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Weyl tensor

western boundary currents Ocean currents flowing on the western boundary at speeds much higher than those in the rest of an ocean basin. This western intensification of ocean currents is a combined effect of rotation and surface curvature of the Earth, or the so-called beta-effect. The Gulf Stream and Kuroshio Current are the surface western boundary currents of the North Atlantic and North Pacific, respectively. In the Atlantic, deep western boundary currents are also observed, transporting deep water formed in the Nordic and Labrador Seas.

western boundary intensification The intensification of current toward the western boundary of the ocean due to the variation of the Coriolis parameter with latitude (the β-effect).

west Greenland current A branch of the east Greenland current that branches northwestward along the southwest coast of Greenland.

westward drift The movement of features of the geomagnetic field to the west. This was first noticed by Halley in the 17th century, who hypothesized that the Earth’s magnetic field emanates from magnetized layers in the Earth’s interior that are in rotation with respect to the surface. It is now understood that the magnetic field is generated in a molten iron outer core, and that changes in the magnetic field observed at the Earth’s surface are related either to flows at the top of the core, or diffusion of the magnetic field. If the former is responsible for the westward drift, then it would be related to a general westward flow of the core’s surface. The magnitude of the drift is around 0.2 to 0.4◦ per year, although it appears to vary with time and latitude. It is better determined in some places than others: for example, the field in the Pacific region is relatively plain, so it is difficult to tell whether or not it is drifting, and some models of core flow show flows to the east there. Part of the time variation of the westward drift may be related to torsional oscillations. See core flow.

wet-bulb temperature The temperature to which a parcel of air is cooled by evaporating water into it gradually, adiabatically, and at constant pressure until it is saturated. It is measured

directly by a thermometer whose bulb is covered by a moist cloth over which air is drawn.

wetness (or degree of saturation) (S)

The

proportion of pore space that contains water:

S =

Vw

=

θ

Va + Vw

φ

where Vw is the volume of water in the sample, and Va is the volume of air in the sample. When a soil or rock sample is saturated with water the volume of air goes to zero, the volumetric water content equals porosity, and wetness equals one.

wetted perimeter A linear measure of the length of a river or canal cross-section that is wetted by flowing fluid.

Weyl space-times (1917) The static and axially symmetric vacuum metrics of the form

ds2 = −e2U dt2 + e−2U

e2γ (dρ2 + dz2) + ρ2dφ2

where ρ and z are generalized cylindrical coordinates. The function U = U(ρ, z) satisfies the Laplace equation )U = 0 as follows from the vacuum Einstein equations. The linearity of the Laplace equation allows a superposition of solutions. The function γ = γ (ρ, z) is determined by the line integral in the (ρ, z) plane

γ =

ρ

∂ρ

 

−

∂z

dρ

 

 

 

 

 

∂U

 

2

 

∂U

2

 

+2

∂z

 

∂ρ dz .

 

 

 

 

 

∂U

 

∂U

 

 

 

 

 

The solution U = m/(ρ2 + z2)1/2 yields the axisymmetric Curzon metric (rather than the spherical Schwarzschild solution).

Weyl tensor A tensor, components of which are the following linear combination of the components of the curvature tensor:

2

Cαβρσ =Rαβρσ − −

n 2

Rα[ρ gσ ]β − Rβ[ρ gσ ]α

2R

+(n − 1)(n − 2) gα[ρ gσ ]β .

© 2001 by CRC Press LLC

Wheeler– DeWitt equation

The Weyl tensor shares all the permutation symmetries with the Riemann tensor Rαβρσ . Moreover, Weyl tensor possesses some additional fea-

tures. It is traceless Cαβασ = 0 and does not transform under a local conformal transforma-

tion

g

g

e2σ(x) ,

Cα

=

Cα .

 

µν = ¯µν ·

 

βρσ

¯ βρσ

Hence, it is also called conformal tensor. The minimal dimension of a manifold in which the Weyl-tensor exists is four. (The conformal structure of a 3-manifold is carried by the three-index Cotton tensor.) In the theory of general relativity, the stress-energy tensor of a given matter distribution determines the traces of the curvature locally by Einstein’s gravitational equations, and the Weyl tensor is the carrier of the intrinsic degrees of freedom of the gravitational field. See curvature tensor, Rie- mann tensor.

Wheeler–DeWitt equation The quantized form of the Hamiltonian constraint obtained by replacing the classical canonical variables γij and their momenta πij with operators acting on a Hilbert space of states. The simplest way to employ quantization is to map

γij

→

γˆij = γij ×

∂

 

 

πij

→

πˆ ij = −i h

 

,

 

 

∂γ

 

 

 

¯

 

ij

in analogy with what is done with coordinates and momenta of a particle in ordinary quantum mechanics.

This procedure is, however, plagued by several formal problems, e.g., the operator ordering ambiguity: both the super-Hamiltonian and the super-momentum densities (which should be quantized as well) contain products of elements of the metric tensor times momenta. But no unique way of writing the quantized version of such products is given a priori, since any choice would lead to the same classical expressions. This reflects the possibility of defining canonical operators differing from the ones shown above and thus different quantum theories for the same 3+1 splitting of space-time.

Whatever form of quantization is chosen, the

Wheeler–DeWitt equation can be formally writ-

ten as

Hˆ 5 = 0 ,

© 2001 by CRC Press LLC

514

where 5 = 5[γ , φ] is often referred to as the wave function of the universe and is a functional of the metric tensor γ and the matter fields φ on the hypersurface 6t . This equation is in the form of a time-independent Schrödinger equation which selects out the zero energy eigenstates as physical states. Indeed, one of the most intriguing puzzles of this approach to quantum gravity is the lack of a time variable. One possible way of solving this puzzle is to resort to the semiclassical approximation (see time in semi- classical gravity).

Physical states 5, which satisfy both the Wheeler–DeWitt equation and the quantized super-momentum constraints, are functionals of the 3-metric of space (see Superspace). In order to build a Hilbert space out of them, one must also face the problem of defining a conserved scalar product (the constraints are functional differential equations). Since solutions of the constraints are not available in general and because of all the formal problems mentioned above, one usually imposes some symmetry in order to reduce the number of degrees of freedom of the metric before quantizing. In some cases this is sufficient to reduce the metric tensor and the matter fields to a form that contains only a finite number N of functions of time only (see minisuperspace). Therefore, the quantized secondary constraints become ordinary differential equations for 5 as a function of those N variables.

It is by now generally accepted that the Wheeler–DeWitt approach to quantizing gravity cannot lead to the ultimate quantum theory of the gravitational interaction and better candidates are to be found among the theories of extended objects (strings and n-dimensional branes). See Hamiltonian and momentum constraints in general relativity, semiclassical gravity, wave-function of the universe.

whimper A space-time singularity not accompanied by infinities in physical quantities like density and pressure (such infinities are essential in a Big Bang singularity), though some curvature components may be infinite.

whistlers Pulses of electromagnetic wave energy with frequencies in 0.3 to 35 kHz (VLF) range. When converted to audio signals, they are

Wien’s displacement law

heard as “whistles” with steadily falling pitch that lasts for a few tenths of a second (short whistlers) to several seconds (long whistlers). Whistlers are generated by lightning sources or atmospherics. The electromagnetic energy produced in the lightning sources travels along the Earth’s magnetic lines in the ionosphere almost without any attenuation to the other hemisphere. A part of this energy in the other hemisphere is reflected back along the same magnetic lines producing whistlers in a radio receiver in the hemisphere of origin, and another part enters the ionosphere and propagates in the Earth – ionosphere waveguide mode to the hemisphere of origin producing short whistlers. Since the losses in the ionosphere are small, many echoes may occur in what is known as an “echo” train before they are lost in the background noise. The changing pitch indicates that the path time “t” along the magnetic line of force is related to the

frequency “f” by the dispersion relation

√

D = t f

where D is the dispersion and may range from

√

about 10 to 200 sec. Since t will depend on the line of force along which the energy travels and the electron density which decreases with height, D is found to increase with increase in electron density. The whistler properties can be explained by the application of the magnetoionic theory to low frequency wave propagation in the ionosphere. Whistlers are used as a research tool to study the electron density and associated properties in the Earth’s magnetosphere.

whistler waves A low-frequency plasma wave which propagates parallel to the magnetic field at a frequency less than the electroncyclotron frequency. Whistler waves are circularly polarized, rotating in the same sense as the electrons in the plasma. Whistlers are so-named because of their characteristic audio frequency tone.

white dwarf A star supported entirely by electron degeneracy pressure, formed by contraction when internal fusion energy sources are exhausted.

white hole The time-reverse of a black hole, i.e., a region of the spacetime that can only eject

matter and radiation but can accrete none of it. A white hole would be a naked singularity, with a past event horizon, which does not shield it from our view. Light from a white hole would be blue-shifted. A white hole can be imagined as a Big Bang taking place in a limited volume of space, while the remaining part of the space already exists. White holes are natural in inhomogeneous models of the universe — they are places where the Big Bang occurs later and was still going on while it had already been completed elsewhere. Since literally everything can come out of these holes, thus strongly violating causality, a cosmic censorship principle has been invoked to avoid their existence in our universe. There is currently no evidence that they exist. See black hole, future/past event horizon, naked singularity.

white light Solar radiation integrated over the visible portion of the spectrum (4000 to 7000 Å) to produce a broad-band (white light) signal. Best used to observe the photosphere and also in eclipse or coronagraph observations of the solar corona.

white light flare A major flare in which compact regions become visible in white light. Such flares are usually strong X-ray, radio, and particle emitters.

Widmanstä tten pattern A distinctive pattern of crystal faces found in nickel-rich iron meteorites. When an iron-nickel mixture is heated to near melting and slowly cooled, two types of crystals are formed: a nickel-poor phase (kamacite) and a nickel-rich phase (taenite). The pattern formed by the growth of these two crystal phases can be seen when the meteorite is polished and etched with acid.

Wien’s displacement law (Wien law) The wavelength λm at which the maximum radiation intensity (ergs/sec/cm2/steradian/wavelength interval) of a black body radiator occurs is inversely proportional to the absolute temperature T , so:

T λm = C ,

where C is a constant, equal to 0.28978 cm-K.

© 2001 by CRC Press LLC

515

Wien distribution law

Wien distribution law A relation between the monochromatic emittance F (erg/sec/wavelength/cm2/steradian) of an ideal black body and that body’s temperature T :

F λ−5f (T λ) .

Wien’s law

See Wien’s displacement law.

wiggle (cosmic string)

After cosmological

phase transitions in which cosmic strings are produced, the strings will trace regions where the Higgs field departs from the low temperature vacuum manifold of the theory. These regions have random spatial locations and, therefore, the ensuing string has no reason to be straight. The string will, in general, possess an irregular shape and small scale structures in the form of wiggles will accumulate all along its length. Further interactions will make the number of these small irregularities increase. Universal expansion does not eliminate these wiggles; hence, wiggly strings are the most natural outcome during network evolution.

This structure can be analytically approximated by an effective equation of state describing how a distant observer would perceive the

string. The effective energy per unit length ˜

U

(larger than the Goto–Nambu energy U, as there is more string matter in a given segment due to the wiggles) and the effective tension T˜ (smaller than the tension of a Goto–Nambu string) will

UT

U2

.

satisfy the relation ˜ ˜ =

 

Furthermore, the space around the wiggly string is no longer locally flat, and thus the string will behave like a standard gravitational attractor. Now particle geodesics in the vicinity of a wiggly string will be deflected by two mechanisms: the conical deficit angle deviation (from far away the string looks pretty straight and featureless) and the standard gravitational attraction. The relative velocity between two test particles on different sides of the string will be

δv = 8

 

˜

 

sγs + 4πG

 

πGUv

 

 

 

˜

− ˜

 

/

(vsγs)

U

T

 

 

 

where vs is the velocity of the string, γs is the corresponding Lorentz factor (1 − vs2/c2)−1/2, and G is Newton’s constant. The first of these

effects dominates for fast moving strings and would lead to the formation of wakes. The second one is relevant for slow strings and could be at the root of the generation of filamentary distributions of astrophysical large scale structures. See cosmic phase transition, cosmic string, cos- mic topological defect, deficit angle (cosmic string), wake (cosmic string).

Wilson cycle The Atlantic ocean is presently “opening”, growing wider due to the creation of new sea flow at the Mid-Atlantic Ridge. It is expected that in the future, subduction zones will form on the boundaries of the Atlantic, and the ocean will close resulting in a continental collision between the Americas and Europe and Africa. This opening and closing of the Atlantic has happened twice in the past and is known as the Wilson cycle, in honor of J. Tuzo Wilson who first proposed this behavior. The last closing of the Atlantic Ocean created the Appalachian Mountains and occurred about 200 million years ago.

wind The general term for moving air, typically driven by naturally arising pressure gradients depending on altitude differences, solar heating, surface temperature, and the Coriolis force due to the Earth’s rotation.

wind chill factor The perceived sensation of temperature, TWC, in the presence of a wind. Based on studies of the rate of water freezing under different conditions.

The following equations can be used to determine the wind chill factor TWC:

Wind speed V given in mph:

TWC =Ts − ((Ts − T ) (.474266

√

+ .303 V −.02 V ))

Wind speed in knots:

TWC =Ts − ((Ts − T ) (.474266

√

+ .325518 V −.0233 V ))

Wind speed in m/s:

TWC =Ts − ((Ts − T ) (.474266

√

+ .4538 V −.045384 V ))

© 2001 by CRC Press LLC

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