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.

The value of knowing the Laplace transforms of the basic functions described in the previous chapter is greatly enhanced by certain properties of the Laplace transform. That is because these properties allow you to determine the transform of much more complicated time-domain functions by combining and modifying the transforms of simple functions such as those discussed in .

Chapter 1 told the history of ASTP from the perspective of the principal political and diplomatic actors who attempted to use the supposedly neutral sphere of space science and engineering to reset superpower relations. This chapter covers similar terrain but reverses the order of analysis by examining the perspective of the engineers and how they attempted to design a technical solution to the political challenges of détente. Put another way, I discuss the engineering and design of technological fixes to solve the political problem of averting mutual assured destruction (MAD). The goal is to determine how well those fixes functioned politically (as a way to de-escalate tensions between the superpowers) and technically (by enhancing the safety and effectiveness of human habitation in space).

The term “technological fix” was coined in the 1960s by the director of Oakridge National Laboratories, Alvin Weinberg. The basic idea was hardly new. Modern faith in technology had produced a mania for technological fixes, a belief that “solutions founded on technological innovation may be innately superior for addressing issues traditionally defined as social, political, or cultural.” The main attraction of the technological fix is that it promises to bypass the cultural and political challenges of changing behaviors and attitudes by shifting the problem to the supposedly objective realm of technical problem-solving, and to the experts and engineers who supposedly have only technical rather than partisan goals. For example, advocates of nuclear power in the 1960s, like solar or wind power today, presented it as a solution to the economic and political dilemmas of fossil-fuel dependence. If it worked as planned, politicians would avoid the hard work of changing deeply entrenched behaviors of energy consumption, providing a cheap way to produce and consume power that would also protect the environment. It was a case of having your cake (energy independence and a cheap power source) and eating it too (blissfully tapping into the electric grid without destroying the environment).

ASTP was a technological fix designed to make superpower relations less dangerous and more secure, and it had the added benefit of advancing the cause of space exploration. Up to that point, with US troops mired in Vietnam and Soviet troops blasting away hopes of reforming communism in Czechoslovakia, little else seemed to be working to mitigate the literally explosive potential of superpower relations.