±âº» ÆäÀÌÁö Æ÷Æ®Æú¸®¿À ´ëÇѹα¹ÀÇ ÀüÅë°ÇÃà Áß±¹°ú ÀϺ»ÀÇ ÀüÅë°ÇÃà ¼­À¯·´°ú ¹Ì±¹ÀÇ °ÇÃà ±¹¿ª û¿À°æ Çö´ë ¿ìÁÖ·Ð ´ëÇѹα¹ÀÇ »êdz°æ ¹éµÎ´ë°£ Á¾ÁÖ»êÇà ³×ÆÈ È÷¸»¶ó¾ß Æ®·¹Å· ¸ùºí¶û Áö¿ª Æ®·¹Å· ¿ä¼¼¹ÌƼ ij³â µî Ƽº£Æ® ½ÇÅ©·Îµå ¾ß»ý »ý¹° ÆÄ³ë¶ó¸¶»çÁø °¶·¯¸® Ŭ·¡½Ä ·¹ÄÚµå °¶·¯¸® AT Æ÷·³ Æ®·¹Å· Á¤º¸ ¸µÅ©


 ·Î±×ÀÎ  È¸¿ø°¡ÀÔ

Ư¼ö»ó´ë¼º ¿ªÇÐ II-2. ÃøÁö¼±; ±¤¼± µî
    ±è°ü¼®  2018-07-05 06:34:56, Á¶È¸¼ö : 973
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5.4 ÀÚÀ¯ÀÔÀÚ ¿îµ¿ÀÇ º¯ºÐ ¿ø¸®(Variational Principle for Free Particle Motion)

ù¹øÂ° ±×¸²Àº ½Ã°£¼ºÀ¸·Î ºÐ¸®µÈ(timelike seperated) µÎ Á¡µé »çÀ̸¦ ¿òÁ÷ÀÌ´Â ÀÚÀ¯ÀÔÀÚÀÇ ¼¼°è¼±Àº ±×µé »çÀÌÀÇ °íÀ¯½Ã°£(proper time)À» ±ØÄ¡È­ÇÑ´Ù(extremize)´Â °ÍÀÔ´Ï´Ù.
´ºÅæ ¿ªÇÐÀÇ ¹ýÄ¢Àº ±ØÄ¡È­µÈ ÀÛ¿ëÀÇ ¿ø¸®(the principle of extremal action)·Î ºÒ¸®¿ì´Â º¯ºÐ ¿ø¸®(variational principle)·Î¼­ °ø½ÄÈ­ µÇ´Âµ¥, »ó´ë·ÐÀû ¿ªÇп¡¼­µµ °°Àº ¹æ½ÄÀÌ Àû¿ëµË´Ï´Ù.
Àß ÀÌÇØÇØ¾ß Çϴµ¥, ±× °úÁ¤Àº ¸ÕÀú °íÀ¯½Ã°£ÀÇ ½ÄÀ» ÀÛ¼ºÇÑ ´ÙÀ½¿¡ ±× ½ÄÀ» ¿ÀÀÏ·¯-¶ó±×¶ûÁÖ ¹æÁ¤½Ä(Euler-Lagrange equatuion)À» ÀÌ¿ëÇØ ¿îµ¿ ¹æÁ¤½Ä(motion eqation)À» ãÀ¸¸é µË´Ï´Ù.

¥óAB = ¡ò(from A to B) d¥ó = ¡ò(from A to B) [dt©÷ - dx©÷ - dy©÷ -dz©÷]1/2                                     <5-42>
¥óAB = ¡ò(from A to B) d¥ò = [(dt/d¥ò)©÷  - (dx/d¥ò)©÷- (dy/d¥ò)©÷ - (dz/d¥ò)©÷]1/2                         <5-43> <- ¥ò = 0 at A point, ¥ò = 1 at B point
d(¡Ó𝐹/¡Ó𝑦')/d𝑥 - ¡Ó𝐹/¡Ó𝑦 = 0  [¿ÀÀÏ·¯-¶ó±×¶ûÁÖ ¹æÁ¤½Ä(Euler-Lagrange equatuion)]    <5-44> <-  𝐹(𝑥, 𝑦, 𝑦') ÇÔ¼öÀÇ ±ØÄ¡È­(exremization), »ó¼¼ÇÑ ³»¿ë°ú Áõ¸íÀº À§Å°¹é°ú¸¦ ÂüÁ¶
𝐹 ↦ L = [(dt/d¥ò)©÷  - (dx/d¥ò)©÷- (dy/d¥ò)©÷ - (dz/d¥ò)©÷]1/2,   𝑥 ↦ ¥ò,  𝑦 ↦ x¥á,  𝑦' ↦ dx¥á/d¥ò    <5-45>
d(¡ÓL/¡Ódx¥á/d¥ò)/d𝑥 - ¡ÓL/¡Óx¥á = 0,  ¡ÓL/¡Óx¥á = 0,  d(¡ÓL/¡Ódx¥á/d¥ò)/d𝑥 = 0                                <5-46> <-  ¡ñ LÀº x¥á¿Í µ¶¸³Àû(independant)À̹ǷÎ
d(¡ÓL/¡Ódx¥á/d¥ò)/d𝑥, L ↦ ¡î (dx¥á/d¥ò)©÷, d[(1/L) * (dx¥á/ d¥ò)] = 0                                        <5-47> <-  m ↦ dx¥á/d¥ò,  d[(1/2 ¡î m©÷) * m©÷/d¥ò]/d¥ò,  d[(1/2 ¡î m©÷) * 2m * (dm/d¥ò)]/d¥ò
d©÷x¥á/d¥ó©÷  = 0  [AB»çÀÌÀÇ Á÷¼± ¼¼°è¼±(Straight world line between A and B)]       <5-48> <-  L = d¥ó/d¥ò À̹ǷÎ, ¾çº¯¿¡ (d¥ò/d¥ó)©÷ ¸¦ °öÇÏ¿© d¥ò¸¦ ¼Ò°ÅÇÔ.

Áï, ÀÚÀ¯ÀÔÀÚÀÇ ÆòÆòÇÑ ½Ã°ø°£¿¡¼­ÀÇ ¼¼°è¼±Àº °íÀ¯½Ã°£À» ±ØÄ¡È­ÇÑ °î¼±ÀÎ °ÍÀÔ´Ï´Ù. (À̸¦ °ü¼º ÁÂÇ¥°èÀÇ ½Ã°ø°£¿¡¼­ÀÇ  ÃøÁö¼±-geodesic-À̶ó ºÎ¸¦ ¼ö ÀÖ½À´Ï´Ù.)

5.5 ±¤¼±(Light Rays)

Á¤Áö Áú·® Á¦·Î ÀÔÀÚµé(Zero Rest Mass Particles)

Á¤Áö Àß·®ÀÌ ¾ø°í 𝑉 = 1ÀÎ ±¤¼ÓÀ¸·Î ¿òÁ÷ÀÌ´Â ÀÔÀÚ¿¡ ´ëÇØ »ý°¢ÇØ º»´Ù¸é, ºûÀÇ ¾çÀÚ(quanta of light)¿Í Áß·Â(gravity)-±¤ÀÚ(photons), Áß·ÂÀÚ(gravitons)¿Í ¸î¸î Áß¼º¹ÌÀÚ(nutrinos)°¡ ¿¹ÀÔ´Ï´Ù.
¿©±â¼­´Â ºñ¾çÀÚÀûÀÎ ¾ç»ó(nom-quatum aspects)¿¡¼­ÀÇ ±¤¼±(light rays)À̶ó°íµµ ºÒ¸®¿ì´Â ±¤ÀÚµé(photons)¿¡ ´ëÇØ¼­¸¸ ÃÐÁ¡À» ¸ÂÃßÁö¸¸ ±¤¼ÓÀ¸·Î ¿òÁ÷ÀÌ´Â ´Ù¸¥ ÀÔÀڵ鿡µµ Àû¿ëµË´Ï´Ù.

𝑥 = 𝑡,  𝑉 = 1,  ¥ó(proper time) = -¡î ds©÷ = 0              <5-48>  <- ¡Å °íÀ¯½Ã°£ ¥óÀÌ 0ÀÌ¶ó¼­ ¸Å°³º¯¼ö(parameter)·Î »ç¿ëÇÒ ¼ö ¾øÀ½.
𝑥¥á = 𝑢¥á ¥ë                                                           <5-49>  <-  ¥ë ´Â ´Ù¸¥ ¸Å°³º¯¼ö(parameter),  𝑢¥á = (1,1,0,0) [t = 1, x = 1]
𝐮 ∙ 𝐮 = 0                                                           <5-50>  <-  ¡ñ Á¢¼±º¤ÅÍ(tangent vector) 𝐮 ÀÇ ¿ä¼ÒÀÎ 𝑢¥á = d𝑥¥á/ d¥ë Àε¥,  𝐮 ´Â ³Î º¤ÅÍ(null vector)À̹ǷÎ
d𝐮 /d¥ë = 0                                                        <5-51>  <-   ±¤¼±(light ray)ÀÇ ¿òÁ÷ÀÓÀÇ ¹æÁ¤½Ä, ÀÔÀÚÀÇ ¿îµ¿ ¹æÁ¤½ÄÀÎ <5-26> d𝐮/d¥ó = 0 °ú À¯»çÇÔ..

¿©±âÀÇ ±¤¼±¿¡ ´ëÇÑ ¿îµ¿ ¹æÁ¤½Ä¿¡ »ç¿ëµÇ´Â ¸Å°³º¯¼ö ¥ë ´Â ÀÔÀÚ¸¦ À§ÇÑ ¸Å°³º¯¼ö¿Í °°Àº °ÍÀ¸·Î¼­ ¾ÆÇÉ ¸Å°³º¯¼ö(affine parameters)¶ó°í ºÎ¸¨´Ï´Ù.  

¿¡³ÊÁö, ¿îµ¿·®, Á֯ļö ±×¸®°í ÆÄµ¿ º¤ÅÍ(Energy, Momentum, Frequency and Wave Vector)

±¤ÀÚ(photons)¿Í Áß¼º¹ÌÀÚ(neutrinos)´Â ¿¡³ÊÁö¿Í 3-¿îµ¿·®(three momentum)À» °¡Áý´Ï´Ù. ÇöûÅ©-¾ÆÀν´Å¸ÀÎ °ü°è½Ä¿¡ ÀÇÇØ¼­ °¢ °ü¼º°è¿¡¼­ ÀÇ ±¤ÀÚ¿¡³ÊÁö 𝐸 ´Â °¢Áøµ¿¼ö ¥ø ¿Í ¿¬°üµË´Ï´Ù.

𝐸 = h 𝑣 = ©¤ ¥ø   [Planck-Einstein relations]            <5-52>  <-   h: ÇöûÅ© »ó¼ö,  𝑣: Á֯ļö,  ©¤: µð·¢ »ó¼ö(Dirac's constant), ¥ø: °¢Áøµ¿¼ö(angular frequency),  𝑣 = ¥ø/2¥ð,  ©¤ = h/2¥ð
𝑃 = ©¤ 𝐾                                                             <5-53>  <-  <5-36>¿¡¼­ 𝑃 = 𝐸 𝑽, ∣𝑽∣ = 1,  ¡Å ∣𝑃∣ = 𝐸,  𝐾: ÆÄµ¿ 3-º¤ÅÍ(wave three vector), ∣𝐾∣ = ¥ø,
p¥á = (𝐸, 𝑃) = (©¤ ¥ø, ©¤ 𝐾 ) = ©¤ k¥á                             <5-54>  <-  𝐤: ÆÄµ¿ 4-º¤ÅÍ(wave four vector)
𝐩 ∙ 𝐩 = 𝐤 ∙ 𝐤 = 0                                                  <5-55>
d𝐩 /d¥ë = 0,  d𝐤 /d¥ë = 0                                       <5-56>  <-   ¥ë: ¾ÆÇÉ ¸Å°³º¯¼ö(affine parameter)

±¤ÀÚ(photons)´Â 0 ÀÇ Á¤Áö Áú·®À» °®°í, 4-º¤ÅÍ 𝐩, 𝐤 ´Â ³Î(null) ¼¼°è¼±ÀÇ Á¢Çϸç, Á¢¼±º¤ÅÍ 𝐮 ´Â ¾ÆÇÉ ¸Å°´º¯¼ö ¥ë ÀÇ ±Ô°ÝÈ­(normalization)¸¦ Á¶Á¤ÇÔÀ¸·Î½á 𝐩, 𝐤 ¿Í ÀÏÄ¡½Ãų ¼ö ÀÖ°Ô µË´Ï´Ù.

µµÇ÷¯ ÆíÀÌ¿Í »ó´ë·ÐÀû ºñ¹Ö[ºÐ»çÃâ](Doppler shift and Relativistic Beaming)

»ç¹æÀ¸·Î ¥ø ÀÇ (°¢)Áøµ¿¼ö·Î ±¤ÀÚ¸¦ ºÐÃâÇϰí ÀÖ´Â ±¤¿øÀÌ ÀÖ´Â °è¿Í ÀÌ¿Í x'Ãà ¹æÇâÀ¸·Î ¼Óµµ V·Î ¿òÁ÷ÀÌ´Â °ü¼º°è¸¦ »ý°¢Çغ¸¸é ¿ª½Ã ·Î·»Ã÷ ºÎ½ºÆ®(Lorentz Boost)·Î¼­ °ü°è½ÄÀÌ ¼º¸³ÇÕ´Ï´Ù.
ÀÌ °æ¿ì¿¡  x'Ãà°ú ¥á'ÀÇ °¢µµ¸¦ ÀÌ·ç´Â ÇÑ ±¤ÀÚ(a phonton)ÀÇ Áøµ¿¼ö(Á֯ļö frequency)´Â ¾î¶°Çұ »ìÆìº¸±â·Î ÇսôÙ.

¥ø = ¥ã (¥ø' - V kx' )                                              <5-57>  <-  <5-54>¿¡ µû¶ó ÆÄµ¿ 4-º¤ÅÍ 𝐤 ÀÇ k¥á = (¥ø, 𝐾),  k'¥á = (¥ø', 𝐾') ¶ó°í Çϸé
¥ø' = ¥ø ¡î (1 - V©÷) / (1- V cos ¥á')                         <5-58>  <-  »ó´ë·ÐÀû µµÇ÷¯ ÆíÀÌ(Relativistic Doppler Shift}
¥ø' ≈ ¥ø (1 + V cos ¥á')                                        <5-59>  <-  V°¡ ÀÛÀ» ¶§, 1/(1 - x) = 1 + x + x©÷ + ...,   (∣x∣<1),  ¸ÅŬ·Î¸° ±Þ¼ö(Maclaurin series)

¥á'= 0 ÀÌ¸é ¥Ä¥ø = + V ¥ø À̹ǷΠû»öÆíÀÌ(blue shift)ÀÌ, ¥á'= ¥ð ÀÌ¸é ¥Ä¥ø = - V ¥ø À̹ǷΠÀû»öÆíÀÌ(red shift)ÀÌ, ¥á'= ¥ð/2 À̸é <5-58>¿¡ µû¶ó Ⱦ´Ü Àû»öÆíÀÌ(tansverse red shift)°¡ µË´Ï´Ù!
À§ÀÇ µÎ¹øÂ° ±×¸²Àº V = 0.75c ÀÇ »ó´ë·ÐÀû ºñ¹Ö Çö»óÀ» º¸¿©ÁÝ´Ï´Ù. ¿ø·¡´Â °°Àº ±æÀÌÀÎ È­»ìÇ¥ÀÇ º¤Å͵éÀÌ µµÇ÷¯ È¿°ú¶§¹®¿¡ ª¾ÆÁö°í(Àû»öÆíÀÌ), ±æ¾îÁö¸ç(û»öÆíÀÌ), °¢°¢ ¹æÇâµéµµ º¯È­ÇÕ´Ï´Ù.

cos ¥á'= (cos ¥á + V) / (1 + V cos ¥á)                     <5-60>  <-   x(x')Ãà°ú ¥á(¥á')ÀÇ °¢µµ¸¦ ÀÌ·ç¸ç ¹æÃâµÇ´Â ±¤ÀÚÀÇ cos ¥á = kx/ ¥ø,  cos ¥á' = kx'/ ¥ø',  ·Î·»Ã÷ º¯È¯(Lorentz Transformation)µÊ.

±ÕÀÏÇÏ°Ô ºûÀ» ³»´Â ¹°Ã¼°¡ ¿ì¸®¿¡°Ô ´Ù°¡¿Ã ¶§°¡ ¸Ö¾îÁú ¶§º¸´Ù µµÇ÷¯ È¿°ú¿¡ ÀÇÇØ °­µµ(intencity)°¡ ´õ ÁýÁßµÇ¾î ´õ ¹à°Ô º¸ÀÌ´Â °ÍÀ» »ó´ë·ÐÀû ºñ¹Ö(Relativistic Beaming)*À̶ó ºÎ¸¨´Ï´Ù.

5.6 °üÂûÀÚ¿Í °üÃø(Observers and Observations)

À§ÀÇ ¼¼¹øÂ° ±×¸²¿¡¼­ °¡¼ÓµÇ°í ÀÖ´Â ¼¼°è¼±ÀÇ ÇÑ ÁöÁ¡(world point)¿¡¼­ ±¹¼Ò ½ÇÇè½Ç(local laboratory)·Î¼­ Á÷±³ÇÏ´Â ½Ã°£ Â÷¿ø°ú 3¹æÇâÀÇ °ø°£ Â÷¿øÀÇ 𝐞𝟘, 𝐞𝟙, 𝐞𝟚, 𝐞𝟛 ±âÀú 4-º¤Å͸¦ °®½À´Ï´Ù.

𝐞𝟘 = 𝐮obs                                                            <5-61>   <-   𝐮obs: °üÃøÀÚÀÇ 4-¼Óµµ º¤ÅÍ(Observer's four-velocity vector),  ¼¼°èÁ¡¿¡¼­ÀÇ ´ÜÀ§ Á¢¼±º¤ÅÍ(unit tangent vector at world point)
𝐩 = p¥á𝐞¥á                                                             <5-62>
p𝟘 = -𝐩 ∙ 𝐞𝟘,  p𝟙 = 𝐩 ∙ 𝐞𝟙,  p𝟚 = 𝐩 ∙ 𝐞𝟚,  p𝟛 = 𝐩 ∙ 𝐞𝟛       <5-63>   <-   ±âÀú º¤Å͵éÀº »óÈ£°£ Á÷±³(orthoonal)ÇϹǷΠÁ¡°ö(dot product)À» ÇÏ¸é ½º½º·ÎÀÇ ¿ä¼Ò¸¸ÀÌ ³²À½.

À§ÀÇ ³×¹øÂ° ±×¸²Àº ÇÑ ÀÔÀÚ°¡ ±×´ë·Î Á¤ÁöÇØ ÀÖÀ¸¸é¼­(at rest) °üÃøÀÚ´Â ¼Óµµ V·Î ¿òÁ÷ÀÏ ¶§, °üÃøÀÚÀÇ 4-¼Óµµ º¤Å͸¦ µû¶ó 4-¿îµ¿·® ÁßÀÇ ÀÔÀÚÀÇ ¿¡³ÊÁö(energy)¸¦ ÃøÁ¤ÇÏ´Â °æ¿ìÀÇ µµÇØÀÔ´Ï´Ù.
                                                          
𝐩 = (m, 0, 0, 0)                                                    <5-63>   <-   m: Á¤Áö Áú·®(rest mass)
𝐞𝟘 = 𝐮obs = (¥ã, V¥ã, 0, 0)                                         <5-64>  
𝐸  = -𝐩 ∙ 𝐞𝟘 = -𝐩 ∙ 𝐮obs = m ¥ã                                  <5-65>  
  
Áï, ¿òÁ÷ÀÌ´Â °üÂûÀÚ°¡ ÃøÁ¤ÇÑ Á¤ÁöÇÑ ÀÔÀÚÀÇ ¿¡³ÊÁö´Â °á±¹ °üÃøÀÚÀÇ ½Ã°£Ãà ±âÀúº¤ÅÍ 𝐞𝟘¸¦ µû¸£´Â ±× ÀÔÀÚÀÇ ¿¡³ÊÁö-¿îµ¿·® 4-º¤ÅÍ(energy- momentum 4-vector)ÀÇ ¿ä¼ÒÀÎ °ÍÀÔ´Ï´Ù..

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. Ư¼ö»ó´ë¼ºÀÇ ¼öÇÐÀº ±âÃÊ ¼öÁØÀ̹ǷΠ'°³³ä'À» Àß ÆÄ¾ÇÇÏ¸é µÇÁö¸¸, ÀϹݻó´ë¼ºÀº '°íµî ¼öÇÐ'-¹ÌºÐ±âÇÏÇаú ÅÙ¼­Çؼ®À» ¹ÙÅÁÀ¸·Î ÇÑ´Ù°í ...
       ¿©±â¼­´Â ÃøÁö¼±À» ã´Â º¯ºÐ¹ýÀÇ '¿ÀÀÏ·¯-¶ó±×¶ûÁÖ ¹æÁ¤½Ä'ÀÌ °¡Àå ³ôÀº ³­À̵µ¸¦ Áö´Ñ ¼ö¸®¹°¸®ÇÐÀ̹ǷΠº°µµÀÇ ÇнÀÀÌ ÇÊ¿äÇÔ.
* ÃÊ±Þ ¿µ¾îÀÎ beamingÀ» Àü°øÀÚ¸¸ ¾Ë ¼ö ÀÖ´Â 'ºÐ»çÃâ(ÝÄÞÒõó)'·Î ¹ø¿ªÇؼ­ ±â¾ïÇÏ´Â °ÍÀº ³Í¼¾½ºÀÎ µíÇØ¼­ ¹ßÀ½´ë·Î Ç¥±âÇÔ.


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