Brian Nelson 1/09/2024
The Universe is round.
What does that even mean?
Well, to get started we need to understand some basics. Present physical theories treat the universe as a four-dimension space-time entity, with these dimensions as the x-direction, y-direction, and z-direction (space) and t-direction (time). This model was first championed by Albert Einstein at the turn of the twentieth century. He used it as his framework for defining how gravity was not a force, but was a way that masses affected the universe around them.
For Einstein, the universe was conceptualized as a tesseract (a 4-dimensional cube) with infinite sides. Infinite in space and time. As a tesseract, each dimension was orthogonal to the other three (formed right angles). Somewhere in this infinite space-time was a cluster of masses that was referred to as the Milky Way Galaxy.
When Albert Einstein tackled the question of how to mathematically describe his General Theory of Relativity, he came up with the following equation to describe the effects of gravity on the space-time of the universe. It is now referred to as the Einstein Field Equation (EFE):
Where,
- Rμν is the Ricci curvature tensor
- R is the scalar curvature
- gμν is the metric tensor
- G is Newton’s gravitational constant
- c is the speed of light
- Tμν is the stress-energy tensor
The left-hand side of the equation represented the “Einstein Tensor” which described the curvature (collapse) of space-time. The right-hand side described the effect of matter.
Einstein quickly realized that this relationship would result in a continually collapsing universe if any matter was present. Since he believed that the universe was infinitely old and steady-state, he added a value of – Λ to the right-hand side of the equation to balance it out. He called Λ his cosmological constant and set it equal to the average value of the effects of all matter.
This raster equation was mathematically rigorous, but was too generalized to be easy to work with.
Soon afterwards (in 1922) Alexander Friedmann, as well as a number of other physicists, derived a scalar equation from the EFE by making a few simplifying assumptions. Their first assumption was that space is homogeneous and isotropic (the cosmological principle). This assumption is made by everybody who studies cosmology. Their second assumption was that the density of matter and energy in the universe was equal throughout. This second assumption is obviously incorrect at small scales (the density of matter in your body is much greater than the density in the voids between galaxies), but is approximately correct at large scales (Megaparsec sizes). The third assumption that Friedmann made was that inter-stellar space could be modeled as a perfect homogenous fluid.
With those assumptions Friedmann derived the following equations:
where:
a = a scale factor (any chosen length) (m)
á = the decrease in the scale factor with time (m/s)
k = a shape factor to take into account the natural curvature of space
k is positive if space is hyperspheric in the dimension of time
k is negative if space is saddleshape in the dimension of time
k is zero if space is flat in the dimension of time
c = the speed of light in a vacuum (300,000,000 m/s)
G = Newton’s gravitational constant (6.674 x 10 –14 m3 /g-s2)
ρ = the average density of matter and energy (g/m3 )
Λ = Einstein’s cosmological constant
Note, that Friedmann originally derived this equation for a collapsing universe.
While Einstein was publishing his new model, a group of astronomers working at the Wilson Observatory under Edwin Hubble were discovering some new facts about the universe. First off, they determined that the Milky Way Galaxy was not the only galaxy in the universe. In fact, there were thousands, if not millions, of galaxies in space; some of which were much more massive than our own. Second, they determine that all of these galaxies were moving away from us, at approximately speeds directly related to their distance from us in all directions. This lead them to conclude that the universe was expanding as it moved in the positive time direction.
This derivation of the EFE, as modified for expansion, is the mathematical definition that forms the basis for the modern STANDARD MODEL of the Universe (also called the ΛCDM Model). Although the equation is usually rearranged to make it more usable.
A common way to write this equation is:
where: (á / a) equals the Hubble value (H)
If the universe is expanding with time, then it cannot be infinite. Otherwise if it is infinite today, it would be infinite + something tomorrow (a more infinite infinite?), and would have been infinite – something yesterday (a less infinite infinite?). And the mass density and energy density of the universe would never change from the present value. So, Einstein had to abandon the idea of an infinite sized tesseract model. However, with the universe now being unstable, he could also abandon the idea of a Cosmological Constant to balance things out. Instead, a new model containing some sort of initial expansion force, that originally overcame gravity, could be conceived. And gravity was simply slowing down the rate of expansion.
Present Model of the Universe
Over the last one hundred years, astronomers and cosmologist have been adding to and refining the Einstein/Hubble model of the universe.
For one thing, cosmologists have postulated that soon after the beginning of time (tmin ) a large sudden expansion of space occurred. This happened in the first billionth of a trillionth of a trillionth of a second of existence, and expanded the total space in the universe from the size of a marble to 1030 cubic lightyears. During this expansion, referred to as Inflation, all of the electrons, protons, neutrons, and photons came into existence.
Secondly, after this one billionth of a trillionth of a trillionth of a second Inflation, the expansion rate of the universe dramatically slowed down as the space was filled with a plasma of atomic nuclei, free electrons, free photon, and neutrinos.
A half a million years after Inflation, the temperature of the universe had cooled enough that the electrons and protons combine to form stable atoms. This has been named Combination (or Recombination) and marks the beginning of the universe that we are able to observe. Combination also resulted in the production of homogeneous photons of light with a wavelength centered around 600 μm. This light we can still see today in every direction, although it has red-shifted to the microwave wavelength.
Since Combination, the universe has continued to expand (going in the past-to-present- to-future direction) and seems to have been expanding at an accelerating rate.
It is difficult, if not impossible, to represent a four-dimensional space on a two dimension medium, so depictions of the universe tend to eliminate one or two space dimensions to show a representation of either an x-t plane or an x-y-t space. One example of the present model is shown below, from US NASA’s WMAP website (2022).
Time is in the left-to-right direction, starting at a singularity (quantum fluctuation) at the left, and ending at the present on the right. x and y are show vertically, and z is omitted.
There are some serious flaws in this picture.
First, it shows a gradual increase (a few million years) from the time that the universe is the size of a marble to the time it is 2 billion light-years in diameter. Present theories predict that this inflation event occurred within one billionth of a trillionth of a trillionth of a second. This would be effectively a flat surface in the picture. Secondly, we know that Combination (the afterglow event at 400,000 years after Inflation) occurred when the volume of the universe was 1100 times smaller than it is today. This would mean that the radius of the universe at that time was 1/(1100)1/3 = 0.097 times the radius today. The picture shows the radius at that time to be more than ½ the radius today. Thirdly, the picture shows galaxies and proto-galaxies as distinct dots occurring at discrete times throughout the last 13.7 billion years. They would be better shown as threads existing from left to right, since they exist as mass and energy throughout the entire history of the universe.
But there are four more serious problems with this condom-shaped depiction of the x-y-time universe. The first is that it shows the x-y dimensions having an outer wall enclosing the universe. Now, since astronomers have determined that the x, y, and z dimensions are not infinite, then there must be some limit on these dimensions. But then we need to define what happens when matter or light reaches this wall. Does it bounce off? Does it simply stop? Or does it fall off the edge of the universe into nothingness? Most cosmologist just wave their hands at this problem and say that the universe is expanding so fast that nothing ever reaches the wall. But, masses and energies near enough to the edge will in fact reach it eventually. Also, if there is a wall around the universe, then space near the wall would act differently than space in the “middle” of the universe; which would mean that space cannot be homogeneous and isentropic.
The second problem has to do with the volume of space-time that the universe is expanding into. This large volume of “Nothingness” (apologies to Michael Ende) can be defined in the x-y-z-t hyperspace, but literally contains nothing. No matter, no energy, no vacuum, nothing. The question then becomes: is this some other universe, or is it where spirits exist, or what?
A third problem is that the Nothingness extends to the left, to a time before time. This factor suggests that there was something that existed before the beginning of time. A number of cosmologist (and theologians) have speculated as to what could have been occurring before the beginning of time. One is that there was another universe that underwent a Big Crunch, which gave birth to our universe. Other theories are just as speculative. However, nothing in the Einsteinian model of the universe contains any interpretation of a negative time.
The fourth problem with this model of the universe is probably the most important. All models of the universe assume that the four dimensions are orthogonal. That is to say that each of the x-, y-, z-, and t-dimensions form right angles with each of the other dimensions. All our theories about the reality of the universe are based on this assumption.
If we look at the present model of the universe, and simplify our analysis to just two dimensions (x and time), we start with the Einsteinian idea that the universe was cubic.
time direction
x direction
With the Hubble data, cosmologists determined that the universe was not a tesseract but was a hypercone.
x direction
time direction
With the discovery that the expansion rate of the universe was increasing, and that Inflation had occurred at the earliest time, this shape was modified to:
x direction
time direction
In this shape for the x-t plane of the universe, for the center of the universe the x direction and the time direction are orthogonal. But for any other point in the universe, orthogonality does not occur. This creates a situation where modern physics is not applicable in most of the universe.
x direction
time direction
So we need to find a way to make every point on the x surface orthogonal to time. And the only way to do that is to curve the x direction in a positive arc.
x direction
time direction
And once it has been determined that the x dimension must be positively curved in relationship to the time dimension, you can extend the x dimension around until it meets up with itself on the other side, forming a circle.
So, what this means is that for orthogonality to exist, the x, y, z, time universe is not a hypercone but is a hypersphere with time as the radial direction.
This actually helps to solve the other problems associate with the present model of the universe. Now there is no “wall” limiting the x, y, and z directions. And there is no “center point” on the x, y, and z directions. The x-y-z-t universe fills the whole hyperspace, so there is no hypervolume of Nothingness that the universe is expanding into. And since the beginning of time is at the center of the hyperspace, there is no time before the beginning of time. If you go in the negative time direction, you only end up going in a positive direction on the other side of the x-y-z volume.
Now the x-t plane looks like:
time
direction
x direction
Applying the Einstein Field Equation to a Hypersherical Universe
Although the EFE was originally developed to apply to physics on a planetary or smaller scale, in general, astronomers and cosmologists agree that it can be used to describe the universe in totality. The Friedmann Equation (or Friedmann-Lemaître-Robertson-Walker metric) is more easily solvable. This FLRW metric is generally considered to be THE STANDARD model that describes the Universe.
The FLRW metric is usually written as:
or, rearranged as:
Now, close to 100 % of astronomers and cosmologist have concluded that the space (the x-, y-, z-dimensions) is flat in relationship to time. This means that k (the shape constant) = 0; and the last value on the right side of the equation can be erased. As we have just seen, space needs to be positively curved, so the value for k must be greater than zero. It is more appropriate to erase the first value on the right side of the equation, since nobody in science can explain what this mysterious “dark energy” is.
So, the equation reduces down to:
Now this equation has four unknowns: ȧ, a, ρ, and k.
ȧ cannot be determined from astronomical observations or any other scientific means. However, ȧ/a equals the Hubble value. For the present time (t0) the Hubble value (H0) has been calculated a number of times. It is either 2.22 x 10 -18 sec -1 (68 km/s/MPc) or 2.39 x 10 -18 sec -1 (74 km/s/MPc), mattering on how it is calculated. For this analysis we will use the 2.2 x 10 -18 sec -1 value.
For the present time, a value for the total density of matter/energy (ρ0) can be estimated from astronomical observations. We will analyze this later.
The value for k is an interesting challenge. k represents the shape relationship of a (the total length of x) to t (the total time). If we go back to our hyperspherical model of the universe as illustrated above, a equals the circumference of the circle and t equals the radius of the circle. The geometric relationship therefore is a = 2π t. Except that a is measured in meters and t is measured in seconds. A conversion factor is needed, and one is readily apparent – the speed of light. When we look at the FLRW metric, we see that this has already been included in the equation. The second point that has to be taken care of is that in the metric the values of a, t and c are squared, so the shape relationship will also need be squared. k then is equal to (2π)2 or equal to 4π2 .
So, for a given value of average density ρ0 , we can now use the Friedmann equation to calculate a0 (the total circumference of the universe today). And, from that we can calculate:
(a0 ) 3 = the total volume of the universe today (V0 )
ȧ0 = the present expansion rate of the universe
and t 0 = the present apparent age of the universe
Estimating Total Density
The definition for density is a little less straight-forward. Generally, ρ is defined as the sum of the density of dark matter, baryonic energy and baryonic matter. (Dark energy is taken into account by the lambda value)
The estimate of the total density of mass and energy of the universe consists of three things:
ρ (baryonic or normal matter) = approximately 1.66 x 10-24 g/m3
ρ (baryonic or normal energy) = p/c2 = approximately 3.87 x 10-24 g/m3
ρ (dark matter) = 17.3 times normal matter = 2.87 x 10-23 g/m3
However, in the modified Friedmann equation, the value for ρ is not the sum of these three values:
ρ = ρ(baryonic energy) + ρ(dark matter) + ρ(baryonic matter)
but is:
ρ = ρ(baryonic energy) – ρ(dark matter) – ρ(baryonic matter)
Thought Experiment to Test This Observation
Now to give an example of this model we will examine a small volume of space that is not under the influence of any local mass or energy. So we can locate it out in the middle of one of the great voids between galactic clusters.
We will define a distance “a” to equal 1.00 meters, and our space will be:
2a x 2a x 2a = 2.00m x 2.00m x 2.00m = 8.00m3 in volume
We will further define two points in the center of our volume, B and C that are 1a distance apart. Our volume therefore looks like this:
The values for a are for a given time, since we know that the universe is expanding. If the value of a is 1.00 meters right now, today; then we can calculate what it will be tomorrow (exactly 24 hours from now).
So, let’s do that : a(tomorrow) = a(now) + á/a x a(now) x Δtime
Since there is no mass or energy effect: á/a = (Λc2 / 3)1/2 = 2.01 x 10 – 18 s – 1
Δtime = 3600 x 24 = 86,400 seconds
a(now) = 1.00 meters
So: a(tomorrow) = 1.00 m + 1.74 x 10-13 m = 1.00000000000174 m
This makes sense.
B and C are slightly further apart due to the effects of dark energy.
But this does not tell us about the effects of gravity, since there is no mass in our experiment. Now, let us add some mass.
At each point B and Point C we add a 1.00 g pebbles, and again leave the volume for one day.
Now, the average density becomes, ρ = 2.00 g / 8.00 m3 = 0.25 g/m3
and: ( á/a )2 = Λc2 / 3 + 8/3 x 3.1415 x 6.674 x 10 –8 m3 /g-s2 x 0.25 g/m3
= 0.000000140 s-2
or: á/a = 0.00037 s-1
So: a(tomorrow) = 1.00 m + (0.00037 s-1) x (1.00 m) x 86,400 s
So: a(tomorrow) = 1.00 m + 32.3m = 33.3 meters
This means that in one day our two pebbles repulsed each other by 30 times their original distance due to gravity.
There must be something wrong with our equation if it results in gravity being a repulsive force.
It can be argued that the Friedmann equation originally assumed uniform matter distribution – and our experiment included two point masses. However, even if we assume the 2 grams of mass was evenly distributed as dust throughout our 8 meter volume, the results would have been the same. The distance between the two points B and C would still have increased, and the mass of dust would have dispersed outward, and not condensed inward like we know occurs.
Therefore, although energy (both baryonic and dark) causes the distance between two points in space to expand with time; matter (both baryonic and dark) causes the distance to decrease. (I.e. gravity is an attractive force for matter.) So, our equation for density is:
ρ = ρ(baryonic energy) – ρ(dark matter) – ρ(baryonic matter)
We will run our calculation both with the inclusion of the dark matter density and without it. This is due to our incomplete knowledge of what dark matter is, and whether it effects space the same way baryonic matter does.
ρ = 3.87 x 10-24 g/m3 – 2.87 x 10-23 g/m3 – 1.66 x 10-24 g/m3 = – 2.65 x 10-23 g/m3
or, without dark matter having a gravitational effect on space:
ρ = 3.87 x 10-24 g/m3 – 1.66 x 10-24 g/m3 = 2.21 x 10-24 g/m3
So, the modified Friedmann Equation becomes:
4.93 x 10 -36 sec -2 = (5.59 x 10-13 m3/g-s2 )( -2.65 x 10-23 g/m3 ) + (3.55 x 1018 m2/s2) / a2
And therefore: a = 4.24 x 1026 m = 47.3 billion light years
Or:
4.84 x 10 -36 sec -2 = (5.59 x 10-13 m3/g-s2 )( 2.21 x 10-24 g/m3 ) + (3.55 x 1018 m2/s2) / a2
And therefore: a = 9.81 x 1026 m = 109 billion light years
Since a = 2πc · t, we can calculate the present apparent age of our hyperspheric universe, with and without the influence of dark matter as:
with dark matter: t(present) = 7.5 Billion Years
without dark matter: t(present) = 17.4 Billion Years
We know that the age of the universe is greater than 8 billion years. (Astronomers have detected galaxies and stars much older than that.) Therefore we have to assume that dark matter (whatever it may turn out to be) cannot have a gravitational influence on space-time. This does not preclude the existence of dark matter; it still can influence the universe and baryonic matter/energy in all the other ways we have detected. It just cannot have a gravitational influence.
Our calculations also show that a hyperspherical universe is slightly older than a flat universe, 17.5 BY vs. 13.6 BY.
Note: If we assume the present Hubble Value is 74 km/s/MPc instead of 68km/s/MPc,
the values calculate to:
with dark matter: a = 46.2 BLYr, t = 7.35 Billion Years
without dark matter: a = 99.0 BLYr, t = 15.7 Billion Years
Slightly smaller values than for the smaller value of H.
With the present value for a (which equals the total distance in the x, y, and z directions) established, we can calculate estimates for other characteristics of the universe at the present time.
Total Volume of the universe = a3 = 1.3 x 10 33 cubic lightyears (9.8 x 10 80 m3 )
Total baryonic mass of the universe = 1.6 x 10 54 Kg
Total mass equivalent of baryonic energy in the universe = 3.8 x 10 54 Kg
The expansion rate of the universe: ( ȧ ) = H · a
= 2.2 x 10 9 m/s = 7.34 c for 68km/s/MPc, or
= 2.1 x 10 9 m/s = 7.11 c for 74km/s/MPc