Hopefully this pdf will attach so you can check out some of the details of Dr. Hartnett's theory.

My copying the paper into NOSTR doesn't do a good job. Maybe I can share with you on Minds.

"Finite bounded expanding white hole universe

without dark matter

John G. Hartnett

School of Physics, the University of Western Australia,

35 Stirling Hwy, Crawley 6009 WA Australia

john@physics.uwa.edu.au

October 17, 2018

Abstract

The solution of Einstein’s field equations in Cosmological General Relativity (CGR), where the Galaxy is at the center of a finite yet bounded

spherically symmetrical isotropic gravitational field, is identical with the

unbounded solution. This leads to the conclusion that the Universe may

be viewed as a finite expanding white hole. The fact that CGR has been

successful in describing the distance modulus verses redshift data of the

high-redshift type Ia supernovae means that the data cannot distinguish

between unbounded models and those with finite bounded radii of at least

cτ . Also it is shown that the Universe is spatially flat at the current epoch

and has been at all past epochs where it was matter dominated.

Keywords: Cosmological General Relativity, high redshift type Ia supernovae, dark matter

1 Introduction

In an interview with Scientific American George Ellis once said [9]

“People need to be aware that there is a range of models that could explain

the observations, . . . For instance, I can construct you a spherically symmetrical

universe with Earth at its center, and you cannot disprove it based on observations. . . . You can only exclude it on philosophical grounds. In my view there

is absolutely nothing wrong in that. What I want to bring into the open is the

fact that we are using philosophical criteria in choosing our models. A lot of

cosmology tries to hide that.”

This paper proposes a model where the Galaxy is at the center of a spherically symmetrical finite bounded universe. It contends that fits to the magnituderedshift data of the high-z type Ia supernovae (SNe Ia) [12, 13, 1], are also consistent with this model. That is, providing that the radius of the Universe (a

spherically symmetrical matter distribution) is at least cτ where c is the speed

1

of light and τ ≈ 4.28 × 1017 s (or 13.54 Gyr).[8] Here τ is the Hubble-Carmeli

time constant, or the inverse of the Hubble constant evaluated in the limits of

zero gravity and zero distance.

This model is based on the Cosmological General Relativity (CGR) theory

[5] but explores the motion of particles in a central potential. In this case

the central potential is the result of the Galaxy being situated at the center

of a spherically symmetrical isotropic distribution comprising all matter in the

Universe.

This paper is preceded by Hartnett [10] that forms the basis of the work

presented here. Also Oliveira and Hartnett [8] progressed the work by developing

a density function for higher redshifts. Those paper assumed the unbounded

model. The reader should be familiar with Hartnett [10] at least before reading

this.

1.1 Cosmological General Relativity

The metric [2, 3, 5] used by Carmeli (in CGR) in a generally covariant theory

extends the number of dimensions of the Universe by the addition of a new

dimension – the radial velocity of the galaxies in the Hubble flow. The Hubble

law is assumed as a fundamental axiom for the Universe and the galaxies are

distributed accordingly. The underlying mechanism is that the substance of

which space is built, the vacuum, is uniformly expanding in all directions and

galaxies, as tracers, are fixed to space and therefore the redshifts of distant first

ranked galaxies quantify the speed of the expansion.

In determining the large scale structure of the Universe the usual time dimension is neglected (dt = 0) as observations are taken over such a short time

period compared to the motion of the galaxies in the expansion. It is like

taking a still snap shot of the Universe and therefore only four co-ordinates

x

µ = (x

1

, x2

, x3

, x4

) = (r, θ, φ, τv) are used – three of space and one of velocity.

The parameter τ, the Hubble-Carmeli constant, is a universal constant for all

observers.

Here the CGR theory is considered using a Riemannian four-dimensional

presentation of gravitation in which the coordinates are those of Hubble, i.e.

distance and velocity. This results in a phase space equation where the observables are redshift and distance. The latter may be determined from the

high-redshift type Ia supernova observations.

1.2 Phase space equation

The line element in CGR [6]

ds2 = τ

2

dv2 − e

ξ

dr2 − R

2

(dθ2 + sin2

θdφ2

), (1)

represents a spherically symmetrical isotropic universe, that is not necessarily

homogeneous.

It is fundamental to the theory that ds = 0. In the case of Cosmological

Special Relativity (see chap.2 of [5]), which is very useful pedagogically, we can

2

write the line element as

ds2 = τ

2

dv2 − dr2

, (2)

ignoring θ and φ co-ordinates for the moment. By equating ds = 0 it follows

from (2) that τ dv = dr assuming the positive sign for an expanding universe.

This is then the Hubble law in the small v limit. Hence, in general, this theory

requires that ds = 0.

Using spherical coordinates (r, θ, φ) and the isotropy condition dθ = dφ = 0

in (1) then dr represents the radial co-ordinate distance to the source and it

follows from (1) that

τ

2

dv2 − e

ξ

dr2 = 0, (3)

where ξ is a function of v and r alone. This results in

dr

dv = τe−ξ/2

, (4)

where the positive sign has been chosen for an expanding universe.

2 Solution in central potential

Carmeli found a solution to his field equations, modified from Einstein’s, (see

[10] and [2, 5, 6]) which is of the form

e

ξ =

R′2

1 + f(r)

(5)

with R′ = 1, which must be positive. From the field equations and (5) we get a

differential equation

f

′ +

f

r

= −κτ 2

ρef f r, (6)

where f(r) is function of r and satisfies the condition f(r)+ 1 > 0. The prime is

the derivative with respect to r. Here κ = 8πG/c2

τ

2 and ρef f = ρ − ρc where ρ

is the averaged matter density of the Universe and ρc = 3/8πGτ 2

is the critical

density.

The solution of (6), f(r), is the sum of the solution (2GM/c2

r) to the homogeneous equation and a particular solution (- κ

3

τ

2ρef f r

2

) to the inhomogeneous

equation. In [5] Carmeli discarded the homogeneous solution saying it was not

relevant to the Universe, but the solution of a particle at the origin of coordinates, or in other words, in a central potential.

Now suppose we model the Universe as a ball of dust of radius ∆ with us, the

observer, at the center of that ball. In this case the gravitational potential written in spherical coordinates that satisfies Poisson’s equation in the Newtonian

approximation is

Φ(r) = −

GM

r

(7)

3

for the vacuum solution, but inside an isotropic matter distribution

Φ(r) = −G

4πρ

r

Z r

0

r

′2

dr′ + 4πρ Z ∆

r

r

′

dr′

!

=

2

3

Gπρr2 − 2Gπρ∆2

, (8)

where it is assumed the matter density ρ is uniform throughout the Universe.

At the origin (r = 0) Φ(0) = −2Gπρm∆2

, where ρ = ρm the matter density at

the present epoch. In general ρ depends on epoch. Because we are considering

no time development ρ is only a function of redshift z and ρm can be considered

constant.

From (8) it is clear to see that by considering a finite distribution of matter

of radial extent ∆, it has the effect of adding a constant to f(r) that is consistent

with the constant 2Gπρ∆2

in (8), where f(r) is now identified with −4Φ/c2

.

Equation (5) is essentially Carmeli’s equation A.19, the solution to his equation A.17 from p.122 of [5]. More generally (5) can be written as

e

ξ =

R′2

1 + f(r) − K

, (9)

where K is a constant. This is the most general form of the solution of Carmeli’s

equation A.17. So by substituting (9) into Carmeli’s A.18, A.21 becomes instead

1

RR′

(2R˙ R˙′ − f

′

) + 1

R2

(R˙ 2 − f + K) = κτ 2

ρef f . (10)

Therefore (9) is also a valid solution of the Einstein field equations (A.12 -

A.18 [5]) in this model. Making the assignment R = r in (10) yields a more

general version of (6), that is,

f

′ +

f − K

r

= −κτ 2

ρef f r. (11)

The solution of (11) is then

f(r) = −

1

3

κτ 2

ρef f r

2 + K. (12)

From a comparison with (8) it would seem that the constant K takes the form

K = 8πGρef f (0)∆2/c2

. It is independent of r and describes a non-zero gravitational potential of a finite universe measured at the origin of coordinates. There

is some ambiguity however as to which density to use in Carmelian cosmology

since it is not the same as Newtonian theory. Here ρef f is used and evaluated

at r = 0.

In the above Carmelian theory it initially assumed that the Universe has

expanded over time and at any given epoch it has an averaged density ρ, and

hence ρef f . The solution of the field equations has been sought on this basis.

4

However because the Carmeli metric is solved in an instant of time (on a cosmological scale) any time dependence is neglected. In fact, the general time

dependent solution has not yet been found. But since we observe the expanding

Universe with the coordinates of Hubble at each epoch (or redshift z) we see the

Universe with a different density ρ(z) and an effective density ρef f (z). Carmeli

arrived at his solution with the constant density assumption. I have made the

implicit assumption that the solution is also valid if we allow the density to vary

as a function of redshift, as is expected with expansion.

Now it follows from (4), (9) and (12) that

dr

dv = τ

s

1 + 

1 − Ω

c

2τ

2



r

2, (13)

where Ω = ρ/ρc. This compares with the solution when the central potential is

neglected (i.e. ∆ → 0). In fact, the result is identical as we would expect in a

universe where the Hubble law is universally true.

Therefore (13) may be integrated exactly and yields the same result as

Carmeli,

r

cτ

=

sinh( v

c

√

1 − Ω)

√

1 − Ω

. (14)

Since observations in the distant cosmos are always in terms of redshift, z,

we write (14) as a function of redshift where r is expressed in units of cτ and

v/c = ((1 + z)

2 − 1)/((1 + z)

2 + 1) from the relativistic Doppler formula. The

latter is appropriate since this is a velocity dimension.

What is important to note though is that regardless of the geometry of the

Universe, provided it is spherically symmetrical and isotropic on the large scale,

(14) is identical to that we would get where the Universe has a unique center,

with one difference which is explored in the following section. For an isotropic

universe without a unique center, one can have an arbitrary number of centers.

However if we are currently in a universe where the Galaxy is at the center of

the local isotropy distribution this means the Universe we see must be very large

and we are currently limited from seeing into an adjacent region with a different

isotropy center.

3 Gravitational Redshift

In Hartnett [10] the geometry in the model is the usual unbounded type, as

found in an infinite universe, for example. In a finite bounded universe, an

additional effect may result from the photons being received from the distant

sources. The gravitational redshift (zgrav) resulting from the Galaxy sitting at

the unique center of a finite spherically symmetrical matter distribution must

be considered. In this case we need to consider the difference in gravitational

potential between the points of emission and reception of a photon. Now the

00th metric component, the time part of the 5D metric of coordinates x

k =

t, r, θ, φ, v (k = 0 − 4), is required but it has never been determined for the

5

cosmos in the Carmelian theory. In general relativity we would relate it by

g00 = 1 − 4Φ/c2 where −4Φ is the gravitational potential. The factor 4 and

minus sign arise from the Carmelian theory when (12) and (8) are compared.

So the question must be answered, “What is g00 metric component for the large

scale structure of the universe in CGR?”

First note from (5) and (6) the g11 metric component (considered in an

unbounded universe for the moment)

g11 = −



1 +

1 − Ω

c

2τ

2

r

2

−1

(15)

in CGR we can write a scale radius

R =

cτ

p

|1 − Ω|

. (16)

Hence we can define an energy density from the curvature

ΩK =

c

2

h

2R2

=

c

2

τ

2

R2

, (17)

which, when we use (16), becomes

ΩK = 1 − Ω. (18)

This quantifies the energy in the curved spacevelocity.

In the FRW theory the energy density of the cosmological constant is defined

ρΛ = Λ/8πG hence

ΩΛ =

Λ

3H2

0

. (19)

Even though the cosmological constant is not explicitly used in CGR, it follows

from the definition of the critical density that

ρc =

3

8πGτ 2

=

Λ

8πG, (20)

when the cosmological constant Λ is identified with 3/τ 2

. Therefore in CGR it

follows that

ΩΛ =

Λ

3h

2

= Λ 

τ

2

3



= 1. (21)

This means that in CGR the vacuum energy ρvac = Λ/8πG is encoded in the

metric via the critical density since ρef f = ρ−ρc principally defines the physics.

So ΩΛ = 1 identically and at all epochs of time. (The determination of ΩΛ in

[10] was flawed due to an incorrect definition.) Also we can relate ΩΛ to the

curvature density by

ΩK = ΩΛ − Ω, (22)

which becomes

Ωk = ΩΛ − Ωm, (23)

6

at the present epoch (z ≈ 0). Here Ω = Ωm(1 + z)

3 and hence ΩK → Ωk as

z → 0.

Finally we can write for the total energy density, the sum of the matter

density and the curvature density,

Ωt = Ω + ΩK = Ω + 1 − Ω = 1, (24)

which means the present epoch value is trivially

Ω0 = Ωm + Ωk = Ωm + 1 − Ωm = 1. (25)

This means that the 3D spatial part of the Universe is always flat as it expands.

This explains why we live in a universe that we observe to be identically geometrically spatially flat. The curvature is due to the velocity dimension. Only

at some past epoch, in a radiation dominated universe, with radiation energy

density ΩR(1 + z)

4

, would the total mass/energy density depart from unity.

Now considering a finite bounded universe, from (12), using Ω = ρ/ρc, I

therefore write g00 as

g00(r) = 1 + (1 − Ωt)r

2 + 3(Ωt − 1)∆2

, (26)

where r and ∆ are expressed in units of cτ. Equation (26) follows from g00 =

1 − 4Φ/c2 where Φ is taken from the gravitational potential but with effective

density, which in turn involves the total energy density because we are now

considering spacetime.

Clearly from (24) it follows that g00(r) = 1 regardless of epoch. Thus from

the usual relativistic expression

1 + zgrav =

s

g00(0)

g00(r)

= 1, (27)

and the gravitational redshift is zero regardless of epoch. As expected if the

emission and reception of a photon both occur in flat space then we’d expect

no gravitational effects.

In an unbounded universe, though no gravitational effects need be considered, the result g00 = 1 is also the same. Therefore we can write down the

full 5D line element for CGR in any dynamic spherically symmetrical isotropic

universe,

ds2 = c

2

dt2 −



1 +

1 − Ω

c

2τ

2

r

2

−1

dr2 + τ

2

dv2

. (28)

The θ and φ coordinates do not appear due to the isotropy condition dθ = dφ =

0. Due to the Hubble law the 2nd and 3rd terms sum to zero leaving dt = ds/c,

the proper time. Clocks, co-moving with the galaxies in the Hubble expansion,

would measure the same proper time.

Since it follows from (26) that g00(r) = 1 regardless of epoch, g00(r) is not

sensitive to any value of ∆. This means the above analysis is true regardless

7

of whether the universe is bounded or unbounded. The observations cannot

distinguish. In an unbounded or bounded universe of any type no gravitational

redshift (due to cosmological causes) in light from distant source galaxies would

be observed.

However inside the Galaxy we expect the matter density to be much higher

than critical, ie Ωgalaxy ≫ 1 and the total mass/energy density can be written

Ω0|galaxy = Ωgalaxy + Ωk ≈ Ωgalaxy, (29)

because Ωk ≈ 1, since it is cosmologically determined. Therefore this explains

why the galaxy matter density only is appropriate when considering the Poisson

equation for galaxies.[11]

As a result inside a galaxy we can write

g00(r) = 1 + ΩK

r

2

c

2τ

2

+ Ωgalaxy

r

2

c

2τ

2

, (30)

in terms of densities at some past epoch. Depending on the mass density of the

galaxy, or cluster of galaxies, the value of g00 here changes. As we approach

larger and larger structures it mass density approaches that of the Universe as

a whole and g00 → 1 as we approach the largest scales of the Universe. Galaxies

in the cosmos then act only as local perturbations but have no effect on ΩK.

That depends only on the average mass density of the whole Universe, which

depends on epoch (z).

Equation (30) is in essence the same expression used on page 173 of Carmeli

[5] in his gravitational redshift formula rewritten here as

λ2

λ1

=

s

1 + ΩKr

2

2

/c2τ

2 − RS/r2

1 + ΩKr

2

1

/c2τ

2 − RS/r1

. (31)

involving a cosmological contribution (ΩKr

2/c2

τ

2

) and RS = 2GM/c2

, a local

contribution where the mass M is that of a compact object. The curvature

(ΩK) results from the averaged mass/energy density of the whole cosmos, which

determines how the galaxies ‘move’ but motions of particles within galaxies is

dominated by the mass of the galaxy and the masses of the compact objects

within. Therefore when considering the gravitational redshifts due to compact

objects we can neglect any cosmological effects, only the usual Schwarzschild

radius of the object need be considered. The cosmological contributions in (31)

are generally negligible. This then leads back to the realm of general relativity.

4 White Hole

Now if we assume the radial extent of a finite matter distribution at the current

epoch is equal to the current epoch scale radius, we can write

∆ = 1

√

Ωk

=

1

p

|1 − Ωm|

, (32)

8

expressed in units of cτ. In such a case, ∆ = 1.02 cτ if Ωm = 0.04 and

∆ = 1.01 cτ if Ωm = 0.02.

It is important to note also that in Carmeli’s unbounded model (14) describes

the redshift distance relationship but there is no central potential. In Hartnett

[10] and in Oliveira and Hartnett [8] equation (14) was curve fitted to the SNe

Ia data and was found to agree with Ωm = 0.02 − 0.04 without the inclusion of

dark matter or dark energy. Therefore the same conclusion must also apply to

the finite bounded model suggested here.

In order to achieve a fit to the data, using either the finite bounded or

unbounded models, the white hole solution of (6) or (11) must be chosen. The

sign of the terms in (12) means that the potential implicit in (12) is a potential

hill, not a potential well. Therefore the solution describes an expanding white

hole with the observer at the origin of the coordinates, the unique center of the

Universe. Only philosophically can this solution be rejected. Using the Carmeli

theory, the observational data cannot distinguish between finite bounded models

(∞ > ∆ ≥ cτ) and finite (∆ = 0) or infinite (∆ = ∞) unbounded models .

The physical meaning is that the solution, developed in this paper, represents

an expanding white hole centered on the Galaxy. The galaxies in the Universe

are spherically symmetrically distributed around the Galaxy. The observed

redshifts are the result of cosmological expansion alone.

Moreover if we assume ∆ ≈ cτ and Ωm = 0.04 then it can be shown [8] that

the Schwarzschild radius for the finite Universe

Rs ≈ Ωm∆ = 0.04 cτ. (33)

Therefore for a finite universe with ∆ ≈ cτ it follows that Rs ≈ 0.04 cτ ≈

200 M pc. Therefore an expanding finite bounded universe can be considered to

be a white hole. As it expands the matter enclosed within the Schwarzschild

radius gets less and less. The gravitational radius of that matter therefore

shrinks towards the Earth at the center.

This is similar to the theoretical result obtained by Smoller and Temple [14]

who constructed a new cosmology from the FRW metric but with a shock wave

causing a time reversal white hole. In their model the total mass behind the

shock decreases as the shock wave expands, which is spherically symmetrically

centered on the Galaxy. Their paper states in part “...the entropy condition

implies that the shock wave must weaken to the point where it settles down

to an Oppenheimer Snyder interface, (bounding a finite total mass), that eventually emerges from the white hole event horizon of an ambient Schwarzschild

spacetime.”

This result then implies that the earth or at least the Galaxy is in fact close

to the physical center of the Universe. Smoller and Temple state [15] that “With

a shock wave present, the Copernican Principle is violated in the sense that the

earth then has a special position relative to the shock wave. But of course, in

these shock wave refinements of the FRW metric, there is a spacetime on the

other side of the shock wave, beyond the galaxies, and so the scale of uniformity

of the FRW metric, the scale on which the density of the galaxies is uniform, is

no longer the largest length scale”[emphasis added].

9

Their shock wave refinement of a critically expanding FRW metric leads to

a big bang universe of finite total mass. This model presented here also has a

finite total mass and is a spatially flat universe. It describes a finite bounded

white hole that started expanding at some time in the past.

5 Conclusion

Since the Carmeli theory has been successfully analyzed with distance modulus

data derived by the high-z type Ia supernova teams it must also be consistent

with a universe that places the Galaxy at the center of an spherically symmetrical isotropic expanding white hole of finite radius. The result describes particles

moving in both a central potential and an accelerating spherically expanding

universe without the need for the inclusion of dark matter. The data cannot be

used to exclude models with finite extensions ∆ ≥ cτ.

References

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ΩΛ and w from the first year data set”, Astron. Astrophys. (2005)

arXiv:astro-ph/0510447

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[4] M. Carmeli, “Is galaxy dark matter a property of spacetime?”, Int. J.

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[5] M. Carmeli, Cosmological Special Relativity (World Scientific, Singapore,

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Found. Phys. Lett. 19(6):519-535 (2006) arXiv: astro-ph/0603500

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10

[10] J.G. Hartnett, “The distance modulus determined from Carmeli’s cosmology fits the accelerating universe data of the high-redshift type Ia

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[11] Hartnett, J.G. “Spiral galaxy rotation curves determined from Carmelian

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[12] R.A. Knop, et al, “New constraints on ΩM , ΩΛ and w from an independent set of 11 high-redshift supernovae observed with the Hubble Space

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