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ÀϹݻó´ë¼º 3. Ư¼ö»ó´ë¼º; ÃøÁö¼±
    ±è°ü¼®  2019-09-04 15:20:26, Á¶È¸¼ö : 564
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* Special Relativity Review

     ∘  Invariance of the Interval <- Figure II-8 ÂüÁ¶
        Laboratory °üÂûÀÚ S¿Í S¿¡ ´ëÇØ »ó´ëÀûÀ¸·Î ¼Óµµ ¥â·Î ¿òÁ÷ÀÌ´Â rocket °üÂûÀÚ S'¸¦ °¡Á¤ÇÑ »ç°í ½ÇÇè¿¡¼­ Figure 2-8ó·³
        ·ÎÄÏ ¾ÈÀÇ ÁÂÇ¥ ¿øÁ¡¿¡¼­ y'ÃàÀ¸·Î °Å¸® L¸¸Å­ ¶³¾îÁø °Å¿ïÀ» ÇâÇØ¼­ ºûÀÇ beamÀ» ºñÃß¸é ¹Ý»çÇÏ¿©  𝛥t' = 2L ºû À̵¿½Ã°£ ÈÄ¿¡
        µÇµ¹¾Æ¿É´Ï´Ù. rockert °üÂûÀÚ´Â ¿øÁ¡¿¡ ÀÖ´Â ÇÑ single clock À¸·Î emmision(event A)¿Í reception(event B) »çÀÌÀÇ
        time interval 𝛥t'¸¦ ÃøÁ¤ÇÒ ¼ö ÀÖ½À´Ï´Ù. ±×·¸Áö¸¸ laboratory °üÂûÀÚ S´Â ÀÌ interval µ¿¾È rocket°ú ±× ¼ÓÀÇ °Å¿ïÀÌ ¿ìÃøÀ¸·Î
        °Å¸® 𝛥x¸¦ ¿òÁ÷ÀÌ´Â °ÍÀ» º¾´Ï´Ù. µû¶ó¼­ °üÂûÀÚ S¿¡°Ô ºûÀº rockect °üÂûÀÚ¿¡°Ô ÀÎ½ÄµÈ °Íº¸´Ù ±ä figur 2-8(b)ÀÇ A¡æM¡æB¸¦ 
        À̵¿ÇÏ¿´½À´Ï´Ù. (∆AMBÀÇ ³ôÀÌ´Â LÀ̸ç, »ó´ëÀû ¿îµ¿ÀÇ ¹æÇâ¿¡ ¼öÁ÷ÀÎ ±æÀÌ´Â ºÒº¯ÇÕ´Ï´Ù.) ±¤¼ÓÀº µÎ °üÂûÀÚ¿¡°Ô µ¿ÀÏÇÕ´Ï´Ù.
        𝛥t'/2 = [(𝛥t/2)2 - (𝛥x/2)2]1/2  or   (𝛥t')2 =  (𝛥t)2 - (𝛥x)2   [II-77]
        𝛥t' = (1 - ¥â2)1/2 𝛥t  or 𝛥t = (1 - ¥â2)1/2 𝛥t' <- S¿¡ ´ëÇÑ S'ÀÇ ¼Óµµ ¥â = 𝛥x/𝛥t,  ÈÄÀÚ´Â S¿¡¼­ event A, B°¡ ÀϾ °æ¿ì   [II-78]
        (𝛥t')2 - (𝛥x')2 = (𝛥t)2 - (𝛥x)2  <-  𝛥x' = 0 À̹ǷΠ  [II-79]
        𝛥𝜏 = [(𝛥t)2 - (𝛥x)2]1/2 = [(𝛥t')2 - (𝛥x')2]1/2 <- interval  between event A and B: ÁÂÇ¥°èÀÇ º¯°æ¿¡ invariant ÜôܨÇÔ.   [II-80]
        𝛥𝜏 = [(𝛥t)2 - (𝛥x)2  - (𝛥y)2  - (𝛥z)2]1/2 <- timelike interval: proper time between the events    [II-81]
        𝛥𝜎 = [(𝛥x)2 + (𝛥y)2 +  (𝛥z)2 - (𝛥t)2 ]1/2 <- spacelike interval: proper distance between the events   [II-82]
        L' = ¥â 𝛥t' =  ¥â 𝛥t (1 - ¥â2)1/2 = L (1 - ¥â2)1/2  <- L: Á¤Áö »óÅÂÀÇ ¹°Ã¼ ±æÀÌ,  L': µî¼Ó ¥â·Î ¿òÁ÷ÀÌ´Â ¹°Ã¼ ±æÀÌ(Ãà¼ÒµÊ)   [II-83]  
 
    ∘  The Lorentz Transformation <- Figure II-8, II-9 Âü°í
        x = a11x' + a12y' + a13z' + a14t',  y = y',  z = z', t = a41x' + a42y' + a43z' + a44t'  <-  x = x' + ¥ât   [II-84]
        x = a11x' + a14t',  t = a41x' + a44t'  <-  °ø°£ÀÇ µî¹æ¼º(isotropy)À¸·Î ÀÎÇØ¼­ a12 = a13 = 0,  a42 = a43 = 0   II-85,86]
        x = a11x' + ¥â (1 - ¥â2)-1/2 t'  <- S' ¿øÁ¡ÀÌ when x' = 0, x = ¥â t,  t = (1 - ¥â2)-1/2 t'   [II-87]
        t = a41x' + (1 - ¥â2)-1/2 t'  <- À§¿Í °°Àº ¹æ½ÄÀ¸·Î   [II-88]
        [a41x' + (1 - ¥â2)-1/2 t']2 - [x = a11x' + ¥â (1 - ¥â2)-1/2]2 =  t'2 - x'2  <-  eq. (87)(88) and  t2 -  x2 =  t'2 - x'2
        (a412 - a112) x'2 + 2(1 - ¥â2)-1/2 (a41 - ¥â a11) t' x' + t'2 =  t'2 - x'2,  ¡Å  (a412 - a112) = -1, a41 - ¥â a11 = 0
        x = (x' + ¥â t')(1 - ¥â2)-1/2,  y = y',  z = z',  t = (¥â x' + t')(1 - ¥â2)-1/2  <- called as Lorentz Transformation   [II-89]
        x' = (x - ¥â t')(1 - ¥â2)-1/2,  t = (-¥â x' + t')(1 - ¥â2)-1/2  <- the inverse transformation: ¥â replaced by -¥â   [II-90]
        𝛥x = (𝛥x' + ¥â 𝛥t')(1 - ¥â2)-1/2,  𝛥t = (¥â𝛥x' + 𝛥t')(1 - ¥â2)-1/2  <- for pair of envents   [II-91]  

    ∘  Lorentz Geometry
        L(𝛼) = ¡ò𝛼 ds = ¡ò𝛼 [(dx)2 + (dy)2 + (dz)2]1/23 in 𝔼3; L(𝛼) = ¡ò𝛼 d𝜏 = ¡ò𝛼 [(dt)2 - (dx)2 - (dy)2 - (dz)2]1/2 in 𝓡4  [II-92]
        (d𝜏)2 = (dt)2 - (dx)2 - (dy)2 - (dz)2]1/2  <- proper time of 𝛼  or spacetime length;  𝓡4: Minkowski space   [II-93]
        Lorentz coordinates : °ü¼º °üÂûÀÚ°¡ »ç¿ëÇÏ´Â °Å¸® ÃøÁ¤°ú µ¿±âÈ­µÈ ½Ã°è(synchronized clock)ÀÇ °üÁ¡¿¡¼­ Á¤ÀÇµÈ ÁÂÇ¥°è 
        u0 = u0(t, x, y, z), u1 = u1(t, x, y, z), u2 = u2(t, x, y, z), u3 = u3(t, x, y, z) <- smooth, non-singular Jacobian matrix   {II-95] 
        (d𝜏)2 = 𝑔𝑖𝑗 du𝑖 du𝑗;  If  d2ur/(d𝜏)2 + 𝛤r𝑖𝑗 du𝑖/d𝜏 du𝑗/d𝜏 = 0,  r = 0,1,2,3, a curve u𝑖(𝜏), 𝑖 = 0,1,2,3  is called geodesic   [II-96,97]
        ÀÌó·³ ÀϹÝÈ­µÈ Çü½ÄÀÇ Ç¥Çö¿¡ ÀÇÇØ¼­, Chapter I ¿¡¼­¿Í °°ÀÌ Christoffel ±âÈ£¿Í curvature tensor¸¦ Á¤ÀÇÇÒ ¼ö ÀÖ½À´Ï´Ù!

6. Geodesics(ÃøÁö¼±)

     ∘  ½Ã°ø°£Àº ´ÙÀ½ÀÇ metric formÀ» °®´Â semi-Riemannian 4-manifold·Î ÆÄ¾ÇµË´Ï´Ù. (d𝜏)2 = 𝑔𝜇𝜈dx𝜇dx𝜈,  in  (x0, x1, x2, x3)
         ¸¸ÀÏ <𝐯, 𝐯> = 𝑔𝜇𝜈du𝜇du𝜈 °¡ °¢°¢ ¾ç, ¿µ, À½ÀÏ ¶§, vector 𝐯 = v 𝜇 ¡Ó/¡Ó𝐱𝜇 ¸¦ °¢°¢ timelikelightlike, spacelike ¶ó°í ºÎ¸¨´Ï´Ù.  

     ∘  Definition III-1
        ¸¸ÀÏ ÇÑ ½Ã°ø°£ °î¼±(spacetime curve) 𝛂°¡ ´ÙÀ½À» ¸¸Á·ÇÏ´Â ÇÑ paramatrization x𝜆(𝜌)¸¦ °®´Â´Ù¸é, ±×°ÍÀº ÇÑ geodesicÀÌ´Ù.
        d2x𝜆/(d𝜌)2 + 𝛤𝜆𝜇𝜈 dx𝜇/d𝜌 dx𝜈/d𝜌 = 0,  𝜆 = 0,1,2,3 <- This definition is independent of a choice of coordinate system.   [6-120]

     ∘  ¹æÁ¤½Ä (120)Àº ´ÙÀ½ ÇÔ¼ö°¡ »ó¼öÀÓÀ» ¾Ï½ÃÇÕ´Ï´Ù. Áï, <𝛂', 𝛂'> = (d𝜏/d𝜌)2 = 𝑔𝜇𝜈 dx𝜇/d𝜌 dx𝜈/d𝜌 = C2 <- C2: constant
        C2°¡ °¢°¢ ¾ç, ¿µ, À½ÀÏ ¶§, 𝛂¸¦ °¢°¢ timelike, lightlike, spacelike ¶ó°í ºÎ¸¨´Ï´Ù.
        ¸¸ÀÏ 𝛂°¡ timelikeÀ̶ó¸é, 𝜌 = a𝜏 + b (a, b´Â ½Ç¼ö)·Î Ãß·ÐÇÕ´Ï´Ù. ¿ì¸®´Â 𝛂°¡ "future-directed"µÇµµ·Ï, a>0·Î °¡Á¤ÇÒ °ÍÀÔ´Ï´Ù.    
            -> ±×·¯¸é 𝛂¸¦ proper timeÀ¸·Î½á °£´ÜÇÏ°Ô Àç¸Å°³È­(reparamatrization)ÇÔÀ¸·Î½á Eq. (120)ÀÇ 𝜌¸¦ 𝜏·Î ´ëÄ¡ÇÒ ¼ö ÀÖ½À´Ï´Ù.
        ¸¸ÀÏ 𝛂°¡ lightlikeÀ̶ó¸é, 𝜏´Â 𝛂¸¦ µû¶ó ÀÏÁ¤Çϰí, <𝛂', 𝛂'> = 0 À̹ǷÎ, ¿ì¸®´Â proper timeÀ» ¸Å°³º¯¼ö·Î »ç¿ëÇÒ ¼ö ¾ø½À´Ï´Ù.
        ¸¸ÀÏ 𝛂°¡ spacelikeÀ̶ó¸é, d𝜏/d𝜌´Â Çã¼ö°¡ µÇ°í °î¼±Àº proper distance·Î Àç¸Å°³È­ÇÏ¿© 𝜌 = a𝜎 + b°¡ µË´Ï´Ù.
            -> ¾Æ¹«·± ½ÅÈ£³ª ¹°Áú ´ë»óµµ spacelike °æ·Î·Î À̵¿ÇÒ ¼ö ¾ø±â ¶§¹®¿¡ ¿ì¸®´Â ÀÌ °æ¿ì´Â ÇÊ¿äÇÏÁö ¾Ê½À´Ï´Ù.  
        ±×·¡¼­ ¸¸ÀÏ °î¼± 𝛂°¡ °¢ Á¡¿¡¼­ <𝛂', 𝛂'>°¡ ¾ç¼öÀ̶ó¸é, timelike À̶ó°í ºÎ¸¨´Ï´Ù.
 
    ∘  Theorem III-2
        𝛂°¡ ¾ç³¡Á¡ °£ÀÇ ½Ã°ø°£ °Å¸®(proper time °£°Ý)À» ±ØÄ¡È­ÇÏ´Â(extremize) timelike °î¼±À̶ó°í Çϸé, 𝛂´Â ÇÑ geodesicÀÌ´Ù.

     ∘  Theorem III-3
        ÇÑ event 𝐏¿Í 𝐏¿¡¼­ÀÇ non-zero vector 𝐯°¡ ÁÖ¾îÁø´Ù¸é, 𝛂(0) = 𝐏 ±×¸®°í 𝛂'(0) = 𝐯ÀÎ À¯ÀÏÇÑ  geodesic 𝛂(𝜌)°¡ Á¸ÀçÇÑ´Ù.


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