## Section 1. Geometry of visibility

Let’s examine an object or particle that was at a point **A** and it had speed **v** in a point **A** (we don’t care about its speed before or after). Let’s say we have two observers: a stationary observer at a point **B**and another observer moving at constant speed **v**: the second observer was at the point **A** when the observed object/particle was there, and later it was at some point in time **e** at the moment when the first observer located in **B,** noticed/saw the object at the point **A **(after some 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 the square root, and in his frame of reference (where he thinks of himself as stationary) the light has not traveled from **A** to **B**but from **e** to **B**and that’s what he took **t×sqrt(1-v²/c**²) time which is less than **T**. Since we know all the sides in the triangle **ΔABE**we can find an angle **∠EAB** value **i**:

**θ = Arccos (v/c)**

Only at an angle **Arkos (v/c)**particle **A** moving at speed **v** seen by the first observer! We will discuss the nuances of this seemingly strange limitation. Such a line of sight can be geometrically represented as **AC** line below where dot **° С** is constructed as the intersection point between the perpendicular to the velocity vector **v** at its end and radial distance **° С **(299,792,458 m) because for an angle **i** in the picture below we have **Cos(θ) = v/c**.

Here we see the constant speed of light **° С** located on the hypotenuse of the triangle **ΔAvc**and **acc** there is the line of sight. Now we can imagine all possible speeds **v** at the object **A** to be seen on that line of sight. All such velocities terminate on the circle of diameter **° С**, and this diameter matches the line of sight: check the right image above. Here is a summary of such a circle and the line of sight:

Stationary (means **v=0**) object/particle **A** always visible. The longest vector **v=c** it is not attainable for objects/particles with mass as according to Einstein for such objects/particles **v .**

But in real life we don’t experience such visibility limitations, why is that? This is because real objects are made up of atoms wrapped in electrons that are always oscillating. And their vibrations, imperceptible to us, often, from time to time, offset the speed of the object, making the combined speed of vibration of an atom plus the speed of the object equal to **0**. This is because the speed of vibration of electrons is much higher than the speed of an object. For example, the speed of an electron knocked out of an atom in the photoelectric effect is approx **600 km/sec**. Assuming that atoms/electrons vibrate at such a speed, any object at a speed much lower **600 km/sec** is visible: from time to time for a blink, atoms that are still at that moment are visible (because the combined **speed+vibration=0** very often, thousands of times per second).

Let us consider the case when the speed of vibration **Δv** value is less than the object’s velocity **v** value. Then in the picture below the line of sight is not just fixed **AB1** or **AB2**but varies from **AB1** to **AC1** and by **AB2** to **AC2** for combined speed **v1=v+Δv** instead of just **v**:

Thus, an angle **bows (v/c)**at which a non-vibrating object is seen changes over a range of angles **arccos(sqrt(v²-Δv²)/c) ± arcsin(Δv/v)**, in which a vibrating object is visible. In reality, this field of view/observability is three-dimensional — between two cones:

**PS** About the speeds **v** close to the speed of light, we must keep in mind that the addition of velocities **v±Δv** the operation must be adjusted to the Einstein velocity addition operator **v⨁Δv,** so that the sum does not exceed the speed of light **° С**. For example, for collinear velocities **v** and **Δv**their real amount is **v⨁Δv = (v+Δv)/[1+v×Δv/c²]**which never exceeds **° С**. Even for **v=c**,** c⨁Δv **remains** ° С**:

**c ⨁Δv = (c+Δv) / [1+c×Δv/c²] = (c+Δv) / [1+Δv/c] = (c+Δv) / [(c+Δv)/c] = c**.

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