From my book Star Gate under construction (following Hawking & Ellis, "The Large Scale Structure of Space-Time", Cambridge.

The intrinsic space-time geometry is only in the non-gravity force-free geodesic structure. There is no objective intrinsic gravity force in Einstein's theory of gravity. There is such an objective gravity force in Newton's older theory. Einstein eliminates Newton's pure gravity force acting at a distance replacing it with local variable curvature, i.e. geodesic deviations of neighboring passive test particles. Any attempt to restore Newton's gravity in Einstein's theory as another valid way of interpreting Einstein's theory is crackpot, inconsistent and displays a fundamental lack of understanding of what John Archibald Wheeler called "Einstein's Vision." Non-geodesics have no fundamental relation to the intrinsic geometry of spacetime. You cannot have non-geodesic paths of test particles without a non-gravity force like the long-range electromagnetic force. You cannot "pull g's" i.e. feel "weight" without a non-gravity force pushing you off a timelike geodesic inside the local light cone.

Geodesics are the straightest EXTREMAL paths in curved space-time.

Note that timelike geodesics are the longest paths in experienced proper time along them in the sense of the action principle of particle mechanics from the calculus of variations, i.e. neighboring bundle of paths with same starting and ending points. Virtual nongeodesic paths without non-gravity forces in this sense are quantum gravity zero point vacuum fluctuations in the Feynman path integral expression for the quantum amplitude of a classical geodesic path. Spacelike geodesics are the shortest straightest paths outside the light cone. Null geodesics have zero length. Real on-mass-shell faster than light tachyons do not exist in globally flat quantum field theories. They signal a vacuum instability as in the Higgs mechanism for the macro-quantum coherent origin of inertia and gravity as curvature. The Haisch-Puthoff model for the origin of inertia and gravity from random locally incoherent electromagnetic zero point fluctuations is wrong IMO.

A useful local nongeodesic "LNIF" frame in curved spacetime is the HOVERING non-geodesic "shell frame" at a fixed distance from a source. However such a frame does not always exist, e.g. inside the black hole null surface event horizon one-way membrane trapped surfaces containing null geodesics. Misunderstanding the contingent nature of the this admittedly useful frame, when it exists, leads one to delusionary ideas that, for example, with the SSS source M

g = GM/r^2

is an objective gravity force even in Einstein's GR. That is not true at all. That formula simply tells you how much rocket thrust you need in space to keep at a fixed distance from the source. It also tells you your weight per mass if you stand on a rigid surface of circumference 2pir encircling mass M.

The "Hilbert error" claim that there are no SSS event horizons is
also crackpot IMO.

1. Globally flat Minkowski space-time
The metric in Cartesian coordinates is (c = 1)

ds^2 = -(dx^4)^2 + (dx^1)^2+ (dx^2)^2 + (dx^3)^2

Here the geodesics of maximal proper time are

x^a(affine parameter) = b^a(affine parameter) + c^a

Theorem: any two points in globally flat spacetime are connected by a unique geodesic.
Proof needs fancy formal stuff about exponential map of tangent space to manifold. See Hawking & Ellis for details.

Use the global coordinate transformation
to spherical polar coordinates

x^4 = t
x^3 = rcostheta
x^2 = rsinthetacosphi
x^3 = rsinthetasinphi

theta is latitude on celestial sphere centered at r = 0 phi is longitude

Do the differential calculus with product rule to get

ds^2 = - dt^2 + dr^2 + r^2[(dtheta)^2 + sin^2theta(dphi)^2]

There is a non-physical coordinate singularity at r = 0 where the two
angles theta & phi are undefined.

Theta ranges from 0 to pi, phi from 0 to 2pi, r from zero to infinity.

The choice of r = 0 in this unstable pre-inflationary pre Big Bang
globally flat false vacuum is arbitrary here of course.

2. Wheeler-Feynman type of trick

Use the past to future retarded and future to past advanced light
cone radial "null coordinates"

v = t + r i.e. advanced destiny wave back from the future (retro-
causal future light cone)

w = t - r i.e. retarded history wave toward the future (past light cone)

Both v & w range from -infinity to + infinity

the GLOBAL FRAME INVARIANT metric field is then

ds^2 = -dudv - (1/4)(v - w)^2[(dtheta)^2 + sin^2theta(dphi)^2]

note that [(dtheta)^2 + sin^2theta(dphi)^2] describes a unit 2D
spherical surface S2. Every "point" in the t-r plane is actually an
S2 with radius r.

The v = constant, w = constant hypersurfaces are made from light cone null geodesics. Their intersection is a sphere

i.e. all the tangent vectors inside those hypersurfaces are null
because no dv^2 & dw^2 terms.

v(w),av(w),bn^a^b = 0

,a is ordinary partial derivative

Penrose diagram for globally flat Minkowski spacetime to make infinity finite.

v = tanp

w = tanq

both p & q range from - pi/2 to + pi/2

Therefore, algebra & trig give

ds^2 = sec^2p sec^2q{-dpdq + (1/4)sin^2(p - q)[(dtheta)^2 + sin^2theta (dphi)^2]}

There exists a conformal map to

ds*^2 = -4dpdq + sin^2(p - q)[(dtheta)^2 + sin^2theta(dphi)^2]

i.e.

ds^2 = (1/4)sec^2(t' + r')sec^2(t' - r')ds*^2

Where we define

t' = p + q

r' = p - q

t' + r' ranges from -pi to pi

t' - r' ranges from -pi to pi

r' > 0

Therefore, algebra demands

ds*^2 = - (dt')^2 + (dr')^2 + sin^2r'[(dtheta)^2 + sin^2theta(dphi)^2]

which is LOCALLY a piece of the Einstein static universe.

Thus the WHOLE OF GLOBALLY FLAT MINKOWSKI SPACE-TIME is given by the FINITE REGION

t' + r' ranges from -pi to pi

t' - r' ranges from -pi to pi

r' > 0

of

ds^2 = (1/4)sec^2(t' + r')sec^2(t' - r')ds*^2

The original coordinates t & r are in

2t = tan[(1/2)(t'+r')] + tan[(1/2)(t'-r')]

2r = tan[(1/2)(t'+r')] - tan[(1/2)(t'-r')]

Suppress theta & phi, then the 1 + 1 string analog Einstein universe
is globally equivalent to the unit circle S1, i.e. x^2 + y^2 = 1
imbedded in ANYONIC quantum well 2 + 1 Minkowski with

ds^2 = - dt^2 + dx^2 + dy^2

Note we really mean (ds)^2, (dt)^2 etc.

The full 4D Einstein static universe with his original cosmological
constant /\ is the unit S^3

x^2 + y^2 + z^2 + w^2 = 1

imbedded in Kaluza-Klein 5D hyperspace

ds^2 = - dt^2 + dx^2 + dy^2 + dz^2 + dw^2

"One therefore has the situation: the whole of Minkowski space-time
is conformal to the region

t' + r' ranges from -pi to pi

t' - r' ranges from -pi to pi

r' > 0

of the Einstein static universe, that is the shaded region of Fig 14. The boundary of this region may therefore be thought of as representing the conformal structure of infinity of Minkowski spacetime. It consists of the null surfaces p = +pi/2 labeled I^+ and q = -pi/2 labeled I^-, together with the points (p = pi/2,q = pi/2) (labeled i^+), (p = pi/2,q = - pi/2) (labeled i^0) and (p = -pi/2,q = -pi/2) (labeled i^-). Any future directed timelike geodesic in Minkowski space approaches i^+(i^-) for large positive (negative) values of its affine parameter, so one can regard any timelike geodesic as originating at i^- and finishing at i^+. Similarly one can regard null geodesics as originating at I^- and ending on I^+, while spacelike geodesics both originate and end at i^0. Thus one may regard i^+ and i^- as representing future and past timelike infinity, I^+ and I^- as representing future and past null infinity, and i^0 as representing spacelike infinity. (However nongeodesics do not obey these rules; e.g. nongeodesic timelike curves may start on I^- and end on I^+.) Since any Cauchy surface intersects all timelike and null geodesics, it is clear it will appear as a cross-section of the space everywhere reaching the boundary at i^0. One can also represent the conformal structure at infinity by drawing a diagram of the (t',r') plane. Each point of this diagram represents a sphere S2 and radial null geodesics are represented as lines at +- pi/4. In fact, the structure of infinity of any spherically symmetric spacetime can be represented by a diagram of this sort, which we call a Penrose diagram. On such diagrams, we shall represent infinity by single lines, the origin of polar coordinates by dotted lines, and irremovable singularities of the metric by double lines. ... Finally, ... one can obtain spaces locally identical to (Minkowski) but with different large scale topological properties by identifying points which are equivalent under a discrete isometry without fixed point ..." 5.1 Hawking and Ellis.

© Dr. Jack Sarfatti, Ph.D. [Sat, 19 Aug 2006]