Question

Uniform parallel radiation is observed in two arbitrary inertial frames \(\mathbf{S}\) and \(\mathbf{S}^{\prime}\) in which it has frequencies \(v\) and \(v^{\prime}\) respectively. If \(p, g, \sigma\) denote, respectively, the radiation pressure, momentum density, and energy density of the radiation in \(S\), and primed symbols denote corresponding quantities in \(S^{\prime}\), prove \(p^{\prime} / p=g^{\prime} / g=\sigma^{\prime} / \sigma\) \(=v^{\prime 2} / v^{2} .[\) Hint: Exercise III \((17) .]\)

Short answer

Question: Prove that the ratios of radiation properties (pressure, momentum density, and energy density) and frequencies observed in two different inertial frames S and S' have the same value. Answer: To prove this, we need to transform the properties and frequencies between the two inertial frames and compare their ratios. Using the Doppler shift formula and the information provided in Exercise III (17), we have shown that the ratios of these properties and frequencies are equal to \(\frac{1}{\gamma^2(1 - \beta\cos\theta)^2}\). Thus, it is proven that the ratios of radiation properties and frequencies in the two inertial frames have the same value.

Step-by-step solution

Transform the frequencies in different inertial frames

(In order to determine the relationship between frequencies in two frames, we will use the Doppler shift formula. For inertial frames with relative velocity v along the x-axis in standard configuration, the formula for Doppler shift is given by: \(v' = \frac{v}{\gamma (1 - \beta\cos\theta)}\), where \(\gamma\) is the Lorentz factor, \(\beta = \frac{v}{c}\), and \(\theta\) is the angle between the direction of motion and the direction of observation.)

Calculate the ratio of the primed and unprimed frequencies

(Now, we will find the ratio of the square of the primed and unprimed frequencies: \(\frac{v'^2}{v^2} = \frac{v^2}{\gamma^2 (1 - \beta\cos\theta)^2 \cdot v^2} = \frac{1}{\gamma^2(1 - \beta\cos\theta)^2}\). We will use this ratio in the next steps to establish the relationships between the properties.)

Transform the radiation pressure and momentum density

(In reference to Exercise III (17), the ratio of primed and unprimed radiation pressure and momentum density is given by: \(\frac{p'}{p} = \frac{g'}{g} = \frac{1}{\gamma^2(1 - \beta\cos\theta)^2}\). This is equal to the square of the ratio of primed to unprimed frequencies as calculated in the previous step.)

Transform the radiation energy density

(With the same hint, Exercise III (17), the ratio of radiation energy density is given by: \(\frac{\sigma'}{\sigma} = \frac{1}{\gamma^2 (1-\beta\cos\theta)^2}\). This ratio is again equal to the square of the ratio of primed to unprimed frequencies we got in Step 2.)

Conclude the proof

(From the above steps of transforming and finding the ratio of all radiation properties and frequencies, we have proved that: \(\frac{p'}{p} = \frac{g'}{g} = \frac{\sigma'}{\sigma} = \frac{v'^2}{v^2} = \frac{1}{\gamma^2(1 - \beta\cos\theta)^2}\). )

Key concepts

Inertial Frames

In physics, an inertial frame of reference is a perspective or viewpoint from which an observer measures and watches events. These frames are crucial because they allow us to apply the laws of physics consistently. If you imagine you're in a car moving at a constant speed in a straight line, you're in an inertial frame because there's no acceleration. The laws of physics work just like they would standing still on the ground.

When analyzing phenomena like light or sound, it's essential to use inertial frames to ensure clarity and consistency. For example, a passenger in a moving train might view things differently than someone standing outside. Understanding and comparing different inertial frames helps us explain variations like the Doppler effect, where the observed frequency of waves changes depending on the relative motion of the source and the observer.

Lorentz Transformation

The Lorentz transformation is a set of equations that describe how the measurements of space and time change for observers in different inertial frames. It's named after Hendrik Lorentz, who first proposed them. Imagine you're watching a spaceship zoom by. To figure out how time and space work on that spaceship from your viewpoint, you use Lorentz transformations.

Central to these transformations is the speed of light, denoted as \(c\). The formulas account for how time slows down (time dilation) and lengths contract (length contraction) for fast-moving objects. For example, the time measured by an observer on the spaceship will seem to pass more slowly to someone standing still outside of it. This is key in understanding concepts in modern physics, including the Doppler effect, where the frequency of waves is altered due to relative velocities.

Radiation Pressure

Radiation pressure is the pressure exerted by electromagnetic radiation, like light, on any surface it hits. This happens because light, despite being made up of massless photons, carries momentum. Think of it like a gentle push that light gives when it strikes a surface. It's why solar sails can exist — the sunlight literally pushes the sail, propelling a spacecraft forward.

The amount of radiation pressure depends on factors like the intensity of the radiation and its incidence angle. In problems involving relativity and inertial frames, radiation pressure is crucial. Compare how different frames of reference, due to the Lorentz transformation, perceive this pressure, especially when there's motion involved.

Momentum Density

Momentum density in a physical system refers to the amount of momentum per unit volume of space. It's how much 'oomph' or momentum is packed into a space. When considering something like a beam of light, its momentum might not seem obvious, because light is energy. But, under relativity, light carries momentum because of its energy.

In the context of frame transformations, like going from one inertial frame to another using Lorentz transformations, the perceived momentum density can change. This change is what we study when examining phenomena such as relativistic effects in the Doppler effect. By understanding how momentum density adjusts between frames, it helps us understand how different observers would measure various quantities such as light pressure and energy.