Optical Effects of Time Variability

Chapter 2 from Classical Physics Beyond Einstein’s

Alexandre Kassiantchouk Ph.D.

Published in

Time Matters

7 min readJul 8, 2024

📖 Book just came out: read in PDF, on Amazon or Google Books.

2.1. Color Change (Redshift, Blueshift)
2.2. Refraction
2.3. Reflection
2.4. Multiple Images
2.5. Visibility Angle
2.6. Concave Lensing 💥 — recently added, click to read.

2.1. Color Change (Redshift, Blueshift)

Since Maxwell we know that light is an electromagnetic wave. Waves have frequencies — number of waves per second, which are expressed in the Hertz unit, which is 1 wave per second. Our naked eye can see light at about 400–800 terahertz range (terahertz is a trillion waves per second). Lower end of that frequency range is perceived by our brain as red light, and the higher end is viewed as blue light. Time dilation has an interesting effect on light frequency.

Light received in a faster time area has D-times lower frequency than at its source, where time is slower by time dilation factor D. Astronomers say in such cases, that received light was redshifted by a factor 1+Z, which is the same as D: D = 1+Z. Z value they call “redshift” (which is the same as D–1, with Z=0 for “no redshift” corresponding to D=1 for “no time dilation”). Redshift is just an effect of a different number of seconds passed at the source and for the receiver (number of waves does not change between source and receiver: waves do not appear or disappear in between).

When light is received in a slower time than the time at the light source, then D<1 and light is received at a frequency higher than it was at the source (as the drawing above explains). Astronomers say that such light is blueshifted.

Formula relating frequency of the light received fᵣ to its frequency at its source fₛ is:

fᵣ = fₛ / D,

where D is time dilation at the source from a receiver perspective. The same formula in terms of wavelength λ, which is inversely related to frequency f (by f×λ=c), is:

λᵣ = λₛ / D.

2.2. Refraction

When a light beam crosses borderline between areas having different time speed, then it is refracted by Snell’s law with time dilation D as refractive index:

That can be derived geometrically, exploiting fact that a beam shoulders travel at different time speeds, covering different distances, and that difference is proportional to the time dilation factor D:

P.S. Refraction is different both geometrically and physically from contraction or expansion of space speculated in relativity. For example, contraction/expansion below a borderline will break any line by changing Tan (but not Sin) of the incidence angle by the contraction/expansion factor:

P.P.S. Refraction (and not Einstein’s space curvature) is the real reason for some obscured by the Sun stars being visible to astronomers during Solar eclipses.

2.3. Reflection

When an angle of incidence B > arcsin (1/D), then D×Sin(B)>1, and such beam cannot cross a border from slower time to faster time, because then Sin(R)>1 (real Sin value cannot exceed 1) for the angle of refraction R. Thus, such beam is reflected at the angle of reflection equal to the angle of incidence. Proving that is no different from the classical proof because light beam stays in the same time:

2.4. Multiple Images

Let’s return to the very first drawing in this book, but now let’s consider not a narrow laser beam, but a widely shining light source (a star, for example, or a wide-angled flashlight):

Light from the source 1 can reach the observer directly by blue trajectory in 1 sec (in local to the pink area time).
Light from the source is reflected from the border between the timezones as well, and it reaches the observer by green trajectory in a bit more than 1 sec. We assume that distance to the borderline is negligible in comparison with 299,792,458 m. Then the light source is visible somewhere at “position 2”, besides the real position 1.
Third, the red route is the most interesting, when light is twice refracted by Snell’s law, with refractive index D=2: Sin(90°) = 2×Sin(30°). Left shoulder of the red beam travels in the twice-faster-time area, between refraction points. So, most of that path is covered in about a second of blue time, which is about half a second of pink time. Thus, this beam reaches the observer in a bit more than half a second (in local to the pink area time). Source of the light is visible at “position 3”, in addition to positions 1 and 2.

2.5. Visibility Angle

In 3.6 we will re-derive Einstein’s formula for relativistic time dilation:

It states that time for a moving at speed v object slows down by that factor D. But now, we will use it to prove that a particle moving at velocity v can be seen only at arccos(v/c) angle:

Drawing on the right shows that particle A

Is always visible, when it is at rest;
When its velocity v is small, the particle is visible at an angle almost perpendicular to its velocity vector v (angle of visibility is a bit less than 90°);
When this particle moves at a high speed close to c, line of sight is almost in the way of the particle (angle of visibility is close 0°).

Let’s explore a particle that was at point A and had velocity v at that point (we don’t care about its velocity before or after that). Let’s say, we have two observers: a stationary observer at point B (see the drawing below), and another observer moving at constant velocity v: the second observer was at point A when the observed particle was there, and later he was at some point E at the moment, when the first observer, stationed in B, noticed/saw the particle 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), time slows down by a factor of 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 traveled not from A to B, but from E to B, and that took t×sqrt(1–v²/c²) time. Since we know all sides in the triangle ΔABE, we can find value θ of the angle ∠EAB:

θ = Arccos(v/c)

P.S. In real life we do not experience such restrictions to visibility, why is that? Because we see light reflected from electrons whirling in atoms (and atoms themselves wobble, and electrons’ velocities and thermal velocities of atoms are random). Thus, velocities of these electrons combined with the object’s velocity are very random, therefore, combined velocity is often equal zero or often perpendicular to an observer position, and at such combined velocity they are often visible. Electrons in atoms move at speeds of several hundred km/sec, and the size of an atom is billions of times smaller than a meter, thus, we can see those whirling electrons billions of times per second, which is more than enough for our slow eyes to see the object all the time. No surprise: a century ago people watched movies at about 20 frames per second. But if an object itself moves at a speed of thousands km/sec, then speed of electrons in hundreds of km/sec cannot cancel such a high speed out, and such a fast moving object is visible at one angle, but invisible at another angle to its velocity.

P.P.S. To get physical explanation/intuition (besides geometrical) on why the angle of visibility is 90–°, check 📖 Time Matters chapter 99: