Regarding the center of the universe, there are differences in the coordinate system used between physicists and the public.
When physicists describe the universe, they describe it in 4-dimensional space including the dimension of curvature, or 4-dimensional spacetime including the dimension of time. However, public think in 3-dimensional space (Movement over time in three dimensions of space. Newtonian mechanical world view).
In the problem related to the center of the universe, because the center of a 4-dimensional space exists in the 4th dimension (because it requires 4 coordinates), the center does not exist in the 3-dimensional space that public see. This is also why physicists use the analogy of a balloon or a raisin in bread.
Therefore, physicists always say that the center of the universe does not exist (in the 3-dimensional space we see).
On the other hand, it is difficult for the public to understand and agree with physicists' claims because we can define the center by looking at a 3-dimensional sphere.
Where is the center of the observable universe?
If the observable universe is described in 3 dimensions, the center is me (or Earth), and if the observable universe is described in 4 dimensions, everywhere is not the center. An event depends on the coordinate system it describes.
The public asks in 3D space, and physicists answer in 4D space or 4D spacetime(or 3+1 dimension).
However, there are three things that make physics great, one is assumption, another is approximation, and the last one is experiment. This is because complex real-world phenomena can be simplified through assumption and approximation, and once they reach a level where they are sufficiently useful in reality, that alone is valuable.
In the first place, approximations have meaning because all values, including physical constants, are strictly approximations, and the tools that measure them themselves have measurement errors.
Regardless of whether 4D space including curvature is an accurate description, it is necessary to think about whether this can be approximated in 3D space.
Observations of the universe show that space is nearly flat, which suggests that the fourth dimension of curvature can be ignored or approximated (depending on the accuracy required).
It is estimated that the current universe would be meaningful even if it were described in 3-dimensions.
Now, if we describe the universe as having three spatial dimensions, it is very likely that the center of the universe will lie within the three-dimensional mass distribution. Moreover, gravity is an attractive force, and when a repulsive force exists on a cosmic scale, the repulsive force is likely to be of a type that depends on r.
F=-G(-M)m/r^2 = -G(-(4/3)πr^3ρ)m/r^2 = +(4πGρ/3)r
When this repulsive or anti-gravity force exists, the universe can expand into a uniform sphere.
If the mass distribution in the entire universe is not infinite, the center of the 3D mass distribution must exist in 3D, and we can find its location. However, if the entire universe is infinite, there is no center.
[How to find the 3D space center of the entire universe]
If we extrapolate from the Hubble-Lemaître law,
1)We observe the red-shift of all galaxies that are a certain distance r from the Earth. The observed galaxies lie on the surface of a sphere of radius r.
2)Due to various factors, there will be errors in red-shift. From this data, find the plane (or circumference) with the smallest red-shift error.
So that all of the red lines are the same size. It’s important also that the lengths are proportional, so for example E_2A’/E_1A is equal to E_2B’/E_1B and E_2C’/E_1C since recession velocity from the center needs to be proportional to distance.
E_2A’/E_1A=E_2B’/E_1B=E_2C’/E_1C
V_rel=HD
This is the Hubble-Lemaître law.
3)The 3D space center of the entire universe exists in the direction perpendicular to the plane where the redshift error value is the smallest.
Because the entire universe is larger than the observable universe, it is possible that the three-dimensional space center of the entire universe exists outside the observable universe.
If the mass distribution in the universe is finite, we need to find the 3D space center of the universe and establish an absolute coordinate system (or center of mass coordinate system).