 General Relativity
"Where there is matter, there is geometry." --- Johannes Kepler

We now assume familiarity with Multivector Calculus. The properties of M-curves, notably the Rieman curvature tensor, described there are utilised here. We also include in this section a parallel development of parallel transport and linear connections   more typical of the General Relativistic literature yet also novel (AFAIK) , and provide a coordinate-independant development of the GR Field Equations.

Notations
The notations and conventions used previously and defined in the glossary are retained here, notably ¦Ñ for the differential ¦Ñp(a) º (a¿Ñp)¦(p) ; ¦ for the inverse differential (¦Ñ)-1 ; ¦D for the adjoint ¦Dp(aº Ñp(a¿¦(p)) ; ¦-D for the inverse adjoint.

Material-energy 1-tensor in a flat space
The material-energy 1-tensor is an alternate representation of a 1-vector flow. Supose mp is a (nonunit) timelike 1-vector field over a flat spacetime such that |d4p|mp represents the 4-momentum of the matter in a small 3-simplex (4-volume) element d4p at p. We assume the matter conservation law Ñp¿mp = 0     to apply everywhere other than at well defined "sinks" and "sources" where matter is somehow "introduced" or "removed".
We can express mp as mpup where scalar mp º |mp| and unit timelike 1-vector up º mp~   With regard to an observer e4, mp splits via up = (1-vp2)½(vp+e4). into three-momentum mpvp and "relative" scalar density mp(1-vp2)½ .

Consider a small 2-simplex (surface element) having timelike unit normal a and content (area) d. The flow of matter "into" the simplex is given by (mp¿a). Each "unit" of that matter provides unit 1-vector 4-momentum up so we can regard
Tp(a) º (mp¿a)up    = mp(up¿a)up = -mp¯up(a) = -mp¯mp(a) = -¯°mp(a) .
as the "flow of 4-momentum in direction a".
Tp(up) = mp(up2)up = -mpup = -mp .
Tp(a) is symmetric, and often represented by Tijp = mpuiuj    ;       Tij = mpuiuj .    ;     or Tij  = mpuiuj , The use of 'T' for matter flow is traditional in the GR literature, and should not be confused with 'T' for "Time".
[ Proof :  Tijp º ei¿Tp(ej) = ei¿mp(up¿ej)up = mp(ei¿up)(ej¿up) = mpuiuj  .]

Ñp¿Tp(a) = mp¿(Ñp(a¿up))     and so vanishes when mp=0 (ie. mp=0 corresponding to vacuum) and also when Ñp(a¿up)=0, ie. when a¿up is constant (to first order) in a neighbourhood of p.
[ Proof :  Ñp¿Tp(a) = ((Ñp(a¿up))mp + (a¿up)(Ñpmp))<0> = mp¿(Ñp(a¿up)) by the matter conservation law  .]
Ñp¿Tp(a) is thus an a-directed measure of deviation from a "Gallilean" state of uniform motion.

The contraction of Tp(a) is scalar -mp . [ Proof :  Ña¿Tp(a) = (Ña¿(a¿up)mp) = (Ña¿(a¿up))mp = (up)mp = mp up2  .]

Gravitation as a higher dimensional embedding
"an embedding space introduces elements that are quite irrelevant to our purpose" Wolfgang Rindler

Flat N-D spacetime

We seek to represent a nonhomogeneous "warped" spacetime as a locally isomapped 4D submanifold via a higher dimensional embedding of a homogeneous "flat" 4D Minkowski spacetime Â3,1 into a homogeneous "flat" ND spacetime. We extend Â3,1 by postulating a further nonnull basis vector e0 and potentially further nonnull vectors e5,..,eN-1 into a higherdimensional space UN º Âp,q,r with p+q+r = N ³ 5 ; p ³ 3, q ³ 1 . We assume an identity embedding 1 : Â3,1 ®  UN acting on 1-vectors so that we can move 1-vectors (and, by outtermorphic extension, multivectors) "unchanged" between R3,1 and  (a subspace of) UN. Let (e1,e2,e3,e4) be a fixed orthonormal frame for R3,1 and define unit pseudoscalar i=e1234. Using 1 , we can extend this  into a fixed orthonormal basis (e0,e1,e2,...eN-1) for UN with unit pseudoscalar u=e012..(N-1).
We assume an orthonormal inverse frame (e0,e2,...eN) for UN with e1=e1,e2=e2,e3=e3,e4=-e4. This frame induces N seperate N-D vector differential operators dxi = dei = ei¿Ñp .

Point Embedding
We require a (predominantly) invertible point-embedding ¦ : R3,1 ®  UN taking our homogenous 4D subspace into a 4-curve ("warped" 4D "surface" or "manifold") ¦(R3,1) Ì UN. With each point (event) p Î R3,1 we associate N-D hyperevent p = ¦(p) Î UN.
We assume that ¦ is extended over UN such that ¦Ñ and ¦ exist as nonvansishing invertible UN-point-dependant UN-1-vector valued functions of UN-1-vectors outtermorphically extended into multivector-valued functions of multivectors.
We assume 4-blade Jp º ¦Ñp(i) to be nondegenerate so that ¦Ñp is invertible over Jp and insist further that ¦Ñ(ei)¿Jp = 0 " iÏ{1,2,3,4} so that ¦-Ñp(Jp) = i.
[  We use J ("Jacobian") rather than f for typographical distinctness. Jp should not be confused with the traditional use of J for electric current. ]
This ensures that , for b Î Jp, b¿¦Ñp(a) is maximised for an a Î i so the adjoint ¦D(b) = ¦ÑDp(b) º Ña(b¿¦Ñp(a)) must map Jp®i.
Let scalar determinant jp º |Jp| .

In essence, the manifold is determined by ¦Ñp . ¦ can be considered as the "integrative consequence" of ¦Ñp provided we have as "initial conditions" ¦ defined over a closed hypercurve "seeding surface" . We can form the second differential of ¦-1 at ¦(p) which we denote by ¦-Ñp2 .

¦Ñp generates a point-dependant invertible embedded frame { hip = ¦Ñp(ei) ; 0£i£N } for Jp

Coordinate-based Approach
We can represent the point embedding ¦ over R3,1 by the N scalar functions  yn(x1,x2,x3,x4) defined by
¦(x) = ¦(åixiei) = ån=0N-1 yn(x1,x2,x3,x4)en = ån=0N-1 ynen .
Unless otherwise indicated, integer suffixes i,j and k are henceforth to be considered to range under S from 1 to 4 inclusive.
¦Ñp can be repesented by two associated point-dependant Nx4 matrices:
yn,i p º en¿¦Ñp(ei) = yn/xi     ;     yn,i p º en¿¦Ñp(ei) = ei¿¦Dp(en)     1 £ i £ 4 ; 0 £ n < N .
For brevity we will often ommit the p suffix from such "matrix elements", despite their point dependance. We will also append , i to such scalar elements as a shorthand for /xi ;  , ij for /xi /xj ; and so forth . Thus yn,ij º yn, i , j º yn,i/xj º (/xj)(yn/xi) .
We have yn,i =Sig(en)yn,i     0 £ n < N .
¦Ñp(ei) = ån=0N-1 yn,ien = ån=0N-1 yn,ien     ;     ¦Dp(en) = åi=14 yn,iei .
[ Proof : ¦Dp(en) = åi=14 (¦Dp(en)¿ei)ei = åi=14 (en¿¦Ñp(ei))ei = åi=14 yn,iei  .]

¦Ñ2p(a,b) = ån=0N-1 åij yn,ijaibjen = ån=0N-1 åij yn,ijaibjen     where yn,ij º yn,i/xj .
[ Proof : ¦Ñp+dp(ei) = ån=0N-1  yn,i(p+dp)en = ån=0N-1 (yn,i(p) + åj=14 yn,ijdpj)en + O(dp2)       and the result follows  .] .

Metric Tensor
gp : R3,1 ® R3,1 defined by the composition gp º ¦Dp¦Ñp is comfortingly four-dimensional and also symmetric.
[ Proof :  gpD = (¦Dp¦Ñp)D = ¦ÑpD¦DpD = ¦Dp¦Ñp = gp  .]
gp is known as the metric tensor since ¦Ñp(a)¿¦Ñp(b) = a¿ gp(b) = gp(a)¿b and in particular (¦Ñp(a))2 = a¿(gp(a)) . Physicists often refer to v¿gp(v) =  ¦Ñp(v)2  (rather than v2) as the "magnitude" of v. [ Proof : ¦Ñp(a)¿¦Ñp(b) = a¿¦Dp(¦Ñp(b)) = a¿gp(b)  .]
Thus a¿gp(a) represents the second order approximation of (¦(p+a) - ¦(p))2 with regard to mapspace displacement a .
(¦(p+a) - ¦(p))2 = ¦Ñp(a)2 + O(a2) = a¿gp(a) + O(a2)  .
gp and gp-1 = ¦-Ñp¦-Dp are sometimes known as the fundamental tensors with an associated invariant line element dp¿gp(dp) = ¦Ñp(dp)2.
If ¦Ñp is symmetric, gp = ¦Ñp 2 and ¦Ñp = "Ö gp" is analagous to the "fiducial tensor" and "positional gauge" hp of other treatments.
gp has scalar determinant g = gp º | gp | = jp2 . [  In conventional GR treatments, jp appears as Ög or Ö-g according as to how the metric is signed . ]

Coordinate-based Approach
Rather than representing gp as gij º ei¿gp(ej) we choose the quadratic form representation
gijp º gp(ei)¿ej = ¦Ñp(ei)¿¦Ñp(ej) = hip¿hjp = ån=0N-1 yn, i yn , j    ;     ei º gp(ei) = åk=14 gik ek     and hence     a¿b = åij=14 gijaibj
[ Proof :  a¿b = a¿gp(b) = a¿gp(åi= biei) = a¿åibigp(ei)) = a¿åijbigij ej)  .]
gijp  º gp-1(ei)¿ej     (so that e j º gp-1(ej) = åkgkjek)     is the conventional matrix inverse of gij p.
[ Proof : åk=14 gikgkj   = åk(gp(ei)¿ek)gkj = gp(ei)¿åkekgkj = gp(ei)¿gp-1(ej) = ei¿gp(gp-1(ej)) = di j  .]
Thus gijp = Gijp / |gp| where Gijp is the i,j th cofactor of matrix { gijp } .
Note carefully that gij breaches our usual suffix convention in that it represents ei¿gp-1(ej) rather than ei¿gp(ej). A more appropriate notation would be g-ij but gij is universal in the GR literature and we follow it here.
Ommitting p subscripts, we cite without proof a "determinant derivative rule" g, k = gåi j=14  gij gij,k .

Since gp is symmetric, its 4x4 matrix representation has 10 [  4 on diagonal, 6 off diagonal symmetric ]  independant components which physicists regard as coordinate-dependant "gravitational potentials" defining the "warping" of a 4D spacetime. If gp can be diagnonalised (not guaranteed for symmetric gp in a NonEuclidean space), we have four "core" potentials with gii=1/gii . [  No summation convention applies ]

Example: Spherical Surface in 3D
The classic example of a warped 2D space is the unitsphericalpolar surface mapping p = ¦(p) = ¦(q,f) = ( sinq cosf, sinq sinf, cosq) . As shown in Multivertors ans Manifolds, it has metric gp(dx,dy) = (dx , sin2qdy) with associated line element dp¿gp(dp) = dx2 + cos2qdy2.
gp fully characterises the spherical surface but suppose we were to encounter intelligent 2D bugs living in one?
The 2D bugs would notice that the circumference of circles were always less than p times the diameter, as predicted by Euclidean geometry. Small circles would have circumferences close to theoretical values whereas those with diameters approaching half the 3D "equatorial" circumference would suggest p=2. This would lead them to discover the dx2 + sin2qdy2 metric which they would extend by -t2 to form a 3D spacetime metric. We would probably direct such bugs to look to 3D ¦ and a mysterious new dimension ("height") rather the their 2D spacial metric to fully understand their environment, yet physicists persist in fixating entirely on the 4D "gravitational" metric rather than consider higher dimensions. Projected derivatives and their like are expressed as increasingly complicated formulations of gp and a realm of coordinate based tensor analysis is entered from which few return unscathed.

Covariant Coordinates
A base 4-frame {e1, ..,e4} for R3,1 generates not only a point-independant inverse 4-frame {e1,e2,..,e4}, but also a point-dependant (nonorthormal) covariant 4-frame (aka. covector 4-frame or 4-coframe ) { e i º gp-1(ei) } which itself has inverse 4-frame { ei º gp(ei) } [ Proof :   e i¿ej = gp-1(ei)¿gp(ej) = gp(gp-1(ei))¿ej = ei¿ej  .]
Trivially, ei¿ej = ei¿ej = gij     ;        ei¿e j = ei¿ej = gij .
A given vp Î R3,1 can thus be expressed both as vp = åiviei     ( with contravariant coordinates vi º ei¿vp = e i¿vp ) and as vp = åivi e i     ( with covariant coordinates vi º ei¿vp = ei¿vp = ei¿gp(vp) = åj=14 gijvj ) .
The "motivation" for covariant coordinates is that åi=14 wivi = ¦Ñp(wp)¿¦Ñp(vp) .

Suffix Conventions     We retain here the notational "lowering" of a suffix to indicate inverse (reciprocal) coordinates. Only if a suffix is underlined are covariant coordinates (with regard to a given ¦) indicated. This explicit convention differs from much of the GR literature, in which the precise meaning of lowering or raising suffix is implied by context, with "downstairs" suffixes usually indicating covariant coordinates.
The exception to this would appear to be g, with gij representing the contravariant quadratic form expression gijºei¿gp(ej) as here rather than gijºei¿gp(ej) = gp(ei)¿gp2(ej) = ei¿gp3(ej) ; with gijº ei¿gp(ej) denoting the inverse matrix for { gij } .
gij here represents ei¿gp(ej) whereas in coventional GR treatments it denotes gij=e i¿gp(ej) = d ij = ?(i-j,1,0) = (i=j) , ie. the Kroneka delta matrix representation of the "identity" 1-tensor.

Parallel Transport
Moving b = ¦Ñp(0)(v) along a path p(t) within ¦(R3,1) while continously projecting into Jp(t) is known as parallel transport along the path. Such transport preserves the (¿)-magnitude, but not the direction, of b. The results of parallel transport along differing paths to a common endpoint can vary with the path taken. In particular, the direction can change when transported around a looped path back to p(0).

Let  Gp(b,c)  º ¦-Ñp(¦Ñ2p(b,c)) . [  There is no explicit equivalent for this equation in conventional GR literature, though equivalent coordinate-based expressions are common. It is ,to the limited knowledge of the author, original to this work. ]
G kijp º ek¿Gp(ei,ej) = ek¿¦-Ñp(¦Ñ2p(ei,ej)) = ¦-Dp(ek)¿¦Ñ2p(ei,ej) = ¦Ñp(ek)¿¦Ñ2p(ei,ej)     is called an affinity (aka. a Christophel symbol of the second kind) and represents a linear (affine) connection as discussed in Multivector Manifolds.
Gkij º ek¿Gp(ei,ej)   = ¦-Dp(ek)¿¦Ñ2p(ei,ej) = ¦Ñp(ek)¿¦Ñ2p(ei,ej) = ån=0N-1 yn,k yn , ij     is known as a Christoffel symbol of the first kind.
[ Proof : ¦Ñp(ek)¿¦Ñ2p(ei,ej) = (åm=0N-1 ym ,kem)¿(ån=0N-1 yn ,ijen) = ånyn,k yn ,ij  .]

We can express G explicitly in terms of gp (in the GR literature,   Gkij and G kij are often defined in this way).
Gkij = ½(gki,j+gkj,i - gij,k)       and hence     gki,j = Gkij + Gijk .
[ Proof : gij=yn,i yn,j Þ gij,k = yn,ik yn,j + yn,ik yn, jk = yn,ik yn,j + yn,i yn,jk and the result follows    .] _
G lijp = åk=14 glk Gkij .
[ Proof : åk=14 glk¦Ñp(ek)¿¦Ñ2p(ei,ej) = ¦Ñp(åk=14 glkek)¿¦Ñ2p(ei,ej) = ¦Ñp(gp-1(el)¿¦Ñ2p(ei,ej)  .]

We have the symmetries: Gijk= Gikj    ;     G ijk= G ikj .

We have the coordinate-based connective expressions for parallel transport
- dv » Gp(vp , dp) º åijk G ijk vkdpj ei = -åijk G kijvk dpj e i = -åijk Gkijvkdpje i

To prove these: let up[dp] denote the UN parallel transport of direction up from ¦(p) to ¦(p+dp) with vp[dp] the corresponding map direction. We will assume up[dp] = ¯(up,Jp+dp) + O(dp2) .
Let vi(p,dp) and vi (p,dp) be contra and covariant coordinates for vp[dp] at p+dp
( ie. vi(p,dp) º vp[dp]¿ei  ; vi (p,dp) º gp+dp(vp[dp])¿ei = åj=14 gijp+dpvj(p,dp) ).     Clearly vi(p,0) =   vi p and vi (p,0) =   vi p .
dvi  º vi (p,dp) - vi (p,0) = ¦Ñp(vp) ¿ ¦Ñ2p(ei.dp) + O(dp2)     = åj k Gkijvkdpj + O(dp2) . Hence dv = åi j k Gkijvkdpje i .
[ Proof : vi (p,dp) = gp+dp(vp[dp])¿ei = ¦Ñp+dp(vp[dp])¿¦Ñp+dp(ei) = up[dp]¿¦Ñp+dp(ei) = up(p)¿¦Ñp+dp(ei)
= ¦Ñp(v(p))¿¦Ñp+dp(ei) » ¦Ñp(v(p))¿(¦Ñp(ei)+ ¦Ñ2p(ei,dp)) = gp(v(p))¿ei + ¦Ñp(v(p))¿¦Ñ2p(ei,dp) = vi (p,0) + ¦Ñp(vp)¿¦Ñ2p(ei,dp)  .]

dvi  = åj l G lijvl dpj + O(dp2) . [ Proof :  åj kvk Gkijdpj =  åj k(ålgklvl) Gkijdpj  .]
dvi = - åj k G ijkvkdpj + O(dp2) .
[ Proof : Let map vectors a and b be parallel transported. d(a¿b) = 0 Þ d(åi=14 aibi)=0 Þ åi((dai)bi + ai(dbi))=0
Þ åiai(dbi)) = -åijk G kijak dpj bi = -åijk G ikjai dpj bk = -åiaiåj k G ikjdpj bk .     Since a arbitary , dbi = - G ikjdpj bk  .]

This completes the proof of the connective expressions.

For a diagonalised gp we have Gkij = 0 if i,j,k are distinct ; -½gii ,k if i=j ; ½gii, j if i=k ; ½gjj , i if j=k .
Also G kij = gkk Gkij = gkk-1 Gkij .

We also have the perhaps surprising result åk=14 Gkjk = ( ln(jp)) , j = jp-1 jp , j .
[ Proof : g,j = gåi=14 åk=14  gikgik,j = gåi=14 åj=14  gik (Gkij + Gijk) = 2gåi=14 Giij
Þ åi=14 Giij = ½ g-1 g,j = ½( ln(g)),j = ( ln(jp)),j  .]

Geodesics and Velocity
A geodesic is a path p(t) t Î [a,b] within ¦(L) whose coresponding mapspace path p(t) in L Ì R3,1 ( with "natural parametrisation" condition p'(t)¿p'(t)=k   , a constant) solves the geodesic trajectory equation

p"(t) + Gp(p'(t) , p'(t)) = 0
.
We define the map-velocity of the worldline at p(t) by v(t)=p'(t) and the velocity at ¦(p(t)) by u(t)= ¦Ñp(v(t)) .
The normalisation condition u(p)2 = 1 implies dt2 = dp¿gp(dp) .     [ Proof : (¦Ñp(dp/dt))2=1 Þ dt2 = (¦Ñp(dp))2 = dp¿gp(dp)  .]

A geodesic is thus obtained by parallel transportation of a vector in the direction specified by the vector but a more intuitive alterante equivalent definition is that geodesics are the (minimising) extremal paths for the metric |dp ¿ gp(dp)|½ and so also extremal paths for the scalar Lagrangian L(p,p',t) = p'(t)¿gp(p'(t)) . The above geodesic trajectory equation is just the N-D Euler-Lagrange equation (d/dt)L/xi' = L/xi .
[ Proof :  See Rindler 10.2  .]

Coderivative
On seeking a R3,1 vector associated with a given directional derivative uÑp(¦Ñp(a)) , we have the problem that up+dp-up need not lie within Jp+dp. If we use ¦Ñpp+dp(¯(up+dp-up,Jp+dp)) , effectively replacing up by up[dp] , we obtain the directional coderivative discussed in Multivector Manifolds.
 Ðaðvp = vÑðp(a) º Limd ® 0(¦-Ñp+da(up+¦Ñp(da)-up)/d) = Limd ® 0((¦-Ñp+da(up+¦Ñp(da)-up[dp])/d) = Limd ® 0((vp+da-vp[da])/d) = Limd ® 0((vp+da -vp+vp - vp[da])/d) = vÑp(a) - Limd ® 0((vp[da]-vp)/d) = vÑp(a) + Gp(vp , a) = Ðavp + Gp(vp , a)
where vÑ(a) = Ðavp  is the standard (¿-based) differential in L.
We thus have Ðeiðej = Gp(ej , ei) .
We define Ðaðb = Ðab for scalars, and extend Ðað over map multivectors with the product rule. It is customary to use :i to denote the effect of Ðeið on a coordinate in like manner to ,i denoting Ðei.
The connective expressions give coordinate-expressions for Ðejðvp :
vi :j = vi ,j - åk G kijvk     ;     vi:j = vi, j + åk G kijvk .
These can be generalised to (covariant) coordinate expressions for coderivatives of higher rank tensors such as
Fij :k = Fij ,k - ål( G likFlj + G ljkFil )    ;     F ij :k = F ij ,k + ål( G ilkF lj - G ljkF il ) .
[ Proof :  None here. See Rindler   .]

A key result is Ðaðei = 0     , the ei thus act as "constants" under codifferentation.
[ Proof : The convntional "proof" is to apply the covaraiant coordinate expression for the coderivative of a 1-tensor to gij and obtain zero.
[gij :k = gij ,k - ål( G likglj + G ljkgil) = gij ,k - ( Glikglj + Gljkgil) =...= 0   ]
but is it really legitimate to apply this rule to the essentially contravariant gij rather than to gij ? The traditional low suffixes in gij were not assigned covariantly but as a represention of the quadratic form of the metric with regard to contravarient coordinates.. Is metric gij covariant with regard to itself? Under the notations employed here, gijºei¿gp(ej) =ei¿gp3(ej)  .
[Under Construction. comment appreciated. ]  .]

We can define a mapspace 1-vector codel-operator     Ñðp(a) º åk=14 ek(Ðekða)     but we must take care here, as indicated by the brackets,  because the ek are not "immune" to "scalar" operators Ðejð (as they are to Ðej in the Ñ defintion).
This gives the codivergence of a 1-field as
Ñðp¿vp º  åk=14 ek¿(Ðekðvp) =  åkvk:k =  (Ñp¿(jpvp)) jp-1
[ Proof :   =  åk(vk,k + åj G kjkvj) =  åkvk,k + åjjp-1 jp ,jvj =  åk (jpvk),k jp-1  .]
Thus we have      (Ñðp¿vp) jp = Ñp¿(vpjp) and so Ñðp¿vp = ¯ vpjp( Ñp ) vp .
[ Proof :   ¯ vpjp(Ñp) vp º (Ñp¿(vpjp) )(vpjp)-1 vp = (Ñp¿(vpjp) )jp-1 = Ñp¿(vpjp) = Ñðp¿vp  .]

We have a covariant coordinate expression for the codivergence of a 1-tensor Fp( ) :
Ñð¿Fp(ej) = åk Fkj : k = jp-1 åkÐek(jp Fkj) - ålk GljkF kl
[ Proof : åk Fkj : k = åk ( Fkj , k+ ål( G klkF lj - G ljkF kl ) ) = åk Fkj , k  + å_lk G klk Flj -  ålk GljkF kl
= åk Fkj , k +  åk(jp-1 jp , k) Fkj -  ålk GljkF kl = jp-1 åk(jp Fkj , k + jp , k Fkj) -  ålk GljkF kl  .]

For symmetric Fp( ) this becomes     Ñð¿Fp(ej) = jp-1 åkÐek(jp Fkj) - ½ålkgkl , j F kl .
[ Proof : = ålk GlkjF kl = ½( ålk GlkjF kl +  åkl GljkF kl ) = ½( ålk GlkjF kl +  ålk GkjlF kl ) = ½ålk( Glkj + Gkjl)F kl = ½ålkgkl , j F kl  .]

For antisymmetric Fp( ) it is         Ñð¿Fp(ej) = jp-1 åkÐek(jp Fkj)
[ Proof : ålk GlkjF kl = 0  .]

Material-energy 1-tensor in a curved space
As in the flat space case, a natter flow mp = mpup can be alternatively expressed as via the symmetric 1-tensor
Tp(a) º (mp¿a)up = mp(up¿a)up = -mp¯up(a) = -mp¯mp(a)   =   -¯°mp(a)
as the "flow of 4-momentum in direction a".
We now have covariant coordinate representatuions   Tijp = mpuiuj    ;       Tij  =  åkgjkTijp = mpuiuj .    ;     Tij  =  åkgikTij = mpuiuj ,
[ we can safely ommit the p suffix from the latter two expressions, since an underlined suffix always implies point dependance].

Ñp¿Tp(a) = mp¿(Ñp(a¿up))
[ Proof :  Ñp¿Tp(a) = ((Ñp(a¿up))mp + (a¿up)(Ñpmp))<0> = mp¿(Ñp(a¿up)) by the matter conservation law  .] and so vanishes when mp=0 (no matter, or matter at rest) and also when Ñp(a¿up)=0, ie. when a¿up is constant (to first order) in a neighbourhood of p.
Ñp¿Tp(a) is thus an a-directed measure of deviation from a "Gallilean" state of uniform motion.

The contraction of Tp(a) is scalar mp . [ Proof :  Ña¿Tp(a) = (Ña¿(a¿up)mp) = (Ña¿(a¿up))mp = (up)mp = mp  .]

If we are working in the mapspace. our small simplex has content jp|d4p| in UN and our conservation of matter law becomes
Ñp¿(jpmp) = Ñðp¿mp=0 .
The condition Ñp(a¿up)=0 for "Gallilean" flow is replaced by Ñðp(a¿up)=0 , the condition for geodesic flow.
Ñð¿Tp(ej) = 0       for a matter-conserving flow along geodesics .
[ Proof :   Ñð¿Tp(ej) = jp-1 åkÐek(jp Tkj) - ålk GljkT kl = jp-1 åkÐek(jpmpukuj) - ålk Gljkmpukul
= jp-1 åkÐek(jpmpuk)uj + mpåk(ukuj , k - ål Gljkukul)     [  chain rule ]     = jp-1 ujåkÐek(jpmpuk)     [  if geodesic flow ]     = 0     [  by matter consvervation ]  .]

Slow Flows

We say a flow is slow with regard to e4 if up » e4 " p : mp ¹ 0 so that all e4-percieved spacial speeds of matter are vastly less than 1. We then have mp » mpe4 Þ Tp(a) » mp ¯(a,e4) . All matrix representation components of Tp are then negligible save e4¿Tp(e4) » mp » Ña¿Tp(a) .

Curvature 2-Tensor
[  Note that R is a more common symbol for the curvature tensor in the GR context , we will retain C here, reserving R for rotors. ]
As discussed in Multivector Manifolds, the curvature tensor (aka. Riemann-Christoffel tensor) arises as the skewsymmetrolof the second directional coderivative.
2(Ða×Ðb)Fp = cpaÙb×Fp .
The Rici 1-tensor and curvature 0-tensor are its first and second contractions Ña¿cpaÙb and Ñb¿(Ña¿cpaÙb) .
Conventional GR considers their actions with regard to the map rather than UN     Cx(aÙb) = _f_i(cp_f_i(aÙb) .
Because of the symmetry of the connection, Cx(a)b) is a 4D 2-multiform (ie. a bivector-valued linear-function of a bivector) and so can be represented by 6 bivectors or 36 scalars. It is, however, more commonly regarded in GR as a bivector-specific 1-tensor, ie. a (1;3)-tensor skewsymmetric in two of its "inputs". Only 20 of the  44=256 scalars in map-based matrix-representations are independant.
It follows from the obvious but tiresome algebraic manipulations of operator      Ða =  Ða + Gp(  , a)        (where aÎIp) that the ith covariant coordinate of 2(Ðej×Ðek)vp is given by
ei¿(Cx(ej)ek)(vp)) = ål=14  Ci jkl vl where
Ci jkl  º ei¿(Cx(ej)ek)(el)) = G lik ,j + åm=14  G mik G lmj  - í¬ýj«k .
We also have the representation
Ci jkl  º ei¿(Cx(ej)ek)(el)) = åm=14 glm Ci jkm  =...= Glik ,j - åm Gmlj G mik - í¬ýj«k
[ Rijkl = Cj kli  and Rijkl = Cj kli  are standard GR notations, although some authors differ with regard to suffix order ]
For a diagonalised gp , Ci jkl =0 for distinct i,j,k,l.

The Ricci tensor and scalar curvatures can be expressed, via contraction summations of the Riemann cuirvature, as somewhat grotesque expressions involving the G ijk and derivatives G ijk ,l and hence as even grimmer expressions involving terms of the forms gij , gij,j , gij,k , and gij,kl .

Einstein 1-Tensor
Gp(a) º Cp(a) - ½Cpa   =   (1-½ÑÞ)[Ñ¯]2¯(a) is known as the Einstein 1-tensor. It has zero point-divergence (codivergence) and so can be regarded as a direction-specific 1-potential. Its directional-divergence (contraction) is scalar ½(2-M)Cp = -Cp .
We have Gkn = ½ åij( Cxeij . eijk )
[ Proof :   åij( Cxeij . eijk ).en = åij( (Cxeij Ù en) . eijk )     [ by multivector identity:  (b2.c3).a = (b2Ùa).c3 ]
= åij( ((ålmClmij elm)Ùen) . eijk ) = åijlm  Clmij(elmn . eijk ) = -åijlm  Clmij dijklmn   = ... = åm 2Cmkmn - dknC   = 2Gkn
[ dijklmn º   ± 1 if i j k are an even/odd permutation of distinct l m n ;  0 else , is the primary matrix representation of the unit (identity) 3-multiform that maps any trivector to itself. ] See Girard for a fuller discussion  .]

GR Field Equation
Einstein's insight was to essentially equate the zero-codivergent matter flow 1-tensor with the zero-codivergent Einstein 1-tensor, up to scalar multiplication and the addition of a zero-divergent scaled identity function. He postulated
Gp(a) = kTp(a) + la     " a     for universal constants k and l. To provide agreement with Newtonian gravitation to first order, k must be set to -8pg where g is the Newtonian gravitational constant and observations suggest l to be very small or zero.
The GR Field Equation is thus Ep(a) º   Cp(a) - ½Cpa  + 8pg Tp(a) - la = 0
where

• Cp(a) is the Ricci 1-tensor and Cp its contraction ie. the scalar curvature.
• Tp(a) is the material-energy 1-tensor
• Scalar cosmological constant 0 £ l » 10-49 m-2 » 2-156 m-2 is so small (but positive for a de Sitter Â1,3 spacetime) as to be effectively 0 at subcosmological scales.   For Â3,1 spactime l is negative.
[ Coordinate-based formulations such as     Cij - ½gijC + 8pgTij - lgij = 0     are more common in the GR literature. ]

In the operator notation of Multivector Manifolds chapter, we have
(1-½ÑÞ)[Ñ¯]2¯ = 8pg ¯°mp + l     abbreviating     (1-½ÑÞ)[Ñ¯]2¯(a) = 8pg ¯°mp(a) + la

Contraction of the GR Field Equation yields the scalar equation     -Cp + 8pgTp - 4l = 0     .
Ñðp¿Ep(a) = 0 since Gp(a) º Cp(a) - ½Cpa , Tp(a), and a all have zero codivergence seperately.

Setting l=0 gives the more usual GR equation Gp(a) = -8pg Tp(a) .     In vacuum (mp=0) and taking l=0 we have Gp(a) = 0 ,  which is equivalent to Cp(a) = 0 . The vacuum GR equation can thus be expressed as a formidable second order differential equation in the ten gij metric "coordinates".

Newtonian approximation
We say a gravitational connection is weak if gp(a)= a + hp(a) where |hp| is small. We have gij = dij + hij for small symmetric hij.
We then have Gijk »  G ijk both sufficiently small that Ci jkl  » G lik ,j - G lij ,k » Glik ,j - Glij ,k .
We say a gravitational field is static if ,4 derviatives are negligable so that, in particular, Gi44 »  -½g44 ,i . A field can only be static with regard to a favoured timelike e4.
For a weak, slow, static connection the geodesic trajectory equation p"(t) » -(v4)2Gp(e4 , e4) reduces to Newtonian equations vi/t » ½h44 ,i
[ Proof : vi/x4 x4/t » (v4)2½g44 ,i Þ vi/t v4 » (v4)2½g44 ,i Þ vi/t » v4½g44 ,i » ½g44 ,i = ½h44 ,i  .] so for a weak static connection we have
C44 » -åk Gk44 ,k = -åk=14  (½(gk4,4+gk4,4 - g44,k)),k » åk=13 g44,kk = Ñr2g44     where Ñr is the standard ("flat") 3D Euclidean Ñ acting in e4* in the mapspace.
Since Cp = -8pgTp = mp for a slow flow, e4¿Ep(e4) = C44 + e4¿(4pgTp)e4 + 8pgT44 + l Þ Ñr2g44 + 4pgmp » 0  .
h44 = g44 - 1 thus acts as a Newtonian gravitational potential if we assume g44=1 at infinity.

Yilmaz Gravitation
Yilmaz rejects the matter conseveration law Ñð¿mp = 0 and favours the Freud Identity Ñ¿(JpTp(a)) = 0 . Consequently
Ñð¿Tp(ej) = jp-1 åkÐek(jp Tkj) - ålk GljkT kl = - ålk GljkT kl = - mpåk ukuj , k     [  if geodesic flow ]
Accordingly, our field equation becomes        Cp(a) - ½Cpa  + 8pg Tp(a) + 2tp(a) - la = 0
where tp(a) º - 4pg ålk GljkT kl = - 4pg mpåkukuj , k     is known as the "stress-energy 1-tensor of the gravitational field" .
l is usually taken to be zero in Yilmaz theory. The notation tp(a)º4pg Tp(a) is common.

Schwarschild Solution
Black holes are postulated to arise  a static (time-invariant) spherically symmetric solution to the GR vecuum Field equation known as the   Schwarschild solution. A spherically symmetric metric with gij,4=0 and g4i=0 is found to be
ds2 = (1-2mr-1)dt2 - (1-2mr-1)-1dr2 - r2(dq2 + sin(q)2df2)     [ Rindler 11.13 ]
at a given event te4 + r where 3D spacial point r is expressed in spherical polar coordinates as [q,f,r] with r>r0 and m is a constant of integration having dimension of length (meters) which we associate with GMc-2 where M is the mass of a central body of radius r1 and Gc-2 » 2-231.14 m2 is the (c-scaled) squared-length Gravitational constant.
Taking Gc-2 = 2-90.12 m kg-1 and inserting earth mass M=282.30 kg = 2223.33 m-1 we 2m = 2-6.81 m » 0.88 cm for Schwarzchid radius of an earth-mass black hole while proton and electron masses 2-88.95 kg and 2-99.79 kg give vastly sub-Planck length radii 2-179.0 m and 2-188.9 m respectively.
Whether time-invarient (with regard to which privilidged observer?) solutions actually occurr in nature is unknown to this author. The Schwarschild solution for a point mass has a discontinuity at r=2m (the event horizon) and many scientific careers have been built analysing this singularity.
Such singularities do not arise from the Yilmaz field equations.

The Euclidean-Schwarzschild metric is obtained by substituting Q=2pT-1i-1t and x=4m(1-2mr-1)½ (running from -8m2¥ at r=0 through 0 at r=2m to 4m at r=¥) to obtain ds2 = x2(4m)-2(dQ)2 + (r2(4m2)-1)2 (dx)2 + r2((dq)2+ sin(q)2(df)2) is Euclidean in the sense of a positive definite metric.
Setting R=(1-2mr-1)½ (so that x=4mR ) running monotonically upward from -2m¥ at r=0 through R=0 at r=2m to R=1 at r=¥ gives
ds2 = R2T2(2p)-2(dQ)2 + (4m)2(r2(4m2)-1)2 (dR)2 + r2((dq)2+ sin(q)2(df)2) is Euclidean in the sense of a positive definite metric.

Hawking considers e4-periodic rather than an e4-stationary solutions associating the period T having period T (with period T)  t-periodic stationary

Incorporating nonzero cosmological constant l changes the metric for r>r0  to
ds2 =  a(r) dt2 - a(r)-1dr2 - r2(dq2 + sin(q)2df2)     where a º a(r) º (1-2mr-1 - 3-1lr2) runs from -2m¥ through 0 at ? to - 3-1l_infinity_sqrd ar r=¥ [ Rindler 14.22 ] .
This has infinites when r solves cubic ra(r) = 2m-r+3-1lr3 = 0 which occurs for r slightly below m(2 + 3-18lm(1+2lm2)-1) and (for nonhuge m) slightly below m independant distance 3½l » 282 m which is about 2-5 times the distance of the furthest known galaxy but 216 times the theorised 265.7 m characteristic length of gravity . Since it would take an infinite (externally percieved) time for a graviton emitted by the central mass to cross over this outter horizon , the gravitonic paradigm precludes influence at distances greater than this, making gravity a finite range force.

Geodesics restricted to the qp plane have Lagrangian L(f,r,t,f',r',t',t) = L(f',r',t',t) = at'2 - a-1r'2 - r2f'2 and since this is independant of f and t we have two geodesic-constants, angular momentum h=r2f' and energy k=a(r)t' .
Setting L(f',r',t',t)=1  provides a(r)(ka(r)-1)2 - a(r)-1r'2 - r2(hr-2)2 = 1 Þ k2 - r'2 - ar-2h2   =   a yielding geodesic condition
r'2   =   k2 - a(r) (r-2h2 + 1 )   =   k2 - a(r)r-2(h2 + r2) .
Radial trajectories having constant f and h=0 have Lagrangian L(r',t',t) = at'2 - a-1r'2 with geodesic condition r'2   =   k2 - a(r) and we see that a(r) is acting like a scalar potential energy, zero at both horizons and positive between them.
We thus have dt/dr = ±|k2 - a(r)| and inserting k=a(r0) for a test particle released at rest (t'=1) at r=r0>2m gives dt/dr = ±|a(r0)2 - a(r)| .
For l=0 and setting E=½(k2-1) this is dt/dr = ±|k2 - 1 + 2mr-1|   =   ±2|E + mr-1|   =   ±(2(E + mr-1)) assuming mr-1 > -E along the trajectory.

This differs from Rindler who sets E = -mr0-1 by analogy with the Newton form and so has k=|1+2E|½ = (1-2mr0-1)½ = a(r0)½. [ Rindler 12.1.C ] .
Regardless, the integration òr0r dr (2(E + mr-1)| is finite for any R>0 (though recall that our vacuum metric is only valid for r>0) so a radial geodesic crosses the Schwarschild radius in finite proper time.
However, the path integration recovering external time t has a ka(r)-1 factor and consequently diverges as r®2m so external observers consider an infalling particle to never reach the "event horizon",

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Latest Edit: 14 Jun 2014. 