The importance of being ice

The inner planets and moons with which we are familiar (Earth, Mars, the Moon, etc.) consist of rock-forming minerals that are dominantly silicates. Silicates are composed of silicon and oxygen bonded together, usually in combination with aluminum, magnesium, iron, calcium, sodium, and other metals. On the Galilean satellites (Io excepted), the crust-forming rock is water ice, where hydrogen substitutes for silicon in its bonds with oxygen. Other ices are present in small quantities, such as methane, ammonia, solid nitrogen, carbon monoxide, carbon dioxide or sulfur dioxide, and other more complex molecules. Silicates and carbonaceous material are also present in large quantities but, except for dirtying the surface ices, they tend to reside deep in the interior, the less dense ices in the outer layers. These ices are volatile under average Earth conditions, but at surface temperatures of 100 K (–285 °F) or so, ice is much harder and more durable on Europa than the crunchable ice cubes in our drinks. Thus, the outer layers of these bodies behave much like the terrestrial planets, and are subject to many of the same processes, as well as some peculiar to the icy satellites. The dominance of ice as a crust-forming rock has several important consequences, however, for the geologic processes that occur within these planetary bodies.

The mathematical structure of general relativity makes its equations quite remote from a direct understanding of their content. Indeed, the combination of a covariant four-dimensional description of the physical laws and the need to cope with the relativity of the observations makes a physical measurement an elaborate procedure. The latter consists of a few basic steps:

1. (i) Identify the covariant equations which describe the phenomenon under investigation.

2. (ii) Identify the observer who makes the measurements.

3. (iii) Choose a frame adapted to that observer, allowing the space-time to be split into the observer's space and time.

4. (iv) Decide whether the intended measurement is local or non-local with respect to the background curvature.

5. (v) Identify the frame components of those quantities that are the observational targets.

6. (vi) Find a physical interpretation of the above components, following a suitable criterion such as a comparison with what is known from special relativity or from non-relativistic theories.

7. (vii) Verify the degree of residual ambiguity in the interpretation of the measurements and decide on a strategy to eliminate it.

Clearly, each step of the above procedure relies on the previous one, and the very first step provides the seed of a measurement despite the mathematical complexity.

A physical measurement requires a collection of devices such as a clock, a theodolite, a counter, a light gun, and so on. The operational control of this instrumentation is exercised by the observer, who decides what to measure, how to perform a measurement, and how to interpret the results. The observer's laboratory covers a finite spatial volume and the measurements last for a finite interval of time so we can define as the measurement's domain the space-time region in which a process of measurement takes place. If the background curvature can be neglected, then the measurements will not suffer from curvature effects and will then be termed local. On the contrary, if the curvature is strong enough that it cannot be neglected over the measurement's domain, the response of the instruments will depend on the position therein and therefore they require a careful calibration to correct for curvature perturbations. In this case the measurements carrying a signature of the curvature will be termed non-local.

Observers and physical measurements

A laboratory is mathematically modeled by a family of non-intersecting time-like curves having u as tangent vector field and denoted by Cu; this family is also termed the congruence. Each curve of the congruence represents the history of a point in the laboratory. We choose the parameter τ on the curves of Cu so as to make the tangent vector field u unitary; this choice is always possible for non-null curves.