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Einstein tensor

of Einstein–Cartan gravity is a direct generalization of the Einstein–Hilbert action of General Relativity:

EC = −

16πG

√

−gg

˜µν

µνR

where

µν is the (nonsymmetric) Ricci tensor

with torsion

∂ 8λ

−

∂ 8λ

8λ

8τ

8λ

˜µν =

λ ˜µν

ν ˜µλ

+ ˜

µν

˜λτ

− ˜µλ

˜τν

and ˜ λ is a non-symmetric afﬁne connection

8µµ

with torsion, which satisﬁes the metricity condition ˜ µgαβ = 0. The above action can be rewritten as a sum of the Einstein–Hilbert action and the torsion terms, but those terms are not dynamical since the only one derivative of the torsion tensor appears in the action in a surface term. As a result, for pure Einstein–Cartan gravity the equation for torsion is Tµνλ = 0, and the theory is dynamically equivalent to General Relativity. If the matter ﬁelds provide an external current for torsion, the Einstein-Cartan gravity describes contact interaction between those currents. See metricity of covariant derivative, torsion.

Einstein equations The set of differential equations that connect the metric to the distribution of matter in the spacetime. The features of matter that enter the equations are the stressenergy tensor Tµν, containing its mass-density, momentum (i.e., mass multiplied by velocity) per unit volume and internal stresses (pressure in ﬂuids and gases). The Einstein equations are very complicated second-order partial differential equations (10 in general, unless in special cases some of them are fulﬁlled identically) in which the unknown functions (10 components of the metric gµν) depend, in general, on 4 variables (3 space coordinates and the time):

Gµν ≡ Rµν − 2 gµνR

8πG

=c2 Tµν

Here Gµν is the Einstein tensor, Rµν the Ricci tensor and R its trace, and G is Newton’s constant. The factor c−2 on the right assumes a choice of dimensions for Gµν of [length]−2 and

for Tµν of [mass/length3]. It has been common since Einstein’s introduction of the cosmological constant ; to add ;gµν to the left-hand side. Positive ; produces a repulsive effect at large distances. Modern theory holds that such effects arise from some other (quantum) ﬁeld coupled to gravity, and thus arise through the stress-energy tensor. A solution of these equations is a model of the spacetime corresponding to various astronomical situations, e.g., a single star in an otherwise empty space, the whole universe (see cosmological models), a black hole. Unrealistic objects that are interesting for academic reasons only are also considered (e.g., inﬁnitely long cylinders ﬁlled with a ﬂuid). In full generality, the Einstein equations are very difﬁcult to handle, but a large literature exists in which their implications are discussed without solving them. A very large number of solutions has been found under simplifying assumptions, most often about symmetries of the spacetime. Progress is also being made in computational solutions for general situations. Through the Einstein equations, a given matter-distribution inﬂuences the geometry of spacetime, and a given metric determines the distribution of matter, its stresses and motions. See constraint equations, gauge, signature.

Einstein–Rosen bridge A construction by A. Einstein and N. Rosen, based on the Schwarzschild solution, wherein at one instant, two copies of the Schwarzschild spacetime with a black hole of mass M outside the horizon were joined smoothly at the horizon. The resulting 3- space connects two distant universes (i.e., it has two spatial inﬁnities) each containing a gravitating mass M, connected by a wormhole. Subsequent study showed the wormhole was dynamic and would collapse before any communication was possible through it. However, recent work shows that certain kinds of exotic matter can stabilize wormholes against such collapse.

Einstein summation convention See summation convention.

Einstein tensor The symmetric tensor

Gab = Rab − 2 gabR

Einstein Universe

where Rab is the Ricci tensor, gab is the metric tensor, and R = Raa is the Ricci scalar. The divergence of the Einstein tensor vanishes identically. See gravitational equations, Ricci tensor.

Einstein Universe The historically ﬁrst cosmological model derived by Einstein himself from his relativity theory. In this model, the universe is homogeneous, isotropic, and static, i.e., unchanging in time (see homogeneity, isotropy). This last property is a consequence of the longrange repulsion implied by the cosmological constant which balances gravitational attraction. This balance was later proved, by A.S. Eddington, to be unstable: any small departure from it would make the universe expand or collapse away from the initial state, and the evolution of the perturbed model would follow the Friedmann–Lemaître cosmological models. Because of its being static, the Einstein Universe is not in fact an acceptable model of the actual universe, which is now known to be expanding (see expansion of the universe). However, the Einstein Universe played an important role in the early development of theoretical cosmology

— it provided evidence that relativity theory is a useful device to investigate properties of the universe as a whole.

ejecta In impact cratering, material that is tossed out during the excavation of an impact crater. The ejected material is derived from the top 1/3 of the crater. Some of the ejected material falls back onto the ﬂoor of the crater, but much is tossed outside the crater rim to form an ejecta blanket. Ejecta blankets on bodies with dry surface materials and no atmosphere (like the moon and Mars) tend to display a pattern with strings of secondary craters (craters produced by material ejected from the primary crater) radiating outward from the main crater. Glassy material incorporated into the ejecta can appear as bright streaks, called rays. A radial ejecta blanket is typically very rough within one crater radii of the rim — few individual secondary craters can be discerned. Beyond one crater radii, the ejecta blanket fans out into radial strings of secondary craters. On bodies with a thick atmosphere (like Venus) or with subsurface ice (like Mars and Jupiter’s moons of Ganymede and Callisto), impact craters are

typically surrounded by a more ﬂuidized ejecta pattern, apparently caused by the ejecta being entrained in gas from either the atmosphere or produced during vaporization of the ice by the impact. The extent of the ﬂuidized ejecta blankets varies depending on the state/viscosity of the volatiles and the environmental conditions. In supernova physics, the ejecta are the material blown from the stars as a result of the explosion.

Ekman convergence The stress on the Earth’s surface varies from place to place and hence so does the Ekman transport. This leads to convergence of mass in some places, and hence to expulsion of ﬂuid from the boundary layer, called the Ekman convergence. In other places, the Ekman transport is horizontally divergent, i.e., mass is being lost across the sides of a given area, so ﬂuid must be “sucked” vertically into the boundary layer to replace that which is lost across the sides, called Ekman suction. This effect is called the Ekman pumping.

Ekman layer (Ekman, 1905) The top or bottom layer in which the (surface or bottom) stress acts. The velocity that was driven by the stress is called the Ekman velocity. Typically, the atmospheric boundary Eckman layer is 1 km thick, whereas the oceanic boundary Eckman layer is 10 to 100 m thick.

Ekman mass transport The mass transport by the Ekman velocity within the boundary Ekman layer is called the Ekman mass transport or Ekman transport. In steady conditions, the Ekman transport is directed at right angles to the surface stress. In the atmosphere, the transport is to the left in the northern hemisphere relative to the surface stress. In the ocean, the transport is to the right in the northern hemisphere relative to the surface stress.

Ekman pumping See Ekman convergence.

Elara Moon of Jupiter, also designated JVII. Discovered by C. Perrine in 1905, its orbit has an eccentricity of 0.207, an inclination of 24.77◦, and a semimajor axis of 1.174 × 107 km. Its radius is approximately 38 km, its mass 7.77 × 1017 kg, and its density 3.4 g cm−3. It has a

E-layer screening

geometric albedo of 0.03 and orbits Jupiter once every 259.7 Earth days.

elastic deformation The reversible deformation of a material in response to a force. In a macroscopic sense, if the strain is proportional to stress (Hooke’s law) and falls to zero when the stress is removed, the deformation is deﬁned as elastic deformation. In a microscopic sense, if atomic displacement is small compared to the mean atomic spacing and atoms climb up the potential hill only to a small extent they will always be subjected to a restoring force, and upon the removal of a force, they will come back to the original position. Thus, elastic deformation is characterized by (i) reversible displacement (strain) and (ii) instantaneous response.

elastic limit The greatest stress that a material is capable of sustaining without any permanent strain remaining after complete release of the stress.

elastic lithosphere That fraction of the lithosphere that retains elastic strength over geological time scales. The elastic lithosphere is responsible for holding up ocean islands and volcanos, and it is responsible for the ﬂexural shape of the lithosphere at ocean trenches and at many sedimentary basins. A typical elastic lithosphere thickness is 30 km. Because of this thickness many sedimentary basins have a width of about 200 km.

elastic modulus See Young’s modulus.

elastic rebound The process that generates an earthquake. A fault is locked and tectonic deformation builds up the elastic stress in the surrounding rock. When this stress exceeds the rupture strength of the fault, slip occurs on the fault. The surrounding rock elastically rebounds generating seismic waves.

elastic string model Linear topological defects known as cosmic strings can carry currents and their dynamics are described by an equation of state relating the energy per unit length to the tension. This equation of state can be classiﬁed according to whether the characteristic propagation speed of longitudinal (sound type) perturba-

tions is smaller (supersonic), greater (subsonic), or equal (transonic) to that of transverse perturbations. Corresponding models bear the same names. See Carter–Peter model, current instability (cosmic string), equation of state (cosmic string).

E layer The lowest thermospheric layer of the ionosphere at heights between 85 and 140 km. Ionization is primarily due to solar X-rays and EUVs, and to a much smaller amount, also due to particles with rather low energies (some keV) and to an even smaller extent, due to meteorite dust. Dominant ionized particle species is molecular oxygen O+2 . Formally, the E layer can be described by a Chapman proﬁle.

E-layer screening The process whereby the E layer prevents radio signal propagation from taking place by a higher layer, normally the F region. Generally, E-layer screening occurs when the E layer MUF is greater than the operating frequency. There are two ways in which this can occur. During much of the daytime, a radio wave that would normally have propagated by the F layer will be screened, or prevented, from reaching the F layer because it is ﬁrst reﬂected back to the Earth by the intervening E layer. While the signal may be propagated by successive reﬂections from the E region, the additional hops will lead to sufﬁcient attenuation to prevent the signal from being useful. It is also possible that a signal propagating away from the nighttime terminator will, after being reﬂected by the F region then encounter an increasing E region MUF and be reﬂected back from the top of the E region towards the F region. In this case, the signal is prevented from reaching the receiver, but may be effectively ducted along between the E and F regions until it again reaches the nighttime terminator and can propagate to ground level. In both cases, the E-layer screening effect can be enhanced by the presence of sporadic E. On long (> 5000 km ) multi-hop paths E-layer screening enhanced by sporadic E, rather than absorption, is the low frequency limiting factor. See ionospheric radio propagation path.

Electra

Electra Magnitude 3.8 type B5 star at RA 03h44m, dec +24.06’; one of the “seven sisters” of the Pleiades.

electric drift See drift, electric.

electric ﬁeld, parallel	See parallel electric
ﬁeld.
electric regime (cosmic string)		It has been

known since the seminal work of E. Witten in 1985 that a current can build up in a cosmic string. As current generation proceeds via random choices of the phase of the current carrier, the resulting current can be of two distinct kinds, timelike or spacelike, depending on whether its time component is greater or smaller than its space component (there is also the possibility that they are equal, leading to a so-called lightlike current, but this is very rare at the time of current formation and would only occur through string intercommutation). Explicitly, setting the phase of the current carrier as a function of the time t and the string coordinate z in the form

ϕ = ωt − kz ,

with ω and k arbitrary parameters, one can deﬁne a state parameter through

w = k2 − ω2 ,

the case w < 0 (respectively, w > 0) corresponding to a timelike (respectively, spacelike) current.

In the timelike case, the conﬁguration is said to be in the electric regime, whereas for spacelike currents it is in the magnetic regime. The reason for these particular names stems from the possibility that the current is electromagnetic in nature, which means coupled with electric and magnetic ﬁelds. Then, for the electric regime, one can always ﬁnd a way to locally remove the magnetic ﬁeld and thus one is led to describe solely an electric ﬁeld surrounding a cosmic string. The same is true in the magnetic regime, where this time it is the electric ﬁeld that can be removed and only a magnetic ﬁeld remains. See current carrier (cosmic string), current genera- tion (cosmic string), intercommutation (cosmic string).

electroglow A light emitting process in the upper atmospheres of Jupiter, Saturn, Uranus, and Saturn’s satellite Titan. Sunlight dissociates some H2 and ionizes the hydrogen; the electrons are accelerated and interact with H2, producing the glow.

electromagnetic current meter A device that uses Faraday’s Law of magnetic induction to measure ﬂow velocities. The current meter head establishes a magnetic ﬁeld. A moving conductor (water) creates an electrical potential that is measured by the instrument. The electrical potential is proportional to the speed of the current. Often used for oneor two-dimensional velocity measurements.

electromagnetic induction This is the generation of currents in a conductor by a change in magnetic ﬂux linkage, which produces a magnetic ﬁeld that opposes the change in the inducing magnetic ﬁeld, as described by Faraday’s laws and Lenz’s law. In geophysics, this can be used to study the conductivity of the mantle through measurement of its response to magnetic ﬂuctuations originating in the ionosphere and/or magnetosphere. Maxwell’s equations can be used to show that in the appropriate limit, a magnetic ﬁeld obeys a diffusion equation:

∂t	= − ×	µ0σ × B
∂B		1

where σ is the local conductivity. Fluctuating external ﬁelds of frequency ω diffuse into

a layer at the top of the mantle with a skin depth

√

2/µ0σ ω. The induced ﬁeld is then essentially a reﬂection of this external ﬁeld from the skindepth layer, and it has a phase relative to the external ﬁeld that relates the conductivity structure of the layer to the frequency of the signal (essentially, long period signals penetrate deeper and therefore depend more on deeper conductivity than do short period signals).

electromagnetic radiation Radiation arising from the motion of electric charges, consisting of variations in the electric and magnetic ﬁelds, and propagating at the speed of light. Depending on the wavelength, observable as radio, infrared, visible light, ultraviolet, X-rays, or gamma rays.