The Astonishing Velocity for an Observer to See a Green Light as Red Due to the Doppler Effect

Contrary to the conventional question, which often asks how quickly one must approach a red light to see it as green, a scenario posing the opposite pursuit is both intriguing and somewhat suicidal, if one considers the implications of such high speeds. This phenomenon is governed by the relativistic Doppler shift, a key concept in the realm of astrophysics and theoretical physics. Below, we delve into the specifics of this phenomenon and calculate the exact speed needed for an observer to perceive a green light as red.

Understanding the Doppler Shift

The Doppler effect refers to the change in frequency and wavelength of a wave for an observer moving relative to its source. In the context of light, the Doppler shift can cause the perceived color of light to change based on the observer's relative motion. Typically, if one approaches a red light source, it will appear more blue, and if one moves away, it will appear more red.

Calculating the Observer Speed

For an observer to see a green light (with a wavelength of approximately 570nm) as red (with a wavelength of approximately 650nm), we must calculate the precise velocity required for this transformation. The relativistic Doppler shift equation is used for this purpose:

[ frac{lambda_{shift}}{lambda_{original}} sqrt{frac{1 beta}{1-beta}} ]

Where (beta frac{v}{c}) and (v) is the velocity of the observer relative to the speed of light ((c)).

Step-by-Step Calculation

Given that the green wavelength (570nm) needs to be shifted to approximately 605nm (a midpoint between 570nm and 650nm), we can set up the equation:

[ frac{650 text{nm}}{570 text{nm}} sqrt{frac{1 beta}{1-beta}} ]

Solving for (beta):

[ frac{650}{570} sqrt{frac{1 beta}{1-beta}} ]

Squaring both sides:

[ left( frac{650}{570} right)^2 frac{1 beta}{1-beta} ]

[ frac{422500}{324900} frac{1 beta}{1-beta} ]

[ frac{422500}{324900} 1.3008 ]

[ 1.3008(1-beta) 1 beta ]

[ 1.3008 - 1.3008beta 1 beta ]

[ 0.3008 2.3008beta ]

[ beta frac{0.3008}{2.3008} approx 0.1307 ]

[ beta 0.13 ]

[ v beta c 0.13 times 3 times 10^8 text{m/s} approx 3.9 times 10^7 text{m/s} ]

This corresponds to a velocity of approximately 141 million kilometers per hour, or 87.6 million miles per hour, an extremely high speed that is well beyond the capabilities of current technology or even theoretical physics. This calculation underscores the extreme conditions required for such a shift to occur.

Subtle Considerations

It's important to note that color perception is not solely dependent on a single wavelength but rather on the entire spectrum of light. However, for simplicity, we considered only the peak wavelengths. Even so, the entire spectrum would shift together, not just the green wavelengths. Therefore, while the perceived color may change, the overall shift in the spectrum would make the star appear blue, not green, as it would happen in the common scenario of a star appearing yellow and then becoming green.

Graphical Representation

A graphical representation of the spectra of red and green LEDs shows that the red peak wavelength at 650nm and the green peak wavelength at 570nm. Shifting the green spectrum to a wave length of 605nm using the relativistic Doppler shift formula can confirm our calculations. The green spectrum indeed shifts to a redder wavelength, validating our theoretical findings.

Conclusion

The relativistic Doppler effect offers a fascinating glimpse into the extremes of physics. Observers would need to travel at a velocity of 0.13 times the speed of light to perceive a green light as red, which is at the edge of current understanding and technology. This concept further highlights the fundamental principles of light behavior and the fascinating possibilities within the realm of astrophysics and theoretical physics.