Science of Visibility and Invisibility
Comes from Synchrotrons and GPS
Section 1. Geometry of Visibility
Section 2. Synchrotron Radiation
Section 3. “Black Holes”
Section 4. Planetary Atom Model Is Back
More: Vanishing Stars and “Triple Transient” 80-Year-Long Mystery Solved
- In the first section we explain an uncanny implication of time dilation (slower time) on the angle at which we can see (or cannot see) a moving object. For any moving object at velocity v, time slows down a bit, by factor of square root of 1-v²/c², where c is the speed of light. This factor is basically 1 for small velocities v, meaning there is almost no time dilation, but nevertheless, even infinitesimal time dilation leads to very interesting outcomes / applications. If you are not interested in geometry, you can skip the first section and go directly to other sections, where applications are described.
- The second section explains light emission puzzles observed for accelerated particles in so-called synchrotrons, arguably the most expensive physical devices (only James Webb Space Telescope can compete with them in price tag). This section is important as hands-on experience: these devices and particles’ behavior there are very well studied. We provide a new interesting perspective and verify it in this section, to apply it in the later sections.
- The third section explains how “Black Holes” work: theoretical “Black Holes” do not exist, nor does “Event Horizon” (where time stops and where nothing, even light, can escape from). “Nearly Black Holes”, around which time slows down significantly (but does not stop), exist and we have images of these “Black Holes”. We explain why some light and particles cannot escape from there (that is why they are black), but still, some can escape through quasars jets.
- In the fourth section we rehabilitate the planetary model of the atom and explain why accelerated electrons do not radiate in atoms. A century ago, this conundrum led to the rejection of the planetary atomic model and to the creation of Quantum Mechanics. We solve that puzzle here.
All sections here are about observation (by emitted or reflected light). If you are not a math person, feel free to skip the first section.
Section 1. Geometry of Visibility
Let’s explore an object or a particle that was at point A and had velocity v at point A (we don’t care about its velocity before or after that). Let’s say, we have two observers: a stationary observer at point B, and another observer moving at constant velocity v: the second observer was at point A when the observed object/particle was there, and later he was at some point E at the moment, when the first observer, stationed in B, noticed/saw the object at the point A (after some time delay t for light to travel from A to B). For the moving observer (between A and E), according to Einstein, time slows down by a factor sqrt(1-v²/c²), where sqrt is square root, and in his frame of reference (where he thinks of himself as not moving) light has travelled not from A to B, but from E to B, and that took t×sqrt(1-v²/c²) time, which is less than t. Since we know all sides in the triangle ΔABE, we can find angle ∠EAB value θ:
θ = Arccos(v/c)
Only at the angle Arccos(v/c), particle A moving at speed v is visible to the first observer! We will discuss nuances of this seemingly strange restriction. Such line of sight can be geometrically represented as AC line below, where point C is constructed as the intersection between the perpendicular to the velocity vector v at its end and radial distance c (299,792,458 m), because for angle θ in the picture below, we have Cos(θ) = v/c.
Here we see the speed of light constant c residing on the hypotenuse of the triangle ΔAvc, and Ac there is the line of sight. Now, we can imagine all possible velocities v of the object A it to be seen at this line of sight. All such velocities end up on the circle with the diameter c, and this diameter lines up with the line of sight: check the right image above. Here summary about such circle and the line of sight:
Stationary (meaning v=0) object/particle A is always visible. Longest vector v=c is not achievable for objects/particles having mass, as, according to Einstein, for such objects/particles v<c.
But in real life we do not experience such limitations to visibility, why is that? It is because real objects are composed of atoms, wrapped in electrons, which always wobble. And their vibrations, unnoticeable to us, often, from time to time, compensate for the velocity of the object, making combined velocity of an atom vibration plus the object velocity equal to 0. That is because electron vibration speed is much higher than the speed of an object. For example, speed of electron knocked out of an atom in photoelectric effect is about 600 km/sec. Assuming that atoms/electrons vibrate at such speed, any object at a speed well below 600 km/sec is visible: from time to time for a blink of an eye, stationary at that moment atoms are visible (because the combined velocity+vibration=0 happens very often, thousands times per second).
Let’s explore the case when the vibration velocity Δv value is smaller than the object’s velocity v value. Then, in the picture below, the line of sight is not just fixed AB1 or AB2, but it varies from AB1 to AC1 and from AB2 to AC2 for a combined velocity v1=v+Δv instead of just v:
Thus, angle arccos(v/c), at which non-vibrating object is visible, changes to a range of angles arccos( sqrt(v²-Δv²)/c ) ± arcsin(Δv/v), at which vibrating object is visible. In reality, such area of visibility/observability is 3-dimensional — between two cones:
P.S. For velocities v close to speed of light we should keep in mind that adding velocities v±Δv operation should be adjusted to the Einstein’s velocity-addition operator v⨁Δv, so that sum does not exceed speed of light c. For example, for collinear velocities v and Δv, their real sum is v⨁Δv = (v+Δv)/[1+v×Δv/c²], which never exceeds c. Even for v=c, c⨁Δv stays c:
c ⨁Δv = (c+Δv) / [1+c×Δv/c²] = (c+Δv) / [1+Δv/c] = (c+Δv) / [(c+Δv)/c] = c.
P.P.S. Actually, range of sight is a bit wider than in the drawing above, wider than arccos( sqrt(v²-Δv²)/c ) ± arcsin(Δv/v). Maximum value for θ is arccos[ v/c — Δv/v — Δv²/(c×v) ] in this case:
Minimum value for θ is arccos[ v/c + Δv/v — Δv²/(c×v) ] in this case:
Actual range for visibility is: θ = arccos[ v/c ± Δv/v — Δv²/(c×v) ] :
Section 2. Synchrotron Radiation
Since Maxwell we know that light is an electromagnetic wave. Light emission, or electromagnetic wave emission, is caused by an accelerated charge: any charged particle acceleration changes / destabilizes the electromagnetic field around, and that kink / wave propagates through space at the light speed c. We all know radio transmitters, where electric current causes electrons’ fluctuations / accelerations / decelerations in an antenna. Accelerated electrons in the antenna emit radio waves. The only difference between (visible) light and radio waves is frequency (or wavelength): visible light has higher frequency (shorter waves) than radio waves. Synchrotrons, which we discuss here, emit even shorter waves than we can see. Synchrotrons are circular accelerators of electrons (or other charged particles):
Particles moving at a constant velocity do not radiate (do not emit light / electromagnetic waves). At synchrotron, electrons move at very high speed (in some synchrotrons, at 99.9% of the speed of light), but in a circular orbit, and change in their velocity direction is acceleration directed to the center of the circle. Interesting part is: basically, all light emission from these electrons is confined in a very narrow cone. In physics lectures such confinement is poorly explained: only why the beam is very energetic in this direction (by Lorentz transformation and by Doppler effect). But absence of radiation outside this cone (except some “noise”) is not explained.
We will use this case to sharpen our teeth, since it is a hands-on, well studied case. And in the next sections, we will reuse that experience for even more interesting applications.
In the previous section we saw that particle, moving at velocity v and wobbling at speed Δv, is visible only for an observer between two cones. Here I present the image from the previous section showing these cones of visibility and invisibility:
For a very high velocity v of electrons in synchrotrons (up to 99.9% of c),
θ = arccos[ v/c — Δv²/(c×v) ± Δv/v ] is very close to arccos(1), which is 0. That is why the cone of visibility is very narrow, and there is no radiation outside it. The inner cone of invisibility is not noticeable to an observer, because he receives radiation not just from a single cone, but from several close to each other cones: previous moment beam (shines a ring on the observation screen) and a following beam (shines another ring on the observation screen) fill in the hole in the ring from the beam in between:
Section 3. “Black Holes”
Theoretical “Black Holes” with “Event Horizon”, where time slows down to stop, and where nothing, even light, can escape from, do not exist because stopping time requires infinite energy. Images presented by astronomers are of “nearly Black Holes”, around which time is very slow, but does not stop. Actually, such images are doctored computer simulations — check disclosures / disclaimers below the images from https://www.npr.org/2023/04/13/1169469591/goodbye-fuzzy-donut-the-famous-first-black-hole-photo-gets-sharpened-up:
“Accretion disk” particles radiate; they lose energy and start falling / spiraling down towards the “Black Hole”. There is no “Event Horizon”, as we mentioned, just time slows down significantly around such massive/dense objects (that effect is known as gravitational time dilation): the closer to “Black Hole”, the slower time is. And as we discussed in the previous two sections, light emitted by a fast-moving particle is aligned along / close to the current velocity of the particle. Spiral down particle’s trajectory is nearly tangent to the sphere. Because of that, light beam emitted by a spiraling down particle is closer to a tangent rather than to a perpendicular to the concentric spheres around the “Black Hole”. Radiuses are perpendicular to the spheres, that means, light emitted by the particles spiraling around the “Black Hole” (blue line in the image below) has a significant angle (meaning it is not aligned) with the radiuses of spheres it crosses:
Time dilation causes light refraction, as we proved in the first chapter of Time Matters:
By Snell’s law, angle between the beam and the radius of spheres increases, until beam cannot cross timezone border any more:
At some level light is trapped at that level over the “Black Hole”. Meaning, at some level, the system, comprised of “Black Hole” and low-level particles, stops radiating outside, thus, stops losing energy, unless these particles are picked up by a strong electromagnetic field in quasars, start spiraling up along the poles, and are thrown out as jets (remember, there is no event horizon around “Black Holes” to prevent that):
Section 4. Planetary Atom Model Is Back
From school, we know Bohr’s planetary model of atom, where electrons rotate around the atomic nucleus (which consists of protons and neutrons):
This model has a major flaw. Electrons on a circular (or curved) orbit experience acceleration, and any accelerated charge, by the laws of electrodynamics, radiates electromagnetic waves. For example, it is how radio transmitters work: electrons in an antenna are accelerated back and forth along the antenna — and the accelerated electrons radiate radio waves. Such waves carry away some amount of energy. For the radio transmitter, additional energy is constantly provided from its batteries, but in the case of an electron in the atom, energy is just lost / carried away from the atom (the wavy red line in the image above depicts such energy loss through electromagnetic radiation). And an electron that is losing its energy will spiral down towards the nucleus and eventually fall into it. Because of that, the planetary model of the atom was rejected, and Quantum Mechanics appeared, with many postulates / speculations to justify why electrons in the atom do not fall into the nucleus. But still, to explain that, you need at least 3 Quantum Mechanics postulates/speculations, one of which is: a free electron (an electron outside an atom) changes from being a particle into a “cloud” (of probabilities) when captured by an atom. After that, it has “no real location” inside the atom and does not move around the nucleus. Only now physicists start devising tools that might localize electrons inside the atom.
BUT! Using knowledge that we have gained in the previous sections, we can reinstate the planetary model of atom and overcome the main objection to it, which is based on “accelerated electrons lose energy”. In this book, on many occasions, we explained that time dilates inside atomic nucleus, and around atom, and strong force is no different from gravity, which is just time dilation. The only difference is that rates/gradients of time dilation inside the atom are in large numbers. And it is true that at first, when a free electron starts spiraling around a free proton or around an ion (which is an atom missing some electrons) — it radiates. But what happens when it comes close enough to the nucleus, when time dilation should be accounted for? We already discussed a very similar case and solved it in the previous section: the system (composed of the nucleus and the electron) stops losing energy, because electromagnetic radiation is contained by time dilation (by Snell’s law). Non-radiating atom = stable system, and Bohr’s planetary model makes sense again.
Continued in The Main Observation Problem, Challenging Astronomy, Solved Today (Vanishing Stars and “Triple Transient” 80-Year-Long Mystery).
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