We are now able to deduce the Lorentz transformation, relating two inertial frames. We examine three paradoxes, namely the famous twins paradox, the pole-in-the-barn paradox, and the so-called Bell's spaceships paradox. We also take another look at the relationship with electromagnetism.

We look at the immediate consequences of the two axioms, and discover, qualitatively and then quantitatively, the phenomena of length contraction and time dilation.

We survey relativity's contact with experiment and observation, briefly discussing the classical tests of SR and of GR, and including a discussion of the famous 1919 Dyson-Eddington observations of the bending of starlight during the solar eclipse. In the latter, we look at the historical and social pressures on the scientists involved, and what effect these have on the processes of theory choice.

A brief recap of some mathematical topics.

Inwe saw how observers could make measurements of lengths and times in frames which are in relative motion, and reasonably disagree about the results – the phenomena of length contraction and time dilation. In , we were able to put numbers to this and derive a quantitative relation, Eq. (), between the duration of a ‘tick’ of the light clock as measured in two frames. We want to do better than this, and find a way to relate the coordinates of any event, as measured in any pair of frames in relative motion. That relation – a transformation from one coordinate system to another – is the Lorentz transformation (LT). The derivation inhas a lot in common with the account given in .

In this chapter, I am going to introduce the two axioms of Special Relativity. These axioms are, to an extent, the only new physics introduced in this text: once I have introduced them and made them plausible, the rest of our work is devoted to examining their consequences, and the way in which they change the physics we are already familiar with.

I provide a checklist/recipe for doing relativity calculations.

Another useful transform related to the Fourier and Laplace transforms is the Z-transform, which, like the Laplace transform, converts a time-domain function into a frequency-domain function of a generalized complex frequency parameter. But the Z-transform operates on sampled (or “discrete-time”) functions, often called “sequences” while the Laplace transform operates on continuous-time functions. Thus the relationship between the Z-transform and the Laplace transform parallels the relationship between the discrete-time Fourier transform and the continuous-time Fourier transform. Understanding the concepts and mathematics of discrete-time transforms such as the Z-transform is especially important for solving problems and designing devices and systems using digital computers, in which differential equations become difference equations and signals are represented by sequences of data values.

The previous three chapters were designed to help you understand the meaning and the method of the Laplace transform and its relation to the Fourier transform (), to show the Laplace transform of a few basic functions (), and to demonstrate some of the properties that make the Laplace transform useful (). In this chapter, you will see how to use the Laplace transform to solve problems in five different topics in physics and engineering. Those problems involve differential equations, so the first section of this chapter () provides an introduction to the application of the Laplace transform to ordinary and partial differential equations. Once you have an understanding of the general concept of solving a differential equation by applying an integral transform, you can work through specific applications including mechanical oscillations (), electrical circuits (), heat flow (), waves (), and transmission lines (). Each of these applications has been chosen to illustrate a different aspect of using the Laplace transform to solve differential equations, so you may find them useful even if you have little interest in the specific subject matter. And as in every chapter, the final section () of this chapter has a set of problems you can use to check your understanding of the concepts and mathematical techniques presented in this chapter.

Becoming familiar with the Laplace transform F(s) of basic time-domain functions f(t) such as exponentials, sinusoids, powers of t, and hyperbolic functions can be immensely useful in a variety of applications. That is because many of the more complicated functions that describe the behavior of real-world systems and that appear in differential equations can be synthesized as a mixture of these basic functions. And although there are dozens of books and websites that show you how to find the Laplace transform of such functions, much harder to find are explanations that help you achieve an intuitive understanding of why F(s) takes the form it does, that is, an understanding that goes beyond “That’s what the integral gives”. So the goal of this chapter is not just to show you the Laplace transforms of some basic functions, but to provide explanations that will help you see why those transforms make sense.

The Laplace transform is a mathematical operation that converts a function from one domain to another. And why would you want to do that? As you’ll see in this chapter, changing domains can be immensely helpful in extracting information from the mathematical functions and equations that describe the behavior of natural phenomena as well as mechanical and electrical systems. Specifically, when the Laplace transform operates on a function f(t) that depends on the parameter t, the result of the operation is a function F(s) that depends on the parameter s. You’ll learn the meaning of those parameters as well as the details of the mathematical operation that is defined as the Laplace transform in this chapter, and you’ll see why the Fourier transform can be considered to be a special case of the Laplace transform.