Angular Perspective on What We See
Making an Unbelievable/Counterintuitive Result Believable/Intuitive
When we see something, we actually see light reflected off electrons surrounding atoms of the object we see. In Science of Visibility and Invisibility we discussed an unbelievable but a simple outcome of Einstein’s time dilation, which is that we can see a particle only at a very specific angle to its motion:
- When a particle is motionless, we can see it from any position.
- When a particle moves slowly, it is visible only from a position almost perpendicular to its velocity (see an explanation below the drawing for why we still can see a moving object under other angles, nevertheless).
- When a particle moves extremely fast (at a speed close to the speed of light c), we can see it only with our eye on or close to its path:
The only reason we can see everything around us, even when it is moving at any angle, it is because electrons whirl in atoms and atoms themselves wobble — thermal velocities of atoms are random. Thus, velocities of the electrons combined with the object’s velocity are very random, therefore, equal zero very often, and at that combined velocities they are often visible. Electrons in atoms move at speeds of several hundred km/sec, and the size of an atom is billions times smaller than a meter, thus, we can see those whirling electrons sometimes, billions times per second, that is more than enough for our slow eyes to see the object steadily (no surprise, as some of us watched old movies at about 20 frames per second). But if an object moves at a speed of thousand km/sec, then even random speeds of electrons cannot cancel such a high speed out, and such a fast moving object is visible at one angle, but invisible at another angle, and that solved the 80-year-long mystery of vanishing-without-any-trace stars.
The purpose of this article is to make this unbelievable/counterintuitive result believable/intuitive: Why is a slow moving particle/electron visible only at an angle close to 90° to its velocity, and why does the angle of vision decrease for a faster moving electron?
To explain that physically (not just geometrically by Einstein’s Special Relativity), we will use several known facts from physics:
- Light is an electromagnetic wave, with electric force waving perpendicularly to the wave propagation;
- Light is particle/photon as well, and particles have momentum/impulse;
- Photoelectric effect: light knocks out electrons from atoms;
- Conservation of Momentum law;
- Most laws in physics (especially in mechanics) are reversible (when velocity changes to the opposite direction, laws of mechanics are still true).
Since Maxwell, we know that light is an electromagnetic wave, which propagates in a direction perpendicular to electric and magnetic forces waving/wobbling:
In the animation above, red vector E shows electric force, and blue vector B shows magnetic force. In general, those forces are free to wobble in the direction orthogonal/perpendicular to the wave propagation:
Electric force acts on any charged particle, especially, on an electron (since it is the lightest charged particle, it reacts the most to any force applied). The most studied interaction between light and electrons is photoelectric effect, when light knocks out electrons from atoms:
The part we are interested in the most is the angle between the light and emitted electrons. That subject was well studied, and it is well summarized here:
The most probable direction of initial photo-electron ejection is a little forward of perpendicular to the incoming un-polarized X-ray beam (based on ejection being parallel to the electric vector of the x-ray photons which vector is perpendicular to the beam length).
You can ignore “un-polarized X-ray”: “un-polarized” means that E and B are chaotic/randomly directed, as we mentioned above, with E still perpendicular to the light beam; X-ray means energetic, high-frequency light (because low-energetic light does not knock out electrons from atoms). The angle between the light and emitted electrons is a bit greater than 90°:
Why 90+°? 90° angle is due to the wave nature of light: electric force of the light wave is orthogonal/perpendicular to the light beam. + is due to particle/photon nature of light: it has momentum/impulse, which it transfers to the electron when the photon collides with the electron (by Conservation of Momentum law). Impulse/energy of regular visible light (which has a low frequency) is small. Even an electron is quite heavy for low frequency light to act on it, therefore, velocity of an emitted electron is usually small in comparison with the speed of light. But for x-rays, and, especially, for gamma-rays (and even more energetic cosmic rays), the impulse/energy of a photon is higher, and the angle between the photon and the emitted electron increases further, and so is the speed v of the emitted electron. Thus, physics of light explains the angle increasing from 90° (for low energetic photons) to almost 180° (for highly energetic photons), and the emitted electron velocity increasing to almost the speed of light c. As you can see now, the drawing above is basically the same, but with arrows changed to the opposite directions in comparison with the drawing below, which we wanted to explain without relying on relativity:
Do you still remember “Most laws in physics (especially in mechanics) are reversible”? The only difference between the two last drawings is that arrows in one drawing point in the opposite directions to the arrows in the other drawing. Basically, photoelectric effect and relativistic effect on visibility are the similar outcomes of light duality:
- Light as a wave of electric (and magnetic) force and that wave propagates in a direction perpendicular to the electric force direction;
- Light as a particle has impulse/momentum.
That makes the previously unbelievable/counterintuitive geometrical effect of relativistic time dilation (when time for an object moving at speed v slows down by √(1–v²/c²) factor) on visibility of a particle only at angle arccos(v/c) to its velocity more believable/intuitive now.
P.S. Check related UFO Fuzziness and Sporadic Cloaking Explained.
P.P.S. Free “Time Matters: 10th edition” in PDF, on Amazon, on Google Books.
