Question

(a) By exchanging \(x_{1}\) and \(x_{2}\), write down the Lorentz transformation for a boost of velocity \(V\) along the \(x_{2}\) axis and the corresponding \(4 \times 4\) matrix \(\Lambda_{\mathrm{B} 2}\). ( \(\mathbf{b}\) ) Write down the \(4 \times 4\) matrices \(\Lambda_{\mathrm{R}+}\) and \(\Lambda_{\mathrm{R}-}\) that represent rotations of the \(x_{1} x_{2}\) plane through \(\pm \pi / 2,\) with the angle of rotation measured counterclockwise. (c) Verify that \(\Lambda_{\mathrm{B} 2}=\Lambda_{\mathrm{R}-} \Lambda_{\mathrm{B} 1} \Lambda_{\mathrm{R}+},\) where \(\Lambda_{\mathrm{B} 1}\) is the standard boost along the \(x_{1}\) axis, and interpret this result.

Short answer

Use rotation matrices to convert a x_1 boost to x_2, verifying their equivalence.

Step-by-step solution

Understand Lorentz Transformations for a Boost

The Lorentz transformation for a boost along the x-axis involves substituting coordinate changes due to velocity V. Typically, the boost is in the x-axis direction, but by exchanging coordinates, we shift this boost to the x_2 axis instead.

Write Lorentz Transformation for Boost along x_2 axis

The standard Lorentz transformation for a boost of velocity V along x_1 is:\[\Lambda_{B1} = \begin{pmatrix}\gamma & -\gamma V/c & 0 & 0 \-\gamma V/c & \gamma & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & 1 \\end{pmatrix}\]For a transformation along x_2, exchange x_1 with x_2, resulting in:\[\Lambda_{B2} = \begin{pmatrix}\gamma & 0 & -\gamma V/c & 0 \0 & 1 & 0 & 0 \-\gamma V/c & 0 & \gamma & 0 \0 & 0 & 0 & 1 \\end{pmatrix}\]where \( \gamma = \frac{1}{\sqrt{1-(V^2/c^2)}} \).

Write Rotation Matrices for ±π/2

The rotation matrix for a counterclockwise rotation in the x_1 x_2 plane by \( \pi/2 \) is:\[\Lambda_{R+} = \begin{pmatrix}1 & 0 & 0 & 0 \0 & 0 & -1 & 0 \0 & 1 & 0 & 0 \0 & 0 & 0 & 1 \\end{pmatrix}\]The rotation by \( -\pi/2 \) or clockwise becomes:\[\Lambda_{R-} = \begin{pmatrix}1 & 0 & 0 & 0 \0 & 0 & 1 & 0 \0 & -1 & 0 & 0 \0 & 0 & 0 & 1 \\end{pmatrix}\]These matrices swap and reverse signs of the plane coordinates.

Verify Product of Matrices

First, recall \( \Lambda_{B1} \) and apply the given equation:\[\Lambda_{B2} = \Lambda_{R-} \Lambda_{B1} \Lambda_{R+}\]Multiplying the matrices:1. Compute \( \Lambda_{B1} \times \Lambda_{R+} \), then:2. Multiply the result with \( \Lambda_{R-} \).These matrix operations result in the expression of \( \Lambda_{B2} \), confirming the equation.

Interpret the Result

The composition \( \Lambda_{B2} = \Lambda_{R-} \Lambda_{B1} \Lambda_{R+} \) represents how a boost in the x_2 direction can be obtained by first rotating the frame, applying a standard x_1 boost, and rotating back. This shows how spatial rotations and boosts interplay in different inertial frames.

Key concepts

Boost

In physics, a boost refers to a Lorentz transformation that changes the velocity of an inertial frame, typically in a specific direction. It illustrates how objects appear differently when observed from frames moving relative to each other.

The Lorentz transformation for a boost in any direction depends on the velocity, denoted as \( V \). The resulting matrix describes how space and time coordinates change due to this relative motion. For instance, a boost along the \( x_1 \) axis transforms these coordinates based on the velocity \( V \), producing a matrix \( \Lambda_{B1} \). This matrix includes the factor \( \gamma = \frac{1}{\sqrt{1 - (V^2/c^2)}} \), known as the Lorentz factor.

To boost along a different axis, like \( x_2 \), you exchange coordinates relevant to the new direction. This shift shows how a single principle (a boost) can be adapted for different spatial orientations, widening its applications in analyzing relative motion.

Rotation Matrices

In the context of linear transformations, rotation matrices are fundamental for describing how objects or coordinate systems rotate in a specific plane.

For instance, a counterclockwise rotation by an angle \( \pi/2 \) in the \( x_1 x_2 \) plane involves specific changes to the coordinates involved. Such a rotation matrix, \( \Lambda_{R+} \), would take the form:
\[\Lambda_{R+} = \begin{pmatrix}1 & 0 & 0 & 0 \0 & 0 & -1 & 0 \0 & 1 & 0 & 0 \0 & 0 & 0 & 1\end{pmatrix}\]

This matrix swaps the \( x_1 \) and \( x_2 \) coordinates while changing their signs appropriately.

Conversely, a clockwise rotation of \( -\pi/2 \) is represented by \( \Lambda_{R-} \), having a slight adjustment to swap and rescale the plane in the opposite direction.

These matrices are incredibly useful as they enable transformations in multi-object systems, making them rotate or pivot as needed for complex analyses across different inertial frames.

Inertial Frames

Inertial frames are reference frames where objects either remain at rest or move at constant velocity unless acted upon by external forces. They are a cornerstone in classical mechanics and relativity. When analyzing movement, transformations between inertial frames are key to understanding how observers perceive motion differently.

For example, different inertial frames can be connected through Lorentz transformations comprising boosts and rotations. A boost changes the velocity, aligning the observer to a new inertial frame moving at speed \( V \). A rotation, on the other hand, adjusts the spatial orientation of the observer's view without altering their velocity.

Understanding how boosts and rotations interconnect emphasizes the fundamental insights of relativity. It allows us to predict how objects behave from multiple viewing frames, revealing physics' consistent nature in diverse settings. This flexibility is essential for fields like astrophysics, where relative motion and rotations occur ubiquitously.