Practical Relativity

From First Principles to the Theory of Gravity

Richard N. Henriksen

• Publisher: Wiley
• Year: 2011
• ISBN:
  9781119956341
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Relativity in Physics

The book is intended to serve as lecture material for courses on relativity at undergraduate level. Although there has been much written on special relativity the present book will emphasize the real applications of relativity. In addition, it will be physically designed with the use of box summaries so as to allow easy access of practical results. The book will be composed of eight chapters. Chapter 1 will give an introduction to special relativity that is the world without gravity. Implications will be presented with emphasis on time dilation and the Doppler shift as practical considerations. In Chapter 2, the four-vector representation of events will be introduced. The bulk of this chapter will deal with flat space dynamics. This will require the generalization of Newton's first and second laws. Some important astronomical applications will be discussed in Chapter 3 and in Chapter 4 some engineering applications of special relativity such as atomic clocks will be presented. Chapter 5 will be dedicated to the thorny question of gravity. The physical motivation of the theory must be examined and the geometrical interpretation presented. Chapter 6 will present astronomical applications of relativistic gravity. These include the usual solar system tests; light bending, time delay, gravitational red-shift, precession of Keplerian orbits. Chapter 7 will be dedicated to relativistic cosmology. Many of the standard cosmological concepts will be introduced, being mathematically simple but conceptually subtle. The concluding chapter will be largely dedicated to the global positioning system as an engineering problem that requires both inertial and gravitational relativity. The large interferometers designed as gravitational wave telescopes will be discussed here.

Part I

The World Without Gravity

1

Non-Relativity for Relativists

Dura lex, sed lex (The law is hard but it is the law)

1.1 Vectors and Reference Frames

In this section we discuss our fundamental concepts as drawn from experience. This ends in frustration since experience is approximate, most things are known relative to other things, and our concepts often seem to be defined in terms of themselves. Thus ‘fundamental’ argument resembles the circular snake devouring its tail (the Ouroboros). However we must make a beginning, and so we confront our first definition and its algebraic implications.

What is an inertial reference frame? I prefer to parse this question into two principal questions. By ‘reference frame’ we mean some well-defined system of assigning a measured time and a measured position to an ‘event’. For the moment an ‘event’ is point-like, as for example the time at which a particle or the centre of mass of an extended body takes a particular spatial position. The reference frame also implies an ‘observer’ who records the measurements. The resulting numbers are the ‘coordinates’ of the event in this reference frame. By ‘inertial’ we mean a reference frame in which Newton’s first law of motion¹ applies to sufficiently isolated bodies. This axiom requires not only that the coordinates of a body be determinable from moment to moment, but also that fixed spatial directions be defined. Neither one of these definitions is particularly exact or obvious and yet they are fundamental to our subject. Thus we continue their exploration in the next two sections.

1.1.1 Reference Frames

Although this is not strictly necessary, location is normally specified relative to a set of objects that have no relative motion between them. Some fixed point within this set of objects is chosen as the reference point or ‘origin’ from which all distances are measured. On small enough scales that we can reach continuously, the measurement is made by placing a standard length along a straight line between the points of interest. We call this standard length a ‘ruler’ or a ‘unit’ and we assume that we can determine a ‘straight line’. On larger scales, various more subtle methods are required.

Our most familiar example is the Earth itself. On small scales we have no difficulty in establishing a rigid frame of reference by assuming Euclidean geometry. That is, we assume that the Earth is ‘flat’ so that trigonometry and an accurate ruler suffice to measure distance. When lasers are used we are assuming that even the near space above the surface of the Earth is Euclidean and that light follows the straight lines. On larger scales the Earth is found to be a sphere, so that its surface does not obey Euclidean geometry. Position has to be assigned by latitude and longitude, which requires the use of a combination of accurate clocks and astronomical observations in the measurements. Distance is computed between points using the rules of spherical trigonometry, rather than the Euclidean rule of Pythagoras² (see e.g. Figure 1.1).

Figure 1.1 The three perpendicular axes emanating from O are reference directions. Each axis is rigid and the projections of OP on these axes furnish the Cartesian weights or components. The theorem of Pythagoras gives the distance OP in terms of these

The Earth is not exactly a rigid sphere, but a global reference frame precise enough to detect this fact became generally available only with the advent of the Global Positioning System (GPS) of satellites. This remarkable development, based on multiple one-way radar ranging, has allowed us to measure the ebb and flow of oceans and continents in a non-rigid, spheroidal global frame. However, it assumes principles that we have yet to examine, and that will be the subject of much of this book.

Thus the procedure to define a ‘rigid’ frame of spatial reference always involves assumptions about the nature of the world around us, and it is these that we must carefully examine subsequently. Moreover such a reference frame is always an idealization. Errors are involved in determining practical spatial coordinates on every scale, so that our knowledge of distance is always approximate. Moreover the degree of idealization increases with spatial extent of the reference frame, as it becomes progressively more difficult to maintain rigidity.

In parallel with spatial position, we have managed recently to establish a global measure of time that allows us to say whether or not events occurred simultaneously. This means that a single number can be assigned to a global point-like event (e.g. the onset of an earthquake in China or sunrise at Stonehenge on Midsummer’s Day). The number is assigned by each of a network of synchronized atomic clocks distributed over the reference frame of the Earth. The sequence of such numbers defines ‘coordinate time’ for the Terrestrial Reference Frame. The difference between such numbers that encompass the beginning and end of an extended event (such as a lifetime) may be called a ‘duration’ for brevity. In practice, only durations of finite length are meaningful since no measurement can be made with infinite precision, but we normally assume that they can be arbitrarily small in principle....