/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 Derive the Lorentz transformatio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 33: Problem 44

Derive the Lorentz transformations for time from the transformations for space.

Short Answer

Expert verified
The time component of the Lorentz transformations can be derived from the space components, giving: \(t' = \gamma (t - vx/c^2)\)

Step by step solution

01

Understand the Space Components of the Lorentz Transformation

The Lorentz transformation for space is given by: \(x' = \gamma (x - vt)\), where \(x'\) is the position in the moving frame, \(x\) is the position in the stationary frame, \(v\) is the relative velocity of the frames, \(t\) is the time in the stationary frame, and \(\gamma\) is the Lorentz factor given by \(\gamma = 1 / \sqrt{1 - (v/c)^2}\), with \(c\) being the speed of light.
02

Rearrange to solve for time in the stationary frame

Isolate \(t\) on one side to express time in the stationary frame in terms of the other variables: \(t = (x'/\gamma + v) / v\).
03

Express Relative Velocity is in Terms of the Lorentz Factor

Use the expression for the Lorentz factor to write \(v\) in terms of \(\gamma\). This gives \(v = c \sqrt{1 - 1/\gamma^2}\). Substitute this into the equation for \(t\).
04

Solve to Get the Time Component of the Lorentz Transformation

The result of simplifying gives the time component of the Lorentz transformation: \(t' = \gamma (t - vx/c^2)\), where \(t'\) is the time in the moving frame.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the speed of a particle whose relativistic kinetic energy is \(50 \%\) greater than the Newtonian value calculated for the same speed.

Find the kinetic energy of an electron moving at (a) \(0.0010 c\) (b) \(0.60 c,\) and \((\mathrm{c}) 0.99 c .\) Use suitable approximations where possible.

You're writing a galactic history involving two civilizations that evolve on opposite sides of a \(1.0 \times 10^{5}\) -ly-diameter galaxy. In the galaxy's reference frame, civilization B launched its first spacecraft 45,000 years after civilization A. You and your readers, from a more advanced civilization, are traveling through the galaxy at \(0.99 c\) on a line from \(\mathrm{A}\) to \(\mathrm{B}\). Which civilization do you record as having first achieved interstellar travel, and how much in advance of the other?

Twins \(A\) and \(B\) live on Earth. On their 20 th birthday, twin \(B\) climbs into a spaceship and makes a round-trip journey at \(0.95 c\) to a star 30 light years distant, as measured in the Earth-star reference frame. What are their ages when twin B returns to Earth?

At what speed will the momentum of a proton (mass 1 u) equal that of an alpha particle (mass 4 u) moving at \(0.5 c ?\)
See all solutions

Recommended explanations on Physics Textbooks

Mechanics and Materials

Read Explanation

Scientific Method Physics

Read Explanation

Wave Optics

Read Explanation

Medical Physics

Read Explanation

Physical Quantities And Units

Read Explanation

Classical Mechanics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.