Kinematics of Special Relativity  

General Concepts > s.a. lorentz group; minkowski spacetime; spacetime models; time / reference frames; time [clock hypothesis].
* Interval: A useful expression is ds2 = −c2 t1t2, where... @ Ohanian & Ruffini 94, p70.
* Boost: A Lorentz 'rotation' in a space-time plane in Minkowski space, represented by a transformation of coordinates of the form t' = t cosh u + x sinh u, x' = t sinh u + x cosh u, for a boost along the x-axis (with tanh u = β); The coordinates y and z in this case are unchanged, and the boost vector field is

v:= x ∂/∂t + t ∂/∂x ;

In general relativity, the corresponding conserved quantity (center of energy, > see conserved quantities) for asymptotically flat spacetimes is

M(N) = ∫∞ [N(∂a qbc − ∂b qac) − N,a (qbc − fbc) + N,b (qac − fac)] qac qbd dSd .

* Four-velocity: Defined, for a timelike curve xm(τ), as um = dxm/dτ, a unit timelike vector.
* Velocity composition: For collinear 3-velocities v1 and v2, u = (v1+v2) / (1+v1v2 / c2); It is neither commutative nor associative, due to the presence of Thomas precession.
@ Lorentz "rotations": Terrell PR(59); Weisskopf PT(60)sep; in Taylor & Wheeler 92, 92-93;
@ Simultaneity: Havas GRG(87); Sarkar & Stachel PhSc(99)jun; Giulini gq/00; Mamone FP(01); Janis in(02); Scherr et al AJP(02)dec [student beliefs]; Bolós et al IJTP(02)gq/05 [spacelike vs lightlike]; Minguzzi gq/05-proc [in special and general relativity]; Grünbaum FP(10) [re conventionality of simultaneity]; Besnard FP(12)-a1104 [in Minkowski spacetime]; Leardini a1310 [apparent paradox]; > s.a. EPR paradox.
@ Velocity: Lévy-Leblond AJP(80)may; Mermin AJP(83)dec, 90 ch19 [addition]; Sen JMP(90) [relative velocities]; Ungar FP(00) [composition]; Mathews AJP(05)jan [and acceleration]; O'Donnell & Visser EJP(11)-a1102 [including Wigner rotation and Thomas precession]; Kocik AJP(12)aug-a1408 [geometric diagram for addition of velocities].
> Online resources: see Kristian Evensen's interactive diagram.

Effects
* Time dilation: T = γ T0; Argue with bouncing photon clock; First tested in 1938 to within 1% of predictions in an experiment by Herbert Ives and G R Stilwell on the transverse Doppler effect in an atomic beam; Directly seen in the momentum dependence of the lifetimes of muons in 1941 and kaons in 1952; Verified with cosmic ray muon decays and with GPS tests; 2007, best measurement by Gwinner et al has accuracy quantified by a "Mansouri-Sexl parameter" of less than 8.4 × 10−8.
* Length contraction: L = γ−1 L0; Argue based on time dilation and L/T0 = L0/T (can be illustrated with the example of a long car entering a garage); Has not been directly seen experimentally; > s.a. Lorentz-FitzGerald Contraction.
* Doppler shift: Components of vectors normal to β are invariant, so k' perp = kperp; for other ones,

k' 0 = γ (k0 − β · k) ,   k' 1 = γ (k1 − βk0) ;
for light   ω' = γω (1 − βcosθ) ,   tanθ' = sinθ/[γ(cosθ − β)] .

@ Transformations: Brehme AJP(62)jul [Lorentz and Galilei, geometrical]; Lévy-Leblond AJP(76)mar, Lévy AJP(07)jul [derivation]; Janssen SHPMP(09) [kinematical vs dynamical explanations]; Cazaroto a1307 [quantum mechanical interpretation?].
@ Length contraction: Rindler AJP(61)jun [paradox]; Nikolić AJP(99)nov-phy/98 [accelerated rod]; Pierce AJP(07)jul [and lock and key paradox]; Redžić EJP(08) [interpretations]; Miller AJP(10)jun; Klevgard a1602; Walstad a1711 [without relativity]; Crouse & Skufca L&A-a1803 [in discrete spacetime]; Hoffmann a1804 [and spreading of relativistic probability densities]; Ceresuela & Llosa CJP(19)-a1808 [Bell's accelerated spaceships paradox].
@ Appearance of objects: Penrose PCPS(59) [sphere]; Terrell PR(59); Weisskopf PT(60)sep; in Van Bladel 84; Peres AJP(87)jun; Blackman EJP(98)ap/97; Field AJP(00)apr; Morava a0804 [in terms of groups].
@ Time dilation: Bohm & Hiley AJP(85)aug [and active interpretation of boosts]; Saathoff et al PRL(03) [deviations < 2.2 × 10−7]; Gwinner MPLA(05) [tests, rev]; Harvey & Schücking PT(05)mar [history, and gravitational time dilation]; news pw(07)nov [best measurement yet]; Christov FP(10) [reinterpretation of the Ives-Stilwell experiment]; news pt(10)nov [seen at 10 m/s]; Shields et al PRA(10) [and two-particle interactions]; Friedman & Nowik PS(12) [tests under acceleration using Mössbauer spectroscopy]; Wang TPT(13)mar [and actual appearance of clocks, with relativistic Doppler effect]; Botermann et al PRL(14) + news nat(14)sep [experiment]; news pw(17)jun [test]; Paige et al a1809 [for quantum clocks]; > s.a. gravitational redshift; special relativity.

Twin Paradox > s.a. mach's principle.
* Idea: Identical twins who are separated, undergo different accelerations, and are then reunited will have aged differently; Not really a paradox, it is a kinematical special relativity effect due to differences in proper times.
@ General references: Rodrigues & Rosa FP(89), comment Sachs FP(89); Harpaz EJP(90); Debs & Redhead AJP(96)apr [and simultaneity]; Ghosal et al FPL(05) [and equivalence principle], gq/05; Minguzzi AJP(05)sep [explicit formula]; Grøn EJP(06); Kohler FPL(06) [instantaneous turnaround as limit]; Grandou & Rubin a0704/AJP [as a consequence of causality], IJTP(09) [ingredients]; Székely SL(10)-a0807 [geometrical characterization]; Benguigui a1212; Shuler JModP(14) [rev].
@ In curved spacetime: Dray AJP(90)sep, Barrow & Levin PRA(01)gq [closed universe]; Bansar et al gq/05 [with a compact dimension]; Luminet a0910-ch; Boblest et al EJP(11)-a1011 [de Sitter spacetime]; Sokołowski GRG(12)-a1203 [in Schwarzschild spacetime]; Sokołowski & Golda APPB(14)-a1404 [static spacetimes]; Gasperini MPLA(14); Fung et al EJP(16)-a1606 [computational approach].
@ Variations: Price & Gruber AJP(96)aug; Cranor et al AJP(00)nov, Wortel et al AJP(07)dec-gq [circular]; Iorio FPL(05); Abramowicz et al PRA(07) [on the photon sphere]; Field a0806; Abramowicz & Bajtlik a0905 [version in which the accelerated twin is older]; Lévy-Leblond a1504 [two scenarios]; Ben-Ya'acov EJP-a1701 [relativistic rigid motion version]; Loriani et al a1905 [quantum version with interference of clocks].

Accelerated Observers > s.a. acceleration; Jerk; modified electrodynamics; special relativity [modified].
@ General references: Desloge & Philpott AJP(87)mar; Desloge AJP(89)jul [spatial geometry]; Tartaglia & Ruggiero EJP(03)gq; Mashhoon gq/03-in, IJMPD(05)gq/03-in [non-locality]; Pardy gq/03; Alba & Lusanna gq/03 [3+1 view]; Gourgoulhon 13; Pepino & Mabile a1907 [pedagogical].
@ Rotation: Herrera NCB(00)gq/99; Bashkov & Malakhaltsev gq/00; Rizzi & Ruggiero FP(02), ed-03; Klauber FP(07)gq/06; Herrera mp/06 [and Thomas precession]; > s.a. reference frames [rotating, Ehrenfest paradox]; rotations; Sagnac Effect; time.
@ Related topics: Gruber & Price AJP(97)oct [time dilation]; Chicone & Mashhoon CQG(04)gq [critical speed c/√2].

Related Topics > s.a. classical particles [relativistic]; constants [speed of light]; physics teaching.
@ General references: Christodoulou & O'Murchadha CMP(81) [boost problem in general relativity]; Coll & Morales IJTP(92) [frames]; Mermin AJP(97)jun, AJP(98)dec [spacetime diagrams]; Pierceaux phy/04-conf [Poincaré's kinematics and light fronts]; Price AJP(05)may-gq [projectiles and pendula]; Dryzek et al AJP(06)jan [lab experiment on E2 = p2c2 + m2c4]; Salgado AJP(16)may-a1111 [visual calculations on rotated graph paper]; Dray AJP(13)aug [spacetime diagrams]; Mathews AJP(20)feb [seven formulations]; Kanda et al a2006-conf [logical analysis].
@ Misconceptions: McGlynn & van Kampen EJP(08), Franklin EJP(10), comment Redžić SAJ(14)-a1005.
> Deformed kinematics: see DSR; momentum space.
> Other topics: see clocks; Gyrogroup; Pythagorean Theorem; Rapidity; Rigid Body; Superluminal Motion; tachyons.

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