Science of Visibility and Invisibility | by Alexander Kasianchuk | Time matters | December 2023

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 Band 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 Bbut from e to Band that’s what he took t×sqrt(1-v²/c²) time which is less than T. Since we know all the sides in the triangle ΔABEwe 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 ΔAvcand 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 AB2but 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 Δvtheir 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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