In the theory of relativity space and time loose their individuality and become indistinguishable in a continuous network termed space-time. The latter provides the unique environment where all phenomena occur and all observers and observables live undisclosed until they are forced to be distinguished according to their role. Unlike other interactions, gravity is not generated by a field of force but is just the manifestation of a varied background geometry. A variation of the background geometry may be induced by a choice of coordinates or by the presence of matter and energy distributions. In the former case the geometry variations give rise to inertial forces which act in a way similar but not fully equivalent to gravity; in the latter case they generate gravity, whose effects however are never completely disentangled from those generated by inertial forces.

The space-time

A space-time is described by a four-dimensional differentiable manifold M endowed with a pseudo-Riemannian metric g. Any open set U ∈ M is homeomorphic to ℜ4 meaning that it can be described in terms of local coordinates xα, for example, with α = 0, 1, 2, 3. These coordinates induce a coordinate basis {∂/∂xα ≡ ∂α} for the tangent space TM over U with dual {dxα}.

The concept of space-time brings into a unified scenario quantities which, in the pre-relativistic era, carried distinct notions like time and space, energy and momentum, mechanical power and force, electric and magnetic fields, and so on. In everyday experience, however, our intuition is still compatible with the perception of a three-dimensional space and a one-dimensional time; hence a physical measurement requires a local recovery of the pre-relativistic type of separation between space and time, yet consistent with the principle of relativity. To this end we need a specific algorithm which allows us to perform the required splitting, identifying a “space” and a “time” relative to any given observer. This is accomplished locally by means of a congruence of time-like world lines with a future-pointing unit tangent vector field u, which may be interpreted as the 4-velocity of a family of observers. These world lines are naturally parameterized by the proper time τu defined on each of them from some initial value. The splitting of the tangent space at each point of the congruence into a local time direction spanned by vectors parallel to u, and a local rest space spanned by vectors orthogonal to u (hereafter LRSu), allows one to decompose all space-time tensors and tensor equations into spatial and temporal components. (Choquet-Bruhat, Dillard-Bleick and DeWitt-Morette 1977).

A physical measurement is meaningful only if one identifies in a non-ambiguous way who is the observer and what is being observed. The same observable can be the target of more than one observer so we need a suitable algorithm to compare their measurements. This is the task of the theory of measurement which we develop here in the framework of general relativity.

Before tackling the formal aspects of the theory, we shall define what we mean by observer and measurement and illustrate in more detail the concept which most affected, at the beginning of the twentieth century, our common way of thinking, namely the relativity of time.

We then continue on our task with a review of the entire mathematical machinery of the theory of relativity. Indeed, the richness and complexity of that machinery are essential to define a measurement consistently with the geometrical and physical environment of the system under consideration.

Most of the material contained in this book is spread throughout the literature and the topic is so vast that we had to consider only a minor part of it, concentrating on the general method rather than single applications. These have been extensively analyzed in Clifford Will's book (Will, 1981), which remains an essential milestone in the field of experimental gravity. Nevertheless we apologize for all the references that would have been pertinent but were overlooked.

A test gyroscope is a point-like massive particle having an additional “structure” termed spin. A spinning body is not, strictly speaking, point-like because its average size cannot be less than the ratio between its spin and its mass, in geometrized units. This guarantees that no point of the spinning body moves with respect to any observer at a velocity larger than c.

Rotation is a common feature in the universe so knowing the dynamics of rotating bodies is almost essential in modern physics. Although in most cases the assumption of no rotation is necessary to make the equations tractable, the existence of a strictly non-rotating system should be considered a rare event, and a measurement which revealed one would be of great interest. This is the case for the massive black holes which appear to exist in the nuclei of most galaxies. Detailed measurements are aimed at detecting their intrinsic angular momentum from the behavior of the surrounding medium. A black hole, however, could also be seen directly if we were able to detect the gravitational radiation it would emit after being perturbed by an external field. A direct measurement of gravitational waves is still out of reach for earthbound detectors; nevertheless they are still extensively searched for since they are the ultimate resource to investigate the nature of space-time.

The aim of modern astronomy is to uncover the properties of cosmic sources by measuring their key parameters and deducing their dynamics. Black holes are targets of particular interest for the role they have in understanding the cosmic puzzles and probing the correctness of current theories. Black holes can be considered simply as deep gravitational potential wells; therefore their existence can only be inferred by observing the behavior of the surrounding medium. The latter can be made of gas, dust, star fields, and obviously light, but all suffer tidal strains and deformations which herald, out of the observer's perspective, the black hole's existence and type. Essential tools for the acquisition of this knowledge are the equations of relative acceleration which stand as basic seeds for any physical measurement. We shall revisit them for specific applications, but will always neglect electric charge in our discussion.

Measurements in Schwarzschild space-time

Consider a collection of particles undergoing tidal deformations; we shall deduce how these would be measured by any particle of the collection, taken as a fiducial observer. Let us assume that the test particles of the collection move in spatially circular orbits in Schwarzschild space-time whose metric is given by (8.1). Indeed, the physical measurements which can be made in the rest frame of the fiducial observer in the collection are the most natural to be performed in satellite experiments.

A substantial fraction of our data on many Solar System objects has been obtained by close-up studies conducted by spacecraft. This Appendix starts with a short section on rocketry (how a rocket works). Section F.2 contains tables listing many of the most significant lunar and interplanetary spacecraft and astronomical observations in space. This appendix further includes diagrams of two historically significant spacecraft (Figs. F.1 and F.5), and two historic images (Figs. F.2 and F.6).

Rocketry

The principles of ‘rocket science’ are actually quite simple, although many practical aspects of ‘rocket engineering’ are far more complicated. A rocket accelerates by expelling gas (or plasma) at high velocity. Conservation of momentum implies that the velocity, ν, of the rocket of mass M (which includes propellent), expelling gas at velocity νexp and rate dM/dt satisfies:

where Fext accounts for all external forces on the rocket. Equation (F.1) is known as the fundamental rocket equation.

In a uniform gravitational field that induces an acceleration gp with no other external forces, the rocket equation reduces to

Integrating equation (F.2) and setting v = 0 at t = 0 gives

where M0 is the mass at t = 0, and there is a minus sign in front of the last term in equation (F.3) because the gravitational force is directed downwards. Note that there is a premium to burning fuel rapidly – the shorter the burn time, the greater the velocity for given ejection speed and mass.

A meteorite is a rock that has fallen from the sky. It was a meteoroid (or, if it was large enough, an asteroid) before it hit the atmosphere and a meteor while heated to incandescence by atmospheric friction. A meteor that explodes while passing through the atmosphere is termed a bolide. Meteorites that are associated with observations prior to or of the impact are called falls, whereas those simply recognized in the field are referred to as finds.

The study of meteorites has a long and colorful history. Meteorite falls have been observed and recorded for many centuries (Fig. 8.1). The oldest recorded meteorite fall is the Nogata meteorite, which fell in Japan on 19 May 861. Iron meteorites were an important raw material for some primitive societies. However, even during the Enlightenment it was difficult for many people (including scientists and other natural philosophers) to accept that stones could possibly fall from the sky, and reports of meteorite falls were sometimes treated with as much skepticism as UFO ‘sightings’ are given today. The extraterrestrial origin of meteorites became commonly acknowledged following the study of some well-observed and documented falls in Europe around the year 1800. The discovery of the first four asteroids, celestial bodies of sub-planetary size, during the same period added to the conceptual framework that enabled scientists to accept extraterrestrial origins for some rocks.

In the previous two chapters, we discussed the atmospheres and surface geology of planets. Both of these regions of a planet can be observed directly from Earth and/or space. But what can we say about the deep interior of a planet? We are unable to observe the inside of a planet directly. For the Earth and the Moon we have seismic data, revealing the propagation of waves deep below the surface and thereby providing information on the interior structure (§6.2). The interior structure of all other bodies is deduced through a comparison of remote observations with observable characteristics predicted by interior models. The relevant observations are the body's mass, size (and thus density), its rotational period and geometric oblateness, gravity field, characteristics of its magnetic field (or absence thereof), the total energy output, and the composition of its atmosphere and/or surface. Cosmochemical arguments provide additional constraints on a body's composition, while laboratory data on the behavior of materials under high temperature and pressure are invaluable for interior models. Quantum mechanical calculations are used to deduce the behavior of elements (especially hydrogen) at pressures inaccessible in the laboratory.

In this chapter we discuss the basics of how one can infer the interior structure of a body from the observed quantities.