±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° Æijë¶ó¸¶»çÁø °¶·¯¸® Ŭ·¡½Ä ·¹ÄÚµå °¶·¯¸® AT Æ÷·³ Æ®·¹Å· Á¤º¸ ¸µÅ©


 ·Î±×ÀÎ  È¸¿ø°¡ÀÔ

Ư¼ö»ó´ë¼º ¿ªÇÐ II-1. 4-º¤ÅÍ; µ¿·ÂÇÐ [u. 3/2021]
    ±è°ü¼®  2018-07-05 06:34:14, Á¶È¸¼ö : 22,752
- Download #1 : SR3.jpg (59.2 KB), Download : 1



Special Relativistic Mechanics(Ư¼ö»ó´ë¼º ¿ªÇÐ)

* ¾ÕÀ¸·Î´Â »ó´ë¼ºÀ̷п¡¼­ ÀϹÝÈ­ µÈ ½Ã°£ ´ë½Å ±¤¼Ó c (299 792 458 m/s) = 1 ´ÜÀ§ °Å¸®(units length)·Î ȯ»êÇÏ´Â Áú·®-±æÀÌ ½Ã½ºÅÛ ML(mass-length) systemÀ» ÁÖ·Î »ç¿ëÇÕ´Ï´Ù.*

5.1  4º¤ÅÍ(Four-Vectors)

4-º¤ÅÍ(four-vectors)¶õ 4Â÷¿ø ½Ã°ø°£ º¤ÅÍÀÌ¸ç ±×¸²Ã³·³ ½Ã°£²Ã(timelike), °ø°£²Ã(spacelike), ³Î(null) 4-º¤ÅÍ°¡ ÀÖÀ¸¸ç, ³Î 4-º¤ÅÍ´Â ºñÀ¯Å¬¸®µå ±âÇÏ¿¡¼­ ±æÀÌ°¡ 0ÀÔ´Ï´Ù.
¾ÕÀ¸·Î °£È¤ ±×³É º¤ÅÍ(vector)¶ó°í ÇÒ ¶§¿¡µµ º¸Åë 4-º¤ÅÍ(four-vectors)¸¦, ±âÀúº¤ÅÍ(basis vectors)´Â ±âÀú 4-º¤ÅÍ(basis four-vectors)¸¦ °¡¸£Å°´Â °ÍÀ¸·Î °£ÁÖÇϽñ⠹ٶø´Ï´Ù.

        𝐚 = at𝐞t + ax𝐞x + ay𝐞y + az𝐞z   <5-1>  <-  '𝐚'¿Í °°ÀÌ ±½Àº ±ÛÀÚ(boldface letters)´Â º¤ÅÍ, '𝐞'´Â ±âÀúº¤ÅÍ(basis vectors).
        𝐚 = a0𝐞0 + a1𝐞1 + a2𝐞2 + a3𝐞3   <5-2>  
        ¢²3¥á=0 a¥á𝐞¥á   <5-3>  <-  ¾Æ·¡¿Í °°ÀÌ Einstein summation convention(¾ÆÀν´Å¸ÀÎ ÇÕ»ê ±Ô¾à)À¸·Î Ç¥±â °¡´É
        𝐚 = a¥á𝐞¥á   <5-4>  <-  Greek indices: 0~3 ÇÕ»ê, Roman indices: 1~3 ÇÕ»ê [HartleÃ¥ÀÇ °æ¿ì, Landau-LifshitzÃ¥¿¡¼­´Â ¹Ý´ëÀÓ.]
        at'= ¥ã(at - Vax/c2),   ax'= ¥ã(ax - Vat/c2),   ay'= ay,    az'= az,   ¥ã = (1 - V2/c2)-1/2   <5-9>   <- Lorentz Boost of a Vector)  
        𝐚 ∙ 𝐛 = (a¥á𝐞¥á) ∙ (b¥â𝐞¥â) = (𝐞¥á ∙ 𝐞¥â) a¥á b¥â   <5-11>
        ¥ç¥á¥â ¡Õ 𝐞¥á ∙ 𝐞¥â   <5-13>
        𝐚 ∙ 𝐛 = ¥ç¥á¥â a¥á b¥â   <5-5>
        (¥Äs) ©÷ =  ¥Ä𝑥 ∙ ¥Ä𝑥   <5-14>
                  ¦­-1   0   0   0 ¦­
        ¥ç¥á¥â =     ¦­ 0   1   0   0 ¦­   <5-15>   <- ÆòÆòÇÑ ½Ã°ø°£ÀÇ °è·®(Mertic of Flat Spacetime)
                  ¦­ 0   0   1   0 ¦­
                  ¦­ 0   0   0   1 ¦­
         ds2  = ¥ç¥á¥âdx¥á dx¥â    <5-16>
        𝐚 ∙ 𝐛 = -at bt + ax bx + ay by + az bz   <5-17a>
        𝐚 ∙ 𝐛 = -a0 b0 + a1 b1 + a2 b2 + a3 b3   <5-17b>

Âü°í·Î, Landau-Lifshitz ¡× 6. Four-vectors ¿¡¼­´Â HartleÃ¥°ú ´Þ¸® 4-vectorÀÇ µÎ typeÀÇ ¿ä¼Òµé(elements)À» µµÀÔÇÕ´Ï´Ù.
        x0 = ct,   x1 = x,   x2 = y,   x3 = z  ->   ds©÷ =  (x0)2 - (x1)2 - (x2)2 - (x3)2.     <- Landau-LifshitzÃ¥Àº timelike intervalÀ» »ç¿ë.
        𝐴0 = ¥ã(𝐴'0 + V/c𝐴'1),  𝐴1 = ¥ã(𝐴'1 + V/c𝐴'0), 𝐴2 =         𝐴'2, 𝐴3 = 𝐴'3   (6.1)     <- <5-9>¿Í °°Àº º¤ÅÍÀÇ ·Î·»Ã÷ ºÎ½ºÆ®Àε¥ °ü¼º°è ¼ø¼­°¡ ´Ù¸¥ ½ÄÀÓ.
        (radius of 𝐴)2  ¢¡  (𝐴0)2 - (𝐴1)2 - (𝐴2)2 - (𝐴3)2. ¿©±â¼­  𝐴𝑖 contravariant element(¹Ýº¯ ¿ä¼Ò)¿Í 𝐴𝑖 covariant element(°øº¯ ¿ä¼Ò)¸¦ µµÀÔÇÕ´Ï´Ù.
        𝐴0= 𝐴0,   𝐴1= -𝐴1,   𝐴2= -𝐴2,   𝐴3= -𝐴3   (6.2)
        𝐴𝑖𝐴𝑖 = 𝐴0𝐴0 + 𝐴1𝐴1 + 𝐴2𝐴2 + 𝐴3𝐴3.                                     
        𝐴𝑖𝛣𝑖 = 𝐴0𝛣0 + 𝐴1𝛣1 + 𝐴2𝛣2 + 𝐴3𝛣3.          <-  𝐴𝑖𝛣𝑖: 4-scala,  °ü¼º°è°£¿¡¼­ ºÒº¯ÇÏ´Â ¾ç.

5.2  Æ¯¼ö»ó´ë¼º ¿îµ¿ÇÐ(Special Relativistic Kinematics)

À§ µÎ¹ø° ±×¸²Àº ´Ü¼øÈ÷ °¡¼ÓµÈ(simple accelerated)ÀÇ ¼¼°è¼±ÀÇ ¿¹·Î¼­ °íÀ¯½Ã°£ 𝜏·Î¼­ ¸Å°³º¯¼öÀûÀ¸·Î Ç¥½ÃµÈ ¼¼°è¼±À» º¸¿©ÁÝ´Ï´Ù. [È­»ìÇ¥ 𝐮´Â ¾Æ·¡ÀÇ 4-velocity)]

        x¥á = x¥á(¥ó)  <- 𝜏: °íÀ¯½Ã°£(proper time)   <5-21>
        t(¥ò) = a-1sinh(¥ò),   x(¥ò) = a-1cosh(¥ò)   <5-22>  
        d¥ó2 = -ds2 = dt2 - dx©÷ = [a-1cosh(¥ò)d¥ò]2 - [a-1sinh(¥ò)d¥ò]2 = (a-1d¥ò)2   <5-23>   <- cosh2(¥ò) - sinh2(¥ò) = 1
        t(¥ó) = a-1sinh(a¥ó),   x(¥ó) = a-1cosh(a¥ó)   <5-24>

À§ ¼¼¹ø° ±×¸²Àº ¾Õ¿¡¼­ ¸»ÇÑ ÀÔÀÚÀÇ ¼¼°è¼±(particle's worldline)ÀÇ °î¼±¿¡¼­ µÎ¹ø° ±×¸²¿¡¼­¿Í ¸¶Âù°¡Áö·Î Á¢¼±º¤ÅÍ(tangent vector 𝐮¸¦ º¸¿©ÁÖ°í ÀÖ½À´Ï´Ù.

        u¥á = dx¥á /d𝜏   <5-25>   <-  𝐮: 4-velocity)
        ut = dt/d𝜏 = 1 / ¡î (1 - V2/c2) = ¥ã   <5-26>   
        ux = dx/d𝜏 = dx/dt dt/d𝜏 = 𝑉x / ¡î (1 - V2/c2) = ¥ã 𝑉x  <5-27>
        u¥á = (¥ãc, ¥ã𝑉) = ¥ã(c, 𝑉)  <5-28>   <-  𝑉 = (𝑉x, 𝑉y, 𝑉z): ÀÔÀÚÀÇ 3-velocity
        𝐮 ∙ 𝐮 = -1   [Normalization of four-velocity]   <5-29>   <- 4-vectorÀÇ normalization(±Ô°ÝÈ­)
¿Ö³ÄÇϸé 𝐮 ∙ 𝐮 = -¥ã2 + (¥ã/c)2 V2 = -1/(1- V2/c2) + V2/[c2(1- V2/c2)] = - (1 - V2/c2)/(1 - V2/c2) = -1
        𝐮 ∙ 𝐮 = ¥ç¥á¥â [dx¥á /d𝜏] [dx¥â /d𝜏] = -1   <5-30>
Âü°í·Î, Landau-Lifshitz ¡× 7 Four-dimensional velocity ¿¡¼­´Â 𝐮𝑖 ∙ 𝐮𝑖 = 1   (7.3).     <- Landau-LifshitzÃ¥Àº timelike intervalÀ» »ç¿ë  

5.3  Æ¯¼ö»ó´ë¼º µ¿·ÂÇÐ(Special Relativistic Dynamics)

<¿îµ¿ ¹æÁ¤½Ä(Equation of Motion)>

        d𝐮/d𝜏 = 0   <5-34>  <- Newton's First Law(´ºÅæÀÇ Á¦1¹ýÄ¢) ¡ñ <5-22>¿¡¼­ V°¡ µî¼ÓÀÓÀ¸·Î
        𝐚 ¡Õ d𝐮/d𝜏   <5-36> <- 𝐚: four-acceleration
        𝐟 = 𝑚 d𝐮/d𝜏   <5-37>  <- Newton's Second Law(´ºÅæÀÇ Á¦2¹ýÄ¢)
        𝐟 ∙ 𝐮 = 0   <5-39>  <- d(𝐮 ∙ 𝐮) /d𝜏 = 0,   𝐮 ∙ 𝐚 = 0.
À§ µÎ¹ø° ±×¸²¿¡¼­ °¡¼ÓµµÀÇ Å©±â(the magnitude of acceleration):
        (𝐚 ∙ 𝐚)1/2 = a      <5-30>  <- at ¡Õ dut/d𝜏 = a sinh(a𝜏),  ax ¡Õ dux/d𝜏 = a cosh(a𝜏),  cosh2(a¥ó) - sinh2(a¥ó) = 1.

<¿¡³ÊÁö-¿îµ¿·®(Energy-Momentum)>

        𝐏 = 𝑚0 𝐮 = 𝑚0¥ã(c, 𝑉) = (-𝑚c, 𝑝)     <5-41> <-  𝐏: 4-monentum
        d𝐏/d¥ó = 𝐟   <5-42>
        𝐏2 ¡Õ 𝐏 ∙ 𝐏 = -𝑚2c2 + 𝑝2   <5-43>
        pt = 𝑚¥ãc,  𝑝 = 𝑚¥ã𝑉   <5-44>
        pt = 𝑚c2 + 1/2 𝑚𝑉2 + ...,      𝑃  = 𝑚𝑉 + ...   <5-45> 
¿Ö³ÄÇÏ¸é ºûÀÇ ¼Óµµº¸´Ù ¾ÆÁÖ ´À¸± ¶§, 𝑉 ¡ì 1, ¥ãÀÇ Å×ÀÏ·¯ ±Þ¼ö(Tayler's series): (1+x)-1/2 = 1 - 1/2x + 3/8 x2 - ..., [x = -𝑉2/c2]
        p¥á = (pt, 𝑃) = (𝑚¥ãc,  𝑚¥ã𝑉)   <5-46>  <-  𝐸: ¿¡³ÊÁö(energy) 𝐸 ¡Õ ptc = 𝑚¥ãc2 *
        𝐸 = 𝑚c2 [when 𝑝 = 0]   <5-47> <-  in MLT system, when particle is at rest.  'The Most Famous Equation in Relativity or Physics'    
        𝐸 = (𝑚2c4 + 𝑝2c2)1/2   <5-48>                          
        d𝑃 /dt ¡Õ 𝐹   <5-49> <-  𝐹: 3-force),  Newton's Third Law(´ºÅæÀÇ Á¦3¹ýÄ¢)
        𝐟 = (¥ã 𝐹 ∙ 𝑉, ¥ã 𝐹)   <5-40> <-   d𝑃/d¥ó = (d𝑃/dt) (dt /d¥ó) = 𝐹 ¥ã.
        d𝐸 /dt = 𝐹 ∙ 𝑉   <5-41>

¿ä¾àÇÑ´Ù¸é, ´ºÅæ ¿ªÇÐ(Newtonian mechanics)Àº Ư¼ö»ó´ë¼º ¿ªÇÐ(Special relativistc mechanics)ÀÇ ³·Àº ¼Óµµ¿¡¼­ÀÇ ±Ù»çÀÎ °ÍÀÔ´Ï´Ù.

¡Ø Âü°í·Î, ³ªÁß¿¡ »ó´ë·ÐÀû ¾çÀÚ¿ªÇп¡¼­ ¾ÆÁÖ À¯¿ëÇÏ°Ô ¾²ÀÏ ¿¡³ÊÁö¿Í ¿îµ¿·®°úÀÇ °ü°è½ÄÀ» Á¤¸®Çϸ頠
        𝐏2 = 𝑚02c2 = 𝑚2c2 - 𝑝 2   (S1)   <- Ư¼ö»ó´ë¼º¿¡ ÀÇÇÑ ºÒº¯·®(invariant) 𝐏2, 𝑚0: Á¤Áö»óÅÂÀÇ Áú·®
  ±×¸®°í 𝐸 = 𝑚c2 ¶ó´Â »ç½ÇÀ» ÀÌ¿ëÇÏ¸é ´ÙÀ½À» ¾ò½À´Ï´Ù.
        𝐸2 = 𝑝2c2 + 𝑚02c4,   𝑝2 = c2(𝑚2 - 𝑚02)   (S2)   

* ¿¡³ÊÁö °ü°è½Ä À¯µµ (ÇÐºÎ¿ë ±³Àç A. Beiser Concepts of Modern Physics 6th edition McGraw-Hil 2003 pp. 26-27)    
        KE = ¡ò0s  𝐟 ds
        KE = ¡ò0s d(¥ã𝑚𝑉)/dt ds = ¡ò0𝑉 𝑉 d(¥ã𝑚𝑉) = ¡ò0𝑉 𝑉 d(𝑚𝑉/¡î(1 - V2/c2)
           ÀÌ ºÎºÐÀûºÐ °ø½Ä (¡ò x dy = xy - ¡ò y dx) À» ÀÌ¿ëÇؼ­ Á¤¸®Çϸé,  [¥ã = 1/¡î(1 - V2/c2)]
        KE = 𝑚c2/¡î(1 - V2/c2) - 𝑚c2 = (¥ã - 1)𝑚c2
           ÀÌ °á°ú´Â ÇÑ ¹°Ã¼ÀÇ ¿îµ¿¿¡³ÊÁö KE´Â ¥ã𝑚c2¿Í 𝑚c2ÀÇ Â÷À̸¦ °¡¸£Å°¹Ç·Î Àüü¿¡³ÊÁö 𝐸´Â
        𝐸  = ¥ã𝑚c2 = 𝑚c2 + KE = 𝐸0 + KE  µû¶ó¼­ Á¤ÁöÇÑ ¹°Ã¼ÀÇ ¿¡³ÊÁö 𝐸0
        𝐸0 = 𝑚c2,    𝐸 = 𝑚c2/¡î(1 - V2/c2)
 Ãß°¡·Î ´ëÇпø ¼öÁØÀÇ Landau-LifshitzÀÇ ¶ó±×¶ûÁö¾È(Lagrangian)À» »ç¿ëÇÑ Ãß·ÐÀº [¡× 8 ÂüÁ¶]
        𝑆 = -𝛼 ¡òab ds     <- ÃÖ¼ÒÀÛ¿ëÀÇ ¿ø¸®(principle of least action)
        𝑆 = ¡òt1t2 𝐿 dt      <-  𝐿: ¸®±×¶ûÁö¾ð(Lagrange function)
        ds2 = c2dt'2 = c2dt2 - dx2 - dy2 - dz2
        ds = cdt' = cdt¡î[1 - (dx2 + dy2 + dz2)/c2dt2] = cdt¡î(1 - v2/c2)
        ±×·¯¹Ç·Î 𝐿 = -𝛼c¡î(1 - v2/c2) ≈ -𝛼c + 𝛼v2/2c     <- Å×ÀÏ·¯ ±Þ¼ö »ç¿ë
        À̸¦ °íÀü¿ªÇÐÀû Ç¥Çö 𝐿 = 𝑚v2/2 °ú ºñ±³ÇÔÀ¸·Î½á 𝛼 = 𝑚c ¸¦ ±¸ÇÕ´Ï´Ù. ±×·¯¹Ç·Î,
        𝐿 = -𝑚c2¡î(1 - v2/c2) ≈ -𝑚c2 + 𝑚v2/2     <- 𝐿 = KE - PE  PE: ÆÛÅټȿ¡³ÊÁö
        ¿¡³ÊÁö 𝐸 = 𝐏 • v - 𝐿   (𝐏 • v = 2KE) À̹ǷΠ𝐸 = 𝑚c2/¡î(1 - v2/c2) ≈ 𝑚c2 + 𝑚v2/2
        µû¶ó¼­, v = 0 ÀÏ ¶§ 𝐸0 = 𝑚c2

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


Name
Spamfree

     ¿©±â¸¦ Ŭ¸¯ÇØ ÁÖ¼¼¿ä.

Password
Comment

  ´ä±Û¾²±â   ¸ñ·Ïº¸±â
¹øÈ£ Á¦               ¸ñ À̸§ ¿¬°ü ³¯Â¥ Á¶È¸
123          Waves  3c. Wave packets and dispersion    ±è°ü¼® 8 2024-05-07
09:24:42
1372
122            Waves  4. Mechanical wave equation    ±è°ü¼® 8 2024-05-07
09:24:42
1372
121              Waves  5. Electromagnetic wave equation    ±è°ü¼® 8 2024-05-07
09:24:42
1372
120                Waves  6. Quantum wave equation    ±è°ü¼® 8 2024-05-07
09:24:42
1372
119  Gravirational Collapse and Space-Time Singularities  📚 🔵    ±è°ü¼® 1 2023-06-12
13:50:03
5488
118  Mathematics of Astronomy  1. Gravity; Light  ✅    ±è°ü¼® 4 2023-05-02
08:31:40
1595
117    Mathematics of Astronomy  2. Parallax, angular size etc.      ±è°ü¼® 4 2023-05-02
08:31:40
1595
116      Mathematics of Astronomy  3. Stars    ±è°ü¼® 4 2023-05-02
08:31:40
1595
115        Mathematics of Astronomy  4. Black holes & cosmology    ±è°ü¼® 4 2023-05-02
08:31:40
1595
114   StillÀÇ <ºí·ÏÀ¸·Î ¼³¸íÇÏ´Â ÀÔÀÚ¹°¸®ÇÐ>  ✅    ±è°ü¼® 3 2022-04-14
18:49:01
1115
113    BeckerÀÇ <½ÇÀç¶õ ¹«¾ùÀΰ¡?>    ±è°ü¼® 3 2022-04-14
18:49:01
1115
112      PenroseÀÇ <½Ã°£ÀÇ ¼øȯ> (°­Ãß!) [u. 5/2023]  🌹    ±è°ü¼® 3 2022-04-14
18:49:01
1115
111  ÀϹݻó´ë¼º(GR) ÇнÀ¿¡ ´ëÇÏ¿©..    ±è°ü¼® 1 2022-01-03
09:49:28
372
110  HTML(+) ¸®ºä/ȨÆäÀÌÁö ¿î¿ë^^  [1]  ±è°ü¼® 1 2021-11-08
16:52:09
250
109  PeeblesÀÇ Cosmology's Century (2020)    ±è°ü¼® 1 2021-08-16
21:08:03
459
108  <ÇѱÇÀ¸·Î ÃæºÐÇÑ ¿ìÁÖ·Ð> ¿Ü  ✅    ±è°ü¼® 5 2021-06-06
13:38:14
2283
107    RovelliÀÇ <º¸ÀÌ´Â ¼¼»óÀº ½ÇÀç°¡ ¾Æ´Ï´Ù>    ±è°ü¼® 5 2021-06-06
13:38:14
2283
106      SmolinÀÇ <¾çÀÚ Áß·ÂÀÇ ¼¼°¡Áö ±æ>    ±è°ü¼® 5 2021-06-06
13:38:14
2283
105        SusskindÀÇ <¿ìÁÖÀÇ Ç³°æ> (°­Ãß!)  🌹    ±è°ü¼® 5 2021-06-06
13:38:14
2283
104          ´ëÁßÀû ¿ìÁÖ·Ð Ãßõ¼­ ¸ñ·Ï [u. 9/2021]  [1]  ±è°ü¼® 5 2021-06-06
13:38:14
2283
103  Zel'dovich's Relativistic Astrophysics  ✅    ±è°ü¼® 1 2021-04-01
08:16:42
1355
102  Dirac Equation and Antimatter    ±è°ü¼® 1 2021-03-15
12:49:45
599
101  11/30 žç ÈæÁ¡ sunspots  ✅    ±è°ü¼® 2 2020-11-30
16:14:27
1020
100    Coronado PST ÅÂ¾ç »çÁø^^    ±è°ü¼® 2 2020-11-30
16:14:27
1020
99  Linde's Inflationary Cosmology [u. 1/2021]    ±è°ü¼® 1 2020-11-06
09:19:06
909
98  The Schrödinger Equation (7) Harmonic Oscillator  ✅    ±è°ü¼® 1 2020-09-17
21:43:31
2721
97  Çö´ë ¿ìÁÖ·ÐÀÇ ¸íÀú WeinbergÀÇ <ÃÖÃÊÀÇ 3ºÐ>  ✅    ±è°ü¼® 3 2020-08-09
11:37:44
1443
96    ¹°¸®Çеµ¸¦ À§ÇÑ Çö´ë ¿ìÁַм­´Â?    ±è°ü¼® 3 2020-08-09
11:37:44
1443
95      Çö´ë ¿ìÁÖ·ÐÀÇ ÃÖ°í, ÃÖ½Å, °íÀü¼­.. [u. 10/2024]   [1]  ±è°ü¼® 3 2020-08-09
11:37:44
1443
94   Mathematical Cosmology 1. Overview  🔵    ±è°ü¼® 6 2020-06-07
16:23:00
5767
93    Mathematical Cosmology 2. FRW geometry     ±è°ü¼® 6 2020-06-07
16:23:00
5767
92      Mathematical Cosmology 3. Cosmological models I    ±è°ü¼® 6 2020-06-07
16:23:00
5767
91        Mathematical Cosmology 4. Cosmological models II    ±è°ü¼® 6 2020-06-07
16:23:00
5767
90          Mathematical Cosmology 5. Inflationary cosmology    ±è°ü¼® 6 2020-06-07
16:23:00
5767
89            Mathematical Cosmology 6. Perturbations    ±è°ü¼® 6 2020-06-07
16:23:00
5767
88  Hobson Efstathiou Lasenby GR 11a. Schwartzschild ºí·¢È¦  🔴  [2]  ±è°ü¼® 3 2020-05-13
13:44:21
17707

    ¸ñ·Ïº¸±â   ÀÌÀüÆäÀÌÁö   ´ÙÀ½ÆäÀÌÁö     ±Û¾²±â [1] 2 [3][4][5]
    

Copyright 1999-2025 Zeroboard / skin by zero & Artech