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2018-07-05 06:34:14, Á¶È¸¼ö : 22,752 |
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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 Ư¼ö»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 Ư¼ö»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¤¸®Çϸé
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 |
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