⺻ Ʈ ѹα ߱ Ϻ  ̱  û  ַ ѹα dz δ밣 ֻ   Ʈŷ   Ʈŷ 似Ƽ ij ƼƮ ũε ߻ ij󸶻 Ŭ ڵ AT Ʈŷ  ũ


 α  ȸ

Ϳ ټ 6. ټ [u. 1/2020]
      2020-01-01 19:32:21, ȸ : 592


Vectors and Tensors Ϳ ټ  (1)  (2)  (3)  (4)  (5)

6. Tensor applications ټ (2)

     6.3 The Riemann curvature tensor ټ       
     ∘  Ϲݻ뼺 ߿ tensor Riemann ټε, ٿ Riemann tensor Ȥ Christoffel ߰ θ Ͽ
         Riemann-Christoffel tensor θϴ. tensor ߿伺 Riemann ټ ̸ Ŭ() ʿ ǰ,
          ƴ "־ (the hallmark of curvature)"̶ մϴ. κ å "(parallel transport)"̳
         "ȯ(cummutator)" ϴ ΰ մϴ. ""̶ vector ũ ٲ ʰ ϴ ̵ Īմϴ.
     ∘  ־ " Ű" ϱ ƽϴ. ǥ Ecuador Quito ϴ
         vector ϱ ݴ鿡 ϸ "Ʒ" ŵϴ. ̹ ϵ ̴ ٸ ϸ
          Indonesia , vector Ű ֽϴ. ̵̾, ä ο
          ޶ ֽϴ. , vector ٸ , ־ ٷ ִٰ Ȯ ֽϴ.    
     ∘   ǻ, ""̶ ̺ ǵ˴ϴ. ̺ ΰ , ù° ̺̰ ι°
         Christoffel ȣk ϴ Ͻʽÿ. vector ȯ ϸ鼭 ͸ ϴ Riemann tensor
         ϴ Դϴ.*  
     ∘   Riemann ټ vector ̺ "ȯ(commutator)"κ ڿ ɴϴ. ⼭ "ȯ" ó ,
         ϴ ι ̸ ŵϴ. ϳ A ϰ ٸ B Ѵٸ, ȯڴ  
         [AB] = AB - BA ǵ˴ϴ. Riemann ټ 쿡 ̺ ˴ϴ. װ ̺
         ƹ ̰ Ƿ, ȯ ݵ մϴ. ̺п ȯ ƴ
         ־ Ӽ ֽϴ.
     ∘   ټ ϱ ؼ, 츮 vector 𝐕𝛼 𝒙𝛽 ؼ ̺ մϴ. 𝐕𝛼;𝛽 = 𝐕𝛼 /𝒙𝛽 - 𝛤𝜎𝛼𝛽𝐕𝜎.  (6.21)  <- 𝐕𝛼𝛽 ε ǥ
         ̹ 𝒙𝛾 ؼ 𝐕𝛼𝛽 ٸ ̺ մϴ.  𝐕𝛼𝛽;𝛾 = 𝐕𝛼𝛽 /𝒙𝛾 - 𝛤𝜏𝛼𝛾𝐕𝜏𝛽 - 𝛤𝜂𝛽𝛾𝐕𝛼𝜂.  (6.22)   ⿡ Eq.(6.21) ϸ,
             𝐕𝛼𝛽;𝛾 = 2𝐕𝛼 /𝒙𝛾𝒙𝛽 - (𝛤𝜎𝛼𝛽 /𝒙𝛾)𝐕𝜎 - 𝛤𝜎𝛼𝛽(𝐕𝜎 /𝒙𝛾) - 𝛤𝜏𝛼𝛾(𝐕𝜏 /𝒙𝛽 - 𝛤𝜎𝜏𝛽𝐕𝜎) - 𝛤𝜂𝛽𝛾(𝐕𝛼 /𝒙𝜂 - 𝛤𝜎𝛼𝜂𝐕𝜎).  (6.23)
         ̴ vector 𝐕𝜎 𝒙𝛽 ܰ踦 Ŀ 𝒙𝛾 ܰ踦 ε, Ǵٽ Ųٷ մϴ..  
             𝐕𝛼;𝛾 = 𝐕𝛼 /𝒙𝛾 - 𝛤𝜎𝛼𝛾𝐕𝜎.   (6.24);  𝐕𝛼𝛾;𝛽 = 𝐕𝛼𝛾 /𝒙𝛽 - 𝛤𝜏𝛼𝛽𝐕𝜏𝛾 - 𝛤𝜂𝛾𝛽𝐕𝛼𝜂.  (6.25)
             𝐕𝛼𝛾;𝛽 = 2𝐕𝛼 /𝒙𝛽𝒙𝛾 - (𝛤𝜎𝛼𝛾 /𝒙𝛽)𝐕𝜎 - 𝛤𝜎𝛼𝛾(𝐕𝜎 /𝒙𝛽) - 𝛤𝜏𝛼𝛽(𝐕𝜏 /𝒙𝛾 - 𝛤𝜎𝜏𝛾𝐕𝜎) - 𝛤𝜂𝛾𝛽(𝐕𝛼 /𝒙𝜂 - 𝛤𝜎𝛼𝜂𝐕𝜎).  (6.26)
         , ̺ ϸ, Christoffel ȣ Ʒ -Ī- ϸ,
             𝐕𝛼𝛽;𝛾- 𝐕𝛼𝛾;𝛽= -(𝛤𝜎𝛼𝛽 /𝒙𝛾)𝐕𝜎+ (𝛤𝜎𝛼𝛾 /𝒙𝛽)𝐕𝜎+𝛤𝜏𝛼𝛾𝛤𝜎𝜏𝛽𝐕𝜎-𝛤𝜏𝛼𝛽𝛤𝜎𝜏𝛾𝐕𝜎= (𝛤𝜎𝛼𝛾 /𝒙𝛽 - 𝛤𝜎𝛼𝛽 /𝒙𝛾 +𝛤𝜏𝛼𝛾𝛤𝜎𝜏𝛽-𝛤𝜏𝛼𝛽𝛤𝜎𝜏𝛾)𝐕𝜎.  (6.27)
          ȣ ó մϴ. Riemann ټ:  𝐑𝜎𝛼𝛽𝛾 𝛤𝜎𝛼𝛾 /𝒙𝛽 - 𝛤𝜎𝛼𝛽 /𝒙𝛾 + 𝛤𝜏𝛼𝛾𝛤𝜎𝜏𝛽 - 𝛤𝜏𝛼𝛽𝛤𝜎𝜏𝛾.  (6.28) 
     ∘   Riemann ټ ϳ tensor "ġټ(Ricci tensor)"ε, 𝜎 𝛽 ν ֽϴ.
         ׷ 4 쿡, ġټ:  𝐑𝛼𝛾 𝐑𝜎𝛼𝜎𝛾 = 𝐑1𝛼1𝛾 + 𝐑2𝛼2𝛾 + 𝐑3𝛼3𝛾 + 𝐑4𝛼4𝛾.  (6.30)  <- 4
     ∘   metric tensorν Ricci tensorټ ø ٸ  ϰ, ̸ "Ricci scalar" մϴ.
         ġĮ:  𝐑 𝑔𝛼𝛾𝐑𝛼𝛾 = 𝐑𝛾𝛾 = 𝐑11 + 𝐑22 + 𝐑33 + 𝐑44.  (6.31)  <- 4
     ∘   , "Einstein tensor" ˷ tensor Ricci tensor Ricci scalar metric tensor ν ֽϴ.
         Einstein tensor:  𝐆𝛼𝛾 𝐑𝛼𝛾 - (1/2)𝐑𝑔𝛼𝛾.  (6.32)   tensor Ϲݻ뼺 Ÿϴ.
     ∘   ħ,  "Einstein (Einstein's field equation)": 𝐆𝜇𝜈 + 𝚲𝑔𝜇𝜈 = (8𝜋𝐺/𝑐4)𝐓𝜇𝜈,  (6.33)   <- 𝐺: ߷», 𝑐:
         ⿡ 𝐓𝜇𝜈 "ټ(the energy-momentum tensor)"̰,   𝚲 Einstein ָ ϱ ߾
         "ֻ(the cosmological constant)"Դϴ. ̰ ٷ  " ð η ˷ְ, ð ̴
          ˷ش." Ϲݻ뼺 ù° ϴ Դϴ.  
 
     ∘   Riemann ټ ϱ ؼ, ϳ ǥ ؼ غ մϴ.  ◂
             ds2 = 𝑎2d𝜃2 + 𝑎2sin2(𝜃)d𝜙2,  𝑔𝜃𝜃= 𝑎2,  𝑔𝜃𝜙= 𝑔𝜙𝜃= 0,  𝑔𝜙𝜙= 𝑎2sin2(𝜃).  (6.34);   𝛤𝑙𝑖𝑗 = (1/2)𝑔𝑘𝑙 [𝑔𝑖𝑘 /𝑥𝑗 + 𝑔𝑗𝑘 /𝑥𝑖 - 𝑔𝑖𝑗 /𝑥𝑘]   (5.23)
             𝛤𝜃𝜃𝜃 = (1/2)[𝑔𝜃𝜃(𝑔𝜃𝜃 /𝜃) + 𝑔𝜙𝜃(𝑔𝜃𝜙 /𝜃) + 𝑔𝜃𝜃(𝑔𝜃𝜃 /𝜃) + 𝑔𝜙𝜃(𝑔𝜃𝜙 /𝜃) - 𝑔𝜃𝜃(𝑔𝜃𝜃 /𝜃) - 𝑔𝜙𝜃(𝑔𝜃𝜃 /𝜙)],
             𝛤𝜃𝜃𝜙 = (1/2)[𝑔𝜃𝜃(𝑔𝜃𝜃 /𝜙) + 𝑔𝜙𝜃(𝑔𝜃𝜙 /𝜙) + 𝑔𝜃𝜃(𝑔𝜙𝜃 /𝜃) + 𝑔𝜙𝜃(𝑔𝜙𝜙 /𝜃) - 𝑔𝜃𝜙(𝑔𝜃𝜃 /𝜃) - 𝑔𝜙𝜃(𝑔𝜃𝜙 /𝜙)],
             𝛤𝜃𝜙𝜃 = (1/2)[𝑔𝜙𝜃(𝑔𝜃𝜃 /𝜃) + 𝑔𝜙𝜃(𝑔𝜙𝜙 /𝜃) + 𝑔𝜃𝜃(𝑔𝜃𝜃 /𝜙) + 𝑔𝜙𝜃(𝑔𝜃𝜙 /𝜙) - 𝑔𝜃𝜃(𝑔𝜃𝜙 /𝜃) - 𝑔𝜙𝜃(𝑔𝜙𝜃 /𝜙)],
             𝛤𝜙𝜃𝜃 = (1/2)[𝑔𝜃𝜙(𝑔𝜃𝜃 /𝜃) + 𝑔𝜙𝜙(𝑔𝜃𝜙 /𝜃) + 𝑔𝜃𝜙(𝑔𝜃𝜃 /𝜃) + 𝑔𝜙𝜙(𝑔𝜃𝜙 /𝜃) - 𝑔𝜃𝜙(𝑔𝜃𝜃 /𝜃) - 𝑔𝜙𝜙(𝑔𝜃𝜃 /𝜙)],
             𝛤𝜙𝜃𝜙 = (1/2)[𝑔𝜃𝜙(𝑔𝜃𝜃 /𝜙) + 𝑔𝜙𝜙(𝑔𝜃𝜙 /𝜙) + 𝑔𝜃𝜙(𝑔𝜙𝜃 /𝜃) + 𝑔𝜙𝜙(𝑔𝜙𝜙 /𝜃) - 𝑔𝜃𝜙(𝑔𝜃𝜙 /𝜃) - 𝑔𝜙𝜙(𝑔𝜃𝜙 /𝜙)],
             𝛤𝜙𝜙𝜃 = (1/2)[𝑔𝜃𝜙(𝑔𝜙𝜃 /𝜃) + 𝑔𝜙𝜙(𝑔𝜙𝜙 /𝜃) + 𝑔𝜃𝜙(𝑔𝜃𝜃 /𝜙) + 𝑔𝜙𝜙(𝑔𝜃𝜙 /𝜙) - 𝑔𝜃𝜙(𝑔𝜙𝜃 /𝜃) - 𝑔𝜙𝜙(𝑔𝜙𝜃 /𝜙)],
             𝛤𝜃𝜙𝜙 = (1/2)[𝑔𝜃𝜃(𝑔𝜙𝜃 /𝜙) + 𝑔𝜙𝜃(𝑔𝜙𝜙 /𝜙) + 𝑔𝜃𝜃(𝑔𝜙𝜃 /𝜙) + 𝑔𝜙𝜃(𝑔𝜙𝜙 /𝜙) - 𝑔𝜃𝜃(𝑔𝜙𝜙 /𝜃) - 𝑔𝜙𝜃(𝑔𝜙𝜙 /𝜙)],
             𝛤𝜙𝜙𝜙 = (1/2)[𝑔𝜃𝜙(𝑔𝜙𝜃 /𝜙) + 𝑔𝜙𝜙(𝑔𝜙𝜙 /𝜙) + 𝑔𝜃𝜙(𝑔𝜙𝜃 /𝜙) + 𝑔𝜙𝜙(𝑔𝜙𝜙 /𝜙) - 𝑔𝜃𝜙(𝑔𝜙𝜙 /𝜃) - 𝑔𝜙𝜙(𝑔𝜙𝜙 /𝜙)].
           𝑔𝜙𝜙 /𝜃 ̿ ̺а 0 ǹǷ, Eq. (6.34) Ʈټ Ͽ 0 ƴ ͵ մϴ.
             𝛤𝜙𝜃𝜙 = 𝛤𝜙𝜙𝜃 = (1/2)𝑔𝜙𝜙(𝑔𝜙𝜙 /𝜃) = (1/2){1/𝑎2sin2(𝜃)}{2𝑎2sin2(𝜃)cos(𝜃) = cos(𝜃)/sin(𝜃) = cot(𝜃)
             𝛤𝜃𝜙𝜙 = (1/2)-𝑔𝜃𝜃(𝑔𝜙𝜙 /𝜃) = -(1/2)(1/𝑎2){2𝑎2sin2(𝜃)cos(𝜃) = -sin(𝜃)cos(𝜃)
         ũ ȣ ټ մϴ. 𝐑𝜎𝛼𝛽𝛾 𝛤𝜎𝛼𝛾 /𝒙𝛽 - 𝛤𝜎𝛼𝛽 /𝒙𝛾 + 𝛤𝜏𝛼𝛾𝛤𝜎𝜏𝛽 - 𝛤𝜏𝛼𝛽𝛤𝜎𝜏𝛾; [ 𝜎 = 𝜃, 𝜎 = 𝜙]
             𝐑𝜃𝜃𝜃𝜃=𝛤𝜃𝜃𝜃 /𝜃-𝛤𝜃𝜃𝜃 /𝜃+𝛤𝜃𝜃𝜃𝛤𝜃𝜃𝜃+𝛤𝜙𝜃𝜃𝛤𝜃𝜙𝜃-𝛤𝜃𝜃𝜃𝛤𝜃𝜃𝜃-𝛤𝜙𝜃𝜃𝛤𝜃𝜙𝜃,            𝐑𝜙𝜃𝜃𝜃=𝛤𝜙𝜃𝜃 /𝜃-𝛤𝜙𝜃𝜃 /𝜃+𝛤𝜃𝜃𝜃𝛤𝜙𝜃𝜃+𝛤𝜙𝜃𝜃𝛤𝜙𝜙𝜃-𝛤𝜃𝜃𝜃𝛤𝜙𝜃𝜃-𝛤𝜙𝜃𝜃𝛤𝜙𝜙𝜃,    
             𝐑𝜃𝜃𝜃𝜙=𝛤𝜃𝜃𝜙 /𝜃-𝛤𝜃𝜃𝜃 /𝜙+𝛤𝜃𝜃𝜙𝛤𝜃𝜃𝜃+𝛤𝜙𝜃𝜙𝛤𝜃𝜙𝜃-𝛤𝜃𝜃𝜃𝛤𝜃𝜃𝜙-𝛤𝜙𝜃𝜃𝛤𝜃𝜙𝜙,        𝐑𝜙𝜃𝜃𝜙=𝛤𝜙𝜃𝜙 /𝜃-𝛤𝜙𝜃𝜃 /𝜙+𝛤𝜃𝜃𝜙𝛤𝜙𝜃𝜃+𝛤𝜙𝜃𝜙𝛤𝜙𝜙𝜃-𝛤𝜃𝜃𝜃𝛤𝜙𝜃𝜙-𝛤𝜙𝜃𝜃𝛤𝜙𝜙𝜙,    
             𝐑𝜃𝜃𝜙𝜃=𝛤𝜃𝜃𝜃 /𝜙-𝛤𝜃𝜃𝜙 /𝜃+𝛤𝜃𝜃𝜃𝛤𝜃𝜃𝜙+𝛤𝜙𝜃𝜃𝛤𝜃𝜙𝜙-𝛤𝜃𝜃𝜙𝛤𝜃𝜃𝜃-𝛤𝜙𝜃𝜙𝛤𝜃𝜙𝜃,        𝐑𝜙𝜃𝜙𝜃=𝛤𝜙𝜃𝜃 /𝜙-𝛤𝜙𝜃𝜙 /𝜃+𝛤𝜃𝜃𝜃𝛤𝜙𝜃𝜙+𝛤𝜙𝜃𝜃𝛤𝜙𝜙𝜙-𝛤𝜃𝜃𝜙𝛤𝜙𝜃𝜃-𝛤𝜙𝜃𝜙𝛤𝜙𝜙𝜃,     
             𝐑𝜃𝜙𝜃𝜃=𝛤𝜃𝜙𝜃 /𝜙-𝛤𝜃𝜙𝜃 /𝜙+𝛤𝜃𝜙𝜃𝛤𝜃𝜃𝜃+𝛤𝜙𝜙𝜃𝛤𝜃𝜙𝜃-𝛤𝜃𝜙𝜃𝛤𝜃𝜃𝜃-𝛤𝜙𝜙𝜃𝛤𝜃𝜙𝜃,       𝐑𝜙𝜙𝜃𝜃=𝛤𝜙𝜙𝜃 /𝜃-𝛤𝜙𝜙𝜃 /𝜃+𝛤𝜃𝜙𝜃𝛤𝜙𝜃𝜃+𝛤𝜙𝜙𝜃𝛤𝜙𝜙𝜃-𝛤𝜃𝜙𝜃𝛤𝜙𝜃𝜃-𝛤𝜙𝜙𝜃𝛤𝜙𝜙𝜃,  
             𝐑𝜃𝜃𝜙𝜙=𝛤𝜃𝜃𝜙 /𝜙-𝛤𝜃𝜃𝜙 /𝜙+𝛤𝜃𝜃𝜙𝛤𝜃𝜃𝜙+𝛤𝜙𝜃𝜙𝛤𝜃𝜙𝜙-𝛤𝜃𝜃𝜙𝛤𝜃𝜃𝜙-𝛤𝜙𝜃𝜙𝛤𝜃𝜙𝜙,     𝐑𝜙𝜃𝜙𝜃=𝛤𝜙𝜃𝜙 /𝜙-𝛤𝜙𝜃𝜙 /𝜃+𝛤𝜃𝜃𝜙𝛤𝜙𝜃𝜙+𝛤𝜙𝜃𝜙𝛤𝜙𝜙𝜙-𝛤𝜃𝜙𝜃𝛤𝜙𝜃𝜙-𝛤𝜙𝜃𝜙𝛤𝜙𝜙𝜃,   
             𝐑𝜃𝜙𝜃𝜙=𝛤𝜃𝜙𝜙 /𝜃-𝛤𝜃𝜙𝜃 /𝜙+𝛤𝜃𝜙𝜙𝛤𝜃𝜃𝜃+𝛤𝜙𝜙𝜙𝛤𝜃𝜙𝜃-𝛤𝜃𝜙𝜃𝛤𝜃𝜃𝜙-𝛤𝜙𝜙𝜃𝛤𝜃𝜙𝜙,     𝐑𝜙𝜙𝜃𝜙=𝛤𝜙𝜙𝜙 /𝜃-𝛤𝜙𝜙𝜃 /𝜙+𝛤𝜃𝜙𝜙𝛤𝜙𝜃𝜃+𝛤𝜙𝜙𝜙𝛤𝜙𝜙𝜃-𝛤𝜃𝜙𝜃𝛤𝜙𝜃𝜙-𝛤𝜙𝜙𝜃𝛤𝜙𝜙𝜙,      
             𝐑𝜃𝜙𝜙𝜃=𝛤𝜃𝜙𝜃 /𝜙-𝛤𝜃𝜙𝜙 /𝜃+𝛤𝜃𝜙𝜃𝛤𝜃𝜃𝜙+𝛤𝜙𝜙𝜃𝛤𝜃𝜙𝜙-𝛤𝜃𝜙𝜙𝛤𝜃𝜃𝜃-𝛤𝜙𝜙𝜙𝛤𝜃𝜙𝜃,     𝐑𝜙𝜙𝜙𝜃=𝛤𝜙𝜙𝜃 /𝜙-𝛤𝜙𝜙𝜙 /𝜃+𝛤𝜃𝜙𝜃𝛤𝜙𝜃𝜙+𝛤𝜙𝜙𝜃𝛤𝜙𝜙𝜙-𝛤𝜃𝜙𝜙𝛤𝜙𝜃𝜃𝛤𝜙𝜙𝜙𝛤𝜙𝜙𝜃,                
             𝐑𝜃𝜙𝜙𝜙=𝛤𝜃𝜙𝜙 /𝜙-𝛤𝜃𝜙𝜙 /𝜙+𝛤𝜃𝜙𝜙𝛤𝜃𝜃𝜙+𝛤𝜙𝜙𝜙𝛤𝜃𝜙𝜙-𝛤𝜃𝜙𝜙𝛤𝜃𝜙𝜙-𝛤𝜙𝜙𝜙𝛤𝜃𝜙𝜙, 𝐑𝜙𝜙𝜙𝜙=𝛤𝜙𝜙𝜙 /𝜙-𝛤𝜙𝜙𝜙 /𝜙+𝛤𝜃𝜙𝜙𝛤𝜙𝜃𝜙+𝛤𝜙𝜙𝜙𝛤𝜙𝜙𝜙 - 𝛤𝜃𝜙𝜙𝛤𝜙𝜃𝜙𝛤𝜙𝜙𝜙𝛤𝜙𝜙𝜙.
          0 ƴ Riemann ټ ãƼ, տ ش Christoffel ȣ մϴ.  
             𝐑𝜃𝜙𝜃𝜙 = 𝛤𝜃𝜙𝜙 /𝜃 - 𝛤𝜙𝜙𝜃𝛤𝜃𝜙𝜙  =  [sin2(𝜃) - cos2(𝜃)] - [-cos2(𝜃)] =  sin2(𝜃),                  
             𝐑𝜃𝜙𝜙𝜃 = -𝛤𝜙𝜙𝜙𝛤𝜃𝜙𝜃 + 𝛤𝜙𝜙𝜃𝛤𝜃𝜙𝜙  = -[sin2(𝜃) - cos2(𝜃)] + [-cos2(𝜃)] = -sin2(𝜃),                                                                                             
             𝐑𝜙𝜃𝜃𝜙 = 𝛤𝜙𝜃𝜙 /𝜃 + 𝛤𝜙𝜃𝜙𝛤𝜙𝜙𝜃  = -[1 + cot2(𝜃)] + cot2(𝜃) = -1
             𝐑𝜙𝜃𝜙𝜃 = -𝛤𝜙𝜃𝜙 /𝜃 - 𝛤𝜙𝜃𝜙𝛤𝜙𝜙𝜃. =  [1 + cot2(𝜃)] - cot2(𝜃) =  1   ▮

          [ ؼ]  Riemann curvatire tensor and Gauss curvature ټ 콺   
             ̹ '̺б 4: Riemann ټ' (<-ٷΰ) ؼ Riemann ټ տ нߴ Richard Faber GR
             (Differential Geometry & Relativity Theory)  Gauss 2⺻İ Weingarten Riemann ټ
             Ʈټ Ͽ  𝐑𝑚𝑖𝑗𝑘 = 𝑔𝑚h𝐑h𝑖𝑗𝑘 ϰ, Gauss   𝐊 = 𝑅1212/𝑔  [𝑔: det(𝑔𝑖𝑗] ϰ ֽϴ.
             , κ Ϲݻ å åó ̺(covariant differentiation) (parallel transport) ϹǷ         
              Riemann ټ ü  ԵǾ ִ ǥ ʴ Ϳ ָϿ մϴ.**
                                                                             
*  'A First Corurse in General Relativity' (Schutz 2009, Cambridge University Press) ִٴ ּ .
** Riemann ټ ؼ Ϲ GR å麸 ̺б ؼ Ƹ-- .

p.s. 'A Student's Guide to Vectors and Tensors'(Daniel A. Fleisch 2012, Cambridge University Press) Chapter 6.3 ߰.


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160    Gleick's CHAOS <ī: ο >  ✍🏻     2 2025-05-11
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159  Supplement  Chapter 2e. Problems     3 2025-03-24
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158    Supplement  Chapter 3e. Problems     3 2025-03-24
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157      Supplement  Chapter 4c. Problems    [1]   3 2025-03-24
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156  Appendix  Aa. Elements of GR     2 2025-01-28
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155    Appendix  Ab. Einstein Equation     2 2025-01-28
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154  Baumann's Cosmology  8a. Quantum Conditions     4 2025-01-08
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153    Cosmology  8b. Quantum Fluctuations     4 2025-01-08
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152      Cosmology  8c. Primordial Power Spectra     4 2025-01-08
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151        Cosmology  8d. Obs. Constraints; 9 Outlook     4 2025-01-08
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150  Baumann's Cosmology  7a. CMB Physics  ✅     5 2024-12-13
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149    Cosmology  7b. Primordial Sound Waves     5 2024-12-13
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148      Cosmology  7c. CMB Power Spectrum     5 2024-12-13
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147        Cosmology  7d. Glimpse at CMB Polarization     5 2024-12-13
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146          Cosmology  7e. Summary and Problems     5 2024-12-13
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145  Baumann's Cosmology  6a. Relativistic Perturbation     4 2024-11-08
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144    Cosmology  6b. Conservation Eqs; Initial Conditions     4 2024-11-08
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143      Cosmology  6c. Growth of Matter Perturbations     4 2024-11-08
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142        Cosmology  6d. Summary and Problems     4 2024-11-08
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141  Baumann's Cosmology  4a. Cosmological Inflation     5 2024-10-21
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140    Cosmology  4b. Physics of Inflation     5 2024-10-21
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139      Cosmology  5a. Newtonian Perturbation     5 2024-10-21
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138        Cosmology  5b. Statistical Properties     5 2024-10-21
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137          Cosmology  5c. Summary and Problems     5 2024-10-21
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136  Baumann's Cosmology  3a. Hot Big Bang  ✅      4 2024-09-22
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135    Cosmology  3b. Thermal Equilibrium     4 2024-09-22
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134      Cosmology  3c. Boltzmann Equation     4 2024-09-22
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133        Cosmology  3d. Beyond Equilibrium     4 2024-09-22
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132  Baumann's Cosmology  1. Introduction  ⚫  [1]   5 2024-09-01
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131    Cosmology  2a. Expanding Universe     5 2024-09-01
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130      Cosmology  2b. Dynamics       5 2024-09-01
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129        Cosmology  2c. Friedmann Equations     5 2024-09-01
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128          Cosmology  2d. Our Universe     5 2024-09-01
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