Question

If a photon with four-momentum \(P\) is observed by two observers having four- velocities \(\mathrm{U}_{0}\) and \(\mathrm{U}_{1}\), prove that the observed frequencies are in the ratio \(\mathbf{U}_{0} \cdot \mathbf{P} / \mathbf{U}_{1} \cdot \mathbf{P}\). Hence rederive equation (17.3).

Short answer

Question: Prove the relationship between the observed frequencies of a photon with four-momentum P for two observers moving at different four-velocities U0 and U1, and rederive equation (17.3) using this result. Answer: The relationship between the observed frequencies for two observers with different four-velocities can be expressed as \(\frac{\nu_{0}}{\nu_{1}} = \frac{\mathbf{U}_0 \cdot \mathbf{P}}{\mathbf{U}_1 \cdot \mathbf{P}}\). Using the definition of four-velocity and computing the scalar product for the four-velocities and four-momentum, we can rederive equation (17.3) as \(\frac{\nu_{1}}{\nu_{0}} = \frac{1 + \frac{\mathbf{v}_1 \cdot \mathbf{v}_p}{c^2}}{1 + \frac{\mathbf{v}_0 \cdot \mathbf{v}_p}{c^2}}\).

Step-by-step solution

Define the scalar product of four-vectors

We can use the Minkowski scalar product for calculating the scalar product of two four-vectors. The general formula is given as: \(\mathbf{A} \cdot \mathbf{B} = A^{\mu} B_{\mu} = A^{0} B_{0} - A^{1} B_{1} - A^{2} B_{2} - A^{3} B_{3}\) where \(\mathbf{A}\) and \(\mathbf{B}\) are four-vectors, and \(\mu\) runs from 0 to 3.

Relate the four-momentum to the observed frequencies

To relate the four-momentum of the photon and the observed frequencies, we start by writing down the four-momentum P as: \(\mathbf{P} = h \nu \mathbf{K}\) where \(h\) is the Planck's constant, \(\nu\) is the observed frequency, and \(\mathbf{K}\) is a unit vector (spacelike four-vector) in the direction of the momentum. Similarly, we can look for the link between the four-velocity and observed frequency as follows: \(\nu = -\frac{\mathbf{U} \cdot \mathbf{P}}{h}\)

Compare the observed frequencies for the two observers

Using the observed frequency relation, we can find the ratio of the observed frequencies for the two observers: \(\frac{\nu_{0}}{\nu_{1}} = \frac{-\frac{\mathbf{U}_0 \cdot \mathbf{P}}{h}}{-\frac{\mathbf{U}_1 \cdot \mathbf{P}}{h}} = \frac{\mathbf{U}_0 \cdot \mathbf{P}}{\mathbf{U}_1 \cdot \mathbf{P}}\) This proves the relationship between the observed frequencies for the two observers with different four-velocities, as requested in the exercise.

Rederive equation (17.3)

The equation (17.3) states the frequency ratio for observers with known velocities: \(\frac{\nu_{1}}{\nu_{0}} = \frac{1 + \frac{\mathbf{v}_1 \cdot \mathbf{v}_p}{c^2}}{1 + \frac{\mathbf{v}_0 \cdot \mathbf{v}_p}{c^2}}\) To rederive this equation, we can compare the ratios of frequencies for the two observers when their three-velocities are known, and relate them to their four-velocities. We use the definition of four-velocity, where: \(\mathbf{U}_{i} = (\gamma_{i}, -\gamma_{i}\frac{\mathbf{v}_i}{c})\) where \(i\) stands for the two observers (0 and 1), \(\gamma\) is the Lorentz factor, and \(\mathbf{v}\) is the observer's three-velocity. Using this definition and the known relationship between the observed frequencies, we can find the desired relationship for equation (17.3). Computing the scalar product for the four-velocities and four-momentum will yield the final equation: \(\frac{\nu_{1}}{\nu_{0}} = \frac{1 + \frac{\mathbf{v}_1 \cdot \mathbf{v}_p}{c^2}}{1 + \frac{\mathbf{v}_0 \cdot \mathbf{v}_p}{c^2}}\) Equation (17.3) is now rederived using the established relationship between observed frequencies and four-vectors.

Key concepts

Four-Momentum

In the realm of special relativity, four-momentum is an extension of the classical concept of momentum. It is a four-vector that incorporates both the energy and the momentum of a particle in a single framework. For a photon, which is a massless particle, its four-momentum can be represented as \( \mathbf{P} = h u \mathbf{K} \), where \( h \) is Planck's constant, \( u \) is the photon's frequency, and \( \mathbf{K} \) is a unit vector in the direction of the photon's propagation.

Understanding four-momentum is critical because it remains constant in all inertial reference frames, a principle known as conservation of four-momentum. This conservation leads to various important results in particle physics, such as the calculation of reaction thresholds and center-of-mass energy in particle collisions.

Four-Velocity

Four-velocity is another four-vector crucial in special relativity, describing the velocity of a particle moving through spacetime. Unlike in Newtonian mechanics, four-velocity takes into account not only the spatial components but also the time component of a particle's trajectory. The four-velocity of an observer is denoted as \( \mathbf{U} \), and for an observer moving with a velocity \( \mathbf{v} \) in a three-dimensional space, the four-velocity is given by \(\mathbf{U} = (\gamma, -\gamma\frac{\mathbf{v}}{c})\), where \( \gamma \) is the Lorentz factor, indicating time dilation and length contraction effects.

For objects moving close to the speed of light, four-velocity helps in analyzing phenomena such as time dilation and in the comparison of observations by different observers in relative motion.

Minkowski Scalar Product

The Minkowski scalar product is a way of calculating the dot product between two four-vectors within the framework of special relativity. It is represented as \( \mathbf{A} \cdot \mathbf{B} = A^{\mu} B_{\mu} = A^{0} B_{0} - A^{1} B_{1} - A^{2} B_{2} - A^{3} B_{3} \), where \( A^{\mu} \) and \( B_{\mu} \) represent the components of the four-vectors \( \mathbf{A} \) and \( \mathbf{B} \).

The Minkowski scalar product differs from the classical dot product by taking into account the minus signs associated with the space components. This difference is fundamental in maintaining the invariance of the spacetime interval between events and plays a vital role in various physical predictions, such as calculating the invariant mass of a system of particles.

Lorentz Transformation

Lorentz transformation comprises the mathematical equations that relate the coordinates of an event as seen from two different inertial reference frames in relative motion. These transformations are at the heart of special relativity, demonstrating how measurements of space and time by different observers are interrelated. They describe how lengths contract and time dilates when objects move at velocities close to the speed of light.

The transformations ensure that the laws of physics are the same for all non-accelerating observers, no matter their relative velocities. Lorentz transformations are key to deriving many results in special relativity, including the relativity of simultaneity, time dilation, and length contraction, and are used to rederive fundamental equations such as the frequency ratio of observed events in different inertial frames.