PACS numbers: 03.70.+k, 04.62.+v, 11.10.-z

On The Covariant Tangent Vector Coordinate Systems

Haewon Lee ^∗

Department of Physics, Chungbuk National University, Cheongju 361-763, Korea (Received 19 December 2014 : revised 9 January 2015 : accepted 12 January 2015)

A method that can be used to express covariant derivatives and metric tensors by using tangent vectors in a curved space-time is introduced. In addition to this, parallel transport along geodesics is incorporated. As a result, the new differential operators can be expanded as polynomials of the tangent vectors and the covariant local functions at a given point. This result can be used to find the metric tensor in the tangent-vector’s coordinate system. This metric tensor can also be expanded by using tangent vectors and Riemann curvature tensors. The covariant derivatives and the metric tensor are calculated up to local functions of dimension 6.

PACS numbers: 03.70.+k, 04.62.+v, 11.10.-z

Keywords: Tangent vector, Covariant derivative, Curved space-time

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/

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i  . s    õ   H ] X ‚  ý a³ ð> \ " f B jà Ôa Ë : J $ ™" f\  ¦ ½ ¨   H X <\ • ¸  6   x| ¨ c à º e ”  . s  B jà Ôa Ë : J $ ™" f  H ] X 

‚

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% i  .

PACS numbers: 03.70.+k, 04.62.+v, 11.10.-z Keywords: ] X ‚   7 ˜' , / B N  p ì  r, Ï ã L“ É r r / B N ç ß –

∗

E-mail: [email protected]

170

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“ É

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€   7 á §  8 ¼ # o  > 
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f ´ òõ  Œ •6   x | ¾ Ó (effective action)`  ¦ / B N   p ì  r  © œ[ þ t – Ð „  > h (derivative expansions)   H X < ¼ # o  >   6   x ) a   [4].

s

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f_  ] X ‚   7 ˜'  (X)\  ¦ r / B N ç ß –`  ¦ l Õ ü t   H D h– Ðî  r ý a³ ð– Ð



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:_  s  & V , _  ' Ÿ 1 l x (behavior)“ É r < É ª p _  @ / © œs  . $ 



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†

½ Ó[ þ t`  ¦ > í ß – % i % 3   [1,3].

s

 ~ ½ ÓZ O \ " f p ì  r ƒ  í ß – [ þ t“ É r r / B N ç ß – ý a³ ð x \ " f / B N



 & h “   † < Êà ºü < ] X ‚   7 ˜' [ þ t ¢ ¸  H ] X ‚   7 ˜' \  @ /ô  Ç p ì  r

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 r ƒ  í ß – [ þ t“ É r ] X ‚   7 ˜'  ý a³ ð\  @ /ô  Ç p ì  rƒ  í ß – – Ð ^  ¦ Ã

º e ” “ ¦ x    H   à º\  ¦ ° ú   H / B N  & h  † < Êà º[ þ t“ É r  © œÃ ºü < ° ú   s

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|

¾ Ó " é ¶ s  6“   / B N  † < Êà º\  ¦ Ÿ í† < Ê   H † ½ Ó t  ½ ¨ % i  .

¢

¸ô  Ç s  > í ß –_   Òí ß –Ó ü t – Ð B jà Ôa Ë : J $ ™" f\  ¦ ] X ‚   7 ˜'  ý a³ ð

>

\ " f O(X ⁶ )  t  ½ ¨ % i  .

II. Œ Ÿ «
ì Å ± n ɍ Ò Ås ð ' [ Ò ÷ƒ »4 

€

 $  / B N   ] X ‚   7 ˜' ý a³ ð> \  @ /K  ç ß –é ß –y  4 Ÿ ¤_ þ v`  ¦ # Œ

˜

Ðl – Ð  . / B N  p ì  rƒ  í ß –  ∇ µ   H ∇ µ = ∂ _ν − iA µ + Γ _µ õ

 ° ú  s  j þ t à º e ”  . # Œl " f A ^µ ≡ A ^a _µ T ^a   H Yang-Mills



© œ`  ¦ Ø Ôv “ ¦ Γ ^µ   H r / B N ç ß – J $ ™" f
 Û ¼x  -\   Œ •6   x   H o

ë ß – s 6 £ § (connection) s  . €  $  r / B N ç ß –_  l ï  r& h  x

1

] X ‚   7 ˜' ý a³ ð>   H normal coordinate system s  “ ¦• ¸ ô  Ç .

2

ƒ  í ß –  M\  @ /K  & V , “ É r hy| M |xi \  ¦ _ p ô  Ç .

\

 ¦ ‚  × þ ˜ô  Ç . r / B N ç ß –_    É r & h  ˜ x   H & h  x \ " f_  ] X ‚   7 ˜'  X µ – РÒ'    & ñ | ¨ c  כ s  . 7 £ ¤ & h  x \ " f ] X ‚   7 ˜'  X _µ ~ ½ ӆ ¾ ÓÜ ¼– Ð U  ´s  pX _µ X _ν g ^µν (x) “   t 2 £ §U  ´`  ¦ Õ ªo €   ì

ø Í@ /¼ #  = å Q& h s  ˜ x   ) a  . s  Qô  Ç ý a³ ð> \  ¦ ³ ðï  r  o ³ ð (normal coordinate system)  “ ¦• ¸  ҏ É r  . ˜ x õ  X ^µ ü <_ 

› '

a >   H  s Û ¼º ú ˜  σ(˜ x, x) – Ð ¸ ú ˜
 è ­ qà º e ”  . [5,6]   s

Û ¼º ú ˜  σ(˜ x, x)   H x \ " f ˜ x  t _  þ jé ß – o (geodesic distance) _  ] jY  L _  ì ø ÍÜ ¼– Ð & ñ _ ÷ &“ ¦ ˜ x õ  X µ ü <_  › ' a

>

  H  6 £ § d ” Ü ¼– Ð Å Ò# Q”   .

X µ = −∇ µ σ(˜ x, x) = − ∂

∂x ^µ σ(˜ x, x). (1) σ(˜ x, x)  H Hamilton-Jacobi ~ ½ Ó& ñ d ” 

X µ X ^µ = 2σ (2)

\

 ¦ ë ß –7 á ¤ r †   . ¢ ¸ô  Ç lim _{x→x ˜} ∇ ˜ µ ∇ ν σ(˜ x, x) = −g µν (x) s 



. · ú ¡Ü ¼– Ѝ  H ŠҖ Ð / B N  p ì  r l   ñ ∇ @ /’  \  & h (.) `  ¦   6

 

x ½ + É  כ s  . ¢ ¸ô  Ç “   oÛ ¼ 0 A\  ∼  e ” Ü ¼€   ˜ x \ " f_  p  ì

 r s 
 J $ ™" f “   oÛ ¼\  ¦ _ p ô  Ç . 7 £ ¤ σ. _{µν ˜ ˜ λ} = ˜ ∇ λ ∇ ν ∇ ˜ µ σ ü

< ° ú   .

] X

‚   7 ˜'  ý a³ ð> \ " f  H & h _  ý a³ ð ˜ x @ /’  \  X ^µ \  ¦   6

 

x ô  Ç . p ì  r ƒ  í ß –  M • ¸ ] X ‚   7 ˜'  X ^µ \  ¦  6   x # Œ   r

 æ ¼# Œ4 R  ô  Ç . s „  \  €  $  l ï  r& h  x \ " f & h  ˜ x  t  _

 t 2 £ §U  ´`  ¦   " f ¨ î ' Ÿ s 1 l x (parallel transportation)`  ¦

# Œ˜ Ð . s M :_  ¨ î ' Ÿ s 1 l x ' Ÿ § > = (matrix) I(˜ x, x)   H   6

£

§ _  d ” `  ¦ ë ß –7 á ¤ ô  Ç .

X ˜ _µ ∇ ˜ ^µ I(˜ x, x) = 0 = X _µ I(˜ x, x) ← −

∇ ^µ I(x, x) = 1,

I(˜ x, x ⁰ )I(x ⁰ , x) = I(˜ x, x). (3)

#

Œl " f ˜ X _µ = ˜ ∇ _µ σ(˜ x, x) = _{∂ ˜} ^∂ _x

_µ

σ(˜ x, x) s “ ¦ I(˜ x, x) ← −

∇ ^µ =

∂

∂x

^µ

I(˜ x, x) − I(˜ x, x)(−iA µ + Γ µ ) \  ¦ _ p ô  Ç . ¢ ¸ô  Ç 0 A\ 

"

f [ j & h  x, x ⁰ Õ ªo “ ¦ ˜ x   H ° ú  “ É r t 2 £ §U  ´  © œ\  e ” # Q  ô  Ç



.

{ 9

ì ø Í& h Ü ¼– Ð € ª œ  © œ : r \ 
š ¸  H p ì  r ƒ  í ß –  M “ É r / B N



 p ì  rƒ  í ß –  ∇ µ ü < C  ⠁ © œ (background field) φ – Ð
 

? /t   H X < s \  ¦ M ( ˜ ∇, φ(˜ x)) ü < ° ú  s  j þ t à º e ” `  ¦  כ s  .

#

Œl \  ¨ î ' Ÿ s 1 l x ' Ÿ § > =`  ¦  6   x # Œ D h– Ðî  r ƒ  í ß –  M\  ¦



6 £ § õ  ° ú  s  & ñ _   .

M ≡ I(x, ˜ x)M ( ˜ ∇, φ(˜ x))I(˜ x, x) = M (∇, φ). (4)

#

Œl " f ∇ = I(x, ˜ x) ˜ ∇ I(˜ x, x) s “ ¦ φ =

I(x, ˜ x) φ(˜ x) I(˜ x, x) s  . 7 á §  8 ½ ¨^ ‰& h “   ³ ð‰ & ³“ É r / B I

˜

Ð# Œ×  ¦  כ s  . s \  ¦ 0 AK " f  H  A _  d ” `  ¦ ë ß –7 á ¤   H  

 É

r  s J $ ™" f(bi-tensor) g µν (˜ x, x) \  ¦ • ¸{ 9  # Œ  ô  Ç .

˜

x µ ∇ ˜ ^µ g αβ (˜ x, x) = 0 = X µ ∇ ^µ g αβ (˜ x, x), g µν (x, x) = g µν (x),

g µν (˜ x, x) = g νµ (x, ˜ x)

g µν (˜ x, x ⁰ )g ^ν _λ (x ⁰ , x) = g µλ (˜ x, x). (5)

#

Œl " f• ¸ [ j & h  x, x ⁰ Õ ªo “ ¦ ˜ x   H ° ú  “ É r t 2 £ §U  ´  © œ\  e ” 

#

Q  ô  Ç . g ^µν (˜ x, x) \ " f ' Í   P : “   oÛ ¼ µ   H ' Í   P : ý a

³

ð ˜ x \  5 Å q “ ¦ ¿ º  P : “   oÛ ¼ ν   H ¿ º  P : ý a³ ð x \  5

Å

q ô  Ç .   " f / B N   p ì  r`  ¦ ½ + É M :\   H o ë ß – s 6 £ §“ É r K  {

© œ÷ &  H “   oÛ ¼\ ë ß –  Œ •6   x ô  Ç . € ª œA á ¤ = å Q& h \ " f B jà Ôa Ë : J $ ™

"

f\  ¦ Y  L # Œ “   oÛ ¼\  ¦ `  ¦ o “ ¦ ? /w n = à º e ”  . \ V\  ¦ [ þ t€   g ^µ _ν (˜ x, x) = g ^µλ (˜ x)g λν (˜ x, x) s  . 0 A\ " f s p  ƒ  / å LÙ þ ¡1 p w s

 _ p  ì  r" î t  · ú §“ É r  â Ä º y Œ • J $ ™" f “   oÛ ¼ 5 Å q ô  Ç & h 

`

 ¦ ì  r" î y  l  0 AK  “   oÛ ¼0 A\  ∼ `  ¦ ³ ðl \  ¦ ½ + É  כ s 



. 7 £ ¤ g µν ˜ = g µν (˜ x, x) ü < ° ú   .

g µν (˜ x, x)  # Qb  G>   6   x ÷ &  H t  · ú ˜l  0 AK  \ V\  ¦ [ þ t # Q p

ì  r ƒ  í ß –  B ^{µ ˜} ∇ ˜ ^µ \  ¦ “ ¦ 9K ˜ Ð .

I(x, ˜ x)  B µ ˜ ∇ ˜ ^µ 

I(˜ x, x)

= I(x, ˜ x)B _{µ ˜} I(˜ x, x)I(x, ˜ x) ˜ ∇ ^µ I(˜ x, x)

= g _µ ^ν (x, ˜ x)I(x, ˜ x)B ν ˜ I(˜ x, x)g ^µ _λ (x, ˜ x)I(x, ˜ x) ˜ ∇ ^λ I(˜ x, x)

= B _µ ∇ ^µ . (6)

#

Œl " f ∇ ^µ = g ^µ _λ (x, ˜ x)I(x, ˜ x)( ˜ ∇ ^λ )I(˜ x, x) s “ ¦ B µ = g _µ ^ν (x, ˜ x)I(x, ˜ x)B _{ν ˜} I(˜ x, x) s  .

s

] j ý a³ ð ˜ x @ /’  \  ] X ‚   7 ˜'  X \  ¦  6   x # Œ ƒ  í ß – 

 C  ⠁ © œ[ þ t`  ¦ — ¸¿ º x ü < X – Ð ³ ð‰ & ³ô  Ç . €  $  ∇ ^µ   H   6

£

§ õ  ° ú  s  j þ t à º e ”  .

∇ ^µ = g ^{µ˜ ν}



I(x, ˜ x)I(˜ x, x). ν ˜ − σ(˜ x, x). να ˜

∂

∂X α

 (7) C

 ⠁ © œ φ _   â Ä º  H 0 A_  \ V\ " f s p  ¶ ú ˜( R‘ : r  ü < ° ú   .

C

 ⠁ © œ φ  J $ ™" f €   y Œ • J $ ™" f “   oÛ ¼\  @ /K  g ^µν (x, ˜ x)

\

 ¦ Y  L K   ô  Ç . \ V\  ¦ [ þ t€  

φ _µ = g _µ ^{ν ˜} I(x, ˜ x)φ ν (˜ x)I(˜ x, x). (8)

· ú

¡Ü ¼– Ѝ  H ] X ‚   7 ˜'  X – Ð ³ ð‰ & ³ ) a ƒ  í ß –  M `  ¦ M x – Ð

 ? /l – Ð  . M ^x _  & ñ _ – РÒ'  M ^x  & h  x \  @ /K  /

B N  & h e ” `  ¦ ~ 1 >  · ú ˜ à º e ”  .



 " f M ^x \  ¦ X µ ü < ∂X ^∂

_µ

– Ð „  > h\  ¦ €    6 £ § õ  ° ú  

“ É

r g 1 J – Ð j þ t à º e ” `  ¦  כ s  .

M x = X

a α···βµ···ν (x)X ^α···β ∂

∂X _µ · · · ∂

∂X _ν , (9)

#

Œl " f X ^α···β ≡ X ^α · · · X ^β s “ ¦ a α···βµ···ν (x)   H / B N  & h 

“

  x _  J $ ™" f ' Ÿ § > = † < Êà ºs  . ƒ  í ß – _  à º (order) d s

 €   X \  @ /ô  Ç p ì  r ƒ  í ß – _  à º• ¸ d \  ¦  Å t  · ú §

`

 ¦  כ s  . Ä ºo   H / B I 0 Aü < ° ú  “ É r „  > h\  ¦ > í ß –   H ~ ½ ÓZ O 

`

 ¦ ™ è> h½ + É  כ s  . 0 A_  y Œ • † ½ Ó\ " f / B N  † < Êà º a α···βµ···ν

_

 | 9 | ¾ Ó " é ¶`  ¦ Dim(a) – Ð
 ? /“ ¦, X\  @ /ô  Ç à º\  ¦

#(X) – Ð ∂X ^∂

_µ

\  @ /ô  Ç à º\  ¦ #(∂) – Ð
  · p €   d = Dim(a) − #(X) + #(∂)   ) a  . ¢ ¸ô  Ç s p  ƒ  / å L ô  Ç@ /– Ð

#(∂) ≤ d s  .

III. ± n ɍ Ò Å s ð ' [z º  ¹ Å4  M 

d ”

 (9) ü < ° ú  “ É r „  > h\  ¦ l  0 AK   6 £ § õ  ° ú  “ É r ƒ  í ß – 

\

 ¦ & ñ _ ô  Ç .

D ≡ X _µ

 ∂

∂X _µ



x

= X _µ ∂ ˜ x ^ν

∂X _µ

 ∂

∂ ˜ x ^ν



x

= ˜ X ^ν

 ∂

∂ ˜ x ^ν



x

= σ(˜ x, x). ^{ν ˜}

 ∂

∂ ˜ x ^ν



x

. (10)

#

Œl " f  t } Œ • ×  ¦ _  d ” [ þ t“ É r ˜ X _µ X ˜ ^µ = 2σ(˜ x, x) _  € ª œ  `  ¦ x ^ν – Ð p ì  r “ ¦ › ' a > d ”   _∂X

µ

∂ ˜ x

^ν



x

= −  _{∂ ˜ X}

ν

∂x

^µ



˜ x

`

 ¦  6   x 

#

Œ 7 £ x" î ½ + É Ã º e ”  . X – Ð „  > h “ ¦ z  ·“ É r † < Êà º e ” `  ¦ M :

#

Œl \  D \  ¦  Œ •6   x r †   . X _  k   † ½ Ód ” \   Œ •6   x r v 

€

  “ ¦Ä »u  k 
“ : r  . \ V\  ¦ [ þ t€   X _  1   † ½ Ód ” `  ¦ ™ è



  9€   D − 1 `  ¦  Œ •6   x r v “ ¦ 1-2   † ½ Ód ” `  ¦   ™ è 

 9€   (D − 1)(D − 2) \  ¦  Œ •6   x r †   . s  Qô  Ç ~ ½ ÓZ O `  ¦   6

 

x # Œ d ”  (7) î ß –_  † ½ Ó[ þ t`  ¦ X – Ð „  > hô  Ç . s  Qô  Ç ~ ½ ÓZ O 

`

 ¦  6   x €   σ. µ ˜ ü < ° ú  “ É r “   [ þ t s  > 5 Å q
š ¸>   ) a  . ô  Ç

¼

#  X → 0 { 9  M : σ. ^{µ ˜} → X µ  ÷ &Ù ¼– Ð " é ¶   H à º ë ß –

p

u _  σ. µ ˜ 
“ : r  €   > í ß –s  ç ß –é ß –  ’ xt ë ß – † ½ Ó © œ Õ ªX O t 

· ú

§ .

d ”

 (7) \  Å Ò# Q”   ƒ  í ß –  ∇ ^µ \  ¦ X – Ð „  > h # Œ ˜ Ðl – Ð

 . ' Í P : † ½ ӓ É r g _µ ^{ν ˜} I(x, ˜ x)I(˜ x, x). ˜ ν Ü ¼– Ð X _  1 † ½ ÓÂ Ò '

 r  Œ •  ) a  . 0 A\ " f ƒ  / å L ô  Ç ~ ½ ÓZ O @ /– Ð D \  ¦  Œ •6   x r & • ¸ σ. _{µ ˜} ü < ° ú  “ É r “   
š ¸t  · ú §  H X < @ /’  \  D + 1 `  ¦  Œ •6   x r

v €  

(D + 1)g _µ ^{ν ˜} I(x, ˜ x)I(˜ x, x). ˜ ν = σ. ^{λ ˜} g _µ ^{˜ ν} I(x, ˜ x)R λ˜ ^˜ ν I(˜ x, x)

= −σ ^{∗˜ λ} _{ν ˜} g _µ ^{ν ˜} I(x, ˜ x)I(˜ x, x). λ ˜

(11)

  ) a  . # Œl " f R ^αβ ≡ [∇ α , ∇ β ]   H / B GÒ  ¦J $ ™" f ƒ  í ß – 

–

Ð + '\  ¦  Ø Ô  H J $ ™" f\ • ¸  Œ •6   x ô  Ç . ¢ ¸ô  Ç σ _µ˜ ^∗ _{˜ ν} ≡ σ. _{µ˜ ˜ ν} − g _µν (x)   H O(X ² ) s  .   " f g µ ^{ν ˜} I(x, ˜ x)I(˜ x, x). _{ν ˜} =

1

2 X ^λ R λµ + O(X ² ) e ” `  ¦ · ú ˜ à º e ”  . ¢ ¸ô  Ç D = σ. ^{µ ˜} ∂ ˜ µ

\

 ¦ ˜ x \ " f Û ¼º ú ˜  € ª œ“   † < Êà º\   Œ •6   x r ~  ´ M :\   H @ /’  \  D ≡ σ. µ ˜ ∇ ˜ ^µ \  ¦  Œ •6   x r & • ¸  ) a  . · ú ¡Ü ¼– Ð
š ¸  H  â Ä º  H

—

¸¿ º s \  K { © œ  ) a  . ¢ ¸ô  Ç Dσ. _{µ ˜} = σ. _{µ ˜} “   $ í | 9 s   Å Ò   6

 

x ) a  . D \  ¦ ì ø Í4 Ÿ ¤& h Ü ¼– Ð  Œ •6   x r &  | 9 | ¾ Ó " é ¶ s  6 s  

“

  / B N  † < Êà º\  ¦ Ÿ í† < Ê   H † ½ Ó`  ¦ ½ ¨½ + É Ã º e ” % 3   H X <   õ \  ¦

&

ñ o  €    6 £ § õ  ° ú   .

g _µ ^{ν ˜} I(x, ˜ x)I(˜ x, x). _{ν ˜} = 1

2 R ⁽¹⁾ µ + 1

3 R ⁽²⁾ µ + 1

8 R ⁽³⁾ µ − 1

16 σ ^{(2)˜ λ} _{µ ˜} R ⁽¹⁾ λ

+ 1

30 R ⁽⁴⁾ µ − 13

600 σ ^{(3)˜ λ} _{µ ˜} R ⁽¹⁾ λ − 1

30 σ ^{(2)˜ λ} _{µ ˜} R ⁽²⁾ λ

+ 1

144 R ⁽⁵⁾ µ − 7

864 σ ^{(3)˜ λ} _{µ ˜} R ⁽²⁾ λ − 1

96 σ ^{(2)˜ λ} _{µ ˜} R ⁽³⁾ λ

− 1

288 σ ^{(4)˜ λ} _{µ ˜} R ⁽¹⁾ λ + 23 4800 σ ^{(2) ˜ η} _˜

λ σ ^{(2)˜ λ} _{µ ˜} R ⁽¹⁾ η . (12)

#

Œl " f  6 £ § õ  ° ú  “ É r » ¡ ¤ €  • ³ ð‰ & ³`  ¦  6   x % i  .

R ⁽¹⁾ µ = X ^α R αµ , R ⁽²⁾ µ (X) = X ^αβ R αµ . β ,

R ⁽³⁾ µ (X) = X ^αβγ R αµ . _βγ , · · · (13)

¢

¸ô  Ç σ ^{(2)˜ λ} _{µ ˜} = X ^αβ σ. ^{λ ˜}

˜

µ ˜ α ˜ β | x=x ˜ , σ ^{(3)˜ λ} _{µ ˜} = X ^αβγ σ. ^{˜ λ} _{µ ˜ ˜ α ˜ β ˜ γ} | x=x ˜ s  .

ƒ

 í ß –  ∇ µ _  ¿ º  P : † ½ ӓ É r g ^{µ˜ ν} σ. _{να ˜ ∂X} ^∂

α

– Ð" f ° ú  “ É r ~ ½ Ó Z O

Ü ¼– Ð q “ §& h  ~ 1 >   6 £ § _    õ \  ¦ % 3   H  .

−g ^{µ˜ ν} σ. _{νλ ˜} = g _µλ (x) − 1

2! σ ⁽²⁾ _{λ ˜ µ} − 1

3! σ ⁽³⁾ _{λ ˜ µ} − 1

4! σ _{λ ˜} ⁽⁴⁾ _µ + · · · (14)

#

Œl " f• ¸ q 5 p w ô  Ç » ¡ ¤ €  • ³ ð‰ & ³`  ¦  6   x % i  . 7 £ ¤ σ _{λ ˜} ⁽²⁾ _µ = X ^αβ σ. _{λ ˜ µ ˜ α ˜ β} | x=x ˜ , σ _{λ ˜} ⁽³⁾ _µ = X ^αβγ σ. _{λ ˜ µ ˜ α ˜ β ˜ γ} | x=x ˜ 1 p x õ  ° ú   . d ”  (8) _  C  ⠁ © œ_   â Ä º• ¸ ~ 1 >  „  > h÷ &  H X <  6 £ § õ  ° ú   .

φ _{µ ˜} = φ _µ + X ^α φ _µ . _α + 1

2! X ^αβ φ µ . αβ + 1

3! X ^αβγ φ µ . αβγ · · · (15) s

 ] X \ " f  t } Œ •Ü ¼– Ð ] X ‚   7 ˜'  ý a³ ð> \ " f_  B jà Ôa Ë : J

$ ™" f\  @ /K  · ú ˜ ˜ Ð . d ”  (14)   H  – Ð s  B jà Ôa Ë : J $ ™" f

\

 ¦   & ñ ô  Ç . 7 £ ¤ X ý a³ ð> \ " f_  B jà Ôa Ë :J $ ™" f\  ¦ g ^µν  

€  

g ^µν = ∂X ^µ

∂ ˜ x ^α

∂X ^ν

∂ ˜ x ^β g ^αβ (˜ x)

= σ. ^µ _{α ˜} σ. ^{ν ˜ α}

= σ. ^µ _{α ˜} g ^{αβ ˜} σ. ^{ν ˜ γ} g γβ ˜ (16) s

l  M :ë  H s  .

IV. Coincident Limit

„

  ] X \ " f % 3 “ É r   õ \  ¦ ˜ Ѐ   σ. _{µ ˜ α ˜ β···} õ  σ. µ ˜ ˜ α ˜ β··· ü < ° ú  

“ É

r € ª œ[ þ t _  coincident limit (˜ x = x) \  ¦ ½ ¨   H  כ s  € 9 כ ¹

 . σ. µ ˜ ˜ α ˜ β··· _   â Ä º  H d ”  σ. µ ˜ σ. ^{µ ˜} = 2σ \  ˜ ∇ α \  ¦ > 5 Å q

&

h

6   x r &  ½ ¨½ + É Ã º e ”   [5,6]. Õ ª   õ \  ¦ כ ¹€  • €    6 £ § õ

 ° ú   .

σ _µ˜ ⁽²⁾ _{˜ ν} = 2 3 R ⁽²⁾ _µν σ _µ˜ ⁽³⁾ _{˜ ν} = 3R ⁽³⁾ _µν σ _µ˜ ⁽⁴⁾ _{˜ ν} = ( 12

5 R ⁽⁴⁾ − 8

15 R ⁽²⁾ R ⁽²⁾ ) µν (17)

#

Œl \ " f• ¸

R _µν ⁽²⁾ ≡ X ^αβ R _αµβν , R ⁽³⁾ _µν ≡ X ^αβγ R _αµβν . _γ , · · · (18) 1

p

x õ  ° ú  “ É r » ¡ ¤ €  • ³ ð‰ & ³`  ¦  6   x % i  . ¢ ¸ô  Ç R αµβν “ É r o ë ß – /

B

GÒ  ¦J $ ™" f– Ð R µν A λ = R µνδλ A ^δ – Ð & ñ _   ) a  .

Õ

ª Q
 σ. µ ˜ α ˜ β··· _   â Ä º  H ˜ ∇ α \  ¦ σ. µ σ. ^µ = 2σ \  > 5 Å q & h  6

 

x r & " f coincident limit \  ¦ ½ ¨½ + É Ã º \ O  .  s J $ ™" f g µ˜ ν



 H t 2 £ §U  ´`  ¦    7 ˜' \  ¦ ¨ î ' Ÿ s 1 l x`  ¦ K Å ÒÙ ¼– Ð σ. ^µ g µ˜ ν =

−σ. ν ˜   ) a  . Õ ª QÙ ¼– Ð  6 £ § › ' a > d ” s  $ í w n ô  Ç .

−σ. ^µ g _{µ˜ ν} σ. ^{˜ ν} _λ = σ. _{ν ˜} σ. ^{˜ ν} _λ = 1

2 (σ. _{ν ˜} σ. ^{˜ ν} ). _λ = σ. _λ (19) σ. µ = −X µ s Ù ¼– Ð −X ^µ \  ¦ d ”  (14) _  € ª œ  \  Y  L “ ¦ » ¡ ¤

€



•(contraction) €  

−X λ = −X λ − 1

2! X µ σ ⁽²⁾ _{λ ˜ µ} − 1

3! X µ σ ⁽³⁾ _{λ ˜ µ} − 1

4! X µ σ ⁽⁴⁾ _{λ ˜ µ} + · · · (20)

  ) a  . s  › ' a > d ” “ É r — ¸Ž  H X \  @ /K  $ í w n  Ù ¼– Ð Ä º8 £ ¤ _

 y Œ • † ½ Ós  0 s  ÷ &“ ¦   ² D G  6 £ § _  € ª œ[ þ t s  — ¸¿ º ˜ x = x { 9  M

: 0   ) a  .

X σ. λ ˜ µ ˜ α , X

σ. λ ˜ µ ˜ α˜ γ , · · · (21)

#

Œl " f P “ É r ∼  e ”   H — ¸Ž  H “   oÛ ¼  Ë ¨l \  @ /K  @ / g As  ÷ &• ¸2 Ÿ ¤ @ /g A o(symmetrization) ô  Ç   H  כ `  ¦ _ p  ô

 Ç . s – Ð Â Ò'   A _    õ \  ¦ % 3 `  ¦ à º e ”  .

σ _µ˜ ⁽²⁾ _ν = 1 3 R ⁽²⁾ _νµ σ _µ˜ ⁽³⁾ _ν = 1

2 R ⁽³⁾ _νµ σ _µ˜ ⁽⁴⁾ _ν = ( 3

5 R ⁽⁴⁾ − 7

15 R ⁽²⁾ R ⁽²⁾ ) _νµ σ _µ˜ ⁽⁵⁾ _ν = ( 2

3 R ⁽⁵⁾ − R ⁽²⁾ R ⁽³⁾ − 4

3 R ⁽³⁾ R ⁽²⁾ ) _νµ σ _µ˜ ⁽⁶⁾ _ν = ( 5

7 R ⁽⁶⁾ − 96

35 R ⁽⁴⁾ R ⁽²⁾ − 25

7 R ⁽³⁾ R ⁽³⁾

− 11

7 R ⁽²⁾ R ⁽⁴⁾ + 62

15 R ⁽²⁾ R ⁽²⁾ R ⁽²⁾ ) _νµ (22)

V. 4   ˜ m

/

B N   p ì  r ƒ  í ß –  (7) `  ¦ X – Ð „  > h   H X < ¿ º d ”  (12) ü

< (14) \  d ”  (14), (18) õ  (22) \  ¦ & h 6   x €    ) a  . s X O 

>

 €   | 9 | ¾ Ó " é ¶ s  6 s  “   / B N  † < Êà º\  ¦ Ÿ í† < Ê   H — ¸

Ž

 H † ½ Ó[ þ t`  ¦ ½ ¨½ + É Ã º e ”  . Õ ª Q
 „  ^ ‰   õ   H B Ä º ´ ú §“ É r

†

½ Ó`  ¦ Ÿ í† < Ê “ ¦ e ” # Q # Œl " f  H | 9 | ¾ Ó " é ¶ s  4 s  “   † ½ Ó [

þ

t ë ß –`  ¦ ˜ Ð# ŒÅ Òl – Ð ô  Ç .

∇ µ = (1 − 1

6 R ⁽²⁾ − 1

12 R ⁽³⁾ − 1

40 R ⁽⁴⁾ + 7

360 R ⁽²⁾ R ⁽²⁾ ) _µ ^α ∂

∂X ^α + 1

2 R ⁽¹⁾ µ + 1

3 R ⁽²⁾ µ + 1

8 R ⁽³⁾ µ − 1

24 R ^(2)ν _µ R ⁽¹⁾ ν + O(X ⁵ ) (23) ] X

‚   7 ˜'  ý a³ ð> \ " f_  B jà Ôa Ë : J $ ™" f g ^µν \  ¦ ½ ¨ l  0 AK " f  H d ”  (16) \  d ”  (14) ü < d ”  (22) \  ¦ & h 6   x r v €    ) a  .

Σ µν ≡ −g _µ ^{ν ˜} σ. ˜ νλ = (1 − 1

6 R ⁽²⁾ − 1

12 R ⁽³⁾ − 1

40 R ⁽⁴⁾ + 7

360 R ⁽²⁾ R ⁽²⁾ ) µλ + O(X ⁵ ) (24) s

Ù ¼– Ð

g ^µν = (Σ ^> Σ) _µν = (1 − 1

3 R ⁽²⁾ − 1

6 R ⁽³⁾ − 1

40 R ⁽⁴⁾ + 1

15 R ⁽²⁾ R ⁽²⁾ ) ^µν + O(X ⁵ ) (25)

  ) a  .

VI. + s Ç Â ] Ø

Ï ã

L # Q”   r / B N ç ß –\ " f & ñ _   ) a { 9 ì ø Í& h “   / B N  p ì  rƒ  í ß – 

\

 ¦ ] X ‚   7 ˜' ý a³ ð> `  ¦  6   x # Œ ³ ð‰ & ³ ÷ & ¨ î ' Ÿ s 1 l x`  ¦ # î '

Ÿ    H ~ ½ ÓZ O `  ¦ ] jr  % i  . Õ ª   õ  / B N  p ì  rƒ  í ß –  / B N



 $ í s  ì  r" î ô  Ç + þ AI – Ð  r  ³ ð‰ & ³÷ &# Q”     H  כ `  ¦ ˜ Ð% i 



. / B N   p ì  r ƒ  í ß –   H l ï  r& h  x \ " f_  / B N  † < Êà ºü < ] X 

‚

  7 ˜'  X – Ð s À Ò# Qt   H  † ½ Ód ” Ü ¼– Ð „  > h½ + É Ã º e ”   H X <

| 9

| ¾ Ó " é ¶ Ü ¼– Ð 6  † ½ Ó t  ½ ¨ % i  . ¢ ¸ô  Ç ] X ‚  ý a³ ð> \ 

"

f_  B jà Ôa Ë : J $ ™" f• ¸ ½ ¨½ + É Ã º e ” % 3  .

P

c p 8 ý ò k >

s

  7 Hë  H“ É r 2013 † < Ƹ  • ¸ Ø  æ· ¡ ¤ @ /† < Ɠ § † < ÆÕ ü tƒ  ½ ¨t " é ¶  \ O  _

 ƒ  ½ ¨q t " é ¶ \  _  # Œ ƒ  ½ ¨÷ &% 3 _ þ v m  .

REFERENCES

[1] H. Lee and S. W. Lee, Sae Mulli, 58, 654 (2009).

[2] H. Lee, New Phys.: Sae Mulli 60, 44 (2010).

[3] H. Lee, P. Y. Park and H. K. Shin, Phys. Rev. D 35, 2440 (1987).

[4] H. Lee, P. Y. Park and H. K. Shin, Phys. Rev. D 40, 4202 (1989).

[5] B. S. DeWitt, Dynamical Theory of Groups and Fields (Gordan and Breach, New York, 1965).

[6] B. S. DeWitt, Phys. Rep. 19C, 295 (1975).