M-field Ip can be expressed in UN via NCM scalar fields
iij..mp = eij...m¿Ip though the normalisation
condition renders one such field redundnat apart from sign. In the mapspace ¦Ñ-1(Ip) is everywhere the map pseudoscalar.
In UN the projector 1-tensor ¯ can be expressed with regard to a universal basis as N2 scalar fields although with regard to
an extended tangential frame at p it can be regarded as an identity matrix having the last N-M terms in the lead diagonal zeroed.
The Riemann curvature tensor cpaÙb, as a bivector valued function of bivectors, constarined within and upon Ip requires MC22 = ¼M2(M-1)2 scalar fields in theory although the Bianchi symmetries reduce the generality considerably. At a fixed given p Î CM, mapping bivectors in Ip to bivectors in Ip induces the obvious Cx(aÙb) = ¦-Ñp(cp¦Ñp(aÙb)) 2-tensor acting in the mapspace which we can represent with ¼M2(M-1)2 scalar fields Cklij º ekl¿Cx(eij)
The alternate view of the Riemann curvature as a bivector specific 1-tensor , or a (1;3)-tensor
antisymmetric in two of its arguments, provides the alternate coordinate form
Ckijl º ek¿(Cx(eij)×el)
cpaÙb and Cx(aÙb) are alternate representations of the same "thing",
although cp defined over subspace Ip of UN can more readily be
"extended" out of CM.
We can contract the curvature tensor to obtain
the Ricci 1-tensor)
cpb º Ña¿cpaÙb
= åi=14 ei¿cpeiÙb
= åi=14 ei¿
(( Я(b)wei) - ( Я(ei)wb) + (wb×wei))
, a point-dependant symmetric 1-vector-valued function of 1-vectors
. Note that the divergence is with respect to direction rather
than a point and is computable entirely "at" x.
The mapspace representation of the Ricci 1-tensor consists of M 2 scalars
Ckj º ek¿ Cx(ei)
= åj=1M Ckjij
or with Cij º
åk=1M Ckijk
which itself contracts to the scalar curvature
R = åi=14 Cii .
The mapspace Einstein tensor is given by M2 scalar fields
Gkj º
Ckj - ½R .
Streamline Coordinates
Introduction
The following is a more traditional approach to manifold calculus which we include for completeness.
This section can safely be skipped by readers interested only in multivector methods.
Suppose we have a 1-field vp defined over M-curve CM such that
vp Î Ip .
It follows from a mathematical result concerning the unique solvability of ordinary differential equations
that, provided vp is sufficiently smooth and is nowhere 0-valued, for every p0ÎCM
there is a unique "tangent matcher" path in UN for vp :
Ci p =
{ p(t) : t Î [t0,t1] } [ where t0<0<t1 ]
with p(0)=p0 ;
p(t) Î CM and (¶/¶t)p(t) = vp(t) " t Î [t0,t1] .
( p(0)=p0 ensures that the parameterisation is unique up to choice of domain range [t0,t1])
Such paths are called integral curves or streamlines for vp.
We will use both terms
interchangeably here. Streamlines cannot intercept eachother
(other than at points with vp=0 which we have disallowed) .
Since every pÎCM has such a curve, the set of all
integral curves for each vp (known as a congruence for vp) "fills" and effectively defines CM.
It is important to note that the 1-curve streamlines of field vp are independant of the magnitude of vp,
(given vp2 ¹ 0) since it is the unit tangent that defines a 1-curve.
Suppose now we have M not-necessarily mutually orthogonal unit 1-fields ui p defined over M-curve CM satisfying
u1(p)Ùu2(p)Ù..uM(p) = apIp
" p Î CM where scalar field ap > 0 " p .
An obvious example is ui p=¦Ñx(ei)= hip where CM=¦(Map) is a point embedding specification for CM.
Given a general multivector-valued function F(p) defined over CM ,
ui p generates
at each pÎCM a specific "streamlined" derivative of F which is the tangential derivative
within the 1-curve ui p streamline. We will write this as
dli(F(p)) º ¶F(pi(li))/¶li . We will sometime ommit the
brackets and write dliF(p).
If we so differentiate the identity map F(p)=1(p)=p we recover ui p
and, indeed, one can think of 1-field ui p as being the
streamlined derivative operator field dlip .
We will accordingly use the symbol
Я(ui p) interchangeably with dli
.
We introduce the notation Dli¬i p to indicate the point pi(Dli) Î CM reached
by moving from p by arclength Dli along the li (ie. the ui p) streamline through p.
Dli¬i p º
òCi p [0,Dli] 1 dp = ò0Dli pi'(li) dli
We can extend Dli¬i to act on general multivector fields by
Dli¬i F(p) º F( Dli¬i p) .
This defintion is integrative, good for all Dli within the neighbourhood of ¦ applicability.
For small Dli=e we can apply Taylor's Theorem to the streamline li-parameterisation and obtain
F( e¬i p) = eedliF(p)
and in particular
[ Set F(p)=p ]
we have
e¬i p = eedli(p) which we can write as
e¬i = eedli .
We might be tempted to try to parameterise CM using y:Map®CM defined by
y(Dl1, Dl2, ..Dlm, q0) º
Dlm¬M Dlm-1¬M-1 ... Dl2¬2 Dl1¬1 q0 =( )=
Dlm¬M ( Dlm-1¬M-1 (... Dl2¬2 ( Dl1¬1 q0))..) .
[
That is, starting from a fixed reference point q0ÎCM
we follow the v1(p) stream line for arclength Dl1 to a new point
Dl1¬1 q0 from which we follow the v2(p) streamline for arc length Dl2
to Dl2¬2 Dl1¬1 p0 and so forth.
After M such steps we arrive a final point
qMÎCM which we can associate with parameters
(Dl1,..,Dlm).
]
Any M scalars (sufficiently small that we stay within the neighbourhood) interpreted in this way define a point in CM,
but not necessarily uniquely. If it is possible to reach the same point by more than
one such route, then the mapping is noninvertible and does not provide a true coordinate system for CM.
Only if the dli (and hence the Dli¬i ) everywhere commute with eachother do
the li provide a true coordinate system.
[ Proof :
...
.]
( e¬i × e¬j ) is a 1-field defined over CM returning
a UN 1-vector
( e¬i × e¬j )(p)
= (eedli × eedlj)(p)
= ...
= e2(dli×dlj)(p) + O(e3) .
Taking e®0 we can visualise (dli×dlj)(p) º [ ui p,ujp ](p)
as the 1-vector difference at p between following i and then j as opposed to j and then i streamlines
by small arclength e.
Lie Derivative
We make very little use of Lie derivatives in this work. They are described here only for completeness.
This section can safely be skipped by readers interested only in multivector methods.
Lie Bracket
The Lie Bracket is a highly general mathematical construct. With regard to
1-fields in an M-curve we define it by
[ui p,ujp](v) º dli(dlj[v])
- dlj(dli[v])
= 2(dli×dlj)v
which we can write as
[ui p,ujp] = dliujp - dljui p .
The use of × above is semantically legitimate if we extend the definition of
commutator multivector product × to operators
in the obvious appropriate way.
Lie Drag
Any one of our ui p generates a congruence for CM so let us fix on ui p as a "preferred" congruence
with associated dervivative field dli .
Consider two points p0 and p0 in CM with p0= Dli¬i p0 for small Dli>0.
Clearly p0= -Dli¬i p0 .
For j¹i, the specific ujp streamline through p0 (i¹1) at p with
associated directed derivative dlj generates a distinct path
pj*(lj) = -Dli¬i pj(lj) passing through point p0 = -Dli¬i p0
known as the Lie drag
of ujp .
[ Here * indicates "new" or "modifed" rather than a multivector dual
]
. This path need not be the streamline at p0
of any of our uk p 1-fields and, in particular, it need not be the
lj streamline there .
It is however, a streamline having at
p0 a tangent vector written (iÝ-Dli(ujp))(p0) fully within Ip0 and an associated
streamlined differential operator
dlj* = (iÝ-Dli(dlj))(p0) .
Though dlj* is not one of the dlk, it is not general. Because of its construction it commutes with
dli (ie. dlj*(dli(F))) = dli(dlj*(F))) .
[ Proof :
....
.]
We refer to
(iÝ-Dli(ujp))(p0) ( or (iÝ-Dli(dlj))(p0) ) as
the Lie drag of ujp ( or dlj ) from p0 to p0= -Dli¬i p0 .
We define the Lie drag of a scalar field f(p) defined over CM by
(iÝ-Dli(f))(p) º f( -Dli¬i p)
so that
(iÝe(f))(p) = eedlif
. 7
Lie Derivative
The 1-vector Lie derivative of a scalar field is simply £dlif(p) = dlif(p) .
Since the Lie drag of ujp (or dlj) from p0 to p0 can
be meaningfully subtracted from ui p (or dli) at p0 .
The Lie dervivative at p0 of ujp with regard to
1-field ui p is an operator defined (by its action on
a general 1-field F) as
( £dli(dlj))F(p)
º LimDli ® 0 [ ( (iÝ-Dli(dlj))(p0)(F(p)) - dlj(F(p)) ) / Dli ] ïp0
= LimDli ® 0 [2(dli×dlj)F ïp0 ]
[ Proof :
dlj*F ïp0
= dlj*F ïp0 - Dli(dli(dlj*(F))))ïp0 + O(Dli2)
= dlj*F ïp0
+ Dli(dli(dlj(F))ïp0 + O(Dli2)
- Dli(dlj*(dli(F))ïp0 + O(Dli2)
Hence
LimDli ® 0 [ ( (iÝ-Dli(dlj))(p0) - dlj )F / Dli ] ïp0
= LimDli ® 0 [((dlj* - dlj )F ïp0 ) / Dli ]
= LimDli ® 0 [(dli(dlj(F)) - dlj*(dli(F)) ) ïp0 ]
which provides the result assuming dlj* can be safely replaced in the limit by dlj
.]
Omitting the brackets , we thus have the 1-vector operator identity
£ui pujp = £dlidlj = 2 dli×dlj = [ ui p,ujp ] .
Covariant Frame
Let us now identify the spaces UN and VN and adopt fixed frames
{ei} and {ei} in both.
With regard to ¿ within VN , {ei} is not a reciprocal frame for {ei}. Rather, we need
the point-dependant (nonorthormal) covariant frame
{e i º gx-1(ei)} which itself has ¿ reciprocal frame
{ei º gx(ei)}
[ Proof :
e i¿ej = gx(gx-1(ei))ej = ei¿ej
Also e i¿ej = gx-1(ei)¿gx(ej) = gx(gx-1(ei))¿ej = ei¿ej
.]
For brevity we ommit an x subscript from ei , e i , and ¿ .
The underline serves to remind us
of the point-dependance.
Covectors
Any "inner product" ¿ is a (0;2)-tensor (a potentially point-dependant bilinear scalar-valued function
of two 1-vectors in VM) . For any 1-field ( (1;0)-tensor) vp , ¿ induces a scalar-valued directional function
( a (0;1)-tensor)
vp(a) º a¿vp known as a covector field.
We can recover vp at a given p from its covector as vp = Ñavp(a) . A vector
and its covector are essentially alternate representations of the same "thing". If we "feed"
vp "to" its covector we get "squared magnitude" 0-tensor vp(vp) = vp¿vp
For ¿ in ÂN , if 1-vector vp is regarded as a 1×N matrix
"column vector"
, then
its covector is the N×1 matrix "row vector" (or transpose) vpT .
For ¿ in Âp,q , we take the transpose
of the conjugate vp .
With regard to ¿, we will here define the 1-covector of vp = åi=1N viei as the 1-vector
vp£ º åi=1N viei so that vp£ ¿+ a = vp¿a .
In a Euclidean space, vp£ = vp .
(1;0)-tensor vp has "contravariant" coordinate representation åiviei where vi º ei¿vp .
(0;1)-tensor vp has "covariant" coordinate representation åivie i where vi º vp(ei) = ei¿vp
and e i is the 1-covector of ei which we associate with the 1-vector e i.
Parallel transport
Suppose we were to drag heavy iron girders around the (assumed perfectly spherical) Earth for a bet.
We might start at the
North Pole, with two girders girder (assumed straight and with ends clearly distinguished) lying "horizontal" on the ground ; perpendicularto a
"vertical" flagpole we assume to mark the Pole and parallel to eachother.
Since the globe is spherical, the girder is actually balanced tangentially on a single point
but the radius is so vast that the ground is for all girder-related purposes locally flat.
Both gripping one end we start to drag the girder along a meridian, always pulling due South, that is, allowing no rotation in
the local "horizontal" tangent plane. When we reach the equator, or girder has retained its length
and is still facing South (local coordinates) but, speaking 3-dimensionally, it is now parallel to the flagpole we leftbehind.
If we then push the second girder sideways from the North Pole without rotation the girder maintains an East-West
alignment. Once we reach the equator (at
a different point to previously) we then drag the girder "lengthwards" ¼ of the way
round the equator to join the first girder, to which it is now perpendicular.
Moving a 1-vector along a curve while keeping it "held within" an M-curve in this way is known
as parallel transport in the M-curve. The example shows that it preserves length but not direction. If we parallel
transport a 1-vector from p to q within an M-curve then the result is dependant on the path taken.
Parallel transport is invoked to provide a notion of "parallelism" for 1-vectors in different tangent spaces
(ie. 1-vectors "at" different points of an M-curve) that does not require a higher-dimensional embedding space.
The "distance" between two points on a manifold can be meaningfully defined as the arclength of a minimial-arclength
path joining the points (a geodesic), but to compare distant directions we need for every qÎCM
a function
qp C(a) Î Ip giving the tangent vector at pÎCM "parallel" to aÎIq at q
ÎCM ; as obtained by
"parallel transportation" of a "along" a given path C from q to p (eg. along a minimal length geodesic).
Let iDlip0(a) denote the parallel transport of a from p0 for arclength Dli along the ui p streamline.
We obviously want
iDlip0(a)ÙI Dli¬i p0 = 0 and we might
therefore think of defining
iDlip(a) = ¯I Dli¬i p0(ujp)~ whenever ¯I Dli¬i p0(ujp) is nonnull
but instead we simply postulate a 1-vector
iDlip(a) within I Dli¬i p0 that satifies
iDlip(a)2 = a2 . We will henceforth assume a to be a unit vector.
The normalisation condition for iDlip(a) means that the right-connection ba,p(Dli) º a-1iDlip(a) for iDli at p satisfies ba,pba,p§ = 1 and is thus a 1-rotor expressable as a 2-spinor ewui p a p(Dli) , where wui p a p(Dli) = cos-1(a-1¿iDlip(a) (a-1ÙiDlip(a)~ is a UN 2-blade. [ we assume Dli is sufficiently small and CM sufficiently smooth that the subtended angle is comfortably below ½p ]
We would like the angle subtended by two 1-vectors parallelly transported along the same streamline to remain the same, ie.
iep(a) ¿ iep(b) = a¿b
and we can acheive this by insisting that ui p¿wui p a p(e) = O(e2) .
iep(a) º
aewui p a p(e)
= a + a¿wui p a p(e) + O(e2)
where for given scalar e, wui p a p(e) is a pure UN-bivector 2-field defined at every
pÎCM
for all aÎIp .
Parallel displacement along the ith streamline thus preserves orthonormality and corresponds to a Lorentz rotation
having UN rotor Rp i(e) which we assume to tend to a well defined limit as e®0.
M such rotor fields fully define parallel transport and effectively define CM.
Linear Connection
gp(ujp,dli,e)
º
ujp - iep(ujp)
= -ujp¿wui p ujp p(e) + O(e2)
is a 1-vector lying in general neither
in Ip nor I e¬i p. It cannot be properly regarded as a tangent 1-field
of the M-curve (traditionally speaking, it is not a "tensor").
We make the continuity assumption that
Lime ® 0[ e¬i ujp - i-e e¬i p( e¬i ujp)] = -gp(ujp,ui p).
[ We use the symbol g here in accordance with traditional GR use of G for the gravitational connection acting in
the mapspace. Conflict with the widespread use
in multivector literature of g for a (typically Â1,3) fiducial frame is unfortunate.
]
Taking gp(ujp,ui p) º Lime ® 0[e-1gj p(ui p,e) ] we obtain
the connection, the 1-vector component of dli ujp ïp that exists outside Ip.
gp(ujp,ui p) = -ujp¿wui p ujp = wui p ujp ×ujp
.
gp(ujp,ui p) is orthogonal to both ui p and ujp at p .
A linear connection (aka. affine connection) derives from an assumed
bilinearity of gp which can then
can be fully defined by M2 1-fields
gp( hjp , hip )
which then effectively define qp C(a).
Indeed, some authors define iDli as the consequence of a given linear connection.
We say the connection is symmetric if
gp(ui p,ujp)
= gp(ujp,ui p) .
Suppose now that the ui p are everywhere mutually orthogonal so that {ui p} provides
an orthonormal frame for Ip with inverse frame {uip}. Because wui p ujp is a 2-blade perpendicular to Ip, we have
wui p ujp ×ui p = 0 i¹j . Thus we can define M pure bivectors
wui p p º åj=1M wui p ujp which satisfy
gp(ujp,ui p) = wui p p×ujp
The connection is symmetric if ujp¿wui p p = ui p¿wujp p " 1£i,j£M .
A linear connection arises if wui p a is linear in a.
[ Proof :
Since the ui p are orthogonal
gp((åj=1M bjujp),(åi=1M aiui p))
= (åj=1M bjujp)¿wui p åi 1M= aiui p)}
= (åj=1M bjujp)¿(åi=1M aiwui p ui p )
= åi=1M åj=1M aibj(ujp¿wui p ui p )
= åi=1M åj=1M aibjgp(ujp,ui p)
.]
Within the mapspace
Parallel transport in the mapspace
iex(b) º ¦Ñx+eei-1( ie¦(x)(¦Ñp(b)) )
preserves ¿ instead of ¿.
[ Proof :
iep(a) ¿x+eei iep(b)
= iep(¦Ñx(a))¿iep(¦Ñx(b))
= ¦Ñp(a)¿¦Ñx(b)
= a¿xb
.]
Let b be a map direction at xÎMap with corresponding CM direction a=¦Ñx(b)
at p=¦(x).
It may not make any geometrical "sense" to subtract a map 1-vector "at" x+eei from one "at" x
but we can still do it!
Gx(b,ei,e)
º
b - iex(b)
= b - ¦Ñx+eei-1(iep(¦Ñx(b)))
= b - ¦Ñx+eei-1(¦Ñp(b) - gp(¦Ñp(b),dli,e))
is, for given ei and scalar e, a map-1-vector-valued function of map-1-vector b, linear in b if
the CM connection gp is linear. We
then have Gx(ei,ej,e) = åk=1M Gkijx(e)ek .
Letting
Gx(b,ei) = Lime ® 0(e-1Gx(b,ei,e))
we achieve an affine connection for the mapspace, symmetric if g is.
Gx(a,b) = ¦-Ñx(¦Ñ2x(a,b))
[ Proof :
See General Relativity Chapter
.]
We can think of G as a rule for moving map 1-vectors around.
In the chaper on
General Relativity
we derive an expression for G in terms of the metric.
Directed Coderivative
The directed coderivative of b(p) with respect to streamline ui p at p
is defined by parallel transporting b( e¬i p) back from e¬i p to p and there subtracting
b and dividing by e, in the limit as e®0. It is thus a 1-vector within Ip.
Ðui p(b) = dlib + gp(b,dli)
= dlib + wui p ujp p×b
[ Proof :
Ðui p(b(p)) º
Lime ® 0[
i-e e¬i p(b( e¬i p)) - vj(p) / e ]
= Lime ® 0[
( i-e e¬i p(b( e¬i p)) - b( e¬i p)) + ( b( e¬i p) - b(p)) / e ]
= gp(b,dli) +dliujp
.]
In the limit, since li is the natural (proper) parameterisation, we can
consider dli to be equivalent to the N-D directional derivative Ðui p
Accordingly we can define a directed coderivative operator
Ða º Ða + wa×
for a given direction aÎIp which when applied at p to a 1-field over and within CM
returns a 1-vector in Ip.
We then have
iDlip(ujp) = eDliÐli e¬i ujp
= eDliÐli eDlidli ujp .
[ Proof :
iDlip(ujp) =
vj( Dli¬i p)
+ DliÐlivj( Dli¬i p)
+ Dli2Ð (ui p)2vj( Dli¬i p) + ...
.]
Ða can be regarded as a differential over and "within" CM, although its component parts Ða and wa× cannot.
Ða generates a 1-vector codel-operator
Ñp º åk=1M ukpÐuk p
= Ñ[CM] + åk=1M ukp(wuk p×)
.
Within the mapspace
Ða induces a differential operator within M-D parameter space MapÌVM
Ðað(B(x)) = ¦-Ñp(ЦÑp(a)(b(p)))
where b(p) º B(¦-1(p)) .
Ðað(vp) = Ða(vp) + G(a,vp)
where G is linear as a consequence of the linearity of ¦-Ñp .
In particular,
Gp(ej,ei)
º Ðeiðej =
åk=1M Gkjiek
where point-dependant scalar
Gkji
is known as a Christoffel symbol.
Geodesics
We say a 1-field ui p is geodesic if
iDlip(ui p) = ui p( e¬i p) " pÎCM, Dli
where Dli is assumed small enough to remain in the neighbourhood.
Geodesic fields are thus invarient under parallel transport and satisfy
Ðui pui p = 0.
The physical interpretation of geodesics are the potential trajectories of "free falling" particles
and (assuming a natural streamline parameterisation pi'(li)2 = 1)
we can then regard the
pi"(li) + g(pi'(li),pi'(li)) = 0
derived from Ðui p(ui p) = 0 as
relating the "acceleration" of a "particle" to its velocity and the local "geometry" g.
An alternate definition of a geodesic is the "shortest subcurve between two points". If we fix
two points p and q on a geodesic streamline then the arclength from p to q is stationary (minimal)
for small changes in the path from p to q that keep it within CM. The equivalence of the two
defintions is intuitively reasonable and the proof (ommitted here) arises from fairly straightforward calculus
Curvature
[Under Construction]
1-Curvature
The traditional curvature of a 2D 1-curve p(t) (ie. a 1-curve confined to a 2-plane
) is defined
as
C(t) º Limd ® 0[ d-1 qÐ(p'(t+d) , p'(t)) ]
where t is the natural parameterisation - ie. the instantaneous angular tangent change per unit arclenth.
C(t)-1 is known as the raduius of curvature and coresponds to the radius of the circle
matching the 1-curve to second order at p(t) . For an N-D 1-curve we have 1-curvature
p"(t) which is a 1-vector orthogonal to p'(t) ("normal" to the 1-curve) as a consequence of
the constancy of p'(t)2 . Its magnitude C(t) is the curvature of the 1-curve confined in
the limit to 2-plane p(t).
For a 3D 2-curve we have tangent unit vectors h1,h2 and normal n=h1×h2 forming the SYMBOL">t) is the curvature of the 1-curve confined in
trihedron) at p. Taking p as the origin, the induced coordinates (x1,x2,x3) satisfy
x3 = ½ (¶2x3/¶(x1)2) x12 +
(¶2x3/¶x1¶x2) x1x2 + ½ (¶2x3/¶x2¶) x2 |2
By rotation about e3 we can diagnonalise this form as
x3 = ½ (k1(x1)2 + k2(x2)2)
where k1,k2 are known as principle curvatures.
If the surface is parameterised as p(l1,l2)
with h1p = ¶p/¶l1 , h2p = ¶p/¶l2
then
¶2p/¶li¶lj º ¶ hip /¶lj =
åk=12 Gkij hkp + bij h3p i,j,k Î {1,2}
¶ h3p /¶li = - åk l=12 gklbli hkp i Î {1,2}
2-Curvature
The second directional coderivative operator
ÐliÐlj is not symmetric in i,j.
We can take the skewsymmetrol of it to obtain the
Rieman Curvature operator
cpui pÙujp(u) = cpui pÙujp×u =
2(Ðli×Ðlj)u =
( Я(li)wujp - Я(lj)wui p
+ wui p×wujp) . u
[ Proof :
.]
ÐbÐau =
( Я(b) + wb×)( Я(a) + wa×)u
=
( Я(b) + wb×)( Я(a)u + ½(wau-uwa))
= ( Я(b) Я(a)u + ½( Я(b)wau- Я(b)uwa)
+wb×( Я(a)u + ½(wau-uwa))
=
Я(b) Я(a)u + ½( Я(b)wau - Я(b)uwa
+ wb Я(a)u - Я(a)uwb )
+ ½wb×(wau-uwa)
=
Я(b) Я(a)u +
½( Я(b)(wau) + wb Я(a)u
- Я(a)(uwb)
- Я(b)(uwa) )
+ ¼(
wbwau - wbuwa
- wauwb + uwawb )
Þ 2(¶b×¶a)u =
2( Я(b)× Ð¯(a) + Я(b)×(wa×) + (wb×)× Ð¯(a) +
(wb×)×(wa×))u
=
Я(b)(wau)- Я(a)(wbu) + wb Я(a)u -wa Я(b)u
+ (wb×wa)×u
= ( Я(b)wa)u - ( Я(a)wb)u + (wb×wa)×u
=
( Я(b)wa)×u - ( Я(a)wb)×u + (wb×wa)×u
Given a linear connection, the second-derivative curvature operator cpaÙb(v) is thus a
geometric commutatator product with a particular bivector cpaÙb .
The more general definition of the curvature operator is 2(Ða×Ðb) - Ð[a,b] but we are most interested in the case when [a,b]=0 ( ie. Ða×Ðb =0 ) .
cpaÙb º
Я(b)wa - Я(a)wb + wb×wa
=( )= ( Я(b)wa) - ( Я(a)wb) + (wb×wa)
is known as the curvature 2-tensor (aka. Riemann-Christoffel tensor).
It is a CM-point-dependant Ip-bivector-valued function of Ip-bivectors acting as a "second order correction"
for the linear approximator ¦(p0) + ¦Ñp(dp) for ¦ near p0.
Geometric interpretation
If [ui p,ujp]=0 so that we have a loop
-Dli¬j -Dli¬i Dlj¬j Dli¬i p = p
then
cpui pÙujp(uk p) =
(DliDlj)-1 j-Dlj( i-Dli( jDlj( iDli(uk p) ))))
which is clearly dependant solely on value of uk p and the geometry of CM . In particular, it is independant
of any differential of uk p.
We thus have the geometric interpretation of cpui pujp)(uk p) as the
1-vector change in uk p resulting from parallel
transport of uk p around (the projection into CM of) a tiny planar loop through p in ui pÙujp
, divided by the "area" of that loop. This must lie in Ip so we have
cpui pÙujp(uk p) = ål=14 clijkulp
where scalar clijk º ulp¿cpui pÙujp(uk p) .
[ Proof :
j-Dlj i-Dli jDlj iDli(uk p)
=
e-DljÐlj
e-DliÐli
eDljÐlj
eDliÐliuk p
= ...[tedious expansion]...
= uk p + 2DliDlj(Ðli×Ðlj)uk p +
O(Dl3)
.]
Symmetries and Bianci Relations
It is possible to express Cxeij in terms of the gij and their derivatives
and doing so reveals some more symmetries:
Clkij º
el¿(Cxeij(ek))
= ei¿(Cxelk(ej))
= -ek¿(Cxeij(el))
.
The last of these reflects the reversability of parallel transport.
We also have (from the G expression for Cijkl)
the first Bianci identity
ek¿Cxeij + ej¿Cxeki + ei¿Cxejk = 0 .
which can also be expressed as
Cxeij ¿ ekl
= Cxekl ¿ eij
.
It follows that
a¿Cx(bÙc))
+ b¿Cx(cÙa)
+ c¿Cx(aÙb) = 0 " 1-vector a,b,c Î Ip.
As a result, the M 4 element matrix representation of Cx
has only M 2(M 2-1)/12 independant elements (20 out of 256 for N=4).
ÐeiðCxejk + ÐejðCxeki + ÐekðCxeij = 02 is known as the second Bianci identity.
The codivergence of the Ricci tensor
Ñðx¿ cpa
º åiei¿Ðei cpa
= ½ÐaCx
[ Proof :
Ñðx¿ cpb
=
Ñðx¿(Ña¿cpab))
Þ ... ???
.]
Tortion
The differential operator Ða Я(b) - Ðb Я(a) - 2( Я(a)× Ð¯(b))
is known as the tortion operator. It is 0 for a symmetric connection when we thus have
£ui pujp = Ðui p Я(ujp) - Ðujp Я(ui p)
.
Suppose now that ui p is a geodesic field and 1-vector bp is Lie dragged along geodesic ui p congruence so
that £ui pbp = 0 . We must then have
Ðui p2 Я(bp) =
2(Ðui p×Ðbp) Я(ui p) .
[ Proof : Ðui p Я(bp) = Ðbp Я(ui p)
Þ Ðui p2 Я(bp) = Ðui pÐbp Я(ui p)
= 2(Ðui p×Ðbp) Я(ui p) +
ÐbpÐui p Я(ui p)
= 2(Ðui p×Ðbp) Я(ui p)
since ui p geodesic Þ Ðui p Я(ui p) = 0
.]
Thus we have a second geometric interpretation of the curvature tensor as the second derivative of
a vector dragged along a geodesic confluence, informally: a measure of geodesic "splay".
[Under Construction]
Extremal M-Curves
A standard problem in classical mechanics is to determine the form of the curve assumed by a
rope of length a hanging motionless between two fixed points a distance less than a apart
under a uniform gravitational field, which is an extremal 1-curve in N=2 dimensions.
If we assume the rope to hold a curve shape p(s)=(x(s),y(x))
(the Y axis being gravitationally vertical) that does not loop "over" itself vertically, we have
ds2 = dx2 + dy2 so that the length constraint is
a = ò01 ds = òx0x1 dx(1+y'2)½
where y' º dy/dx.
The mechanical constraint is that total gravitational potential energy
ò01 ds rgy(s) = rg òx0x1 dx y(1+y'2)½
be minimised,
where g is uniform vertical gravitational acceleration and r is the mass of unit length of rope.
More generally, we might seek to minimise
òx0x1 dx¦(x,y,y') subject to a constraint
òx0x1 dxg(x,y,y') = a.
A standard approach is to form h(x,y,y') º ¦(x,y,y') + lg(x,y,y')
where scalar l is known as a Lagrange multiplier. One then embodies freedom to vary the
path by means of two
z0 and z1 such that
y(x0,z0,z1)=y(x0) ; y(x1,z0,z1)=y(x1) " z0,z1 (boundary condition)
; y(x,0,0)=y(x) ; and y(x,z0,z1) is twice continuosuly differentiable in all parameters.
Forming K(z0,z1) = ò+x0x1 dx h(x,y(x,z0,z1),y'(x,z0,z1)) the extremal condition requires
¶K/¶z0 and ¶K/¶z1 to vanish at z0=z1=0 and this leads (via integration by parts)
to the Euler-Lagrange equation
¶h/¶y = d/dx(¶h/¶y') .
In the case of the hanging rope,
h(x,y,y') = (rgy-l)(1+y'2)½ and setting l = -rgy0 we have
h(x,y,y') = rg(y-y0)(1+y'2)½
= rg(y-y0)(1+(y-y0)'2)½ for some constant y0 .
Setting h = y-y0 the Euler-Lagrange equation is
rg(1+h'2)½ =
d/dx rghh'(1+h'2)-½
with first order simplification
h' rghh'(1+(y-y0)'2)-½ -
rgh(1+h'2)½ = b
Þ
rg(1+h'2)-½h = b
having solution h = b cosh(x-c)/b) so the rope has shape
y0 + b cosh((x-c)/b) where c,y0 and b are chosen to
match the given endpoints.
We say an M-curve CM is M-extremal for an action functional
¦CM:UN ® UN if some particular magnitude measure (content, scalar part, square, modulus or whatever) of
òCM dM-1p ¦CM(p) is maximised (or minimised) by CM in the sense that
integrating other any M-curve which deviates only slightly from CM will produce
a result no higher (or lower) than the CM integral. CM is a "locally optimal"
M-curve for a particular integration ¦CM. We specify the dependancy of ¦ on CM to allow
action functionals dependant on the geometric properties of the "sampling curve" CM , such as tangent
or normal vectors , to allow "direction dependant sampling" or "velocity dependance".
Euler-Lagrange Equations
For M=1 we have a path p(s) 1-extremal for FC1 = F(s,p(s),p'(s)).
Hamilton's principle provides that the state of a system characterised at time t by k multivector variables
x1(t),..,xk(t) varies from time t0 to t0 so as to 1-extremise
a (usually real scalar) integral measure
S(t) º S(t0) + òt0t0 dt L(t,x1(t),..,xk(t),x'(t),...,xk'(t))
with the action functional L(t,x1(t),..,x1'(t)) known as the Lagrangian
of the system.
L is usually assumed to be real-scalar valued and independant of p"(t) and higher derivatives and is
traditionally assumed to seperate into "kinetic" and "potential" componenets independent of time and velocity respectively
as
L(t,p(t),p'(t)) = T(p(t),p'(t)) - V(t,p(t)) but more generally we might postulate
a multivector-valued Lagrangian of k multivector-valued variables and their first temporal derivatives
L(t,x1(t),..,xk(t),x1'(t),..,xk'(t)).
If the Lagrangian L is itself an integral over some spacial M-curve (typically an
M=N-1 hypercurve reprenting a contemporal slice of a Base space)
L = òBase(t) dN-1p L(t,p,p')
that spacially integrated function is known as a Lagrangian density. Note that such requires a "velocity" p' be associated with every point p in Base(t) .
Extremal Paths
A standard approach for M=N=1 to 1-extremise ¦(s,x,x')
is to embody freedom to vary the path by means of two scalar parameters
z0 and z1 such that
x(s0,z0,z1)=x(s0) ; x(s1,z0,z1)=x(s1) " z0,z1 (boundary condition)
; x(s,0,0)=x(s) ; and x(s,z0,z1) is twice continuosuly differentiable in all parameters.
Forming K(z0,z1) = òs0s1 ds¦(s,x(s,z0,z1),x'(s,z0,z1)) the extremal
condition requires
¶K/¶z0 and ¶K/¶z1 to vanish at z0=z1=0 and this leads (via integration by parts)
to the Euler-Lagrange equation
¶¦/¶x = (d/ds)(¶¦/¶x') .
In particular cases where ¦(s,x,x') = ¦(x,x') so that ¶¦/¶s=0 and there is no explicit s
dependence,
we have d/ds(x'¶¦/¶x' - ¦)= 0
giving a first order differential equation
x'¶¦/¶x' - h = constant .
[ Proof : y" ¶h/¶y' + y'd/dx¶h/¶y' - ¶h/¶x-¶h/¶yy' - ¶h/¶y'y"
= y'(d/dx¶h/¶y' - ¶h/¶y) - ¶h/¶x
= 0
.]
A nongeometric generalisation is to extremise òs0s1 ds¦(s,x1,x2,..xk,x1',x2',..xk')
subject to M scalar constraints gi(s,x1,x2,...xk)=0 i=1,2,..,M
(Note the absence of any g dependance on the xi').
To do so we form
h(s,x1,..,_xK,x1',..,xk') = ¦(s,x1,..,_xK,x1',..,xk')
+ åj=1M lj(s)gj(s,x1,..,xk,x1',..,xk')
where lj(s) are M arbitary functions of s
and obtain Euler-Lagrange formulae
¶h/¶xi = (d/ds)(¶h/¶xi') for i=1,2,..k .
Generalising geometrically, we have F(s,x1,x2,..,xk,x1',x2',..,+xk')
with geometric Euler-Lagrange Equations
¶F/¶xi = (d/ds)(¶F/¶xi') " i=1,2,..k
and first order equation
åi=1k (xi'*Ñxi')F - F = constant (independant of s)
if ¶F/¶s = 0.
Considering the xi as seperate grades of a single multivector argument x we can regard the Euler-Lagrange equations
as individual coordinate terms of a single geometric equation
¶xF(s,x,x') = (d/ds)(¶x'F(s,x,x'))
where ¶x = åijk.. eijk..¶/¶xijk..
over all blades comprising x space so that, for example,
¶x = åi=1N ei(¶/¶xi).
Generalised Momentum
When F is a scalar-valued Lagrangian we have (xi'*¶xi')L =
xi'*(¶xi'L) and multivector m = Ñxi'L(x1,..xk,x1',..,xk)
is known as generalised spacial momenta or cononical momenta.
Typically xi' is spacial 1-vector valued and so momenutum is a spacial 1-vector.
If multivector L is independant of xi so that ¶L/¶xi=0 then the ith Euler-Lagrange equation provides
(d/dt) ¶L/¶xi' = 0 so Mi = ¶L/¶xi' is constant, ie. unchanging with t,
Thus the constants of a system are consequences of absent dependendencies (aka. symmetries) in the Lagrangian.
Energy is conserved when L depends on t only indirectly via x(t) and it derivatives.
Momentum is conserved when L depends on x only indirectly via x'.
For L(t,x,x') = ½mx'2 - f(x,t) we obtain classical momentum
Ñx'½mx'2 = mx' , constant if f(x,t)=f(t).
Adding an electromagnetic term qc-1ax¿x' to L introduces
qc-1ax to the momentum
If
L(t,x1(t),..,xk(t),x1'(t),..,xk'(t))
= L(x1(t),..,xk(t),x1'(t),..,xk'(t))
so that there is no explicit t dependance then
the Hamiltonian H(x1(t),..,xk(t),x1'(t),..,xk'(t))
º åi=1k (xi'*Ñxi')L - L is a constant,
the "energy" of the system.
We can regard the Hamiltonian as generalised temporal momentum. Conservation of energy but varying spacial momentum
resulting from only non-relativistic potentials f(x,t)=f(x) .
But H = - ¶S/¶t .
Even though, like the Lagrangian, S(t) cannot strictly be regarded as a function of position since it is only defined over
our given extremal path, we can nontheless postualte an action field S(t,x) an it can be shown that
d/dx ( åi=1k yi' ¶¦/¶yi' - ¦)
= åi=1k yi" ¶¦/¶yi'
+ åi=1k yi' d/dx ¶¦/¶yi'
- ¶¦/¶x
- åi=1k ¶¦/¶yiyi'
- åi=1k ¶¦/¶yi'yi"
=
åi=1k yi' d/dx ¶¦/¶yi'
- ¶¦/¶x
- åi=1k ¶¦/¶yiyi'
=
åi=1k yi'( d/dx(¶¦/¶yi') - ¶¦/¶yi )
- ¶¦/¶x
m = (xi'*¶xi')L = ÑxiS(t,xi) and in particular we have the Hamilton-Jacobi equation
H(xi,xi',t) = -¶S/¶t
[ Proof : For any ¦(x,x')
d/dt ( åj=1k xi' ¶¦/¶xj' - ¦)
= åj=1k xj" ¶¦/¶xj'
+ åi=1k xj' d/dt ¶¦/¶xj'
- ¶¦/¶t
- åj=1k ¶¦/¶xjxj'
- åj=1k ¶¦/¶xj'xj"
=
åj=1k xj' d/dt ¶¦/¶xj'
- ¶¦/¶t
- åj=1k ¶¦/¶xjxj'
=
åj=1k xi'( d/dt(¶¦/¶xj') - ¶¦/¶xj )
- ¶¦/¶t
=
åj=1k xj'( d/dt(¶¦/¶xj') - ¶¦/¶xj )
if ¶¦/¶t = 0
= 0
if d/dt(¶¦/¶xj') = ¶¦/¶xj j=1,2,..,k
which are the Euler-Lagrange equations.
=
åi=1k yi'( d/dx(¶¦/¶yi') - ¶¦/¶yi )
if ¶¦/¶x = 0
= 0
if d/dx(¶¦/¶yi') = ¶¦/¶yi i=1,2,..,k
which are the Euler-Lagrange equations.
.]
Theoretical physics then becomes a quest for the One True Action Lagrangian density , usually assumed
real scalar and generating kinematic equations involving zeroth, first, and second differentials only.
Geometric (multivector-valued) Lagrangians extermised in the sense that each multivector coordinate is stationary under variation.
This is an intrinsically nonrelativistic approach in the case of multiple particulate systems since it requires
a favoured temporal parametrisation t with L(t,p,p') = L(p,p')
As "locally shortest routes", Geodesics are 1-extremals for the scalar path length
of ò0t ds|d1p¿g(d1p)|½ . Integrating a square root can be messy
but fortunately such paths also extremise
ò0t ds ds p'(s)¿gp(s))(p'(s))
where p'(s) denotes the (d/ds)p(s)
. _Be Thus we set
L(p,p',s) =
p'¿gp(p')|½ and
minimise scalar integral ò0t ds ds L(p(s),p'(s),s) by solving the N Euler-Lagrange equations
d/ds(¶L/¶pi') =
¶L/¶pi) for i=1,2,...N .
L(p,p',s) will generally depend on p via _gp but if _gp has a symmetry such that
¶L/¶pi=0 for a particular coordinate i then
¶L/¶pi' is constant along any geodesic path.
Geodesic flow can consequently be viewed
as arising from a Lagrangian density
L(p(t),p'(t)) = |p'(t)¿_gp(p'(t))|½ .
with particles following timelike trajectories that extremise (minimally) their proper time (arclength).
If t is proper time parameterisation then
L(p(t),p'(t)) = |p'(t)2|½ = 1 along a geodesic.
Electrodynamical forces due to a 1-vector four-potential ap (typically with Ñp¿ap=0) are introduced by adding scalar
-q p'(t)¿ ap(t)
to the Lagrangian density leading to the Lorentz force law
mp"(t) = qfp.p'(t)
where fp = ÑpÙap .
[ Proof :
L = m(-p'2)½ - q p'¿ap
Þ ÑpL =
Ñp(m(-p'2)½ - q p'¿ ap)
= -qÑp(p'¿ap) ;
Ñp'L =
-m(-p'(t)2)-½ p'(t) - qap
so the Euler-Lagrange equation is
qp'¿Ñpap =
(d/dt)(m(-p'2)-½ p' + qap )
= mp" + q(p'¿Ñp)ap
Þ mp" = q(Ñp(ap¿p') - (p'¿Ñp)ap))
= q(ÑpÙap)¿p'
.]
Extremal surfaces
A more geometric generalistaion is to consider the problem of finding the M-curve of given content
maximising a particular boundary or interior integral. We might, for example, seek the loop
starting and ending at a given point a and constrained to lie in a given 2-curve (surface) containing a that maximises
enclosed content (area) for a given boundary content (pathlength)
Of fundamental importance is the fact that spacial Lagrangian density
(Ñ[e123]yp)2 is extremised for integration over CM
(subject to constraint of given boundary values over dCM) if yp
satisfies the spacial Laplace equation Ñp [e123]2yp = 0
over CM .
In electrodynamics we have L = - 2-4p-1 fp2
+ c-1jp¿ap
where c is scalar lightspeed and fp = ÑpÙap extremised by solutions to
Maxwell equations.
Next : 4D Spacetime
×××××××××××××××××××××××××××××××××××××××
References/Source Material
David Hestenes "New Foundations For Mathematical Physics" Websource
Bernard Schutz
"Geometrical Methods of Mathematical Physics"
Cambridge University Press 1980
[Amazon US UK]
[ Traditional presentation of manifold derivatives]
David Hestenes, Garret Sobczyk
"Clifford Algebra to Geometric Calculus"
D. Reidel Publishing 1984,1992
[Amazon US UK]
Since this document draws heavily from and frequently cites
this work we clarify the notational differences between them.
Their P(a) is our ¯Ip(a) º ¯(a) ("projector").
Their Pb(a) is our ¯Ñ(a,b) ("differential of projector").
Their Sa is our [ÑÙ¯](a) ("curl").
Their Ñ is our Ñ ("coderivative").
Their
¶
is predominantly our Ñ ("tangential derivative")
but also our Ñ ("derivative") early in the work.
Their da is our
Ðaß º ¯Ð¯aß ("extensor coderivative").
Their (a.¶) is our Ða=(a¿Ñ) ("directed derivative").
Their ei is our hip (potentially nonorthonormal "tangent frame").
Their gi is our ei (orthonormal "fiducial (basis) frame").
Glossary
Contents
Author
Copyright (c) Ian C G Bell 1998
Web Source: www.iancgbell.clara.net/maths or
www.bigfoot.com/~iancgbell/maths
Latest Edit: 18 May 2007.