Can a car change its color due to the Doppler effect?

Science memes have become extremely popular among the general public and scientists. Some achieved such wide recognition that they became iconic, blending science and humor into an irresistible mix of intellectual enjoyment. Perhaps the most notable is the one where a car changes its color from blue to red as it drives past a pedestrian. The meme uses a well-known example from sound to illustrate the Doppler effect in light, playing out the analogies in a humorous yet informative manner. Drawing parallels between light and sound through this meme can help make a fundamental physics concept easier to grasp, turning the learning process into a fun experience. We will determine if a car can change its color due to the Doppler effect, the speed required for such a transformation, and the true color of the vehicle.

Fig. 1. This meme illustrates the Doppler effect in light using a familiar example from sound.

What is the Doppler effect? The Doppler Effect is associated with wave phenomena, with light and sound being the most common examples. Waves are defined by their frequencies, which determine the color we see and the pitch we hear. Our eyes have photoreceptor cells, and our ears contain hair cells that detect frequency variations and relay this information to the brain, where it is converted into colors and tones. Thus, we can experience the world in all its diversity because our visual and auditory systems are capable of differentiating between wave frequencies.

Although our sensory system possesses remarkable abilities, the accuracy of the information is affected if the wave source moves relative to the observer. For example, when a car approaches us, we perceive the frequency as higher, resulting in a higher pitch. Conversely, if the car moves away, the frequency appears to be lower, resulting in a lower pitch. Despite the appearance, the frequency of the sound produced by the car remains constant, which is why the passengers in the car don't notice any change. In science, when we explore various phenomena, we are the observers whose measurements are affected by this effect. The Doppler formulas allow us to calculate the Doppler shift and understand the actual situation versus a perceived reality.

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Fig. 2. The car exhibits a frequency shift due to the Doppler effect while approaching and receding from the observer.

The difference between the Doppler effect in sound and light. The Doppler effect applies to all types of waves, including mechanical, sound, and light waves. The primary difference lies in the speed at which the waves travel. For our eyes and ears to detect a Doppler shift, the speed of the wave source must be comparable to the speed of the wave itself. Since light travels almost a million times faster than sound, a light source must also move nearly a million times faster than a sound source, like a car, to show an equivalent optical shift. Textbooks often use sound examples because, on Earth, light sources can't achieve such extremely high, relativistic speeds. As a result, we don't observe the visual manifestation of the Doppler effect in everyday life. However, astronomers observe this phenomenon as galaxies display a blueshift when moving toward us and a redshift when moving away (Fig. 3).

Fig. 3. The star exhibits a frequency shift due to the Doppler effect, appearing blue while approaching observers and red while receding from them.

When a car drives past and its pitch changes, its color changes too. However, this change is imperceptible to the human eye because the Doppler shift in light is negligible in the moving cars. It would require high-precision instruments to detect it. For the Doppler shift to be visible, the vehicle would need to travel at a relativistic speed, unachievable under Earth's conditions. But if it could reach this astronomical speed, some strange things would happen to the color of the car and to its headlights.

The Doppler effect on the headlights. Headlight is similar to starlight. Starlight comprises various wavelengths that form a continuous spectrum, spanning the entire visible range from 400 nm to 700 nm (Fig. 4). This mix of wavelengths causes stars to appear white, with shades of red, yellow, and blue, depending on a star's temperature and chemical composition. The Doppler effect shifts each wavelength by the same fractional amount v/c, where c is the speed of light and v is the speed of the light source. Consequently, the entire spectrum of blue, green, yellow, and red components shifts as a whole toward shorter wavelengths when a star or a car is approaching (blue shift) and toward longer wavelengths when they are receding (red shift).

Fig. 4. The wavelengths and frequencies of the visible spectrum, ranging from red to blue.

For white light to demonstrate the full span Doppler effect, changing color from bluish to reddish (Fig. 3), a star and a car would need to travel at nearly 20% of the speed of light (v/c = 0.2). As their speed increases further, some light components would slide into the invisible ultraviolet and infrared bands. Once they surpass half the speed of light, v/c > 0.5, the entire visible spectrum slides into the invisible bands. A car would play a dangerous game of "Now you see me, now you don't" with a pedestrian attempting to cross the road. The car would only briefly become visible when it was directly in line with an observer. While approaching and moving away, the car would turn into a ghost, undetectable by human photoreceptors.

The Doppler effect on the car's color.   We perceive the car as red because it is coated with paint that absorbs all wavelengths except for red when white sunlight illuminates its surface. By reflecting red components to our eyes, the car acts as a secondary light source, with reflected wavelengths being subject to the Doppler effect. The Doppler effect is consistently symmetrical, shifting each wavelength, λ, by the same proportion, Δλ, toward the blue and red ends of the spectrum. Let the red wavelength be λ = 670 nm (Fig. 5). The Doppler shift Δλ in relation to the car speed, v, can be calculated using a simple formula, where we neglect the relativistic component for simplicity:

Given that the speed of light is approximately 300,000 km/s (186,000 miles per second), the car must travel at a speed of 4,500 km/s (2,800 miles/s) to experience a Doppler shift of Δλ = ±10 nm. If you were a pedestrian attempting to cross the road, you would perceive the color as a shade of red corresponding to the wavelength λ = 660 nm when the car is approaching, and as a shade of red corresponding to the wavelength λ = 680 nm when it is moving away from you (Fig. 5). The true color of the car, corresponding to the rest wavelength λ = 670 nm, would briefly appear when the car was directly beside you. As you can see, even at this astronomical speed, the shift would remain subtle.

Fig. 5. Due to the Doppler effect, the red light with the rest wavelength of 670 nm experiences a 10 nm shift, as a result of the light source moving at a speed of 0.015c.

Referring back to our meme, we now possess all the necessary information to determine the car's actual color and the speed it needs to display a full-span Doppler shift from blue to red. Let the color appear blue with a wavelength of λ = 470 nm when the car is approaching an observer, and red with a wavelength of λ = 650 nm when it is receding. Given the symmetry of the Doppler shift, this combination will give a rest wavelength of λ = 565 nm, (470 + 650)/2, identifying the car's actual color as a shade of green.

By plugging in a Doppler shift of Δλ = 85 nm (650 nm - 565 nm) into the formula above, we calculate that the car would need to travel at about 15% of the speed of light (0.15c), which is far beyond the capabilities of current technology, even for rockets in space. Still, the practicality of this scenario is not our concern. The meme inspired us to dive into the core of the Doppler effect, explore it like physics pros, and enjoy the colorful experience. Isn't science incredible? The thought experiments it provokes can outshine the most vivid imagination! The car's length was humorously contracted and stretched to emphasize the shorter blue and longer red wavelengths. In reality, the Doppler effect does not alter the length of objects.

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I will conclude with a physics joke that is believed to have inspired this meme. A physicist, caught running a red light, devised a scheme to avoid a penalty. In court, he claimed that the red light appeared green due to the Doppler shift. Unfortunately for him, the judge was a physics fan. Without an argument, he accepted the physicist's explanation and doubled the fine for driving at a relativistic speed in the residential area. Trapped by his own lie, the physicist had no option but to accept the punishment.