Those familiar with other treatments of relativistic mathematics, even the few multivector based ones, will find unorthodox notations here.
In particular, brackets are used in preference to precedence conventions (in accordance with programmer morality)
and "subscripts" emphasising positional (event) dependence are seldom ommitted.
Explicit S symbols ares used in preference to evoking a "summation convention"
for repeated ("dummy") indices.
We retain use of the
contractive inner product
( ¿ )
in preference to the conventional (Hestenes) inner product ( . ) where possible.
Since a¿b = a.b for 1-vector a whenever b has a zero scalar component,
the products are equivalent in many contexts.
Minkowski and Hestenes SpaceTimes
Minkowski SpaceTime Â_{3.1}
Let (e_{1},e_{2},e_{3}) be an orthonormal orientational reference frame for a Euclidean
3D space Â^{3}. We
embody the concept of "time" by postulating a fourth vector e_{4} orthogonal to e_{1},e_{2},e_{3}
with e_{4}^{2}=-1 thus creating a basis (coordinate frame)
E=(e_{1},e_{2},e_{3},e_{4}) for
Minkowski spacetime Â^{3,1} .
Note that e_{4} is sometimes given nonunit magnitude c
» 3 × 10^{8} meter/second » 2^{28} meter/second
(the "speed of light")
making a light year approx 10^{16}m » 2^{53}m
but we retain "natural units" with c=1 here.
It is all too easy, encouraged by the name, to confuse lightspeed with photon speed. Some photons travel
very slowly in some mediums, and reference to a speed "in vacuum" requires an unrealistic
notion of vacuum uncluttered by "quantum foam".
When thinking about lightspeed one should forget about photons and consider c as the e_{4}-independant
e_{4}-observed instantaneous speed of a null trajectory. Physicists think of "massless particles"
in the sense of null four-momentum but this too
is misguided for we must ultimately associate mass with accelaration rather than momentum, and "lightspeed" trajectory need not be straight.
The current human estimate of c was obtained by measuring the speed of photons through nonvacuum, and became "exact"
by redefining the meter in terms of that estimate. Consequently, to the extent that humans underestimate the geodesic lightspeed, the "meter" is longer than our meter rulers.
We can extend E
to a basis for Â_{3,1} ( spacetime multivectors).
i=e_{1}e_{2}e_{3}e_{4} , the unit pseudoscalar spanning spacetime,
(anti)commutes with all (odd)even multivectors and satisfies i^{2} = -1 ;
i^{§} = i .
Â_{3,1} <0;4>-multivectors a + bi having only scalar and pseudoscalar parts
form a subalgebra under the geometric product isomorphic to "complex numbers" Â_{1,1+}
and commute with all even multivectors ( Â_{3,1+} ).
Note that
(a + bi)(a + bi)^{§} =
(a + bi)^{2} = a^{2}-b^{2} + 2abi .
We say a spacetime 1-vector v is spacelike if
v^{2} > 0 , timelike if v^{2} < 0, and null (aka lightlike)
if v^{2} = 0.
We say a 1-vector is forward if v¿e_{40} < 0 for some universally agreed timelike e_{40}.
Null vectors other than 0 are called proper null.
Let spacelike V satisfy V¿e_{4} = 0 (so that V Î e_{4}^{*}).
If V^{2} < 1 then e_{4}+v is forward timelike (so not null) and we can define another forward timelike unit vector
f_{4} = (e_{4}+v)~ = g_{V}(e_{4}+v)
where positive scalar relativity factor
g_{V} º (1-V^{2})^{-½}
is 1 for V=0 increasing to ¥ for V^{2}=1.
We also have
e_{4}¿f_{4} = -g_{V} ;
(e_{4}Ùf_{4})^{2} = g_{V}^{2} - 1
; and
v = -^(f_{4},e_{4}) / (f_{4}¿e_{4}) = (f_{4}Ùe_{4})e_{4} /(f_{4}¿e_{4}) .
If we apply the rotational rotor
R_{e4®f4} = f_{4}(e_{4}+f_{4})~
to {e_{1},e_{2},e_{3},e_{4}} we obtain a new
orthonormal basis F = {f_{1},f_{2},f_{3},f_{4}} for spacetime which is no less "fundamental"
than {e_{1},e_{2},e_{3},e_{4}}.
We use e_{4} rather than e_{0} for the timelike extendor of Â^{3} as
a matter of subjective notational preference. It is mathematically irrelevant but computationally prudent
as wish to "retain" e_{0} for use in a higher dimensional embedding.
Minkowski TimeSpace Â_{1.3}
Many authors, notably Hestenes, consider three basis vectors
g_{1},g_{2},g_{3}
of negative signature and
one, g_{0} of positive signature and so work in
Minkowski timespace algebra
( aka. Hestenes spacetime algebra or S.T.A. )
Â_{1,3} rather than Â_{3,1}. Their meanings
of the terms "spacelike" and "timelike" are consequently the reverse of ours here
with regard to signature signs.
As for Minkowski spacetime algebra Â_{3,1} , the pseudoscalar
i=g_{0}g_{1}g_{2}g_{3}
(anti)commutes with (odd) even multivectors and has i ^{2}=-1 .
The phrase "Minkowski signature"
is frequently used for both {+,+,+,-} and {+,-,-,-} but these two approaches (aka. the east coast and west coast metric )
, while substantively similar in some
regards, are not equivalent.
Null <1;4>-vectors
have 1-vector components of positive square in both alegebras, so have
"timelike" 1-vector component in Â_{1,3} but "spacelike"
1-vector component in Â_{3,1}.
This author favours that spatial "squared distance"
be considered as "positive", temporal "squared distance" as "negative" since this
segues better with standard non-relativistic models - and we will
accordingly work with Minkowski spacetime algebra Â_{3,1} here.
Our "timelike" e_{4} thus has e_{4}^{-1} = e^{4} = -e_{4} whereas
g_{0}^{-1} = g^{0}
= g_{0} .
Nullvector Decomposition
Given an arbitary nonzero 1-vector s with s^{2}=0 then for any unit minussquare e_{-} we can express s as
-(s¿e_{-})e_{-} + (s+(s¿e_{-})e_{-}) as the sum of orthogonal 1-vectors each of square
± (s¿e_{-})^{2} . Assume that l = (s¿e_{-}) <0 , ie. that s is "forward" with regard to e_{-} .
Taking e_{-}' = (e_{-} + ds)(1-2dl)^{-½} for any
d > ½l^{-1}
we obtain
l' = s¿e_{-}' =
l (1-2dl)^{-½} which can be made arbitarily small
for large d (taking e_{-}' "closer" to s) or arbitarily large (negative) by taking d close to
½l^{-1} .
Thus means we can rechoose e_{-} to ensure l<-1 and having done so, setting
postive d=½l^{-1}(1-l^{2}) yields s¿e_{-}' = -1 .
Thus we can express any non zero forward nullvector as the sum of a unit forward minussquare vector and an orthogonal
unit plussquare vector.
This is profoundly important from a physical perspective since it implies null vectors cannot be meaningfully thought of as scaled.
We might think that the nullvector l(e_{-}+e_{+}) for l>1 is somehow "bigger" than e_{-}+e_{+}
but as we can find unit othonormal e_{-}' and e_{+}' such that l(e_{-}+e_{+}) = e_{-}'+e_{+}' , in what sense it bigger
"in kind"?
We can only speak of scale of nullvectors with referenece to another favoured (timelike, spacelike or null)
direction d not parallel to s, since d¿(l(e_{-}+e_{+}))= ld¿(e_{-}+e_{+}) then provides a d-dependant scalar measure of the "length" of s.
Note that the decomposition into orthonormal components is not unique, for by our construction above e_{-}' lies in the 2-plane spanned by
sÙe_{-} . If however we are given a nullvcetor s and an associated nonparallel 1-vector a this induces a unique decomposition of s
into e_{-}+e_{+} where e_{-} and e_{+} are orthornormal 1-vectors in sÙa.
Alternative spacetime representations
We can represent Â^{1,3} 1-vectors
in Â_{3} as x = x^{1}e_{1} + x^{2}e_{2} + x^{3}e_{3} + x^{4}
provided we use x^{»}¿x as our inner product
where ^{»} is a conjugation negating 1-blades but preserving scalars known in this context as space reversal
or conjugation. Clifford conjugation ^{§}^{#} negating only odd-grade blades is suitable.
A disadvantage of this approach is that it is that differing observers
disagree on the grades of blades. By incorporating a fourth 1-vector of opposite signature to the spacial
ones, we can "make geometry invarient" and keep conjugations such as ^{§} and ^{D} observer-independant.
Points as Events
We refer to points in spacetime as events. If we wish to specify a particular event
as being a spacetime location where something does (or did) "happen" we will capitalise as Event.
Note that a basis (N-frame) E provides only an orientation of spacetime.
It does not embody any form of "origin" or "base event".
We say an event p is "percieved" by "observer" E at 0 to have occured at "place"
^_{e4}(p) at "time" ¯_{e4}(p). That is, we decompose p as p=P+te_{4} where pÎe_{4}^{*} is perpendicuular to e_{4}.
[ Note that a realworld observer typically "becomes aware of", or
registers, an event
significantly after its "perceived" occurance due to finite signal speeds.
We attribute our observers with sufficient knowledge and computational acumen to calculate for
such delays and by "perceive" we really mean
"retrospectively deduces to have occured".
The theory of special relativity, now supported by significant experimental evidence, follows directly from the Minkowski model, without need of any reference to "light".
A 2D Minkowski Diagram (from Rindler). The dotted line represents the putative worldine of supralight (v^{2}>1) traveller. The indicated event is perceieved as (a,b) by observer E with frame (x,t) but as (a',b') by an observer E' with frame (x',t'). Hyperbolae denoted h and x_{i} represent events "unit seperated" (p^{2}=±1) from the common "origin" event agreed by E and E'. |
A worldine form of particular interest is
p(t) = (½tb)^{↑}p(0)(-½tb)^{↑}
with p'(t) = b×p(t) , where
b is constant (independant of t).
Picturing Minkowski Geometry
Planes and Simplexes
The first thing to note is that k-planes are still "flat". The solution set to
a^{1}x^{1}
+a^{2}x^{2}
+a^{3}x^{3}
-a^{4}x^{4}
= d
in Minkowski spacetime
looks a lot like the solution space to
a^{1}x^{1}
+a^{2}x^{2}
+a^{3}x^{3}
+(-a^{4})x^{4}
= d
in a Euclidean space.
Thus k-planes are still essentially slices of slices.
To imagine, say, a 2-simplex (a_{0},a_{1},a_{2}} _in Minkowski spacetime first ask what a 2-simplex is
geometrically. It is the convex hull containing 3 points, ie. a flat trianglar "facet" "hanging" in spacetime. The extended simplex is the 2-plane containing the triangle.
Let us first assume ((a_{1}-a_{0})Ù(a_{2}-a_{0}))^{2}<0 corresepnding to spacially seperated a_{0},a_{1} amd a_{2}.
We can then postulate an observer
Stationary Stan travelling at e_{4} normal to this 2-plane so that Stan percieves all three events
as simultanous. Stan sees a triangle appear instantanously and then vanish immediately. Imagine this
momentary triangle and colour the edges red and the interior area green. Put white dots at each vertex.
Because this triangle exists in a 2D subpace of 3D space we can forget about "height" and imagine
a stack of 2D timeslices only one of which is non empty. Imagine slowly sweeping a 3D 2-plane representing the
perception of another observer Olly whose worldline is only slightly deviated from e_{4}. Olly sees a white dot appear (event a_{0} say) which immediately diverges into
two red dots connected by a green line. These red dots drift apart
, the green line lengthening
, till suddenly one of dots flashes white and revereses its direction (event a_{1} say) and drifts back to rejoin
the other red dot at event a_{2}. Both dots vanishing in a white flash. It is important to recognise that
Olly sees these red dots as moving "faster than light".
We colour timelike edges pink rather than red and so imagine a spacial 2-simplex as appearing to any observers as an instantaneous white flash, a pink dot travelling
at sublight speed for a while before vanishing in a white flash. Particular observers see the pink dot as stationary.
Simultaneity and Chronological ordering
In the above, Olly also sees events a_{0},a_{1}, and a_{2} as spacially seperated,
but as non-simultaneous. He considers a_{0} to occur "before" a_{1} and both to occur "before" a_{2}. Another observer Pete
with a different velocity to both Stan and Olly might also see a_{0},a_{1} and a_{2} as non-simultaneou, but disagree on their choronolical ordering.
Perhaps percieving a_{1} as the "initial white flash", a_{2} as the "bounce", and a_{0} as the "merge and vanish" .
Observers differ on the chronological ordering of spacelike seperated events, but agree
on the chronological ordering of timelike seperated events.
Impossibility of FTL Signalling
The essential problem here is that any given spacelike vector is regarded as temporally "forward" by some observers and "backward" by others.
Consider two spacelike sperated events 0 and ve_{1} + e_{4} with
scalar v > d > 1. As percieved by E, it would require a speed v
for an object to pass through both events on a straight line trajectory.
Suppose an object does so, then explodes at ve_{1} + e_{4} as the result of a hidden bomb.
Now consider an origin-coincident observer F with f_{4} = (e_{4}+d^{-1}e_{1})~.
[ f_{4} is timelike since f_{4}^{2} = -1 +d^{-2} < 0 ]
F percieves the object as being intact at F-time 0 and the explosion as occuring at F-time
f^{4}¿(ve_{1} + e_{4}) < 0 prior to the time of intactness.
[ -(e_{4}+d^{-1}e_{1})~¿(e_{4}+ve_{1})
= (1 - vd^{-1})/|(e_{4}-d^{-1}e_{1})| < 0 ]
F does not have prior knowledge of the explosion at event 0, note. He cannot warn the object of it's doom
as they pass at _tE=_tF=0 . F only becomes aware of
the explosion having occurred when light from ve_{1} + e_{4} reaches him at a later positive F-time.
But what F deduces
to have occurred, figuring for the finite signal speeds, violates physical laws as he understands them;
a clock travelling with the object appears to F to run backwards. Thus anything capable of travelling faster than light
would have to be unchanging, or obey temporally reversible physical laws.
Suppose now that E sends sends a signal from 0 to ve_{1} + e_{4} recieved by origin-displaced
observer F having worldine
ve_{1} + e_{4} + t(e_{4}+d^{-1}e_{1})~ . Since F regards event 0 as happening at time
f^{4}¿(-(ve_{1} + e_{4})) >0 then if F is also allowed to send an FTL "responce" signal he could potentially relay
E's signal to event 0-de_{4} for a small positive d, causing E to recieve the "responce" before sending the original.
This is known as a temporal loop.
One way to prohibit such loops is to disallow FTL signalling. Another is to impose
a favoured temporal direction on space (eg. radially outwards from an agreed centre)
and allow only "forward" FTL signalling with regard to this direction, so preventing looping back.
This has the interesting effect of making the average "outward" velocity far greater than the average "inward" velocity,
leading to an expanding universe in the sense that matter gets further from the agreed centre,
Spacetime Multivectors
Hyperbolic Spinor Representation of 1-vectors
Suppose 1-vector x = x^{1}e_{1}+x^{2}e_{2}+x^{3}e_{3}+x^{4}e_{4} is forward timelike, ie. x^{2}
= -|x|^{2} < 0 ; x^{4} > 0 .
Let X
º X_{x,e4} º ^_{e4}(x) =
x^{1}e_{1}+x^{2}e_{2}+x^{3}e_{3}
so X^{2} = |X|^{2} = ^_{e4}(x)^{2} = (xÙe_{4})^{2} .
x^{~}^{4} º e^{4}¿x^{~} =
x^{4}|x|^{-1} = x^{4}((x^{4})^{2}-(x^{1})^{2}-(x^{2})^{2}-(x^{3})^{2})^{-½} ³ 1
has (x^{~}^{4})^{2} - 1 = |X|^{2}|x|^{-2} .
For x^{2} £ 0,
Ñ_{X}|x|^{k} = -kX|x|^{k-2}
and in particular Ñ_{X}x^{2} = 2X
Pure plussquare scaled 2-blade x º xÙe_{4} = Xe_{4} is sometimes refered to as a relative vector.
It anticommutes with e_{4} .
We define the natural magnitude of x by k_{x} º k º ln(|x|) = ¼ ln(x^{4}) whenever x^{2}¹0 .
We define the e_{4}-dependant scalar hyperbolic parameter or rapidity of
a forward timelike 1-vector x by
c º c_{x,e4} º
cosh^{-1}(x^{~}^{4})
= ln(x^{~}^{4} + (x^{~}^{4}^{2}-1)^{½})
= ln(x^{~}^{4} + |X||x|^{-1})
= ln(x^{4} + |X|) - ln(|x|)
[ ln(a)ºlog_{e}(a) denotes the scalar natural logarithm function ]
so that
x^{4} = cosh(c)|x|
; |X| = |x| sinh(c) .
x = |x|(-cx^{~})^{↑} e_{4}
= (k - cx^{~})^{↑} e_{4}
= (|x|^{½}(-½cx^{~})^{↑})_{§}(e_{4})
= (½k -½cx^{~})^{↑}_{§}(e_{4})
is the e_{4} specific hyperbolic spinor form
representation of events inside forward null cone L^{+}_{0} by
by positive scalar "length" |x|=(-x^{2})^{½}, positive scalar rapidity c,
and a unit plussquare 2-blade
x^{~} º (xÙe_{4})^{~} = X^{~}e_{4} = (Xe_{4})^{~} .
Ñ_{x}x^{~} = Ñ_{x}X^{~}e_{4} = (N-2)|X|^{-1}e_{4} where N=4 unless we add more spacial dimensions.
For spacelike x we have minusquare x^{~} and phase l º l_{x,e4} º cos^{-1}(x^{~}^{4}) and trigonometric spinor form x = |x|(-lx^{~})^{↑} e_{4} .
For null x we have c=¥ and require either |X| or x^{4} to recover x from
x^{~} .
c_{x4e4,e4}=0 and c_{x}®¥ as x^{2}®0.
c is a fundamental parameter for x because timelike 1-vector
Ñ_{x}c = |x|^{-2}(x^{4}X^{~} + |X|e_{4})
= |x|^{-1}(-cx^{~})^{↑} X^{~}
= |x|^{-2}xx^{~}
= |x|^{-2}x¿x^{~}
is normal to x ,
with x(Ñ_{x}c) = x^{~} .
Furthermore
Ñ_{x}^{2}c = (Ñ_{x}c)^{2} = |x|^{-2} is independant of our choice of e_{4}.
Ñ_{X}c = |x|^{-2}x^{4}X^{~}
has (Ñ_{X}c)^{2} = Ñ_{X}^{2}c = |x|^{-4}(x^{4})^{2} .
[ Proof :
¶ cosh^{-1}(a)/¶a = (a^{2}-1)^{-½} and
Ð_{e4}x^{~}^{4}
= |x|^{-1}(1-|x|^{-2}(x^{4})^{2})
= -|X|^{2}|x|^{-3} so
Ð_{ei}c = (x^{~}^{4}^{2}-1)^{-½}Ð_{ei}x^{~}^{4}
= (|X|^{2}|x|^{-2})^{-½}x^{4}Ð_{ei}|x|^{-1}
= (|X|^{-1}|x|) x^{4}x^{i}|x|^{-3}
= |X|^{-1} |x|^{-2} x^{4}x^{i}
for i=1,2,3
Ð_{e4}c = (x^{~}^{4}^{2}-1)^{-½}Ð_{e4}x^{~}^{4}
= -(X^{2}x^{-2})^{-½}|X|^{2}|x|^{-3}
= -|X| |x|^{-2} .
Ñ_{x}c = å_{i=1}^{3} e^{i}|X|^{-1} |x|^{-2} x^{4}x^{i} - e^{4}|X||x|^{-2}
= |x|^{-2}(|X|^{-1} x^{4}X + |X|e_{4})
= |x|^{-1}( cosh(c)X^{~} + sinh(c)e_{4})
= |x|^{-1}( - cosh(c)x^{~} + sinh(c))e_{4}
= |x|^{-1}( - cosh(c) + sinh(c)x^{~})x^{~}e_{4}
= |x|^{-1}(-cx^{~})^{↑} X^{~}
x(Ñ_{x}c)
= |x| (-cx^{~})^{↑} e_{4} |x|^{-1}(-cx^{~})^{↑} X^{~}
= e_{4}X^{~} = x^{~}
Ñ_{x}^{2}c
= å_{i=1}^{3} (|X|^{-1} |x|^{-2} x^{4}x^{i})^{2} - (|X||x|^{-2})^{2}
= (|x|^{-2}x^{4})^{2} - (|X||x|^{-2})^{2}
= (|x|^{-2})^{2}(x^{4}^{2} - |X|^{2})
= |x|^{-2}
.]
The e_{4}-speed of a 1-vector x is (e^{4}¿x)^{-1} ^_{e4}(x)^{2} . It is xero
for x=±e_{4}, and 1 for null x, it is independant of the scale of x. It is 1 for null x and less than 1 for x^{2}<0.
Canonical Rotor Form of Even Spacetime Multivectors
Before discussing the physical implications of Minkowski spacetime we will make an important
mathematical observation. The even subalgebra Â_{3,1 +} is equivalent (isomorphic) to Â_{3} since the former is generated by
orthornormal anticommuting bivector "basis" {e_{14},e_{24},e_{34}} with
e_{14}^{2}=e_{24}^{2}=e_{34}^{2}=1 .
Any nonnull even R_{3,1} multivector b can be uniquely expressed as
b = R(re^{ib})^{½} = (re^{ib})^{½}R
where
(re^{ib})^{½} º r^{½}( cos(b/2)+i sin(b/2))
for scalar r, b
and R=b(bb^{§})^{-½} is a unit rotor (ie. an even multivector
satisfying R^{§}=R^{-1}).
[ Proof :
(bb^{§})^{§} = bb^{§} Þ (bb^{§})_{<2>}=0 .
For even bÎ R_{3,1} , bb^{§}
= b_{<0>}^{2} + b_{<4>}^{2} - b_{<2>}^{2}
which remains even so the absence of a bivector component allows
bb^{§} = re^{a+ib}
Þ (bb^{§})^{-½} = r^{-½}e^{-½(a+ib)}
which commutes with all even
multivectors and so with R
.]
This is known as the canonical form or invariant decomposition
of b .
Since i^{§}=i and i (anti)commutes with (odd)even multivectors we have
b_{§}(a) º bab^{§}
= re^{ib}R_{§}(a_{<+>})
+ rR_{§}(a_{<->})
" a .
For nonnull pure bivector b_{2} we have b_{2}b_{2}^{§} = -b_{2}^{2} so that
b_{2} =c_{2}e^{if}
with timelike bivector c_{2} = r^{½}R and scalar f=½b.
A null bivector b_{2} can be expressed as b_{2} = sce^{if}
where s is a null 1-vector and c a unit spacelike (c^{2}=1) 1-vector orthogonal to s .
Canonical Exponentiated Form of Even Spacetime Mutivectors
In Â_{3,1} we have three distinct forms of 1-spinor:
e^{qa} =
cosq + a sinq if a^{2} = -1 ;
coshq + a sinhq if a^{2} = 1 ;
and 1+a if a^{2} = 0.
Most Â_{3,1} rotors can be expressed as
R=e^{½a2} for nonnull (possibly nonunit) bivector a_{2}, the remainder require
R=±e^{½a2}
=±(1+½a_{2})
with a_{2}^{2}=0.
For a_{2}^{2} ¹ 0 we have a_{2}=l(ae_{1234})^{↑}c_{2} = l( cos(a)c_{2} + sin(a)e_{1234}c_{2})
where l=|a_{2}|, decomposing a_{2} into commuting plussquare and minussquare 2-blades.
for scalar a and unitsquare 2-blade
c_{2} .
Most Â_{3,1+} multivectors can thus be expressed
in spinor-factored or canonical exponentiated form
as
b= (r(ib)^{↑})^{½}R
= r^{½}(½bi)^{↑}(½l(ia)^{↑}c_{2})^{↑}
= (½(r^{↓} + bi + l cos(a)c_{2} + l sin(a)ic_{2}))^{↑}
for 2-vector c_{2} with c_{2}^{2}=1 and positive scalars r,l,a.
= (½(r^{↓} + bi + qc_{2} + fic_{2}))^{↑}
where
q = l cos(a) and f = l sin(a) .
Spinor (½l cos(a)c_{2})^{↑} is known as the boost factor of the rotor
since its effect taking c_{2}=e_{34} is an accelartion in the e_{3} direction.
[ we can safely "combine" the exponentiations since all terms commute, c_{2} being even.
]
We then have
b^{↓} = ½(r^{↓} + bi + l cos(a)c_{2} + l sin(a)ic_{2})
A "degenerate" subclass of Â_{3,1+} multivectors
has the form
b = a(½bi)^{↑}(1+a_{2}) = (a^{↓}+½bi+a_{2})^{↑}
for null 2-vector a_{2} and two scalars a,b
and satisfies
b^{k} = a^{k}e^{½kbi}(1+2^{k}a_{2})
;
b^{-k} =
a^{-k}e^{-½kbi}(1-2^{k}a_{2})
for integer k³0 .
Â_{4.2} Genralised Homegenised Minkowski Spacetime
We can extend Â_{3,1} by orthogonal null vectors e_{0},e_{¥} in like manner to
the Generalised Homegenised extension
Â^{N}^{%} Ì Â^{N,1}
of Li et al and the arguments and proofs in our discussion of the Generalised Homeogenised Euclidean Space
carry into the Minkowski case Â_{3,1}^{%} Ì Â_{4,2} without difficulty to give
Generalished Homogenised Minskowski Spacetime or GHMST aka.
conformally extended spacetime.
Most generally, we extend N-D space U_{N} with basis { e_{1},...,e_{N} }
by two orthogonal vectors e_{+} and e_{-} satisfying
e_{+}^{2} = 1;
e_{-}^{2} = -1;
e_{+}¿e_{-} = 0
to form an (N+2)-D space U_{N}^{%} with basis
{ e_{0},e_{¥},e_{1},...,e_{N} } where
e_{0} = ½(e_{-} - e_{+}) ;
e_{¥} = e_{-} + e_{+}
are null vectors. We have bivector
e_{¥0} º e_{¥}Ùe_{0} = e_{+}Ùe_{-} =
e_{+}e_{-} with properties e_{¥0}^{2}=1 ;
e_{¥0}^{§} = e_{¥0}^{§}^{#} = -e_{¥0} ;
and "absorbtion" by the nullvectors:
e_{¥0}e_{¥} = -e_{¥}e_{¥0} = -e_{¥} ;
e_{¥0}e_{0} = -e_{0}e_{¥0} = e_{0} .
We define e^{0} º -e_{¥} ; e^{¥} º -e_{0}. These satisfy
e^{0}¿ e_{0} = e^{¥}¿e_{¥}=1 but also e^{0}Ùe_{0} = - e_{¥0} ; e^{¥}Ùe_{¥} = e_{¥0}
so serve only as "quasi-inverses" for e_{0} and e_{¥}.
If i=e_{1}Ù...Ùe_{N} is a unit pseudoscalar for U_{N}
then e_{¥0}i is a unit pseudoscalar for U_{N}^{%}.
We can express any 1-vector x Î U_{N}^{%} as
x + ae_{0} + be_{¥} where x Î U_{N}.
Our embeding is once again
x = ¦(x) = x + e_{0} + ½x^{2}e_{¥}
with inverse mapping
x = (e^{0}¿x)^{-1} ^_{e¥0}(x) =
(e_{¥}¿x)^{-1} (e_{¥0}Ùx)e_{¥0} when (e_{¥}¿x)¹ 0 ;
and x=^_{e¥0}(x) when (e_{¥}¿x)= 0 .
This time, for U^{N}=Â^{3,1}, we are mapping into three distinct geometric spaces.
Spacelike x map to the e^{¥}¿x>0 half of the horosphere ;
timelike x map to the e^{¥}¿x<0 half of the horosphere ; while
null x map to L_{0} + e_{0} as x ® x+e_{0}.
In particular, e_{0} corresponds to an arbitary U_{N} origin 0, while
e_{¥} represents a hypothetical U_{N} point at infinity ¥.
If a,b are the null U_{N}^{%} 1-vectors associated with U^{N} points (events) a and b, we have:
a¿b =a.b = -½(a-b)^{2}
= -½(a-b)^{2} ;
aÙb = aÙb +½(a^{2}b-b^{2}a)e_{¥}
+ (a-b)e_{0} - ½(a^{2}-b^{2})e_{¥0}.
As for Â^{N}^{%}, if a_{0},a_{1},...a_{k} are k+1 1-vector points (events) in U^{N} then
e_{¥}Ùa_{0}Ùa_{1}Ù...Ùa_{k}
= e_{¥}Ù(a_{0} + e_{0})Ùa_{k}
represents the U_{N} extended simplex {a_{0},a_{1},...a_{k}}.
For k=3 the simplex extends into the 3-plane through a_{0} with tangent a_{3} = (a_{1}-a_{0})Ù(a_{2}-a_{0})Ù(a_{3}-a_{0}) .
In Â_{3,1}, if a_{3}^{2}<0 with then the dual
a_{3}^{*} is a timelike 1-vector representing the wordline direction of an observer percieving
all four events as cotemporaneous. The simplex itself is percieved by this observer as a convex hull of 4 cotemporaneous points, ie. an instantanoues tetrahedral volume.
Further,
(e_{¥}Ùa_{0}Ùa_{1}Ù...Ùa_{k})^{2}
=... = ((a_{1}-a_{0})Ù...Ù(a_{k}-a_{0}))^{2} giving the square of k! times the content of the simplex.
As for Â^{N}^{%}, a U_{N}^{%} (k+1)-blade a_{k} = a_{0}Ùa_{1}Ù...Ùa_{k} derived from events a_{0},a_{1},..,a_{k} has geometrical interpretations:
We attach no physical significance to the e_{+} and e_{-} dimensions, considering them as wholly abstract constructs
facilitating the computation of intersections.
Â_{5.2} Spacially Extended GHMST
For reasons which will become clear later, we will extend Minkowski space time by a fourth spacal dimension
to which we will regard as physically significant rather than a mere computational aid. We thus regard the physical universe as being modelled by
a structure in Â_{4,1} and we add general homogenising dimensions e_{0} and e_{¥} as before
to represent k-spheres and k-planes in Â_{4,1} as (k+2)-blades in Â_{5,2}.
We will denote the fourth spacial dimension with e_{5}, retaining e_{4} for the timelike axis, rather than
taking e_{4} spacelike and e_{5} timelike.
The Â_{4,1} unit pseudoscalar e_{12345} commutes with everything in Â_{4,1}, squares to -1, and reverses to itself.
The Â_{5,2} unit pseudoscalar e_{¥0}e_{12345} commutes with everything in
Â_{5,2}, squares to -1, and is negated by reversion.
Since any multivector in Â_{4,1} can be expressed as a+be_{12345}=c+de_{1234} for a,b in Â_{4} and c,d in Â_{3,1} we have Â_{4,1} isomorphically equivalent to both C_{4} and C_{3,1} with e_{12345} taking the role of central complex scalar i. Note that such a "dimension reducing complexifications" are e_{4}- or e_{5}-dependant.
It is worth explicitly noting that Â_{4,1} "contains" Â_{1,3} as the subgroup generated by the four anticommuting 4-blades { e_{1}^{*},e_{2}^{*},e_{3}^{*},e_{4}^{*} } where e_{i}^{*} = e_{i}e_{12345}^{-1} .
For now we will ignore our "fifth dimension" and assume that all multivectors have zero
e_{5}¿a=0 unless expliticly stated otherwise, ie. the coefficient of any blade including e_{5} is to be assumed zero.
k-spheres and Nullcones
Nullcones
We refer to L^{-}_{c} = { r : (r-c)^{2} = 0 & (r-c)¿e_{4} > 0 }
as the rear null cone (aka. rear lightcone) at c. If we extended U_{N} into U_{N}^{%} using generalised hompgenised coordinate
extenders e_{0} and e_{¥} in the usual way, then L^{-}_{c} is represented by null (N-1)-blade
c^{*} = (e_{0}+c+½c^{2}e_{¥})^{*}
We refer to L^{-}_{c}^{·} = { r : (r-c)^{2} £ 0 & (r-c)¿q > 0 }
as the filled rear nullcone at c with regard to an understood timelike direction q.
We refer to L^{-}_{c,q,l}^{·} = { r : (r-c)^{2} = 0 & (r-c)¿q = l }
as the l-past at c for timelike direction q.
In a flat (homegeneous) Â_{3,1} space all rear nullcones are displacements of
the origin rear nullcone L^{-}_{0}.
L_{p} = (p+e_{0}+½p^{2})^{*} in Â_{3,1}^{%} embodies the nullcone at p
and in particular L_{0} = e_{0}^{*} = e_{0}i^{-1} respresents the nullcone at 0 .
We can intersect with this nullblade by lifting into Euclidean space, but projection is problematic.
¯_{_wigwog}(p) simply rescales p such that p^{2}=d so
Lim_{d ® 0} ¯_{ Od,c}(p) = c.
Given a timelike unit d nonparallel with v=p-c we have v=ad + V
where a=-d¿v and V=v-ad is nonzero and orthogonal to d. We can then decompose
v = (a+|V|)(d+V^{~}) - (aV^{~}+ad) into orthogonal components and
define _prl3[p,c,d] = ¦(c + (a+|V|)(d+V^{~})) and
_Rflct2[ L_{c},d](p) = ¦(c + |V|d + aV^{~}) as d-directed projections and reflections in L_{c}.
Dirac D Function
The Dirac D function Â_{p,1} ® Â
is defined by
D(x) º 2d(x^{2}) S_{ign}(e^{4}¿x) , ie. (somewhat informally) zero off the nullcone
L_{0} , +¥ on the forward nullcone L^{+}_{0}, -¥ on the rear null cone L^{-}_{0} and zero at 0.
Though we have defined it with reference to a particular e_{4}, D(x) is frame-independant in the sense that any
other "forward" e_{4}' with e_{4}¿e_{4}'< 0 generates the same D(x).
S_{ign}(x) = x|x|^{-1} is defined to be 1 for x>0, -1 for x<0, and 0 for x=0.
d(x) is the scalar Dirac delta function which can be loosely defined
as d(x) = _epslon^{-1}(1-e^{-1}|x|) for |x|£e and 0 for |x|³e
for very small e.
D(x) = |X|^{-1}( d(x^{4}-|X|) - d(x^{4}+|X|) )
[ Proof :
ò_{-¥}^{¥} dx¦(x)d(x^{2}-a^{2})
=
ò_{-¥}^{0} dx¦(x)d(x^{2}-a^{2})
+ ò_{0}^{¥} dx¦(x)d(x^{2}-a^{2})
= ò_{¥}^{-a2} dy¦(-(y^{2}+a^{2})^{½})½(-(y^{2}+a^{2})^{½})^{-1}d(y)
+ ò_{-a2}^{¥} dy¦((y^{2}+a^{2})^{½})½((y^{2}+a^{2})^{½})^{-1}d(y)
= ½(¦(-a)a^{-1}
+ ¦(a)a^{-1}) = ½a^{-1}(¦(a)+¦(-a) .
Hence since ¦ is arbitary we have
d(x^{2}-a^{2}) = ½a^{-1}(d(x-a) + d(x+a)) .
Whence d(x^{2}) = d(-x^{2}) =
½|X|^{-1}(d(x^{4}-|X|) + d(x^{4}+|X|)) and the result follows.
.]
It can be shown [ Dirac 75.23 ] that for p=3 we have
ò d^{4}x D(x) (i k¿x)^{↑}
= 4p^{2}i D(k)
with geometric form
ò d^{4}x D(x) (b k¿x)^{↑}
= 4p^{2} D(k) ib for any b with b^{2}=-1.
Since k^{2} D(k)=0 we obtain
Ñ_{x} D(x) = 0.
Tspheres and Tballs
We will here refer to the instersection of a "filled" hypersphere or "ball" { p : (p-p_{0})^{2} £ r^{2}} in a Minkowski space with a nonspacial k-plane (ie. a k-plane containing a timelike 1-vector) as a k-tball. We will refer the the boundary of a k-tball (the "surface" of the k-sphere) over which (p-p_{0})^{2} = r^{2} as a k-tsphere . We will reserve the terms k-sphere amd k-ball for structures in Euclidean (sub)spaces.
A (N-1)-tsphere or hypertsphere in Â_{N,1} with centre event c and
positive radius r is a hypercurve, the set of all events satisfying
(p-c)^{2}= r^{2}.
All observers percieve it in the same way. Simultaneous with event c they percieve a spacial 2-sphere
(ie a 3D "globe") of radius
r centre c and surface area 4pr^{2} and volume (4/3)pr^{3} which then increases in radius at an ever increasing sublightspeed. At time ±t,
they percieve a (spacial) 2-sphere of radius (r^{2}+t^{2})^{½}
centre c±te_{4} . A 3-tsphere centre c is thus percieved by all observers as
a 2-sphere of infinite radius at time t=-¥ that contracts to radius r at time t=0 and then expands again
as t increases, always centred at ¯_{e4*}(c) where e_{4} is observer specific. The content of a
tsphere is infinite.
A k-tsphere of radius R is the intersection of a hypertsphere (p-c)^{2}=R^{2}
with a nonspacial (k+1)-plane
. For any r³R we
can find a hypertsphere of that radius which will serve but it is natural to specify r=R so that
the k-tsphere is a "maximal slice" of the hypertsphere. We thus regard a k-tsphere of radius R centre c as the
intersection of a hypertsphere of radius R centre c and a (k+1)-plane through c.
An observer with timelike e_{4} normal to the plane will observe an instantaneous spacial (k-1)-sphere contemporaneous with centre event c.
An observer with spacelike e_{3} normal to the plane will observe a phenomena constrained spacially to plane e_{124}
through c.
The intersection of a k-tsphere with a spacial (l+1)-plane (ie. a (l+1)-plane
whose tangent (l+1)-blade spans no timelike 1-vectors) for m<l is a l-sphere.
A 3-tsphere of negative radius -r<0 and centre c is the set { p : (p-c)^{2}=-r^{2} } .
This has two components or pieces and divides spacetime into three regions, wherwas positive r hypertsphere have one component and divide spacetime into two regions.
O^{+}_{-r,c} is seen by all nonaccelerating observers as a sphere
of radius ¥ at t=-¥ that contracts at increasing supralightspeed to radius 0 at
t=-r before event c whereupon nothing exists until
t=+r after c when a spacial 2-sphere of zero radius appears, expanding at decreasing supralightspeed to infinite radius at t=¥.
At t=r_{0}±d the radius is (2r_{0}d+d^{2})^{½}.
We can regard O^{+}_{-r,c} as the set of all points reachable by a straight-line travelling particle
starting from event c with any timelike unit velocity v after time r as measured by the particle.
Although O^{+}_{-r,c} approaches the nullcone L^{+}_{c} for large e^{4}¿(p-c) its content is infinite.
We have a more general form of the Dirac Delta function
D_{r,c}(x)
º 2d((x-c)^{2} + r^{2}) S_{ign}(e^{4}¿(x-c))
= = (X^{2}+r^{2})^{-½}(
d(x^{4}-(X^{2}+r^{2})^{½})
- d(x^{4}+(X^{2}+r^{2})^{½} )
which is zero off O^{+}_{-r,c} and ±¥ in the forward/backward components of the hypertsphere.
Extending to Â_{4,1} with x^{5}=_cu5=0 we have
D_{r,c}(x) = D(x-c-re_{5}) .
The 3-tsphere of zero radius and centre c is of course the nullcone at c so O_{0,c} = L_{c} . A k-tsphere of zero radius, centre c is the projection of L_{c} into a k-plane through c.
1-tsphere
A 1-sphere is a ring in spacetime with timelike axis n. There are observers who percieves an instantaneous red hoop but most see two red particles (no white flash) which appear together but fly apart following two symmetic trajectories at FTL speeds to recombine slightly more than 2S from their appearance point a moment later. S
A 1-tsphere of postive radius is a timelike 1-curve
p(t) =
S(S^{-1}te_{43})^{↑}e_{3}
= S(½S^{-1}te_{43})^{↑}_{§}(e_{3})
= S( cosh(S^{-1}t)e_{3} + sinh(S^{-1}t)e_{4}) .
p(t)^{2} = S^{2} ; p'(t)^{2}=-1
which is the natural parameterisation of a
timelike path of a particle maintaining a constant
seperation S from a particular event 0. It is e_{4}-percieved as a trajectory coming from
¥e_{3} at time t=-¥ having nearlight velocity towards O ,
subject to a uniform repulsive bivector forcefield
W_{p} = W_{p} = p"(t)p'(t) = S^{-1}e_{34} and passing
spacially closest to O at Se_{3} at t=0 before moving back off towards ¥e_{3} again, approaching the lightcone L^{+}_{0}
as it constantly accelerates towards lightspeed.
The future hemitspherical half of a 1-tsphere of negative radius
is the naturally paremeterised spacelike path
p(c) = S(S^{-1}ce_{43})^{↑}e_{4}
= S(½S^{-1}ce_{43})^{↑}_{§}(e_{4})
= S( cosh(S^{-1}c)e_{4} + sinh(S^{-1}c)e_{3})
has p(c)^{2} = -S^{2}
;
p'(c) = (S^{-1}ce_{43})^{↑}e_{3}
;
p'(c)^{2} = 1 and can be viewed either as the path of a FTL particle maintaining a constant
seperation S from event 0 (passing through event Se_{4} at c=0)
or (for -¥ < c < ¥ ) as the loci of all possible points
reachable by a particle travelling in a straight timelike trajectory from event 0 after time S
as measured by the particle.
A (N-1)-tball or hypertball of centre c and radius S is the set
{ p : (p-c)^{4} <S^{4} } . It is accordingly a "solid" rather than a "surface"
and acts like a "thick null cone" for small S. It contains the frame-dependant "hermitian sphere"
p^{†}p = |p|_{+}^{2} < R^{2} where ^{†} is Hermitian conjugation
associated with any given frame and so can be thought of as a "small event" .
Imposing a time frame e_{4}, at time t a hyperball is percieved as { pÎÂ^{3} :
(p^{2} - t^{2})^{2} £ S^{4} } . For t>S this is a 2-spherical shell of inner radius
(t^{2}-S^{2})^{½} and outter radius (t^{2}+S^{2})^{½} and consequent thickness
t((S/t)^{2} + 24(S/t)^{6} + _{O}((S/t)^{10}))
= t^{-1}S^{2} + _{O}(t^{-5}S^{6}).
The content of this shell is approximately 4pt^{2} t^{-1}S^{2}
= 4ptS^{2} .
Next : Helices