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 ·Î±×ÀÎ  È¸¿ø°¡ÀÔ

ÀϹݻó´ë¼º 4. Einstein À广Á¤½Ä ***
    ±è°ü¼®  2019-09-04 15:23:20, Á¶È¸¼ö : 1,182


7. The Field Equations íÞÛ°ïïãÒ  ***

7-1. The Vacuum Field Equation *

     ∘  ÀÔÀÚµéÀÌ ½Ã°ø°£ geodesicÀ» µû¸£´Â °ÍÀ» ¸»Çϸ鼭, ¿ì¸®´Â ½Ã°ø°£ÀÇ ±âÇÏÇÐÀÌ ¾î¶»°Ô ¹°Áú¿¡ ÀÛ¿ëÇϴ°¡¸¦ ±â¼úÇϰíÀÚ ÇÕ´Ï´Ù.
        ÇÏÁö¸¸ À̰ÍÀº ´ÜÁö ¹ÝÂÊÀÇ À̾߱âÀÏ »ÓÀÔ´Ï´Ù. ÀÌ ÀÌ·ÐÀ» ¿Ï¼ºÇÏ·Á¸é ¶ÇÇÑ ¾î¶»°Ô ¹°ÁúÀÌ ±× ±âÇÏÇÐÀ» °áÁ¤Çϴ°¡¸¦ ±â¼úÇÒ Çʿ䰡
        ÀÖÀ¾´Ïµð. Áï, ¿ì¸®´Â ¹°ÁúÀÇ ºÐ¹è¿¡ metric °è¼ö 𝑔¥ì¥í¸¦ °ü·Ã½ÃŰ´Â ¹æÁ¤½Ä ÇÑ ¼¼Æ®°¡ ´õ ÇÊ¿äÇÕ´Ï´Ù. [¾Æ·¡ '7-2' ÂüÁ¶]
        ÀÌÁ¦ 1916³â ³í¹® '»ó´ë¼ºÀÇ ÀÏ¹Ý ÀÌ·ÐÀÇ ±âÃÊ'¶ó´Â ³í¹®¿¡ óÀ½ °ÔÀçµÈ À¯¸íÇÑ EinsteinÀÇ Field Equations íÞÛ°ïïãÒÀ» ޱ¸ÇÕ´Ï´Ù.
        ¿ì¸®´Â ÀÌ section¿¡¼­ EinsteinÀ» ÀÌ ¹æÁ¤½Ä¿¡ µµ´ÞÇϵµ·Ï ÇÑ ¾î¶°ÇÑ Ãß·Ð(some of reasoning)ÀÇ À±°ûÀ» »ìÆìº¼ ¿¹Á¤ÀÔ´Ï´Ù. ¸ÕÀú,
        ¿ì¸®´Â ¿©±â¼­ °í¸³µÈ ±¸´ëĪ Áú·® MÀÇ ¹Û-Áø°ø(vacuum)-¿¡¼­ Ç༺ÀÇ ¼·µ¿ È¿°ú´Â ¹«½ÃÇÑ Å¾ç Áß·ÂÀå °°Àº °æ¿ì¸¸À» °í·ÁÇÕ´Ï´Ù.

     ∘  [´Ü°è 1] ¸ÕÀú NewtonÀÇ ¹ýÄ¢À¸·ÎºÎÅÍ Áß·ÂÀÇ Æ÷ÅÙ¼È ÇÔ¼ö¸¦ À¯µµÇÑ ÈÄ, ÀÌ¿¡ »óÀÀÇÏ´Â ÀÏ¹Ý »ó´ë·ÐÀÇ tensor ÇÔ¼ö¸¦ ã½À´Ï´Ù.
        Let 𝐗 = (x, y, z),  r = (x2 + y2 + z2)1/2 = ¡«𝐗¡«.  Let  𝐮r = (1/r)𝐗 <- unit "radial vector"
        By Newton's law, the force 𝐅 on a particle m located at 𝐗 is (in geometric units),  𝐅 = - (Mm/r2)𝐮r
        Combining this with Newton's Second Law,  𝐅 = m (d2𝐗 /dt2), We have  d2𝐗 /dt2 = - (M/r2)𝐮r.
        Defining the potential function 𝚽 = 𝚽(r) by  𝚽 = - M/r, r > 0   [7-121]
        We have  d2x𝑖/dt2 = - ¡Ó𝚽/¡Óx𝑖,  𝑖 = 1,2,3 <- ¡Ór/¡Óx𝑖 = ¡Ó(𝐗 ∙ 𝐗)1/2/¡Óx𝑖 = 2 x𝑖/2 (𝐗 ∙ 𝐗)1/2 = x𝑖/r,  𝑖 = 1,2,3,  - 𝛁𝚽 = - M/r2 𝐮r   [7-122]
        Differentiating and then summing, we obtain Laplace's equation,  𝛁2 𝚽 = ¡Ó2𝚽/¡Ó x2 + ¡Ó2𝚽/¡Ó y2 + ¡Ó2𝚽/¡Ó z2 = 0   [7-123]
        For a continous distribution of matter throughout a region of space (not our case),  𝛁2 𝚽 = 4𝜋𝜌 [Poisson's equation]   [7-124]
        The function 𝚽 are replaced in general relativity by the equation  d2x𝜆/d𝜏2 + 𝛤𝜆¥ì¥í (dx¥ì/d𝜏) (dx¥í/d𝜏) = 0,  𝜆 = 0,1,2,3    [7-125]
        which contaion the first partial derivatives of metric coefficent 𝑔¥ì¥í,  𝛤𝜆¥ì¥í = (1/2)𝑔𝜆¥â(¡Ó 𝑔¥ì¥â/¡Óx¥í + ¡Ó 𝑔¥ì¥â/¡Óx¥ì - ¡Ó 𝑔¥ì¥í/¡Óx¥â)   [7-126]

     ∘  [´Ü°è 2] ¾Õ¼­ÀÇ 𝑔¥ì¥í¸¦ Æ÷ÅÙ¼È ÇÔ¼ö¿¡ ´ëÀÀ½ÃŲ ÈÄ, ±× ÀÌÂ÷ Æí¹ÌºÐÀÌ µé¾î°£ tensor ½ÄÀÌ Áú·® ¹Û ºó°ø°£¿¡¼­ '0'ÀÌ µÇµµ·Ï ÇÕ´Ï´Ù.
        We might expect the field equation in empty space to be a sytem of equations of the form  𝐺 = 0.   [7-127]
        Now we have the only tensors with metric coefficients 𝑔¥ì¥í, 𝑅𝜆¥ì¥í𝜎 = ¡Ó𝛤𝜆¥ì𝜎/¡Óx¥í - ¡Ó𝛤𝜆¥ì¥í/¡Óx𝜎 + 𝛤¥â¥ì𝜎 𝛤𝜆¥í¥â -  𝛤¥â¥ì¥í 𝛤𝜆¥â𝜎   [7-128]
        In case of flat spacetime of special relativity because 𝑔¥ì¥í are constant, the only solution: 𝑅𝜆¥ì¥í𝜎 = 0,  𝜆,¥ì,¥í,𝜎 = 0,1,2,3   [7-129]
        To allow for essential gravitational field that cannot be trnsformed away, if we set 𝜎 = 𝜆  in Eq. (129) and then sum over 𝜆,
        we get so-called Ricci tensor  𝑅¥ì¥í = 𝑅𝜆¥ì¥í𝜆 = ¡Ó𝛤𝜆¥ì𝜆/¡Óx¥í - ¡Ó𝛤𝜆¥ì¥í/¡Óx𝜆 + 𝛤¥â¥ì𝜆 𝛤𝜆¥í¥â -  𝛤¥â¥ì¥í 𝛤𝜆¥â𝜆   [7-130]
        Einstein chose, as the field equation for gravitational filed in empty space(a vacuum),  𝑅¥ì¥í = 0,  ¥ì,¥í = 0,1,2,3  ▮   [7-131]
        (In the case of some matter thoughout space, the right side is replaced tensors of energy, density and pressure of matter.)
  
    ∘  [´Ü°è 3] ¼±ÅÃµÈ Ricci tensor°¡ weak static Áß·ÂÀå¿¡¼­ õõÈ÷ ¿òÁ÷ÀÌ´Â ÀÔÀÚÀÇ °æ¿ì¿¡ °íÀü ¿ªÇаúÀÇ ÀÏÄ¡ÇÏ´Â Á¶°ÇÀ» ã½À´Ï´Ù.
        Let (x0, x1, x2, x3) be a locally Lorentz coordinate system in a neighborhoodof an event on a moving particle's world-line.
        If the particle's velocity is very samll (¦­dx𝑖/dt¦­¡ì 1 for 𝑖 =1,2,3)  d2x𝜆/d𝜏2 = - 𝛤𝜆¥ì¥í (dx¥ì/d𝜏) (dx¥í/d𝜏) ≈ - 𝛤𝜆00 (dt/d𝜏)2   [7-132]
       Since the graviational field is static (time derivative of the metric coefficients vanish),  𝛤𝜆00 = -(1/2)𝑔¥â𝜆 ¡Ó𝑔00 /¡Óx¥â
       Since the graviational field is waek, we may choose locally Lorentizan coordinates satisfying
                             ⌈ 1   0    0   0 ⌉  
       𝑔¥ì¥í = ¥ç¥ì¥í + h¥ì¥í,  (¥ç¥ì¥í) = ¦­0   -1    0   0¦­ = (¥ç¥ì¥í),  h¥ì¥í: small compared to unity. 𝑔¥ì¥í = ¥ç¥ì¥í + k¥ì¥í,  k¥ì¥í: small compared to unity.          
                            ¦­0    0   -1   0¦­
                             ⌊ 0   0    0  -1 ⌋        
       Accordingly, we have  𝛤𝜆00 = -(1/2)¥ç¥â𝜆 ¡Óh00 /¡Óx¥â  <- ¡ñ ¥ç¥ì¥í = constant, ¡Ó ¥ç00/¡Óx¥â = 0. ¡Å ¡Ó𝑔00/¡Óx¥â = ¡Óh00/¡Óx¥â   [7-133]
       Substituting Eq. (133) into (132), we obtain (to a clse approximation),  d2x𝜆/d𝜏2 = (1/2) (dt/d𝜏)2 ¥ç𝛼𝜆 ¡Óh00 /¡Óx𝛼   [7-134]

    ∘  [´Ü°è 4] ÃÖÁ¾ ´Ü°è·Î¼­ Newtonian Æ÷ÅÙ¼È ÇÔ¼ö 𝚽¿Í ÀÏ¹Ý »ó´ë¼ºÀÇ metric coefficient 𝑔¥ì¥í¿ÍÀÇ °ü°è½ÄÀ» µµÃâÇÕ´Ï´Ù.
       For 𝜆 = 0, this yields d2t/d𝜏2 = 0  (¡ñ ¥ç𝑖0 = 0, 𝑖 =1,2,3,  since it is static, ¡Óh00/¡Óx0 = 0,) or dt/d𝜏 = constant. Consequently,
       dx𝑖/dt = (dx𝑖/d𝜏) (d𝜏/dt),  d2x𝑖/dt2 = (d2x𝑖/d𝜏2) (d𝜏/dt)2, for 𝜆 ¡Á 0, d2x𝑖/dt2 = (1/2)¥ç𝛼𝜆 ¡Óh00 /¡Óx𝛼 = -(1/2)¡Óh00 /¡Óx𝑖,  𝑖 =1,2,3
       If we compare the latter with the Newtonian, d2x𝑖/dt2 = - ¡Ó𝚽/¡Óx𝑖 = -(1/2)¡Óh00 /¡Óx𝑖,  𝑖 =1,2,3  or h00 = 2𝚽 + C [C: constant].
       Since h¥ì¥í should vanish at infinity, we must  have C = 0 , and so h00 = 2𝚽 and ¥ç00 = 1, 𝑔00 = 1 + 2𝚽  ▮   [7-135]  

    ∘  Lemma III-4  
       For each ¥ì,  𝑔𝜆¥â ¡Ó𝑔𝜆¥â/ ¡Ó x¥ì = (1/𝑔) (¡Ó𝑔 / ¡Ó x¥ì) =  (¡Ó / ¡Ó x¥ì) ln¦­𝑔¦­
    ∘  Setting ¥í = 𝜆 in Eq. (126) and summing over 𝜆 and 𝑔𝜆¥â = 𝑔¥â𝜆,  we obtain,
        𝛤𝜆¥ì𝜆 = (1/2)𝑔𝜆¥â(¡Ó 𝑔¥ì¥â/¡Óx𝜆 + ¡Ó 𝑔𝜆¥â/¡Óx¥ì - ¡Ó 𝑔¥ì𝜆/¡Óx¥â) = (1/2) 𝑔𝜆¥â¡Ó 𝑔𝜆¥â/¡Óx¥ì = (1/2) (¡Ó / ¡Ó x¥ì) ln¦­𝑔¦­=  (¡Ó / ¡Ó x¥ì) ln¦­𝑔¦­1/2
    ∘  Substituting the result to Eq. (130) gives,
       𝑅¥ì¥í = (¡Ó2/¡Ó x¥ì¡Ó x¥í) ln¦­𝑔¦­1/2 - ¡Ó𝛤𝜆¥ì¥í /¡Óx𝜆 + 𝛤¥â¥ì𝜆 𝛤𝜆¥í¥â -  𝛤¥â¥ì¥í (¡Ó /¡Óx¥â) ln¦­𝑔¦­1/2. -> 𝑅¥ì¥í = 𝑅¥í¥ì ¡ñ  𝛤¥â¥ì¥í =  𝛤¥â¥í¥ì  ▮   [7-140]
 
7-2. Einstein's Field Equation (Supplement) **

    ∘  [´Ü°è 1] Newton ¹ýÄ¢ÀÇ Æ÷ÅÙ¼£ ¹æÁ¤½Ä 𝛁2 𝚽 = 4𝜋𝐺𝜌 [Poisson's equation]¿¡ »óÀÀÇÏ´Â »ó´ë·ÐÀû ÇÔ¼ö¸¦ ã½À´Ï´Ù.
        ÀÏ´ÜÀº Áº¯À» Ricci tensor·Î, ¿ìº¯Àº energy-momentum tensor 𝑇¥ì¥í¸¦ ¼±ÅÃÇÏ´Â °ÍÀÌ ÇÕ¸®ÀûÀÎ µí ÇÕ´Ï´Ù.  𝑅¥ì¥í = 𝜅𝑇¥ì¥í   (4.37)
        ½ÇÁ¦·Î EinsteinÀº ÀÌ·¸°Ô ÇÑ ÀûÀÌ ÀÖ¾ú½À´Ï´Ù¸¸ ¿©±â¿¡´Â ¿¡³ÊÁö º¸Á¸ ¹ýÄ¢°ú ¹®Á¦°¡ ÀÖÀ½À» ¹ß°ßÇß½À´Ï´Ù.  
        For the energy-momentum conservation in curved space,  𝛁¥ì𝑇¥ì¥í = 0, which then imply 𝛁¥ì𝑅¥ì¥í = 0.   (4.38)(4.39)
        ±×·±µ¥ Bianchi identity (3.94)¿¡ ÀÇÇϸé 𝛁¥ì𝑅¥ì¥í = (1/2) 𝛁¥í𝑅, ÇÑÆí Á¦½ÃµÈ field equationÀº 𝑅 = 𝜅𝑔¥ì¥í𝑇¥ì¥í = 𝜅𝑇 À̹ǷÎ, (4.40) *** 
        𝛁¥ì𝑇 = 0, scalar 𝑇 = constant °¡ µÇ´Âµ¥, Áø°ø¿¡¼­´Â 𝑇 = 0 À̳ª ¹°Áú¿¡¼­´Â  𝑇 > 0 À̹ǷΠÀÌ´Â ´ë´ÜÈ÷ ºÒÇÕ¸®ÇÕ´Ï´Ù.  (4.41)
        ±×·¡¼­ EinsteinÀº Ricci tensor·ÎºÎÅÍ ÀÚµ¿À¸·Î ¿¡³ÊÁö º¸Á¸ ¹ýÄ¢À» ¸¸Á·Çϵµ·Ï ´ÙÀ½°ú °°Àº Einstein tensor ¸¦ ¸¸µé¾ú½À´Ï´Ù.
        The Einstein tensor 𝐺¥ì¥í= 𝑅¥ì¥í- (1/2) 𝑅𝑔¥ì¥í,  𝑅: Ricci scalar, where 𝛁¥ì𝐺¥ì¥í = 0. We are led to propose  𝐺¥ì¥í = 𝜅𝑇¥ì¥í   (4.42)(4.43) 

    ∘  [´Ü°è 2] Einstein tensor Áß  Ricci tensor¸¸À» ºÐ¸®Çؼ­ energy-momentum tensor¿ÍÀÇ °ü°è½ÄÀ¸·Î º¯È¯ÇÑ ÈÄ, 𝑅00¸¦ ±¸ÇÕ´Ï´Ù.
        (4.43)ÀÇ ¾çº¯À» Ãà¾àÇϸé 4Â÷¿ø¿¡¼­  𝑅 = 𝜅𝑇, ´Ù½Ã (4.43)À» ¾²¸é  𝑅¥ì¥í = 𝜅{𝑇¥ì¥í - (1/2)𝑇𝑔¥ì¥í}.   (4.44)(4.45)   
        Áö±ÝºÎÅÍ Faber's Eq. (133)(134)(135)¸¦ ºñ±³ ÂüÁ¶Çϴµ¥, Carroll's Lorentz metric (¥ç¥ì¥í)ÀÇ + - ±âÈ£°¡ ¹Ý´ëÀÓ¿¡ ÁÖÀÇÇØ¾ß ÇÕ´Ï´Ù.
        In weak-field limit, 𝑔¥ì¥í = ¥ç¥ì¥í + h¥ì¥í,  ¦­h¦­¡ì1.  𝑔¥ì¥í = ¥ç¥ì¥í - h¥ì¥í,   where h¥ì¥í = ¥ç¥ì𝜌¥ç¥í𝜎h𝜌𝜎   (4.13)(4.14)  
        Because 𝑔00 = -1 +  h00,   𝑔00 = -1 -  h00,  𝑇 = 𝑔00𝑇00 ≈ - 𝑇00. Hence we get  𝑅00 = (1/2)𝜅𝑇00   (4.46)(4.47)(4.48)  
        𝑅00 = 𝑅¥ë0¥ë0,  We need only  𝑅𝑖0𝑗0, since 𝑅0000 = 0,  We have  𝑅𝑖0𝑗0 = ¡Ó𝛤𝑖00/¡Óx𝑗 - ¡Ó𝛤𝑖𝑗0/¡Óx0 + 𝛤𝑖𝑗𝜆 𝛤𝜆¥í¥â -  𝛤𝑖0𝜆 𝛤𝜆𝑗0   (4.49)   
        Since the second term vanishes for static fields and the third and fourth tems can be neglected because it is the form (𝛤)2
        if we compare them with the first one of 𝛤 . We are left with  𝑅𝑖0𝑗0 ≈ ¡Ó𝛤𝑖00/¡Óx𝑗 = ¡Ó𝑗𝛤𝑖00.
        𝑅00 = 𝑅𝑖0𝑗0 ≈ ¡Ó𝑖[(1/2)𝑔𝑖𝜆{¡Ó0𝑔𝜆0 + >{¡Ó0𝑔0𝜆 - ¡Ó𝜆𝑔00}] = -(1/2)¥ç𝑖𝑗¡Ó𝑖¡Ó𝑗h00 = - (1/2)𝛁2h00. Therefore  𝛁2h00 = - 𝜅𝑇00.   (4.50)(4.51)   

     ∘  [´Ü°è 3] ¾Õ¿¡¼­ À¯µµÇÑ 𝚽¿Í  𝑔¥ì¥í¿ÍÀÇ °ü°è½ÄÀ¸·ÎºÎÅÍ Einstein tensor¿Í energy-momentum tensor¿ÍÀÇ ºñ·Ê»ó¼ö¸¦ ã½À´Ï´Ù.
        From Newtonian potential 𝛁2𝚽 = 4𝜋𝐺𝜌, replacing 𝚽 by -(1/2)h00 and replacing 𝜌 by 𝑇00, we get  𝛁2h00 = -8𝜋𝐺𝑇00   (4.36)  .
        Then we have 𝜅 = 8𝜋𝐺. We can present Einstein's equations for general relativity:  𝑅¥ì¥í - (1/2)𝑅𝑔¥ì¥í = 8𝜋𝐺𝑇¥ì¥í. ▮   (4.52)        
     ∘  Áø°ø¿¡¼­´Â 𝑇¥ì¥í = 0 À̹ǷÎ, '7-1' [7-131]¿¡¼­Ã³·³ Vacuum Field Equations Áø°ø À广Á¤½ÄÀº 𝑅¥ì¥í = 0. ▮   (4.53)       
        
       À§ Equation (4.52)¸¦ ±âÁ¸ cgs ´ÜÀ§°è·Î Ç¥±âÇÏ¸é ³Î¸® ¾Ë·ÁÁø Einstein íÞÛ°ïïãÒ '𝑅¥ì¥í - (1/2)𝑅𝑔¥ì¥í =  (8𝜋𝐺/c4)𝑇¥ì¥í'ÀÌ µË´Ï´Ù!

      * Richard L.Faber Differential Geometry and Relativity TheoryÀÇ ¼öÇÐÀû rigor¸¦ ±¸ºñÇÑ Field EquationÀº '7-1' »ÓÀ̶ó¼­...  
      ** Sean M. Carroll Lecture Notes on General Relativity (1997, UC Santa Barbara)¸¦ Âü°íÇÏ¿©, ÇÊÀÚ°¡ ÀÛ¼ºÇÏ¿© º¸¿ÏÇÔ~
      *** °¡Àå Áß¿äÇÑ chapterÀÔ´Ï´Ù!; Bianchi identity(3.94) °üÇØ¼­´Â Dirac's GR section 13-14 Âü°í ¹Ù¶÷. (u. 3/10/2020)  

p.s. Einstein°ú ´ë¼öÇÐÀÚ Hilbert¿ÍÀÇ À广Á¤½Ä priority ´ÙÅùÀÌ ÀÖ¾ú´Âµ¥, HilbertÀÇ ÀÚÀÎÀ¸·Î Einstein ¿ì¼±±ÇÀÌ ÀÎÁ¤µÇ¾ú´Ù°í ÇÔ. 


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