Question

A meteor made of pure kryptonite (Yes, we know: There really isn't such a thing as kryptonite ...) is moving toward Earth. If the meteor eventually hits Earth, the impact will cause severe damage, threatening life as we know it. If a laser hits the meteor with wavelength \(560 \mathrm{nm}\), the entire meteor will blow up. The only laser powerful enough on Earth has a \(532-\mathrm{nm}\) wavelength. Scientists decide to launch the laser in a spacecraft and use special relativity to get the right wavelength. The meteor is moving very slowly, so there is no correction for relative velocities. At what speed does the spaceship need to move so the laser has the right wavelength, and should it travel toward or away from the meteor?

Short answer

Answer: The spaceship should move away from the meteor with a speed of approximately 14,987,690 m/s.

Step-by-step solution

Recall the formula for Doppler effect in special relativity.

To find the relationship between the observed wavelength and the emitted wavelength when considering special relativity, we use the formula for relativistic Doppler effect: \[ \lambda_{observed} = \lambda_{emitted} \sqrt{\frac{1 + \beta}{1 - \beta}} \] where \(\lambda_{observed}\) is the observed wavelength, \(\lambda_{emitted}\) is the emitted wavelength, and \(\beta = \frac{v}{c}\), with \(v\) being the relative speed between the observer and the emitter, and \(c\) being the speed of light.

Identify the target and initial wavelengths.

We are given the initial wavelength of the laser as \(532\,\text{nm}\) and the target wavelength as \(560\,\text{nm}\). Thus, the relationship between these wavelengths can be expressed as: \[ 560\,\text{nm} = 532\,\text{nm} \sqrt{\frac{1 + \beta}{1 - \beta}} \]

Solve for \(\beta\).

To find the value of \(\beta\), which will allow us to find the speed \(v\), we'll rearrange the equation from Step 2 by dividing by \(532\,\text{nm}\), and then squaring both sides: \[ \frac{560\,\text{nm}}{532\,\text{nm}} = \sqrt{\frac{1 + \beta}{1 - \beta}} \] \[ \left(\frac{560}{532}\right)^2 = \frac{1 + \beta}{1 - \beta} \] Now, we'll solve for \(\beta\) by multiplying both sides by \((1 - \beta)\) and then rearranging: \[ \left(\frac{560}{532}\right)^2 (1 - \beta) = 1 + \beta \] \[ \beta \left(\frac{532^2 + 560^2}{532^2}\right) = 2 \cdot \frac{560^2}{532^2} \] \[ \beta = \frac{2 \cdot (560^2)}{(532^2 + 560^2)} \]

Calculate the value of \(\beta\).

Plugging in the numbers, we can find the value of \(\beta\): \[ \beta = \frac{2 \cdot (560^2)}{(532^2 + 560^2)} \approx 0.04996 \]

Find the speed \(v\) using \(\beta\).

Now that we have \(\beta\), we can find the speed \(v\) using the formula \(\beta = \frac{v}{c}\). Rearranging this formula, we have: \[ v = \beta \cdot c \] Since \(c \approx 3 \times 10^8\,\text{m}/\text{s}\), the speed of the spaceship is: \[ v \approx 0.04996 \cdot 3 \times 10^8\,\text{m}/\text{s} \approx 14,987,690\,\text{m}/\text{s} \]

Determine if the spaceship should move towards or away from the meteor.

Since the target wavelength (560 nm) is greater than the initial wavelength (532 nm), the laser's wavelength needs to be redshifted. This means that the spaceship should move away from the meteor. So, the spaceship should move away from the meteor with a speed of approximately \(14,987,690\,\text{m}/\text{s}\).

Key concepts

Understanding Special Relativity

Special relativity is a fundamental theory in physics developed by Albert Einstein that transformed our understanding of space and time. At its core, it addresses how observers moving at different velocities measure the same phenomena differently, particularly those moving at significant fractions of the speed of light. According to special relativity, the laws of physics are the same in all inertial frames of reference, but time and space are relative concepts that depend on the observer's state of motion. Importantly, the theory posits that the speed of light in a vacuum is constant for all observers, regardless of their relative motion.

One of the startling outcomes of special relativity is time dilation, where time appears to pass at different rates from different frames of reference. Similarly, length contraction occurs—the length of objects in the direction of motion contracts as their speed approaches the speed of light. These effects are negligible at everyday speeds but become significant at relativistic speeds—close to the speed of light. This foundational concept is crucial for understanding phenomena like the relativistic Doppler effect, which comes into play when dealing with objects moving at high velocities, such as a spacecraft aiming to hit a meteor with a laser beam.

The Doppler Shift in Space

The Doppler shift, or Doppler effect, refers broadly to the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. It is a common phenomenon that we experience in everyday life, such as when an ambulance passes by and its siren's pitch seems to change. However, in the realm of astronomy and high-speed travel, such as the scenario with the meteor and the spacecraft, we encounter the relativistic Doppler effect.

In the relativistic context, the formulas used to describe the Doppler effect need to account for the effects of special relativity. This is particularly important when dealing with objects moving at speeds comparable to the speed of light. As the spacecraft moves towards or away from an object, such as our hypothetical meteor made of kryptonite, the observed wavelength of the light it emits or reflects can be shifted to shorter wavelengths (blueshift) if moving towards the observer, or longer wavelengths (redshift) if moving away from the observer.

Understanding the Doppler shift is not only crucial for cosmic phenomena but also for navigating high-stakes interstellar challenges. By applying the concept of the Doppler shift accurately, one can determine the proper wavelengths needed to avert a disaster, much like scientists aiming to adjust a laser's wavelength to prevent a meteor impact.

Wavelength Calculation Under Relativistic Conditions

Wavelength calculation is an essential aspect of physics, optics, and astronomy, enabling us to determine the size and distance of waves from their sources. When dealing with high velocities, it's crucial to incorporate the principles of special relativity into these calculations to achieve accurate results. The relativistic Doppler effect formula is one such adaptation for high-speed scenarios, allowing us to understand how emitted wavelengths appear when observed from different reference frames.

When a spaceship attempts to adjust the wavelength of a laser to destroy an incoming meteor, as in the example provided, scientists must calculate the precise speed at which the spaceship should travel to achieve the necessary wavelength shift. This involves a comparison between the emitted wavelength (from the laser) and the observed wavelength (needed to destroy the meteor), and the subsequent determination of the factor by which the wavelength will change, represented by the term 'beta' (β) in the formulas. By using these relativistic formulas, we can solve for variables such as the spacecraft's necessary speed to ensure that the wavelength of the laser, once adjusted by the motion, will be exactly what is required to neutralize the threat of the meteor.

The action of moving away or towards a target to alter wavelength through the Doppler effect is a direct application of wavelength calculation principles and demonstrates the profound intersection of theoretical physics with practical, potentially life-saving technology.