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, Tend ] where the Earth-based observer is located today (x=0, Tend = 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).
