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Chapter 2: Problem 8

The rapidity \(\phi\), of a particle moving with velocity \(u\), is defined by \(\phi=\tanh ^{-1}(u / c)\) [cf. Exercise I (12)]. Prove that collinear rapidities are additive, i.e. if A has rapidity \(\phi\) relative to B, and B has rapidly \(\psi\) relative to \(\mathrm{C}\), then \(\mathrm{A}\) has rapidity \(\phi+\psi\) relative to \(\mathrm{C}\).

Short Answer

Expert verified
Question: Prove the additivity of collinear rapidities, meaning that if A has rapidity \(\phi\) relative to B, and B has rapidly \(\psi\) relative to C, then A has rapidity \(\phi+\psi\) relative to C. Answer: We used the definition of rapidity, properties of hyperbolic trigonometric functions, and the relativistic velocity addition formula to prove that \(\phi_{AC} = \phi + \psi\), showing the additivity of collinear rapidities.

Step by step solution

01

Write out the given definition of rapidity

We are given the definition of rapidity \(\phi\) as \(\phi = \tanh^{-1}(u / c)\). Similarly, we can define rapidities \(\phi\) and \(\psi\) as follows: 1. \(\phi = \tanh^{-1}\left(\frac{u_{AB}}{c}\right)\), where \(u_{AB}\) is the velocity of A relative to B. 2. \(\psi = \tanh^{-1}\left(\frac{u_{BC}}{c}\right)\), where \(u_{BC}\) is the velocity of B relative to C. Our goal is to show that the rapidity of A relative to C, which we will call \(\phi+\psi\), is equal to the sum of \(\phi\) and \(\psi\).
02

Find expressions for the velocities

Let's find expressions for the velocities \(u_{AB}\) and \(u_{BC}\) in terms of rapidities \(\phi\) and \(\psi\): 1. \(u_{AB} = c\tanh\left(\phi\right)\) 2. \(u_{BC} = c\tanh\left(\psi\right)\)
03

Use relativistic velocity addition formula

To find the velocity of A relative to C, \(u_{AC}\), we will use the relativistic velocity addition formula: \(u_{AC} = \frac{u_{AB} + u_{BC}}{1 + \frac{u_{AB}u_{BC}}{c^2}}\) Substitute the expressions for velocities from Step 2 into this formula: \(u_{AC} = \frac{c\tanh\left(\phi\right) + c\tanh\left(\psi\right)}{1 + \frac{c\tanh\left(\phi\right)c\tanh\left(\psi\right)}{c^2}}\) Simplify the formula by cancelling out \(c\): \(u_{AC} = c\frac{\tanh\left(\phi\right) + \tanh\left(\psi\right)}{1 + \tanh\left(\phi\right)\tanh\left(\psi\right)}\)
04

Use the addition formula for the hyperbolic tangent

The addition formula for the hyperbolic tangent states that \(\tanh\left(\phi+\psi\right) = \frac{\tanh\left(\phi\right) + \tanh\left(\psi\right)}{1 + \tanh\left(\phi\right)\tanh\left(\psi\right)}\). Then, the expression for \(u_{AC}\) becomes: \(u_{AC} = c\tanh\left(\phi+\psi\right)\)
05

Find the rapidity of A relative to C

To find the rapidity of A relative to C, we can use the definition of rapidity with \(u_{AC}\): \(\phi_{AC} = \tanh^{-1}\left(\frac{u_{AC}}{c}\right) = \tanh^{-1}\left(\tanh\left(\phi+\psi\right)\right)\) Since the inverse hyperbolic tangent function and the hyperbolic tangent function are inverses of each other, we have: \(\phi_{AC} = \phi + \psi\) This proves that the collinear rapidities are additive, meaning if A has rapidity \(\phi\) relative to B, and B has rapidly \(\psi\) relative to C, then A has rapidity \(\phi+\psi\) relative to C.

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