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Lorentz transformation

frames moving relatively along the x-axis is:

 

t

 

 

 

 

 

 

γ

−γ v

0

0

 

t

 

 

 

 

−γ v

c2

0

0

x

 

x

γ

z

 

 

=

 

0

 

0

0

1

z

 

 

y

 

 

 

 

0

 

0

1

0

 

y

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

where γ

=

 

 

 

, the unprimed coordinates

 

 

 

 

1− v2 c2

relate to the first frame, and the origin 0 (x = 0, y = 0, z = 0) in the primed frame is moving at speed v in the +x direction, as measured in the unprimed frame. This is typically written in a more compact way in matrix notation as

x = F(v)x .

Here x stands for the column vector, and F is the Lorentz matrix. The inverse to F is the transformation with opposite velocity −v:

x = F(−v)x .

By writing out the transformations we obtain:

x

 

=

γ (x − vt)

t

=

γ t −

v

x .

c2

The first of these is like the Newtonian expres-

sion except for the factor γ

=

1

, which

 

 

 

1− c2

 

 

 

 

v2

 

is very near unity for most terrestrial motions (v c). The very surprising feature of the second is that time as measured in relatively moving frames has an offset that differs at different positions. Further, there is a rate offset proportional to γ . Again, these effects are noticed only when dealing with relative motions near the speed of light v c.

For general motion not along an axis:

t

=

γ t −

v · x

 

 

 

 

 

 

c2

·

 

−

 

 

 

=

 

+

 

 

v2

 

x

 

 

x

 

v

(γ − 1)

(v

 

x)

 

t .

 

 

 

 

 

 

Here the x are 3-vectors, and the complicated form of the expression for x is just a separation and separate transformation of components parallel and perpendicular to the motion.

If one chooses differential displacements dxi and dt corresponding to motion at the speed of light, then

0 = −c2dt2 + δij dx dxi i, j = 1, 2, 3 ;

© 2001 by CRC Press LLC

286

summed on i, j; (δij = 1 if i = j; 0 otherwise). The Lorentz transformations ensure that this statement remains true, and of exactly the same form (and c has the same value) when expressed in the “primed” frame. Thus, the speed of light

is

c2 = δij dxi dxj . dt dt

In this situation, the numerical value is unchanged when carrying out such transformations:

c = 29979245620 × 1010 cm/sec

in every observation. See coordinate transformation in special relativity.

Lorentz transformation [electric and mag-

netic fields]

See electromagnetism, Lorentz

transformation.

 

 

Lorentz transformation [implications]

An

important implication is that time is not an absolute variable; that is, its value varies for different reference systems. This leads towards a four-dimensional space-time coordinate system. That is, when transforming observations from one reference system to another, not only must one transform the three spatial coordinates but also time. Direct consequences of the Lorentz transformation are Time Dilation (see time dilation) and Length Contraction (see Lorentz–Fitzgerald contraction). Another important aspect of Lorentz transformations is that the equations that describe the relationship between electric and magnetic field, their causes and effects, are invariant under them. Special relativity postulates that physical laws must remain unchanged with respect to two uniformly moving reference systems, and thus they must be invariant under Lorentz transformations. This postulate leads to Relativistic Mechanics, which on one hand reduce to Newton’s Mechanics at low velocities, and on the other hand are invariant under Lorentz transformations. Thus, one finds that special relativity presents a consistent theoretical groundwork for both mechanics and electromagnetism.

Love numbers These are numbers that relate the elastic deformation of the Earth to applied

Lowest Useable Frequency (LUF)

deforming forces. If a potential Vn is applied to the Earth in the form of a spherical harmonic of degree n (for example, the tidal potential due to the moon, which is a zonal harmonic of degree 2), then if the Earth were a fairly inviscid fluid it would quickly adopt the shape of the new equipotential: locally, the Earth’s surface would move vertically and sideways. The new potential is not the simple sum of the old potential and the deforming potential: There is an additional potential due to the deformation of the Earth by the applied potential. hn is the ratio of the height of the resultant elastic deformation to that of the deforming potential, ln is the equivalent ratio for the horizontal displacement, and kn is the ratio of the additional potential associated with the Earth’s deformation to the deforming potential. There are additional Love numbers hn, kn, and ln associated with surface loading (e.g., by an ice sheet), and also hn, kn, and ln that represent the effect of shear forces on the surface of the Earth (such as winds). The fact that the Earth rotates and is not spherically symmetric introduces complications.

Love wave A horizontally polarized (SH ) elastic surface seismic wave. Love waves propagating near the free surface of a two-layer medium, in which the shear wave speed in the lower layer is greater, are a result of the multiple reflection of the horizontally polarized shear wave from the layer interface beyond the critical angle of refraction and from the free surface. Love waves are dispersive, with the phase velocities ranging between the shear wave speeds of the two layers. Love waves also occur as guided waves in an embedded layer of low shear wave speed.

lower hybrid waves Electrostatic ion oscillations at a frequency intermediate to the electron extraordinary wave (high frequency) and the magnetosonic wave (low frequency). Nonlinear wave processes at the lower hybrid resonance frequency are extremely important in transferring energy between different particle populations and fields in astrophysical plasmas.

lower mantle The region of rock in the Earth’s interior reaching roughly 3500 to 5700 km in radius.

Lowes power spectrum The power of the geomagnetic field, as a function of spherical harmonic degree. The magnetic field is commonly written as the gradient of a potential B = − V which is then expanded in Schmidt quasi-normalized spherical harmonics:

∞

n

n

+

1

 

 

 

V =re n 1 m 0 rre

 

 

 

 

 

 

 

 

 

 

 

 

 

 

=

=

 

 

 

 

 

 

 

 

gnm cos mφ + hnm sin mφ Pnm (cos θ)

 

 

∞ n

 

 

r

 

n

 

+ re n 1 m 0

re

 

 

 

 

 

 

 

 

 

 

 

= =

qnm cos mφ + snm sin mφ Pnm (cos θ)

where re is the radius of the Earth and the coordinate system is the spherical system (r, θ, φ). Terms in gnm and hmn represent fields generated by internal sources, and terms in qnm and snm represent fields generated by external sources. The radial portion of the field is Br = −∂V /∂r. The Lowes power spectrum Rn is usually defined separately for the internal and external fields in terms of the mean square value of the field at the Earth’s surface. For the internal field:

 

n

+ hnm2

Rn = (n + 1) m 0 gnm2

 

 

 

 

 

=

 

and for the external field:

 

 

n

 

 

Rn = n

 

 

 

 

 

m=0

The Lowes power spectrum for the internal field is consistent with roughly equal power for each degree between degrees 2 and 13 when the power spectrum is extrapolated down to the core-mantle boundary, and roughly equal power for degrees greater than around 15 at the Earth’s surface.

Lowest Useable Frequency (LUF) Defined as the minimum operating frequency that permits acceptable performance of a radio circuit between given terminals at a given time under specified working conditions. The LUF is determined by the absorption, the radio noise background, the radio frequency interference, and system parameters such as the transmitted power

© 2001 by CRC Press LLC

Low frequency radio emission from planets

and antenna gains. See ionospheric radio propagation path.

Low frequency radio emission from planets 1. Mercury: No radio emissions have been detected because there is only a very weak magnetic field.

2.Venus: No radio emissions have been detected coming from Venus due to a lack of a magnetic field. Venus does have an ionosphere, however, and orbiters have detected “whistlers” there.

3.Earth: In addition to all the manmade radio emissions, Earth emits a natural radio emission called Auroral Kilometric Radiation (AKR). It is triggered by the interaction of Earth’s magnetic field with the solar wind and ranges in frequency from about 30 to 800 kHz (wavelength 10 to 0.4 km).

4.Mars: Mars does not have a global magnetic field, only a patchy remnant field at places near the surface. Mars does not have radiation belts; therefore, it does not emit any low frequency radio emission.

5.Jupiter: Four well-established bands of planetary radio emissions have been established which correspond to spectral peaks in the Jovian emission: kilometer wavelength radiation (KOM), hectometer wavelength (HOM), decameter wavelength (DAM), and decimeter wavelength (DIM). The KOM frequencies range from 10 to 1000 kHz corresponding to wavelengths of 30 to 0.3 km, respectively. The DAM frequencies range from about 3 to 40 MHz (100 to 7.5 m); the HOM ranges from 300 to 3000 kHz (1 to 0.1 km); the DIM ranges from 100 MHz to 300 GHz (3 m to 1 mm).

The spectral peak intensity of the radiation occurs at about 8 MHz in the powerful and bursty DAM radio emissions. These emissions occur in the plasmasphere surrounding Jupiter from interactions with the satellite Io and with the sun. They are the only extraterrestrial planetary radio sources capable of being observed with ground-based radio telescopes. All other sources must be observed from space due to the absorptive properties of the Earth’s ionosphere. The HOM is more continuous and is triggered by particle-magnetic field interactions in the plasmasphere surrounding Jupiter. The KOM emission is found to be coming from plasma in-

teractions within Jupiter’s magnetosphere and from the relatively dense torus of plasma that surrounds Jupiter at the orbit of the satellite Io. The torus arises from volcanic eruptions from the surface of Io and the ejected particles get ionized by the solar UV radiation and interact with Jupiter’s magnetic field triggering various radio emissions. The DIM radio emission is caused by relativistic electrons in Jupiter’s inner magnetosphere. These particles are trapped in belts that are similar to Earth’s Van Allen radiation belts. The high magnetic field strengths cause the particles to be accelerated to high speeds and to emit radio waves at high frequencies.

6.Saturn: Saturn emits radio waves at kilometric wavelengths and the emission is called Saturnian Kilometric Radiation (SKR). SKR is believed to be similar to Earth’s AKR triggered by interactions of the planet’s magnetic field with the solar wind. The emission occurs over a frequency range of about 20 to 1200 kHz.

7.Uranus: Uranus also has a magnetic field and interactions with the solar wind cause Uranian kilometric radiation (UKR) over a frequency range of about 60 to 850 kHz.

8.Neptune: The Neptunian magnetic field has a surface field strength similar to that of Uranus and the Earth (approx. 0.1 to 1.0 Gauss). Because of the presence of the magnetic field, and thus a magnetosphere, interactions with the solar wind trigger Neptunian Kilometric Radiation (NKR) from about 20 to 865 kHz.

9.Pluto: No spacecraft have flown near Pluto; therefore, no magnetic field has been directly measured. No radio emissions have been detected nor are any expected.

low-velocity zone The region of the Earth beneath the lithosphere where seismic velocities are low. The asthenosphere is a low-velocity zone.

luminosity class The classification of a star based on the appearance of its spectrum, usually the relative strengths of emission and absorption lines, compared to the spectra of standard stars of the classification system. The classical Morgan and Keenan (MK) system is a twodimensional system: spectral type and luminosity class. In the MK system, classification was

© 2001 by CRC Press LLC

lunar highlands

defined at moderate (3 Å) resolution in the blue region (4000 to 5000 Å).

The primary luminosity classes range from I (supergiant) to III (giant) to V (dwarf or mainsequence), with classes II and IV as intermediate cases. The spectral criteria that define luminosity class are primarily a function of surface gravity (reflecting atmospheric density and envelope size), with the larger, supergiant stars having a lower surface gravity and less presure broadening of the lines.

luminosity distance Distance to an astronomical object obtained from the measured flux once its intrinsic luminosity is known: if L is the luminosity of the object and F its measured flux, then dL = √L/4πF . Other definitions of distances exist. They all coincide for a nonexpanding Euclidean universe, but they differ in the real expanding universe at redshifts close to unity and larger. Luminosity distance can be easily related to dA, the angular diameter distance: dL = dA(1 + z)2. See magnitude.

luminosity function of galaxies A function specifying the number density of galaxies per unit luminosity (or, equivalently, per unit magnitude). From counts and measurements of the integrated magnitude of galaxies in rich clusters, P. Schechter derived the following law:

8(L) = const × L/L α exp −L/L ,

where L is the galaxy luminosity, L ≈ 3 × 1010L is a turnover luminosity in units of solar luminosity, and α is found to be in the range ≈ −1.0 to − 1.5. This law suggests that the most luminous galaxies are the rarest, and that the number of galaxies increases with decreasing luminosity. According to Schechter’s law, a galaxy population in a magnitude limited sample — where galaxies are counted down to a fixed limit of brightness — is dominated by galaxies of luminosity near to L . On the contrary, in a volume limited sample — where ideally all galaxies are identified up to a fixed distance — the faintest galaxies would be by far the most numerous, and would contribute to the vast majority of light. Recent results suggest that Schechter’s law predicts even fewer faint galaxies than observed.

luminous blue variables The brightest known single stars, near absolute magnitude −10. They have already evolved slightly off the main sequence and typically have vigorous stellar winds that will gradually reduce their masses. The extreme luminosities and winds result in erratic variability, including unpredictable outbursts. A well-known example is Eta Carinae, which, in 1843, brightened to become the second brightest star in the sky. It has been fading ever since (though with occasional recoveries) and is no longer a naked-eye star. Luminous blue variables probably evolve to Wolf Rayet stars.

luminous efficiency Commission de l’Eclairage (CIE), 1924. A roughly Gaussian curve centered at 555 nm with value unity there, decreasing to zero at 425 nm and at 700 nm, meant to represent the response of human vision to the same physical flux at different wavelengths.

Luminous power (lumens) For monochromatic radiation, 683 times radiant power(watts) times luminous efficiency. For a mixture of wavelengths, the sum of the luminous powers for the individual wavelengths. See Abney’s law of additivity, luminous efficiency.

lunar eclipse A darkening of the full moon because the Earth is directly between the sun and the moon; the shadow of the Earth darkens the moon.

lunar highlands The lighter-color areas on the moon, classically called the “Terrae” (plural of the Latin, terra, “land”), which is usually translated as “highlands” or uplands. Portions of the highlands were removed in massive meteor impacts and were subsequently filled with younger, low lying volcanic flows to form the lunar maria. Compared to the maria, the highlands are an older surface, not altered since the heavy cratering era of planetary formation, rough and broken on a large scale. Highlands rise higher than the maria. (When the moon is in a phase where sunlight hits it at an angle, such as First Quarter phase, it is possible to see, using binoculars, the highlands casting shadows on the lower maria, and inside craters.) Chemically, the lunar highlands differ from the lunar lowlands

© 2001 by CRC Press LLC

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