The Minkowski, de Sitter and anti-de Sitter space-times described in Chapters 3–5 are the only conformally flat solutions of Einstein's vacuum field equations with a possibly non-zero cosmological constant. However, there also exist non-vacuum conformally flat space-times. The most important of these are the perfect fluid FLRW cosmologies that were reviewed in the previous chapter. Conformally flat radiative space-times with pure radiation will be discussed in Chapters 17 and 18. Another important conformally flat space-time is the Bertotti–Robinson universe which contains a uniform non-null electromagnetic field.

Interestingly, the Minkowski, de Sitter and anti-de Sitter solutions are constant-curvature 4-spaces (with ten isometries), the FLRW cosmologies (having six isometries) are foliated by constant-curvature 3-spaces, while the Bertotti—Robinson space-time is a direct product of two 2-spaces of constant curvature (and so also has six isometries). In fact, it belongs to a larger family of geometries of this type, which also include the Nariai, anti-Nariai and Plebański–Hacyan direct-product space–times. In general, these are electrovacuum solutions, which apart from the Bertotti—Robinson solution are of algebraic type D. These will all be described in this chapter.

It is also natural here to include a short description of the Melvin universe, which is another interesting electrovacuum type D solution.

The Bertotti–Robinson solution

The conformally flat solution of the Einstein–Maxwell equations for a nonnull electromagnetic field was obtained independently by Bertotti (1959) and Robinson (1959), see also Levi-Civita (1917).

The previous nine chapters have been mainly devoted to various black hole space-times and other solutions that at least contain stationary regions. The present chapter, and the following four, will concentrate on some of the most important exact solutions which represent gravitational waves in general relativity. It is convenient to start here with the simplest case of non-expanding waves, known as pp-waves, which are also important in the context of higher-dimensional theories, and their subclass of plane waves. General classes of solutions with non-expanding and expanding waves will be described, respectively, in Chapters 18 and 19, and both types of impulsive waves will be reviewed in Chapter 20. The collision and interaction of plane waves will be covered in Chapter 21. These may all be associated with other forms of pure radiation. Cylindrical gravitational waves will be more briefly described in Section 22.3.

The class of pp-waves describes plane-fronted waves with parallel rays. They are defined geometrically by the property that they admit a covariantly constant null vector field k, and may represent gravitational waves, electromagnetic waves, some other forms of matter, or any combination of these.

Using (2.15) and the properties described in Section 2.1.3, it follows from the defining property kμ;v = 0 that the vector field k is tangent to an expansion-free, shear-free and twist-free null geodesic congruence. Since this congruence is non-twisting, there exists a family of 2-surfaces orthogonal to k which may be considered as wave surfaces (see Kundt, 1961). It is also implied by the fact that k is covariantly constant that these wave surfaces are indeed planar and that rays orthogonal to them are parallel.

Physical situations exist in which different regions of space-time have different matter contents. These can be modelled by compound space-times. (For example, in Subsection 9.5.2, a space-time was discussed in which a region represented by the Vaidya metric is sandwiched between a Minkowski and a Schwarzschild region.) In such cases, we have followed the approach of Lichnerowicz (1955) and used a global metric form that is at least C3 everywhere except on junctions, represented by hypersurfaces N, on which the metric is only C. Since the curvature of the space-time involves second derivatives of the metric, such situations give rise to discontinuities in the curvature across N. When N is null, these may represent various forms of shock waves which propagate with the speed of light.

More extreme situations may also be considered for which the metric is still (at least) C3 almost everywhere, but merely C0 on some hypersurface N. In such situations, some components of the curvature of the space-time will formally contain a δ-function. When N is null, these may be interpreted as impulsive waves. They are regarded as impulsive gravitational waves when the δ-function components occur in the Weyl tensor, or impulsive components of some kind of null matter when they occur in the Ricci tensor.

The geometry of impulsive waves in flat space was first described in detail by Penrose (1972). However, some particular examples of exact solutions which include impulsive gravitational waves or thin sheets of null matter were known before then. Many further examples have subsequently been obtained, and involve a variety of backgrounds.

Summary

The Channeled Scabland comprises a regional anastomosing complex of overfit stream channels that were eroded by Pleistocene megaflooding into the basalt bedrock and overlying sediments of the Columbia Plateau and Columbia Basin regions of eastern Washington State, USA. Immense fan complexes were emplaced where sediment-charged water entered structural basins. The cataclysmic flooding produced macroforms eroded into the rock (coulees and trenched spur buttes) and sediment (streamlined hills and islands). Several types of depositional bars also are scaled to the channel widths. The erosional mesoforms (scaled to flow depth) include longitudinal grooves, butte-and-basin scabland, potholes, inner channels and cataracts. These make up an erosional sequence that is scaled to levels of velocity, power per unit area and depths achieved by the cataclysmic flooding. Giant current ‘ripples’ (dunes) developed in the coarse gravel bedload, and large-scale scour marks were formed around various flow obstacles, including rock buttes and very large boulders.

Introduction

The Channeled Scabland region (Figure 5.1) is that portion of the basaltic Columbia Plateau and Columbia Basin that was subjected to periodic cataclysmic flooding during the late Pleistocene, resulting in a distinctive suite of flood-related landforms. Bretz (1923a, pp. 577–578) defined ‘scablands’ as ‘lowlands diversified by a multiplicity of irregular and commonly anastomosing channels and rock basins eroded into basalt…’ The term was in local use in reference to chaotically eroded tracts of bare basalt which occur in relatively large channels that the floods cut through the loess cover on the plateau.

Summary

Megafloods from glacial lakes were common along the margin of the Laurentide Ice Sheet during the last deglaciation. These outbursts resulted in a complex network of spillways with characteristic erosional and depositional forms. Meltwater was impounded in front of and beneath the ice margin as the Late Wisconsin Laurentide Ice Sheet melted back from its maximum extent. Lakes that formed in this dynamic environment drained completely or partially. Various factors aided in the impoundment of meltwater lakes, including isostatic depression of the land surface in the vicinity of the retreating margin, topography that sloped toward the ice margin, glacial erosion of trough-like depressions by ice streaming in major outlet lobes, and moraine ridges formed when the ice was more extensive. Subaerial megafloods were triggered probably by the failure of ice-cored or sediment-cored moraine dams or rapid incision of outlets caused by incoming subaerial and/or subglacial megafloods. Complete drainage of lakes was likely where non-resistant glacial or bedrock materials made up the outlet region.

Proglacial megaflood discharges were typically 0.1–1.0 Sv, short in duration, and, in some places, achieved velocities in excess of 10 ms−1. Outburst flows were highly erosive and carved a suite of small-scale to large-scale erosional forms, including potholes, longitudinal grooves, streamlined hills, transverse bedforms, anastomosing channels and spillways. Not all these forms are present along all megaflood pathways, except for spillways, which are ubiquitous; they are trench-shaped, 1–4 km wide, and tens of metres deep.

Summary

Jökulhlaups in Iceland may originate from marginal or subglacial sources of water melted by atmospheric processes, permanent geothermal heat or volcanic eruptions. The release of meltwater from glacial lakes can take place as a result of two different conduit initiation mechanisms and the subsequent drainage from the lake occurs by two different modes. Drainage can begin at pressures lower than the ice overburden in conduits that expand slowly over days or weeks. Alternatively, the lake level may rise until the ice dam is lifted and water discharges rise linearly, peaking in a time interval of several hours to 1–2 days. The linearly rising floods are often associated with large discharges and floods following rapid filling of subglacial lakes during subglacial eruptions or dumping of one marginal lake into another. Jökulhlaups during eruptions in steep ice and snow-covered stratovolcanoes are swift and dangerous and may become lahars and debris-laden floods. The jökuhlaups can be seen as modern analogues of past megafloods on the Earth and their exploration may improve understanding of ice–volcano processes on other planets.

Introduction

The Icelandic word jökulhlaups means glacier-related floods ranging from small bursts to megafloods of enormous landscaping impact. They may originate from marginal or subglacial sources of water melted by atmospheric processes, permanent geothermal heat or volcanic eruptions. They may range from floods consisting almost entirely of water to hyperconcentrated fluid–sediment mixtures and even more destructive gravity-driven mass flow of mostly volcanic materials mixed with water and ice.

Summary

The flow of subaerial jökulhlaups is in principle similar to other subaerial water floods, such as dam-break floods, although some jökulhlaups may carry so much suspended sediment and ice fragments that they would be more appropriately described as rapidly flowing debris flows or lahars. Many subaerial jökulhlaups start out as subglacial floods and propagate as subaerial floods below an outlet at the glacier terminus. Other jökulhlaups, in particular many outburst floods caused by volcanic eruptions, lead to a partial or almost complete breakup of the glacier along the flow path and become subaerial after flowing only a short distance subglacially. The dynamics of the subaerial part of jökulhlaups differs fundamentally from the dynamics of the part of the flood that flows along the bed of the glacier or ice cap. The estimated discharge of jökulhlaups observed at many locations in Iceland during the twentieth century ranges from 0.1 to 300 × 103 m3 s−1 and prehistoric jökulhlaups have been estimated to have reached on the order of 106 m3 s−1. The subaerial propagation of jökulhlaups can cause widespread damage to buildings, roads, communication lines and farmland. Two-dimensional numerical modelling, based on a depth-integrated formulation of the dynamics of shallow water flow, has been used to study the flow of subaerial jökulhlaups at four locations in Iceland, two of which are described in this chapter. Model results include estimates of travel times, the most probable flood routes and the extent of lowland areas that might be flooded.