Photons and Their Relative Speed: A Relativistic Perspective

When discussing the behavior of photons, especially their relative speed as they approach each other, we are delving into the fascinating realm of special relativity. Photons are particles of light, which travel at the constant speed of light in a vacuum, denoted as c. However, their interaction and relative speed in close proximity are governed by the principles of relativistic physics rather than classical mechanics.

Classical Intuition vs. Special Relativity

It is intuitive to think that if two photons are moving towards each other, each at speed c, their relative speed would be the sum of their individual speeds. This intuitive approach, however, leads to the incorrect conclusion that their relative speed would be 2c. But this violates one of the fundamental principles of physics: no object can travel faster than the speed of light.

Relativistic Velocity Addition Formula

According to the principles of special relativity, the formula for adding velocities is:

vrel (v1 v2) / (1 (v1 * v2) / c2)

If we substitute v1 c and v2 -c (since they are moving towards each other), the equation becomes:

vrel (c (-c)) / (1 (c * (-c)) / c2) 0 / (1 - 1)

This results in an indeterminate form (0/0). However, the more appropriate understanding is provided by the concepts of special relativity, which state that photons do not have a well-defined relative speed in the traditional sense.

Quantum Mechanics and Relativity

The behavior of photons is best understood through the lens of quantum mechanics and relativity. According to these theories, from any inertial frame, both photons always travel at speed c. Their interaction is more accurately described by quantum electrodynamics rather than classical mechanics.

The Short Answer

The relative speed of two objects moving at v1 and v2 and heading towards each other is given by the formula vrel (v1 v2) / (1 (v1 * v2) / c2). If you set v1 v2 c in this formula, you get c.

Special Cases and Fun Facts

When both photons are traveling in the same direction, their relative velocity with respect to each other is zero. This is quite intuitive and consistent with our everyday experiences. However, a fascinating aspect is that even if photons are traveling in opposite directions, their relative velocity is still c, the speed of light. This is because the speed of light is an absolute limit imposed by the laws of special relativity, and no object can exceed it.

It should be emphasized that the relative velocity of photons cannot be determined by using the classical or Galilean velocity addition rule. Instead, the Einstein's velocity addition rule, a principle of special relativity, governs their interactions. This rule ensures that the speed of light remains constant in all inertial frames, a cornerstone of modern physics.

Conclusion

The relative speed of two photons approaching each other is a compelling example of the deep and intricate nature of relativistic physics. Understanding this concept requires a shift from classical intuition to the laws of special relativity and quantum mechanics. This example highlights the profound impact of these theories on our perception of space, time, and the fundamental nature of light itself.