Question

Consider motion in one spatial dimension. For any velocity \(v_{n}\) define the parameter \(\theta\) via the relation \(v=c \tanh \theta,\) where \(c\) is the speed of light in vacuum. This quantity is called the velocity parameter or the rapidity corresponding to velocity \(v\). a) Prove that for two velocities, which add via a Lorentz transformation, the corresponding velocity parameters simply add algebraically, that is, like Galilean velocities. b) Consider two reference frames in motion at speed \(v\) in the \(x\) -direction relative to one another, with axes parallel and origins coinciding when clocks at the origin in both frames read zero. Write the Lorentz transformation between the two coordinate systems entirely in terms of the velocity parameter corresponding to \(v\) and the coordinates.

Short answer

a) \(v = c \tanh \theta\) b) \(v = c \sinh \theta\) c) \(v = c \cosh \theta\) d) \(v = c\) Answer: a) \(v = c \tanh \theta\)

Step-by-step solution

Express \(v\) as a function of \(\theta\)

We are given the relation between rapidity \(\theta\) and velocity \(v\): $$v=c \tanh \theta.$$ Using this expression, we can find other useful relationships with the help of the hyperbolic function properties. ##Step 2: Find the expression for \(\gamma\)##

Find the expression for \(\gamma\)

Recall the definition of \(\gamma\) as $$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}},$$ We will now express \(\gamma\) in terms of \(\theta\). Using the expression for \(v\) and the relation between \(\cosh\) and \(\sinh\) functions, we have $$\gamma = \frac{1}{\sqrt{1-\frac{(c \tanh{\theta})^2}{c^2}}}= \frac{1}{\sqrt{1-\tanh^2 \theta}} = \frac{1}{\mathrm{sech} \theta}=\cosh \theta$$. ##Step 3: Prove that rapidities add algebraically##

Prove that rapidities add algebraically

Let \(v_1 = c\tanh\theta_1\) and \(v_2 = c\tanh\theta_2\) be two velocities. Using the Lorentz velocity transformation, the sum of velocities in the x-direction is given by $$v' = \frac{v_1+v_2}{1+\frac{v_1 v_2}{c^2}}.$$ Now, let \(\theta'=\theta_1+\theta_2\). We want to show that \(v' = c\tanh(\theta')\). Expressing \(v'\) in terms of \(\theta_1\) and \(\theta_2\), we have $$v' = \frac{c\tanh\theta_1+c\tanh\theta_2}{1+\frac{c^2\tanh\theta_1\tanh\theta_2}{c^2}} = \frac{\tanh\theta_1+\tanh\theta_2}{1+\tanh\theta_1\tanh\theta_2}.$$ Using the addition formula for hyperbolic tangents, \(\tanh(\theta_1+\theta_2) = \frac{\tanh\theta_1+\tanh\theta_2}{1+\tanh\theta_1\tanh\theta_2}\), we find that indeed, $$v' = c\tanh(\theta')$$. ##Step 4: Write the Lorentz transformation in terms of rapidity##

Write the Lorentz transformation in terms of rapidity

We will now express the Lorentz transformation for the given scenario in terms of rapidity \(\theta\). The Lorentz transformation equations are given by: $$x' = \gamma(x-vt)$$$$t' = \gamma\left(t-\frac{vx}{c^2}\right),$$ where \(\gamma=\cosh\theta\) and \(v=c\tanh\theta\). Substituting these expressions, we obtain: $$x' = \cosh\theta(x-c\tanh\theta t) = \cosh\theta(x-\sinh\theta ct)$$$$t' = \cosh\theta\left(t-\frac{c\tanh\theta x }{c^2}\right) = \cosh\theta\left(t-\frac{\sinh\theta x}{c}\right).$$ These are the Lorentz transformation equations in the x and t coordinates in terms of rapidity.

Key concepts

Velocity Parameter

In the study of special relativity, understanding the concept of a velocity parameter is crucial. The velocity parameter, often denoted by \(\theta\), provides an alternative way to express the velocity of an object and is intricately related to the concept of rapidity. By defining velocity as \(v = c \tanh \theta\), where \(c\) represents the speed of light, we gain a powerful tool that simplifies many relativistic calculations.

The equation \(v = c \tanh \theta\) comes from utilizing hyperbolic trigonometric functions to represent velocities. Hyperbolic functions show up frequently in relativistic physics because they satisfy the same sort of relations that trigonometric functions do for rotational motion but are suited for hyperbolic geometry, which is the geometry of special relativity. The use of the velocity parameter facilitates the understanding of how velocities combine under the laws of special relativity, as opposed to classical physics, where velocities simply add up.

Rapidity

Rapidity is a concept directly linked to the velocity parameter and is used to provide a more intuitive understanding of relative motion at near-light speeds. While common speeds simply add or subtract, relativistic speeds do not follow such simple arithmetic due to the effects predicted by Einstein's theory of relativity. Rapidity allows us to 'add' velocities in a way that aligns with the Lorentz transformation.

Working with rapidity involves using the hyperbolic tangent function, which is where the connection to hyperbolic functions becomes evident. Considering two objects, each with its own rapidity, when they move relative to each other, their combined rapidity is the sum of their individual rapidities. This is due to the properties of the hyperbolic tangent function, allowing for the algebraic addition of rapidities, a significant simplification over the more complex Lorentz velocity addition.

Hyperbolic Functions

Hyperbolic functions, such as \(\sinh(x)\), \(\cosh(x)\), and \(\tanh(x)\), are analogs of the familiar trigonometric functions and arise from definitions similar to the exponential function. These functions are sometimes referred to as the 'trigonometry of special relativity.' Just as the circle defines the geometry of trigonometric functions, a hyperbola defines the geometry for hyperbolic functions.

In the context of Lorentz transformations, hyperbolic functions are used to relate time and space coordinates of events as observed in different inertial frames. They are particularly useful because they inherently encode the relativistic effect of time dilation and length contraction. The function \(\cosh\theta\) relates to the Lorentz factor \(\gamma\), which is key to transforming time and space coordinates, while the function \(\tanh\theta\) is used to express the relationship between velocity and rapidity. When students delve into the addition formulas for hyperbolic functions, they uncover the elegant structure underlying relativity and its effects on the fabric of spacetime.