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 ¦^Ñ_p(a) º (a¿Ñ_p)¦(p) ; ¦^-Ñ for the inverse differential (¦^Ñ)^-1 ; ¦^D for the adjoint ¦^D_p(a) º Ñ_p(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 m_p is a (nonunit) timelike 1-vector field over a flat spacetime such that |d⁴p|m_p represents the 4-momentum of the matter in a small 3-simplex (4-volume) element d⁴p at p. We assume the matter conservation law Ñ_p¿m_p = 0     to apply everywhere other than at well defined "sinks" and "sources" where matter is somehow "introduced" or "removed".
    We can express m_p as m_pu_p where scalar m_p º |m_p| and unit timelike 1-vector u_p º m_p^~   With regard to an observer e₄, m_p splits via u_p = (1-v_p²)^½(v_p+e₄). into three-momentum m_pv_p and "relative" scalar density m_p(1-v_p²)^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 (m_p¿a). Each "unit" of that matter provides unit 1-vector 4-momentum u_p so we can regard
     T_p(a) º (m_p¿a)u_p    = m_p(u_p¿a)u_p = -m_p¯_up(a) = -m_p¯_mp(a) = -¯°_mp(a) .
as the "flow of 4-momentum in direction a".
    T_p(u_p) = m_p(u_p²)u_p = -m_pu_p = -m_p .
    T_p(a) is symmetric, and often represented by T^ij_p = m_puⁱu^j    ;       Tⁱ_j = m_puⁱu_j .    ;     or T_ij  = m_pu_iu_j , The use of 'T' for matter flow is traditional in the GR literature, and should not be confused with 'T' for "Time".
[ Proof :  T^ij_p º eⁱ¿T_p(e^j) = eⁱ¿m_p(u_p¿e^j)u_p = m_p(eⁱ¿u_p)(e^j¿u_p) = m_puⁱu^j  .]

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

    The contraction of T_p(a) is scalar -m_p . [ Proof :  Ñ_a¿T_p(a) = (Ñ_a¿(a¿u_p)m_p) = (Ñ_a¿(a¿u_p))m_p = (u_p)m_p = m_p u_p²  .]

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 Â^3,1 into a homogeneous "flat" ND spacetime. We extend Â_3,1 by postulating a further nonnull basis vector e₀ and potentially further nonnull vectors e₅,..,e_N-1 into a higherdimensional space U^N º Â^p,q,r with p+q+r = N ³ 5 ; p ³ 3, q ³ 1 . We assume an identity embedding 1 : Â_3,1 ®  U_N acting on 1-vectors so that we can move 1-vectors (and, by outtermorphic extension, multivectors) "unchanged" between R_3,1 and  (a subspace of) U_N. Let (e₁,e₂,e₃,e₄) be a fixed orthonormal frame for R_3,1 and define unit pseudoscalar i=e₁₂₃₄. Using 1 , we can extend this  into a fixed orthonormal basis (e₀,e₁,e₂,...e_N-1) for U_N with unit pseudoscalar u=e_012..(N-1).
      We assume an orthonormal inverse frame (e⁰,e²,...e^N) for U^N with e¹=e₁,e²=e₂,e³=e₃,e⁴=-e₄. This frame induces N seperate N-D vector differential operators d_xⁱ = d_ei = e_i¿Ñ_p .

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

    In essence, the manifold is determined by ¦^Ñ_p . ¦ can be considered as the "integrative consequence" of ¦^Ñ_p 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 ¦^-Ñ_p2 .

    ¦^Ñ_p generates a point-dependant invertible embedded frame { h_ip = ¦^Ñ_p(e_i) ; 0£i£N } for J_p

Coordinate-based Approach
    We can represent the point embedding ¦ over R^3,1 by the N scalar functions  yⁿ(x¹,x²,x³,x⁴) defined by
     ¦(x) = ¦(å_ixⁱe_i) = å_n=0^N-1 yⁿ(x¹,x²,x³,x⁴)e_n = å_n=0^N-1 yⁿe_n .
    Unless otherwise indicated, integer suffixes i,j and k are henceforth to be considered to range under S from 1 to 4 inclusive.
    ¦^Ñ_p can be repesented by two associated point-dependant Nx4 matrices:
    yⁿ_{,i p} º eⁿ¿¦^Ñ_p(e_i) = ¶yⁿ/¶xⁱ     ;     y_{n,i p} º e_n¿¦^Ñ_p(e_i) = e_i¿¦^D_p(e_n)     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 ¶/¶x^j ; and so forth . Thus yⁿ_,ij º yⁿ_{, i , j} º ¶yⁿ_,i/¶x^j º (¶/¶x^j)(¶yⁿ/¶xⁱ) .
     We have yⁿ_,i =Sig(e_n)y_n,i     0 £ n < N .
      ¦^Ñ_p(e_i) = å_n=0^N-1 yⁿ_,ie_n = å_n=0^N-1 y_n,ieⁿ     ;     ¦^D_p(e_n) = å_i=1⁴ y_n,ieⁱ .
[ Proof : ¦^D_p(e_n) = å_i=1⁴ (¦^D_p(e_n)¿e_i)eⁱ = å_i=1⁴ (e_n¿¦^Ñ_p(e_i))eⁱ = å_i=1⁴ y_n,ieⁱ  .]

    ¦^Ñ2_p(a,b) = å_n=0^N-1 å_ij yⁿ_,ijaⁱb^je_n = å_n=0^N-1 å_ij y_n,ijaⁱb^jeⁿ     where y_n,ij º ¶y_n,i/¶x^j .
[ Proof : ¦^Ñ_p+dp(e_i) = å_n=0^N-1  yⁿ_,i(p+dp)e_n = å_n=0^N-1 (yⁿ_,i(p) + å_j=1⁴ yⁿ_,ijdp^j)e_n + _O(dp²)       and the result follows  .] .

Metric Tensor
    g_p : R_3,1 ® R_3,1 defined by the composition g_p º ¦^D_p¦^Ñ_p is comfortingly four-dimensional and also symmetric.
[ Proof :  g_p^D = (¦^D_p¦^Ñ_p)^D = ¦^Ñ_p^D¦^D_p^D = ¦^D_p¦^Ñ_p = g_p  .]
    g_p is known as the metric tensor since ¦^Ñ_p(a)¿¦^Ñ_p(b) = a¿ g_p(b) = g_p(a)¿b and in particular (¦^Ñ_p(a))² = a¿(g_p(a)) . Physicists often refer to v¿g_p(v) =  ¦^Ñ_p(v)²  (rather than v²) as the "magnitude" of v. [ Proof : ¦^Ñ_p(a)¿¦^Ñ_p(b) = a¿¦^D_p(¦^Ñ_p(b)) = a¿g_p(b)  .]
    Thus a¿g_p(a) represents the second order approximation of (¦(p+a) - ¦(p))² with regard to mapspace displacement a .
    (¦(p+a) - ¦(p))² = ¦^Ñ_p(a)² + _O(a²) = a¿g_p(a) + _O(a²)  .
    g_p and g_p^-1 = ¦^-Ñ_p¦^-D_p are sometimes known as the fundamental tensors with an associated invariant line element dp¿g_p(dp) = ¦^Ñ_p(dp)².
    If ¦^Ñ_p is symmetric, g_p = ¦^Ñ_p ² and ¦^Ñ_p = "Ö g_p" is analagous to the "fiducial tensor" and "positional gauge" h_p of other treatments.
    g_p has scalar determinant g = g_p º | g_p | = j_p² . [  In conventional GR treatments, j_p appears as Ög or Ö-g according as to how the metric is signed . ]

Coordinate-based Approach
    Rather than representing g_p as gⁱ_j º eⁱ¿g_p(e_j) we choose the quadratic form representation
    g_ijp º g_p(e_i)¿e_j = ¦^Ñ_p(e_i)¿¦^Ñ_p(e_j) = h_ip¿h_jp = å_n=0^N-1 yⁿ_{, i} y_{n , j}    ;     e_i º g_p(e_i) = å_k=1⁴ g_ik e^k     and hence     a¿b = å_ij=1⁴ g_ijaⁱb^j
[ Proof :  a¿b = a¿g_p(b) = a¿g_p(å_i= bⁱe_i) = a¿å_ibⁱg_p(e_i)) = a¿å_ijbⁱg_ij e^j)  .]
    g^ij_p  º g_p^-1(eⁱ)¿e^j     (so that e ^j º g_p^-1(e^j) = å_kg^kje_k)     is the conventional matrix inverse of g_{ij p}.
[ Proof : å_k=1⁴ g_ikg^kj   = å_k(g_p(e_i)¿e_k)g^kj = g_p(e_i)¿å_ke_kg^kj = g_p(e_i)¿g_p^-1(e^j) = e_i¿g_p(g_p^-1(e^j)) = d_{i j}  .]
    Thus g^ij_p = G^ij_p / |g_p| where G^ij_p is the i,j ^th cofactor of matrix { g_ijp } .
    Note carefully that g^ij breaches our usual suffix convention in that it represents eⁱ¿g_p^-1(e^j) rather than eⁱ¿g_p(e^j). A more appropriate notation would be g^-ij but g^ij is universal in the GR literature and we follow it here.
    Ommitting _p subscripts, we cite without proof a "determinant derivative rule" g_{, k} = gå_{i j=1}⁴  g^ij g_ij,k .

    Since g_p 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 g_p can be diagnonalised (not guaranteed for symmetric g_p in a NonEuclidean space), we have four "core" potentials with gⁱⁱ=1/g_ii . [  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 g_p(dx,dy) = (dx , sin²qdy) with associated line element dp¿g_p(dp) = dx² + cos²qdy².
    g_p 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 g_p and a realm of coordinate based tensor analysis is entered from which few return unscathed.

Covariant Coordinates
    A base 4-frame {e₁, ..,e₄} for R_3,1 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 g_p^-1(eⁱ) } which itself has inverse 4-frame { e_i º g_p(e_i) } [ Proof :   e ⁱ¿e_j = g_p^-1(eⁱ)¿g_p(e_j) = g_p(g_p^-1(eⁱ))¿e_j = eⁱ¿e_j  .]
    Trivially, e_i¿e_j = e_i¿e_j = g_ij     ;        eⁱ¿e ^j = eⁱ¿e^j = g^ij .
    A given v_p Î R_3,1 can thus be expressed both as v_p = å_ivⁱe_i     ( with contravariant coordinates vⁱ º eⁱ¿v_p = e ⁱ¿v_p ) and as v_p = å_iv_i e ⁱ     ( with covariant coordinates v_i º e_i¿v_p = e_i¿v_p = e_i¿g_p(v_p) = å_j=1⁴ g_ijv^j ) .
    The "motivation" for covariant coordinates is that å_i=1⁴ wⁱv_i = ¦^Ñ_p(w_p)¿¦^Ñ_p(v_p) .

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 g_ij representing the contravariant quadratic form expression g_ijºe_i¿g_p(e_j) as here rather than g_ijºe_i¿g_p(e_j) = g_p(e_i)¿g_p²(e_j) = e_i¿g_p³(e_j) ; with g^ijº eⁱ¿g_p(e^j) denoting the inverse matrix for { g_ij } .
    gⁱ_j here represents eⁱ¿g_p(e_j) whereas in coventional GR treatments it denotes gⁱ_j=e ⁱ¿g_p(e_j) = d ⁱ_j = ?(i-j,1,0) = (i=j) , ie. the Kroneka delta matrix representation of the "identity" 1-tensor.

Parallel Transport
    Moving b = ¦^Ñ_p(0)(v) along a path p(t) within ¦(R^3,1) while continously projecting into J_p(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  G_p(b,c)  º ¦^-Ñ_p(¦^Ñ2_p(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 ^k_ijp º e^k¿G_p(e_i,e_j) = e^k¿¦^-Ñ_p(¦^Ñ2_p(e_i,e_j)) = ¦^-D_p(e^k)¿¦^Ñ2_p(e_i,e_j) = ¦^Ñ_p(e^k)¿¦^Ñ2_p(e_i,e_j)     is called an affinity (aka. a Christophel symbol of the second kind) and represents a linear (affine) connection as discussed in Multivector Manifolds.
    G_kij º e_k¿G_p(e_i,e_j)   = ¦^-D_p(e_k)¿¦^Ñ2_p(e_i,e_j) = ¦^Ñ_p(e_k)¿¦^Ñ2_p(e_i,e_j) = å_n=0^N-1 yⁿ_,k y_{n , ij}     is known as a Christoffel symbol of the first kind.
[ Proof : ¦^Ñ_p(e_k)¿¦^Ñ2_p(e_i,e_j) = (å_m=0^N-1 y^m _,ke_m)¿(å_n=0^N-1 y_{n ,ij}eⁿ) = å_nyⁿ_,k y_{n ,ij}  .]

    We can express G explicitly in terms of g_p (in the GR literature,   G_kij and G ^k_ij are often defined in this way).
    G_kij = ½(g_ki,j+g_kj,i - g_ij,k)       and hence     g_ki,j = G_kij + G_ijk .
[ Proof : g_ij=yⁿ_,i y_n,j Þ g_ij,k = yⁿ_,ik y_n,j + yⁿ_,ik y_{n, jk} = y_n,ik yⁿ_,j + y_n,i yⁿ_,jk and the result follows    .] _
    G ^l_ijp = å_k=1⁴ g^lk G_kij .
[ Proof : å_k=1⁴ g^lk¦^Ñ_p(e_k)¿¦^Ñ2_p(e_i,e_j) = ¦^Ñ_p(å_k=1⁴ g^lke_k)¿¦^Ñ2_p(e_i,e_j) = ¦^Ñ_p(g_p^-1(e^l)¿¦^Ñ2_p(e_i,e_j)  .]

    We have the symmetries: G_ijk= G_ikj    ;     G ⁱ_jk= G ⁱ_kj .

    We have the coordinate-based connective expressions for parallel transport
    - dv » G_p(v_p , dp) º å_ijk G ⁱ_jk v^kdp^j e_i = -å_ijk G ^k_ijv_k dp^j e ⁱ = -å_ijk G_kijv^kdp^je ⁱ

    To prove these: let u_p[dp] denote the U_N parallel transport of direction u_p from ¦(p) to ¦(p+dp) with v_p[dp] the corresponding map direction. We will assume u_p[dp] = ¯(u_p,J_p+dp) + _O(dp²) .
       Let vⁱ(p,dp) and v_i (p,dp) be contra and covariant coordinates for v_p[dp] at p+dp
    ( ie. vⁱ(p,dp) º v_p[dp]¿eⁱ  ; v_i (p,dp) º g_p+dp(v_p[dp])¿eⁱ = å_j=1⁴ g_ijp+dpv^j(p,dp) ).     Clearly vⁱ(p,0) =   vⁱ _p and v_i (p,0) =   v_{i p} .
    dv_i  º v_i (p,dp) - v_i (p,0) = ¦^Ñ_p(v_p) ¿ ¦^Ñ2_p(e_i.dp) + _O(dp²)     = å_{j k} G_kijv^kdp^j + _O(dp²) . Hence dv = å_{i j k} G_kijv^kdp^je ⁱ .
[ Proof : v_i (p,dp) = g_p+dp(v_p[dp])¿e_i = ¦^Ñ_p+dp(v_p[dp])¿¦^Ñ_p+dp(e_i) = u_p[dp]¿¦^Ñ_p+dp(e_i) = u_p(p)¿¦^Ñ_p+dp(e_i)
        = ¦^Ñ_p(v(p))¿¦^Ñ_p+dp(e_i) » ¦^Ñ_p(v(p))¿(¦^Ñ_p(e_i)+ ¦^Ñ2_p(e_i,dp)) = g_p(v(p))¿e_i + ¦^Ñ_p(v(p))¿¦^Ñ2_p(e_i,dp) = v_i (p,0) + ¦^Ñ_p(v_p)¿¦^Ñ2_p(e_i,dp)  .]
    dv_i  = å_{j l} G ^l_ijv_l dp^j + _O(dp²) . [ Proof :  å_{j k}v^k G_kijdp^j =  å_{j k}(å_lg^klv_l) G_kijdp^j  .]
    dvⁱ = - å_{j k} G ⁱ_jkv^kdp^j + _O(dp²) .
[ Proof : Let map vectors a and b be parallel transported. d(a¿b) = 0 Þ d(å_i=1⁴ a_ibⁱ)=0 Þ å_i((da_i)bⁱ + a_i(dbⁱ))=0
Þ å_ia_i(dbⁱ)) = -å_ijk G ^k_ija_k dp^j bⁱ = -å_ijk G ⁱ_kja_i dp^j b^k = -å_ia_iå_{j k} G ⁱ_kjdp^j b^k .     Since a arbitary , dbⁱ = - G ⁱ_kjdp^j b^k  .]
    This completes the proof of the connective expressions.

    For a diagonalised g_p we have G_kij = 0 if i,j,k are distinct ; -½g_{ii ,k} if i=j ; ½g_{ii, j} if i=k ; ½g_{jj , i} if j=k .
    Also G ^k_ij = g^kk G_kij = g_kk^-1 G_kij .

    We also have the perhaps surprising result å_k=1⁴ G^k_jk = ( ln(j_p)) _{, j} = j_p^-1 j_{p , j} .
[ Proof : g_,j = gå_i=1⁴ å_k=1⁴  g^ikg_ik,j = gå_i=1⁴ å_j=1⁴  g^ik (G_kij + G_ijk) = 2gå_i=1⁴ Gⁱ_ij
    Þ å_i=1⁴ Gⁱ_ij = ½ g^-1 g_,j = ½( ln(g))_,j = ( ln(j_p))_,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 R^3,1 ( with "natural parametrisation" condition p'(t)¿p'(t)=k   , a constant) solves the geodesic trajectory equation
   
     p"(t) + G_p(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)= ¦^Ñ_p(v(t)) .
    The normalisation condition u(p)² = 1 implies dt² = dp¿g_p(dp) .     [ Proof : (¦^Ñ_p(dp/dt))²=1 Þ dt² = (¦^Ñ_p(dp))² = dp¿g_p(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 ¿ g_p(dp)|^½ and so also extremal paths for the scalar Lagrangian L(p,p',t) = p'(t)¿g_p(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 v^Ñ(a) = Ð_av_p  is the standard (¿-based) differential in L.
    We thus have Ð_ei^ðe_j = G_p(e_j , e_i) .
    We define Ð_a^ðb = Ð_ab for scalars, and extend Ð_a^ð over map multivectors with the product rule. It is customary to use _:i to denote the effect of Ð_ei^ð on a coordinate in like manner to _,i denoting Ð_ei.
    The connective expressions give coordinate-expressions for Ð_ej^ðv_p :
    v_{i :j} = v_{i ,j} - å_k G ^k_ijv_k     ;     vⁱ_:j = vⁱ_{, j} + å_k G ^k_ijv_k .
    These can be generalised to (covariant) coordinate expressions for coderivatives of higher rank tensors such as
    F_{ij :k} = F_{ij ,k} - å_l( G ^l_ikF_lj + G ^l_jkF_il )    ;     F ⁱ_{j :k} = F ⁱ_{j ,k} + å_l( G ⁱ_lkF ^l_j - G ^l_jkF ⁱ_l ) .
[ Proof :  None here. See Rindler   .]

    A key result is Ð_a^ðe_i = 0     , the e_i 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 g_ij and obtain zero.
[g_{ij :k} = g_{ij ,k} - å_l( G ^l_ikg_lj + G ^l_jkg_il) = g_{ij ,k} - ( G_likg_lj + G_ljkg_il) =...= 0   ]
but is it really legitimate to apply this rule to the essentially contravariant g_ij rather than to g_ij ? The traditional low suffixes in g_ij were not assigned covariantly but as a represention of the quadratic form of the metric with regard to contravarient coordinates.. Is metric g_ij covariant with regard to itself? Under the notations employed here, g_ijºe_i¿g_p(e_j) =e_i¿g_p³(e_j)  .
[Under Construction. comment appreciated. ]  .]
    We can define a mapspace 1-vector codel-operator     Ñ^ð_p(a) º å_k=1⁴ e^k(Ð_ek^ða)     but we must take care here, as indicated by the brackets,  because the e^k are not "immune" to "scalar" operators Ð_ej^ð (as they are to Ð_ej in the Ñ defintion).  
    This gives the codivergence of a 1-field as  
    Ñ^ð_p¿v_p º  å_k=1⁴ e^k¿(Ð_ek^ðv_p) =  å_kv^k_:k =  (Ñ_p¿(j_pv_p)) j_p^-1
[ Proof :   =  å_k(v^k_,k + å_j G ^k_jkv^j) =  å_kv^k_,k + å_jj_p^-1 j_{p ,j}v^j =  å_k (j_pv^k)_,k j_p^-1  .]
    Thus we have      (Ñ^ð_p¿v_p) j_p = Ñ_p¿(v_pj_p) and so Ñ^ð_p¿v_p = ¯ _vpjp( Ñ_p ) v_p .
[ Proof :   ¯ _vpjp(Ñ_p) v_p º (Ñ_p¿(v_pj_p) )(v_pj_p)^-1 v_p = (Ñ_p¿(v_pj_p) )j_p^-1 = Ñ_p¿(v_pj_p) = Ñ^ð_p¿v_p  .]

    We have a covariant coordinate expression for the codivergence of a 1-tensor F_p( ) :
    Ñ^ð¿F_p(e_j) = å_k F^k_{j : k} = j_p^-1 å_kÐ_ek(j_p F^k_j) - å_lk G_ljkF ^kl
[ Proof : å_k F^k_{j : k} = å_k ( F^k_{j , k}+ å_l( G ^k_lkF ^l_j - G ^l_jkF ^k_l ) ) = å_k F^k_{j , k}  + å_{_lk} G ^k_lk F^l_j -  å_lk G_ljkF ^kl
    = å_k F^k_{j , k} +  å_k(j_p^-1 j_{p , k}) F^k_j -  å_lk G_ljkF ^kl = j_p^-1 å_k(j_p F^k_{j , k} + j_{p , k} F^k_j) -  å_lk G_ljkF ^kl  .]

    For symmetric F_p( ) this becomes     Ñ^ð¿F_p(e_j) = j_p^-1 å_kÐ_ek(j_p F^k_j) - ½å_lkg_{kl , j} F ^kl .
[ Proof : = å_lk G_lkjF ^kl = ½( å_lk G_lkjF ^kl +  å_kl G_ljkF ^kl ) = ½( å_lk G_lkjF ^kl +  å_lk G_kjlF ^kl ) = ½å_lk( G_lkj + G_kjl)F ^kl = ½å_lkg_{kl , j} F ^kl  .]

    For antisymmetric F_p( ) it is         Ñ^ð¿F_p(e_j) = j_p^-1 å_kÐ_ek(j_p F^k_j)
[ Proof : å_lk G_lkjF ^kl = 0  .]

Material-energy 1-tensor in a curved space
    As in the flat space case, a natter flow m_p = m_pu_p can be alternatively expressed as via the symmetric 1-tensor
     T_p(a) º (m_p¿a)u_p = m_p(u_p¿a)u_p = -m_p¯_up(a) = -m_p¯_mp(a)   =   -¯°_mp(a)
as the "flow of 4-momentum in direction a".
    We now have covariant coordinate representatuions   T^ij_p = m_puⁱu^j    ;       Tⁱ_j  =  å_kg_jkT^ij_p = m_puⁱu_j .    ;     T_ij  =  å_kg_ikTⁱ_j = m_pu_iu_j ,
[ we can safely ommit the _p suffix from the latter two expressions, since an underlined suffix always implies point dependance].

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

    The contraction of T_p(a) is scalar m_p . [ Proof :  Ñ_a¿T_p(a) = (Ñ_a¿(a¿u_p)m_p) = (Ñ_a¿(a¿u_p))m_p = (u_p)m_p = m_p  .]

    If we are working in the mapspace. our small simplex has content j_p|d⁴p| in U_N and our conservation of matter law becomes
    Ñ_p¿(j_pm_p) = Ñ^ð_p¿m_p=0 .
    The condition Ñ_p(a¿u_p)=0 for "Gallilean" flow is replaced by Ñ^ð_p(a¿u_p)=0 , the condition for geodesic flow.
    Ñ^ð¿T_p(e_j) = 0       for a matter-conserving flow along geodesics .
[ Proof :   Ñ^ð¿T_p(e_j) = j_p^-1 å_kÐ_ek(j_p T^k_j) - å_lk G_ljkT ^kl = j_p^-1 å_kÐ_ek(j_pm_pu^ku_j) - å_lk G_ljkm_pu^ku^l
    = j_p^-1 å_kÐ_ek(j_pm_pu^k)u_j + m_på_k(u^ku_{j , k} - å_l G_ljku^ku^l)     [  chain rule ]     = j_p^-1 u_jå_kÐ_ek(j_pm_pu^k)     [  if geodesic flow ]     = 0     [  by matter consvervation ]  .]

Slow Flows

    We say a flow is slow with regard to e₄ if u_p » e₄ " p : m_p ¹ 0 so that all e₄-percieved spacial speeds of matter are vastly less than 1. We then have m_p » m_pe₄ Þ T_p(a) » m_p ¯(a,e₄) . All matrix representation components of T_p are then negligible save e⁴¿T_p(e₄) » m_p » Ñ_a¿T_p(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(Ð_a×Ð_b)F_p = c_paÙb×F_p .
    The Rici 1-tensor and curvature 0-tensor are its first and second contractions Ñ_a¿c_paÙb and Ñ_b¿(Ñ_a¿c_paÙb) .
    Conventional GR considers their actions with regard to the map rather than U_N     C_x(aÙb) = _f_i(c_{p_f_i(aÙb}) .
    Because of the symmetry of the connection, C_x(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      Ð_a =  Ð_a + G_p(  , a)        (where aÎI_p) that the i^th covariant coordinate of 2(Ð_ej×Ð_ek)v_p is given by
    e_i¿(C_x(e_j)e_k)(v_p)) = å_l=1⁴  C_i jk^l v_l where
    C_i jk^l  º e_i¿(C_x(e_j)e_k)(e^l)) = G ^l_ik ,j + å_m=1⁴  G ^m_ik G ^l_mj  - í¬ý_j«k .
     We also have the representation
    C_i jkl  º e_i¿(C_x(e_j)e_k)(e_l)) = å_m=1⁴ g_lm C_i jk^m  =...= G_lik ,j - å_m G_mlj G ^m_ik - í¬ý_j«k
    [ Rⁱ_jkl = C_j klⁱ  and R_ijkl = C_j kli  are standard GR notations, although some authors differ with regard to suffix order ]
    For a diagonalised g_p , C_i jk^l =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 ⁱ_jk and derivatives G ⁱ_jk ,l and hence as even grimmer expressions involving terms of the forms g^ij , g^ij_,j , g_ij,k , and g_ij,kl .

Einstein 1-Tensor
    G_p(a) º C_p(a) - ½C_pa   =   (1-½Ñ_Þ)[ѯ]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)C_p = -C_p .
    We have G^k_n = ½ å_ij( C_xeij . e^ijk )
[ Proof :   å_ij( C_xeij . e^ijk ).e_n = å_ij( (C_xeij Ù e_n) . e^ijk )     [ by multivector identity:  (b₂.c₃).a = (b₂Ùa).c₃ ]
    = å_ij( ((å_lmC^lm_ij e_lm)Ùe_n) . e^ijk ) = å_ijlm  C^lm_ij(e_lmn . e^ijk ) = -å_ijlm  C^lm_ij d^ijk_lmn   = ... = å_m 2C^mk_mn - d^k_nC   = 2G^k_n
[ d^ijk_lmn 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
    G_p(a) = kT_p(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 E_p(a) º   C_p(a) - ½C_pa  + 8pg T_p(a) - la = 0
where

• C_p(a) is the Ricci 1-tensor and C_p its contraction ie. the scalar curvature.
• T_p(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 Â_1,3 spacetime) as to be effectively 0 at subcosmological scales.   For Â_3,1 spactime l is negative.

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

    Contraction of the GR Field Equation yields the scalar equation     -C_p + 8pgT_p - 4l = 0     .
    Ñ^ð_p¿E_p(a) = 0 since G_p(a) º C_p(a) - ½C_pa , T_p(a), and a all have zero codivergence seperately.

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

Newtonian approximation
    We say a gravitational connection is weak if g_p(a)= a + h_p(a) where |h_p| is small. We have g_ij = d_ij + h_ij for small symmetric h_ij.
     We then have G_ijk »  G ⁱ_jk both sufficiently small that C_i jk^l  » G ^l_ik ,j - G ^l_ij ,k » G_lik ,j - G_lij ,k .
    We say a gravitational field is static if _,4 derviatives are negligable so that, in particular, G_i44 »  -½g_44 ,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)²G_p(e₄ , e₄) reduces to Newtonian equations ¶vⁱ/¶t » ½h_44 ,i
[ Proof : ¶vⁱ/¶x⁴ ¶x⁴/¶t » (v⁴)²½g_44 ,i Þ ¶vⁱ/¶t v⁴ » (v⁴)²½g_44 ,i Þ ¶vⁱ/¶t » v⁴½g_44 ,i » ½g_44 ,i = ½h_44 ,i  .] so for a weak static connection we have
    C₄₄ » -å_k G_k44 ,k = -å_k=1⁴  (½(g_k4,4+g_k4,4 - g_44,k))_,k » å_k=1³ g_44,kk = Ñ_r²g₄₄     where Ñ_r is the standard ("flat") 3D Euclidean Ñ acting in e₄^* in the mapspace.
    Since C_p = -8pgT_p = m_p for a slow flow, e₄¿E_p(e₄) = C₄₄ + e₄¿(4pgT_p)e₄ + 8pgT₄₄ + l Þ Ñ_r²g₄₄ + 4pgm_p » 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 Ñ^ð¿m_p = 0 and favours the Freud Identity Ñ¿(J_pT_p(a)) = 0 . Consequently
    Ñ^ð¿T_p(e_j) = j_p^-1 å_kÐ_ek(j_p T^k_j) - å_lk G_ljkT ^kl = - å_lk G_ljkT ^kl = - m_på_k u^ku_{j , k}     [  if geodesic flow ]    
    Accordingly, our field equation becomes        C_p(a) - ½C_pa  + 8pg T_p(a) + 2t_p(a) - la = 0
    where t_p(a) º - 4pg å_lk G_ljkT ^kl = - 4pg m_på_ku^ku_{j , 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 t_p(a)º4pg T_p(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 g_ij,4=0 and g_4i=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=2^82.30 kg = 2^223.33 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 3^½l^-½ » 2⁸² m which is about 2^-5 times the distance of the furthest known galaxy but 2¹⁶ times the theorised 2^65.7 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 q=½p 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)|^-½ 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)|^-½ .
    For l=0 and setting E=½(k²-1) this is dt/dr = ±|k² - 1 + 2mr^-1|^-½   =   ±2^-½|E + mr^-1|^-½   =   ±(2(E + mr^-1))^-½ 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 ò_r0^r dr (2(E + mr^-1)|^-½ 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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Copyright (c) Ian C G Bell 1999
Web Source: www.iancgbell.clara.net/maths
Latest Edit: 14 Jun 2014.