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 |d^{4}p|m_{p} represents the 4-momentum of the matter in a
small 3-simplex (4-volume) element d^{4}p 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_{p}u_{p} where scalar m_{p} º |m_{p}|
and unit timelike 1-vector u_{p} º m_{p}^{~}
With regard to an observer e_{4}, m_{p} splits
via u_{p} = (1-v_{p}^{2})^{½}(v_{p}+e_{4}).
into three-momentum m_{p}v_{p}
and "relative" scalar density m_{p}(1-v_{p}^{2})^{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_{p}^{2})u_{p} = -m_{p}u_{p} = -m_{p} .
T_{p}(a) is symmetric, and often represented by
T^{ij}_{p} = m_{p}u^{i}u^{j}
;
T^{i}_{j} = m_{p}u^{i}u_{j} .
; or
T_{ij } = m_{p}u_{i}u_{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 e^{i}¿T_{p}(e^{j})
= e^{i}¿m_{p}(u_{p}¿e^{j})u_{p}
= m_{p}(e^{i}¿u_{p})(e^{j}¿u_{p})
= m_{p}u^{i}u^{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_{p}m_{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_{p}^{2}
.]
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 e_{0} and
potentially further nonnull vectors e_{5},..,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 (e_{1},e_{2},e_{3},e_{4}) be a fixed orthonormal frame for R_{3,1} and define
unit pseudoscalar i=e_{1234}.
Using 1 , we can extend this into a fixed orthonormal basis
(e_{0},e_{1},e_{2},...e_{N-1}) for U_{N} with unit pseudoscalar
u=e_{012..(N-1)}.
We assume an orthonormal inverse frame
(e^{0},e^{2},...e^{N}) for U^{N} with e^{1}=e_{1},e^{2}=e_{2},e^{3}=e_{3},e^{4}=-e_{4}.
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 ¦^{-Ñp2} .
¦^{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 y^{n}(x^{1},x^{2},x^{3},x^{4})
defined by
¦(x) = ¦(å_{i}x^{i}e_{i}) = å_{n=0}^{N-1} y^{n}(x^{1},x^{2},x^{3},x^{4})e_{n}
= å_{n=0}^{N-1} y^{n}e_{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:
y^{n}_{,i p} º e^{n}¿¦^{Ñ}_{p}(e_{i})
= ¶y^{n}/¶x^{i} ;
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 ¶/¶x^{i} ; _{, ij} for
¶/¶x^{i} ¶/¶x^{j} ; and so forth .
Thus y^{n}_{,ij} º
y^{n}_{, i , j} º
¶y^{n}_{,i}/¶x^{j} º
(¶/¶x^{j})(¶y^{n}/¶x^{i}) .
We have y^{n}_{,i} =Sig(e_{n})y_{n,i}
0 £ n < N
.
¦^{Ñ}_{p}(e_{i}) = å_{n=0}^{N-1} y^{n}_{,i}e_{n}
= å_{n=0}^{N-1} y_{n,i}e^{n} ;
¦^{D}_{p}(e_{n}) = å_{i=1}^{4} y_{n,i}e^{i} .
[ Proof :
¦^{D}_{p}(e_{n})
= å_{i=1}^{4} (¦^{D}_{p}(e_{n})¿e_{i})e^{i}
= å_{i=1}^{4} (e_{n}¿¦^{Ñ}_{p}(e_{i}))e^{i}
= å_{i=1}^{4} y_{n,i}e^{i}
.]
¦^{Ñ2}_{p}(a,b) =
å_{n=0}^{N-1} å_{ij} y^{n}_{,ij}a^{i}b^{j}e_{n}
= å_{n=0}^{N-1} å_{ij} y_{n,ij}a^{i}b^{j}e^{n}
where
y_{n,ij} º ¶y_{n,i}/¶x^{j} .
[ Proof :
¦^{Ñ}_{p+dp}(e_{i}) = å_{n=0}^{N-1} y^{n}_{,i}(p+dp)e_{n}
= å_{n=0}^{N-1} (y^{n}_{,i}(p) +
å_{j=1}^{4} y^{n}_{,ij}dp^{j})e_{n}
+ _{O}(dp^{2}) 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 v^{2}) 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}(a^{2}) = a¿g_{p}(a) + _{O}(a^{2}) .
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_{p}^{2} .
[ 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
g^{i}_{j} º e^{i}¿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} y^{n}_{, i }y_{n , j }
;
e_{i} º g_{p}(e_{i}) = å_{k=1}^{4} g_{ik }e^{k}
and hence
a¿b = å_{ij=1}^{4} g_{ij}a^{i}b^{j}
[ Proof : a¿b = a¿g_{p}(b)
= a¿g_{p}(å_{i=}^{} b^{i}e_{i})
= a¿å_{i}b^{i}g_{p}(e_{i}))
= a¿å_{ij}b^{i}g_{ij }e^{j})
.]
g^{ij}_{p} º g_{p}^{-1}(e^{i})¿e^{j}
(so that e^{ j} º g_{p}^{-1}(e^{j}) = å_{k}g^{kj}e_{k})
is the conventional matrix inverse of g_{ij p}.
[ Proof :
å_{k=1}^{4} g_{ik}g^{kj}
= å_{k}(g_{p}(e_{i})¿e_{k})g^{kj}
= g_{p}(e_{i})¿å_{k}e_{k}g^{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_{ij}_{p} } .
Note carefully that g^{ij} breaches our usual suffix convention in that
it represents e^{i}¿g_{p}^{-1}(e^{j}) rather than e^{i}¿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 g^{ii}=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 , sin^{2}qdy) with associated line element
dp¿g_{p}(dp) = dx^{2} + cos^{2}qdy^{2}.
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 dx^{2} + sin^{2}qdy^{2} metric which they
would extend by -t^{2} 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
{e_{1}, ..,e_{4}} for R_{3,1}
generates not only a point-independant inverse 4-frame {e^{1},e^{2},..,e^{4}}, but also
a point-dependant (nonorthormal) covariant 4-frame (aka. covector 4-frame
or 4-coframe )
{ e^{ i} º g_{p}^{-1}(e^{i}) }
which itself has inverse 4-frame
{ e_{i} º g_{p}(e_{i}) }
[ Proof :
e^{ i}¿e_{j} = g_{p}^{-1}(e^{i})¿g_{p}(e_{j}) = g_{p}(g_{p}^{-1}(e^{i}))¿e_{j} = e^{i}¿e_{j} .]
Trivially, e_{i}¿e_{j} = e_{i}¿e_{j} = g_{ij} ;
e^{i}¿e^{ j} = e^{i}¿e^{j} = g^{ij} .
A given v_{p} Î R_{3,1} can thus be expressed both as
v_{p} = å_{i}v^{i}e_{i}
( with
contravariant coordinates v^{i} º e^{i}¿v_{p} = e^{ i}¿v_{p}
)
and as
v_{p} = å_{i}v_{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=1}^{4} g_{ij}v^{j} ) .
The "motivation" for covariant coordinates is that
å_{i=1}^{4} w^{i}v_{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_{ij}ºe_{i}¿g_{p}(e_{j}) as here
rather than g_{ij}ºe_{i}¿g_{p}(e_{j}) = g_{p}(e_{i})¿g_{p}^{2}(e_{j}) = e_{i}¿g_{p}^{3}(e_{j}) ;
with g^{ij}º e^{i}¿g_{p}(e^{j}) denoting the inverse matrix for
{ g_{ij} } .
g^{i}_{j} here represents e^{i}¿g_{p}(e_{j}) whereas
in coventional GR treatments it denotes
g^{i}_{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}_{ij}_{p}
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} y^{n}_{,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}_{ ,k}e_{m})¿(å_{n=0}^{N-1} y_{n ,ij}e^{n})
= å_{n}y^{n}_{,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}=y^{n}_{,i }y_{n,j}
Þ g_{ij,k}
= y^{n}_{,ik }y_{n,j} + y^{n}_{,ik }y_{n, jk}
= y_{n,ik }y^{n}_{,j} + y_{n,i }y^{n}_{,jk}
and the result follows
.]
_
G^{ l}_{ij}_{p}
= å_{k=1}^{4} g^{lk} G_{kij}
.
[ Proof :
å_{k=1}^{4} g^{lk}¦^{Ñ}_{p}(e_{k})¿¦^{Ñ2}_{p}(e_{i},e_{j})
= ¦^{Ñ}_{p}(å_{k=1}^{4} g^{lk}e_{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^{k}dp^{j} e_{i}
= -å_{ijk} G^{ k}_{ij}v_{k }dp^{j} e^{ i}
= -å_{ijk} G_{kij}v^{k}dp^{j}e^{ 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}(dp^{2}) .
Let v^{i}(p,dp) and v_{i} (p,dp)
be contra and covariant coordinates for v_{p}[dp] at p+dp
( ie. v^{i}(p,dp) º v_{p}[dp]¿e^{i} ;
v_{i} (p,dp) º g_{p+dp}(v_{p}[dp])¿e^{i} =
å_{j=1}^{4} g_{ijp+dp}v^{j}(p,dp) ).
Clearly v^{i}(p,0) = v^{i}_{ 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}(dp^{2})
= å_{j k} G_{kij}v^{k}dp^{j}
+ _{O}(dp^{2}) .
Hence dv
= å_{i j k} G_{kij}v^{k}dp^{j}e^{ 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}_{ij}v_{l }dp^{j}
+ _{O}(dp^{2}) .
[ Proof : å_{j k}v^{k } G_{kij}dp^{j}
= å_{j k}(å_{l}g^{kl}v_{l}) G_{kij}dp^{j}
.]
dv^{i} =
- å_{j k} G^{ i}_{jk}v^{k}dp^{j}
+ _{O}(dp^{2}) .
[ Proof :
Let map vectors a and b be parallel transported.
d(a¿b) = 0 Þ d(å_{i=1}^{4} a_{i}b^{i})=0
Þ å_{i}((da_{i})b^{i} + a_{i}(db^{i}))=0
Þ
å_{i}a_{i}(db^{i})) = -å_{ijk} G^{ k}_{ij}a_{k }dp^{j} b^{i}
= -å_{ijk} G^{ i}_{kj}a_{i }dp^{j} b^{k}
= -å_{i}a_{i}å_{j k} G^{ i}_{kj}dp^{j} b^{k} .
Since a arbitary , db^{i} = - G^{ i}_{kj}dp^{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=1}^{4} G^{k}_{jk} = ( ln(j_{p}))_{ , j}
= j_{p}^{-1} j_{p}_{ , j}
.
[ Proof :
g_{,j}
= gå_{i=1}^{4} å_{k=1}^{4}
g^{ik}g_{ik,j}
= gå_{i=1}^{4} å_{j=1}^{4}
g^{ik}
(G_{kij} + G_{ijk})
= 2gå_{i=1}^{4} G^{i}_{ij}
Þ
å_{i=1}^{4} G^{i}_{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
dt^{2} = dp¿g_{p}(dp)
.
[ Proof :
(¦^{Ñ}_{p}(dp/dt))^{2}=1
Þ dt^{2}
= (¦^{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/¶x^{i}' = ¶L/¶x^{i} .
[ Proof : See Rindler 10.2 .]
Coderivative
On seeking a R_{3,1} vector associated with a given directional derivative
u^{Ñ}_{p}(¦^{Ñ}_{p}(a))
, we have the problem that u_{p}+dp-u_{p} need not lie within J_{p+dp}.
If we use ¦^{Ñp}_{p+dp}(¯(u_{p}+dp-u_{p},J_{p+dp})) , effectively replacing
u_{p} by u_{p}[dp] , we obtain
the directional coderivative
discussed in
Multivector Manifolds.
Ð_{a}^{ð}v_{p} = v^{Ñð}_{p}(a) | º | Lim_{d ® 0}(¦^{-Ñ}_{p+da}(u_{p+¦Ñp(da)}-u_{p})/d) = Lim_{d ® 0}((¦^{-Ñ}_{p+da}(u_{p+¦Ñp(da)}-u_{p}[dp])/d) = Lim_{d ® 0}((v_{p+da}-v_{p}[da])/d) |
= | Lim_{d ® 0}((v_{p+da} -v_{p}+v_{p} - v_{p}[da])/d) = v^{Ñ}_{p}(a) - Lim_{d ® 0}((v_{p}[da]-v_{p})/d) | |
= | v^{Ñ}_{p}(a) + G_{p}(v_{p} , a) = Ð_{a}v_{p} + G_{p}(v_{p} , a) |
A key result is Ð_{a}^{ð}e_{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}_{ik}g_{lj} + G^{ l}_{jk}g_{il}) =
g_{ij ,k} - ( G_{lik}g_{lj} + G_{ljk}g_{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_{ij}ºe_{i}¿g_{p}(e_{j})
=e_{i}¿g_{p}^{3}(e_{j}) .
[Under Construction. comment appreciated. ]
.]
We can define a mapspace 1-vector codel-operator
Ñ^{ð}_{p}(a) º å_{k=1}^{4} e^{k}(Ð_{ek}^{ð}a)
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=1}^{4} e^{k}¿(Ð_{ek}^{ð}v_{p})
= å_{k}v^{k}_{:k}
= (Ñ_{p}¿(j_{p}v_{p})) j_{p}^{-1}
[ Proof :
= å_{k}(v^{k}_{,k} + å_{j} G^{ k}_{jk}v^{j})
= å_{k}v^{k}_{,k} + å_{j}j_{p}^{-1} j_{p}_{ ,j}v^{j}
= å_{k} (j_{p}v^{k})_{,k} j_{p}^{-1}
.]
Thus we have (Ñ^{ð}_{p}¿v_{p}) j_{p} = Ñ_{p}¿(v_{p}j_{p})
and so
Ñ^{ð}_{p}¿v_{p} = ¯_{ vpjp}( Ñ_{p} ) v_{p} .
[ Proof :
¯_{ vpjp}(Ñ_{p}) v_{p}
º (Ñ_{p}¿(v_{p}j_{p}) )(v_{p}j_{p})^{-1} v_{p}
= (Ñ_{p}¿(v_{p}j_{p}) )j_{p}^{-1}
= Ñ_{p}¿(v_{p}j_{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_{k}Ð_{ek}(j_{p} F^{k}_{j}) - å_{lk} G_{ljk}F^{ kl}
[ Proof :
å_{k} F^{k}_{j : k}
= å_{k} ( F^{k}_{j , k}+ å_{l}( G^{ k}_{lk}F^{ l}_{j}
- G^{ l}_{jk}F^{ k}_{l} ) )
= å_{k} F^{k}_{j , k} + å_{_lk} G^{ k}_{lk} F^{l}_{j}
- å_{lk} G_{ljk}F^{ kl}
=
å_{k} F^{k}_{j , k}
+ å_{k}(j_{p}^{-1} j_{p}_{ , k}) F^{k}_{j}
- å_{lk} G_{ljk}F^{ kl}
= j_{p}^{-1} å_{k}(j_{p} F^{k}_{j , k} + j_{p}_{ , k } F^{k}_{j})
- å_{lk} G_{ljk}F^{ kl}
.]
For symmetric F_{p}( ) this becomes
Ñ^{ð}¿F_{p}(e_{j}) =
j_{p}^{-1} å_{k}Ð_{ek}(j_{p} F^{k}_{j})
- ½å_{lk}g_{kl , j }F^{ kl} .
[ Proof :
= å_{lk} G_{lkj}F^{ kl}
= ½( å_{lk} G_{lkj}F^{ kl} + å_{kl} G_{ljk}F^{ kl} )
= ½( å_{lk} G_{lkj}F^{ kl} + å_{lk} G_{kjl}F^{ kl} )
= ½å_{lk}( G_{lkj} + G_{kjl})F^{ kl}
= ½å_{lk}g_{kl , j }F^{ kl}
.]
For antisymmetric F_{p}( ) it is
Ñ^{ð}¿F_{p}(e_{j}) =
j_{p}^{-1} å_{k}Ð_{ek}(j_{p} F^{k}_{j})
[ Proof :
å_{lk} G_{lkj}F^{ kl} = 0
.]
Material-energy 1-tensor in a curved space
As in the flat space case, a natter flow m_{p} = m_{p}u_{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_{p}u^{i}u^{j}
;
T^{i}_{j } =
å_{k}g_{jk}T^{ij}_{p}
= m_{p}u^{i}u_{j} .
;
T_{ij } =
å_{k}g_{ik}T^{i}_{j }= m_{p}u_{i}u_{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_{p}m_{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}|d^{4}p| in U_{N} and our
conservation of matter law becomes
Ñ_{p}¿(j_{p}m_{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_{k}Ð_{ek}(j_{p} T^{k}_{j})
- å_{lk} G_{ljk}T^{ kl}
= j_{p}^{-1} å_{k}Ð_{ek}(j_{p}m_{p}u^{k}u_{j})
- å_{lk} G_{ljk}m_{p}u^{k}u^{l}
=
j_{p}^{-1} å_{k}Ð_{ek}(j_{p}m_{p}u^{k})u_{j}
+ m_{p}å_{k}(u^{k}u_{j , k} - å_{l} G_{ljk}u^{k}u^{l})
[ chain rule ]
=
j_{p}^{-1} u_{j}å_{k}Ð_{ek}(j_{p}m_{p}u^{k})
[ if geodesic flow ]
= 0
[ by matter consvervation ]
.]
Slow Flows
We say a flow is slow with regard to e_{4} if u_{p} » e_{4} " p : m_{p} ¹ 0
so that all e_{4}-percieved spacial speeds of matter are vastly less than 1.
We then have m_{p} » m_{p}e_{4} Þ T_{p}(a) » m_{p} ¯(a,e_{4}) .
All matrix representation components of T_{p} are then negligible save e^{4}¿T_{p}(e_{4}) » 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 4^{4}=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=1}^{4} 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=1}^{4} 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=1}^{4} g_{lm} C_{i}_{ jk}^{m }
=...=
G_{lik ,j} - å_{m} G_{mlj} G^{ m}_{ik}
- í¬ý_{j«k}
[
R^{i}_{jkl} = C_{j}_{ kl}^{i }
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_{p}a
= (1-½Ñ_{Þ})[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)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: (b_{2}.c_{3}).a = (b_{2}Ùa).c_{3} ]
= å_{ij}( ((å_{lm}C^{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}_{n}C
= 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_{p}a + 8pg T_{p}(a) - la = 0
where
In the operator notation of Multivector Manifolds chapter, we have
(1-½Ñ_{Þ})[Ñ¯]^{2}¯ = 8pg ¯°_{mp} + l abbreviating (1-½Ñ_{Þ})[Ñ¯]^{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_{p}a
, 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 e_{4}.
For a weak, slow, static connection the geodesic trajectory equation
p"(t) » -(v^{4})^{2}G_{p}(e_{4} , e_{4})
reduces to Newtonian equations
¶v^{i}/¶t » ½h_{44 ,i}
[ Proof :
¶v^{i}/¶x^{4} ¶x^{4}/¶t » (v^{4})^{2}½g_{44 ,i}
Þ ¶v^{i}/¶t v^{4} » (v^{4})^{2}½g_{44 ,i}
Þ ¶v^{i}/¶t » v^{4}½g_{44 ,i}
» ½g_{44 ,i}
= ½h_{44 ,i}
.]
so for a weak static connection we have
C_{44} » -å_{k} G_{k44 ,k}
= -å_{k=1}^{4}
(½(g_{k4,4}+g_{k4,4} - g_{44,k}))_{,k}
» å_{k=1}^{3} g_{44,kk}
= Ñ_{r}^{2}g_{44}
where Ñ_{r} is the standard ("flat") 3D Euclidean Ñ acting in e_{4}^{*} in the mapspace.
Since
C_{p} = -8pgT_{p} = m_{p} for a slow flow,
e_{4}¿E_{p}(e_{4})
= C_{44} + e_{4}¿(4pgT_{p})e_{4} + 8pgT_{44}
+ l
Þ
Ñ_{r}^{2}g_{44} + 4pgm_{p} » 0 .
h_{44} = g_{44} - 1
thus acts as a Newtonian gravitational potential if we assume g_{44}=1 at infinity.
Yilmaz Gravitation
Yilmaz rejects the matter conseveration law Ñ^{ð}¿m_{p} = 0 and favours the Freud Identity
Ñ¿(J_{p}T_{p}(a)) = 0 .
Consequently
Ñ^{ð}¿T_{p}(e_{j}) =
j_{p}^{-1} å_{k}Ð_{ek}(j_{p} T^{k}_{j})
- å_{lk} G_{ljk}T^{ kl}
= - å_{lk} G_{ljk}T^{ kl}
= - m_{p}å_{k} u^{k}u_{j , k} [ if geodesic flow ]
Accordingly, our field equation becomes
C_{p}(a) - ½C_{p}a + 8pg T_{p}(a)
+ 2t_{p}(a) - la = 0
where
t_{p}(a) º
- 4pg å_{lk} G_{ljk}T^{ kl}
= - 4pg m_{p}å_{k}u^{k}u_{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
ds^{2}
= (1-2mr^{-1})dt^{2} - (1-2mr^{-1})^{-1}dr^{2} -
r^{2}(dq^{2} + sin(q)^{2}df^{2})
_{[ Rindler 11.13 ]}
at a given event te_{4} + r
where 3D spacial point r is expressed in spherical polar coordinates as [q,f,r]
with r>r_{0}
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 r_{1} and
Gc^{-2}
» 2^{-231.14} m^{2} 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^{-1}i^{-1}t and
x=4m(1-2mr^{-1})^{½}
(running from -8m^{2}¥ at r=0 through 0 at r=2m to 4m at r=¥)
to obtain
ds^{2} = x^{2}(4m)^{-2}(dQ)^{2} + (r^{2}(4m^{2})^{-1})^{2} (dx)^{2}
+ r^{2}((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
ds^{2} = R^{2}T^{2}(2p)^{-2}(dQ)^{2}
+ (4m)^{2}(r^{2}(4m^{2})^{-1})^{2} (dR)^{2}
+ r^{2}((dq)^{2}+ sin(q)^{2}(df)^{2}) is Euclidean in the sense of a positive definite metric.
Hawking considers e_{4}-periodic rather than an e_{4}-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>r_{0} to
ds^{2}
= a(r)
dt^{2} - a(r)^{-1}dr^{2} -
r^{2}(dq^{2} + sin(q)^{2}df^{2})
where a º a(r) º (1-2mr^{-1} - 3^{-1}lr^{2})
runs from -2m¥ through 0 at ? to - 3^{-1}l_infinity_sqrd ar r=¥
_{[ Rindler 14.22 ]} .
This has infinites when r solves cubic
ra(r) = 2m-r+3^{-1}lr^{3} = 0 which occurs for r slightly below
m(2 + 3^{-1}8lm(1+2lm^{2})^{-1}) and (for nonhuge m)
slightly below m independant distance 3^{½}l^{-½}
» 2^{82} m which is
about 2^{-5} times the distance of the furthest known galaxy but 2^{16} 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^{-1}r'^{2} - r^{2}f'^{2}
and since this is independant of f and t we have two geodesic-constants, angular momentum h=r^{2}f'
and energy k=a(r)t' .
Setting L(f',r',t',t)=1 provides
a(r)(ka(r)^{-1})^{2} - a(r)^{-1}r'^{2} - r^{2}(hr^{-2})^{2} = 1
Þ k^{2} - r'^{2} - ar^{-2}h^{2} = a
yielding geodesic condition
r'^{2} = k^{2} - a(r) (r^{-2}h^{2} + 1 )
= k^{2} - a(r)r^{-2}(h^{2} + r^{2}) .
Radial trajectories having constant f and h=0 have Lagrangian
L(r',t',t) = at'^{2} - a^{-1}r'^{2} with geodesic condition
r'^{2} = k^{2} - 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 = ±|k^{2} - a(r)|^{-½}
and
inserting k=a(r_{0}) for a test particle released at rest (t'=1) at r=r_{0}>2m
gives dt/dr = ±|a(r_{0})^{2} - a(r)|^{-½} .
For l=0 and setting E=½(k^{2}-1) this is
dt/dr = ±|k^{2} - 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 = -mr_{0}^{-1} by analogy with the Newton form
and so has k=|1+2E|^{½}
= (1-2mr_{0}^{-1})^{½} = a(r_{0})^{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",
Next : Referances and Source Material