þ (d/dt) ò_{St} dp^{3} j^{4}_{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} d^{3}p 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_{p}_{Ñ} = j_{p}¿w_{p} + Ñ_{p}(½j_{p}^{2})
where w_{p} = Ñ_{p}Ùj_{p} and j_{p}=|j_{p}| .
[ Proof : ½Ñ_{p}j_{p}^{2} = ½(Ñ_{p}j_{p}_{Ñ}j_{p} + Ñ_{p}j_{p}j_{p}_{Ñ})_{<1>}
= ½((Ñ_{p}j_{p})j_{p} + Ñ_{p}(j_{p}_{Ñ}j_{p} - 2j_{p}_{Ñ}Ùj_{p}))_{<1>}
= (Ñ_{p}j_{p}).j_{p} - Ñ_{p}¿(j_{p}_{Ñ}Ùj_{p})
= w_{p}.j_{p} - Ñ_{p}¿(j_{p}_{Ñ}Ùj_{p})
= -j_{p}¿w_{p} + (Ñ_{p}¿j_{p})j_{p}_{Ñ} - (Ñ_{p}¿j_{p}_{Ñ})j_{p})
= -j_{p}¿w_{p} + (j_{p}¿Ñ_{p})j_{p}_{Ñ}
.]
For a static 3D flow with Ñ_{P}¿J_{P}=0 we have classical Â_{3} form
(J_{P}¿Ñ_{P})J_{P} = (Ñ_{P}×J_{P})×J_{P} + Ñ_{P}(½J_{P}^{2}) .
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 e_{4}-relative charge j^{4}|j_{x}|^{-1} .
For t=x^{4}>R , the intersection of 3-tsphere O^{+}_{-R,0} and the e_{4}-cotemporal 2-plane
W_{e4,te4} = { p : e^{4}¿p=x^{4}=t} is a spacial 2-sphere of centre x^{4}e_{4} and radius
|X| = (t^{2}-R^{2})^{½}
= ((e^{4}¿p)^{2} + p^{2})^{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) = (t^{2}-R^{2})^{½}
; x^{4}=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_{p}^{2} = k_{p} for some scalar field
k_{p} is satisfied by multivectors of the form
m_{p} = m_{p} + m_{p}e_{1234} provided m_{p}^{2} = m_{p}^{2} + k_{p}.
Mobiles
The Rigid body kinematics of U_{N} carry readily into R_{3,1} as follows:
Let P = { p(t) : t Î [t_{0},t_{1}] } be the proper time formulation
of a worldline.
Taking a_{4}(t) = p'(t)~ = p'(t), we associate three (spacelike) orthonormal vectors
a_{1}(t),a^{2}(t),a_{3}(t) Î a_{4}(t)^{*} with each "particle event" p(t) corresponding to
the spatial orientation of the particle.
A = { A(t) = (a_{1}(t),a^{2}(t),a_{3}(t),a_{4}(t))
: t Î [t_{0},t_{1}] }
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}(t_{0})) = 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_{t0}R^{§} .
Given p(t_{0}) and A_{t0}, these R_{t}
completely specify the spacetime history of the orientated particle since
p(t) = p(t_{0}) + ò_{t0}^{t}a_{4}(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_{t0}R^{§})'
= R'A_{t0}R^{§} + RA_{t0}(R^{§})'
= R'R^{§}A + AR(R^{§})'
= R'R^{§}A - A(R'R^{§})
= (R'R^{§})×A
= (2R'R^{§}).A
= W.A
.]
In particular a_{4}' = W.a_{4} 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®a4}ºa_{4}(f_{4}+a_{4})~ and L is a "spatial" rotation satisfying
L(a_{4}) = a_{4}.
Next : Spacetime Potential Theory