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April, 2007 PROGRESS IN PHYSICS Volume 2

Cosmological Redshift Interpreted as Gravitational Redshift

Franklin Potter∗and Howard G. Preston†

∗Sciencegems.com, 8642 Marvale Dr., Huntington Beach, CA 92646, USA

†15 Vista del Sol, Laguna Beach, CA 92651, USA

E-mail: ∗drpotter@lycos.com and †hpres@cox.net

Distant redshifted SNe1a light sources from the Universe that are usually interpreted

as cosmological redshifts are shown to be universal gravitational redshifts seen by

all observers in the quantum celestial mechanics (QCM) approach to cosmology. The

increasingly negative QCM gravitational potential dictates a non-linear redshift with

distance and an apparent gravitational repulsion. No space expansion is necessary.

QCM is shown to pass the test of the ﬁve kinematical criteria for a viable approach

to cosmology as devised by Shapiro and Turner, so the role of QCM in understanding

the behavior of the Universe may be signiﬁcant.

1 Introduction

The observed redshift from distant sources can be interpreted

as (1) a velocity redshift called the Doppler Eﬀect, (2) a cos-

mological redshift in which space itself is expanding during

the transit time of the photons, and/or (3) a gravitational

redshift as introduced by the General Theory of Relativity

(GTR). High-zredshifts from distant SNe1a light sources

in galaxies are presently being interpreted as cosmological

redshifts, apparently providing observational evidence for

the expansion of the Universe.

A new theory, Quantum Celestial Mechanics(QCM), de-

veloped from GTR by H. G. Preston and F. Potter [1, 2],

accurately predicts the observed SNe1a redshifts from near

and distant galaxies. For the Universe, there exists in QCM

a previously unknown gravitational potential that is used to

derive all of the observed SNe1a redshifts. In addition, QCM

predicts no mass currents in any coordinate direction, i.e., no

galaxies moving away anywhere. These results eliminate the

need for a space expansion. The presently known average

baryonic density of the Universe is suﬃcient for QCM to

explain the critical matter/energy density of the Universe.

Observations of galaxies and distributions of galaxies are

beginning to suggest conﬂicts with the standard concept of

an expanding Universe and its interpretation of a high-z

redshift as a cosmological redshift. For example, galaxies

at z=2.5 are reported [3] to be extremely dense when using

the expanding Universe assumptions and standard galaxy

modeling. However, if the Universe is not expanding, the

linear scales of these galaxies would be much larger, elimi-

nating the high density conﬂict and revealing galaxies much

similar to galaxies seen locally.

Theoretical approaches are also beginning to inquire

about what we really know about cosmic expansion and

its acceleration. In an interesting paper, C. A. Shapiro and

M. S. Turner [4] relax the assumption of GTR but retain

the weaker assumption of an isotropic and homogeneous

space-time described by a metric theory of gravity. Using

the Robertson-Walker metric to describe the Universe and

accepting the dimming and redshifting of a gold set of SNe1a

data [5], they determine the cosmic acceleration kinematic-

ally and provide a list of ﬁve kinematical criteria that must

be met by any approach to cosmology.

In this paper, we compare the QCM predictions for the

state of the Universe to the ﬁve criteria provided by Shapiro

and Turner. Our new result is that QCM agrees with the

ﬁve criteria. Therefore, SNe1a redshifts can be interpreted

as universal gravitational redshifts instead of cosmological

redshifts. There is no need for space expansion.

2 Reviewing the QCM potential

In a series of papers [1, 2, 6] we derived and applied QCM

to the Solar System, to other solar system-like systems such

as the satellites of the Jovian planets and exoplanet systems,

to the Galaxy, to other galaxies in general, and to clusters of

galaxies [7]. In all these cases there is reasonable agreement

with the observational data, i.e., the predicted QCM states of

the gravitationally-bound systems were shown to be actual

states of the systems without the need for dark matter. Recall

that the QCM general wave equation derived from the gene-

ral relativistic Hamilton-Jacobi equation is approximated by

a Schr¨

odinger-like wave equation and that a QCM quantiza-

tion state is completely determined by the system’s total

baryonic mass Mand its total angular momentum HΣ.

These agreements with the data strongly suggest that

QCM applies universally and that all gravitationally-bound

systems should obey the quantization conditions dictated by

QCM. Therefore, not only should the large-scale gravitation-

ally bound systems like a solar system exhibit QCM behav-

ior, but even a torsion balance near an attractor mass should

have quantization states. And the largest gravitationally-

bound system of all, the Universe, should also be describable

by QCM. The QCM states of a torsion bar system will be

F.Potter, H. G. Preston. Cosmological Redshift Interpreted as Gravitational Redshift 31

Volume 2 PROGRESS IN PHYSICS April, 2007

discussed in a future paper. In this paper we concentrate on

the QCM Universe.

For gravitationally-bound smaller systems, we found that

the Schwarzschild metric approximation produced an eﬀfect-

ive gravitational potential for a particle of mass μin orbit

Veﬀ =−GM

r+l(l+ 1)H2c2

2r2,(1)

where Gis the gravitational constant, cis the speed of light

in vacuum, the characteristic length scale H=HΣ/Mc, the

angular momentum quantization number loriginates from

the θ-coordinate symmetry, and ris the r-coordinate dis-

tance from the origin in spherical coordinates. Therefore, in

QCM the total angular momentum squared is l(l+1)μ2H2c2

instead of the classical Newtonian expression. Consequently,

the quantization of angular momentum dictates which parti-

cular circular orbit expectation values <r> in QCM corres-

pond to equilibrium orbital radii, in contrast to Newtonian

gravitation for which all radii are equilibrium radii.

In the case of the Universe we used the GTR interior

metric approximation, which is directly related to the general

Robertson-Walker type of metric. Omitting small terms in

the r-coordinate equation, we derived a new Hubble rela-

tion that agrees with the SNe1a data. At the same time we

showed that our QCM approach produced the required aver-

age matter/energy density of about 2×10−11 J/m3, corres-

ponding to the critical density ρc=8×10−27 kg×m−3, with

only a 5% contribution from known baryonic matter, i.e.,

without needing dark energy.

The QCM eﬀective gravitational potential for all observ-

ers inside a static dust-ﬁlled, constant density universe with

no pressure is

Veﬀ ≈ − kr2c2

2 (1 −k r2)2+l(l+ 1)H2c2

2r2(1 −kr2),(2)

where k= 8 πGρc/3c2. Figure 1 shows this QCM gravita-

tional potential for an r-coordinate distance up to about 10

billion light-years.

If the total angular momentum of the Universe is zero or

nearly zero, H can be ignored and then the negative gradient

of the ﬁrst term in Veﬀ produces an average positive radial

acceleration

<¨r> =k c2r(1 + k r2)

(1 −kr2)3(3)

from which we derive a new Hubble relation

<˙r> =rc√k

1−kr2.(4)

For r-coordinate distances up to about one billion light-

years, when k r21, we recover the standard Hubble rela-

tion and have a Hubble constant h∼2×10−18 s−1, about

62 km per second per megaparsec, an acceptable value [8].

Without the kr2in the denominator, v/c →1at about 14.1

0 2 4 6 8 10

r-coordinate distance (billions of light-years)

-10

-8

-6

-4

-2

0

QCM gravitational potential

Fig. 1: QCM gravitational potential to 10 billion light-years.

billion light-years; otherwise, the maximum visible coordi-

nate distance r=8.74 billion light-years, with more of the

Universe beyond this distance.

Notice that the QCM eﬀective gravitational potential is

negative (when H can be ignored) but produces an apparent

repulsive gravitational radial acceleration! Each observer

anywhere in this Universe will determine that the incoming

photons are redshifted. Why? Because the photons originate

in a source that is in a more negative gravitational potential

where the clock rates are slower than the clock rates at the

observer. And this redshift increases non-linearly because the

potential becomes more negative more rapidly with increas-

ing distance away. There is no need for expansion of space

and its cosmological redshift to explain the SNe1a data.

There is no need for dark energy to explain the accelerated

expansion.

3 The kinematical criteria

Our QCM approach to cosmology and an understanding of

the behavior of the Universe must meet speciﬁc kinematical

criteria. By analyzing the gold set of SNe1a data, Shapiro

and Turner list these ﬁve kinematical criteria to be met by

any viable approach to a cosmology:

1. Very strong evidence that the Universe once accele-

rated and that this acceleration is likely to have been

relatively recent in cosmic history.

2. Strong evidence that the acceleration qwas higher in

the past and that the average dq/dz is positive, where

zis the redshift.

3. Weak evidence that the Universe once decelerated, but

this result may be a model-dependent feature.

4. Little or no evidence that the Universe is presently

accelerating, i.e., it is diﬃcult to constrain qfor z < 0.1

with SNe1a data.

5. No particular models of the acceleration history pro-

vide more acceptable ﬁts to the SNe1a data than any

32 F.Potter, H. G. Preston. Cosmological Redshift Interpreted as Gravitational Redshift

April, 2007 PROGRESS IN PHYSICS Volume 2

others, i.e., several diﬀerent kinematic models ﬁt the

data as well as the cold dark matter hypotheses called

ΛCDM and

w

CDM.

The QCM eﬀective gravitational potential Veﬀ and the

new Hubble relation provide QCM explanations for these

ﬁve criteria:

1. The light now just reaching us from farther and farther

away exhibits an increasing redshift because the Veﬀ is

increasingly more and more negative with increasing

distance. Without QCM, the interpretation would be

that the acceleration is recent.

2. The Veﬀ is increasingly more and more negative with

increasing distance. Without QCM, a higher accelera-

tion in the past is required for the space expansion

approach to cosmology.

3. QCM shows no deceleration at the level of mathemat-

ical approximation we used.

4. The new Hubble relation of QCM reduces to the linear

dependence of the standard Hubble relation for zsmall,

agreeing with there being no acceleration presently.

5. Our QCM approach ﬁts the SNe1a data as well as

the other approaches, producing about a 12% increase

from the linear Hubble when k r2∼0.11, consistent

with the data.

QCM explains the ﬁve criteria in its unique way because

the SNe1a redshift now originates in the properties of the

static interior metric and its QCM gravitational potential.

The important consequence is that QCM cannot be elimi-

nated by any of the ﬁve criteria and must be considered as a

viable approach to cosmology.

4 Final comments

The existence of a repulsive gravitational potential in the

QCM wave equation for the Universe removes the necessity

for invoking dark matter and dark energy. According to

QCM, the Universe is not expanding and does not require

dark energy in order for us to understand its behavior. Pre-

viously labelled cosmological redshifts are actually gravita-

tional redshifts of the photons reaching us from distant

sources in the Universe that are in greater negative gravita-

tional potentials than the observer. Each and every observer

experiences this same behavior. This static Universe is

always in equilibrium.

Submitted on January 22, 2007

Accepted on January 24, 2007

References

1. Preston H. G. and Potter F. Exploring large-scale gravitational

quantization without ˉhin planetary systems, galaxies, and the

Universe. arXiv: gr-qc/0303112.

2. Potter F. and Preston H. G. Quantum Celestial Mechanics:

large-scale gravitational quantization states in galaxies and the

Universe. 1st Crisis in Cosmology Conference: CCC-I, Lerner

E.J. and Almeida J.B., eds., AIP CP822, 2006, 239–252.

3. Zirm A. W. et al. NICMOS imaging of DRGs in the HDF-S:

A relation between star-formation and size at z∼2.5. arXiv:

astro-ph/0611245.

4. Shapiro C. A. and Turner M.S. What do we really know about

cosmic acceleration? arXiv: astro-ph/0512586.

5. Riess A. G. et al. (Supernova Search Team). Observational

evidence from supernovae for an accelerating universe and a

cosmological constant. arXiv: astro-ph/9805201.

6. Potter F. and Preston H. G. Gravitational lensing by galaxy

quantization states. arXiv: gr-qc/0405025.

7. Potter F. and Preston H. G. Quantization states of baryonic

mass in clusters of galaxies. Progress in Physics, 2007, v. 1,

61–63.

8. Bonanos A. Z.et al. (DIRECT Project). The ﬁrst DIRECT

distance determination to a detached eclipsing binary in M33.

arXiv: astro-ph/0606279.

F.Potter, H. G. Preston. Cosmological Redshift Interpreted as Gravitational Redshift 33