Physics 26: 1-5 Special Relativity

Homework

Reading Preparation
Key Equations
WebLecture
Study Activity
Chat Preparation Activities
Chapter Quiz
Lab Work

Reading Preparation

Text Reading: Giancoli, Physics - Principles with Applications, Chapter 26: Sections 1 to 5

Study Points

Section 1: An inertial reference frame travels at a constant speed; it does not experience acceleration. The laws of physics are the same in all inertial reference frames — or to put it another way, all inertial reference frames are equivalent. This is extremely useful because it means that we can legitimately compare the observations of two scientists. Maxwell's equations predict the speed of light will be c in any reference frame. While the speed of light changes as it passes from one medium to another, it does not change based on the speed of its source.
Section 2: Einstein proposed a special theory of relativity for non-accelerating frames of reference, with two principles:
- All inertial references frames are equivalent.
- The speed of light is independent of its source or the speed of the observer.
Section 3: Because time can no longer be considered an absolute quantity, we cannot claim that any two events which appear simultaneous to two sets of observers will appear simultaneous to all sets of observers. However, the observations of the observers will be valid.
Section 4: In order to explain phenomena at relativistic speeds above 0.1c, we need to express the difference in duration experienced by different observers as time dilation (see below). The factor 1/√(1 - v²/c²) is often written as γ (gamma). Events occurring in a frame of reference moving relative to an observer will appear to move more slowly than they do to an observer who is in the moving frame. This gives rise to the twin paradox.
Section 5: Time dilation will affect distance as well. The length of a moving object will be shorter (as measured by an observer at rest) when it is moving than when it is at rest. L = L₀√(1-v²/c²) = L₀/γ.

Key Equations

Principles	Equation	Variables
Time Dilation	$Δ t = \frac{Δ t_{0}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} = γ Δ t_{0}$	Δt: time interval measured by stationary observer t₀: time interval measured by moving observer v: velocity of moving observer c: the speed of light in the medium γ: Lorenz factor = 1/√(1 - v²/c²)
Length contraction	$l = l_{0} \sqrt{1 - \frac{v^{2}}{c^{2}}} = \frac{l_{0}}{γ}$	l: length observed by moving observer l₀: length observed by stationary observer (proper length) v: speed of moving observer γ: Lorenz factor

Web Lecture

Read the following weblecture before chat: Relativity in Classical and Modern Mechanics

Study Activity

Activities A in the real world happen in four dimensions, three of space (x, y ,z) and one of time (t):

A = A (x, y, z, t)

For their description we need a four dimensional coordinate system. As our imagination is only capable of grasping three dimensional objects, we must use two or three dimensional projections to visualize them. Most often we restrict graphic illustrations to events of a single point object moving along one coordinate axis (x). Then we can represent it by a two dimensional graph of position over time:

x = x(t)

In contrast to the familiar way-over-time scheme of allocating position to the ordinate and time to the abscissa of a plane coordinate system, in special relativity is has become habitual to allocate abscissa to time, and ordinate to position (Minkowski Diagram).

t = t(x)

This kind of presentation is of special interest when objects move at a speed not small compared to the speed of light (c = 2.99 792 458 *108 m/s), measured relative to a resting observer at the origin of the system. Using ct instead of t for the ordinate, both axes get the same dimension of length.

ct = ct(x)

A trajectory (curve) in this scheme is called a wordline. For t < 0 it shows the entire past of the event, for t > 0 its future. Any point on the world line is called an event.

To achieve reasonable scaling for fast objects 2.99 792 458 *108 m/s * (1 unit of time) is used as unit for the x axis. If time is measured in seconds the x- unit will be 2.99 792 458 *108 m ≈ 300 000 km = 1 lightsecond.

With this scaling of space−time geometry a light signal passing the origin will appear as a straight line at 45 degrees to the axes (a light cone if one includes two spatial directions).

The simulation will demonstrate the movement of a particle under constant acceleration at its worldline.

Under the laws of classical mechanics there would be no limit to the speed that an object can achieve under constant acceleration, relative to an observer resting at its starting place. It would follow a parabola in space−time.

Special relativity theory tells us that this is not possible. A real, accelerated object can approach the speed of light at most. As it approaches this range, from the standpoint of the resting observer its mass increases while the speed increment decreases.

In the view of an observer moving with the object, speed continues to increase. The resting observer interprets this impression as caused by dilatation of the time scale in the moving object.

The base of special relativity is the experimentally proven fact that light (a photon, which has no rest mass) travels with constant velocity c in any system. Its world line is a diagonal both for the resting and the moving observer. A consequence is that no observed object can travel faster than light. Therefore any events that have a causal connection lie above the light cone. The "classical" path becomes unreal when it reaches the gradient of the cone, independent of how great the acceleration would be.

— Instructions from the Simulator Source at the Tableau website.

Run the Worldline simulation:

How does changing the acceleration (b) change the observations?

Chat Preparation Activities

Forum question: The Moodle forum for the session will assign a specific study question for you to prepare for chat. You need to read this question and post your answer before chat starts for this session.
Mastery Exercise: The Moodle Mastery exercise for the chapter will contain sections related to our chat topic. Try to complete these before the chat starts, so that you can ask questions.

Chapter Quiz

The chapter quiz is not yet due.

Lab Work

If you want lab credit for this course, you must complete at least 18 labs; you may complete more if you are preparing for the AP exam.. One or more lab exercises are posted for each chapter as part of the homework assignment. We will be reviewing lab work at regular intervals, so do not get behind!

Lab Instructions: Measuring Relativity with GPS Data (Two Week lab)

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Physics

Chapter 26: 1-5 Assignment