Do stars in a spiral galaxy simulate 4-dimensional movement in a 3-dimensional space?

Abstract

The stars in the spiral galaxies move in such a way that they increase their distance from the center of the black hole and make a mathematical series. We purpose a mathematical model for predicting the path of the stars in the spiral galaxies. The complex dimensional model is a novel mathematical series for illustrating the relationship between 4-dimensional movement in a 3-dimensional space. We show that arms of spiral galaxies are the path of movement of stars when they try to keep their distance from the center of the galaxy in a 4-dimensional movement.

Full text

Do stars in a spiral galaxy simulate 4-dimensional movement in a 3-dimensional space?
Bahram Kalhor
1
, Farzaneh Mehrparvar1, Behnam Kalhor1
Abstract
The stars in the spiral galaxies move in such a way that they increase their distance from the
center of the black hole and make a mathematical series. We purpose a mathematical model for
predicting the path of the stars in the spiral galaxies. The complex dimensional model is a novel
mathematical series for illustrating the relationship between 4-dimensional movement in a 3-
dimensional space. We show that arms of spiral galaxies are the path of movement of stars when
they try to keep their distance from the center of the galaxy in a 4-dimensional movement.
Keywords: Spiral galaxy, Complex dimensional model (CDM)
Introduction
CDM series is a novel mathematical series for illustrating the relationship between the distance
of moving objects from their start point. In the CDM model distance of movement in each state
is equal to movement distance in the previous state while the direction of movement is
perpendicular to the line between the last state and the first state (or its image on the movement
surface). We describe a movement in which moving objects in each new state move in a new
direction which is perpendicular to the line between state zero and previous state (or its image on
the movement surface). In the CDM model distance of movement in all states are equal and the
angle between movement direction in each state and line between state zero (or its image on the
movement surface) and its previous state is 90 degree.
We can restrict the movement in a 2-dimensional space just in a certain surface which the path
would be approximately like the Fibonacci series, or let the moving object to move in multi-
dimensional space, or even combine them.
Series are used for describing changes of one starting quantity after adding new quantities one by
one [1]. One of the most popular types of time series models is State-space models (SSMs).
SSMs are a type of hierarchical model [2] that are used for predicting movement in a 2-
dimensional surface [3]. These models have been used for modeling population dynamics [4],
stock market prediction [5] and a wide variety of ecological systems. Also, time series models
have been used in Apollo's mission for correcting the path of a spacecraft toward the moon [6].
Despite the SSMs model, in which the state at time t depends only on the state at the previous
time t−1, sometimes we need to model a movement that the state at time t, depends on state at
1
Independent researcher form Alborz, IRAN
Corresponding author. Email: bahram.kalhor@kiau.ac.ir

the previous time t−1 and state at time t0. These new models could be useful in physics
especially when we try to model the path of a movement based on its start point and calculate the
distance of moving objects from their start point in the space-time series [7,8,9,10]. Finding out
the path of movement in multidimensional space-time for a certain moving object with a speed v
in an area with gravity g would be a good sample of these types of movement, such as movement
path of stars in the arms of spiral galaxies.
Although in mathematics we do not have any restriction for moving in upper dimensions, it
seems that because of some restrictions in physics, most of the real moving objects after
completing their path in a 3-dimensional world, continue their path in 2- dimensional surface.
In fact, for simulating our observable universe and galaxies, we suggest a complex movement
area which is a combination of a 3-dimensional movement at the first and continuing in a 2-
dimensional surface. In this case, moving objects save their moving rules, equal distance in each
state and new direction which are perpendicular to the line between state zero and previous state
(or its image on the movement surface).
Nowadays, Physicists are using a wide variety of series for modeling phenomena [11,12].
Although in mathematics, there is no limit for the number of dimensions in multi-dimensional
models, we are living in a 3-dimensional world while in some popular theory in physics such as
String theory and M theory we have 10 ,11 or even more dimensions [13,14,15].
The best examples of these shapes are spiral galaxies with a few numbers of arms. In the spiral
galaxies, the height is approximately 10% of its diameter [16, 17,18]. On the other hand, we
know that it is proven that the shape of the whole universe is approximately flat [19,20], then this
model would be a good model to explain this shape.
Investigating the MDM (Multi-Dimensional Model) movement path in our 3-dimensional world
is important while we have limited to move in a 3-dimensional space and establishing upper
dimensions not allowed. Basic questions are: What would be happened if the moving object
unable to continue its moving into the upper dimensions? Which direction would be chosen for
the next movement? What would be the final shape of the movement path? Is there any rule to
shows its distance to the start point?
The purpose of this paper is to represent a mathematical model for modeling the shape of spiral
galaxies and using novel CDM series for approximating the diameter of multi-dimensional spiral
galaxies or even the universe. On the other hand, according to the suggested mathematical
model, we show that there is a significant relationship between the height of galaxies and their
diameter. Meanwhile, we try to solve some unsolved physics questions based on one
mathematical model.
Methodology
There are four rules:

1. In the multi-dimensional model, moving object in each state moves in a direction which
is perpendicular to all previous state directions while in 2- dimensional model moving
object in each state moves in a direction which is perpendicular to the line between state
zero and its previous state.
2. Distance and average speed of movement in all states are equal
3. In 2- dimensional model moving object uses left-hand or right-hand rule for choosing the
direction in all new states.
4. In complex-dimensional model which start point is not on the moving surface, we use the
image of it on the movement surface as the start point.
MDMs: Multi-Dimensional Model series
The first element of the MDM series is equal to zero and has been described by a point which
is shown in Fig.1(a). It shows that moving objects is at the start point and has not started its
movement, therefore the distance between moving objects and the start point is equal to zero.

The second element of the MDM series is equal to L.
 
 make a one-dimensional line. Distance between the points nd is equal to L which is
the constant value that the moving object after reaching to it must change its direction, Fig.1(b).
  
After reaching to point , the moving object must choose right-hand or left-hand direction to
and move in a new direction, this rule would be permanent for the rest of the states. This
direction would have chosen according to rule number 3. After choosing a direction, we have a
2-dimensional surface that concludes the first three elements of MDM series Fig.1(c). Distance
between  and  calculate by the right triangle formula. 󰇛󰇜󰇛󰇜󰇛󰇜
= 
= 



Fig.1. Schematic view of start point and first elements of MDM series
Fourth point is . The points ,  and make a right triangle and length of line is equal
to  . , ,  and are shown in a three-dimensional cube in Fig.2.
󰇛󰇜󰇛󰇜󰇛󰇜
=
=
=

is Fifth point andmake a fifth-dimensional shape. Although imagine more
dimensions is hard, we can calculate the length of according to right angle .


Fig.2. A three-dimensional path of the movement in the Multi-dimensional model

If we continue these steps, the distance between the nth point from would be calculated by this
formula:
   
and the total MDM series expression would be:
 

   


         
TDMs: Two-dimensional model series
If we restrict moving objects to move in a 2-dimensional surface, the new direction could not be
perpendicular to all previous directions. In a 2-dimensional model, the new direction is
perpendicular to the line between  and previous state. Also moving objects in the state three
and next states, have two options to choose, left-hand or right-hand direction. Regardless of what
direction has been chosen, the moving object must use the same rule in the next states. Fig.3
shows the path of movement on a 2-dimensional surface.
is start point

is location of state 2 which its distance from is L.
 
is 3rd state and in this path moving object has chosen left-hand direction and has continued in
the next states.

Fig.3. Two-dimensional path of movement in the TDM and CDM
At the 3rd state we have a right triangle.
󰇛󰇜󰇛󰇜󰇛󰇜
= 
= 
=

As shown in Fig.4, if we continue these rules, the distance of moving object in the state n would
be calculated by this formula:
= 
and we have a mathematical series that is the same as the MDM series. TDM series expression
would be:
 

   


       
The similarity between MDMs and TDMs shows that moving object can continue its movement
in a 2-dimensional surface while saving its distance rules from previous state and start point at
the same time.

TDM Angles:
In the TDM movement, after 3rd state in each new state, we have a new right triangle. Angle of
 is equal to 󰇛
󰇜  󰇛
 ) Fig.1.f.
CDMs complex dimensional model series
The CDMs is a combination of one 3-dimensional and one 2-dimensional movement path. In
CDMs, moving objects at point  in Fig.2 which is the end of state 4, end its MDMs movement
path and establish a new TDMs path. New TDMs movement surface is parallel to  surface
Fig.4 and uses point  as start point which is image of  on the new surface Fig.4.
In fact, the complex dimensional model is a combination model of a 3-dimensional movement at
the first and continuing in a 2-dimensional surface. In CDMs, moving object tries to save its
distance rules from the start point and previous state like MDMs while it has prevented to
continue its movement in the upper dimensions.
The first four elements are the same first four elements in MDMs.

 


Also, the movement path in the first four states is the same as the movement path in MDM Fig.4.
At the point , moving object has been prevented to create new dimension then it has to choose
to stop moving or continue in a new path. Rules of continuing movement are having distance L
from the previous state and having the same distance from start point like MDMs. In this case,
the moving object starts its movement on a new surface which is parallel to the surface.
Point is out of the surface of and moving object cannot choose next direction by
drawing a line perpendicular to the line between and previous state like TDMs, thus it uses
point as start point which is image of on the new surface Fig (4) and choose movement
direction perpendicular to the line between and .
At this stage, we calculate the distance of and compare it by fifth element in the MDM
series.
===L
and is a right triangle, so
=

Fig.4. First 5 elements in the CDM and location of point 
in right triangle Fig.5(a)
=
= L , hence
=
In right triangle Fig.5(b)
=L
= , so
=
is the 5th element of the CMD series and is equal to the 5th element of the MDM
series.
If moving object continues its direction perpendicular to line , in next right angle
Fig.5.b
=
=L
hence
=
and in right triangle 
=L
=
hence
=


is the 6th element of the CMD series and is equal to the 6th element of the MDM series.
Finally, we have obtained the CDM series formula which is the same as the MDM series formula
 

   


         
Fig.5. The distance of the 5th element from the start point in the CDM
Fig.6 and Fig.7 show right triangles from the latest state to points and and Fig.9 shows the
final path of movement in the CDM model.
Fig.6. The distance of the moving object from the point in the CDM
Fig.7. The distance of the moving object from the start point in the CDM

Fig.8. Path of movement in the CDM
CDM and spiral galaxies
It is proven that the speed of all galaxy's stars in spiral galaxies are almost equal. For instance,
the speed of stars in the Milky Way galaxy is between 210 to 240 kilometer per second. The
equal speed causes all stars to move the same distance in the same time duration. on the other
hand, most spiral galaxies have two major arms. Real pictures of galaxies have shown that the
start points of both arms are in the center of the galaxy while the first parts of both arms are in
reverse directions. Another important thing is the ratio of the height of spiral galaxies to their
diameter. Physicists have observed the central height of Milky Way galaxy, its approximately 12
thousand light-years while the diameter of the Milky Way is almost 100,000 light-years. Thus its
ratio is near 12 percent.
In the CDM, if we start two individual moving objects in two reverse directions with the same
speed, we will have a 3-dimensional shape like Fig.9. If we look at this shape from the top, we
see a 2-dimensional view of a path that is similar to the central map of spiral galaxies Fig.10.
We continued to calculate more elements of the CDM series (expanding arms), results would be
like Fig.11.

Fig.9. Three-dimensional path and their surfaces in the CDM in two reverse directions
Fig.10. Top view of Three-dimensional path in the CDM in two reverse directions

Fig.11. Top view of first 15 elements in the CDM (reverse directions)
If we continue the CDMs series to the first 64 elements, the distance of the 64th element from
the start point is 8L. In this case, the ratio of central height to diameter would be almost 12.5
percent.
In Fig.9 if the second object chooses the same direction as the first moving object in the
movement, then the height of shape would be half and after just 16 movements the ratio of
height to diameter would be 12.5% which is near the real ratio of Milky Way galaxy and the
surface CMD would be flatter.
Conclusion
Regardless of what physics’s rules cause the equal speed of stars in spiral galaxies, the CDMs
shows that the arms of spiral galaxies are the simulation of movement in upper dimensions in a
2-dimensional surface. CDM series provides a formula for predicting the next direction, next
distance from the center of the galaxy and the ratio of height to diameter. CDM explains the flat
shape of spiral galaxies and would be used for predicting its expansion ratio and describing flat
universe.
Abbreviations
TDMs: Two-Dimensional Model series
CDMs: Complex Dimensional Model series
MDMs: Multi-Dimensional Model series
SSMs : State-space models series

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