cosmic everything charts compared

8/8/2019 Cosmic Everything Charts Compared

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GravitationLab.com

Cosmic Everything Charts Compared: General Relativity vs. the

Space Generation Model of Gravitation and Cosmology

R. Benish(1)

(1) Eugene, Oregon, USA, [email protected]

Abstract.

Using logarithmic scales, it is possible to compactly show on a graph how the massesand sizes of all known physical ob jects in the universe relate to one another. Suchgraphs are necessarily skeletal. By filling in more detail than usual (e.g., Kraus, [1]Barrow, [2] and Hartle [3]) and by adding density and gravitational acceleration tothe axes, the meaningfulness of these relationships becomes more strikingly per-ceptible. Organization of the first chart (Figure 1) is entirely consistent with thestandard paradigm, with data obtained from the standard literature. The secondchart (Figure 2) is organized the same way. It uses the same data for mass, but itdiffers in that nine objects are placed inside the Schwarzschild horizon line, which

thereby reflects the absence of black hole horizons and singularities. A new modelof gravity, the Space Generation Model, is presented in support of the second chart.Most importantly, a relatively simple laboratory experiment is proposed whose re-sult would decisively refute either general relativity or the new model, and therebyindicate which chart is closer to the truth.

PACS 04.80.Cc Experimental tests of gravitational theories.

1. Introduction

One of the most well known predictions of Einsteins theory of gravity, general rela-tivity (GR) is that sufficiently large and compact bodies of matter form black holes. Agraph that plots mass vs. radius on logarithmic scales so as to include a wide range ofmasses and sizes, shows such bodies lying on a straight lineas seen for nine points inFigure 1 (Chart 1). Figure 2 (Chart 2) is essentially the same graph except that thesenine points have been moved to the left of the black hole line. This essay concerns thereasons why the latter placement of these points makes more sense, which is tantamountto proposing a new model of gravity.

For reasons that will be explained in detail, I call this the Space Generation Modelof gravitation and cosmology (SGM). A simple laboratory experiment would be the bestway to decide between GR and the SGM. The difference in predictions is not subtle;

c Richard Benish 2010 1


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2 R. BENISH

it is dramatic. Even the weak field regime of Newtons theory of gravity is challenged.The SGM is thus radical in many ways. Yet it is consistent with all physical factsthat I know of. In what follows the SGMs prediction for the proposed experiment willbe supported 1) by a critical assessment of not only GR, but of various foundationalconcepts of physics and 2) by showing that certain alternative concepts are logicallymore coherent. By developing these new concepts we will see that some of the persistentenigmas of contemporary physics disappear. We will venture far and deep and wide. Weseek to discover whether Chart 1 or Chart 2 serves as a better map of the world.

Therefore, well begin by pointing out a few general features of the charts. (Notethat, to facilitate printing or viewing at a larger size, they are available as stand-alonedocuments. [4,5] ) The charts may be thought of as globes that allow the whole worldto be seen at a glance, rather than having to mentally piece it together with scattered bitsof information. We immediately notice that human beings are located near the middle.

Familiar bodies of atomic matter surround us along an orderly arrangement of points.Since the density of atomic matter has a fairly narrow range, masses increase very nearlyas the cubes of the radii of physical bodies. The mass vs. radius slope in this region ofthe charts is thus 3, the density vs. radius slope is 0, and since the acceleration dueto gravity varies by the inverse square law, its slope is 1.

Moving toward the microcosm along our trail of points, measurements are no longerso straightforward as they are for bulk atomic matter. With sophisticated machinerythe sizes and masses of molecules, atoms, nuclei and particles can be deduced. Thelightest thing whose mass has been reliably measured is an electron. The measurementsare tricky, but at least they yield a definite result for an electrons mass. The size ofan electron, on the other hand, is best thought of as more of a theoretical thing thana physical thing. Without going into the reasons for this, suffice it to say that, thoughthe two electron radii shown on the graphs are widely recognized as being of theoreticalimportance, it would be erroneous to think of one electron as having such a definite size.If we include the approximate size of an electron cloud in a ground state hydrogen atom,then these three radii are related to one another by powers of the fine structure constant,, and the Bohr radius, a0.

A hydrogen atom without an electron is a proton, whose mass is nearly the same as aneutron. For the purposes of the charts (which take no account of electric charge) bothprotons and neutrons are essentially indistinguishible nucleons. Muons have the sameelectrical charge as electrons, but are 207 times heavier. Though rare, atoms whoseelectrons are replaced by muons have been created in laboratories. Thus we include muo-nium (which lies between the electronic hydrogen atom and the lone proton [nucleon]).With the exception of the lightest elements, nuclei in atoms have nearly the same den-sity, known as nuclear saturation density. Thus we find another region where the mass

vs. radius slope 3, the density vs. radius slope 0, and the slope of the accelerationdue to gravity is 1. Note that on these charts the radius and density of one nucleonare derived from the radii and densities actually measured from collections of them. Wewill find that nuclear saturation density bears a curious relation to atomic density andother key densities when we consider their occurrence in astrophysical phenomena.

Which brings us, then, to the opposite direction along our scale of size. In the realmof planets and stars, because of gravity, we get two branches in the pattern: Unlike thecase of smaller bodies of atomic matter where gravitys role seems negligible, in thisregion, along one branch gravity causes bodies to be more compressed; adding massactually makes the objects get smaller. Along this branch we encounter brown dwarfs,white dwarfs, neutron stars and finally, extremely compressed stellar objects commonly


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COSMIC EVERYTHING CHARTS COMPARED: GR vs. SGM 3

Planck Acceleration

M32Nucleon& Nuclei

Electron(classical

radius 2a0)

HydrogenAtom (a

0)

Electron(barred Compton

wavelength c = a

0)

J0917+46

J1650-500

3C 66B Galaxy Core

OJ287 Galaxy Core

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PlanckMass

Copyright Richard Benish 2010. With this copyrighttag in place, free unaltered distribution is encouraged.

Note that the gravitational accelerations corresponding to objects

on the Schwarzschild horizon were calculated using Newtonsequation. Whereas, according to general relativity, the acceleration

of a stationary body at the horizon would be infinite . (See, e.g.,

Rindler, Essential Relativity, 2nd Ed., Springer-Verlag, 1977; p. 149.

Or Hartle, Gravity, Addison-Wesley, 2003; p. 435.)

Also, the densities of these objects were calculated as an average

as though the mass were distributed throughout a sphere of radius

= 2GM/c2. Whereas, according to general relativity, the matter would

quickly collapse to a central singularity of infinite density.

An alternative treatment of these extreme and undesirable conse-

quences is presented on the otherwise identical graph: Cosmic

Everything Chart (SGM).** This alternative is physically more

reasonableat least insofar as the Post-Newtonian accelerations and

densities are all finite; there are no singularities. See also Cosmic

Everything Charts Compared** which gives a more thorough

comparison.

**Documents referred to immediately above are accessible at

GravitationLab.com or at Scribd.com, under Benish.

To minimize clutter, many points are not labeledespecially those

for acceleration. But everything is labeled at least once. Identi-

fication may require finding the point corresponding to a different

quantity along the same vertical line. Moon, Earth, Sun, Milky Way

and Milky Way Core have be en colored to facilitate finding these

particular points.

*

log(

3M/4r

3)(kgm3)

logg(GM/r

2)(m

s2)

log r (meters)

logm(

kg)

Coma Cluster

Virgo Cluster

Earth

Sun

OJ286 Galaxy Core

Planetary Nebula

Electron(classical

radius2a0)

NeutronStars

Nucleon& Nuclei

Muonium

VISIBLE

OBJECT

SIZE GAP

Osmium

A

S

T

E

R

O

I

D

S

3C 66B Galaxy Core

Sirius B

Sirius B

White Dwarf (Typical)

J0917+46

Sirius B

Milky Way Core

Milky WayN3379

M87Nearby Group

M32 Nucleus

N4486-B

M32Omega Centauri

Giant Molecular Clouds

M31 Nucleus

Shapley Supercluster

Bok Globules

M31

Planck Density is wayoff the Chart: at log 96.7

Brown Dwarfs

Radiationc2

within CosmicRc =c/H

Radiationc2 withinCosmicRc =c/H

Electron(classical

radius 2a0)

Earth

Center of M15Center of G1

Center of Omega Centauri

Center of M15

HLX-1

HLX-1Center of G1Center of Omega Centauri

Sun & Other MainSequence Stars

Jupiter

Red Giants & Planetary Nebulae

BetelgeuseRigel

Betelgeuse

Red Giant

Rigel

NeutronStars

J1650-500IC 10 X-1 HLX-1

EXO 0748-676J0437-4715

Supra-NuclearDensity Stars


Brown Dwarfs

White Dwarfs

Moon

Moon

Ceres

16 Psyche

433 Eros

1999 KW

2000 UG

Hydrogen Atom

Oxygen AtomHelium Atom

Hydrogen Atom (a0)

Nucleon(radius derived from

nuclear saturation density)

Muonium

Buckyball (C60)

Bacterium

Sand Grain

Blueberry

Elephant

Human

Melon

Virus

Iron

Osmium

ShapleySupercluster

ComaCluster

ComaCluster

Nearby GroupM87

N3379M31

M31 Nucleus Omega Centauri

Omega Centauri

Milky Way

Milky Way

N4486-B3C 66B Galaxy Core

Milky Way Core

N U C L E A R S A T U R A T I O N D E N S I T Y

SCHW

ARZ

SCHIL

DHO

RIZ

ON

(E

dge

of

the

Worl

d)

W A T E R D E N S I T Y

OJ287 Galaxy Core

M32

M32 Nucleus

M32 NucleusM31 Nucleus

Giant MolecularClouds


Bok Globules

Bok Globules

Virgo Cluster

Mass withinCosmicRc =c/H

Electron(barred Comptonwavelength c =

a0)


wavelength c = a

0)

Densityof

Bubbles,Atmospheres

&Giant Stars

Gases at STP(radii derivedfrom densities)

UraniumCalciumAtomic NucleiCarbon

Complex Molecule

Grape Seed

Mass withinCosmicRc =c/H

COSMIC EVERYTHING CHART (standard*)

Fig. 1. Chart 1. Logarithmic scales of mass, radius, density and gravitational acceleration ofobjects spanning the range of size of the known universe. Due to the fuzziness of some objects,deviations from spherical shape and uncertainties of measurement, some values are rougherapproximations than others. Being on a logarithmic scale, all quantities shown are neverthelessfairly accurate. This chart is thus a generally reliable representation of key physical magnitudesin our universe, with the possible exception of objects on the Schwarzschild horizon line.


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4 R. BENISH

M32Nucleon& Nuclei

Electron(classical

radius 2a0)

HydrogenAtom (a

0)


wavelength c = a

0)

J1650-500

IC 10 X-1HLX-1

Sirius B

J0917+463C 66B

OJ287

ComaCluster

Omega CentauriMilky Way

M32 Nucleus

M31 Nucleus

Bok Globules

EXO 0748-676

J0437-4715

Planck Acceleration

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Copyright Richard Benish 2010. With this copyright

tag in place, free unaltered distribution is encouraged.

If the Schwarzschild horizon were real, it would entail infinite central densities,

infinite accelerations and a singularity, at r = 0 (where the laws of physics break

down). Peter Bergmann has suggested that a theory with such properties carries

within itself the seeds of its own destruction. Unfortunately, most physicists ignore

this prediction. Furthermore, it is inappropriate to expect quantum theory, whose

proper domain is small and light objects, to rescue general relativity in this domain

of large and heavy objects. It is physically unreasonable to conceive that bodies of

matter collapse to zero volume behind a barrier to all communication (horizon).

This Chart presents key features of an alternative model of gravity according to

which volumes and densities remain positive and finite; concentrations of matter

may be compressed by gravity so much that spacetime curvature significantly

reduces the frequency and the amount of light that can escape, while densities and

accelerations remain well-behaved (so that there are no singularities).

This model predicts the existence of a density regime about as far removed from

nuclear density as nuclear density is removed from atomic density:

SGM stands for Space Generation Model of gravitation and cosmology. For details

on the motivation for this chart, see Cosmic Everything Charts Compared, and

other documents at GravitationLab.com and under Benish at Scribd.com.

*

Though it varies slightly from one atomic species to another, nuclear saturation

density is closely approximated by

Though it varies considerably, an illustrative

approximate measure of the density of most

familiar substances can be expressed as the mass

of a proton within a spherical volume whose

size is given by the Bohr radius:

Note: To minimize clutter, many points

are not labeledespecially those for

acceleration. But everything is

labeled at least once. Identification

may require finding the point corres-

ponding to a different quantity along

the same vertical line. Moon, Earth, Sun,

Milky Way and Milky Way Core have

been colored to facilitate finding these

particular points.

log(

3M/4r

3)(kgm3)

logg[GM/(r+2GM/c

2)2](ms2)

log r(meters)

lo

gm(

kg)

Electron(classical

radius 2a0)

Earth

Center of M15Center of G1

Center of Omega Centauri

Sun & Other MainSequence Stars

Jupiter

Red Giants & Planetary Nebulae

BetelgeuseRigel

Neutr

onSta

rs

J1650-500

J1650-500

IC 10 X-1

IC 10 X-1HLX-1

HLX-1

EXO 0748-676J0437-4715


Brown DwarfsWhite Dwarfs

Moon

Ceres

433 Eros

1999 KW

2000 UG

Hydrogen Atom

Oxygen AtomHelium Atom

Hydrogen Atom (a0)

Nucleon(radius derived from

nuclear saturation density)Muonium

UraniumCalciumAtomic NucleiCarbon

Buckyball (C60)

Complex Molecule

Bacterium

Sand Grain

Grape Seed

Blueberry

Elephant

Human

Melon

Virus

Iron

Osmium

ShapleySupercluster

ComaCluster

Nearby GroupM87

N3379M31

M31 Nucleus Omega Centauri

Milky Way

N4486-B

3C 66B Galaxy Core

Milky Way Core

N U C L E A R S A T U R A T I O N D E N S I T Y

H Y P E R - S U P E R D E N S I T Y

SCHW

ARZ

SCHIL

D E

dge

of

the

Worl

d

A T O M I C M A T T E R D E N S I T Y

M I N I M U M B A C K G R O U N D

M A T T E R D E N S I T Y

M I N I M U M B A C K G R O U N D

R A D I A T I O N D E N S I T Y

OJ287 Galaxy Core

M32

M32 Nucleus

Giant MolecularClouds

Bok Globules

Virgo Cluster

Mass withinCosmicRc = 3c/H=GM/c

2

Electron(barred Comptonwavelength c =

a0)

Densityof

Bubbles,Atmospheres

&Giant Stars

Gases at STP(radii derivedfrom densities)

Planck MassIf this chart is areliable indicator,then this theoreticalconcoction is of no

immeidate physicalsignificance.

Mass withinCosmicRc = 3c/H=GM/c

2

Coma Cluster

Virgo Cluster

Radiationc2

within CosmicRc = 3c/H=GM/c

2

Earth

Sun

Planetary Nebula

Electron(classical

radius 2a0)

Nucleon &Nuclei

Muonium

VISIBLE

OBJECT

SIZE GAP

Osmium

A

S

T

E

R

O

I

D

S

Sirius B

White Dwarf (Typical)

J0917+46Brown Dwarfs

Sirius B

Milky Way Core

Milky WayN3379

M87Nearby Group

M32 Nucleus

N4486-B

M32Omega Centauri


M31 Nucleus

ShapleySupercluster

Bok Globules

M31

Planck Density is wayoff the Chart: at log 96.7

Betelgeuse

Red Giant

Rigel

Moon



wavelength c = a

0)

NeutronStars

Radiationc2

within CosmicRc= 3c/H=GM/c

2

COSMIC EVERYTHING CHART (SGM*)

( is the fine structure constant.)

Fig. 2. Chart 2. Essentially the same as Chart 1, except that objects that were on theSchwarzschild horizon line are now interpreted in terms of the Space Generation Model (SGM).The masses of these objects are the same as in Chart 1, but their densities and accelerationshave been adjusted to reflect their new proposed radii. According to the SGM radii smaller than2GM/c2 are allowed without entailing division by zero; i.e., without predicting the stoppage oflight and clocks or infinite accelerations. As a result, we find within a region of relatively smallmass differences (horizontal segment of the S-curve) profound transformations in size anddensity. If the pattern initiated from atomic density to nuclear density is continued across theSchwarzschild line, we expect to find two more key densitieseach one corresponding to an

-folding in size and an 3

-folding in density. (See 11 for more details.)


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thought of as black holes. Well return to this branch below.Presently, note that the other branch consists, initially, of stars whose densities aresmall compared to that of familiar matter (except, perhaps, air). Then, after crossing agap corresponding to a range of size that few if any visible bodies of matter adopt, wecome to regions of space so large that they contain many starsor perhaps only cloudsof extremely tenuous molecular matter. Continuing on to larger masses and sizes, wefind globular clusters, galaxies and clusters of galaxies. The largest local structures thatare still identifiable as such are superclusters of galaxies.

At the extremity of this branch we find another length that is not so much a physicalthing as a theoretical thing. In standard cosmology (big bang inflation) this scale lengthis known as the Hubble radius (big bang): RC = c/H, where c is the light speed constantand H is the Hubble constant. The latter constant plays a key role in the law whichexpresses how much the light from distant galaxies is redshifted for increasing distances.

In the alternative SGM cosmology RC is similarly related to the redshift-distance law, butits size is three times greater than the standard Hubble radius (SGM): RC = 3c/H. (Thisis explained in 10 and, in more detail, in SGM Cosmic Numbers and DE. [6]) By eithermodel, on this scale of size it is no longer meaningful to think of being at the boundaryor center of a particular structure. In other words, on this scale the universe appears tobe homogeneous. Therefore, though we may calculate a gravitational acceleration dueto the matter within RC, by contrast with all smaller structures, the calculated value nolonger corresponds to a positive accelerometer reading. All smaller structures that areidentifiable as such manifest inhomogeneities that would result in positive accelerometerreadings because of these inhomogeneities. But on the scale ofRC, the gravitational effectis the same from all directions, so it cancels out to zero.

Now consider the line of slope = 1 cutting across the upper portion of the charts.In Chart 1 the region to the left of (and above) the line is shaded. This boundaryrepresents the Schwarzschild event horizon, whose radius for any given mass, M is r =2GM/c2, where G is Newtons constant. Most physicists consider this line to be ofphysical significance, as it follows from GR, which is otherwise fairly well supported byempirical evidence. By definition the line is unobservable: Everything at or within thehorizon is beyond communication with the outside world. Hence, such objects have beengiven the name, black hole. The existence of black holes is deduced from theory and fromevidence indicating the presence of extremely massive, yet dark, bodies of matter. Theseobservationally deduced masses, as shown on the charts, span the wide range of about(4 1010)M, where M is the mass of the Sun, which is 2 1030 kg.

Considering the fact that we really dont know the sizes of these objects, it is naturalto ask whether this trail of points should in fact be located on the theoretical horizon line.Are there any reasons to think the radii corresponding to these masses should have some

other valuesvalues that do not abide by the standard black hole paradigm? In whatfollows we will consider three kinds of reason to question the standard model: aesthetic,mathematical, and physical. The aesthetic argument, which is brief and presented in2, is by itself not intended to be nearly as convincing as the mathematical or physicalarguments. But in conjunction with them, I think most readers will be able to see whichmodel looks better.

The mathematical and physical arguments are not always distinguishable as such.The most patently mathematical argument, briefly, is that black holes arise in the mindsof physicists from the dubious practice of dividing by zero and accepting the result asphysically reasonableor perhaps acknowledging the unreasonableness but not persist-ing to find a reasonable alternative. A few of the key elements of contemporary black


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6 R. BENISH

hole theory are presented in 3. Most physicists have come to regard the inevitablesingularities in the black hole paradigm as not a problemat least, not a big enoughproblem to discourage the enormous output of academic publications concerning thesehypothetical, unobservable ob jects. Evidently, they are gambling that quantum theorywill someday rescue GR from the predicted singularities. The more physical argumentsare too numerous to mention them all. In 4 we briefly discuss six physical issues thatshould, perhaps, cast some doubt on the black hole paradigm. If authoritative opinionwere the scale by which we judge, the most weighty objection could be the doubts castin 1979 by the prominent physicist, Peter Bergmann. His impression was that admittingthe occurrence of a singularity amounts to going overboard. [7]

The most important reason for caution, from the present point of view, is that ourempirical knowledge of gravity suffers from a large gap that exists not only in the exoticrealm of black holes, but extends to the centers of massive bodies in our common expe-

rience. This gap could be filled by performing a relatively simple laboratory experiment.The basic idea of the experiment and what standard physics predicts should be its out-come, are presented in 5. Throughout this essay, we question standard physics; or moreaccurately, we insist on questioning Nature to see if standard physics rings true. Theresult of the experiment might be such as to invalidate the interior solution of GRandthereby, the whole black hole paradigm.

Although the proposed experiment is disarmingly simple, its potential import is hardto overestimate. If the result supports the standard theories, very little would change;wed simply have confirmed a prediction that physicists have presumed to be true (with-out empirical evidence) for generations. But if the prediction presented in this essay wereto be confirmed, many of the foundations of physics would be overturned. Therefore, in6 we will begin to take a detailed look at the foundations of the model that serves asthe basis for the new prediction. At its core, the whole argument is based on a strict in-terpretation of measurements obtained from motion sensing devices: accelerometers and

clocks. Simple logic quickly shows that this interpretation is untenable if there are onlythree dimensions of space. Rather than prematurely reject the model because of this, weinstead infer this to be an indication that the world actually has four dimensions of space(and one of time). Special emphasis is given to the importance of this inference. Thenew models proposal that there are four instead of three spatial dimensions is put inthe context of the history and current state of other investigations into the possibility ofextra space dimensions. Unlike the earlier conceptions, the new one explicitly relatesthe space dimensions to both matter and time, such that the whole manifold is in a stateof perpetual stationary motion. All three of the fundamental elements of physics, mass,space and time, thus combine interdependently via gravity so as to necessitate a fourthdimension of space.

Since extra-dimensional space is a key feature of various theories of quantum gravity,our purpose in 7 will be to see how these theories compare with the present model. In 8we look at the relationship between higher-dimensional space and the inverse square lawof gravity. The proven validity of the law down to very small sizes is regarded by standardphysicists as evidence that, if higher dimensions of space exist, they are very well hidden.According to the model presented in 6 this is not true: the fourth dimension of spacecould be infinitely large. If it is hidden, it is hidden in plain view.

In 9 we reinforce the implications of 6 and establish the foundations of the newmodel in more detail by deriving equations that explicitly deny the total blackness ofastronomical objects that are currently regarded as black holes. The derivation appealsto three things: 1) the consequence of special relativity according to which the velocities


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of material bodies never exceed that of light; 2) Einsteins equivalence principle; and3) the inverse-square law. These results, which mean that light and clocks do not stopbecause of gravity, include an equation which expresses a (mass-dependent) limit on theacceleration due to gravity, as seen in Chart 2.

An outline of the cosmological consequences of the new gravity model is presented in10. Two of the most prominent points here are: 1) agreement with observations; and2) that the scheme implies a deep significance to and connection between the variousdensity regimes in the universe. Among them are the nuclear saturation density, atomicdensity and the two cosmic background densities: matter and radiation. The SGM cos-mology predicts a stationary universe. That is, a universe that expands but maintainsthe same proportions because matter increases along with space. It has sometimes beensuggested that, especially in a universe whose proportions remain constant, we shouldexpect a connection between the constants arising in atomic theory and those arising

in astrophysics and cosmology. Beginning with a small number of assumptions, we findan extraordinary order arising amongst atomic and cosmological constants. For exam-ple, we end up with an extremely simple definition of Newtons constant and intriguingconnections between the energy density of the cosmic background radiation, nuclear sat-uration density, the fine structure constant, the proton-to-electron mass ratio and othernotoriously mysterious numbers.

In 11 we explore the implications of the cosmological model as they apply to ourCosmic Everything Charts. Within a small range of mass, profound transformationsoccur to stellar matter because the force of gravity begins to be comparable to nuclearand electromagnetic forces. The first of these transformations is the formation of whitedwarf stars. A more extreme transformation is the formation of neutron stars. Alongour trail of points we see that a class of stars with a little more mass than the Sun arefound to have densities of the order of Nuclear saturation density. This is near what thestandard paradigm regards as an astrophysical limit, beyond which a black hole wouldform. Because of this imagined limit, the trail of points is here supposed to take a sharpturn, as shown on Chart 1. But various physical facts as well as the locations of thenon-controversial points outside the horizon, suggest not a sharp turn to hug it, but acontinuation across the Schwarzschild line. This then implies the existence of two morekey density regimes. The first SGM-specific regime would be that of supra-nucleardensity stellar objects. And the last would be that of super massive and hyper-superdensity collapsed cluster cores. What distinguishes Chart 2 from Chart 1 is the logicthat motivates the rearrangement of these nine points.

An application of the SGM to observational data suggesting the existence of blackholes is presented in 12. Objects that are regarded as strong candidates for being bonafide black holes acquire this status by revealing certain characteristics that standard

models in physics are incapable of explaining any other way. It is shown that the observedcharacteristics are entirely consistent with those predicted by the SGM for objects thatare very compact and very dark, but do not have horizons or singularities.

One of the recurrent themes of this essay is the importance of accelerometer readings.In the standard literature we sometimes find discussions that imply a similar importanceand yet come to completely different conclusions. Our final section, 13, begins bycomparing the logic displayed in two of these standard discussions with the logic of theSGM. This launches us into a summary of the essays main points. It is shown thatthe logic that leads to the new arrangement of the nine contentious points on Chart2, converges from two directions. The first appeals to immediate human experienceand empirical evidence obtained on or near Earth. It involves the strict interpretation of


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8 R. BENISH

accelerometer readings, clock rates, the limiting speed of light and the inverse square law,which suggest the existence of a fourth spatial dimension. Along this route we find cogentreasons for doubting the existence of bona fide black holes. Even at this point we wouldsee fit to place the nine contentious points somewhere inside the Schwarzschild horizon.Independently of the black hole question, however, this route leads to certain cosmologicalconsequences. Exploration of these consequences reveals that the well known ubiquity ofthe fine structure constant, at smaller scales extends also to cosmology and astrophysics.This reinforces our doubt (about black holes) borne of physics at smaller scales andprovides the basis for the conjecture as to (approximately) where inside the Schwarzschildhorizon the nine contentious points should be placed. Thus local and global reasoningconverge to yield Chart 2. Emphasized here, as elsewhere is that this model as a wholestands or falls depending on the result of the experiment discussed in 5, 6 and 9.

2. Aesthetic Argument: an Edgeless World

A quick inspection of Figure 1 may naively suggest that the absence of any bodiesof matter (red dots) to the left of the Schwarzschild horizon means this line represents,in a sense, the edge of the world. Many distinguished scholars believe in the reality ofthis edge, whose properties well discuss below. Before considering these properties, letus simply inspect the trail that leads to the edge, i.e., the string of points from dwarf

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-40 -30 -20 -10 0 10 20 30

-40 -30 -20 -10 0 10 20 30

log r (meters) log r (meters)

logdensity

Standard SGM

logacceleration

logmass

logdensity

logacceleration

logmass

Fig. 3. Simplification of Charts 1 and 2. The aesthetic argument, basically, is that Natureis beautiful, which, to my minds eye means also that Nature is fundamentally continuous.Therefore, we should expect not to find an edge to the world where, for example, time stops andbodies of matter collapse to zero volume. These properties characterize the wall that the trailof points runs into in the Standard graph above (left)which causes the trail to bounce backat an acute, unnatural, ugly angle, dictated by a geometrical theory far removed from physicalexperience. Whereas the SGM graph (right) reflects what happens when the imaginary wall isremoved. Whether it really is truer and therefore more beautiful, depends on the result of theexperiment discussed in 5, 6 and 9.


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stars to neutron stars. This trail meets the Schwarzschild horizon line so as to form avery sharp corner. To more clearly see this, refer to Figure 3, which shows the generalshapes of the three lines in both charts with a minimum of detail. The standard idea isthat the trail then actually makes this abrupt, acute turn. Is this natural? My aestheticjudgment tells me it is not. A natural extension of the trail, I suspect, would be to crossthe Schwarzschild line (beginning with objects, J1650-500 and IC 10 X-1) on a smoothcurve.

Of course it would be foolish to try to decide which path is actually taken by Naturebased on just this aesthetic criterion, or by the meager empirical evidence that we have.Nobody knows for sure, either way. The distinguished scholars believe in the sharp cor-nered path because of their faith in GR. They draw the chart as in Figure 1 (and the leftside of Figure 3) because they have confidence in the validity of a variety of assumptions,assumptions that we will question in what follows and assumptions that we ultimately

propose to test with a simple experiment. (Interior Solution Gravity Experiment [8])Until this experiment is performed, defense of the smoothly curved trail of points argu-ment can appeal only to logical reasoning, which certainly includes mathematical andphysical facts as well as, if it may be referred to as such, at least one aesthetic fact,i.e., the look of Figure 3.

3. Einstein, Standard Schwarzschild Coordinates and Their Strong Field

Extension

In 1939, before the expression, black hole was invented and before astronomical ob-servations revealed the presence of extremely massive, yet compact and dark objects,Einstein wrote a paper arguing that the Schwarzschild horizon (which he refers to asa singularity) cannot exist. Given how the black hole industry has blossomed in morerecent decades, it is obvious that most physicists regard Einsteins argument (about hisown theory) as faulty. In simple terms, the singularity arising in this problem is theresult of dividing by zero, so that the answer blows up, becomes infinite, is undefined.This mathematical operation (dividing by zero) is performed within GRs Schwarzschildsolution, which involves coefficients representing the degree of spacetime curvature withrespect to a spherical mass as a function of distance. This is GRs most famous and mostwidely used solution because it is supposed to describe the static spacetime geometryaround a spherically symmetrical body like the Earth or Sun.

In the usual (standard) form of the solution, the coefficient (1 2GM/rc2) appliesto the time coordinate and its inverse applies to the radial length coordinate, r. Thedifference between the square root of these coefficients and unity represents the magnitudeby which clock rates and the lengths of radially oriented rods change due to gravity. (See

9.) For most bodies of matter, the length 2GM/c2

is very small compared to thedistance, r, so the coefficient is usually 1. A black hole results from the case that allthe matter of a body of radius r lies within r = 2GM/c2. This makes the coefficients zerofor time and infinity for length. Consequently, for this extreme case, the length 2GM/c2

corresponds to the spherical surface at which clocks stop ticking, light stops moving andaccelerations become infinite. [9-11]

Einstein opened his argument by asserting that this surface constitutes a place wherethe field is singular. At the end of a lengthy analysis, Einstein concluded that TheSchwarzschild singularity does not appear for the reason that matter cannot be concen-trated arbitrarily. And this is due to the fact that otherwise the constituting particleswould reach the velocity of light. [12] The main fault in Einsteins analysis, from the


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modern point of view, is that he failed to recognize that the spherical surface representsonly a coordinate singularity. Whereas, it has been shown that, by adopting a differentsystem of coordinates this singularity can be removed. Two of the most common systemsof coordinates used for this purpose are known as the Eddington-Finkelstein coordinatesand the Kruskal-Szekeres coordinates. [13]

The transformation from standard Schwarzschild coordinates to extended coordinateshas sometimes been likened to removing the singularity arising on a map of a spherewhose equator is designated as the zero of latitude. [14] It is thereby implied that theremovable Schwarzschild singularity is no more troublesome (shall we say) than thissimple cartographic singularity. One is well-motivated to remove the singularity in theSchwarzschild solution in standard coordinates because, as it is commonly described, thesolution otherwise exhibits pathological or bad behavior at the horizon. This is dueto the possibility of zero appearing in the denominator. The coordinate extension is a

method for transforming the badness out of the equation. Though goodness appearsto be restored by preventing the zero from appearing, it must be pointed out that thisgood behavior applies only to the character of the trajectories (geodesics) representingfalling objects or light rays that cross the horizon from the outside or to falling objectsor light rays within the horizon that cant possibly get out. The good behavior does notrefer to the object itself; i.e., to the source of the gravitational field. Nevertheless, byestablishing that geodesics need not come to an end, but can safely cross the horizon,at least inwardly, such coordinate transformations are typically presented as a kind ofassurance that this behavior is not only geometrically, but also physically reasonable. Sorelativists seem pleased to have tamed a significant portion of the otherwise unruly blackhole; geodesics from 0 < r < are now all in perfect geometrical order.

4. Six Issues of Concern

Notice, however, that some troubling physical questions arise or remain, in spite ofthis clever display of mathematics. We can identify six issues.

1. Under normal circumstances, when we contemplate gravitational problems we havenot only geodesics (falling or light ray trajectories) but a physical body which serves as areference system for observations. We can draw or visualize the body and the geodesicstogether in the same picture. Whereas, in discussions of black holes this contact withphysical reality is missing. Introducing a physical object to represent the location of thehorizon is deemed unacceptable because such a stationary object cannot exist. This isnot because of the force of gravity. It has often been pointed out that the gravitationalforce actually diminishes as the size of a black hole horizon increases. This is reflected inChart 1 by the (descending) points for the acceleration corresponding to the (ascending)

objects on the Schwarzschild line. What causes the impossibility of the physical objectat the horizon is thus nothing but geometry. Its as though physical matter is at themercy of abstract mathematics. Thus, it is not uncommon to find physicists proclaimingthat gravity is just geometry. [15] The good behavior of geodesics engendered by theextended coordinates means that an object can fall past the horizon without experiencingany abrupt change in its physical state. But a stationary physical manifestation of thatwhich has created the horizon is forbidden. It is only right that we should questionthis drastic state of affairs. Possibly, physicists are just imagining their mathematicalabstractions to be more concrete than they really are. Possibly, matter itself does notconform to their geometry.

2. The coordinate transformations used to extend the standard Schwarzschild solution


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inside the horizon involve the change of timelike quantities to spacelike quantities andvice versa. This is sometimes depicted (in Eddington-Finkelstein coordinates) as thetipping of light cones on a spacetime diagram. The cones change from opening upwardto opening to the left (toward r = 0) as the horizon line is passed. This is what happensto the geodesics that enter the horizon, even though an observer on such a geodesicwould not notice. Inside the horizon space turns to time and time turns to space. Sosays geometry. But does this make physical sense? Surely the intuitive answer is no.If our intuition is wrong, then we should accept it as such only after all reasonablealternatives have been considered and tested. Thus, maintaining skepticism (and a searchfor alternatives) is clearly advisable.

3. Issue (1), i.e., the loss of contact with stationary physical matter, may be amplifiedby pointing out again the state such matter would be in at the horizon: clocks would stopticking, light would stop moving, and acceleration would become infinite. These circum-

stances are not changed by extending the coordinates; this patently bad behavior is notmade good. So removing the coordinate singularity by a mathematical transformationin the case of black holes is far from the benign operation of removing a cartographicsingularity from a map. At Earths equator light and clocks do not stop, accelerationsdo not go to infinity and physical objects may still exist there. Faced with the primafacie unphysical predictions of GR, two choices arise: Most physicists have simply optedto disallow the physical body: geometry rules. The alternative is to suspend belief in thetheory: physicality rules. That the theory should make such predictions, that it shouldhave such trouble representing a physical object, is perhaps an indication that the theoryitself is flawed, at least insofar as it applies to extremely compact, massive bodies. Thisin fact was Einsteins view.

4. Issues (1 3) are all concerned with the Schwarzschild horizon, i.e., with thecoordinate singularity. Physicists are happy to have rendered this singularity removable.As is well known, however, this is impossible to do for the singularity at r = 0. This iswhere GR is helpless to avoid infinite density, infinite curvature and the breakdown ofphysical law. This helplessness was made official, in a sense, by the work of Penrose andHawking, who demonstrated mathematically the inevitability of singularities in GR. [16]Unfortunately, most physicists were not deterred by this work. Indeed, it seemed to setoff a flurry of work along similar lines and work that explored the further consequences ofhorizons and singularities, whose physical reality was rarely questioned. Modern physicshas two main pillars: GR and quantum theory. So the failure of GR with regard tothe black hole singularity simply means, to most physicists, that this is where quantumtheory must step in to make everything okay. This logic is questionable because quantumtheory is widely regarded as the proper description of micro-physics involving very smalland light objects. But GR got into trouble on account of very large and heavy objects. Is

it reasonable to expect quantum theory to rescue GR in this domain? Possibly, this kindof reasoning played a part in one of the cautious responses to these developments, quotedbelow. One of Einsteins former assistants, Peter G. Bergmann, voiced his concerns at ameeting in 1979 to celebrate Einsteins centenary. Upon opening a discussion followingthe lectures of Stephen Hawking (and others) Bergmann said:

Singularities . . . are intolerable from the point of view of classical field the-ory because a singular region represents a breakdown of postulated laws ofnature . . . A theory that involves singularities and involves them unavoidably,moreover, carries within itself the seeds of its own destruction . . . It is conceiv-able that the energy condition . . . could be violated. It seems, in some respect,


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that if you are going overboard and admit serious exceptions to what we con-sider the conventional behavior of nature, a violation of the energy conditionmight be swallowed just as much as the occurrence of a singularity . . . Thewhole situation looks like one in which a completely new idea is required. [7](Emphasis added.)

Unfortunately, such remarks concerning black holes are rare among such illustrious au-thors. As the 21st century moves into its second decade, it seems the novelty and theoutrageousness of black holes has all but worn off. Most physicists have grown used tothem.

5. My various critiques and this list of concerns is directed at what has become thestandard interpretation of GR. It is pertinent to mention a small but vocal minority ofphysicists who argue that the proper interpretation of GR denies the existence of black

holes. In the work of Stephen J. Crothers, for example, it is pointed out that, whathas come to be called the Schwarzschild solution is not the solution actually derivedby Karl Schwarzschild. [17] The solution most commonly presented in textbooks andjournals is a significantly different misrepresentation of Schwarzschilds original work.This claim is patently true, as one can see by looking at the originals. Also, many ofthe conclusions drawn by Crothers on the basis of this difference appear true and wellargued. He asserts, for example, that Schwarzschilds original solution exhibits neitherhorizons, infinite densities, nor any reasonable possibility of swapping time for space andspace for time inside r = 2GM/c2. But some of Crothers conclusions, especially thosethat defend the basic idea of GR that gravity should be represented by a static geomet-rical equation, I strongly disagree withand I propose to test my disagreement with alaboratory experiment. In other words, the differences between Crothers interpretationof GR and the SGM still greatly outweigh any mitigating similarities. Furthermore, hiswork is obscure and unjustly ignored by the mainstream. For these reasons, the rest ofthis essay (including concern #6) will proceed on the basis that the standard ideas traceback to the origins claimed of themeven though this is evidently not true. Since myown work is presently even more obscure than his, I can only salute Crothers and thoseothers who have tried to set the record straight.

Our last concern in this section serves as a fitting transition to the next one, becauseit involves the experiment whose result could bear on the black hole paradigm by finallyfilling in a conspicuous gap in our empirical knowledge of gravity.

6. Extensions to the Schwarzschild solution have the curious feature of encompassingthe whole domain 0 < r < . What makes this curious is that the whole thing isan exterior solution. For material bodies under normal conditions the GR descriptionrequires two separate solutions. The most commonly used pair of solutions were both

originally derived by Karl Schwarzschild in 1916: his static exterior and static interiorsolution (for an incompressible fluid sphere). [18] The strange behavior discussed aboveis found at and within r = 2GM/c2. So there is an inside and an outside with respect tothis geometrical horizon; the horizon lies between the extremes. But the whole thing isexterior to the source, to the matter responsible for the field. All the matter is at r = 0.So the whole thing is just a geometrical object. Its physicality is not allowed. Thereis no inside to the physical object because its matter occupies a space of zero volume.By contrast, for normal matter we have either a surface (stars, planets, etc.) or a moregradual transition of changing density (clouds and cluster systems). In the first casethere is a distinct difference between being inside or outside the body. In the second casethe distinction is somewhat blurred, but we can still determine the degree of immersion


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by measuring the change in density. Both situations are quite physical.What is common to both the geometrical extensions of Schwarzschilds exterior solu-tion and the physical insides of real bodies is that we dont really know what happens atr = 0. For black holes this will be forever true. In the case of real bodies the only seriousimpediment to finding out what happens at r = 0 is the lack of desire to look. Supposewe drill a hole through the center of a real body (e.g., a uniformly dense sphere) and dropa test ob ject into it. What happens? The theoretical answer, the standard prediction forsuch an experiment is well known. But the prediction has never been tested, so nobodyreally knows what happens. The alternative gravity model on whose basis we surmisethat nine of the objects on Chart 2 lie within the imaginary horizon also predicts novelbehavior for this circumstance of radial falling inside ordinary matter. Various reasonsfor this novel prediction will be discussed in 6 and 9.

5. Interior Solution Gravity Experiment: Standard Prediction

The interior falling experiment mentioned above has so far been described only interms of an ideally simplified, but impractical circumstance. That is, as a relativelyisolated spherical mass in outer space, as schematically shown in Figure 4. But it couldbe done in an Earth-based laboratory with a modified Cavendish balance. See Figure 5.(For more details about this experiment and its feasibility, see Interior Solution GravityExperiment [8].) The prediction for the result of this experiment, in Newtonian terms, isthat the test mass is supposed to oscillate from one end of the hole to the other becauseof the conservative, attractive force of gravity. According to GR, essentially the samemotion is predicted, but it is supposed to be understood in terms of the curvature ofspacetime. The key to producing the predicted oscillation is that the rates of stationaryclocks inside the sphere are supposed to diminish going inward; the rate of a clock atthe center is supposed to be a local minimum. In other words, there is a very definitecorrelation between how clock rates vary inside matter and how test objects fall insidematter. This is true for both GR and the SGM.

Nobody has ever compared the rates of clocks inside matter with those outside matteras a function of gravity. Therefore doing this experiment would test Newtons attractionhypothesis and, indirectly, Einsteins local minimum clock rate hypothesis. According tothe ideals of science this experiment ought to be done simply because it hasnt been donebefore. The gap in empirical knowledge exists. The gap is of deep physical significancebecause it cutsliterally and figurativelyright through the middle of the object of ourstudy.

Black holes are often referred to as exotic objects. The singularities which makethem so have been defined as regions of the spacetime into which the geodesics of

the spacetime do not enter (i.e., r = 0). Geodesics are simply the trajectories of falling

Fig. 4. Ideal experimental setup: Relatively isolated spherical mass with test object.


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Fig. 5. Practical experimental setup: Modified Cavendish balance in Earth-based laboratory.

objects or light rays. Since geodesics going into a black hole are incomplete, since theyonly go in past the edge of the world and never come back out, the spacetime is singular.Ironically, in the mundane (to relativists) context of ordinary matter (e.g., a rock ora ball of lead) nobody has checked to see whether the predicted geodesics conform toreality. If a hole is drilled through the ball of lead, GR says the geodesic of an objectradially falling into the hole passes r = 0. But nobody has ever seen this happen in thephysical world. It is impossible to see what happens to an object dropped into a blackhole. But in the world we actually live in, empirical evidence is much easier to gather.The experiment calls only for an ordinary body of matter through which is drilled anordinary hole. Therefore, as scientists, we are well-advised to do it. We should notpresume to know its result based only on theory, analogy or extrapolation. The resultneeds to be learned directly from Nature. And this should be our course of action, Iwould argue, even in the absence of a model that predicts a non-standard result.

6. Interior Solution Gravity Experiment: SGM Prediction Derived from

Accelerometers, Clocks, Rotation Analogy and Hyper-Dimensional Space

This section is lengthy and organized into subsections because in it we cover a lot ofground concerning the physical basis of the Space Generation Model (SGM). In manyways this model conflicts with standard ideas. Among them is a novel interpretation ofhyper-dimensional space. To clarify this interpretation and other reasons why the SGMmakes a new prediction for the interior solution gravity experiment, we will appeal lib-

erally to direct physical experience and to various analogies. To keep our perspective wewill often compare the SGM-based reasoning with reasoning borne of standard physics.It should be emphasized that all arguments based on the SGM trace back to the concreteempirical evidence provided by accelerometers and clocks. This will be especially impor-tant to bear in mind when, in spite of this simple basis, the SGM becomes a challenge tovisualize. By contrast, the comparative ease of visualizing certain aspects of GR comesonly at the cost of dubious physical reasoning.

6.1. Logical Strategy and the SGM Prediction. According to the Space Generation

Model, oscillation of the test mass through the larger mass, as predicted by standardtheory, does not happen because gravity is not a force of attraction and the rates of


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clocks inside the body do not diminish to a local minimum, they increase to a local max-imum. The basis for this prediction is a strict interpretation of motion-sensing devices,in particular, accelerometers and clocks. Outside the context of gravity, it has been em-pirically established that 1) Accelerometers give non-zero readings when they are forcedto accelerate, either by rotation or by linear propulsion; and 2) The rates of properlyfunctioning clocks slow down due to their velocity. The most meaningful measurementsof this latter effect are those involving rotational motion, because cyclic motion allowsan absolute measurement, whereas linear motion entails some ambiguity (that we neednot go into).

What do accelerometers and clocks tell us about our common experience of gravity?Our strategy for answering this question is to abide by advice contained in Newtons Rules for Reasoning in Philosophy (whose essence is also captured by the aphorism known asOccams razor). Quoting Newtons Rules I and II:

We are to admit no more causes of natural things than such as are both

true and sufficient to explain their appearances . . . for Nature is pleased withsimplicity and affects not the pomp of superfluous causes.

Therefore to the same natural effects we must, as far as possible, assign the

same causes. [19]

Fact number one is that we live on the surface of a 5.971024 kg sphere of matter. Wehave found at this surface that accelerometers give readings of 9.8 m s2 in the upwarddirection and clocks are slow compared to clocks at infinity by

1 2GM/rc2. (See

9.) This is how slow our clocks would be if we were moving with a velocity

2GM/r.Therefore, the first possibility that we should consider (according to Newtons Rules)is that this acceleration and velocity correspond to our actual state of motion, which isevidently caused by the huge mass beneath our feet. (This motion, in turn, appears to bethe cause of spacetime curvatureabout which more later.) Let us add one more fact:When an accelerometer is dropped (as into a hole through a larger body) its readingimmediately goes to zero. By Newtons Rules this would evidently mean that a fallingaccelerometer is not accelerating.

Now let us relate these facts and deductions to our experiment. Since motion sensingdevices indicate that the large mass moves outward rather than that the test object movesinward, we have no reason to expect the test object to pass the center. Corresponding tothis prediction is the blue curve in Figure 6, which is compared to the red cosine curvepredicted by standard theory.

Even though we arrive at our prediction by strictly abiding by a well-worn logicalstrategy, this interpretation of the facts contradicts other assumptions most of us have

made about physical reality. Upon first encountering it, the description we have arrivedat is likely to evoke the image of some kind of expansion in three-dimensional space.Such imagery quickly runs into conflict with experience, and on the basis of this conflict,one may prematurely conclude that the SGM makes no sense. Among the contradictedstandard assumptions are that gravity is a force of attraction, that energy is strictlyconserved, and that the world possesses only three dimensions of space. But what ifthe world possesses four dimensions of space? What if the motion indicated by ourinstruments is due to a process whereby matter perpetually generates space and thisspace can be shown to have four dimensions? If this were the case then it should bepossible to prove that the standard theories conflict with the facts of Nature, not theSGM. The most cogent proof or disproof would consist in the result of the interior


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0 15 30 45 60 t

+R

SGM

Standard

0

R

Fig. 6. Standard (red) and SGM (blue) predictions for interior solution gravity experiment.

solution falling experiment. If the test object oscillates, standard theories would b esupported. Whereas if the test object does not oscillate, much of standard physics wouldhave been falsified.

I want nothing more than to find out the result of the experiment, but the meansto do so are presently beyond me. Meanwhile, to establish the plausibility of the SGMprediction, it is crucial to address the dimensionality of space question without furtherdelay. Everything else hinges on it. Therefore, in what follows we will approach thequestion from several different angles. We will briefly mention the history of hyper-space and point out its role in some current academic investigations that also concerngravity. By most accounts, the question is open and important. Unfortunately, modelsborn of standard physics that involve extra dimensions are nowhere near as conduciveas the SGM is to empirical testing. If gravity is related to the dimensionality of space,then isnt the most natural place to begin to observe gravitational phenomena at r = 0,

i.e., the center of a massive body? Unlike standard theorists, we will not presume toknow what happens here before we look. Task #1 should therefore be to find out whetherNature abides by the blue curve or the red curve in Figure 6.

6.2. Basics of Hyper-Dimensionality and the Importance of Time in the SGM.

Having no choice, for the moment, but to postpone task #1, let us note that attempts toconceive of a spatial dimension beyond the familiar three have a long history. Althoughsome of these explorations in hyper-space explicitly involve gravity, [20-23] the relation-ship is conceived in these cases quite differently from how it is conceived in the SGM. Wecan begin to see whats involved in any geometrical conception of four-dimensional spaceby considering a schematic representation of each of the familiar dimensions consideredseparately, as in Figure 7. It is easy to see how the first three spatial dimensions may bebuilt up by progressive perpendicular extensions of simpler fundamental geometric enti-

ties: point (0 dimensions), line (1 dimension), plane (2 dimensions), solid (3 dimensions).As soon as the point moves, we have not only a one dimensional extension of it (line) wealso have time. I am aware of only one academic author who has seen fit to point thisout. The philosopher and inventor, Arthur M. Young wrote: In terms of dimensions,the line is extension and the birth of time. [24]

It is important to recognize that time is a dimension unto itself and that, even thoughit goes with the dimensions of space, it has a distinctly different character. It is measuredwith clocks, not rulers. To facilitate keeping this distinction in mind, we will frequentlyadopt the common notation that the one temporal dimension is added to the number ofspatial dimensions in parentheses. Thus the moving point which gives us a line is a (1+1)-dimensional entity (one spatial + one temporal). Similarly, as soon as the line moves


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Fig. 7. Schematic of hierarchy of space dimensions.

perpendicular to itself we have a (2+1)-dimensional entity (two spatial + one temporal)

i.e., a plane. When the plane moves perpendicular to itself we have the common idea of(3+1)-dimensional spacetime.

Also shown in Figure 7 is a possible view of the next step in the progression, i.e.,an extension from the third to the fourth dimension. The idea is that, to geometricallyrepresent the fourth spatial dimension, the three-dimensional object must, to continue thepattern, extend itself perpendicularly in every direction at the same time. It is importantto notice that this last step marks a significant departure from the previous three. Eachof the first three steps was accomplished by a linear extension. A single arrow sufficesto indicate the new direction at each step. After three dimensions have been tracedout, to keep going suddenly requires a volumetric extension. A whole family of arrowsis now needed. The right-most part of the figure is sometimes called a hypercube ora tesseract; it is often found in discussions about hyper-dimensional space. [25-32] InRuckers book, The Fourth Dimension, he states: we can represent a hypercube by

drawing a small cube inside a large cube. The idea is that the small cube is fartheraway in the direction of the fourth dimension. Referring to the same image, ThomasBanchoff calls it a head-on view and a central projection of the hypercube.

Now recall our initial assumption that accelerometer readings indicate an actual stateof motion, and that to generate the next spatial dimension from a given spatial dimen-sion requires time, time to extend all of the given dimension into the next one. Ruckerscomment then implies that further away in the fourth [spatial] dimension correspondsto a different moment in time. In particular, the smaller cube is further and earlier. Itis enlightening to consider another perspective. That is, to imagine ourselves as residingat the center of the hypercube. This way we have no choice but to think of it as avolumetric thing instead of as a flat picture. In this case, larger would mean further andlater.

6.3. Hyper-Dimensionality and the Importance of Matter in the SGM. It is im-portant to recognize that the hyper-dimensionsional explorations cited above take noaccount of time or matter. It is presumed to be sufficiently logical to draw conclusionsabout the dimensionality of space without reference to the other fundamental physical di-mensions.(1) (This is a symptom of fragmentation, about which more will be said later.)

(1) Two distinct meanings of the word, dimension come into play here. In physics the termrefers to the physical units used to represent a quantity. For example, energy has the dimensionsML2/T2, where M, L, and T are generic mass, length, and time. In geometry, the term refersto the degree of space, as in one-dimensional line, two-dimensional surface, etc.


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Space dimensions are not conceived as anything that necessarily involves or requires mo-tion; nor as anything that depends on the cause of motion, i.e., matter. So we get a kindof static, abstract picture of strange spatial behavior having no discernible connection toexperience. In the present conception, by contrast, matter must be mentioned becausewithout it we do not get a positive accelerometer reading. Without matter there is noevidence of motion. And without motion all we get is a zero dimensional point. Thisway of looking at it suggests that space is (4+1)-dimensional because matter is (4+1)-dimensional. The three fundamental elements of physical reality, matter, space, and timeare interdependent. According to the SGM, their combined essence is gravity.

Moreover, their combined essence is also inertia. For centuries physicists have strug-gled to explain the origin of inertia, inertia being that property of matter manifestingitself as a resistance to acceleration. Note that this definition means a resistance to lin-ear acceleration. According to the SGM resistance to linear acceleration is due to that

property of matter manifesting itself as omnidirectional acceleration; i.e., gravity. In thesimplest terms, the more space a body is generating in every direction, the more difficultit is to move in any particular direction. When gravity is thought of as a force of attrac-tion, inertia remains an enigma; when it is thought of as the process of the generation ofspace we can begin to understand why it must be so. Well have more to say about thedimensionality of space after reasoning out in more detail the SGM prediction for theinterior falling experiment.

6.4. Rotation Analogy. According to the inverse square law the acceleration due to

gravity outside of a body varies as the inverse square of the distance. Lets say the bodyis a uniformly dense sphere. Then, moving outward from the center, since the amountof matter increases as the cube of the distance, acceleration inside the body will increasedirectly as the distance. Everybody agrees (and empirical evidence supports the idea)that the acceleration calculated this way corresponds to what accelerometers would sayif they were placed at various locations inside the sphere.

Another situation in which acceleration varies directly as the distance is uniform ro-tation. Due to certain similarities between gravitational fields and uniformly rotatingbodies, in the early development of GR Einstein proposed an analogy between them toclarify the new idea (at the time) of expressing gravitation in terms of non-Euclideangeometry. Because of Einsteins preconception that matter is static, however, his con-clusions based on this analogy differ from ours. We start with the same facts, but nowwe are open to the possibility that matter is not static because accelerometer readingstell us that matter moves.

All three of the circumstances described above are depicted in Figure 8. Clocks areshown in the figure as well as accelerometers. Lets suppose that the clock readings

also correspond to their relative rates. In all three cases everybody agrees about theaccelerometer readings; everybody also agrees about the clocks on the rotating body andabout clocks in the gravity exterior picture. For example, the slowest clock in both ofthese circumstances is the one at the perimeter (surface) of each body. What is unknown(and where predictions differ) is how clock rates vary inside the gravitating body. If wewere to guess, based on the analogy with rotation, and with the assumption that motionsensing devices indicate an actual state of motion, then we would say that the rate of aclock should be a maximum at the spheres center, just as it is a maximum at the centerof the rotating body. To appreciably slow down the rate of a clock we would need tomove it at high velocity. The axis of a rotating body and the center of a large gravitatingbody are places where there is an obvious minimum of velocity so we expect clocks in


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Rotation GravityInterior

GravityExterior

Fig. 8. Stationary motion. Left: A uniformly rotating body causes accelerometer readings tovary directly as the distance (stationary inward acceleration); and causes clock rates to decreasewith distance from the axis (stationary tangential velocity). Middle: Inside a uniformly densesphere accelerometer readings also vary directly as the distance. By analogy with rotation, theSGM predicts that clock rates also decrease with distance, having a minimum at the surfaceand a maximum at the center. Right: Outside a gravitating body accelerometer readings varyas the inverse square of the distance (stationary outward acceleration); and clock rates are aminimum at the surface (stationary outward velocity).

both places to have maximum rates.Here we have a conflict with GR. GR assumes that the sphere is a static thing. The

clock rates found outside the sphere are used, in effect, to deduce a geometrical equationcorresponding to this assumption of staticness. This assumption then further leads toan equation for the interior of the body which predicts that the clock at the centerruns slower than all the others. Notice the disconnect from physical reality. If we ask,what makes the central clock tick slow?, relativists have no answer except geometry. Thegeometry of spacetime makes the clock tick slow. If its true, as Newton advised, thatNature affects not the pomp of superfluous causes, then we should be suspicious.

In the case of rotation the cause of the slow clocks is motion (velocity) just as motion

is the cause of the accelerometer readings (acceleration). Even though all parts of arotating body beyond its axis are moving, it maintains essentially the same appearanceover time. Therefore it has often been referred to as a system which, though moving, isstationary. [33-35] The acceleration is toward the axis and the velocity is perpendicularto this. So we can say it is undergoing stationary inward acceleration and stationarytangential velocity.

Upon contemplating these same facts, Einstein saw the analogy with a gravitatingbody (which he presumed to be static) as meaning that it was justifiable to also regardthe rotating body as static. [36-38] Could Einstein have had it backwards? From an-tiquity Einstein inherited the practice of giving priority to his visual impressions. Sohe presumed that a gravitating body is static, which presumption entails disbelieving


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accelerometer readings. Feeling justified to disbelieve accelerometer readings in the caseof gravity, he maintained that he could do so also in the case of rotation. If Einsteinhad it backwards, then the forwards interpretation of the facts is that our ancient visualimpressions are illusory, that accelerometers and clocks are telling us not that rotatingbodies are at rest, but that gravitating bodies move. Massive bodies are undergoing sta-tionary motion: stationary outward acceleration and stationary outward velocity. In bothcases it is not geometry, but matter that is responsible; matter and space in motion.Therefore we predict that the rate of the central clock is a maximum (as indicated inFigure 8).

A common account of the zero accelerometer reading at r = 0 appeals to the inversesquare law. The accelerometer is at the center of spherical symmetry; gravity is thesame from all directions so the effect cancels. Since the effect on an accelerometer iscanceled by symmetry why should the effect not also be canceled for the rate of a clock?

The symmetry argument would reasonably apply here too, so the clocks rate should beunaffected, i.e., it should be a maximum, not a minimum. If this reasoning is incorrectit can be proved to be so by experiment. Meanwhile, it is important to realize that GRdoes not offer any physical explanation for its prediction that the central clock shouldrun slow.

As mentioned in 5, the way clock rates vary inside matter corresponds to how test ob-jects fall inside matter. If it were true, in spite of our suspicion, that geometry explainsclock rates and makes things move, then a central clock with a local minimum rate (asper GR) would mean that, in our experiment the test object would have a maximumvelocity at the center. It would oscillate, as predicted by Newton and Einstein.

Whereas, if our suspicion that geometry is a superfluous cause is well founded, ifthe fact that motion causes slow clocks in the case of rotation means motion shouldalso be the cause in the case of gravity, then the rate of the central clock will be a localmaximum and the test object dropped from the surface will not pass the center. Nothingever forces it to move.(2) What moves are those points of space connected to the spherewhere motion sensing devices detect motion. These points move past the test object evermore slowly for points closer to the center.

Thus Newtons Rules of Reasoning have led us to the SGM prediction for the result ofthe interior solution gravity experimentwhether we think of it in terms of clock ratesor the behavior of falling test objects. If somebody would only just do this experiment,then we could tell whether physics is in need of an overhaul or if, in this case, NewtonsRules have led us astray.

6.5. MotionThrough Space vs. MotionOf Space. It is interesting to consider the

implications of using rotational motion in the course of building up our hierarchy of

dimensions. After the point moves and we have a (1+1)-dimensional line, the rest of theprogression might look something like Figure 9. On paper, this makes sense. But theprocedure is perhaps even more abstract than using only linear motion, because alreadyin going from a line to a circle, we have introduced acceleration. (Rotational motionis accelerated motion.) Therefore we ask, if the line begins to rotate, what keeps it

(2) Note that something was forcing the object to move before it was released. It was beingforced to move upward by gravity. So when it is released, the falling object actually possesses aninitial outward velocity. Therefore, though the acceleration upon release will immediately dropto zero, the outward velocity diminishes gradually and asymptotically approaches zero near thecenter.


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Fig. 9. Hierarchy of dimensions generated by rotational motion.

intact? Why dont its constituent points instead just fly off on tangents? The samequestions clearly apply also to the subsequent steps.

The implication is that to bring this operation closer to physical reality to preventdisintegration, we need something to provide coherence. We need forces that are charac-teristic of molecular matter. To produce a system that is capable of cohering with itselfwe need a kind of acceleration that would counteract the otherwise disintegrating accel-eration of rotation. Evidently this must be an acceleration operating from the inside out.This certainly appears to be the case for gravitation (which gives accelerometer readingsthat are always positive, never negative). If its true that gravity is the process wherebymatter generates space, then this must connect to the forces more directly responsiblefor the microscopic coherence of matter; i.e., nuclear forces and electromagnetism. Theprocesses must ultimately be continuous with each other so as to maintain proportions.We will find that the SGMs implications for nuclear and atomic matter are more clearlyperceived in the context of cosmology, as the story unfolds in 10. For now, it willsuffice to think of accelerometer readings and clock rates as giving us a measure of the

macroscopically stationary, moving continuum, matter+space.In every case where we find non-zero accelerometer readings, we find that matter is

responsible. Though superficially trivial, deeper exploration of this observation revealsthe more profound fact that we can identify three distinct kinds of acceleration:

1. Linear acceleration, as produced by a rocket. This kind of accelerationconsumes energy. It is intrinsically temporary, as it cannot b e maintainedwithout burning some kind of fuel. (Entropic acceleration.)

2. Rotational acceleration, as discussed above. This kind of acceleration,once initiated, does not consume energy. A rotating body left to itself in freespace will continue rotating forever (in principle). The resulting non-zeroaccelerometer readings depend on the coherence of matter, without which a

rotating body would fly apart.

3. Gravitational acceleration. This kind of acceleration, evidently, is the(anti-entropic) production of energy. It perpetually emanates from all massivebodies. On large scales the omnidirectional acceleration very clearly providesinside out coherence (planets and stars) where microscopic forces would not.

To these key characteristics we now point out an extremely important distinction asbetween gravitation and the first two kinds of acceleration. The first two are clearlyconceivable as motion through space, through pre-existing, seemingly three-dimensionalspace. Thanks to the microscopic coherence of matter, rotation is similar to gravitation in


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that it exhibits a range of velocities and accelerations that nevertheless leave the systemintact (stationary). Gravitation is distinct from the other two types of acceleration,however, because it is not motion through pre-existing space; it is the motion of space.It is the process whereby the background for linear motion and rotation is created bythe active extension of matter, from three to four spatial dimensions. Generation ofspace is evidently synonymous with production of energy. Matter is the inexhaustiblesource. Gravitational acceleration is the stationary motion of space. Without gravity noother kind of acceleration would exist, for this is the essence of both space and matter.There would be no space to accelerate through (or around) were it not generated in thefirst place by gravity.

With regard to accelerometer readings, all three kinds of acceleration can contributeto the net total. And it is often important to disentangle them. But in no case should anon-zero reading be regarded as a state of rest. Nothing is static.

The distinction between motion through space and motion of space will be anotherrecurring theme in what follows.

6.6. Visualization by Analogy: Lower Dimensional Creatures. Wrapping ones mind

around the SGM is not likely to happen in a flash. Though simple in principle, it requireslots of unlearning at a primal level. Having pointed out the distinction between motionthrough space and motion of space, and the importance of time and matter in our newconception of space dimensionality, another analogy should help to bring these thingsinto better focus.

As noted in 6.4, it was Einstein who first introduced the rotation analogy. Recon-sidering this analogy while having a broader range of possibilities in mind, we come upwith a radically different interpretationone that is equally consistent with the facts andis in in better accord with Newtons Rules of Reasoning. So too, the following analogyhas often been introduced by others, but with a broader field of view, well find newsignificance and logical coherence.

The idea is to imagine sentient beings who inhabit a (2+1)-dimensional world. Letscall them Twodees. Having no perception of any extension into the third spatial dimen-sion, Twodees are flat and see only the linear edges of things in their world, which isnothing but surface. We might sympathize with practical-minded Twodees who claimthat it is meaningless or impossible to conceive of a third spatial dimension becauseall points of their world are perfectly locatable with a two-coordinate map. Twodeeswith livelier imaginations may nevertheless come to conceive the existence of a higherdimension by at least two different lines of thought. Well first imagine ourselves in theTwodees shoes, as they grapple with the idea of a third dimension. And then well extendthe problem up to the next higher dimension.

The first way that Twodees could conceive of another space dimension involves sup-posing that their world can be intersected or penetrated by (3+1)-dimensional objects.Suppose the penetrating object is a three-dimensional sphere. Then, as it goes throughtheir plane, the Twodees see a succession of linear cross-sections of changing size. Ifthe sphere travels all the way through their plane, they would see it appear suddenlyas a point, as a growing and then shrinking circle, then disappear. Since we who areimagining this scenario are at least (3+1)-dimensional beings ourselves, this is easy forus to conceive because we are accustomed to visualizing in three spatial dimensions.The intersection of a volume and a plane causes no conceptual difficulties. If the sce-nario described above were to happen, it may convince the practical-minded Twodee ofthe existence of a third dimension. But perhaps not; he might dream up some other


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explanation.The second clue from which the Twodees could deduce the existence of a higherdimension is especially applicable in the case that their world is not perfectly flat buthas some curvature. Suppose, for example, that their world is the surface of a spherewhose curvature could easily have escaped their notice because it is very large. Thegeometry on the sphere would locally appear to obey Euclidean laws of a plane. TheseEuclidean laws would become increasingly violated as the Twodees surveyed ever largerportions of the world. On a Euclidean plane the sum of the interior angles of a triangleequal 180. But the sums of angles of large spherical triangles exceed 180, the more soas the length of a side approaches the length of a circumference. Also, if a straight lineis pursued far enough, the surveyor will come back to her starting point. It is importantto note that, regardless of the sphericity or of whatever other curvature that we couldsee from our perspective, this will not prevent the Twodees from drawing a perfectly

functional map of their world using only two coordinates; i.e., two dimensionseven asthe system thus drawn out does not conform to planar Euclidean geometry.

One school of thought among the Twoodees is that the non-Euclidean propertiesof their world indicate that their world is intrinsically curved. This means that onlytwo coordinates are needed to make a complete map of it. Even though the geometryof the map is not that of a Euclidean plane, this does not prove the existence of ahigher dimension that their world curves into. The way a coordinate grid lies on thesurface would account for the failure of Euclidean geometry, and this would compelthe creatures to acknowledge the curvature of their world. But a mathematician wouldargue that postulating the existence of another dimension to accommodate the curvatureadds nothing to the Twodees actual knowledge of their surface. The alternative schoolof thought, however, emphasizes the possible existence of one more spatial dimension

as a way ofexplaining

the curvature of their world, even though it cannot be directlyvisualized. The Twodees could perhaps make more sense of the curvature if it wereconceived as turning into a new spatial direction (dimension). This latter approach ishow we humans would typically view the sphere: as a (3+1)-dimensional object whosesurface is extrinsically curved.

The relationship between an intrinsically curved space of lower dimension and an ex-trinsically curved space of higher dimension is often described in terms of an embeddingspace. In this case the two-dimensional spherical surface is embedded in our space ofthree spatial dimensions. It is enlightening to compare this imaginary scenario with theapproach to the spacetime curvature arising in GR, as expounded in a modern text-book. Hobson, Efstathiou and Lasenby first clarify the distinction between intrinsic andextrinsic much as we have:

It is important to make a distinction . . . between the extrinsic propertiesof the surface, whch are dependent on how it is embedded into a higher-dimensional space, and properties that are intrinsic to the surface itself. . . .Properties of the geometry that are accessible to the [lower-dimensional crea-ture] are called intrinsic, whereas those that depend on the viewpoint of ahigher-dimensional creature (who is able to see how the surface is shaped inthe three-dimensional space) are called extrinsic. [39]

Then, just like our practical-minded Twodees who are satisfied with the intrinsic descrip-tion, Hobson, et al advise their readers to not trouble themselves with the possibility ofa highe