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Ư¼ö »ó´ë¼º(SR) I-1: Intervals; Time Delay
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Ư¼ö »ó´ë¼º ¿ø¸®(Principles of Special Relativity)

<ÀüÁ¦ ¹× ¿ë¾î>

Ư¼ö ¹×  ÀÏ¹Ý »ó´ë¼º À̷п¡ °üÇÑ ±ÛÀº Âü°í¹®ÇåÀÇ ¹Ì±¹ ´ëÇб³Àç Hartle(2003)À» ±âº»À¸·Î ÇϵÇ, Âü°í¼­ Landau-Lifshitz(1939/1980)·Î º¸¿ÏÇÏ·Á°í ÇÕ´Ï´Ù.
James Hartle(1939~)µµ ÈǸ¢ÇÑ ÇÐÀÚÀÌÁö¸¸ Lev Landau(1908-1968)´Â ·¯½Ã¾ÆÀÇ ÃµÀç °úÇÐÀڷμ­ [E. Lifshitz(1919-1985)´Â Á¦ÀÚ] ¿øÀüÀ» ±â¼úÇÑ °ÍÀÔ´Ï´Ù.
Hartle Ã¥Àº ÇкλýÀ» À§ÇÑ ±³°ú¼­·Î Ãâ¹ßÇÏ¿´À¸³ª Landau-LifshitzÀÇ Ã¥Àº Àü 10±ÇÀÇ ¿ªÇÐ(mechanics) Á¦2±ÇÀ¸·Î¼­ ´ëÇпø»ýÀ» À§ÇÑ ´õ ³ôÀº ¼öÁØÀÔ´Ï´Ù.
´ëÇÐ ¼öÁØÀÇ 'ÃÖ´ëÇÑÀÇ °£°á¼º(simplicity)°ú ¼öÇÐÀû ¾ö¹Ð¼º(rigor)À» ÁöÇâÇϴ ª°í Á¤È®ÇÑ »ó´ë¼ºÀÌ·Ð ÇØ¼³'À» ¸ñÇ¥·Î ÇÏ¿© ¼ö½Ã ¾÷µ¥ÀÌÆ®ÇϰڽÀ´Ï´Ù~

⦁ °è(frame): ¿©±â¼­´Â ÁÂÇ¥ ½Ã½ºÅÛ(a system of coordinate)°ú µ¿ÀǾî
⦁ ±âÁذè(reference frame): °ø°£(space)¿¡¼­ ÀÔÀÚ(particle)ÀÇ À§Ä¡¿Í ½Ã°£À» ¾Ë¸®´Â ½Ã°è(clock)¸¦ °®´Â °è(frame)
⦁ °ü¼º°è(inertial frame): ¿Ü·Â(external force)¸¦ ¹ÞÁö ¾Ê°í »ó´ëÀûÀÎ µî¼ÓÀ¸·Î(uniformly) ¿òÁ÷ÀÌ´Â ±âÁذè(reference frame)
⦁ ¸Æ½ºÀ£ÀÇ ¹ýÄ¢°ú ¸¶ÀÌÄ̽¼-¸ô¸®(Michelson-Morley) ½ÇÇè¿¡ µû¶ó ¸ðµç °ü¼º°è¿¡¼­ ±¤¼Ó c ≅ 2.998*1010cm/s.
⦁ ´ëºÎºÐÀÇ »ó´ë¼ºÀÌ·ÐÀÇ µµÃâÀº »ç°í ½ÇÇè(thought experiment)[Gedankenexperiment]¿¡ ÀÇÇÔ.

1. »óÈ£ÀÛ¿ëÀÇ ÀüÆÄ ¼Óµµ(Velocity of propagation of interaction)

»ó´ë¼º¿ø¸®(principle of relativity)¶õ ¸ðµç °ü¼º°è(inertial frame)¿¡¼­ ÀÚ¿¬ÀÇ ¹ýÄ¢(laws of nature)ÀÇ µ¿ÀϼºÀ¸·Î, ÀÌ´Â ½ÇÇè¿¡ ÀÇÇØ Áõ¸íµÇ¾ú½À´Ï´Ù.
±×·¯¸é °è(frame)¾È¿¡¼­ ¹°Áú ÀÔÀÚµé(material particles)ÀÇ »óÈ£ÀÛ¿ë(interaction)ÀÌ ÀÖ´Ù°í ÇßÀ» ¶§ ÀüÆÄ ¼Óµµ(velocity of progaation)´Â °ú¿¬ ¾î¶°ÇÒ±î¿ä?
°¥¸±·¹¿ÀÀÇ »ó´ë¼º ¿ø¸®(principal of rellaativity of  Galileo)¿¡¼­´Â ¹«ÇÑ ¼Óµµ(infinite velocity)·Î ÀüÆÄµÈ´Ù°í °¡Á¤ÇÏ¿´¾úÀ¸³ª, ½ÇÁ¦ÀÇ »óÈ£ÀÛ¿ëÀº ...
¾ÆÀν´Å¸ÀÎÀÇ »ó´ë¼º ¿ø¸®(principal of relativity of Einstein)¿¡ ÀÇÇØ ±¤¼ÓÀ¸·Î ÀüÆÄµÊÀÌ ¹àÇôÁ³½À´Ï´Ù!


¿ì¸®´Â º¸Åë »ó´ë¼ºÀ̷п¡ ÀÇÇÑ ¿ªÇÐ(mechanics)À» ´ºÅæ ¿ªÇÐ(Newtonian mechanics)°ú ´ëºñÇØ »ó´ë·ÐÀû ¿ªÇÐ(relativisic mechanics)À̶ó ºÎ¸¨´Ï´Ù.
À§ ±×¸²fig.1.°ú °°ÀÌ xyz ÁÂÇ¥°è¸¦ °¡Áø K °ü¼º°è¿Í ÀÌ¿Í xÃà ¹æÇâÀ¸·Î ÀÏÁ¤ÇÑ ¼Óµµ·Î À̵¿ÇÏ´Â x'y'z' ÁÂÇ¥°èÀÇ K¡¯ °ü¼º°è¸¦ »ý°¢ÇØ º¸±â·Î ÇսôÙ.
K¡¯ °ü¼º°èÀÇ A ÁöÁ¡¿¡¼­ ½ÅÈ£(signal)°¡ ¾çÂÊÀ¸·Î Ãâ¹ßÇß´Ù°í °¡Á¤ÇÏ¸é ¸ðµç °ü¼º°è¿¡¼­ ±¤¼ÓÀº  cÀ̹ǷΠ°°Àº °Å¸®ÀÇ B¿Í C¿¡ µ¿½Ã¿¡ µµ´ÞÇÕ´Ï´Ù.
ÇÏÁö¸¸ K °ü¼º°è¿¡ ÀÖ´Â °üÂûÀÚ(observer)¿¡°Ô´Â B´Â A¸¦ ÇâÇØ¼­ °¡°í C´Â A¿¡¼­ ¸Ö¾îÁö¹Ç·Î B¿¡ ¸ÕÀú µµ´ÞÇϰí C¿¡´Â ³ªÁß¿¡ µµ´ÞÇÏ°Ô º¸ÀÔ´Ï´Ù.
ÇÑ °ü¼º°è¿¡¼­ µ¿½ÃÀÎ »ç°ÇÀÌ »ó´ëÀû µî¼ÓÀ¸·Î ¿òÁ÷ÀÌ´Â ´Ù¸¥ °ü¼º°è¿¡¼­´Â µ¿½ÃÀÎ »ç°ÇÀÌ ¾Æ´Ï¹Ç·Î ´ºÅæÀÇ ½Ã°£ °³³äÀº Æó±âÇØ¾ß ÇÕ´Ï´Ù.
¿¹¸¦ µéÀÚ¸é žçÀÇ ÇÑ »ç°ÇÀÌ ¾ÆÀ̽´Å¸ÀÎ »ó´ë¼ºÀ̷п¡¼­´Â ±¤¼Ó °Å¸®ÀÎ ¾à 8ºÐ µÚ¿¡ ÀÛ¿ëÇÏÁö¸¸ °¥¸±·¹¿ÀÀÇ »ó´ë¼º¿¡¼­´Â Áï½Ã ÀÛ¿ëÇÑ´Ù´Â Â÷ÀÌÀÔ´Ï´Ù.
ÇÏÁö¸¸ °ü¼º°è°£ °Å¸®³ª ¼Óµµ°¡ ±¤¼Ó c¿¡ ºñ±³ÇØ Å©Áö ¾ÊÀ» ¶§¿¡´Â ½Ç¿ëÀû ±Ù»ç½ÄÀÎ ´ºÅæÀÇ ¿ªÇÐÀ» ±×´ë·Î Àû¿ëÇÒ ¼ö ÀÖ´Â °ÍÀÔ´Ï´Ù. [Landau-Lifshitz p.3]

2. °£°Ý(Intervals)

»ç°Ç(event)Àº ¹ß»ýÇÑ ½Ã°£°ú Àå¼Ò·Î ±â¼úµÇ¹Ç·Î °¡»óÀÇ 4Â÷¿ø °ø°£À» 3 °ø°£Ãà(three space axis)[x]¿Í ½Ã°£Ãà(time axis)[t]·Î ÆíÀÇ»ó ÀÚÁÖ Ç¥±âÇé´Ï´Ù.
±× ½Ã°ø°£¼Ó¿¡¼­  »ç°Ç(events)µéÀº Á¡À¸·Î Ç¥±âÇÏ¿© ¼¼°èÁ¡(world points)À̶ó°í ºÎ¸£¸ç  ±× Á¡ÀÇ ¿òÁ÷ÀÓÀ» ¼¼°è¼±(world line)À̶ó°í ºÎ¸¨´Ï´Ù.
ÀÌÁ¦ À§ ±×¸²ÀÇ K¿Í K¡¯ °ü¼º°è·Î ±¤¼Ó ºÒº¯ÀÇ ¿ø¸®(the principle of the invariance of the velosity of light)ÀÇ Àû¿ëÀ» ¼öÇÐÀûÀ¸·Î »ìÆì º¸°Ú½À´Ï´Ù.
°ü¼º°è KÀÇ ½Ã°£À» t, °ü¼º°è K¡¯ÀÇ ½Ã°£À» t¡¯¶ó ÇÏ°í »ç°ÇÀÇ °£°Ý(interval)À» ∆s¶ó°í ÇÏ¸é ´ÙÀ½ÀÇ '°£°Ý ºÒº¯(invariance of intervals) ½Ä'ÀÌ ¼º¸³ÇÕ´Ï´Ù.
 
   ½Ã°è(clock)´Â L°Å¸®ÀÇ A°Å¿ï(mirror)°ú  B°Å¿ï»çÀ̸¦ ¿Õº¹ÇÏ´Â ºû ÆÞ½º(light pulse)ÀÇ ½Ã°£ °£°Ý(time interval) : ∆t = 2L/c À» ÃøÁ¤ÇÔ.
   °ü¼º°è K¿¡¼­ ∆t = 2L/c,  ∆x = ∆y = ∆z = 0 ÀÏ ¶§, xÃà ¹æÇâ µî¼Ó V·Î À̵¿ÇÏ´Â °ü¼º°è K'ÀÇ ∆t' = (2/c) *¡î[L©÷ +( ∆x'/2)©÷],  ∆x' = V∆t',   ∆y' = ∆z' = 0
   -(c∆t')©÷ + (∆x')©÷ = - 4[L©÷ + (∆x'/2 )©÷ ] + (∆x')©÷ = -4L©÷ = -(c∆t)©÷  <-  ∆x = ∆y = ∆z = 0,  ∆y' = ∆z' = 0 ¸¦ ¾çº¯¿¡ ´õÇϰí Á¤¸®Çϸé,
   -(c∆t)©÷ + (∆x)©÷ + (∆y)©÷ + (∆z)©÷ = -(c∆t')©÷ + (∆x')©÷ + (∆y')©÷ + (∆z')©÷ = -4L©÷ <- ¸ðµç °ü¼º°èÀÇ ÇÑ ºÒº¯·®(an invariant)À» ½Äº°ÇÏ´Â ¿­¼è.
 
  (∆s)©÷ ¡Õ -(c∆t)©÷ + (∆x)©÷ + (∆y)©÷ + (∆z)©÷        <2-1a>
  ds©÷ = -c©÷dt©÷ + dx©÷ + dy©÷ + dz©÷                     <2-1b> <-  ¹ÌºÐ ¹öÀü
  
3. °íÀ¯½Ã°£(Proper Time)°ú ½Ã°£Áö¿¬(Time Dilation)

  (∆s)©÷ > 0  °ø°£¼º ºÐ¸®(spacelike seperated)
  (∆s)©÷ = 0  ³Înull ºÐ¸®(null seperated or lightlike seperated)
  (∆s)©÷ < 0  ½Ã°£¼º ºÐ¸®(timelike seperted)


°ø°£¼º °£°Ý(spacelike interval): ds©÷ = -c©÷dt©÷ + dx©÷ + dy©÷ + dz©÷       <2-1c> <-  HartleÃ¥ÀÌ interval·Î »ç¿ëÇÔ. * default
½Ã°£¼º °£°Ý(timelike interval):   ds©÷ =  c©÷dt©÷ - dx©÷ - dy©÷ - dz©÷        <2-1d>  <- Landau-LifshitzÃ¥ÀÌ interval·Î »ç¿ëÇÔ.

À§ ±×¸²Ã³·³ ³Înull ºÐ¸®µÈ Á¡µéÀÌ ±¤Ãß[ÎÃõÞ-ºû¿ø»Ô](light cone)°¡ µÇ¸ç Áú·®À» °¡Áø ¹°Ã¼´Â ±¤Ãß ³»ºÎÀÇ.½Ã°£¼º(timelike) ¼¼°è¼±À» µû¶ó ¿òÁ÷ÀÔ´Ï´Ù.
¿©±â¼­ ½Ã°è(clock)´Â ½Ã°£¼º °Å¸®(timelike distance)¸¦ Àç´Â µµ±¸(device)À̰í, ÀÚ(ruler)´Â °ø°£¼º °Å¸®(spacelike distance)¸¦ Àç´Â µµ±¸ÀÔ´Ï´Ù.

½Ã°£¼º ¿µ¿ª¾ÈÀÇ °î¼±(curve)À» µû¶ó ¿òÁ÷ÀÌ´Â °Å¸®ÀÎ ¥ó´Â ½ÇÁ¦ ½Ã°£ÀÌ¸ç °íÀ¯½Ã°£(proper time)À̶ó ÁöĪÇÕ´Ï´Ù.[Hartle p.60-63]
 
 d¥ó©÷ ¡Õ - ds©÷/c©÷                             <3-1>  <- ¥ó (Tau Ÿ¿ì) ¼Ò¹®ÀÚ

±¤Ãß-ºû¿ø»Ô ³»ºÎÀÇ ½Ã°£¼º ¿µ¿ª¾ÈÀÇ ¼¼°è¼±world line »óÀÇ µÎÁ¡ A¿Í B°£ °íÀ¯½Ã°£Àº ¥óAB¸¦ À§ÀÇ µÎ ½Ä¿¡ ÀÇÇØ¼­ °è»êÇϸé...
 ¥óAB = ¡ò(trom A to B)d¥ó = ¡ò(trom A to B) ¡î {dt©÷-(dx©÷+dy©÷+dz©÷)/c©÷} =  ¡ò(trom A to B) dt ¡î {1-(dx©÷+dy©÷+dz©÷)/dt©÷c©÷} = ¡ò(trom A to B) dt ¡î (1-V©÷/c©÷)

 d¥ó = dt ¡î (1 - V©÷/c©÷)                      <3-2>   <- °íÀ¯½Ã°£ÀÇ Áö¿¬(proper time dilation)

Á¤ÁöÇÑ °ü¼º°è K¿Í V·Î ¿òÁ÷ÀÌ´Â °ü¼º°è K'¿¡ °¢°¢ ¼ÓÇÑ ½Ã°èµéclocksÀ» ÅëÇØ dt'°¡ ¹«¾ùÀ» ÀǹÌÇϴ°¡¸¦ ¾Ë¾Æº¸±â·Î ÇսôÙ. [Landau p.7-9]
°ü¼º°è K'¿¡ ÀÖ´Â ½Ã°è´Â ½Ã°£°£°Ýl dt µ¿¾È ¡î (dx©÷ + dy©÷ + dz©÷) °Å¸®¸¦ À̵¿ÇÏ¸ç ±× °ü¼º°è ³»¿¡¼­ Á¤ÁöÇØ ÀÖÀ¸¹Ç·Î dx' = dy' = dz' = 0 ÀÔ´Ï´Ù.

ds©÷ = -c©÷dt©÷ + dx©÷ + dy©÷ + dz©÷ =-c©÷dt'©÷
dt' = dt ¡î [1- (dx©÷ + dy©÷ + dz©÷ )/c©÷dt©÷],  (dx©÷ + dy©÷ + dz©÷ )/dt©÷ = V©÷

 dt' = ds/c = dt ¡î (1 - V©÷/c©÷)        <3-3>   <- °ü¼º°è K'¿¡ ¼ÓÇÑ ½Ã°èÀÇ ½Ã°£Áö¿¬(time dilation)

Âü°í¹®Çå Landau, L.D.; Lifshitz, E.M. (1980)[1939] The Classical Theory of Fields (4th ed.) Butterworth-Heinemann            
             Hartle, J.B. (2003) Gravity: An Introduction to Einstein¡¯s General Relativity, Addison-Wesley

p.s. À§ Landau-Lifshitz(1962)´Â Hartle(2003)ÀÇ Âü°í¹®Çå Áß Ã¹¹øÂ°·Î¼­ ±× ¾Æ·¡¿¡ ´ÙÀ½ÀÇ ÄÚ¸àÆ®°¡ ÀÖÀ½.
      'The 150 pages of the text devoted to general relativity give a concise introduction  to the basics of the subject
       in the clear and straightfoward Landau and Lifshitz style, although few application are covered in any depth.'
       Landau, L. D.¿Í  Lifshitz, E.M.ÀÇ »ó±â Ã¥Àº 1939³â Russian ÃÊÆÇº»ÀÌ·¡ ·¯½Ã¾Æ 7ÆÇÀÌ Lifshitz¿¡ ÀÇÇØ ÃâÆÇµÊ.
      1951³â, 1962³â, 1971³â, 1980³â ³×¹ø ¿µ¾î ¹ø¿ªÆÇ(¿ªÀÚ´Â ¸ðµÎ M. Hamermesh)ÀÌ ³ª¿ÔÀ½.


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