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general circulation model (GCM)

gas. It is either a constant volume thermometer, in which the pressure is measured, or a constant pressure thermometer in which the volume is measured. The measurements are based on the ideal gas law:

nT R = P V ,

(where P is the pressure, V is the volume, T is the temperature, R is the gas constant, and n is the quantity of the gas, measured in grammoles), or on some calibration using the actual real gas.

gauge In general relativity, a statement of the behavior of the coordinates in use. In the ADM form, these may be explicity statements of the metric components α and βi. Or a differential equation solved by these quantities. See ADM form of the Einstein–Hilbert action.

gauge pressure The pressure in excess of the ambiant (typically local atmospheric) pressure. A tire gauge reads gauge pressure.

Gauss The unit of magnetic induction in the cgs system of units. 1 Gauss ≡ 10−4 Tesla.

Gauss–Bonnet topological invariant A quantity formed by integrating over all space time a certain skew-square of the Riemann tensor, which in four dimensions is independent of variations in the metric. Even in higher dimensional theories, it does not contribute to the propagator of the gravitational perturbations. Thus, despite the fact that the Gauss–Bonnet invariant contains fourth derivatives, it does not give rise to the massive spin-2 ghosts and therefore does not spoil the unitarity of (quantum) gravity and thus can be regarded as an acceptable ingredient in quantum gravity. See higher derivative theories, massive ghost.

Gauss coefﬁcients In an insulator, the magnetic ﬁeld may be written in terms of a potential that satisﬁes Laplace’s equation (i.e., B = where 2 = 0). For the Earth’s magnetic ﬁeld, because the planet is nearly spherical it is convenient to expand in terms of the spherical

harmonic solutions to Laplace’s equation:

∞ n	n	1
=re n 1 m 0 rre +

= =

gnm cos mφ + hmn sin mφ Pnm (cos θ)

+ re

∞ n n

n=1 m=0 re

qnm cos mφ + snm sin mφ Pnm (cos θ)

where r is the distance from the center of the Earth, re is the radius of the Earth, θ is the geocentric colatitude, φ is the east longitude, and the Pnm are Schmidt partially normalized associated Legendre polynomials. The gnm, hmn , qnm, and snm are termed the Gauss coefﬁcients. gnm and hmn represented ﬁelds of internal origin, as their associated radial functions diverge as r → 0, while qnm and snm represent external ﬁelds.

gegenschein A band of diffuse light seen along the ecliptic 180◦ from the sun immediately after sunset or before sunrise. It is created by sunlight reﬂecting off the interplanetary dust particles, which are concentrated along the ecliptic plane. To see the gegenschein, you must have very dark skies; under the best conditions it rivals the Milky Way in terms of brightness. The material within the gegenschein is slowly spiraling inward towards the sun due to the Poynting– Robertson Effect; hence, it must be continuously replaced by asteroid collisions and the debris constituting comet tails. The gegenschein is the name given to the band of light seen 180◦ from the sun while zodiacal light is the term applied to the same band of light located close to the sun.

gelbstoff See colored dissolved organic matter.

general circulation model (GCM) A set of mathematical equations describing the motion of the atmosphere (oceans) and budgets of heat and dynamically active constituents like water vapor in the atmosphere and salinity in the oceans. These equations are highly nonlinear and generally solved by discrete numerical methods. Due to computer resource limits, grid size in a GCM typically measures on the order of 100 km. Therefore, many sub-grid

general relativity

processes such as turbulence mixing and precipitation have to be parameterized based on physical considerations and/or empirical relations. Atmospheric GCMs have been successfully applied to weather forecasts for the past three decades. Efforts are being made to couple the ocean and atmospheric GCMs together to form coupled climate models for the purpose of predicting the variability and trends of the climate.

general relativity Description of gravity discovered by Einstein in which the curvature of four-dimensional spacetime arises from the distribution of matter in the system and the motion of matter is inﬂuenced by the curvature of spacetime. Described mathematically by

8πG Gµν = c2 Tµν ,

where Gµν is the Einstein tensor, constructed from the Ricci tensor, and has dimensions of inverse length, and Tµν is the four-dimensional stress-energy tensor and has dimensions of mass per unit volume. General relativity is the theory of mechanical, gravitational, and electromagnetic phenomena occurring in strong gravitational ﬁelds and involving velocities large compared to the velocity of light. General relativity generalizes special relativity by allowing that the spacetime has nonzero curvature. The observable manifestation of curvature is the gravitational ﬁeld. In the language of general relativity every object moving freely (i.e., under the inﬂuence of gravitation only) through space follows a geodesic in the spacetime. For weak gravitational ﬁelds and for objects moving with velocities small compared to c, general relativity reproduces all the results of Newton’s theory of gravitation to good approximation. How good the approximation is depends on the experiment in question. For example, relativistic effects in the gravitational ﬁeld of the sun are detectable at the sun’s surface (see light deﬂection) and out to Mercury’s orbit (see perihelion shift). Farther out, they are currently measurable to the orbit of Mars, but are nonsigniﬁcant for most astronomical observations. They are important for very accurate determination of the orbits of Earth satellites, and in the reduction of data from astrometric satellites near the Earth.

General relativity differs markedly in its predictions from Newton’s gravitation theory in two situations: in strong gravitational ﬁelds and in modeling the whole universe. The ﬁrst situation typically occurs for neutron stars (see binary pulsar) and black holes. In the second situation, the nonﬂat geometry becomes relevant because of the great distances involved. This can be explained by an analogy to the surface of the Earth. On small scales, the Earth’s surface is ﬂat to a satisfactory precision, e.g., a ﬂat map is perfectly sufﬁcient for hikers exploring small areas on foot. However, a navigator in a plane or on a ship crossing an ocean must calculate his/her route and determine his/her position using the spherical coordinates. A ﬂat map is completely inadequate for this, and may be misleading. See separate deﬁnitions of notions used in general relativity: black hole, Brans– Dicke theory, cosmic censorship, cosmological constant, cosmology, curvature, de Sitter Universe, Einstein equations, Einstein Universe, exact solution, Friedmann–Lemaître cosmological models, geodesic, gravitational lenses, horizon, light cone, light deﬂection, mass-defect, metric, naked singularity, nonsimultaneous Big Bang, relativistic time-delay, scale factor, singularities, spacetime, topology of space, white hole, wormhole.

GEO Laser interferometer gravitational wave detection being constructed near Hanover Germany; a joint British/German project. An interferometer is a L-conﬁguration with 600-m arm length. Despite its smaller size compared to LIGO, it is expected that GEO will have similar sensitivity, due to the incorporation of high quality optical and superior components, which are deferred to the upgrade phase (around 2003) for LIGO. See LIGO.

geocentric Centered on the Earth, as in the Ptolemaic model of the solar system in which planets orbited on circles (deferents) with the Earth near the center of the deferent.

geocentric coordinate time	See coordinate
time.

geocentric latitude Latitude is a measure of angular distance north or south of the Earth’s

geodesic

equator. While geodetic latitude is used for most mapping, geocentric latitude is useful for describing the orbits of spacecraft and other bodies near the Earth. The geocentric latitude φ for any point P is deﬁned as the angle be-

tween the line OP from Earth’s center O to the point, and Earth’s equatorial plane, counted positive northward and negative southward. See the mathematical relationships under latitude.

Geocentric latitude φ is the complement of the usual spherical polar coordinate in spherical geometry. Thus, if L is the longitude, and r the distance from Earth’s center, then righthanded, earth-centered rectangular coordinates (X, Y, Z), with Z along the north, and X intersecting the Greenwich meridian at the equator are given by

X= r cos φ cos (L)

Y= r cos φ sin (L)

Z= r sin φ

geocorona The outermost layer of the exosphere, consisting mostly of hydrogen, which can be observed (e.g., from the moon) in the ultraviolet glow of the Lyman α line. The hydrogen of the geocorona plays an essential role in the removal of ring current particles by charge exchange following a magnetic storm and in ENA phenomena.

geodesic The curve along which the distance measured from a point p to a point q in a space of n ≥ 2 dimensions is shortest or longest in the collection of nearby curves. Whether the geodesic segment is the shortest or the longest arc from p to q depends on the metric of the space, and in some spaces (notably in the spacetime of the relativity theory) on the relation between p and q. If the shortest path exists, then the longest one does not exist (i.e., formally its length is inﬁnite), and vice versa (in the latter case, quite formally, the “shortest path” would have the “length” of minus inﬁnity). Examples of geodesics on 2-dimensional surfaces are a straight line on a plane, a great circle on a sphere, a screw-line on a cylinder (in this last case, the screw-line may degenerate to a straight line when p and q lie on the same generator of the cylinder, or to a circle when they lie in

the same plane perpendicular to the generators). In the spacetime of relativity theory, the points (called events) p and q are said to be in a timelike relation if it is possible to send a spacecraft from p to q or from q to p that would move all the way with a velocity smaller than c (the velocity of light). Example: A light signal sent from Earth can reach Jupiter after a time between 30odd minutes and nearly 50 minutes, depending on the positions of Earth and Jupiter in their orbits. Hence, in order to redirect a camera on a spacecraft orbiting Jupiter in 10 minutes from now, a signal faster than light would be needed. The two events: “now” on Earth, and “now + 10 minutes” close to Jupiter are not in a timelike relation. For events p and q that are in a timelike relation, the geodesic segment joining p and q is a possible path of a free journey between p and q. (“Free” means under the inﬂuence of gravitational forces only. This is in fact how each spacecraft makes its journey: A rocket accelerates to a sufﬁciently large initial velocity at Earth, and then it continues on a geodesic in our space time to the vicinity of its destination, where it is slowed down by the rocket.) The length of the geodesic arc is, in this case, the lapse of time that a clock carried by the observer would show for the whole journey (see relativity theory, time dilatation, proper time, twin para- dox), and the geodesic arc has a greater length than any nearby trajectory. The events p and

qare in a light-like (also called null) relation if a free light-signal can be sent from p to q or from q to p. (Here “free” means the same as before, i.e., mirrors that would redirect the ray are not allowed.) In this case the length of the geodesic arc is equal to zero, and this arc is the path which a light ray would follow when going between p and q. The zero length means that if it was possible to send an observer with a clock along the ray, i.e., with the speed of light, then the observer’s clock would show zero timelapse. If p and q are neither in a timelike nor in a light-like relation, then they are said to be in a spacelike relation. Then, an observer exists who would see, on the clock that he/she carries along with him/her, the events p and q to occur simultaneously, and the length of the geodesic arc between p and q would be smaller than the length of arc of any other curve between p and

q. Given a manifold with metric g, the equa-

geodesic completeness

tions of the geodesics arise by variation of the action S = g(x,˙ x)ds˙ and are

x¨c + abc x˙ax˙b = λ(x)x˙c

where x˙a = dxa/ds is the tangent of the curve x = x(s). The parameter s chosen such that λ = 0 is called an afﬁne parameter. The proper time is an afﬁne parameter.

geodesic completeness A geodesic γ (τ) with afﬁne parameter τ is complete if τ can take all real values. A space-time is geodesically complete if all geodesics are complete. Geodesic completeness is an indicator of the absence of space-time singularities.

geodesy The area of geophysics concerned with determining the detailed shape and mass distribution of a body such as Earth.

geodetic latitude The geodetic latitude φ of a point P is deﬁned as the angle of the outward directed normal from P to the Earth ellipsoid with the Earth’s equatorial plane, counted positive northward and negative southward. It is zero on the equator, 90◦ at the North Pole, and −90◦ at the South Pole. When no other modiﬁer is used, the word latitude normally means geodetic latitude. See equations under latitude.

geodynamics The area of geophysics that studies the movement of planetary materials. Plate tectonics is the major subdiscipline of geodynamics, although the study of how rocks deform and how the interior ﬂows with time also fall under the geodynamics category.

geodynamo The interaction of motions in the liquid, iron rich outer core of the Earth with the magnetic ﬁeld, that generate and maintain the Earth’s magnetic ﬁeld. Similar mechanisms may (may have) exist(ed) on other planetary bodies.

Geographus 1620 Geographus, an Earthcrossing asteroid. Discovered September 14, 1951. It is named in honor of the National Geographic Society, which funded the survey that found it. It is a very elongated object, with dimensions approximately 5.1 × 1.8 km. Its mass is estimated at 4 × 1013 gm. Its rotation period

has been measured as 5.222 hours. Its orbital period is 1.39 years, and its orbital parameters are semimajor axis 1.246 AU, eccentricity 0.3354, inclination to ecliptic 13.34◦.

geoid A reference equipotential surface around a planet where the gravitational potential energy is deﬁned to be zero. On Earth, the geoid is further deﬁned to be sea level.

geoid anomalies The difference in height between the Earth’s geoid and the reference spheroidal geoid. The maximum height of geoid anomalies is about 100 m.

geomagnetic activity Routine observations of the Earth’s geomagnetic ﬁeld exhibit regular and irregular daily variations. The regular variations are due to currents ﬂowing in the upper E region resulting from neutral winds due to solar heating effects. The regular variations show repetitive behavior from one day to the next and have repeatable seasonal behavior. The small differences observed in the regular behavior can be used to learn more about the nature of the upper atmosphere. The irregular variations (daily to hourly) in the geomagnetic ﬁeld are called geomagnetic activity and are due to interactions of the geomagnetic ﬁeld and the magnetosphere with the solar wind. While these variations show no regular daily patterns, there is usually a global pattern so that common geomagnetic latitudes will show similar levels of disturbance. This behavior is described by Ap index, geomagnetic indices, geomagnetic storm, K index, Kp index, magnetosphere.

geomagnetic dip equator The locus of points about the Earth where the geomagnetic ﬁeld is horizontal to the surface of the Earth. It is where the geomagnetic inclination is zero. The geomagnetic dip equator, often referred to as the dip equator, is offset with respect to the geographic equator because the Earth’s magnetic ﬁeld is best described by a tilted dipole approximation. See geomagnetic ﬁeld.

geomagnetic disturbance Any type of rapidly varying perturbation to Earth’s magnetic ﬁeld induced by variations in the solar magnetic ﬁeld and its interaction with the magnetic ﬁeld of