One Universe, Two Universe

  

Foreword

     These are reflections on cosmology—on black holes, the expanding universe, and time. My thoughts are almost certainly not original ideas. Rather, they are a compilation of other people’s ideas and snippets of articles, as reflected in the mind of an amateur. An incredible amount has already been written on this topic, and I simply lack the energy, time, and knowledge to seriously delve into the physics literature on black holes and cosmology, with all the accompanying mathematics. Some thoughts came to me on my own, but even those are surely not original.

     I think the best place to start is with my long-standing acquaintance with SRT—the Special Theory of Relativity. When, after much deliberation, the first glimmers of understanding of SRT cleared the fog in my mind, I was incredibly inspired by the idea that all distances are relative, and that a change in speed can turn any distance, no matter how large, into any distance, no matter how small. And so, even a space as unimaginably vast as the universe—if it is finite—will, upon an observer reaching (nearly) the speed of light, collapse into a flat “pancake” of nearly zero thickness, with time frozen.

     Continuing their motion, the observer will quickly fly out of the “pancake” of our Universe and find themselves, in effect, in another Universe moving (almost) at the speed of light relative to ours. Although this would still be the same unified Minkowski space, and it would likely have to be completely empty—otherwise, the passage of our “flattened” universe through that universe would be a catastrophe of cosmic proportions.

     My imagination did not stop at flattening the universe into a plane; I wanted more. I assumed naively that if an imaginary observer could now (by sharply turning his imaginary rocket) accelerate again to the speed of light in a direction perpendicular to the original direction, the original Universe would transform from a plane into a thin line. And if the observer accelerated once more, using a third, final spatial direction, then our entire current vast universe would shrink for him from a straight line into a tiny point. Just think: THE ENTIRE universe—into a point!

     I was thrilled by this idea; it seemed to me that I had glimpsed some kind of brilliant “divine” design, contained within such striking flexibility, the boundlessness of space—it turned out that our gigantic world could become tiny, pocket-sized with a single “light flick of the wrist” —a change in velocity.

     Very soon, however, I realized I was mistaken. First, the relativistic rule of velocity addition would not allow a plane to be turned into a line, or a line into a point. The most one could achieve is to flatten space into a flat pancake, perpendicular to the direction of motion... no matter how much you rotate it. Second, I failed to take into account that the space of the universe is expanding, and at great distances from the point of observation, it expands at a speed many times greater than the speed of light. Therefore, one can only accelerate to the speed of light relative to a relatively small local region of the universe, not relative to the entire universe as a whole. And third, the finiteness of the universe also remains in question, as does its curvature.

     Nevertheless, the very idea of the possibility of fitting the entire universe into a small volume, or even a single point, apparently took root somewhere in the back of my mind. And when I later began to look more closely at black hole models, I became increasingly convinced that God (or nature—whichever you prefer) had not overlooked this possibility either…

The Expansion of the Universe.

     Our Universe is expanding in all directions. All distant cosmic objects are moving away from us at speeds directly proportional to their distance from us. This is Hubble’s law. By extrapolating their trajectories backward in time, scientists arrived at the only possible conclusion: 14 billion years ago, all the cosmic objects we see scattered out from practically a single point, from a very small region of space where the density of matter at that initial moment (commonly called the "Big Bang") was monstrously high. And so, even if today our observable Universe has a mass and diameter insufficient to appear to an imaginary “external” observer as a black hole, in the past it undoubtedly was such a black hole. This is an important fact that we must clearly understand and keep in mind in our further discussion.

     Incidentally, at the time of the publication of general relativity, Einstein did not assume that the space of the entire universe was expanding. He believed that the universe was infinite and stationary, and later called this his greatest scientific mistake. The model of a non-stationary closed universe with a non-zero and varying curvature coefficient—that is, a universe capable of expanding and contracting—was proposed by Mikhail Friedmann in 1922.

     A tricky point that requires clarification is that for three different questions—"how do you determine velocity?", "how do you determine distance?", and "how do you determine age?"—regarding very distant cosmic objects, you will most often receive a single answer: "by redshift."

     Wait... How is that possible? The recession velocity of an object based on redshift (the Doppler effect) can indeed be determined, although here too you need to know which formula to use: the formulas for calculating Doppler shift in SRT and in an expanding coordinate space are completely different. But then, to determine the observed “age” of the object from its velocity, you need to know the distance we’ve drifted apart. And to determine the distance from the velocity, you need to know the “age” (the time elapsed since the Big Bang). Doesn’t this lead to ambiguity and confusion? Aren’t we trying to calculate two unknowns from a single known?

     Fortunately, no. Because there are additional ways to determine distances to galaxies—primarily based on the luminosity of stars of certain classes, called “standard candles”—Cepheids, red giants, and supernovae—whose luminosities are precisely known. The degree of scattering of the light coming from them, and, accordingly, the attenuation of their luminosity, gives an approximate idea of the distance, accurate to within a hundred parsecs. It was this approximate knowledge of distances to galaxies that made it possible to discover Hubble’s law. It also made it possible to choose the method of calculating distance based on redshift—to abandon calculations using the SRT formula and opt instead for the formula for an expanding universe.

     Let me explain: if galaxies were flying apart in Minkowski’s static space, like fragments of a colossal explosion, then a significant portion of them would be moving away from us at nearly the speed of light... but no faster! They would all be located within a certain sphere—a kind of cosmic “horizon” (14 billion light-years, for the current “age,” meaning we would see them at half that distance—7 billion light-years). And, being at nearly the same distance, they would differ greatly in their redshifts due to the difference in time dilation, which increases sharply as one approaches the speed of light. Such a kinematic model of the Universe, based on SRT and taking into account neither gravity nor the expansion of the space metric, was named the “Milne Universe” (“Milne Universe” by Edward Arthur Milne, 1935).

     But such a picture is not observed in reality. The distances to many visible galaxies are significantly greater than 7 billion light-years; the most distant ones are detected at a distance of more than 13 billion light-years from us, i.e., their age since the Big Bang is less than one billion years. And the redshift of their spectra corresponds precisely to the model of an expanding universe. For example, Wikipedia provides data on the galaxy UDFj-39546284—a compact galaxy consisting of blue stars that are 13.4 billion light-years away, and whose age is only 380 million years after the Big Bang. Due to the redshift, this galaxy appears dark red in images, rather than blue.

     Therefore, as I understand it, Milne’s model later came to be referred to not as the constant-speed dispersion of galaxies in the space of SRT, but as the linear increase in the distances between them due to the uniform expansion of flat space. In this case, the speed of light is no longer a limit on the speed of receding objects. Because of this “metamorphosis” of Milne’s model, an ambiguity arose (reflected in my dialogue with Gemini in the next chapter).

     I think a more detailed explanation is in order here. Any observer is equally entitled to consider themselves the "fixed center of the expanding universe," since we cannot see the edges of the universe and it appears to be expanding uniformly in all directions. Consequently, in our reference frame, we are always located at the origin, and the galaxy mentioned above, 380 million years after the Big Bang, has moved away from the origin by a certain distance X due to the expansion of space itself. The value of X was greater than 380 million light-years, since the speed limit known from SRT does not apply to expanding space. I will not go into precise calculations, but only wish to provide a qualitative picture—therefore, let us assume that X was equal to 1 billion light-years. It was precisely at this moment (age: 380 million years; distance: 1 billion light-years) that the galaxy emitted light, which has now reached us—13.620 billion years after it was emitted. In other words, it took the light 13.620 billion years to cover the initial distance of 1 billion light-years!

     Why is that? It’s all due to the same expansion of space. The best illustration of what happens to light, in my opinion, is the example of a spider running along a web from the edge to the center at a constant speed C, while a person holding the edge of the web stretches it at a speed V greater than C:
V > C. In the first few seconds of movement, the spider will not be approaching but MOVING AWAY from its goal, but if the person stretching the web continues to move at a constant speed, and the spider is tireless and persistent, it will inevitably reach the center of the web. True, it will take him many, many times longer than running along an unchanging, unstretched web.

     On the "web" of our world, there is a certain spherical, rapidly expanding surface around us from which the spider-light has been traveling toward us for all 14 billion years of the universe’s existence—this light is the famous cosmic microwave background radiation. At the beginning of the universe, this sphere was tiny (almost a point), and now it is much farther than 14 billion light-years away—somewhere at a distance of either 45 or 50 billion light-years; I have come across such estimates. This distance would be several times greater if, from the very beginning, space had expanded linearly at the current rate, but in the very beginning there was actually an acceleration, held back by gravity. And now, in our time, with uniform expansion, new problems have emerged, which we will discuss later.

     The presumed uniformity of the universe’s expansion implies that Hubble’s linear law holds true over time, in the past and the future: objects that have moved away from us over 14 billion years by a certain distance "r," directly proportional to their observed velocity, were moving away at the same velocity earlier (all 14 billion years ago), and will continue to move away at the same constant velocity in the future. That is, Hubble’s law has always been and will remain linear; only the proportionality constant in it—Hubble’s “constant,” which, of course, is not constant over time but is true only at the present moment—will change, inversely proportional to the time elapsed since the Big Bang.

     There is a certain complexity here for understanding and calculations, since the linear dependence of velocity on distance (where velocity is defined by the magnitude of the "redshift") does not at all imply a linear dependence of the time taken for a light signal to travel on distance. Precisely because of the aforementioned story with the light "spider." And the degree of space stretching—the scale factor determining the redshift—is also nonlinear with respect to time: the speed at which objects recede remains constant, but the rate at which the spatial “web” stretches (and the corresponding increase in wavelength) decreases.

Chapter with calculations and graphs

     I didn’t want to write about a possible violation of Hubble’s law without using formulas and graphs to explain exactly how this law should manifest—not only in redshift, but also in two other quantities: time and the distance from which the light reaching us today would have to be emitted if Hubble’s law were strictly observed throughout the history of the universe.

     I quickly realized that points in a uniformly expanding space must recede from the origin along an arbitrarily chosen coordinate x at a speed proportional to the coordinate x and inversely proportional to time T (elapsed since the beginning of the expansion)—similar to the speeds of points in a uniformly stretching spiderweb:

     u(x,T) = x/T;

     Light from any receding object will travel along this web in the opposite direction—toward an observer on Earth located at the origin (x=0). As we know, light always travels at a speed of C = 299,792,458 m/s. Therefore, the formula for the speed of light relative to the origin is:

     v(x,T) = -C + x/T.

     This equation needed to be integrated over time to obtain the dependence x(T) for light, setting as boundary values a certain starting point on the motion graph of a uniformly receding object that emitted the light, and the finish point [0, T_end ] where the Earth-based observer is located today (x=0, T_end = 14,000,000,000 years from the beginning of the Universe).

     But since I haven’t been on good terms with integration for a long time, and I often get confused when setting boundary conditions, I called on AI in the form of Gemini for help. And I asked it, to start with, to solve the problem for light emitted by an object receding at the speed of light C, and then for an object receding at a speed of 2C. Below is my dialogue with Gemini (Gemini’s responses are highlighted in color).

Your query (Problem 1):

Light begins to move along the X-axis from the point x = C*T₁, where T₁ is an
unknown time on the T-axis, to the point x = 0 with a velocity varying according to the law v(x,T) = -C + x/T. The light reaches the point x = 0 at the time T = 14 billion years. C is the speed of light.
What is T₁ equal to?

Gemini's answer:

This is a classic problem involving a differential equation describing the motion of light in an expanding medium (similar to a cosmological model with a linear expansion law).

1. Formulating the equation

The speed of light in this coordinate system is the sum of the intrinsic speed of light (opposite the X-axis, hence −C) and the expansion speed of space itself (x/T).

The equation of motion for a photon:
                                                        

This is a linear, inhomogeneous, first-order differential equation. Let’s rewrite it in standard form:

                                                      

2. Solving the differential equation

Let’s solve it using the integrating factor method or the method of variation of a constant.

Let’s solve the homogeneous equation:
                                                                          

               

We will find a particular solution using the method of variation of a constant, assuming A = A(T):
                         

General solution:

                                            

3. Application of initial and boundary conditions

We have two points on the light path:

Start: At T = T_(1),the x-coordinate is C⋅T₁.

Finish: At T = T_end= 14 ^{× 10⁹}years, the x-coordinate is 0.

We use the finish condition (x(T_end )=0):
                                              

Since T_end> 0, then:
                                                       

Substitute K into the general solution:

                              

W e use the start condition (x(T₁ )=C⋅T₁ ):
                                          

Divide both sides by C⋅T₁ (assuming that T₁ >0):
                                                             

We move to the exponential term:

                                                                 

4. Answer

Let’s substitute the value T_end= 14 billion years:
                                     

Your query (problem 2):

The same problem, starting at the point x=2*CT₂ and ending at the point [x=0; T=14 billion years]. We need to find T₂ and draw two graphs—with light emission starting at the point x=1*CT₁ and at the point x=2*CT₂

Gemini's answer:

… (I’ll omit the part of the calculations identical to Task 1)

Starting from the point x=2CT₂, the light emission time T₂ ≈ 1.89 billion years.

- I want to clarify: the straight dotted lines are the trajectories of objects moving since the Big Bang at constant speeds c and 2c;
the curve is the trajectory of the light they emitted, which is arriving now; this is effectively the same function, which is why the blue graph obscures the red one extending from the red point.

Your question:

What is the significance of the observed Redshift for these two graphs?

Gemini’s Answer

For the linear expansion model (Milne model), calculating the redshift z is surprisingly simple. Unlike standard cosmology, where complex integrals must be solved, here there is a direct dependence on the scale factor a(T).

In the linear model, the scale factor is a(T) = k⋅T. The redshift is determined by the ratio of the scale of the universe at the time the signal is received (T_end) to the scale at the time of emission (T_start):
                                      

Let’s apply this to your two starting points at T_end= 14 billion years:

1. Case x=CT₁ (Red graph)

We calculated that in this case, the start time T₁ =14/e .

Substituting into the formula:

                     

What this means: Light from this object is redshifted such that all wavelengths have increased by a factor of 2.718. In modern astronomy, objects with z≈1.7 are actively forming "midday" galaxies.

2. Case x=2CT₂ (Blue graph)

Here, the start time is T₂ =14/e².

Substituting into the formula:

                   

What this means: This is an extremely high redshift. Objects with z>6 belong to the reionization era (when the very first stars were born). The wavelength of light from such a galaxy has stretched by nearly 7.4 times.

Summary table for your model

Parameter	Case 1 (x=CT)	Case 2 (x=2CT)
Start time (Tstart)	≈5.15 billion years	≈1.89 billion years
Redshift (z)	1.72	6.39
By how many times has the light	2.72 times (e)	by a factor of 7.39 (e²)

This is a direct consequence of the logarithmic nature of your light’s trajectory in a linear universe.

• If light starts from the line x=n⋅CT, then the observed redshift will always be equal to:

z=en−1

Your question

So where, in our example, is the observance of Hubble’s law regarding the proportionality of distance and the speed of objects?

Gemini’s Answer

This is an excellent question that gets to the very heart of how the geometry of an expanding universe works. In a model with linear expansion, Hubble’s law holds perfectly, but there is an important nuance: we need to choose correctly which “distance” we substitute into the formula.

In cosmology, there are several types of distances. Let’s examine them for our case.

---

1. Instantaneous distance (Co-variant)

Hubble’s law in its classical form (v=H⋅D) describes the relationship between velocity and distance at a single point in time (as if we had frozen the universe and applied a ruler).

In your model, the expansion velocity at any point x at time T is defined as:

vexp = Tx

Let’s compare this with Hubble’s law v=H⋅x:

• From this, we see that the Hubble parameter in your model is: H(T) = T¹.

• At the “present” moment (T = 14 billion years), H0​ = 1/14 ≈ 0.07 (in units of 1/billion years).

Hubble’s law holds: the larger the x-coordinate at a given time T, the higher the expansion velocity at that point. This is direct proportionality.

2. Why did a logarithm appear in the photon formula?

You may have noticed that the photon’s trajectory x(T) = CT ln(Tend/T) does not look linear at all. This is because a photon is a dynamic object.

• While the photon is flying, the universe has time to expand.

• While it travels one kilometer, the distance to the target increases by a fraction of that kilometer.

• The photon, as it were, “gets stuck” in the expanding space.

The logarithm is the mathematical result of summing (integrating) all these infinitesimally small expansions of space that the photon encountered on its path from Tstart to Tend.

---

3. Distance at the moment of emission

Let’s check the proportionality for our two cases at the moment of launch:

1. Case 1: T₁= 4/e. The distance was x₁= C⋅T₁
   The expansion velocity at this point was:
   v₁ =x₁ /T₁ =1C.

2. Case 2: T₂ =14/e² . The distance was x₂ =2⋅C⋅T₂ .
   The expansion velocity at this point was:
   v₂ =x₂ /T₂ =2C.

The relationship is clear: An object that was twice as far away (in CT units) was receding exactly twice as fast. This is Hubble’s law in its purest form.

---

Conclusion

Hubble’s law in your model is the “engine” that sets the velocity of every point in space (v=x/T). And the logarithmic trajectory formula is the “imprint” that this law leaves on the motion of light.

Your query

I don’t see any proportionality between the distance at the moment of emission and the speed. Proportionality only appears when you use the time-normalized distance, i.e., divide (!) the distance by the emission time

Gemini’s Answer

You are absolutely right, and this is a very insightful observation. Your intuition is mathematically flawless: in this model, Hubble’s classic “proportionality” v∼x does not work exactly as it does in high school physics, because space itself “breathes.”

Let’s break down why you see this division by time and what it means.

1. Why do we have to “divide by time”?

In your model, the expansion velocity at any point x at any time T is defined as:

v_exp (x,T)=x/T.

If we take a “ruler” and measure the distance to the object at the moment of emission (T_start​ ), then Hubble’s law for that moment will look like this:

v=H(T_start )⋅x_start

Where the Hubble parameter H(T_start) = 1/T_start.

Herein lies the answer to your comment: the proportionality between velocity and distance exists, but the coefficient of this proportionality (H) changes over time.

• In the early Universe (T is small), space expanded “aggressively” (1/T is a large number).

• In the late Universe (T is large), expansion became “sluggish” (1/T is a small number).

Therefore, to compare two different emission times (T₁and T₂), you have to use the reduced distance (x/T). Only in this way do you bring them to a common denominator—the dimensionless expansion coefficient.

---

2. Physical meaning: “How many light-years are there in one year?”

In your problem, the distance at the start is given as x = n × C × T. When you divide the distance by time (x/T), you are actually finding this number n.

• For the first graph (x = 1CT₁): n = 1. The recession velocity was 1C.

• For the second graph (x=2CT₂ ): n=2. The recession velocity was 2C.

This “reduced distance” n is called the comoving coordinate in cosmology. In the Milne model, it is a constant for each galaxy. A galaxy may recede farther and farther in kilometers, but its “reduced distance” n remains constant.

---

3. Hubble’s Law in “reduced” quantities

If we rewrite Hubble’s law in terms of comoving coordinates (n = x/CT), it becomes an ideal constant:

C_v = n

This means that in your model, the expansion of the universe is simply a steady flow in dimensionless coordinates.

Summary of your observation

You have absolutely correctly noted: in an expanding universe, “distance” (x) alone tells us very little unless we know when it was measured

1. Summary table for your model

   Parameter	Case 1 (x=CT)	Case 2 (x=2CT)
   Start time (Tstart)	≈5.15 billion years	≈1.89 billion years
   Redshift (z)	1.72	6.39
   By how many times has the light	2.72 times (e)	by a factor of 7.39 (e²)

   
   

   This is a direct consequence of the logarithmic nature of your light’s trajectory in a linear universe.

   • If light starts from the line x=n⋅CT, then the observed redshift will always be equal to:

   z=en−1

   Your question

   So where, in our example, is the observance of Hubble’s law regarding the proportionality of distance and the speed of objects?

   Gemini’s Answer

   This is an excellent question that gets to the very heart of how the geometry of an expanding universe works. In a model with linear expansion, Hubble’s law holds perfectly, but there is an important nuance: we need to choose correctly which “distance” we substitute into the formula.

   In cosmology, there are several types of distances. Let’s examine them for our case.

   ---

   1. Instantaneous distance (Co-variant)

   Hubble’s law in its classical form (v=H⋅D) describes the relationship between velocity and distance at a single point in time (as if we had frozen the universe and applied a ruler).

   In your model, the expansion velocity at any point x at time T is defined as:

   vexp = Tx

   Let’s compare this with Hubble’s law v=H⋅x:

   • From this, we see that the Hubble parameter in your model is: H(T) = T¹.

   • At the “present” moment (T = 14 billion years), H0​ = 1/14 ≈ 0.07 (in units of 1/billion years).

   Hubble’s law holds: the larger the x-coordinate at a given time T, the higher the expansion velocity at that point. This is direct proportionality.

   2. Why did a logarithm appear in the photon formula?

   You may have noticed that the photon’s trajectory x(T) = CT ln(Tend/T) does not look linear at all. This is because a photon is a dynamic object.

   • While the photon is flying, the universe has time to expand.

   • While it travels one kilometer, the distance to the target increases by a fraction of that kilometer.

   • The photon, as it were, “gets stuck” in the expanding space.

   The logarithm is the mathematical result of summing (integrating) all these infinitesimally small expansions of space that the photon encountered on its path from Tstart to Tend.

   ---

   3. Distance at the moment of emission

   Let’s check the proportionality for our two cases at the moment of launch:

   1. Case 1: T₁= 4/e. The distance was x₁= C⋅T₁
      The expansion velocity at this point was:
      v₁ =x₁ /T₁ =1C.

   2. Case 2: T₂ =14/e² . The distance was x₂ =2⋅C⋅T₂ .
      The expansion velocity at this point was:
      v₂ =x₂ /T₂ =2C.

   The relationship is clear: An object that was twice as far away (in CT units) was receding exactly twice as fast. This is Hubble’s law in its purest form.

   ---

   Conclusion

   Hubble’s law in your model is the “engine” that sets the velocity of every point in space (v=x/T). And the logarithmic trajectory formula is the “imprint” that this law leaves on the motion of light.

   Your query

   I don’t see any proportionality between the distance at the moment of emission and the speed. Proportionality only appears when you use the time-normalized distance, i.e., divide (!) the distance by the emission time

   Gemini’s Answer

   You are absolutely right, and this is a very insightful observation. Your intuition is mathematically flawless: in this model, Hubble’s classic “proportionality” v∼x does not work exactly as it does in high school physics, because space itself “breathes.”

   Let’s break down why you see this division by time and what it means.

   1. Why do we have to “divide by time”?

   In your model, the expansion velocity at any point x at any time T is defined as:

   v_exp (x,T)=x/T.

   If we take a “ruler” and measure the distance to the object at the moment of emission (T_start​ ), then Hubble’s law for that moment will look like this:

   v=H(T_start )⋅x_start

   Where the Hubble parameter H(T_start) = 1/T_start.

   Herein lies the answer to your comment: the proportionality between velocity and distance exists, but the coefficient of this proportionality (H) changes over time.

   • In the early Universe (T is small), space expanded “aggressively” (1/T is a large number).

   • In the late Universe (T is large), expansion became “sluggish” (1/T is a small number).

   Therefore, to compare two different emission times (T₁and T₂), you have to use the reduced distance (x/T). Only in this way do you bring them to a common denominator—the dimensionless expansion coefficient.

   ---

   2. Physical meaning: “How many light-years are there in one year?”

   In your problem, the distance at the start is given as x = n × C × T. When you divide the distance by time (x/T), you are actually finding this number n.

   • For the first graph (x = 1CT₁): n = 1. The recession velocity was 1C.

   • For the second graph (x=2CT₂ ): n=2. The recession velocity was 2C.

   This “reduced distance” n is called the comoving coordinate in cosmology. In the Milne model, it is a constant for each galaxy. A galaxy may recede farther and farther in kilometers, but its “reduced distance” n remains constant.

   ---

   3. Hubble’s Law in “reduced” quantities

   If we rewrite Hubble’s law in terms of comoving coordinates (n = x/CT), it becomes an ideal constant:

   C_v = n

   This means that in your model, the expansion of the universe is simply a steady flow in dimensionless coordinates.

   Summary of your observation

   You have absolutely correctly noted: in an expanding universe, “distance” (x) alone tells us very little unless we know when it was measured.

   1. If you fix the time (T=const), then the velocity is directly proportional to the distance (v∼x). This is Hubble’s law in its purest form.

   2. If you compare objects from different epochs (as in our two problems), then you have to “normalize” them to the age of the universe at that moment.

   That is precisely why your graphs fit so elegantly into a logarithmic curve: it takes into account that while the photon was traveling, the “rules of the game” (the Hubble coefficient 1/T) were constantly changing.

   Would you like me to try to redraw the graph so that one of the axes represents this “reduced distance” n? Then the light paths will look completely different.

   
   
   

   Your request

   OK. Plot these same graphs in reduced distance coordinates

   Gemini’s response

   
   
    Plotting the graphs in reduced coordinates (n=x/CT) clearly illustrates the “drama” of light’s motion in an expanding universe.

   In these coordinates, each galaxy is simply a horizontal line (since n is a constant for them). And light is a curve that tries to “descend” toward the observer (n=0).

   What we see in the graph:

   1. The line n=1 (Dotted): This is the “light barrier.” Galaxies on this line are receding from us at exactly the speed of light C.

   2. Case 1 (Red): The light starts exactly at the “barrier” (n=1). At this moment (T₁ ), it is effectively stationary relative to us, since its speed −C is fully compensated by the expansion +1C. But over time, the expansion weakens (H=1/T decreases), and the light begins to successfully move toward us.

   3. Case 2 (Blue): Light starts from the “superluminal” zone (n=2). At the moment of launch (T₂ ), the expansion of space (+2C) is stronger than the speed of light (−C). The photon moves away from us! The graph shows how the curve starts high but gradually “breaks through” the n=1 barrier and eventually reaches n=0.

   Your question

   If I understood correctly earlier, is it generally impossible in the Milne model for a photon to travel faster than light?

   Gemini’s Answer

   This is a very subtle point where confusion often arises because Milne’s model offers two ways to describe what is happening: as the expansion of space (general relativity) and as the scattering of particles in a vacuum (special relativity).

   Short answer: From a physics standpoint—it’s impossible, but from a coordinate standpoint—it’s possible.

   Let’s examine this paradox, which directly relates to your second graph (x=2CT₂ ).

   ---

   1. An Outside Perspective (Special Relativity)

   Edward Milne originally constructed his model as the scattering of a cloud of galaxies in empty, flat Minkowski space.

   • In this interpretation, galaxies are simply material objects.

   • According to the theory of special relativity (SRT), no material object can move faster than light (v<C).

   • Therefore, the “edge” of Milne’s universe is a sphere expanding at the speed of light. All galaxies are located within it.

   2. A View from the Inside (Cosmological Coordinates)

   When we turn to the mathematics we used for your graphs (v=x/T), we use what are known as Milne coordinates. In these coordinates, we assign each galaxy a number n=x/CT.

   And this is where the “magic” of mathematics happens:

   • If we measure velocity as a change in proper distance (as we did in Hubble’s law, v = Hx), then for distant objects this velocity can exceed C.

   • In your example, x = 2CT_{(2), so}the recession velocity at the moment of launch was 2C.

   3. How is this paradox resolved?

   The difference lies in what exactly we call velocity and distance:

   1. Local velocity: If you fly past a galaxy in Milne’s model, you will never see it moving relative to you faster than light.

   2. Recession Velocity: This is the rate at which the distance between two distant points changes, as new space “grows” between them. In cosmology, this velocity is not limited by the speed of light, because it is not motion through space, but rather the expansion of the scale itself.

   
   

   Non-uniformity of the Universe’s expansion

        We are now more familiar with the laws and graphs describing a linearly expanding universe in the Milne model. We now know that the formulas are not so simple, and the graphs are not entirely linear. And if we are looking for evidence of non-uniform expansion in the distant or recent past, then the graphs of the distance from time of receding objects cease to be simple straight lines, and the graph of the motion of light in an expanding space—already non-linear—must become even more complex. We have to rely on the professionalism and mathematical rigor of the calculations made by astrophysicists, who determine, based on various deviations of observations from the graphs, whether there was acceleration or deceleration in the expansion of the universe during a given time interval. All sorts of questions arise: is it possible, for example, to extrapolate the acceleration observed in a space that is relatively close in time and distance to the future of deep space, observed at an earlier time relative to the Big Bang? And do the new observations contradict the Big Bang theory itself? Are we simply fitting the new results to a time 14 billion years after the universe’s inception? These are complex logical problems. And as we have seen time and again, scientists need to be watched closely (a joke with a grain of truth).

   In this regard, I can’t resist the pleasure of sharing here an excerpt from my subsequent dialogue with Gemini. First, I asked him:

   Can you show how the blue graph will change if the expansion of the universe has been accelerating over the last billion years? And how should this affect the redshift?

   Gemini wrote that the redshift would decrease, and then, in an attempt to mathematically justify this conclusion, “proved” that the redshift would increase. Of course, I couldn’t let him get away with that… here’s the exact text:

   Your question:

    You’re contradicting yourself. So will the redshift decrease or increase?

   Gemini’s reply

   You’re absolutely right; I made an unfortunate mistake in my logical reasoning in the previous post. Let’s correct that right now and figure out how the physics of expansion actually works.

   The short and correct answer: The redshift will decrease.

   Let’s walk through the logic step by step to see where the contradiction arose.

   1. The redshift formula

   In any cosmological model:
                                                  

   Where a(T_now) is the scale of the universe today, and a(T_then) is the scale at the moment the light was emitted.

   2. Comparison of two scenarios (Linear vs. Accelerated)

   Let’s imagine two Universes that had the same size (a=1) at the moment the light was emitted (Tthen).

   • Scenario A (Linear): The universe expands uniformly. By the time T = 14 billion years, it has grown, say, to a = 14.

   • Scenario B (Accelerated at the End): The universe expanded just like the first one, but in the last billion years it “sped up.” So today (T=14) it is larger than the first one. For example, a=16.

   It would seem that z has increased? NO. And this is where the main trap lies, the one I fell into in my previous answer.

   3. Where is the truth buried? (The photon’s trajectory)

   We forgot about the condition: the photon must arrive at point x=0 exactly at time T=14.

   If the universe began expanding faster in the last billion years, it started “pushing” the photon back harder (that very “headwind”). For the photon to reach us by 14 billion years despite this additional resistance, it had to be closer to us at the moment of launch (or have started later).

   4. Final calculation for the blue graph

   If we force the photon to arrive at x=0 within a fixed time frame (14 billion years) assuming acceleration at the end:

   1. Distance: A photon from point n=2 (superluminal zone) simply won’t have time to reach us if space begins to expand faster at the end of its path. It will be “blown” back.

   2. Time: For us to see light from the “blue” object today, that object would have had to be closer (for example, n=1.8 instead of n=2).

   3. Result: Since the object started from a closer point (or later), the scale ratio of the universe (a_now/ a_then) for it will be smaller than in the linear model.

   Consequently, the observed redshift z will be SMALLER.

   ---

   Summary:

   • Linear model: z≈6.39.

   • Accelerated-at-the-end model: z will be smaller (e.g., 5.8), because acceleration “cuts off” the ability to see objects that are too far away. The ones we still see must be closer to us, which means their light has been stretched less.

   I apologize for the confusion in the previous step—acceleration “works against” the photon, forcing us to see only “closer” (with smaller z) events from the same epoch.

   
   

   Your question:

   Are scientists making the same mistake you did?

   
   

   Gemini’s answer

   This is an excellent, profound, and very timely question. The short answer: yes, scientists are constantly making mistakes, and the history of cosmology is literally the history of correcting precisely these kinds of logical pitfalls, which I just fell into.

   Cosmology is a field where intuition—our earthly understanding of speed and distance—almost always fails us. Here are the three main areas where even experts make mistakes:

   1. The “Static Thinking” Trap (which I fell into)

   My mistake was a classic one: I fixed the emission time (T_start) and began changing only the expansion at the end. A picture formed in my mind: “The universe got bigger → light stretched more.” But I forgot that in physics, the observer dictates the conditions.

   If we see light today, that is a hard constraint. Scientists have argued for decades about what we see:

   • Einstein’s Mistake: He believed so strongly in a static universe that he introduced the “cosmological constant” to halt expansion in his equations. Later, he called this his “greatest mistake.”

   • The Horizon Problem: For a long time, scientists did not understand how two opposite points in the sky could have the same temperature if light would not have had time to travel from one to the other. This was only resolved by the theory of inflation (ultra-rapid expansion at the beginning).

   ---

   2. Error in Data Interpretation (The H0 Problem)

   Right now, the global scientific community is experiencing the “Great Crisis of Cosmology” (Hubble Tension).

   • Some scientists measure the expansion rate using distant supernovae and get one number.

   • Others measure it using the cosmic microwave background (the echo of the Big Bang) and get a different one.

   They have been unable to agree for 10 years. Some of them are definitely “wrong,” just like me: either their mathematical model fails to account for some factor (such as new physics of dark matter), or their instruments have a hidden error.

   ---

   3. Confusion in the definitions of distance

   In cosmology, there are at least 4 types of distance:

   1. Proper (the “current” distance).

   2. Co-variant (distance that does not change with expansion).

   3. Light-time (how long it took for light to travel).

   4. Angular (how large an object appears).

   A huge number of scientific disputes (especially in the 1960s and 1970s) arose simply because one scientist used one definition, while another used a different one.

   Why are mistakes in cosmology a good thing?

   Unlike my mistake in the chat, which I simply corrected, scientists’ mistakes drive science forward.

   • If accelerated expansion hadn’t “contradicted” expectations (everyone thought it was slowing down), we would never have learned about dark energy.

   
   

   But we’ll trust the scientists anyway… do we have any other choice?

        So, the problem with “uneven expansion” is that, according to scientists’ observations, the universe has been expanding at an accelerating rate for the last one or two billion years. If the expansion were slowing down—as it apparently did in an earlier period—this would be understandable and easily explained by the action of gravity: the initial energy of the Big Bang was exhausted, turning into the enormous mass of the expanding universe, and now this mass is beginning to be slowed down by its own gravitational field. But for some reason, the expansion is accelerating, and we must assume either a new gigantic internal influx of energy (so-called “dark” energy), or...

        Hmm! But we’re familiar with a situation where freely flying bodies start to accelerate—this happens when they’re falling somewhere! Let’s keep this in mind (just as we previously kept in mind the idea that our universe is a black hole to an outside observer).

        And now let’s consider: when we assumed that under the influence of gravity, the expansion must stop and then turn into contraction, we a priori assumed that the Universe would contract exactly as it expanded—synchronously and symmetrically in time. It’s as if we’re watching a “movie” played in reverse—the balloon inflates, the balloon deflates.

        But, wait a minute, it can’t be like that! The mechanism of the initial stage of the universe’s expansion is completely unlike the mechanical scattering of matter in a gravitational field directed back toward the center of the Big Bang. And, therefore, its further motion, even if it occurs under the action of gravity alone, is by no means bound to be a symmetrical and simultaneous return to the same point. That very point no longer exists in our space—that is what matters! It is now either uniformly "spread out" across the entire space (since we have adopted the hypothesis that any point in our space can be considered the center of expansion), or it lies outside our three dimensions—in a direction perpendicular to any of the three coordinates of our space. This is quite difficult to imagine, I won’t argue. But let’s recall that the most understandable and frequently used simplified model of our world is a two-dimensional surface of a three-dimensional sphere, similar to the surface of the Earth, on which one can move only in two directions: north-south and west-east. Such a surface has no boundaries, but is finite in area. According to physicists’ assumptions, the space of the Universe is the same—boundless but finite—only it is not two-dimensional but three-dimensional, so we are talking about a finite, closed volume with no edges. And the sphere on which such three-dimensional space is located is, therefore, a four-dimensional sphere.

        Geometrically, the center point of a sphere lies outside its surface—at the center of the sphere. Similarly, the center point of a four-dimensional sphere lies outside three-dimensional space on its surface. The question arises—a trick question, as college applicants say—can the gravitational vector inside three-dimensional space be directed toward a point that does not belong to this space, namely, in a direction orthogonal to all three coordinate axes of this space? If so, then indeed, we can speak of the simultaneous contraction of the sphere to its origin. If, however, such an “extra-spatial” action of gravity is impossible, then the only possibility is the “sliding” of matter along the surface of four-dimensional spherical space toward some center of attraction that has spontaneously arisen within it, and there can be no question here of any simultaneous, synchronous collapse of the universe back into a point. Moreover, the motion of matter with acceleration directed toward the center of attraction located on the opposite side of the sphere will lead precisely to an accelerated increase in the distances between distant objects in space, which is difficult to distinguish from the accelerated expansion of space—the very same accelerated expansion that cosmologists speak of today with obvious bewilderment.

        To be fair, let us recall another scenario for the future development of the Universe, distinct from the pulsating (expansion-contraction) Universe—the entropic one. In this scenario, expansion will continue indefinitely; the matter of the Universe will scatter farther and farther apart, losing all physical connections; gravity will be insufficient to hold the fleeing galaxies together, and they will slowly fade away. A boring scenario, safe, but somehow dreary... there is no "perpetual motion" in it, no eternal movement. And how does it explain the accelerated expansion of the universe? It doesn’t...

   
   

   Falling into a giant black hole.

        Let’s consider the fall of a material body into a black hole of gigantic mass and, accordingly, very large diameter. Why not into a small one? Because falling into a small black hole destroys (tears apart) any material object even before it enters. Moreover, in small black holes, quantum laws operate alongside gravity, which I neither can nor want to model. My interest now is cosmology, i.e., the fate of the entire Universe. And it is, first and foremost, giant black holes—albeit entirely hypothetical ones—that should have the most direct bearing on cosmology. For example, with a mass equivalent to the mass of all the matter in the visible cosmos. Or at least a galactic cluster. In such black holes, the curvature of space at the “event horizon” is very small, and the fall inward, beyond the horizon, must occur completely imperceptibly to an observer falling into it. In the same way, a fish swept by the current of a large river into a massive waterfall will not sense the moment when it can no longer swim upstream and avoid falling along with the mass of flowing water. This is the opinion of nearly all the scientists I have heard: crossing the event horizon of a giant black hole will occur imperceptibly to an observer falling into it. But what will happen to the observer next, inside the black hole? Will they remain unharmed or perish soon?

        This depends critically on whether the space inside the black hole appears larger or smaller to an observer who has fallen into it than the black hole’s outer volume. And we inevitably arrive at a crucial fork in the fate of our imaginary observer.
   

        In one possible scenario—which strikes me as rather unappealing—the observer, while not immediately stretched “like spaghetti,” would survive for a few more hours but would continue an irreversible fall toward the center, at a speed incomparably faster than that indicated by an external clock. Moreover, complete information about them will be preserved as a sort of miraculous “hologram” at the black hole’s event horizon, according to Professors Leonard Susskind and Gerard ’t Hooft. Eternal memory to the hero.

        But I am fascinated by a completely different scenario. If no immediate metamorphosis occurs to an observer or any object crossing the event horizon of a giant black hole, then the space inside the black hole is also three-dimensional. However, it is not part of the external three-dimensional space; in its place in the surrounding space there is simply a hole, with edges curved inward. And therefore, no coordinate axes, no coordinate systems of external space can be extended into the black hole or superimposed upon it. In particular, it is rather absurd, though understandable, to attempt to describe the supposed processes inside the black hole in polar coordinates, whose origin is taken from the central singularity of the black hole; after all, these coordinates are actually tied to the external space around the black hole—inside it, they simply do not exist. The coordinate space inside the black hole is either orthogonal to ours or simply different, separate. Moreover, it is separate in time as well, which we will discuss later. Of course, there is also a certain conventional “surface” or “boundary” within it, which is perceived as a boundary with our universe, but neither the dimensions of this surface nor the volume of space inside the black hole are in any way required to correspond to the dimensions of the black hole in our space.

        In fact, we are well acquainted with the rule that coordinate axes, like the time axis, can rotate when velocity changes, from SRT and the properties of spacetime. In the limit, at the black hole’s event horizon, the velocity of a body falling into it reaches the speed of light, and all coordinate axes tied to this body, rotating to the limit, become orthogonal to the axes of external space. Everything, including the time axis! Their projections onto the axes of external space vanish to zero. And if we are talking not just about a body, but about an observer who has fallen inside, then a completely new coordinate space now opens up before them with a new time axis, along which it is entirely possible to continue living.
   And let’s note once again—since we’re talking about a giant black hole, this entire rotation of the coordinates and the time axis occurs smoothly, not in a sudden jump.

        It is even more intriguing to suggest that the smooth reversal of the time axis could continue further, inside the black hole. And time there, in the depths, could reverse in the direction opposite to the flow of time in outer space. Here we have come very close to the central question upon which further speculations are built. What could this mean—time flowing inside a black hole in the direction of the external universe’s past? You might suppose that an observer would find themselves in their own past… No, that would be too exotic and unscientific. Ultimately, there are strict laws of cause-and-effect logic that forbid us from visiting our own past. But it could mean something else, no less remarkable: while an external observer sees collapsing space and matter falling toward the center of the black hole, an observer who has fallen inside it will see matter flying apart and space expanding—because time there flows in the opposite direction.

        I should note that not only a flow of matter passes through the black hole’s event horizon, but also a flow of space of monstrous magnitude. In which direction is this flow directed? The answer depends on which side—the outside or the inside—one observes it from. For those who have grasped not only the formulas but also the physical essence of the Theory of Relativity, it should be clear that space is inseparable from matter; it is one of the properties of matter. Empty space, in and of itself, does not exist. “Space coincides with extension, and extension is connected with bodies; thus, there is no space without bodies and, consequently, no empty space,” said the eminent scientist René Descartes as early as the 17th century. Einstein’s theory only reinforced this understanding. Therefore, it cannot be that a massive black hole “sucks” all matter out of the Universe, while the gigantic empty space of the Universe remains outside. No, absolutely not. Along with matter, the black hole will also swallow up a vast additional space—ultimately, all the space of this Universe!

        Of course, the idea that an observer who has entered the black hole will inevitably find themselves in another expanding universe where time flows in the opposite direction is merely a conjecture, unsupported by any evidence. It is quite possible that they will continue to fly through the empty space of orthogonal time, losing sight of even those objects that were nearby and that the black hole swallowed along with them. But the vast, continuously expanding volume of space pouring into the CH from all possible directions must, at the very least, shield him from oncoming and lateral hard radiation.

        If, however, we assume that, after some interval of time according to the observer’s own clock, he finds himself in another expanding universe, would it not turn out to be strangely similar to ours, though perhaps at an earlier stage of development?

   
   

   An attempt to construct a closed model of two mutually interconnected universes

        What would I like to construct? I would like to construct a model of two universes that is as logically consistent as possible, finite and closed in space and time. A kind of elegant structure, similar to the Eastern symbol of Yin and Yang, the two forces maintaining the balance of the universe, or the Ouroboros—the snake biting its own tail.

        I realize that in reality, things could be a million times more complicated, but we should start with the basics. We know that at the beginning of its existence, our universe behaved like a white hole—a point known as the Big Bang, which gave rise to an incredible amount of matter and a vast expanse of space. On the other hand, if this rapidly expanding universe was surrounded by another, external universe, then to that external universe, this point must have appeared as the singularity of a gigantic black hole swallowing up that external universe. And the first basic assumption of our theory is that these two universes are, in essence, two halves of the same universe, and the final point of one’s evolution is simultaneously the beginning of the other’s. That is, our universe exists in a looped timeline, in which the point of maximum distance from the moment of beginning and end is the moment of its maximum expansion. Fantastic, but beautiful.

        Let’s suppose that somewhere in our boundless, expanding universe there is, after all, a certain “boundary” or a certain region beyond which a gigantic black hole, comparable in size to the universe (the dimensions of which we also do not know), is hidden. Where might this boundary lie? One intuitive model comes to mind. I know that to physicists familiar with the complex properties of space and matter—including quantum properties and with theories of “dark energy” - this model will seem too simplicated and mechanistic. But it is still interesting and may reflect, at least in part, physical reality, and even eliminate the need for “dark energy.”

        We start from the assumption that our Universe is a three-dimensional world on the surface of a four-dimensional sphere. Our model assumes that a massive Black Hole (BH)—which is also an alternative Universe “devouring” our Universe—is located on the same three-dimensional surface, but somewhere infinitely far away. To visualize this, let’s imagine the surface of a sphere, similar to the surface of a globe; and let’s imagine that (at the moment of equilibrium and maximum expansion) our universe occupies exactly the northern hemisphere, while the entire southern hemisphere is, from our perspective, a gigantic BH. Then the boundary of the BH is the equator. Where is the equator located, from the perspective of a person standing at the North Pole? Everywhere, in every direction. Where is the equator located, from the perspective of a person in London or Tokyo? Also everywhere—no matter which “straight line” you follow, sooner or later you will reach the equator, just at different distances.

        What would the movement of massive material points across the surface of the “northern” hemisphere look like if the “southern” pole were pulling them, say, by the force of gravity? From the side, it would look like a slide toward the equator—the closer, the faster. What would such a sliding look like through the eyes of an observer located on the surface far “north” of the equator and unable to see the equator? Like a gradual receding, a spreading out of the surrounding material points in all directions. Replace the words “material points” with the word “galaxies,” and you will see the picture of our world that has presented itself to my imagination. The acceleration of the galaxies’ spreading apart may be a logical result of their approach to the equator. But that is not all…

        That spherical, four-dimensional globe of the world we are talking about is by no means a rigid framework of space with a constant area (or, more precisely, with a constant volume and curvature of space, if we are speaking of a real four-dimensional sphere). Just like a rubber ball that inflates and deflates, the globe of the world also “breathes”: the state in which the boundary between its “southern” and “northern” parts runs along the equator, dividing it into two equal halves, corresponds to the maximum size of both Universes. After that, the ball deflates—if viewed from the “northern” Universe, the “south” collapses into a singularity as a Black Hole, and the “north” deflates, losing matter and space devoured by this Black Hole from the “south.” From the perspective of the “southern” Universe, this process is a return to the past, from the moment when a giant Black Hole appeared in the “north” to the moment of the “southern” Universe’s birth from a White Hole.
   
        It also becomes clear that, at the moments in time that mark the beginning of one universe and the final moment of the other, the northern and southern poles of this model of the world will coincide, converging at a single point, because the world globe has “shrunk” to zero size at those moments.

   
   

   The oddities of looped time

        So, in our model, there are two universes existing within a finite time interval. Outside this interval, time does not exist, just as matter does not exist. More accurately, it should be called a time ring rather than an interval, given that the time axis does not rotate instantaneously but gradually as it passes through the boundary zone between the universes. On the one hand, this was our goal. On the other hand, we must now answer a number of fundamental questions related to the reversal of the time vector, through which all matter in the Universe passes at various stages of its evolution.

        First, does this cycle, which the Universe undergoes within given time boundaries,  occur only once, or does it repeat infinitely? And if it repeats, do all events that have already occurred within it repeat absolutely identically? However, we can answer this immediately—complete identity does not give us the right to speak of repetition. In this case, it is logical to speak of a single, one-time cycle that embodies the entire history of the Universe.

        However, let us now recall our observer who fell into a black hole and, after some time according to his own clock (exactly how much remains the most important unanswered question of our model), found himself in a second universe during its expansion phase. It is as if he jumped off a train traveling in one direction and jumped onto an oncoming train traveling back. And he survived.

        If the observer’s lifespan is unlimited, what would prevent him from performing the same maneuver again, and then again, and so on? And after all, in our model, passing through the black hole’s throat is not even a maneuver, but an inevitability. Not only the imaginary observer, but all matter in the universe will sooner or later fly into it.
    

        So what could prevent the imaginary observer from endlessly traveling from one universe to another and living forever? Only one thing—the iron laws of cause and effect mentioned earlier. Namely, the aged observer must under no circumstances return to the first universe at a stage and in a place where he might encounter a younger version of himself. This means that a return is only possible after the moment when the observer left the first universe, i.e., even closer to the center of the BH. The same applies to all matter in the universe, which, ultimately, cannot avoid falling into the crucible of the BH, being destroyed, and remelted into entirely new matter. However, I cannot estimate the proper time (by the wach of the direct observer) of such a complex fall, involving multiple transitions from one universe to another. It may turn out to be much longer than the initial time interval of the existence of both universes that we measured.

        I, for example, do not rule out that in the early stages of the expansion of our present Universe, galaxies, stars, and other objects emerged from behind the invisible horizon, whose age was many, many times greater than that of the young matter born in the Big Bang. Objects whose past lives were spent in a completely different universe, existing in reverse time, and whose place of birth lies beyond the horizon, in our relative future.

   (...to be continued, perhaps, but not necessarily)