시공간과 물질, 진동과 파동 상대성 원서 35장 그림 12.1 중심에 있는 은하의 중력렌즈 작용으로 형성된 머나먼 퀘이사의 영상을 보여 주는 아인슈타인의 십자가. 무엇을 배우는가 12.1 에테르 찾기 12.2 아인슈타인의 가설과 기준틀 베타와 감마 보기문제 12.1 아폴로 우주비행선 광원뿔 시공간 간격 12.3 시간팽창과 길이수축 시간팽창 보기문제 12.2 뮤온 붕괴 길이수축 보기문제 12.3 NASCAR 경주차의 길 이수축 쌍둥이 역설 12.4 상대론적 진동수이동 풀이문제 12.1 은하의 적색이동 12.5 로렌츠 변환 불변량 12.6 상대론적 속도변환 풀이문제 12.2 가속기 안의 입자 12.7 상대론적 운동량과 에너지 운동량 에너지 운동량-에너지 관계식 속력, 에너지, 운동량 보기문제 12.4 속력 0.99c의 전자 보기문제 12.5 케이온 붕괴 로렌츠 변환 이체충돌 보기문제 12.6 충돌기 대 고정표적물 가속기 12.8 일반상대성 블랙홀 중력파 12.9 일상생활에서의 상대성: GPS 무엇을 배웠는가/주요내용 문제풀이 능력 키우기 풀이문제 12.3 같은 속도 객관식 문제 설명문제 연습문제 415
416. : (1). (2).......,.. 12.1.,. ( ).,.....,......, 0..,... ( )( ). 15 16.31,.
417 100. 1887,, (aether).?.,..?,.3.,.. 19.... 1887. (34, ). :!!,..... 1905 26... 1.. 1921.36. 2.,,..(.)
418 3.... (inertial reference frame)....,.. 1:. 2: c.. (12.1) 1. SI. c 310 8 m/s.( c 186,000 mi/s c 1 ft/ns.). 29 km/s 65,000 mph.,. ( ).... c/2.. c. c(c/2)1.5c. 3.. c.. c.,,...,, () c " v ".
419 (12.2) (12.3) ". v v " c 12.2 12.3 1 1.. 0. 1. 1. 1 ( 12.2 ).. v " c 1. (1x) 1/2 1 x 2 x 4, ( 1 ) (12.4). 1 2 3 8. 4.010 4 km/h, (10).? SI.,. 12.1 0 1.,...
420.. (a) (11). (b) (21)..,.(,...),...,. 12.3 1. t0, xx 0 ( )., vc vc. 12.3. (0,x 0 )(light cone). t 1 x 1. t 2 x 2. 12.3... (0,x 0 ).. 12.4a..? 12.4b. x y v' ƒv x2 ßvß y2 c (21) x,y,t..,... x.,.. (world line). (space-time). 12.5. x. x0, t0. x, ( 1). x ( 2). x ( 3). x, 45 ( 4). x, 45 ( 5).
421 r., tt 2 t 1., 12 s,. (12.5) s 2,,. (12.6),,.... (proper time). (12.7) s 2 >0 ( ). ( ),.... (12.8) ( )...,..
422. ( ) ( ). (12.9) (time dilation) t 0, t(>t 0 ). t!. 12.2.?. ( 12.6a ). h, 0.( h/c.) v. x v, x ( 12.6b ). 12.6 c. (a). (b) v. (c).. x.. h 12.6. h, 1. xvt, t. h t 0 hct 0 /2 h.. c,!.
423 h x. t. ( 12.6a ) t 0.. ( 12.6b ) tt 0 > t 0..,..,......!,... ( 12.7 ).. 12.1 t 0.. 12.2. 0 2.2 s... 1977 CERN v0.9994c (& 0.9994).. 29.
424.. 2.2 s.,.. CERN...(.). km..?. 1971 4. 5910 ns, 27521 ns... 3(259,200 s) 59 ns 275 ns 10... 0... v L (proper length) L 0 (length contraction). (12.10) ( 12.7 ). v, L 0.
425..,. t 0, Lvt 0. tt 0. v, L 0 vtvt 0. ( )... ( 12.7 ). 12.1 12.2,.,...... v89.4 m/s NASCAR ( 12.8 ). 5.232 m. NASCAR?.. L 0 5.232 m. 12.10 NASCAR.
426,. 12.4 (1x 2 ) 1/2 1 x 2., v/c V 1. 1 2... ( ) ( ) tt 0...20 3.25. 65.0% 1.32. 2(3.25 )6.50. d6.50 /4.92.. td/v(4.92c)/0.650c7.57. 7.6, 7.610.0., 27.6, 30.0.. 65.0%. 1.32. 7.57 (207.57/1.32)25.8...?,.,... x,. ( 12.9 )..
427 12.9 0 x0.. 65.0%(v0.650c). v0.650c. x....,.,.. 5, 3.79. 5. 3.79. 3.79 8. 5 5. 30, 28.,.(. 27.6 ). 7 10.( ).. t0., ( 12.10 ). x v0.650c. 0. 65.0% 3.25 3.79. 12.9 12.10. 12.9 12.10. x. 65.0% 91.4%.( 12.6.), 52. 0.914c, 8., 30 27.6..... 65.0%, 65.0%
428.,. (). (). t0.. 3 65.0%, 65.0%...,..,. f. 16,,.. (relativistic frequency shift)
429.(.) v,( ), F 0 F. (12.11) ( ), ( ).(,.),, cf, (12.12)., (red-shifted)( > 0, ). v >0.. (, < 0 (blueshifted).).. (red-shift parameter)... (12.13), z0.450.?. 12.13. 12.11. 12.13 ( ).
430. v, c.... 12.5%.. v. F( x, y, zt) F( x, y, zt). x v, tt0 xx, yy, zz( ). F
431 x, y, z t ( ) F x, y, zt ( 12.12 ).. (12.14) (Galilean transformation),2 3 3. v c... v (Lorentz transformation),.. (12.15). (12.16) (1, v/c0),,... 2007 J.-M. [American Journal of Physics, Vol. 75, No. 7 (2007), pp. 615-618]. 12.2. 12.12 yy0, zz0, F( x, y, zt) E. x F O E. OE $ (i). $ OO F F, $ OE F E. (i), (i) F F. F (i). OE $ x
432 $ OO vt. xf F. F OE $ x/. F (i) (ii). F( x, y, z t). F $ OE x $ OO vt. F F F OE $ x x/. F (i). (iii).. (ii) vt, (iv). 12.15. (iii) 12.16 (v). (iv) x(v). 1/' 1 v 2 /ßc 2 2 1/(1v 2 /c 2 ) 1 2 v 2 /c 2.. x t(tvx/c 2 ) 12.15.,, ( x ). ( y z ). (invariants).?.. 12.5 s 2 c 2 t 2 r 2.
433. s 2 s 2. s 2,,. r 2.,...,. s 2 s 2.. 12.2. c/2. c(c/2)1.5c.3 " v pgv " psv " sg. p, g, s ( ),,. c.
434 c,. (relativistic velocity addition) x 1. F " u (u x, u y, u z ), FF " v (v,0,0), F " u (u x, u y, u z ). (12.17) F F. (12.18) x. u ", u " v ",..,. x ( v, u ). 12.15 dx dt F..
435 u x dx/dt ( ).,. y, z. dydy y. dt. u x dx/dt v V c, u V c vu x /c 2 1, 1, 12.17 12.18 u x u x v, u x u x v, u y u y, u z u z., ucvc/2 1.5 c,..! 12.17 12.18. " v 1 " v 2 x " v r v " r < v " r v " r v " r ( ) v " r > v " r " v 1 " v 2.. 0.830c 0.750c,.?,. " v 1v " 2.,. x
436.. v, u x, u x 12.17 12.18. 12.13. F v F. u x, u x. 12.17..... x.. 1/(1u x v/c 2 ). 1,..
437,,... " p mv ". c,... (12.19) m (rest mass)( ), " u, " p, 1/' 1ƒu 2 /ßc 2. 12.19m> 0..( m....) 12.14.,. c, mc. c/2, v c. 2. 2. :, :. (12.20), p ".,. 1 2 K mv 2.. m (12.21).(!),..
438 (12.22). (12.23). K mv 2? 1 2 1 2 1 2 1 v 2 /c 2.. (12.24) 1 2 -. x 1.. x 0 t0, t0 0 0. 12.19 dp/dt... dxvdt,. - (5) WK, 0. 12.23. mc 2 v0, 12.21.
439 Kp 2 /2m.. Emc 2 " p mv ". (12.25).. v c.., v 2 c 2 v 2., 2 m 2 v 2 p 2. -. (12.26) 32.25. (12.27) m, c. s 2 c 2 t 2 r 2 E 2 p 2 c 2. 12.27 0.(.) m0, -,. (m0). (12.28)
440 m>0, pmv Emc 2,. (12.29). -. (12.30) (ev), 1 ev1.60210 19 J. E 0 5.1110 5 ev0.511 MeV, m0.511 MeV/c 2. 99.0%,,,?.,,., 99%.. 0.99c. 99.9999999%. 3kmSLAC.
441 Emc 2....,., 12.5.. 39,.? 497.65 MeV/c 2, 139.57 MeV/c 2... 0...,. K....,
442.....,. 40.40. -... 12.3 F " r t F " r t.? 12.19 " p mu ". 12.22, Emc 2. u x (u x v)/(1vu x /c 2 ) 12.17. (12.31).. E 2 p 2 c 2 E 2 p 2 c 2, E 2 p 2 c 2. 12.4,.. (a). (b).. E 2 p 2 c 2. ( 12.15 ). ( 1 2.) S( ).
443 (12.32) E 1, " p 11, E 2, " p 22. E 2 p 2 c 2, S..8. cm. " p 1 cm p " 2 cm (8). S. S. S.. (12.33) 'S.,........,. BNLRHIC 250 GeV.? m p 0.938 GeV/c 2.,.. m 1 m 2 m p K cm 1 K cm 2 K cm (i)
444. 12.32.. 12.27., 1 (p " 1 0, E cm 1 m p c 2 ). 0. ( 2). (ii) S (i) (ii).. -..(.8.) 12.16 2.,.. RHIC 250 GeV, 134,000 GeV(134 TeV)..,.
445. m M. (12.34) m 2. m,...,..,... 1907 ( 12.17 ).,.. (Equivalence Principle)....,..2. 12.18a... 12.18b..,.. (a). (b).
446 1919 5 29...... 1919. 12.19.. 12.20.. 12.20.. 6. 1/5. 1.74",..,.,. HST ( 12.21 ). 12.21a.,. 12.21b.. (Schwarzschild radius). (12.35) (a). (b) 2218.
447. 12 v ' 2 G Mß/ßR.. ( ).. M. 15.. (black hole). ( 1 2 1 2.). 12, 12.4 3.610 6...15 LIGO. LIGO,,, 3002 km.. LISA. 500 km ( 12.22 ).. ( 20 km ), (, ms s )... 1974. 1993.. (m 1.9910 30 kg) LISA.,,.. GPS(Global Positioning System), 20,000 km
448 24 ( 12.23 ). (1 86,164.09074 s (11/365.2425)(86,400 s). 9.3.) GPS. 10 13 1., GPS. 4 GPS. ( ). i r " r t r. (12.36), 12.36.. 1m.,.. 4 km/s, 10 10 1 ( 7 s/ ). 1000.. GPS. c2.9979245810 8 m/s... (i),. L 0. (ii) c. 1 1.,... t 0.
449. E 2 p 2 c 2 m 2 c 4..,.. E 0 mc 2. " p mv ", EE 0 mc 2.. :.. GPS(Global Positioning System) (Galilean transformation) (proper length) (proper time) (inertial reference frame) (light cone) (length contraction) (Equivalence Principle) (Lorentz transformation) (invariants) (black hole) (relativistic velocity addition) (relativistic frequency shift) (world line) (Schwarzschild radius) (time dilation) (space-time) (aether) (red-shifted) (red-shift parameter) (rest mass) (blue-shifted) v " /c, 1/' 1 ß 2, tt 0, LL 0 /, u, u, " p mv ", EE 0 mc 2, E 0 mc 2, m () 'S,,
450 12.1 1 (3.0010 8 m/s)(365.25/)(24/)(3600 s/ )9.4710 15 m 12.2 5% : : 0.31c 31% : 0.32c 32% 50% : : 0.75c 75% 12.3 12.4 ( 2.) (x.) (v.) (.) (.) 1..??,? 12.3,. 2... 3... m p c 2 938 MeV, m e c 2 0.511 MeV. 2.50 GeV,? v.,,... 12.24.. E.
451 E e m e c 2, E p m p c 2.. E e K e m e c 2. E p K p m p c 2......,...
452 12.1? a). b). c). d). e). 12.2 A 70%, B 90%, B? a) A 50% A B b) B 50% B A c) 70% B d) A A B e) B. 12.3 m 0 v0.20c.? a) 0.40c b) 0.10c c) 0.38c d) 0.42c e) 0.99c 12.4?? a) t b) x c) v d) c 2 (t) 2 (x) 2 12.5 A B c/4. A B c/4.,??. a) A. b) B. c). d). 12.6 3.0 GeV/c? a) 0.31c b) 0.33c c) 0.91c d) 0.95c e) 3.2c 12.7 100 m 2 ('3/2)c.? a) '3/2 b) 2 c) 100 m 2. d). 1 2 12.8 0.5c.,? a) 2 b) 2 c) 2 12.9 ( 10 ).. 12.10.. t13 s x0, x2 km, x5km.. t0.?..??.. x km, t x/c,, t? 12.11.,? 12.12 300,000 km/s.. 200,000 km/s 200,000 km/s 400,000 km/s.? 12.13. (27 28 ),....,??(.,,..,.) 12.14 L.( ) L/2.
453 v0.866c., L L/4.? 12.15 x 10. x,? 12.16,..? 12.17 x v., 0. x l. yz0 x0, xl. t t. 12.18.1. 12.19,.? m, E.?. 12.20 80%.?.. 12.1 12.2 12.21 ft/ns. 12.22 g /. 12.23. u. D, D. ( ) v,. -. 12.24 0.8c? 12.3 12.25 0.50c. a)? b)? 12.26 3.8410 8 m. 0.50 c. a)? b)? c)? 12.27 30 10., 40.? 12.28 90.% 2.210 6 s? 12.29 10.0 m 8.00 m ( ).?? a) b) 12.30 80, 81, 80. 80 81?( 1.) 12.31 NASA. 35. NASA, 25. a) 25
454? b)? 12.32 F. xy, x 37. F F x v. a) F x 45? b) F? 12.33 20.0 m, 50.0 m 0.400c.?. 12.4 12.34 (480 nm) (660 nm)? 12.35,., 650 nm 520 nm.,? 12.36 (..).. 560 nm,. 532 nm...?,? 12.37. 32.0 km/h 10.6 GHz.? 12.38 He-Ne. 632.8 nm. 514.5 nm,? z? 12.5 12.6 12.39. (i) A(0,0,0) 0:00:00. (ii) B(500. m,0,0). 0.999c tx. a)? b)? 12.40 c. x 1. 12.41 v50.0 m/s? 12.42 0.90c 0.50c.? 12.43, 3.25 0.65c., (a). (b) (a). 12.44, 3.25 0.650c.. a) 0.650c, 0.914c. b) (a). 12.45 100. m 0.300c. 0.750c.. a) b) c) d) 12.46 1. v, vc tanh, c. v. a).. b) x v. 0. v.
455 12.7 12.47 pmc? 12.48 0.511 MeV/c 2. a) 10? b)? 12.49 RHIC A100. GeV. A (197). 1.00 GeV.?( 12.6.) 12.50 0.997c? 12.51 0.61c. MeV. 12.52. 0.9972c. a). 6 c. b) (MeV )? c) (MeV )? 12.53 0., 5.00 kv. a). b)? c)?(.) 12.54 m 1 m 2 1. 0.700c 1 2. 2 0.200c, 1 0.500c. m 2 /m 1? 12.55 m mc2'2m... a) b) c) 12.56 p x (p x ve/c 2 ), p y p y, p z p z, E(Evp x ). :. 12.57 E 2 p 2 c 2 E 2 p 2 c 2, E 2 p 2 c 2. :. 12.8 12.9 12.58 /c 2.. /c 2. 12.59. a) b). 10 15 m? 12.60 GPS 4.00 km/s.? 12.61. : 12.4. 12.62 50.0 ft 35.0 ft,?. 12.63 2.0010 8 m/s, 2.0010 3 m/s.? 12.64 A 0.75c. B 0.95c.B A,AB? 12.65, 0.9900c 0.9999c. 12.66 A B B. 240 m/s.. A B 3.00.. a) B,? b)? 12.67 2 TNT 15.0.1 4.0010 12 J.. 12.68 90.0 cm? 12.69 0.600c? 12.70, 1/3..
456. 1/3.,? 12.71.,. 20.0 2200.0.? 12.72 2/3.,? a) b) 12.73 100. 0.95c.?? 12.74 1.010 6 V.? 12.75, 2000.0. 10.000%.( 400.00.)? 12.76 0.800c, 1.0010 20 Ns? 12.77, 55.0% 65.0%. 80.0%. a)? b),? 12.78 0.305 14 C... 2.0 m,? 12.79 1.0010 3 20.0.? 12.80. 1.000 GeV/c 2, 1.000 GeV... a). b). 12.81,. dv/dtg(g ). v c.. d, dvgd.(.). a) () 1 v.. b) v t, v(t). c) t t. d) x(t). t0 xc 2 /g. e) x ct ( ). f) g9.81 m/s 2 c 70.7%,??