Question

A space probe has a powerful light beacon that emits 500 nm light in its own rest frame. Relative to Earth, the space probe is moving at \(0.8 c\). An observer on Earth is viewing the light arriving from the distant beacon and detects a wavelength of \(500 \mathrm{nm}\). Is this possible? Explain.

Short answer

Whether it is possible or not will be determined by conducting the calculations in the step-by-step solution, and comparing the calculated observed wavelength to the given value of 500 nm. If they are equal, it is possible; otherwise, it is not.

Step-by-step solution

State the Doppler effect formula

The relativistic Doppler effect formula for the frequency as observed is given by: \(f' = f \sqrt{\frac{1+\beta}{1-\beta}}\), where \(f'\) is the observed frequency, \(f\) is the emitted frequency, and \(β=v/c\) is the ratio of the observer's speed to the speed of light.

Convert wavelength to frequency

In this problem, the wavelength rather than the frequency is given. However, the two are related by the equation: \( f = c/\lambda \), where \(c\) is the speed of light and \(\lambda\) is the wavelength. We can rearrange this equation to convert the frequency back to wavelength: \( \lambda' = c/f' \).

Calculate the observed wavelength

Substitute \( \lambda'=c/f' \), and \( f = c/\lambda \) into \( f' = f\sqrt{\frac{1+\beta}{1-\beta}} \), we can derive that \( \lambda' = \lambda\sqrt{\frac{1-\beta}{1+\beta}} \). Given the space probe is moving at \(0.8c\), \( \beta \) is 0.8. Input these values into the derived equation, the observed wavelength can then be calculated.

Compare the calculated and the observed wavelength

Finally, compare the calculated observed wavelength with the value given in the exercise (500 nm). If they are equal, then it is possible for the observer on Earth to view a wavelength of 500 nm; otherwise, it is not.

Key concepts

Doppler Shift

When you hear the pitch of an ambulance siren change as it passes by you, you're experiencing an audio version of the Doppler effect. In physics, particularly regarding waves, the Doppler shift is a change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source.

For light, this phenomenon occurs when a light source moves towards or away from an observer. If the source is approaching, the light waves are compressed, leading to a higher frequency and a shorter wavelength, called blueshift. Conversely, if the source is receding, the waves are stretched, resulting in a lower frequency and a longer wavelength, known as redshift.

Using the formula for the relativistic Doppler shift, \( f' = f \sqrt{\frac{1+\beta}{1-\beta}} \), where \(\beta = v/c\), the observer can determine the frequency of the light as it's received. This equation accounts for the effects of special relativity, which are significant when dealing with velocities close to the speed of light, denoted as \(c\).

In our space probe example, if the probe emits 500 nm light at rest but is moving relative to Earth at 0.8c, we'll need to apply this equation to see if the observed wavelength could still be 500 nm on Earth.

Light Wavelength

Understanding Wavelength in Light

The wavelength of light is the distance between consecutive crests (or troughs) of a wave, usually measured in meters or its submultiples, such as nanometers (nm), where \(1 nm = 1\times10^{-9} m\). Visible light, which ranges from approximately 380 nm to 750 nm, has a spectrum of colors: Violet has the shortest wavelength, and red has the longest.

To connect wavelength with frequency, we use the equation \( f = c/\lambda \), where \(f\) is frequency, \(c\) is the speed of light, and \(\lambda\) is the wavelength.

In our exercise, the space probe emits light in its rest frame at a wavelength of 500 nm. If not for the probe's high-speed motion, an observer on Earth would detect the light at this same wavelength. However, the motion affects the observed wavelength due to the Doppler shift, which alters the light's frequency as perceived by the observer.

Special Relativity

Albert Einstein's theory of special relativity revolutionized our understanding of space, time, and speed. One of its key concepts is that the speed of light in a vacuum is constant and is not affected by the motion of the source or observer. This has profound implications for how we view phenomena at high velocities, particularly those close to the speed of light.

Special relativity introduces the need to adjust our classical equations, like the Doppler effect, when dealing with relativistic speeds. The modified equation to calculate the observed frequency or wavelength takes into account the factor \(\beta\), which is the ratio of the object's speed to the speed of light.

For our space probe emitting a 500 nm wavelength while traveling at \(0.8c\), special relativity tells us that the observed wavelength cannot remain unaffected. Time dilation and length contraction come into play, thereby altering the observed frequency and wavelength of the emitted light for an observer on Earth. Solving the exercise with relativity in mind, we would find that the observer could not detect a 500 nm wavelength if the probe were indeed moving at \(0.8c\).