Penrose Diagram for Globally Flat Minkowski Space-Time

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.


http://content.answers.com/main/cont...px-Penrose.PNG

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

[...]
And the point of your posting is? Not that it's wrong but it's all been
described in all the standard textbooks and you are adding nothing new
to it. This is not what these NG are for. Put it on your blog instead
where you'd be also able to use proper mathematical typesetting tools.

--
Jan Bielawski

"JanPB" wrote in message
oups.com...
Jack Sarfatti wrote:
From my book Star Gate under construction (following Hawking & Ellis
"The Large Scale Structure of Space-Time", Cambridge.

[...]
And the point of your posting is?

What else? To spam the groups to try to sell his books. ;-)

FrediFizzx

Quantum Vacuum Charge papers;
http://www.vacuum-physics.com/QVC/qu...uum_charge.pdf
or postscript
http://www.vacuum-physics.com/QVC/qu...cuum_charge.ps
http://www.arxiv.org/abs/physics/0601110
http://www.vacuum-physics.com