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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

Nilsen; Phys. Sci. Int. J., vol. 26, no. 9-10, pp. 69-78, 2022; Article no.PSIJ.96694

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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73

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

Nilsen; Phys. Sci. Int. J., vol. 26, no. 9-10, pp. 69-78, 2022; Article no.PSIJ.96694

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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77

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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