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Lets explore an object or a particle that was at point A and had velocity v at point A (we dont care about its velocity before or after that). Lets 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-v/c), 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 tsqrt(1-v/c) time, which is less than t. Since we know all sides in the triangle ABE, we can find angle EAB value :

= 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 in the picture below, we have Cos() = v/c.

Here we see the speed of light constant c residing on the hypotenuse of the triangle Avc, 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

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).

Lets explore the case when the vibration velocity v value is smaller than the objects 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+v instead of just v:

Thus, angle arccos(v/c), at which non-vibrating object is visible, changes to a range of angles arccos( sqrt(v-v)/c ) arcsin(v/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 vv operation should be adjusted to the Einsteins velocity-addition operator vv, so that sum does not exceed speed of light c. For example, for collinear velocities v and v, their real sum is vv = (v+v)/[1+vv/c], which never exceeds c. Even for v=c, cv stays c:

c v = (c+v) / [1+cv/c] = (c+v) / [1+v/c] = (c+v) / [(c+v)/c] = c.

Read the original here:

Science of Visibility and Invisibility | by Alexandre Kassiantchouk | Time Matters | Dec, 2023 - Medium