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Phys. Sci. Int. J., vol. 26, no. 9-10, pp. 69-78, 2022
Physical Science International Journal
Volume 26, Issue 9-10, Page 69-78, 2022; Article no.PSIJ.96694
ISSN: 2348-0130
Gravitational Displacement:
Time Dilation Rooted in Vacuum
Energy
Ivan Nilsen a*
a Insvivia Technologies AS (Norwegian Research Company), Norway.
Author’s contribution
The sole author designed, analysed, interpreted and prepared the manuscript.
Article Information
DOI: 10.9734/PSIJ/2022/v26i9-10768
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Received: 25/10/2022
Accepted: 29/12/2022
Published: 31/12/2022
ABSTRACT
Astronomical findings, particularly from the last decades of research, have confirmed that our
universe either must contain large amounts of an unknown form of matter, called dark matter, or
the laws of gravity must be influenced by undiscovered variables. Although both of the two
approaches contain many candidates with their respective matches and fails, no theories have so
far been able to finally solve the full picture of missing mass at different structural levels with the
relation to several associated problems. In this study, gravity is considered with a new approach,
more specifically not to be a property fundamentally incorporated to space, but something that arise
from the presence of background energy and its responsibility for making time flow at different local
rates. The study suggests that the gravitational constant, G, is only locally constant, and that
gravity itself causes a displacement that decreases the gravitational strength, only to a noticeable
degree for massive astronomical structures like galaxies and more heavy parent structures.
Keywords: Gravity; gravitational displacement; time dilation; general relativity; gravitational constant;
vacuum energy; dark matter; black holes.
Original Research Article
Nilsen; Phys. Sci. Int. J., vol. 26, no. 9-10, pp. 69-78, 2022; Article no.PSIJ.96694
70
1. INTRODUCTION
Newton’s laws of gravity and Einstein’s theory of
relativity respectively stands for some of the most
influential breakthroughs in science, that is
crucial for our understanding of physics and the
universe. It is in the same time known that they,
as classical theories, might be influenced by still
undiscovered physics deeply hidden in the space
itself, possibly with roots in the background
energy so far conceptually known as the vacuum
energy.
According to current acknowledged models, the
observable universe consists of around 25 %
dark matter, 5 % normal matter, and 70 % dark
energy. The exact numbers have varied for
different estimates.
Dark matter [1-5] is typically predicted to be
responsible for more than 80 % of the attractive
gravitational force, and normal matter less than
20 %. It is still unknown what kind of physical
phenomenon that causes the percentage of
normal matter, according to classical theory, to
become that low.
Since dark matter has not yet been identified as
a possible form of matter, it is as a topic of
research considered to include both the research
of potentially undiscovered matter types and the
research of modification of gravitational laws,
where both types of researches aim to solve the
same common problems.
Different theories and models have attempted to
solve the problems associated with dark matter
through the modification of gravitational laws,
known as Modified Gravity (MOG) [6]. Among
them is Modified Newtonian dynamics (MOND)
[7-12], where a range of different models have
been researched. Although MOND has made
successful matches at one structural level, it
commonly breaks down when including two
structural levels. Using the same modifications
for a galactic cluster as for individual galaxies still
shows a missing mass at the galactic-cluster
level. Therefore, the prediction theorized by
MOND, that gravity becomes stronger at large
distances, does not match the full nature of the
universe. An offset of apparent mass distribution
in galactic-cluster collisions is another important
finding [13] that conflicts with MOND.
Due to the inability for present suggested
modifications of gravitational laws to solve the
problems of dark matter, a new approach to
gravity could be crucial to the future research.
The deep nature of time dilation is in this study
particularly considered in order to establish a
new approach.
1.1 Time Dilation
According to the theories of relativity, there are
two types of time dilations:
1) The speed-induced time dilation defined by
Special Relativity and Lorentz-
transformations, and;
2) Gravitational time dilation [14] defined by
General Relativity
The gravitational time dilation is considered to be
fully consistent with the gravitational strength,
meaning that they are two aspects of the same.
The gravitational strength depends on the
amount of gravitational time dilation. Therefore,
some theories also consider time itself, as a
property incorporated to space, to be the
mediator of gravity. Gravity differs in that way
from the other 3 fundamental forces, which are
mediated by particles known as the force
carriers.
1.2 Gravitational Displacement
In the assessment of gravity as something that is
mediated by the background energy of space,
which reflectively is responsible for making time
flow at different local rates, an effect where
gravity itself can displace the local strength of
gravitation is possible. This possible form of
displacement is hereinafter named “gravitational
displacement”.
From intergalactic conditions, the background
energy makes space gravitationally repulsive, as
known from the cosmic expansion and the
subjects associated with dark energy. When
mass is put into the repulsive space, in the form
of astronomical structures like galaxies and
galactic groups, space starts to become locally
attractive instead. This implies that when time
flows at one rate, intergalactically, space is
repulsive, and when it is slowed just a tiny bit, it
becomes attractive.
Earth is known to dilate time by around 7*10-10
second per second, and the sun by around
2,12*10-6 second per second. This tiny bit of time
dilation is what causes the respectively great
strengths of gravitation associated with Earth and
the sun. It implies that just a small bit of time
Nilsen; Phys. Sci. Int. J., vol. 26, no. 9-10, pp. 69-78, 2022; Article no.PSIJ.96694
71
dilation causes energy to be conserved through a
great gravitational force.
As the boundary between a repulsive and a
strong attractive gravitation is rooted in just slight
difference in elapsed time rates, a possible
gravitational displacement may also form within
equivalently slight time dilations.
Although the time dilation near massive black
holes can reach extreme levels, such extreme
levels does only apply to local regions which
border to limited amounts of space. Larger
astronomical volumes of space can only reach a
relatively small time dilation. In the centre of
galaxies, the time dilation might typically be on
the order of 10-4 to 10-6 second per second,
depending on the radius and mass density of the
galactic centre.
A possible cause for gravitational displacement
might be that the background energy of empty
space has the property that time only can be
dilated to a certain degree before energy is
conserved through different gravitational
strengths.
Gravity is caused by the way space conserves
energy. Since energy is measured by time, and
time itself is a part of the picture, it is possible
that the energy-conservation regime responsible
for gravity changes when the time is dilated just
to a slight degree.
Our own solar system is located in a
region where the gravitational displacement
of its parent galaxy is expected to be significant,
and therefore the gravitational constant, G, is
also affected equivalently. In regions farther
away from the galactic centre, the local G is
expected to be higher, and in intergalactic
space, even higher. The orbital velocities in
these regions are thereby expected to be
equivalently higher, in accordance with
observations.
Contrary to MOND, a gravitational displacement
does not make gravity stronger for long
distances, but it implies that gravity by nature is
stronger from the surrounding deep space, as
illustrated in Fig. 1.
Fig. 1. Principal illustration of gravitational displacement
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72
1.3 The Velocity Curves
According to observations, astronomical objects
in the outer regions of galaxies orbits significantly
faster than the gravitational strength of visible
matter are supposed to allow them to do. This is
known as the velocity curves of galaxies, more
specifically the observed and the calculated
velocity curves.
Seen in view of gravitational displacement, this
effect can be divided into two subcomponents:
1) The gravitational strength between the
inner and outer regions of an astronomical
structure becomes higher, not impacted by
distance in itself, but because gravity by
nature is stronger from intergalactic space
whereof the inner regions due to
gravitational displacement contains more
mass than calculated with present models
2) The local gravitational strength between
objects in the outer regions of an
astronomical structure is higher relative to
the inner regions
The velocity curves can, based on the mentioned
two components, be described as an effect of
gravitational displacement. It does not exclude
the possibility that there might exist unidentified
matter additionally, in the form of objects or gas
that is hard to identify or in the form of dark
matter, but it does also open the possibility that
dark matter does not need to exist.
1.4 The Bottom-Level Constant
From an objective viewpoint, the gravitational
strength of a massive object is not determined by
the object itself, but the space surrounding it.
Although the time in black holes can be slowed
to a large degree relative to its surrounding
space, the gravitational strength of the
surrounding space may not be displaceable more
than to a certain degree. A rate at which
time will re-enter a volume occupied by mass, if it
ceased to exist, may form a lower limit to
how much gravitational displacement that is
possible.
A bottom-level constant can be described as the
background energy’s capability to accelerate
time. Where space is occupied by a massive
object, the background energy is forced to
deceleration of time that can only be locally
maintained.
A bottom-level constant must have a value of >1.
Based on present research, a bottom-level
constant cannot be identified to an exact value.
Such identification requires new research. The
lack of a known value makes it difficult to study
gravitational displacement with accuracy. The
use of notional values does however enable
gravitational displacement to be studied
conceptually.
2. METHODS
2.1 The Local Gravitational Constant: The
Local G-formula
In order to calculate the local gravitational
constant G based on gravitational displacement,
two respective reference volumes are required:
1) The reference volume of the known G
2) The reference volume of the relative G
The total mass occupying the space of both
reference volumes are required for the
identification of the gravitational displacement.
The local G for the relative reference volume
may be expressed as:
where Gl is the local gravitational constant, b (=
>1) is the unknown bottom-level constant, d (=
>1) is the displacement factor number, mt1 and
mt2 are the total masses occupying the
respective reference volumes, and v1 and v2 are
the volumes of the respective reference volumes.
By the identification of the respective
displacement factors, Gl is given by:
where d1 and d2 is the respective displacement
factors.
2.2 General Relativity and Newton’s Law
of Gravitation
Gravitational displacement may be accounted for
by the existing laws of gravity by replacing the
gravitational constant G with the local
gravitational constant Gl, derived from the local
G-formula.
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Newton’s law of universal gravitation can be
expressed as:
where F is the gravitational force, m1 and m2 are
the respective masses, r the distance between
their centres, and Gl is the local gravitational
constant.
It can also be expressed, together with the local
G-formula, as:
The Einstein field equations may be written in the
form:
where Gµv is the Einstein tensor, Rµv is the Ricci
tensor, R is the scalar curvature, gµv is the metric
tensor, Tµv is the energy-momentum tensor, Gl is
the local gravitational constant, and c is the
speed of light in vacuum.
3. RESULTS
3.1 The Study of Astronomical Mass
Distribution Profiles
The exact mass density of known galaxies in the
universe is in general hard to identify, and
therefore, such numbers does only exist in the
form of estimates. Additionally, with gravitational
displacement, those numbers will change.
In this study, gravitational displacement is only
demonstrated conceptually, and therefore, it
does not intend to match the mass distribution of
specific known structures.
In the following table, an astronomical structure
with a certain radius and mass distribution profile
is used as a base for studying the gravitational
displacement. The bottom-level constant and the
displacement factor number are assumed to
have certain values. The gravitational constant G
is finally calculated out from the displacement
factors, with a reference volume located between
the inner and outer regions of the astronomical
structure.
G is the gravitational constant, d is the
displacement factor number, the radius is the
distance from the centre of the structure, the
volume is the volume derived from the radius,
and mass total is the overall mass occupying that
volume. Distances are given by kiloparsec (kpc),
volume by cubic-kiloparsec (kpc3), and mass by
billion solar masses (GM☉).
The table illustrates an astronomical structure
with a certain mass distribution profile from a
radius of 1 kpc to 291,93 kpc. The bottom-level
constant is set to 0,5, and the displacement
factor number d to 0,99995. The value of 1
intends to correspond with the deepest space of
the parent structure.
The local gravitational constants are derived from
the local G-formula,
, using the radius of
7,59 kpc as the reference volume d1.
The results demonstrates that the displacement
factor decreases the gravitational constant G to a
significant degree. At a radius of 3,38 kpc and
11,39 kpc, the displacement factor is appr. 0,652
and 0,973, respectively. The gravitational
strength is therefore about 49,2 % higher at
11,39 kpc than 3,38 kpc, which may be
expressed as the displacement factor as
illustrated in Fig. 2.
In view of the total mass numbers, given by
GM☉, the results also demonstrates that a high
mass density alone is not enough to obtain a
gravitational displacement that comes near to the
bottom-level constant. Thereby, no other
astronomical objects than black holes are heavy
enough to locally come near to the bottom level
on its own. Even the largest stars in the universe
do only impact the gravitational constant to a
barely measurable degree near its core based on
these demonstrated values.
The Fig. 3 shows an astronomical structure A
and B, where B is larger than A, both in form of
total mass and volume. The gravitational
displacement is therefore extending to a larger
volume of space for B than A.
For a flat rotating astronomical structure, the
gravitational displacement is expected to extend
similarly in the radial direction as the axial
direction, which may form a spherical halo of
displacement surrounding the flat structure.
Thus, it creates a gravitational lensing effect that
Nilsen; Phys. Sci. Int. J., vol. 26, no. 9-10, pp. 69-78, 2022; Article no.PSIJ.96694
74
Table 1. Astronomical mass distribution profiles
Radius
Mass total
Volume
Displacement
G
d
(kpc)
(GM☉ )
(kpc^3)
factor
(m3 kg-1 s-2)
0,99995
291,9292603
1184,331885
13020008,81
0,999997307
7,20655E-11
0,99995
194,6195068
1166,861041
3857780,388
0,999991176
7,2065E-11
0,99995
129,7463379
1146,307107
1143046,041
0,999971261
7,20636E-11
0,99995
86,49755859
1122,126009
338680,3084
0,999907060
7,2059E-11
0,99995
57,66503906
1093,677657
100349,721
0,999702091
7,20442E-11
0,99995
38,44335938
1060,209008
29733,25067
0,999055763
7,19976E-11
0,99995
25,62890625
1020,834128
8809,852049
0,997051447
7,18532E-11
0,99995
17,0859375
974,5107383
2610,326533
0,990986673
7,14161E-11
0,99995
11,390625
920,0126333
773,4300839
0,973375023
7,01469E-11
0,99995
7,59375
855,8972156
229,1644693
0,926141430
6,6743E-11
0,99995
5,0625
780,4673125
67,9005835
0,819274735
5,90416E-11
0,99995
3,375
691,72625
20,11869141
0,652234559
4,70037E-11
0,99995
2,25
587,325
5,96109375
0,527693158
3,80286E-11
0,99995
1,5
464,5
1,76625
0,501112481
3,6113E-11
0,99995
1
320
0,523333333
0,500028182
3,60349E-11
Fig. 2. Graph of displacement factor
Fig. 3. Principal illustration of the spatial extension of gravitational displacement
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Fig. 4. Principal illustration of a collision offset
Fig. 5. Principal illustration of gravitational displacement profile (GDP)
surrounds the visible matter to a significantly
greater radius.
The kinetic energy of dense regions is, relative to
their resistance through a collision, significantly
higher than for less dense regions. Therefore,
the magnitude of gravitational displacement is
moved offset compared to the distribution of low-
density matter, which are more significantly
slowed in the interaction with each other. The
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76
Fig. 4 illustrates how this effect could appear for
a certain type of collision, similar to the effects
observed through gravitational lensing.
Only one structural level is demonstrated by this
study. Two or several structural levels implies a
higher degree of complexity, and requires further
research.
3.2 Gravitational Displacement Profile
(GDP)
The use of gravitational displacement to redefine
the local gravitational strength in structures
implies that every massive astronomical structure
has its unique gravitational displacement profile
(GDP). The velocity curves are determined by
the GDP, and therefore, astronomical structures
with different mass distribution profiles
are also expected to have different velocity
curves.
Since different astronomical structures are
observed to have different velocity curves,
among other in conflict with MOND, gravitational
displacement demonstrates that this
phenomenon may be an effect of GDP.
The Fig. 5 illustrates an astronomical structure
with a reference volume located between its
inner and outer regions.
If the mass distribution profile of a structure were
changed, for example by adding more mass to its
outer regions than inner regions at a given total
mass, the local gravitational displacement in the
inner regions would decrease significantly. The
GDP and the velocity curves would change
equivalently. This provides an interesting new
type of dynamics that is not accounted for by
present astronomical models.
Fig. 6. Principal illustration of the typical velocity curves
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3.3 The Gap between Velocity Curves
The Fig. 6 illustrates the effect of gravitational
displacement, where the line A is the calculated
velocity curve without accounting for gravitational
displacement, and the line B is the typical
observed velocity curve.
The typical scenario for known galaxies in the
universe is that the gap between the two velocity
curves starts to enlarge after a certain radius,
and then the further enlargement typically takes
off after passing a radius farther away.
Close to the centre of a galaxy, the gap between
the two velocity curves is typically small. As
demonstrated in this study, this might be an
effect of the gravitational displacement nearing
the bottom level of gravitational displacement,
which implies that the effect of gravitational
displacement takes off when the total mass and
mass density becomes high, like in the centre of
galaxies.
4. DISCUSSION
The demonstration of gravitational displacement
as a possible solution to the problems associated
with dark matter, may provide the following 7
consequences:
1) Massive and dense regions in the
universe, like the centre of galaxies,
contain more mass than they appear to
with current laws of gravity;
2) The gravitational strength is higher in
peripheral regions of astronomical
structures than near their centres;
3) The velocity curves of galaxies are
governed by the following two
consequences of gravitational
displacement;
a) an increase of mass in galactic centres
b) a decrease of gravitational strength
relative to intergalactic space
4) The apparent missing mass at galactic-
cluster level, among other in conflict with
MOND-models, is caused by the increased
gravitational strength of intergalactic
space;
5) The variation of velocity curves for different
types of astronomical structures, among
other in conflict with MOND, is caused by
the variation of the gravitational
displacement profile (GDP);
6) The apparent mass-containing halos
surrounding galaxies, among other
observed through gravitational lensing, is
caused by the extension of gravitational
displacement into the space surrounding
the structure;
7) The apparent offset of matter distribution in
galactic-cluster collisions is caused by the
spatial extension of gravitational
displacement
5. CONCLUSION
This study demonstrates that the problems
associated with dark matter may be solvable with
a new approach to gravity, with the
accompanying possibility that dark matter as a
form of matter does not need to exist. Contrary to
present suggested modifications of gravitational
laws, the new approach implies that the apparent
missing mass in astronomical structures, also at
respectively different structural levels, may be an
effect of gravitational displacement. The spatial
extension of gravitational displacement does also
imply an offset in collisions between astronomical
structures, in accordance with the offset
observed through gravitational lensing.
Moreover, it does imply that each astronomical
structure has its unique gravitational
displacement profile, which leads to different
velocity curves for different types of structures.
The clear conclusion is therefore that the study
strongly recommends gravitational displacement
to be further researched as a possible solution to
the problems associated with dark matter.
COMPETING INTERESTS
Author has declared that they have no known
competing financial interests or non-financial
interests or personal relationships that could
have appeared to influence the work reported in
this paper.
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