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Ư¼ö»ó´ë¼º I-2. ·Î·»Ã÷ º¯È¯
    ±è°ü¼®  2018-07-02 00:19:56, Á¶È¸¼ö : 565
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4.1  ·Î·»Ã÷ º¯È¯(Lorentz Transformation)

°ü¼º°è K(K system)¿Í °ü¼º°è K'(K' system)»çÀ̸¦ ¿¬°áÇÏ´Â º¯È¯¹ýÄ¢Àº ds©÷ = -c©÷dt©÷ + dx©÷ + dy©÷ + dz©÷ = -c©÷dt'©÷ + dx'©÷ + dy'©÷ + dz'©÷ ÀÔ´Ï´Ù.
°ü¼º°è°£¿¡ ÀÌ ½Ã°ø°£ 4Â÷¿øÀÇ ºñÀ¯Å¬¸®µåÀûnon-Euclidiean ¼±¿ä¼ÒÀÎ ÀÌ ds¸¦ º¸Á¸ÇÏ´Â º¯È¯(transformation)ÀÇ Áß¿äÇÑ ¿¹°¡ ·Î·»Ã÷ ºÎ½ºÆ®(Lorentz Boosts)ÀÔ´Ï´Ù.
À§ ù¹øÂ° ±×¸²Àº ½Ã°ø°£ ´ÙÀ̾Ʊ׷¥¿¡¼­ Áº¯ÀÇ º¯È­·Î¼­ÀÇ ·Î·»Ã÷ ºÎ½ºÆ®·Î¼­ ÀÌ´Â (ct,x) Æò¸é¿¡¼­ÀÇ È¸Àü»ó»ç(analogs of rotations)¸¦ »ìÆìº¸±â·Î ÇսôÙ.
±×·¡¼­ ÀÌ °ü°è´Â ½Ã°ø°£ÀÇ ºñÀ¯Å¬¸®µåÀûÀÎ ¼º°ÝÀ¸·Î ÀÎÇØ ½Ö°î¼±ÇÔ¼ö(hyperbolic functions)ÀÇ °ü°è·Î ³ªÅ¸³ª°Ô µÇ´Â °ÍÀÔ´Ï´Ù. Áï,

ct' = (cosh¥è)(ct) - (sinh¥è)x            <4-1a> <-  ¥è(Theta ¼¼Å¸) ¼Ò¹®ÀÚ
x' = (-sinh¥è)(ct) + (cosh¥è)x           <4-1b>
y' = y, z' = z                                      <4-1c> <- Æò¸é¿¡¼­ ±âÇÏÇÐÀûÀ¸·Î ³ªÅ¸³»±â À§Çؼ­ y¿Í z´Â º¯ÇÏÁö ¾Ê´Â °æ¿ì¸¦ °í·ÁÇÔ.

* ½Ö°î¼±ÇÔ¼ö´Â 2Â÷¿ø Æò¸é»ó¿¡¼­ ¸Å°³º¯¼ö ¥è¸¦ »ç¿ëÇÑ ÀÚÃë·Î (cosh¥è, sinh¥è)Àº ½Ö°î¼± x©÷ - y©÷ = 1 À» ±×¸®¸ç cosh©÷¥è - sinh©÷¥è= 1 ÀÓ.

(ds)©÷ = -(cdt')©÷ + (dx')©÷ + (dy')©÷ + (dz')©÷
        =  -[cosh¥è(cdt) - sinh¥è(dx)]©÷ +[-sinh¥è(cdt) + cosh¥è(dx)]©÷ + (dy)©÷ + (dz)©÷
        = -(cdt)©÷  + (dx)©÷  + (dy)©÷  + (dz)©÷         

V = c(tanh¥è)              <4-2a> : °ü¼º°è K'ÀÇ ¼Óµµ  <-  x'=0 ÀÏ ¶§ <2-1b>¿¡¼­ 0 = (-sinh¥è)(ct) + (cosh¥è)x,  tanh¥è = sinh¥è/ cosh¥è, V = x/t
tanh¥è = V/c               <4-2b>  <- sinh¥è = V/c /¡î (1-V©÷/c©÷), cosh¥è = 1 /¡î (1-V©÷/c©÷)

¥ã = 1/¡î (1-V©÷/c©÷)        <4-3>   <-  ¥ã(Gamma °¨¸¶) ¼Ò¹®ÀÚ, cosh¥è °ª, Ç¥±â °£¼ÒÈ­¸¦ ·òÇØ µµÀÔµÊ.

t' = ¥ã (t - Vx/c©÷)         <4-4a>
x' = ¥ã (x - Vt)             <4-4b>
y' = y, z' = z              <4-4c>

t = ¥ã (t' + Vx'/c©÷)        <4-5a>
x= ¥ã (x' + Vt')             <4-5b>
y= y', z = z'                <4-5c>

V/c ¡ì 1 (V°¡ ±¤¼Óº¸´Ù ¾ÆÁÖ ÀÛÀ» ¶§):  x = x' + Vt, y = y', z = z', t = t' + (V/c©÷)x'    <4-6>   <- ±Ù»ç½Ä

À§ µÎ¹øÂ° ±×¸²Àº ¿¹·Î¼­ °ü¼º°è K'¿¡¼­ µ¿½ÃÀÎ »ç°Çµé(events) A¿Í B°¡, K syetem¿¡¼­´Â A»ç°Ç ÀÌÈÄ¿¡ B°¡ ÀϾ´Ù´Â °ÍÀ» º¸¿©ÁÖ°í ÀÖ½À´Ï´Ù.

∆t = ¥ã (V/c©÷)∆x'           <4-7>   <-  ∆t' = 0,  ∆x' = x'B - x'A , <4-5a> : µ¿½Ã¼ºÀÇ »ó´ë¼º(the relativity of simultaneity) [Hartle p.73]   

À§ ¼¼¹øÂ° ±×¸²Àº ±æÀÌÀÇ ·Î·»Ã÷ ¼öÃà(Lorentz Contraction)À» º¸¿© ÁÖ´Â °ÍÀ¸·Î K system¿¡¼­ L₀ ÀÎ ¸·´ërod°¡ K' system¿¡¼­´Â ¾î¶»°Ô µÉ±îÇÏ´Â °ÍÀÔ´Ï´Ù.
K system¿¡¼­ L₀ ¶³¾îÁø Á¡ÀÌ K' system¿¡¼­ t' = 0, x'Ãà¿¡ ÀÖÀ¸¸é¼­ c∆t¸¸Å­ À̵¿ÇÑ °Å¸® LÀε¥ ±æ°Ô º¸À̳ª ºñÀ¯Å¬¸®µå ±âÇÏÀÌ¶ó¼­ ½ÇÁ¦·Î´Â ´õ ª½À´Ï´Ù.

L = L₀ ¡î (1-V©÷/c©÷)        <4-8>   <- <4-5b>,  ∆x = L₀ = ¥ã ∆x',  ∆x' = L = 1/¥ã ∆x = 1/¥ã L₀  : ·Î·»Ã÷ ¼öÃà(Lorentz Contraction) [Hartel p.70]                                
∆t' = ∆t ¡î (1-V©÷/c©÷)      <4-9>  <-  <4-5a>, ∆t = t©ü- t©û¸¦ ´ëÀÔÇÏ¿© Á¤¸®ÇØ º¸¸é <3-3>°ú ÀÏÄ¡ÇÔ! ½Ã°£ ÆØÃ¢(time dilation) [Landau-lifshitz p.12]

4.2  ¼ÓµµÀÇ º¯È¯(Transformation of Velocities)

À̹ø¿¡´Â K system¿¡¼­ xÃàÀ» µû¶ó ¼Óµµ V·Î ¿òÁ÷ÀÌ´Â K' system¿¡¼­´Â ¹°Áú ÀÔÀÚÀÇ ¼Óµµ°¡ ¾î¶»°Ô º¸ÀÌ´Â °¡ÇÏ´Â ¼Óµµ º¯È¯ °ø½ÄÀ» »ìÆìº¸±â·Î ÇÕ´Ï´Ù.
K system¿¡¼­ ÀÔÀÚ ¼ÓµµÀÇ °¢ x, y, z ¼ººÐÀ» °¢ 𝑉x, 𝑉y, 𝑉z¶ó Çϰí K' system¿¡¼­ÀÇ ÀÔÀÚ¼Óµµ¸¦ °¢ 𝑉x', 𝑉y', 𝑉z'¶ó Çϸé <4-5a, 5b, 5c>¿¡¼­,
dt = ¥ã (dt' + V dx'/c©÷),  dx= ¥ã (dx' + V dt'),  dy= dy',  dz = dz',  ¥ã = 1/¡î (1 - V©÷/c©÷),  dt/dx = 𝑉x À̹ǷÎ

𝑉x =[𝑉x' + V]/[1 + 𝑉x'* V/c©÷]                   <4-10a>
𝑉y = 𝑉y' ¡î (1 - V©÷/c©÷) /[1 + 𝑉x'* V/c©÷]       <4-10b>
𝑉z = 𝑉z' ¡î ( 1- V©÷/c©÷) /[1 + 𝑉x'* V/c©÷]       <4-10c>

𝑉x' =[𝑉x - V]/[1 - 𝑉x* V/c©÷]                    <4-11a>
𝑉y' = 𝑉y ¡î (1 - V©÷/c©÷) /[1 - 𝑉x* V/c©÷]        <4-11b>
𝑉z' = 𝑉z ¡î (1 - V©÷/c©÷) /[1 - 𝑉x* V/c©÷]        <4-11c>

¿¹·Î¼­ ¸¸ÀÏ K system¿¡¼­ ÇÑ ÀÔÀÚ°¡ xÃàÀ» µû¶ó¼­ ±¤¼Ó c·Î ¿òÁ÷ÀÎ´Ù¸é »ó´ë¼Óµµ V·Î ¿òÁ÷ÀÌ´Â k' system¿¡¼­´Â ¾î¶»°Ô º¸ÀÏ±î »ý°¢ÇØ º¸±â·Î ÇսôÙ.
<5-2a>¿¡ 𝑉x = c ¸¦ ´ëÀÔÇØº¸¸é ¹Ù·Î ¾Ë ¼ö Àִ¹٠𝑉x' = (c - V) / (1 - c* V/c©÷) = c. µû¶ó¼­, ±¤¼ÓÀº ¸ðµç °ü¼º°è¿¡¼­ µ¿ÀÏÇÑ °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. ·Î·»Ã÷ º¯È¯ÀÇ ´õ »ó¼¼ÇÑ °úÁ¤ÀÌ ÇÊ¿äÇϸé ÀÌÀÇ ¿øÀüÀÎ Landau-LifshitzÃ¥ The Lorentzy transformation pp.9-12 À» ÂüÁ¶ ¹Ù¶÷.
       Landau-LifshitzÃ¥Àº Hartle°ú ´Þ¸® ½Ã°£²Ã °£°Ý(timelike interval)ÀÎ ds©÷ = c©÷dt©÷ - dx©÷ - dy©÷ - dz©÷ ¸¦ »ç¿ëÇÏ´Â Â÷À̰¡ ÀÖÀ½.
     


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