Question

Let \(\Lambda_{\mathrm{B}}(\theta)\) denote the 4 \(\times 4\) matrix that gives a pure boost in the direction that makes an angle \(\theta\) with the \(x_{1}\) axis in the \(x_{1} x_{2}\) plane. Explain why this can be found as \(\Lambda_{\mathrm{B}}(\theta)=\Lambda_{\mathrm{R}}(-\theta) \Lambda_{\mathrm{B}}(0) \Lambda_{\mathrm{R}}(\theta)\) where \(\Lambda_{\mathrm{R}}(\theta)\) denotes the matrix that rotates the \(x_{1} x_{2}\) plane through angle \(\theta\) and \(\Lambda_{\mathrm{B}}(0)\) is the standard boost along the \(x_{1}\) axis. Use this result to find \(\Lambda_{\mathrm{B}}(\theta)\) and check your result by finding the motion of the spatial origin of the frame \(\mathcal{S}\) as observed in \(\mathcal{S}^{\prime}\).

Short answer

Use rotations to create a boost along a specific direction using matrix multiplication.

Step-by-step solution

Understanding Boost and Rotation

In our problem, we have ascribe the components of the Lorentz transformation matrix. 'Boost' in special relativity implies movement in proper time; hence, it refers to transformations increasing with velocity. In this structure, our objective is building a boost transformation matrix without predetermined directions.

Rotating the Frame

To establish the boost direction, we use the rotation matrix, denoted \[ \Lambda_{\mathrm{R}}(\theta) = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & \cos \theta & -\sin \theta & 0 \ 0 & \sin \theta & \cos \theta & 0 \ 0 & 0 & 0 & 1 \end{pmatrix} \]This matrix is implemented to rotate the coordinates in 4-dimensional spacetime for a given plane in extant standard setting.

Applying Pure Boost

Instead of delving into an arbitrary direction, apply a direct boost along the x-axis direction, illustrated by the matrix, \[ \Lambda_{\mathrm{B}}(0) = \begin{pmatrix} \gamma & -\beta\gamma & 0 & 0 \ -\beta\gamma & \gamma & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{pmatrix} \]where \( \gamma = \frac{1}{\sqrt{1-\beta^2}} \) and \( \beta \) is the velocity in units of the speed of light.

Re-Rotating Back

The rotation matrix \( \Lambda_{\mathrm{R}}(-\theta) \) feeds into the pure boosted direction paralleling the former rotation, reinstating its original orientation. This equality is established by, \[ \Lambda_{\mathrm{R}}(-\theta) = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & \cos (-\theta) & -\sin (-\theta) & 0 \ 0 & \sin (-\theta) & \cos (-\theta) & 0 \ 0 & 0 & 0 & 1 \end{pmatrix} \]

Combine Transformations

Integrating all procedures, the resultant transformation is determined to be \[ \Lambda_{\mathrm{B}}(\theta) = \Lambda_{\mathrm{R}}(-\theta) \Lambda_{\mathrm{B}}(0) \Lambda_{\mathrm{R}}(\theta) \]Each matrix dynamically works together, orient and reorienting, then boosting capsule while returning back to original disposition.

Verification through Velocity Transformations

To make sure the final position and velocity transformation succeed, consider where the spatial framework observes any frame motion. The matrix calculated aptly affirms pushing through transformations should correspond, as perceivable in observer endpoint.

Key concepts

Boost Matrix

In the realm of special relativity, the boost matrix is an essential tool used to describe how objects move through spacetime, particularly when accounting for high velocities close to the speed of light. A 'boost' refers to a transformation that represents a change in velocity, in contrast to ordinary motion or rotation. In a 4-dimensional spacetime frame, a boost matrix alters the time and space coordinates to reflect the relativistic effects of traveling at significant speeds.

• Key Components: Typically involves a matrix transformation that alters the coordinates, adhering to Einstein's theory of relativity.
• Parameters: Incorporates the Lorentz factor, \( \gamma = \frac{1}{\sqrt{1-\beta^2}} \), where \( \beta \) is the velocity relative to the speed of light, ensuring space and time adjust properly under the theory's constraints.
• Example: A simple boost along the x-axis is represented by the boost matrix \( \Lambda_{\mathrm{B}}(0) = \begin{pmatrix} \gamma & -\beta\gamma & 0 & 0 \ -\beta\gamma & \gamma & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{pmatrix} \).

The boost matrix demonstrates how time and distance in one frame of reference appear modified when transition toward another frame moving at a higher velocity.

Rotation Matrix

The rotation matrix is a fundamental concept used not only in special relativity but also in many other fields to describe how a vector or coordinate system is pivoted around a specific axis. In the context of Lorentz transformations, the rotation matrix \( \Lambda_{\mathrm{R}}(\theta) \) describes how we alter the direction of a boost.

• Application: Used to change the orientation of a system, which is crucial for aligning transformations in a desired direction.
• Structure: A 4x4 matrix that rotates the coordinates around a given angle \( \theta \), specifically in the \( x_1 x_2 \) plane in our scenario.
• Example of Construction: Given by \( \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & \cos \theta & -\sin \theta & 0 \ 0 & \sin \theta & \cos \theta & 0 \ 0 & 0 & 0 & 1 \end{pmatrix} \), where the sine and cosine functions represent the angle of rotation in the two-dimensional plane within the 4D spacetime.

Through rotation matrices, we can effectively redirect the trajectory of boosts, enabling us to evaluate changes in transformations within different orientation frameworks.

Special Relativity

Special relativity, developed by Albert Einstein, revolutionizes our understanding of space and time by introducing key concepts such as the constancy of the speed of light and the relativity of simultaneity. It challenges the pre-existing notions of absolute time and space.

• Key Principles: Major tenets include the invariance of physical laws across all inertial frames and the fact that the speed of light is constant for all observers, regardless of their relative motion.
• Impact on Physics: Redefines time and distance, leading to effects like time dilation and length contraction, which become significant at velocities nearing the speed of light.
• Predictions: From E=mc², describing mass-energy equivalence, to how accelerating frames must account for relativistic velocities, special relativity underpins modern physics' framework.

Understanding special relativity allows us to predict phenomena experienced by objects traveling at great speeds and to correctly compose transformation matrices, like those involved in boosts and rotations.

Velocity Transformation

Velocity transformation is a concept critical in special relativity and provides insight into how speed measurements differ across various reference frames. As objects move at relativistic speeds, space and time undergo alterations that affect observed velocities.

• Definition: It explains the change in observed velocity as one moves from one inertial frame to another under the influence of Lorentz transformations.
• Transformation Equations: Given for velocities composed across two frames, such as \( u' = \frac{u + v}{1 + \frac{uv}{c^2}} \), where \( u \) and \( v \) are velocities in different frames and \( c \) is the speed of light.
• Applications: Utilized to determine how fast an object appears to be moving when the observer is also in motion, essential for formulating the correct boost matrices.

By studying velocity transformations, we ensure that the laws of physics hold true across different frames, preserving the consistent properties seen in the universe, such as the constant speed of light.