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Does matter decelerate expansion of space?2
3
Robert Shour4
5
21st June, 20206
Toronto, Canada7
8
Abstract9
Astronomy infers existence of an unknown force denoted ‘dark en-10
ergy’ accelerating the expansion of space. Astronomy assumes matter11
decelerates expansion. Instead, apply by analogy the method of inverting12
conceptual reference frames and treat expanding space as the inertial ref-13
erence frame. Gravitation appears as an apparent force because matter14
coheres under an apparent but non-existent gravitational force — while15
space expands in a constant way.16
Keywords cosmological expansion, gravitation, reference frames, scaling17
1 Assumptions in conflict18
1.1 Dark energy19
Astronomy has inferred that type IA supernovas have the same luminosity re-20
gardless of distance from Earth. Their luminosities can be used to estimate21
distances from Earth by comparing luminosity of a supernova at unknown dis-22
tance to luminosity of one at a known distance.23
In 1998, astronomers reported measurements of luminosities of type IA su-24
pernovas (Riess et al., 1998; Schmidt et al., 1998; Perlmutter et al., 1998). Lu-25
minosities measured for distant type IA supernovas with known redshift were26
compared to luminosities of nearby supernovas at known distances. Luminosi-27
ties of high redshift supernovas were about 3/4 of what would be expected for28
a universe with no cosmological constant; high redshift supernovas were farther29
away than expected. Astronomy concluded that space itself was accelerating.30
This inferred acceleration was attributed to an energy of unknown character31
denoted dark energy (Turner and Huterer, 1998).32
1.2 Deceleration assumption33
This statement appears in the Introduction to a 1998 article that inferred space34
accelerates:35
While the universe is known to contain a significant amount of or-36
dinary matter, ΩM, which decelerates the expansion, its dynamics37
may also be significantly affected by more exotic forms of energy38
(Riess et al., 1998).39
The deceleration assumption 2
Physics assumes that for space gravitational deceleration opposes accelera-40
tion. Is that assumption valid? To test that assumption, invert the conceptual41
reference frame.42
Implicit in the idea of dark energy accelerating the expansion of space is43
that dark energy, whatever it is, is a kind of force and that gravitation and44
dark energy are two forces with effects opposing each other. What if, instead,45
there is only one effect, not two, and that effect expands space? Can the two46
forces that relate to gravitation and dark energy be reduced to a single effect?47
Wouldn’t that be an advantageous simplification?48
1.3 One object one location assumption49
An individual material object has one location, not two locations, at one and the50
same time. Colloquially, it is impossible to be at two places at the same time.51
Therefore, one might infer, ‘obviously’ the object’s distance from the observer52
must be the same regardless of distance units or means of measurement.53
Apply these observations to cosmology. An individual celestial object has54
one location, not two locations, at one and the same time. The same statements55
about distance that apply to material objects apply to celestial objects including56
those observed using astronomical means.57
While this unremarkable inference may be true in a universe in which there is58
only a single reference frame for physical space, it is not true in all circumstances59
if two distinct but related reference frames are characteristic of the universe.60
Measurement of the observer’s distance to the object must result in the same61
distance whenever conducted in the same reference frame. Different distance62
measurements to the same object imply an error in the means, recording or63
interpretation of the measurement, or in the assumption that different measure-64
ments occur in the same reference frame. I here acknowledge that this seems to65
be a challenging psychological leap, but it is not; there are several examples of66
natural phenomena that have been studied in which different physical reference67
frames are implicit. Some are mentioned in this article and a few are outlined68
towards the end of this article.69
Suppose two reference frames for physical space co-exist. Then distances70
from an observer to a material object might differ when measured in the dif-71
ferent co-existing reference frames. Similarly, measuring distance from Earth72
to a supernova by means of redshift and by luminosity and expecting equality73
assumes that these two different types of distance measurement occur in the74
same reference frame. If redshift and luminosity measure different distances to75
the same object, the assumption that the measurements take place in the same76
reference frame is false absent explanation.77
1.4 Accelerating space conflicts with one location assumption78
The inference that space accelerates is based on measurements of two distances79
to the same supernova: by redshift and by luminosity.80
The deceleration assumption 3
Two different astronomical distances by means of two different methods81
to the same supernova is inconsistent with the assumption of one object one82
location. Possible inferences are:83
A celestial object may be in two places at the same time.84
Reasoning that relies on measurement of a supernova at two different85
distances is flawed, erroneous, or in need of refinement.86
An assumption is invalid.87
Space expands in an accelerating way.88
In quantum theory, the two slit experiment raises the possibility that a89
material object can pass through two distinct openings at the same time. If90
the physics that applies at cosmological scales applies at quantum scales then91
the same physics may apply to the the idea that a material object being in two92
places at the same time, at all scales.93
1.5 Where does conflict lead?94
Astronomical distances occupy at least two different but co-existing ref-95
erence frames, one determined by redshift and the other determined by96
luminosity.97
Each of the redshift and luminosity distance reference frames can be con-98
sidered to be inertial relative to the other.99
There are two reference frames for determination of cosmological distances100
(Shour, 2019a).101
1.6 Is there really a conflict?102
Reasoning that measurement of two distances conflicts with the one object one103
location assumption itself may be flawed or erroneous. This raises the problem:104
can the conflict between two measured astronomical distances and the one object105
one location assumption be reconciled?106
1.7 Is gravitation slowing expansion a faulty assumption?107
If gravitation does not act to slow cosmological expansion, or if gravitation108
does not exist as a fundamental feature in the universe, then the problem of109
how space overcomes gravitation and expands is a problem without foundation.110
Space expands; matter left behind coheres but it is not part of the expansion.111
The position in this paper is that the inference that gravitation slows cosmo-112
logical expansion indirectly conflicts with the one object one location assump-113
tion.114
The deceleration assumption 4
2 Conceptual reference frames115
2.1 Role in physics116
Discussion of reference frames occupies much of this article. First, because117
consideration of the central role reference frames play in fashioning and changing118
theories that model the physical world. Second, because recognition of the119
significant role of reference frames in tackling physics problems that appear120
tough motivated me to consider looking at gravitation in an alternative reference121
frame.122
Historically important advances in physics sometimes depend not on acute123
application of differential equations or on advances in mathematical concepts124
and methods such as differential geometry or Riemannian geometry applied to125
model physical phenomena but instead on changing the conceptual reference126
frame.127
Changing a consensus in physics based on an assumed conceptual reference128
frame to a different or inverted conceptual reference frame presents immense129
challenges (Kuhn, 1970).130
Perhaps foremost among those challenges is that the problem solver is un-131
aware that methods and concepts are situated in a conceptual reference frame.132
It is difficult to change a conceptual reference frame when it is not apparent that133
the relevant physics resides in one. This is particularly so when perception and134
intuition are at odds with underlying physical reality, when what is apparent135
does not in fact correspond to fundamental principles.136
Copernicus alludes to the psychological challenges in the Dedication to Pope137
Paul III preface to his On the Revolutions of the Heavenly Spheres (Copernicus,138
1952) when he writes:139
The crazier my doctrine of the earth’s motion now appeared to most140
people . . .141
Revamping a conceptual reference frame to solve a problem often encounters142
dismissive characterization as ‘crazy’, as was also the case for Wegener’s Earth’s143
moving continents and for Semelweis on the anatomy room origin of sepsis in144
delivering babies. Psychological impediments to acceptance of a new paradigm145
in physics apply at the level of individual physicists, reluctant to be ridiculed146
and to discard acquired learning, and at the level of physics as a collective147
enterprise, resistant to discard what for so long has seemed obviously true and148
is interwoven into the fabric of physics knowledge, teaching and established texts149
(Kuhn, 1970).150
A conceptual reference frame opposite to some other conceptual reference151
frame does not necessarily require that one or the other is right. The advantage152
of one conceptual reference frame over the other may lie in its greater simplicity153
or generality. Greater simplicity or generality may help reveal principles that154
play a more fundamental role in accounting for physical phenomena.155
A reference frame that provides a superior vantage point from which to156
craft general physical principles often reveals what is inertial or invariant. The157
The deceleration assumption 5
historical examples below help illustrate the role of inertia and invariance in158
choosing conceptual reference frames.159
The purpose of describing historical examples is not only for historical con-160
text but also in general to reveal how inversion of conceptual reference frames161
can be deployed as a problem solving technique. In particular the historical162
examples are a guide to inversion of the assumption that matter decelerates ex-163
pansion of space. By analogy, the historical examples imply that gravitationally164
bound unmoving matter and expanding cosmological space are reference frames165
inverse to each other.166
2.2 Conceptual reference frames with inertia or invariance167
In many instances, what makes modeling a natural phenomenon difficult is168
that because of our local reference frame on Earth it is difficult to perceive, to169
notice and sometimes even to imagine that from a different point of view what170
we perceive as moving is inertial, or that what we perceive as inertial is, in a171
physics sense a constant. This is especially a challenge in connection with the172
cosmological constant discussed later below.173
2.3 Inertia, invariance and simplicity174
Finding a reference frame in which motion is inertial, or in which some ratio is175
invariant can lead to a simpler characterization of a physical phenomenon. The176
goal is to identify the conceptual reference frame for which inertia and invariance177
is fundamental, in order to find the simplest description of the world. A simpler178
description provides a simpler concept that is easier for a human problem solver179
to manipulate. Simplicity can remove barriers on other problem solving paths.180
3 Varying a reference frame181
3.1 Varying the setting for a problem182
A problem solving tactic is to vary the conceptual reference frame. Adopt-183
ing a different reference frame can simplify mathematical modeling and reveal184
underlying physical principles.185
In physics, some historically tough problems required for resolution not so-186
phisticated mathematics but rather a change in point of view, a varying of the187
conceptual reference frame.188
We may therefore suppose that in modern times a problem that tests and189
challenges physics may arise because the current default conceptual reference190
frame inhibits resolution. A superior conceptual reference frame is required.191
But before a better conceptual reference frame can be identified, it is necessary192
to perceive that current difficulties arise not in mathematics but in the currently193
adopted point of view.194
For a conceptual reference frame and an inverse of it, the data remains the195
same. What is inverted is the standpoint.196
The deceleration assumption 6
Sir Lawrence Bragg wrote:197
The fun in science lies not in discovering facts, but in discovering198
new ways of thinking about them (Bragg, 1959, p. 124).199
For solving a vexing problem in physics, thinking about it in a new way by200
varying or inverting the conceptual reference can be efficacious.201
3.2 Inverting a conceptual reference frame202
Inverting the conceptual frame is an extreme example of varying a conceptual203
reference frame because a conceptual reference is often preferred when it appears204
to conform to sense and perception. To adopt an inverted reference frame then205
is entirely an achievement of manipulating abstractions through the application206
of reason, difficult to do when the resulting conceptual reference frame appears207
to be contradicted by experience, and by experience not only of the individual208
researcher but by the historical and collective experience of society.209
3.3 Gedankenexperiment210
A gedankenexperiment as it was originally named in German, or thought ex-211
periment in English, involves manipulation of concepts, not material objects.212
Ernst Mach wrote213
. . . the thought experiment introduces the greatest transformation in214
our thinking and reveals the most significant paths of investigation215
(Price and Krimsky, 1973, p. 542).216
Inversion of a conceptual reference frame is a thought experiment. When ap-217
plied to cosmology, a thought experiment can effect a dramatic change because218
the inversion affects assumptions about our place in the universe.219
3.4 As a problem solving method220
George Polya was a master of heuristics in mathematics. He described various221
problem solving strategies for helping to solve mathematical problems, such as222
generalization, specialization, and simplification P´olya (1954, 1962).223
Some problems in mathematics yield to a solution more quickly when the224
conceptual reference frame is varied. Some number problems are more quickly225
solved using modulo arithmetic. Some geometry problems can be solved using226
the complex number plane. A distance problem set in a rectangular room might227
be more quickly solved by opening and flattening all sides of the room. Whether228
Euclid’s parallel or Fifth postulate can be derived from the other postulates229
was resolved by adopting a different conceptual reference frame: the parallel230
postulate is independent of Euclid’s other postulates.231
How far a fly travels flitting between the front wheels of two bicycles ap-232
proaching each other when they are a given distance apart is easier to solve233
The deceleration assumption 7
changing the conceptual reference frame. Instead of considering two bikes in234
motion with the fly traveling diminishing distances summed in a geometric se-235
ries, treat one bike as if it were stationary (Macrae, 1992, p. 10). The conceptual236
reference frame in which a problem sits can also affect ease of solution. Consider237
two ways of stating the problem:238
1. Two bicyclists are 20 kilometers apart approaching each other on a straight239
line. There is a fly on the front of the first one. It flies between from one240
wheel to the other back and forth until the two wheels meet crushing it.241
How far does the fly travel?242
2. A fly travels back and forth between the front of the wheels of two bikes243
2xdistance units apart (an embedded hint that varies the conceptual244
reference frame) until it is crushed. If each bike moves at xdistance units245
per hour in opposite directions towards each other, how far does the fly246
travel?247
Consider the 2xdistance units characterization of the fly problem. Two248
bikes 2xdistance units apart can be simplified to one bike stationary and only249
the other bike moving, towards it, at a total speed of 2xdistance units per250
hour. That characterization of the fly problem is analogous to the situation251
with gravitation and dark energy. Instead of motion being caused by each252
of gravitation and dark energy, what if instead motion is only attributable to253
expanding space?254
In physics, varying and inverting a conceptual reference is of particular conse-255
quence when connected to our perception of the world. It is, I think, a powerful256
problem solving technique in physics that has historically been consequential.257
Its utility has not been exhausted.258
3.5 Contemporaneous dual reference frames259
One approach in this article varies the method of varying or inverting a concep-260
tual reference frame.261
A variety of phenomena can be explained by supposing the contemporary262
existence of two co-existing conceptual reference frames. This varies inverting263
a reference frame as a problem solving method. Instead suppose that two ref-264
erence frames instead of one apply contemporaneously. The idea of two equally265
applicable reference frames is a paradigm shift, contrary to the either-or world266
that logic and sense impression implies we live in.267
The two reference frame solution is a psychological hurdle.268
4 Analogical reasoning269
The approach in this paper is by way of analogy to other cases where inversion270
gave a superior conceptual reference frame to tackle a problem in physics. In271
the existing paradigm gravitation is used to account for and characterize dark272
energy as a kind of anti-gravity. In the thought experiment in this article, invert273
The deceleration assumption 8
the conceptual reference frame. Think of gravity as the anti-expansion of the274
universe to account for gravitation.275
Analogical reasoning saves time, avoiding effort required to solve a problem276
using first principles. If an effective analogy is found, analogy can shorten time277
to problem resolution. (Bearing in mind, however, that all problem resolutions278
are provisional steps to improved but never completed understanding.)279
Approaches using analogy here applied include:280
Flip gravitation’s conceptual reference frame in relation to expanding281
space, by analogy to historical examples in physics where flipping a con-282
ceptual reference frame has been applied to advantage.283
Flip inferences in the original reference frame to the flipped conceptual284
reference frame, and consider the implications of inferences in the flipped285
reference frame.286
Infer that attributes of the existing theoretical framework apply in the287
context of the varied reference frame. For example Bekenstein infers at-288
tributes of black holes based on an analogy of black hole attributes to289
attributes of entropy(Bekenstein, 1972, 1973). This article infers char-290
acteristics the concept of of gravitation by analogy to the technique of291
inverting conceptual reference frames.292
5 Invert expansion assumption293
The inference that a mysterious dark energy exists is premised on acceptance294
that gravitation causes matter to cohere across space. Gravity is thus a default295
assumption. Therefore if space expands, given gravitation, it must be that some296
force or energy is causing space to expand.297
Invert this conceptual reference frame. Discard gravitation as a default as-298
sumption. Suppose that space expanding is the default assumption, and that299
matter stays behind, giving the impression that matter attracts matter.300
If space expands due to something like a cosmological constant, then space’s301
constant rate of expansion is in a sense inertial. Space expands because of some302
fundamental unchanging (invariant) physical relationship. Then, relative to303
expanding space, matter relative to other matter is unmoving, and only relative304
to expanding space does matter give the appearance of attracting other matter.305
6 Conceptual role of type IA supernovas: luminosity306
Astronomy infers that the luminosities of type 1A supernovas are the same.307
Uniformity of luminosity is due to type 1A supernovas all having the same308
underlying physics. The physics being the same implies the luminosity is the309
same. That being the case, the difference in luminosity between two type IA310
supernovas necessarily is due to them being at different distances from Earth.311
The deceleration assumption 9
If the distance to a benchmark type IA supernova is known, then the distance312
to a second type IA supernova can be calculated, by comparing the luminosity313
of the second to the benchmark.314
7 History: what is unchanging in a system?315
7.1 Geocentrism, heliocentrism, general relativity and inertia316
Consider Earth unmoving at the center of the universe to be a conceptual ref-317
erence frame inverse to the conceptual reference frame with the sun unmoving318
at the center of the solar system.319
The Earth can be used as reference frame for charting the movement of320
celestial objects as easily as, and in some cases more conveniently than, the321
sun. For example, to chart how closely an asteroid or comet approaches Earth,322
Earth is the appropriate reference frame. Using the sun as a reference frame323
has advantages for calculation of planetary orbits, for suggesting the origin of324
the solar system from swirling gas, and in revealing physical principles and re-325
lated mathematical models that apply generally to motions of planets and their326
moons in the solar system. A heliocentric model has advantages of generality327
in modeling the motions of solar system planets. Ptolemy’s system of epicycles328
moving on cycles are ad hoc adjustments to circular paths around Earth.329
Ptolemy was certain that330
the earth cannot have any motion in the aforementioned directions,331
or indeed ever move at all from its position at the center (Ptolemy,332
1984, p. 43).333
Ptolemy’s quoted statement omits the roles of reference frames (where the334
observer is located) and of conceptual reference frames (how perception is in-335
terpreted in light of available concepts), likely because the concept of a frame336
of reference did not then exist in relation to perceived immobile Earth. But337
he does remark ‘there is no up and down in the universe with respect to itself’338
(Ptolemy, 1984, p. 44).339
Earth itself, he argues, is indisputably motionless:340
If the earth had a single motion in common with other heavy341
objects, it is obvious that it would be carried down faster than all342
of them because of its much greater size . . . such things are utterly343
ridiculous (Ptolemy, 1984, p. 44).344
On the suggestion by some that Earth ‘revolves’ (using Ptolemy’s terminol-345
ogy in the quotation following) on its axis from west to east:346
one can see that such a notion is quite ridiculous. . . . the re-347
volving motion of the earth must be the most violent of all motions348
associated with it, seeing that it makes one revolution in such a short349
time; the result would be that all objects not actually standing on350
The deceleration assumption 10
the earth would appear to have the same motion, opposite to that of351
the earth: neither clouds nor other flying or thrown objects would352
ever he seen moving towards the east, since the earth’s motion to-353
wards the east would always out run and overtake them (Ptolemy,354
1984, p. 44).355
Invert Ptolemy’s conceptual reference frame of unmoving Earth at the center356
of the universe to the conceptual reference frame used by Aristarchus (Heath,357
1913), Copernicus (Copernicus, 1453), Kepler (Kepler, 1609), Galileo (Galilei,358
1967) and Newton (Newton, 1999).359
Seek the inertial reference frame. Ptolemy’s epicycles are inconsistent with360
the primacy of an inertial reference frame. A planet following a path and then361
moving in a retrograde direction relative to Earth contravenes the primacy of362
inertial reference frames. Ptolemy’s geocentric system must be false if inertia is363
a fundamental physical principle and a fundamental attribute in the universe.364
Modify slightly Ptolemy’s idea of unmoving Earth by having Earth follow365
an inertial path in the curved space described by Einstein’s general relativity.366
Put another way, if we wish to imagine that Earth and the other planets in the367
solar system are are all in inertial reference frames, then it must be that the368
space through which they move is curved. If finding what makes a reference369
frame inertial is accorded the highest priority, in order to detect fundamental370
physical principles, then it must be that space is curved.371
Ptolemy, Aristarchus, Copernicus, Kepler, Galileo, Newton and Einstein372
sought, proposed and applied an invariance that applied to Earth’s motion.373
Copernicus inverted Ptolemy’s conceptual reference frame, shifting the center374
of the universe from Earth to the sun. Advantages ensued, including Newton’s375
theory of universal gravitation ().376
7.2 Sundial shadow and Earth motion377
A sundial gives the approximate time according to where a gnomon’s shadow378
on it is cast. A gnomon is a wood, stone or metal beam attached to the plane379
of the sundial at a right angle to it.380
During daylight, the shadow cast by the gnomon, if the sundial is appropri-381
ately aligned with the apparent motion of the sun in the sky, moves through382
about 180 degrees, depending on the time of year.383
The sundial does not move relative to Earth. By analogical reasoning, as384
the sun’s shadow moves around the edge of the sundial face, so must also the385
sun move around Earth, to which the sundial is immovably fixed.386
The validity of the analogy depends on Earth being immobile. Perhaps387
the sundial influenced philosophers, consciously or subconsciously, to suppose388
that the sun naturally revolved around Earth every day. The sundial is at rest389
relative to Earth. And yet, as part of Earth, it moves.390
In the inverted conceptual reference frame, Earth and everything attached to391
it, including the sundial, revolves around the sun, while Earth spins on its axis392
one rotation per day. If it were obvious that was so, then the inverted reference393
The deceleration assumption 11
frame would not have eluded Ptolemy, geographers and millions of people for394
over 1,400 years. Imagining an inverted reference frame is immensely difficult,395
psychologically and conceptually. The difficulties are under appreciated. So are396
the conceptual and philosophical rewards.397
7.3 Dropped from the top of the mast398
In his Two Chief World Systems (Galilei, 1967) Galileo describes what Aristotle,399
Ptolemy, Tycho and others taught about what happens to a lead ball dropped400
from the top of a ship’s mast. In the words of Salviati, a participant in Galileo’s401
dialogue, they taught:402
Salv. . . . drop a lead ball from the top of the mast of a boat at403
rest, noting the place where it hits, which is close to the foot of the404
mast; but if the same ball is dropped from the same place when the405
boat is moving, it will strike at that distance from the foot of the406
mast which the boat will have run during the time of fall of the lead,407
and for no other reason than that the natural movement of the ball408
when set free is in a straight line toward the center of the earth.409
(Galilei, 1967, p. 126).410
The ancient authors implicitly but not explicitly assume a conceptual refer-411
ence frame in which Earth is motionless. Galileo points out this error:412
Salv. But wasn’t it concluded a little while ago that we could413
not have any knowledge of this fall being straight and perpendicular414
unless it was first known that the earth stood still? Therefore in415
your syllogism, the certainty of the middle term is drawn from the416
uncertainty of the conclusion. Thus you see how, and how badly, it417
is a paralogism. (Galilei, 1967, p. 140).418
If the lead ball moves with Earth’s rotation — follows Earth’s inertial motion419
— it is impossible to determine from where the lead ball hits the ship deck what420
the motion of Earth must be. Galileo continues:421
Therefore the rock at the top of the tower has as its primary422
tendency a revolution about the center of the whole in twenty- four423
hours, and it eternally exercises this natural propensity no matter424
where it is placed. Therefore the rock at the top of the tower has as425
its primary tendency a revolution about the center of the whole in426
twenty- four hours, and it eternally exercises this natural propensity427
no matter where it is placed. (Galilei, 1967, p. 142).428
If instead of assuming motionless Earth, one locates the experiment of the429
dropped lead in a conceptual reference frame that includes inertia, then one430
can only say that whether the ship is motionless in the water or moving at a431
constant speed relative to land, the lead dropped from the top of the mast will432
strike the ship deck in the same place.433
The deceleration assumption 12
For anyone who does will find that the experiment shows exactly434
the opposite of what is written; that is, it will show that the stone435
always falls in the same place on the ship, whether the ship is stand-436
ing still or moving with any speed you please. Therefore, the same437
cause holding good on the earth as on the ship, nothing can be in-438
ferred about the earth’s motion or rest from the stone falling always439
perpendicularly to the foot of the tower. (Galilei, 1967, p. 145).440
Two conceptual reference frames are in Galileo’s analysis are contemplated,441
stationary Earth and Earth following an inertial path. To solve the problem442
of where the lead hits the ship’s deck requires choosing a conceptual reference443
frame that respects inertia.444
7.4 Universal gravitation in the solar system445
Before Copernicus, because of universal gravitation things falling fell toward446
Earth’s center. the ‘fixed stars’ did not fall to Earth’s center. After Copernicus447
and Newton (Newton, 1999), the conceptual reference frame for gravitational448
attraction changed from Earth’s center to, for the planets, the sun’s center.449
7.5 Kleiber’s law and inverting 3/4450
Scientists in the early 19th century knew that the rate of breathing slowed451
for species of animals of larger size (Sarrus and Rameaux, 1838) but did not452
know why. Over time, the problem changed to: what is the exponent bfor453
metabolism Y∝Mbfor animal mass M? An 1838 rate of breathing paper454
implied b= 2/3 (Sarrus and Rameaux, 1838). Max Rubner’s measurements of455
animal metabolism (Rubner, 1883) suggested b= 2/3, but b= 2/3 is doubt-456
ful (Brody, 1945). Max Kleiber’s data suggested b= 3/4 (Kleiber, 1932), a457
hypothesis known known as Kleiber’s Law.458
Mathematical justification for b= 3/4 is assisted by inverting the conceptual459
reference frame from use to supply, which has the effect of inverting exponent460
3/4 to exponent 4/3.461
Instead of 3/4 scaling of animal energy use, find how animal circulatory sys-462
tem capacity supply scales with size. By metabolism slowing by a 3/4 power of463
energy supply, energy supply capacity and energy use requirements are restored464
to being in balance, necessary to maintain heat balance in the animal body.465
The 3/4 scaling is only indirectly a scaling of animal mass M. In fact, the466
3/4 scaling is a scaling of the way energy supply scales with size, E4/3. Since467
energy supply and use for a given animal are in balance, E4/3∝Mand hence468
[E4/3]3/4≡M3/4.469
The 3/4 power of the 4/3 scaled increase in energy supply capacity leads470
to an invariance: [E4/3]3/4=E1. This shows the way to another inversion of471
conceptual reference frames. Scaling up in consequence of an increase in size472
scales up a capacity. But the inverted conceptual reference frame reveals that473
scaling in a certain way occurs in order to preserve an invariance. In the case of474
The deceleration assumption 13
metabolic scaling, locating theory in energy supply capacity instead of energy475
use demand.476
Literally inverting 3/4 implicitly inverts the conceptual reference frame from477
energy use scaling — metabolism — to energy supply scaling. Accounting for the478
scaling of metabolism with animal size has been a problem since before Sarrus479
and Rameaux (Sarrus and Rameaux, 1838, 2017). Inverting the conceptual480
reference frame brings us closer, at least, to an explanation of metabolic scaling481
(Shour, 2019b).482
7.6 Kleiber’s law: cellular instead of animal mass reference frame483
Historically, metabolic scaling has adopted as an analytical conceptual refer-484
ence frame the entire volume and surface area of an animal, as did Sarrus and485
Rameaux (Sarrus and Rameaux, 2017) and Max Rubner (Rubner, 1883), or486
entire animal energy and volume (Kleiber, 1932), or the entire volume of the487
circulatory system compared to an animal’s entire mass (West et al., 1997).488
Instead of aiming for a theory of metabolic scaling from the standpoint of489
metabolism of an entire animal, invert the conceptual reference frame. Begin490
with an average animal cell. The constraint is maintenance of a constant intra-491
cellular temperature. For a cell, the ratio of energy Esupplied to cell mass M492
must be constant. Otherwise the cell would overheat and impair intracellular493
processes. This example illustrates that different aspects of a conceptual ref-494
erence frame can be inverted. For metabolism, the use reference frame can be495
inverted, and the size reference frame can be inverted.496
The dimension of energy distribution in the cell can be represented by497
[M][ ˙
E] = [V][ ˙
E]=[L]4(1)
for generic length L. The dimensions of energy use in the cell is498
[M]=[V] = [L]3.(2)
Since energy supply relative to energy use scales by exponent 4/3, cellular499
metabolism must grow by a 3/4 power of energy supply when the number of500
cells increases, which is to say, when an animal increases in size.501
In the cell, Brownian motion has a 4/3 fractal envelope Lawler et al. (2001).502
Fluid motion in a cell’s three dimensional volume has dimension 4, and cellular503
mass had dimension 3. If energy supply to a cell increases, cellular metabolism504
must change by a 3/4 power of cellular energy supply.505
Scaling up in size from cell to organism, an animal circulatory system has506
dimension 4, where the fourth dimension is blood flow, ˙
E. Since the 4 : 3 ratio507
pertains to an entire organism and to cells individually, 3/4 metabolic scaling508
should be observed over many orders of magnitude; according to one article, 27509
orders of magnitude (West et al., 2002).510
Random intracellular fluidic motion is harnessed by molecular mechanisms511
that can be modeled as Brownian ratchets (Simon et al., 1992).512
The deceleration assumption 14
7.7 What is common in human behavior?513
To account for violent behavior of an individual often involves a psychological514
profile. Instead, invert the conceptual reference frame. Consider what patho-515
logical doctrines might motivate individual behavior. Individual behavior in516
the inverted conceptual reference frame is a result of individuals influenced by517
collective or individual creation of an ideology.518
Varying a conceptual frame of reference is a problem solving tactic not lim-519
ited to physics.520
7.8 Is intelligence individual or collective?521
IQ tests measure individual intelligence. No consensus about what intelligence522
is exists (Neisser et al., 1996).523
The 1970s witnessed average IQ test scores increasing at the rate of about524
3.3% per decade (Flynn, 1987).525
To account for the increase, researchers considered the effects of television,526
improved nutrition, electronics and so on. The conceptual reference frame for527
increasing average IQs was the individual.528
More likely that average IQs increasing is due to an effect that pertains529
to a generation. Invert the individual reference frame to a collective reference530
frame. Ideas shared by societies are persistently appraised, refined, improved531
and sometimes discarded. Lexical change permits compression of information532
— by allusion, contraction short forms, metaphor, idiom and so on. Words533
and phrases more dense with information and allusion take the same amount534
of energy to cognitively manipulate as the same number of less information-535
dense words. Ideas compressed should increase the energy efficiency of problem536
solving. Data supports that inference (Shour, 2009).537
Since energy is required to create intelligence, and energy plays a prominent538
role in physics, IQs and physics are disciplines not so different as as they may539
seem to be.540
7.9 Four dimensional invariance541
Newton’s Principia treated space and time independently:542
Absolute, true and mathematical time, of itself, and from its own543
nature flows equably without regard to anything external, and by544
another name is called duration (Newton, 1845, p. 77).545
Invert the conceptual reference frame which has space and time indepen-546
dent. Unifying them gives space-time. A distance in space-time in an inertial547
reference frame is invariant when situated in a different inertial reference frame548
(Minkowski, 2012).549
The deceleration assumption 15
7.10 Gravitation, acceleration and inertia550
Absent any other applied force, a mass in Earth’s gravitational field much551
smaller than Earth falls towards Earth.552
Invert the conceptual reference frame in this scenario from Earth to the553
falling mass. In the reference frame of the mass, it is difficult to distinguish554
between acceleration due to the effect of gravity and that due to a force applied555
upwards (Einstein and Infeld, 1938, p. 227). This varying of the conceptual556
reference frame reveals connections between a mass experiencing gravitation557
and accelerating. Equivalence led Einstein to a general theory of relativity that558
included accelerating reference frames.559
7.11 Expanding space and inertia versus acceleration560
Physics inferred that the 1998 astronomical observations implied physical space561
is accelerating. The validity of this assumption depends on assuming only one562
reference frame that includes 3 dimensional space. Space stretching in an in-563
creasing way implies acceleration if all matter and motion occurs in one reference564
frame.565
If there are two co-existing cosmological reference frames, there is no accel-566
eration. Each iteration of distance in 4 dimensional space goes 4/3 as far in567
3 dimensional space. Space with light motion has dimension 4: [V][c] = [L]4
568
where Vis a volume of 3 dimensional space and cis the speed of photons.569
Space without light motion has dimension 3: [V]=[L]3. The same energy in 4570
dimensional [V][c] has 4/3 as much energy per dimension in 3 dimensional [V].571
Accordingly, there is no acceleration of space. Space expands in an iner-572
tial way, as there is a cosmic constant, namely an unchanging 4 : 3 ratio of573
dimensions. The 4/3 ratio of dimensions creates a dimensional pressure: when574
energy originating in 4 dimensions must be fitted into 3 dimensional space, the575
3 dimensional space expands in a 4:3 ratio compared to 4 dimensional space. If576
so, then we have found what is unchanging, the missing invariant — the 4/3577
ratio of dimensions — that accounts for space expanding in an unchanging way.578
Invariance and inertial must be sought to understand many physical systems.579
This (I believe) is one of those systems.580
7.12 Analogy between geocentrism and universal gravity581
A human observer on Earth is, relative to Earth, in an inertial reference frame.582
In the reference frame of the celestial objects millions of light years from Earth,583
so far away that they appear over the duration of a day fixed relative to Earth,584
Earth revolves around the sun.585
A human observer on Earth observing the effect of gravity is in the Earth586
reference frame. Is the human observer’s Earth reference frame biased in favor587
of a misleading perception?588
The deceleration assumption 16
7.13 Scaling and dimensional capacity589
In a scaling conceptual reference frame, as size changes parts of the whole may590
change differently, such as areas compared to volumes or weights. That is what591
Galileo observed in connection with the cross-sectional area of bones compared592
to animal weight, in his Two New Sciences (Galilei, 1638; Galileo, 1914). In a593
conceptual reference frame based on dimension, weight occupying 3 dimensions594
(volume) when increases requires that 2 dimensions (the cross-sectional area of595
weight-bearing bone) must increase in size by 3/2 as much as weight to be able596
to bear the increase weight.597
A dimensional capacity conceptual reference frame reveals what invariance598
the different scaling preserves.599
Dimensional capacity also accounts for expanding space.600
8 Is space expanding, not gravity, fundamental?601
8.1 Two reference frames602
There are at least two cosmological reference frames. One reference frame has603
expanding space as an inertial reference frame with space expanding in a con-604
stant way.605
The existence of two reference frames dispenses with some problems in cos-606
mology. Expansion of the universe consistently arises from the 4 : 3 ratio of607
dimensions. The hypothesis of an inflationary universe becomes irrelevant, as608
does accelerating space.609
8.2 Is gravity apparent only?610
Abundant astronomical evidence implies space is expanding. The conclusion611
that space is expanding is recent (Hubble, 1929) as is the evidence on which it612
is based, compared to evidence of gravitation.613
Space moves. Gravity has matter attracting matter. Which of these alter-614
native reference frames should be used to understand the other?615
Einstein modified Newton’s universal gravitation to be a perception of curved616
space. Einstein’s general theory of relativity saved appearances and preserved617
gravitation as a conceptual reference frame. Consider though that gravitation618
is an old concept. Much in mathematics, physics and astronomy has happened619
since Newton. Newton’s deserved reputation inhibits consideration of gravita-620
tion as being apparent only. With dark energy, so called, maybe now is the621
time.622
Consider that a path 4/3 as long in 3 dimensional Vwith the same endpoints623
as a corresponding path in 4 dimensional [V][c] must be curved relative to the624
straight line path in [V][c].625
Suppose space expands in a way that is constant, arising from a constant626
ratio of dimensions. Choose that as our conceptual reference frame. Then what627
is gravity relative to space expanding inertially? Is it not that which does not628
The deceleration assumption 17
move because there are no moving photons to move it: matter? If matter only629
appears to cohere because moving space leaves it behind, then gravitation may630
only be apparent. Einstein’s equations in general relativity are only necessary631
because transformation of space is made much more mathematically difficult if632
in fact there are two co-existing cosmological reference frames.633
9 Astronomical support for two reference frames634
Theoretical and astronomical observations consistent with there being two ref-635
erence frames one of dimension four and the other of dimension 3 include:636
With distance a, matter energy density ρm∝1/a3and radiation energy637
density ργ∝1/a4(Wang, 2010, p. 17). The 4 : 3 ratio of exponents in638
the denominators is consistent with the 4 : 3 ratio of dimensions.639
The ratio of energy densities for dark energy and matter found observa-640
tionally 0.705 : 0.295 is almost exactly the ratio of energy densities 43/33
641
=0.7033 : 0.2967 (Betoule et al., 2014).642
Type 1A supernovas have 3/4 the luminosity expected (Riess et al., 1998;643
Schmidt et al., 1998; Perlmutter et al., 1998), consistent with luminosity644
4/3 farther away in a 3 dimensional reference frame.645
10 Other instances of 4/3 scaling646
In derivation of Stefan’s Law relating to black body radiation (Allen and647
Maxwell, 1948, p, 743) and (Planck, 1914) and in connection with isotropic648
radiation.649
4/3 scaling applies to a variety of phenomena and theories including650
Gas molecular mean path lengths (Clausius, 1860; Shour, 2017),651
Stefan’s Law on black body radiation (Planck, 1914),652
Wind eddies (Richardson, 1926; Kolmogorov, 1991b,a),653
Metabolic scaling (Shour, 2015, 2018),654
Brownian motion (Lawler et al., 2001),655
John James Waterston’s gravitating plane (Waterston, 1892),656
Cell phone tower transmission (Jafar and Shamai, 2007).657
Minkowski space time has 4 dimensions, analogous to [V][c] (Minkowski,658
2012).659
The deceleration assumption 18
11 Concluding remarks660
This article considers the problem solving methods of varying and inverting661
conceptual reference frames.662
Then the conceptual reference applicable to expanding space in relation to663
gravitation is inverted, by supposing that gravitation is an apparent force in-664
duced by expanding cosmological space. The conceptual reference frame that665
assumes gravity holds back space from expanding is inverted. In addition, the666
conceptual reference frame that supposes that we live in a single 3 dimensional667
space is varied to our living in a universe with dual reference frames, one 4668
dimensional and one 3 dimensional.669
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