Faster than light? Relative speed of two photons approaching each other

Do two photons approach each other at twice the speed of light, 2c? This popular question highlights the distinction between relative and closing speeds in Newtonian physics and their interpretation within the context of special relativity. In everyday Newtonian usage, when we mention that two cars are approaching each other at, say, 100 km/h, we typically refer to the closing speed, determined as the sum of the speeds of both cars. However, the relative speed is also calculated by adding the two speeds. Therefore, in Newtonian terms, the result is the same, so we don't need to clarify which speed we mean. This is not the case in special relativity, where closing and relative speeds are handled quite differently. Specifically, the closing speed of approaching photons is determined as 2c, whereas their relative speed is considered undefined.

Fig. 1. In special relativity, when two photons approach each other head-on, their closing speed is determined as 2c, whereas their relative speed is considered undefined.

Before exploring the reasons behind these nuances and their implications in the realm of massless particles, let's clarify the difference between the relative and closing speeds in the case when two cars approach each other head-on:

• Relative speed is the speed of one car as measured by an observer inside the second car. The frame of reference is aligned with the second car.

• Closing speed is the speed at which the distance between two cars is closing. It is calculated by adding the speeds of these cars as they are measured by an observer on the ground. The frame of reference is aligned with the ground.

We begin with Newtonian mechanics. Consider two cars, one red and one blue, approaching each other head-on at speeds V1 and V2 (Fig. 2). An observer on the ground using a radar gun will measure their individual speeds as V1 and V2, just as they appear on the cars' speedometers. The closing speed, Vc = (V1 + V2), represents the rate at which the gap between these cars is closing. It is mainly used to calculate the time and location of the cars' meeting point. For instance, if the cars start their journey with a distance D between them, they will meet in time T = D ÷ (V1 + V2). Knowing their travel time T, we can compute the point at which they converge.

It is important to realize that when we add V1 and V2 to determine the closing speed, nothing is actually traveling at this combined speed. The gap is not moving; it simply shrinks. The distance between the cars decreases because each car reduces it from its side at its own rate.

Fig. 2. The closing speed, Vc = (V1 + V2), allows for calculating the travel time T and the collision point of the red and blue cars.

Let's place an observer with the radar gun inside a blue car. Since the observer is now stationary relative to this car, he will perceive the red car's speed as a relative speed Vrel = (V1 + V2). His radar gun will also measure the red car's speed as Vrel = (V1 + V2), because the radar signal must travel to that car and back to complete the task, effectively measuring the rate at which the gap between them is closing. This is why, in Newtonian physics, the distinction between closing speed and relative speed can become blurred. Shifting the frame of reference from the ground to one of the cars provides an additional viewpoint to help us evaluate the situation from different angles and reach an objective conclusion.

Einstein's relativity was born out of the need to reconcile Newtonian physics with the discovery that the speed of light, c, is a universal constant. In this sense, relativity is not a new law, but rather a reinterpretation of existing laws viewed from the standpoint of the new constant. In Newtonian relativity, velocities add. Therefore, if an observer on the ground measures the speeds of a rocket and an approaching photon as 0.1c and c, respectively, then an observer in the rocket should have measured the photon's speed as 1.1c. However, this was not the case. Experimental data demonstrate that the speed of light remains the constant c, unaffected by the position of the observer.

Guided by the constancy of c and the uniformity of natural laws across space, time, and frames, Einstein modified Newtonian equations to incorporate the new advancements in physics. This led to the development of special relativity, which is founded on two postulates:

The laws of physics remain the same in all inertial frames

The speed of light in a vacuum remains constant for all observers, irrespective of their 

             or the light source motion.

Einstein had no issue with adding up the photon speeds (c + c) in the closing speed because it did not conflict with the constancy of c. The closing speed is determined by summing the speeds of the particles measured from the ground. In the lab (ground) frame, the speed of each photon is measured as c. Though we add these speeds together (Fig. 3), nothing actually travels at the combined speed of 2c, just as nothing traveled at the combined speed (V1 + V2) in the example with two cars in Fig. 2. The speed addition here is merely a mathematical operation, utilising the fact that two particles with equal speeds cover equal distances at the same time, meeting at the midpoint of their journey. No frame transition takes place.

Consider two photons in Fig. 3, one red and one blue, each travelling at a speed of c = 300,000 km/s (186,000 miles per second). The distance between the photons shrinks at a rate of 2c = 600,000 km/sec, because each photon shortens it by 300,000 km every second from its side. We add their speeds (c + c) to determine when and where the photons will meet. However, this addition represents their concurrent motion, not the doubled speed. Nothing is moving at 2c = 600,000 km/sec. We merely have two photons, each independently covering 300,000 km (186,000 miles) every second. Their concurrent motion means that they double the mileage, not the speeds.

Fig. 3. The closing speed between two photons is computed as 2c = c + c. However, nothing is moving at this speed. We merely have two photons, each traveling at c and collectively covering twice the distance every second.

A different approach to relative speed in special relativity arises from the constraints imposed by the constancy of c. While Einstein never specifically discussed the case of two photons traveling toward each other, he conducted a thought experiment in which he envisioned himself running alongside the light beam. He realized that being stationary relative to the beam (Fig. 4) would result in observing a frozen electromagnetic wave, contradicting Maxwell's equations. This insight led to the conclusion that the observer can't “ride a photon” and ask how fast another photon approaches, because a photon doesn’t have a valid rest frame. In a valid frame, such as one aligned with a laboratory on Earth, the observer would see each photon moving at the speed of c. The question of how one photon perceives another remained unanswered.

Fig. 4. The observer placed on the electromagnetic wave will move with it at the speed of light, c., perceiving the wave stationary relative to him.

Einstein's formal stance was that the relative speed between two photons can't be determined in their own frames, as we can't assign a zero value to the speed of light, c. To me, this formalism seems an evasion using scientific terminology. Frames of reference were specifically designed for conducting thought experiments, such as the one Einstein conducted himself. They are a valuable tool in scientific study that allows a researcher to gain an additional viewpoint. When Einstein aligned his perspective with an electromagnetic wave, he gained an insight that had never been considered before, leading to the development of the theory of relativity. This breakthrough in science came from assigning a reference frame to a photon. So, do we really need to abandon this conceptual tool that has served us so well for centuries just because we got a little spooked by what we saw?

Placing an observer on a photon doesn't mean we've stopped the light, just as positioning an observer on the ground doesn't imply that we've stopped the Earth. It merely indicates that the observer has adopted the photon's speed and is stationary relative to it. In classical physics, the relative speed between two bodies is measured from the reference frame of one of them. We will expand this definition into the realm of massless particles by positioning the observer on a photon and exploring the impact of the constancy of c on his measurements. The universe has no special rules for massless particles, as the laws of physics must remain consistent across space, time, and frames for all its constituents. Drawing parallels with Newtonian perspectives will help us develop an intuitive understanding of Einstein's relativity and make it accessible to everyone. Essentially, we will attempt to "ride a photon" and see where it takes us.

In the car example, we used a radar gun to measure the relative speed between the vehicles. In our thought experiment, we will also place an observer on one of the photons, say the blue one, and have him measure the speed of the red photon using the radar gun (Fig.5). The radar gun operates by emitting light signals. When two objects are in relative motion, light waves undergo a frequency shift due to the Doppler effect. When the gun positioned on the first object emits a light signal, the signal travels to the other object, bounces off it, and returns to the emitter. The difference in the emitted and received frequencies determines the relative speed between the objects. If there is no relative motion, the radar gun detects zero Doppler effect.

Fig. 5. The observer on the blue photon attempts to measure the relative speed between two photons with the radar gun..

Einstein would be unable to observe a frozen electromagnetic wave because light can only be detected when it interacts with an eye or a measuring device. Therefore, our thought experiment begins with the observer sending a radar signal to the blue photon to ensure there is no relative motion between them, and the observer is aligned with the photon's frame. The observer then sends a radar signal toward the red photon. Since the speed of light is independent of the motion of the light emitter, the signal will travel at speed c alongside the gun and will only reach the red photon at the collision point. Consequently, no signal will be exchanged between the photons during their journey, meaning no relative speed can be measured using the Doppler effect or any other method.

In a Newtonian framework, we can determine relative speed because we can use signals that travel significantly faster than cars, allowing for continuous exchange of signals between the vehicles. If we had signals traveling faster than light, we could also determine the photons' relative speed and express its value by using Lorentz transformations, where the speed of light, c, would be replaced with the new, higher constant. However, since no such faster signal exists, the information between the photons can't be exchanged faster than their speed, c. As a result, one photon can receive the information about the other only at their meeting point.

We have concluded that the closing speed of two approaching photons is 2c. This finding does not contradict the postulates of special relativity, as nothing is actually traveling at that speed. Each photon travels at the speed of light, c, as observed in the lab frame. The addition (c + c = 2c) is merely used to reflect the fact that the photons cover double the distance due to their concurrent motion. This information is utilized when calculating their travel time and meeting point.

In our thought experiment, we attempted to measure the relative speed of two photons, similar to how we measure the relative speed of two head-on cars in Newtonian relativity. We placed an observer with a radar gun on one photon to gauge the speed of the second photon. We found that the relationship between photons can't be described using the classical concept of relative velocity, because the speed of information can't exceed the speed of light c. Since no data can be exchanged between the photons, the notion of how one object perceives the speed of the other object becomes meaningless.

By assigning a reference frame to the photon, we have arrived at the same conclusion as Einstein’s special relativity, but through a more intuitive approach, drawing parallels with the familiar Newtonian world. We determined that observers in all frames, including both the lab and photon frames, will measure the speed of light as c. The observer in the photon's frame identifies its own speed as c by comparing it to the speed of the light signal emitted by the radar gun.