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2011 1 5 : 2009 6 1

2graddivrot

All Rights Reserved (c) Yoichi OKABE 1997-present.

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ii 2009 6 1 : 2009 6 15 : 2009 7 1 :

1 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 3 7 12

2.1 2.2 2.33

3.1 3.2 3.3 3.44

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 . . . . . . . . . . . . . . . . . . . . . . . 18 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.1 4.2 4.3 4.4 4.55

25 . . . . . . . . . . . . . . . . . . . . . . . 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28 30 . . . . . . . . . . . . . . . . . . . . . . . . . . 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34 37

5.1 5.2 5.3 5.4 5.5

. . . . . . . . . . . . . . . . . . . . . 37 . . . . . . . . . . . . . . . . . . . . . . . . 38 . . . . . . . . . . . . . . . . . . . . . 41 43 . . . . . 44 . . . . . . . . . . . . . . . . . . . . . . . . .

iv6

6.1 6.2 6.37

47 . . . . . . . . . . . . . . . . . . . . . 47 49 51

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1 7.2 7.3 7.4 7.5

. . . . . . . . . . . . . . . . . . . . . . . . .

67 67 . . . . . . . . . . . . . . . . . . . . . . . 68 . . . . . . . . . . . . . . . . . . . . . 68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 . . . . . . . . . . . . . . . . . . . . . . . . .

74 77

1

Riemanian geometry , 1radian 1 graddivrot

2

()

2.1 unit vectorCartesian coordinate

system oblique coordinate system curvilinear coordinate system orthogonal coordinate system orthogonal curvilinear coordinate system (x1 , x2 , ) = (xm | m = 1, 2, . . . ) (xm ) 12m

4

2

xm 1 em (e1 , e2 , ) = (em ) natural basis (r, ) = const r = const () r 1 r 1 r 1

dx = dx1 e1 + dx2 e2 + =

m

dxm em

* dxm dx em 1

dxm em em dxm r |dx1 e1 | = dr 1 dx1 (= dr) [m]e1 [1] dr [1]e1 [m] |dx2 e2 | = d r dx2 (= d) [1] (radian )e2 [m]

Einstein convention

dx = dxm em*1

(2.1)

m m em dx

P

2.1

5

dx distance square distance length square length

ds2 = dx dx = dxm em dxn en = dxm em en dxn = dxm gmn dxngmn metric tensor

(2.2)

gmn = em en

(2.3)

|em | (1 ) em en = |em ||en | cos

m (dx

m 2

)

gmn em em en = en em [gmn ]

gmn = gnm

(2.4)

[gmn ]

em en gmn = 0 gmm em gmm = 1 ds2 = dx2 + dy 2 + dz 2 c2 dt2 dt ds2

ds2 A ( ) A = Am em (2.5)

inner product

A B = (Am em ) (B n en ) = Am em en B n = Am gmn B n

(2.6)

B = A square length A A = (Am em ) (An en ) = Am em en An = Am gmn An (2.7)

6

2

A = dx 2.2

A B = (Am em ) (B n en ) = Am em en B n = Am gmn B n gmn = em en A A

(2.8)

[ 1] mn Kronecker delta

ds2 = dx2 + dy 2 + = dxm mn dxn

(2.9)

gmn = mn gmn

[gmn ] =

( 1 0

0 1

)

(2.10)

1

[ 2] Minkowski space

ds2 = dx2 + dy 2 + dz 2 c2 dt2

(2.11)

y z gmn

(

[gmn ] =

1 0 0 c2

) (2.12)

x4 = ct g44 = 1 x4 = t g44 = c2

e1 e1 = 1 e4 e4 c2 c dx dx t ds2 ds2

d 2 = dt2 dx2 /c2

(= ds2 /c2 )

2.2

7

2.2 (x | = 1, 2, . . . ) = (x ) x = . . . , 1, 0, 1, . . . e dx (dx1 , dx2 , )

dx = dxm m x

( ) x = dxm m x

(2.13)

m transformforward transform m x transform coecient * 2

xm x xm x

dxm = dx xm

(2.14)

dx reverse transform xm dx e

dx = e dx*2

(= em dxm )

8

2

dxm 2.14 dx e

e = xm em = em em

(2.15)

e em em em xm (= xm /x ) xm forward transform coecient dx 2.13 dxm em

em = m x e = e e m

(2.16)

e m m x reverse transform coecient xm (= em ) covariant m x (= e ) contravariant m dx dxm dxm em e

x xn xn n = = m m x x xm m x x x xm m x = = = x xm x m x xn =

(2.17)

2.2 gmn

9

g = e e = xm em en xn = xm gmn xn

(2.18)

gmn gmn [gmn ] [g ]

gmn = em en = m x e e n x = m x g n x

(2.19)

xM xM xm xM xm xm xM xM (, ) r

[ 1] (xm ) = (x, y) (x ) = (r, )

x = r cos y = r sin

(2.20)

xm m

( ) [dxm ]T = dx dy = [dx ]T [ xm ] ( ) ( ) cos sin = dr d r sin r cos

(2.21)

10

2

x2 + y 2 y = tan1 x r= m x m

(2.22)

( [dx ]T = dr

) d = [dxm ]T [m x ] ) ( ( ) x/ x2 + y 2 y/(x2 + y 2 ) = dx dy 2 2 y/ x2 + y 2 x/(x + y ) ( ) ( ) cos sin /r = dx dy (2.23) sin cos /r

m x xm

er = cos ex + sin ey e = r sin ex + r cos ey ex = cos er (sin /r)e ey = sin er + (cos /r)e

(2.24)

(2.25)

g gmn = mn [g ] = [ xm ][gmn ][ xn ]T ( )( )( cos sin 1 0 cos = r sin r cos 0 1 r sin ( ) 1 0 = 2 0 r )T

sin r cos

(2.26)

er e 1 r

[ 2] spherical coordinate system (xM ) =

(x, y, z) a

2.2

11

(xm ) = (, ) m xM M m a

x = a sin cos y = a sin sin z = a cos m xM

(2.27)

( ) [dxM ]T = dx dy dz = [dxm ]T [m xM ] ( ( ) cos cos cos sin = d d a sin sin sin cos M xm

sin 0

) (2.28)

gmn gM N = M N [gmn ] = [m xM ][gM N ][n xN ]T ( ) cos cos cos sin sin = a2 sin sin sin cos 0 ( )T cos cos cos sin sin sin sin sin cos 0 ( ) 1 0 = a2 0 sin2

(2.29)

e e a a sin r

12

2

2.3 A e

A = Am em = Am (m x e ) = (Am m x )e

(2.30)

m x Am A e A

A = Am m x m x

(2.31)

covariant contravariant Am A contravariant component contravariant vector e

A = A e = A xm em A x m

(2.32)m

A em A

Am = A xm

(2.33)

(A B) = (A e ) (B e ) = A e e B = A g B k l A g B = Am m x xk gkl xl n x B n = Am m gkl n B n

(2.34)

= Am gmn B n = A B

2.3

13

2.17 T mn m x n x 2 tensor em en T mn T = T mn em en

T = T mn m x n x

(2.35)

T = T mn em en = T mn m x e e n x = (T mn m x n x )e e (= T e e ) g g = xm gmn xn T 2

T = T mnk em en ek 3 1 0

[ 1] A 1 dx

Ar = Ax cos + Ay sin A = Ax sin /r + Ay cos /r Ax = Ar cos A r sin Ay = Ar sin + A r cos (2.36)

(2.37)

[ 2]

x = (x + ct ) t = (t + x /c)

(2.38)

14 ut

2

ut c ux ux 2.1

A

0

xm m

( [ xm ] =

c

) /c

(2.39)

= 0.6

= 1/ 1 2 = 1/0.8 = 1.25c = 2 3 108 m/s

ex = ex + /cet = 1.25ex + 0.375et et = cex + et = 1.5ex + 1.25et 2.1 m

(2.40)

(

[m x ] =

c

) /c

(2.41)

(3, 1) A

( [A ]T = Ax

( ) ) ( ) /c At = Ax Ay c ( ) ( ) 1.25 0.375 ( ) = 3 1 = 2.25 0.125 1.5 1.25

(2.42)

g [g ] = [ xm gmn xn ] ( )( )( )T /c 1 0 /c = 2 c 0 c c ( )( )T ( ) c /c 1 0 = = c c2 c 0 c2

(2.43)

2.3

15

gmn gmn = mn

[g ] = [ xm mn xn ] ( )( )( )T /c 1 0 /c = c 0 1 c ( )( )T /c /c = c c ( 2 ) 2 2 2 + (/c) c + 2 /c = 2 2 2 2 2 c + /c + (c)

3

1

3.1 1 dual basis (em ) (em ) m em en = en em = n

(3.1)

em en (n = m) em m em en n

em m x xn m m x em en xn = m x n xn

en xn e

18 m x xm = m x em e =

3

e e = e

e = em m x

(3.2)

em = e xm

(3.3)

3.2 em g mn

g mn = em en g mn

(3.4)

g = e e = m x em en n x = m x g mn n xg mn m gn m m gn = em en = en em = n m gn m g = e e = m x em en xn = m x n xn (= )

(3.5)

(3.6)

(3.7)

m gn

m g gmn gn g mn

3.3

19

3.3 A

A = em Am

(3.8)

A = xm Am

(3.9)

A em Am em covariant component covariant vector dx em em

dxm dx = em dxm (3.10)

* f 1

m f = f /xm

f =

f xm f = = x m m f x x xm

(3.11)

m f 3.9 m f

m f = m x f

(3.12)

m f m f

*1

df df /dx df df /dx

20

3

3.4 gmn g mn gmn n gmn An = em en An = em A = em en An = m An = Am

(3.13)

lowering g mn m g mn An = em en An = em A = em en An = n An = Am

(3.14)

raising 2 g mn gmn m gmn g mn gn m m g mn gn = n

[g mn ] [gmn ] [g ] [g ] m Tmn Tn T mn

A em B en

A B = (em Am ) (en Bn ) = Am g mn Bn

(3.15)

A em B en n A B = (Am em ) (Bn en ) = Am m Bn = Am Bm

3.4

21

A B = Am Bm = Am B m* 2

(3.16)

B A A A = Am gmn An = Am g mn An = Am Am (3.17)

m, n , g ||g ||

||g || = || xm gmn xn || = || xm || ||(gmn )|| || xn || = J 2 ||gmn ||J = || xm ||

(3.18)

[ 1]

er = ex cos + ey sin e = ex sin /r + ey cos /r Ar = cos Ax r sin Ay A = sin Ax + r cos Ay g [g ] = [m x g mn n x ] = [m x mn n x ] = [m x m x ] ( )( )T ( cos sin cos sin 1 = = sin /r cos /r sin /r cos /r 0 ) 0 2 1/r (3.21) (3.19)

(3.20)

er 1 e 1/r A 1/r

*2

22

3

r

[ 2] 2.1 m x em ex = ex et c = 1.25ex 1.5et et = ex /c + et = 0.375ex + 1.25et

(3.22)

3.1 (e ) (e )

ut

ut c ux 0 A

ux 3.1

(3, 1) A

( [A ]T = [ xm Am ] = 3

1

)

(

) ( ) 1.25 1.5 = 4.125 5.75 0.375 1.25

(3.23)

A

A A = 2.25 4.125 + 0.125 5.75 = 10 = 12 + 32 g [g ] = [m x g mn n x ] ( )( )( )T c 1 0 c = 2 /c 0 1/c /c ( )( )T ( ) /c c 1 0 = = /c /c2 /c 0 1/c2

(3.24)

(3.25)

3.4 g mn

23

g g g g

[ 3] 3.2 f e1 1 f 3 f /x1 = 3 e2 1 f 1 f /x2 1 (3, 1) e1 3 f 4.125 e2 3 f 5.75 f /x1 4.125f /x2 5.75 (4.125, 5.75) (4.125, 5.75)

ut

ut c ux 0 f =0 3 ux 3.2

A

6

4

4.1 () xM xm xm + dxm ( ) xm 0 10 0 0 10

26

4

180 xM gM N eM xm em em em em

em = m xM eM e m = m x M e M eM = M xm em + M xm em A

A = Am em = Am m xM eM

( ) = AM eM

AM = Am m xM A

A = AM eM = AM M xm em + AM M xm em

(= Am em + n)

Am em n

Am = AM M xm Am = AM M xm

4.1

27

Am x = (xm ) x + dx = (xm + dxm ) em (x + dx)

em (x + dx) = em (x + dx)// (x) + em (x + dx) (x)

(4.1)

em (x + dx)// (x) em (x + dx) x em (x + dx) translation em (x + dx) (x) x em (x + dx)// (x) dx em (x)

< em (x + dx)// (x) = em (x) + dxk nkm (x)en (x)

(4.2)

kmn Christel symbol

{

n km

} (= nkm ) (4.3)

xM eM (x + dx) = eM (x) em (x + dx) dx

em (x + dx) = m xM (x + dx) eM (x + dx) [ ] = m xM (x) + dxk k m xM (x) [M xn (x) en (x) + n] = em (x) + dxk k m xM (x)M xn (x) en (x) + n (4.4)

n 2 n m xM M xn = m xn (= m )

28

4

4.2 4.4

nkm (x) = k m xM (x)M xn (x) =

2 xM (x) xn (x) xk xm xM

(4.5)

nkm (x) = nmk (x)

(4.6)

em (x) = em (x + dx)// (x) em (x) = dxk nkm (x)en (x)

(4.7)

em (x) dem (x) em (x + dx) em (x)

em = dxk m en kn

(4.8)

4.2 4.5 4.5 4.4 em gmn gmn 4.7 x

(x) em el = dxk nkm en el = dxk nkm gnl

k em el = nkm gnl

4.2

29

k gml = k (em el ) = k em el + k el em = kmk gnl + nlk gnm n glk = nlm gnk + kkm gnl l gkm = nkl gnm + nml gnk (4.9)

gmn g 1 +2 3

2nkm gnl = k gml + m glk l gkm 2 gml

nkm =

1 nl g (k gml + m gkl l gkm ) 2

(4.10)

x g gkm m-k-n-m

=

1 g ( g + g g ) 2 1 = n x g nl l x 2[ ] ( xm gml xl ) + ( xk gkl xl ) ( xk gkm xm ) = nkm xk xm n x + xn n x (4.11)

0

xn = xn nkm xk xm

(4.12)

30

4

4.3 A(x)

A =A(x + dx)// (x) A(x) =Am (x + dx)em (x + dx)// (x) Am (x)em (x) =dAm em + Am em = dAm em + dxk nkm Am en 1 0 Am dAm A

dAm covariant derivativeAm Am = A em = dAm + dxk m An kn (4.13)

A = 0 Am = 0 A(x + dx) x A(x + dx)// (x) 0

Am (x + dx)// (x) = Am (x) + Am (x) = Am (x) dxk m An (x) kn

(4.14)

A(x + dx)// (x) 4.13

Am = Am (x + dx) Am (x + dx)// (x) [ ] = [Am (x + dx) Am (x)] Am (x + dx)// (x) Am (x) = dAm + dxk m An kn A dxk Am dxk k Am (=

Am /xk ) covariant dierential k Am =

Am Am = + m An = k Am + m An kn kn xk xk

(4.15)

4.3

31

k Am Am :k k

k Am 2

A = (Am m x ) = Am m x + m x Am = xk k Am m x + xk k n x An = xk k Am m x + xk (m m x k x n x )An kn = xk (k Am + m An )m x A kn 4.12 n Am + m Ak kn 4.15 n Am A m Am div A m Am

div A = m Am = m Am + m An mn

(4.16)

A = (em Am ) = em dAm + em Am = em dAm dxk m en Am kn A Am Am = em A = dAm dxk nkm An k Am k Am =

(4.17)

Am Am = nkm An = k Am nkm An xk xk

(4.18)

k Am Am:k

32 A A = ( xm Am ) = xm Am + xn An

4

= xm xk k Am + ( xn nkm xk xm )An = xk (k Am nkm An ) xm + A 4.18 n Am

4.4

0

An Bk m A B

(A B ) = xm n x xk (An Bk ) mB = xk Bk

A = xm n x An m An m

4.4

33

f = dfk f = k f

(4.19) (4.20)

+dxk m An kn dxk nkm An nk nk nk (Tm Uli ) =Tm (x + dx)// (x)Uli (x + dx)// (x) Tm (x)Uli (x) nk nk nk =(Tm (x) + Tm (dx))(Uli (x) + Uli (x)) Tm (x)Uli (x) nk nk =(Tm )Uli (x) + Tm (Uli (x)) n Tm n Tm um vn

n n n n (Tm um vn ) = (Tm )um vn + Tm (um )vn + Tm um (vn ) n n n = (Tm )um vn + Tm (dum + dxa m ub )vn + Tm um (dvn dxa ban vb ) ab

4.19 n n n n n (Tm um vn ) = d(Tm um vn ) = (dTm )um vn + Tm (dum )vn + Tm um (dvn )

um vn n n l Tm = dTm dxk l km Tln + dxk nkl Tm

dxk

34

4

n n l k Tm = k Tm l km Tln + nkl Tm

(4.21)

gmn k gmn = k gmn l km gln l kn gml = 0

(4.22)

4.9 n n gm = m n n n l k gm = k gm l km gl + nkl gm = 0

(4.23)

0g mn k g mn = k g mn + m g ln + nkl g ml = 0 kl 4.9 ()

(4.24)

k g mn = k (em en ) = em k en + en k em = nkl g ml m g ln kl (4.25)

4.5 geodesic line t xm (t) xm =

dxm /dt I = ds =

t1 t0

gmn (x)dxm (t)dxn (t) =

t1

dt gmn (x)xm xn

t0

4.5 L(x, x)

35

L(x, x) =

gmn (x)xm xn

I x = (xm ) x = (xm ) L Euler equatio ()

d dt L(x, x)

(

L xk

)

L =0 xk

d gkn xn k gmn xm xn dt L 2L gkn xn m gkn xm xn gkn xn (dL/dt) k gmn xm xn = + =0 L L L2 2L L

gkn xn +

1 gkn xn (dL/dt) (m gkn xm xn + n gkm xm xn k gmn xm xn ) = 2 L

t s = ds/dt = L

1dL/dt = 0 g kl l k d2 xk dxm dxn + kmn =0 ds2 ds ds s s v k = dxk /ds

(4.26)

v k =0 ds

v k dv k + kmn v m dxn dv k = = + kmn v m v n = 0 ds ds ds 4.26

5

0 e 0 e 90 e e dxm

5.1 k (l An ) = k (l An + nlm Am )

=(k l An + k nlm Am + nlm k Am ) + nki (l Ai + i lm Am ) i kl (i An + nim Am )

38

5

l (k An ) =(l k An + l nkm Am + nkm l Am )

+ nli (k Ai + i km Am ) i lk (i An + nim Am )

[k , l ]An = k l An l k An = Rnm,kl Am

(5.1)

Rnm,kl Rnm,kl k, l

Rnm,kl = nm,kl nm,lk = (k nlm + nki i lm ) (l nkm + nli i km )

(5.2)

nm,kl = k nlm + nki i lmnm,kl

(5.3)

Rnm,kl Riemann Christoel curvature tensor curvature tensor 5.1

Rnm,kl

5.2

5.2

39

x xP = x + dx = x + dxk ek

nml (xP ) = nml (x + dxk ek ) = nml (x) + dxk k nml (x)

(5.4)

( 4.7 )

em (x) = dxk nkm (x)en (x)

(5.5)

2 em (x) = em (xP ) em (x) =dxl nlm (x + dxk ek )en (x + dxk ek ) dxl nlm (x)en (x) [ ][ ] =dxl nlm (x) + dxk k nlm (x) en (x) + dxk nkn (x)en (x) dxl nlm (x)en (x) =dxk dxl k nlm (x)en (x) + dxk dxl nkn (x)nlm (x)en (x) =dxk dxl nm,kl en (x) (5.6)

5.3 nm,kl

{ n } em (x + dxk ek ) = m + dxk nkm (x) + dxk dxl nm,kl /2 en (x)

(5.7)

OP QR em em (O, P )// em (O, P, Q)// em (O, P, Q, R)//

em (O, P, Q, R, O)// // (O) O xP

xP = x + dal el (x) xQ = xP + dbk ek (xP ) xR = xQ dal el (xQ ) Q R O xO = xR dbk ek (xR ) xO x nm,kl 4 em (O, P )//

[ n ] em (O, P )// = m dak nkm (P ) + dak dal nm,kl /2 en (P )

(5.8)

40

5

em (O, P, Q, R, O)//

em (O, P, Q)// [ i ] em (O, P, Q)// = m dak i km (P ) + dak dal i m,kl /2 ei (P, Q)// [ i ] = m dak i km (Q) + dak dbl l i km (Q) + dak dal i m,kl /2 ei (P, Q)// 5.8

[ ] n ei (P, Q)// = i dbl nl i (Q) + dbk dbl ni,k l /2 en (Q)

em (O, P, Q)// ] [ i = m dak i km (Q) + dak dbl l i km (Q) + dak dal i m,kl /2 ] [ n i dbl nl i (Q) + dbk dbl ni,k l /2 en (Q) [ n = m dak nkm (Q) dbl nlm (Q) ] +dak dal nm,kl /2 + dak dbl nm,lk + dbk dbl nm,kl /2 en (Q) 5.3 em (O, P, Q, R)//

[ j em (O, P, Q, R)// = m dak jkm (Q) dbl jlm (Q)

] +dak dal jm,kl /2 + dak dbl jm,lk + dbk dbl jm,kl /2 ej (Q, R)// [ j = m dak jkm (R) dak dal l jkm dbl jlm (R) dak dbl k jlm ] +dak dal jm,kl /2 + dak dbl jm,lk + dbk dbl jm,kl /2 ] [ n j + dak nk j (R) + dak dal nj,k l /2 en (R) [ n ] = m dbl nlm (R) dak dbl (nm,kl nm,lk ) + dbk dbl nm,kl /2 en (R)

5.3 em (O, P, Q, R, O)//

41

[ j en (O, P, Q, R, O)// = m dbl jlm (R) ( ) ] dak dbl jm,kl jm,lk + dbk dbl jm,kl /2 ej (R, O)// [ j = m dbl jlm (O) dbk dbl k jlm ( ) ] dak dbl jm,kl jm,lk + dbk dbl jm,kl /2 [ ] n j + dbl nl j (O) + dbk dbl nj,k l /2 en (O) ( [ n )] = m dak dbl nm,kl nm,lk en (O)

em (O, P, Q, R, O)// em (O, O)// 5.2 Rnm,kl

em (O, O) = em (O) dak dbl Rnm,kl en (O)

(5.9)

em (O, O)// = em (O) + dak dbl Rm en (O) n,klm em (O, O)// em (O, O)// = m

(5.10)

(5.11)

5.3 A = Am (O, O)// em (O, O)// = Am (O)em (O) A

A =Am (O, O)// em (O, O)// = Am (O, O)// (em (O) dak dbl Rnm,kl en (O)) =Am (O, O)// em (O) dak dbl Rnm,kl Am (O, O)// en (O) ( = Am (O)em (O) )

Am (O, O)// = Am (O) + dak dbl Rm An (O) n,kl

(5.12)

42

5

An (O, O)// An (O)

Am (O, O)// = Am (O) dak dbl Rnm,kl An (O)

(5.13)

5.12

Am (O, O)// Am (O) = [Am (O, O)// Am (O, R)// ] + [Am (O, R)// Am (O, Q)// ] + [Am (O, Q)// Am (O, P )// ] + [Am (O, P )// Am (O)] = Am (O, R) + Am (R, Q) + Am (Q, P ) + Am (P, O) = l Am (R)dbl k Am (Q)dak + l Am (P )dbl + k Am (O)dak = k l Am dak dbl l k Am dak dbl = [k , l ]Am dak dbl 5.1

[k , l ]Am = Rm An n,kl

(5.14)

[k , l ]Am = Rnm,kl An

(5.15)

[k , l ]An = Rni,kl Ai Ri m,kl An m m iRicci formula

5.4

43

5.4

Rmn,kl = gmi Ri n,kl

(5.16)

1. anti-symmetricity Rm = Rm n,kl n,lk Rmn,kl = Rmn,lk () (5.17)

2. anti-symmetricity Rmn,kl = Rnm,kl (5.18)

3. commutation Rmn,kl = Rkl,mn

(5.19)

5.17 : 5.2 5.18 :

[k , l ]gmn = Ri m,kl gin Ri n,kl gmi = Rnm,kl Rmn,kl 0 0 5.19 : 5.2 n, k, l Biannki equation

Rnm,kl + Rnk,lm + Rnl,mk = 0

(5.20)

44

5

Rnm,kl + Rnk,lm + Rnl,mk = 0 5.19 5.17 5.18 n m

Rmn,kl + Rmk,ln + Rml,nk = 0

Rnk,lm + Rnl,mk + Rmk,ln + Rml,nk = 0

Rkn,lm Rln,mk Rkm,ln Rlm,nk = 0

Rkl,mn + Rkm,nl Rln,mk Rkm,ln + Rln,km + Rlk,mn = 0 4 5 2

Rkm,ln Rln,mk = 0

5.5 Ricci tensor

Rmk = Rl m,kl

(5.21)

5.5 scalar curvature

45

R = g mk Rmk = g mk Rl m,kl

(5.22)

Einstein tensor Gik

1 Gij = Rij g ij R 2

(5.23)

0 i Gij = 0

(5.24)

Biannki equation i Rnm,kl + k Rnm,li + l Rnm,ik = 0

(5.25)

Rnm,kl;i + Rnm,li;k + Rnm,ik;l = 0

(5.26)

[i , [k , l ]]Am

[i , [k , l ]]Am = i [k , l ]Am [k , l ] i Am = i k l Am i l k Am k l i Am + l k i Ami, k, l 0

46

5 [i , [k , l ]]Am

[i , [k , l ]]Am = i [k , l ]Am [k , l ] i Am = i (Rnm,kl An ) Rnm,kl i An Rni,kl n Am = (i Rnm,kl )An Rni,kl n Ami Am i, k, l 0

(i Rnm,kl + k Rnm,li + l Rnm,ik )An + (Rni,kl + Rnk,li + Rnl,ik ) n Am = 0 5.20 0 An l gn

0 l l l i (gn Rnm,kl ) + k (gn Rnm,li ) + l (gn Rnm,ik ) = 0

i Rmk k Rmi + l Rl m,ik = 0 g mi g kj i Rij k g kj R + l Rlj = 0 k i l i i

(i

1 Rij g ij R 2

) = i Gij = 0

6

graddivrot

6.1 |(gmn |) gmn

d|(gmn )| = dgmn g mn |(gmn )|

(6.1)

( ) 1 l kmn g mn = |(gmn )|g lk |(gmn )| ( ) 1 m = m |(gmn )| mn |(gmn )|

(6.2)

(6.3)

6.1

48

6

g11 d|(gmn )| = d g21 g31 dg11 = dg21 dg31 = dg11 g12 g22 g32 g22 g32

g12 g22 g32

g13 g23 g33 dg12 dg22 dg32 g13 g11 g23 + g21 g33 g31 g12 g22 g32 dg13 dg23 dg33

g13 g11 g23 + g21 g33 g31

g23 g dg21 12 g33 g32

g13 + g33

= (dg11 g 11 + dg21 g 21 + dg31 g 31 )|(gmn )| + + = dgmn g mn |(gmn )|

g22 g32

1 g23 = 0 g33 0

g12 g22 g32

g 11 g11 g13 g23 = g 11 g21 g 11 g31 g33

g12 g22 g32

g13 g23 = g 11 |(gmn )| g33

6.2

kmn g mn =

1 mn kl (g g m gln + g mn g kl n gml g mn g kl l gmn ) 2 1 = g mn g kl m gln g mn g kl l gmn 2 1 kl l |(gmn )| lk = l g g 2 |(gmn )| ( ) 1 = l |(gmn )|g lk |(gmn )|

m, n 2 k g mn n (g kl gln ) = g mn n n = 0 n g km + g mn g kl n gln = 0 6.1

6.3

m = mn

1 ml 1 1 m |(gmn )| g (m gln + n gml l gmn ) = g ml n gml = 2 2 2 |(gmn )| ) ( 1 |(gmn )| = n |(gmn )|

6.2

49

m, n 6.1

6.2 grad

grad f = M f eM = M f g M N eN

(6.4)

M f g M N eN = (M xm m f )(m xM g mn n xN )(en N xn ) = m f g mn en (6.5)

grad f = m f g mn en

(6.6)

(grad f )n = m f g mn

(6.7)

div A M AM M AM M AM 6.3

( ) 1 k |(gmn )| Ak div A = m Am = m Am + m Ak = m Am + mk |(gmn )| ( ) 1 m = m |(gmn )| A (6.8) |(gmn )| rot A rot A =MNK MNK

M AN eK

(6.9)

N, M, K 1, 2, 3

1 1 0

50mnk

6

1, 2, 3 m, n, k

1 xM m xM n xN k xKmnk

1 x N 2 x N 3 xN

1 x K 2 x K 3 x K (6.10)

=

2 xM 3 xM

(N, M, K) (1, 2, 3) m xM J = (m xM ) J 0 MNL

J

m xM n xN k xKmnk

mnk

=J

MNL

(6.11)

/J M AN 1 J 1 J

rot A =

mnk

m A n ek = m, n

mnk

(m An l mn Al )ek

(6.12)

mnk l

mn

mnk

l mn

rot A =

1 J

mnk

m An ek

(6.13)

(rot A)k =

1 J

mnk

m An2

(6.14)

f f = g mn m n f = g mn m n f + kmn g mn k f2 2

(6.15)

f = div(grad f )

div A A = grad f f = div(grad f ) = 2

1 |(gmn )|

m

) ( |(gmn )| g mn n f

(6.16)

6.3

51

6.3 6.3.1 0 2.3 gmn = em en gmn = gmm mn gmm hm =

gmm scale factor

dx dx =

m

gmm (dxm )2 =

(hm dxm )2m

(6.17)

dxm hm dxm em 1 orthonormal basis

um em = hm um um

um un = mn

(6.18)

dx =

m

dxm em =

m

dxm hm um

(6.19)

em em um = um um 1 um em = (1/hm )um

em en =

1 1 m 1 n u u = mn = g mn hm hn hm hn

(6.20)

em um um em hm 111/hm Am Am Au Am 1/hm 11hm u m

52

6

Am Au um um em u m

dx =

m

dxm em =

m

dxm

1 m u hm

(6.21)

dx dx =

m

g mm (dxm )2 =

( dxm )2m

hm

(6.22)

gM N = M N gmn = m xM n xM gmm = m xM m xM (gM N = M N ) (6.23)M

J = |(m x )| |(gmn )| = |(gmm )| = |(m xM )||(m xM )| = J 2M

J=

|(gmm )| =

m

gmm =

m

hm

(gM N = M N )

(6.24)

graddivrot em Am em um Am Am physical component u

A = Am em = Am um u em = hm um Am = Am /hm u em = um /hm

(6.25)

(6.26)

Am = Au hm m

(6.27)

6.3

53

hm hm

[ 1] x = 0 x = V t = V /c x = x + ct ct = ct + Dx(dx, d(ct))

dx dx = dx2 (cdt)2

dx dx = dx2 (cdt)2 = dx 2 + 2 dx cdt + 2 (cdt )2 (cdt )2 2Ddx (cdt ) D2 dx 2 D =

dx dx = (1 2 )dx 2 (1 2 )(cdt )2 ) 0 (6.28) (1 2 ) hx = ht = 1 2 (t (gmn ) =)xm = xm /hm = xm (xm ) = (xu , ctu ) u u u (x , ct ) (x , ct )

( 1 2 0

6.3.2 grad f grad f = m f g mn en em = hm um

grad f =

n

m f g mn hn un =

m

m f

1 um hm

(6.29)

54

6

(grad f )m =

f 1 xm hm

(6.30)

6.3.3 div A =

|(gmn )| Am )/ |(gmn )| ) ( 1 Am div A = |(gmn )| u m hm |(gmn )| m ( ) 1 (h2 h3 A1 ) (h3 h1 A2 ) (h1 h2 A3 ) u u u = + + (6.31) h1 h2 h3 x1 x2 x3 m

m (

6.3.4

rot A =

1 J

mnk

m An ek

(6.32)

rot A =

1 h1 h2 h3

mnk

u mnk (hn An ) (hk uk ) m x

(6.33)

(rot A)k uk Au = An n u A n

6.3.5 f = m ( f =2 2

|(gmn )| g mn n f )/ |(gmn )|

1 1 m (h1 h2 h3 g mn n f ) = (6.34) h1 h2 h3 h1 h2 h3 [ ( ) ( ) ( )] h2 h3 f h3 h1 f h1 h2 f + + 3 x1 h1 x1 x2 h2 x2 x h3 x3

6.3

55

6.3.6

x = r cos y = r sin ( M ) cos sin r sin r cos

(6.35)

(m x ) = mM

(6.36)

J = |m xM | = r ( gmn = 1 0 0 r2 )

(6.37)

(6.38)

(hm ) = (1, r) g = |gmn | = r2 = J 2

(6.39) (6.40)

6.36

x2 + y 2 y = tan1 x r= m y x 2 x + y2 x2 + y 2 (M xm ) = y x M x2 + y 2 x2 + y 2 ( ) cos sin /r = sin cos /r ( (g mn ) = 1 0 0 1/r2 )

(6.41)

(6.42)

(6.43)

56

6

1 grad f = r f ur + f u r ( ) 1 (rAr ) A u u + div A = r r ( ) (rAr ) 1 A u u rot A = r r [ ( ) ( )] 1 f 1 f 2 f = r + r r r r

(6.44) (6.45) (6.46) (6.47)

rot A h3 = 1u3 = ur u 8

kmn =

1 kl g (m gln + n gml l gmn ) 2

k = 1 = r g kl l = r rmn = 1 1 (m grn + n gmr r gmn ) = r g m n = rm n 2 2

k = 2 = g kl l = mn = 1 1 r r (m gn + n gm gmn ) = (m n + m n ) 2r2 r

3 0

r = r r = r = 1/r

(6.48)

dv r rv v = 0 ds dv 2 + vr v = 0 ds r r2

(6.49)

d(r2 v ) =0 ds

6.3

57

v =

r2

(6.50)

2v r

2v r

dv r 2 = 2 3 vr ds r

d(r2 ) d[(v r )2 ] = 2 ds ds

vr =

dr = ds

2

( )2 r

r 2 r=

2 2 (/r) = s s0

r s

2 + 4 (s s0 )2 = 2

1+

[

]2 2 (s s0 )

(6.51)

6.50

v =

d 2 = 2 ds + 4 (s s0 )2

tan( 0 ) = 6.51 6.52

2 (s s0 )

(6.52)

r=

sec( 0 )

58

6

Rm n,kl m n k l

Rr,r = Rr,r = r r rr = 1 Rr,r = Rr,r = r r rr = 1/r2

(6.53)

Rrr = Rrr,rr + Rr,r = 1/r2 Rr = Rr = 0 R = Rr,r + R, = 1

( (Rmn ) =

1/r2 0

0 1

) (6.54)

(n (Rm ) =

1/r2 0

0 1/r2

) (6.55)

( (Rmn ) =

1/r2 0

0 1/r4

) (6.56)

r R = Rr + R = 0

(6.57)

0

( (Gmn ) =

1/r2 0

0 1/r4

) (6.58)

6.3.7

6.3

59

(xM ) a spherical coordinate system (xm ) m xM M m a

x = a sin cos y = a sin sin z = a cos (m) = (, ) ( (m xM ) = a cos cos cos sin sin sin sin cos ( 1 0 0 sin2 ) (6.61) sin 0 ) (6.60) (6.59)

(gmn ) = a2

(h , h ) = (a, a sin )

(6.62)

x2 + y 2 + z 2 = a2

= tan1 = tan1 (x2 y2

x2 + y 2 z y x y x2 + y 2 x x2 + y 2 0

(6.63)

xz

+ + + yz (M xm ) = (x2 + y 2 + z 2 )x2 + y 2 x2 + y 2 2 x + y2 + z2 cos cos sin / sin 1 = cos sin cos / sin a sin 0

z2)

x2

y2

(6.64)

60 g mn = 1 a2 ( ) 1 0 2 0 1/ sin

6

(6.65)

grad f =

( ) 1 f u + f u sin ( ) (sin A ) A 1 u u + div A = a sin ( ) 1 (sin A ) A u u rot A = a sin [ ( ) ( )] 1 f 1 f 2 f = 2 sin + a sin sin 1 a

(6.66) (6.67) (6.68) (6.69)

rot A h3 = 1u3 = u u k = 1 = g kl l =

mn =

1 1 (m gn + n gm gmn ) = 2 gmn = sin cos m n 2a2 2a 1 (m gn + n gm gmn ) 2a2 sin2 2 sin cos = (m n + m n ) 2 sin2 = sin cos = = cot

k = 2 = g kl l = mn =

3 0

(6.70)

4.26 kmn

dv sin cos (v )2 = 0 ds dv + 2 cot v v = 0 ds

(6.71)

sin2

d(sin2 v ) =0 ds

6.3

61

v =

sin2

(6.72)

2v

2v

dv 2 cos v =2 ds sin3

d[(v )2 ] d[(sin )2 ] = 2 ds ds

v =

d = ds

2

( )2 ( )2 = 1 sin sin ) = s s0

= /

1 cos1

(

cos 1 2

s

cos = (1 2 ) cos[(s s0 )] 6.72

(6.73)

v =

d = ds 1 (1 2 ) cos2 [(s s0 )]

tan

tan( 0 ) =

tan[(s s0 )]

(6.74)

6.73 6.74 (x, y, z) XY X = a cos[(s s0 )]Y = a sin[(s s0 )] Y 0

x = a cos 0 cos[(s s0 )] y = a sin[(s s0 )] z = a sin 0 cos[(s s0 )]

62

6

z = sin 0 cos[(s s0 )] a y 1 tan = = tan[(s s0 )] x cos 0 cos = 0 0 = / = cos 0 6.73 6.74 Rnm,kl m n k l

R, = R, = = sin2 cos2 = 1 R, = R, = = sin2

(6.75)

R = R, + R, = sin2 R = R = 0 R = R, + R, = 1

( (Rmn ) =

sin2 0

0 1

) (6.76)

n (Rm ) =

1 a2

(

sin2 0

0 1/ sin2

) (6.77)

(Rmn ) =

1 a4

(

sin2 0

0 1/ sin4

) (6.78)

R = R + R = [sin2 + (1/ sin2 )]/a2

(6.79)

(Gmn ) =

1 (1/ sin4 ) 2a4

(

sin2 0

0 1

) (6.80)

6.3

63

6.3.8

x = r cos y = r sin z=z (m) = (r, , z) cos sin 0 (m x ) = r sin r cos 0 0 0 1M

(6.81)

(6.82)

J = |(m xM )| = 1/r 1 0 (gmn ) = 0 r2 0 0 (hm ) = (1, r, 1)

(6.83)

0 0 1

(6.84)

(6.85)

x2 + y 2 y = tan1 x z=z r= x x2 + y 2 (M xm ) = y x2 + y 2 0 y + y2 x x2 + y 2 0 x2 0 cos 0 = sin 0 1 0 0 1 0 0 1

(6.86)

sin /r cos /r 0

(6.87)

(g

mn

1 0 ) = 0 1/r2 0 0

(6.88)

64

6

grad f =

f 1 f f ur + u + uz r r z ( ) r 1 (rAu ) Au (rAz ) u div A = + + r r z =

(6.89)

1 (rAr ) 1 A Az u u u + + r r r z [( z ) 1 Au (rA ) u rot A = ur r z ( r ) ( ) ] Az Au (rA ) Ar u u u +r u + uz z r r [ ( ) ( ) ( )] 1 f 1 f f 2 r + + r f = r r r r z z ( ) 1 f 1 2f 2f = r + 2 2+ 2 r r r r z

(6.90)

(6.91)

(6.92)

6.3.9

x = r sin cos y = r sin sin z = r cos (m) = (r, , ) sin cos sin sin (m xM ) = r cos cos r cos sin r sin sin r sin cos J = |(m xM )| = r2 sin 1 0 (gmn ) = 0 r2 0 0 0 0 r2 sin2 cos r sin 0 (6.94) (6.93)

(6.95)

(6.96)

(hm ) = (1, r, r sin )

(6.97)

6.3

65

x2 + y 2 + z 2 x2 + y 2 = tan1 z y = tan1 x r= 1 ) = 0 0 0 1/r2 0 0 0 1/(r2 sin2 )

(6.98)

(g

mn

(6.99)

grad f =

f 1 f 1 f ur + u + u (6.100) r r r sin ( ) 1 (r2 sin Ar ) (r sin A ) (rA ) u u u div A = 2 + + r sin r [ ] 1 (r2 Ar ) 1 (sin A ) A u u u = 2 + + (6.101) r r r sin [( ) 1 (r sin A ) (rA ) u u rot A = 2 ur (6.102) r sin ( r ) ( ) ] (r sin A ) (rA ) Ar Au u u u u + r sin u +r r r [ ( ) ( ) ( )] 1 f f 1 f 2 f = 2 r2 sin + sin + r sin r r sin ( ) ( ) 1 f 1 f 1 2f = 2 r2 + 2 sin + 2 2 (6.103) r r r r sin r sin 2

7

7.1 =0 =0

g R G =0

68

7

7.2 7.3 ( ) (x , y , z , ct ) x (x, y, z, ct) ct = ct = 0 y z a F X = ma dv x /ds = a =

(a/c)cdt /ds s dx /ds = v x = (a/c)ct F ct = (dx /cdt )ma dv ct /ds = (cdt /ds)(dx /cdt )a/c = (a/c)dx /ds x0 v ct (= cdt /ds) = (a/c)(x + x0 ) X = ax/c2 T = act/c2 X = ax /c2 T = act /c2 = acs/c2

7.3

69

dX =T dS dT = X X0 dS

(7.1)

X = cosh S T = sinh S

(7.2)

X0 S = 0 X X0 = 0 S = 0 X = 1 S T S

X 2T

2

=1

(7.3)

X =

1+T

2

(7.4)

X 45 X = T T = 0 X = 1 X X 1 X = 0 X

X X = 1 + T 2 + (X)

( X )

(7.5)

(X) X T = 0 X = X (X) = X 1 T 7.5 7.5 T

dX T = dT 1+T

2

(7.6)

70 T

7

dX = dT 1+T

1+T T

2

(7.7)

T 2

sinh1

1 = X (T ) |T |

( T )

(7.8)

T (T ) 7.5 7.8 (X , T )

1 sinh[(T ) (X)] X = coth[(T ) (X)] + (X) T = 1 sinh2 [(T ) (X)] ( cosh2 [(T ) (X)] (X) cosh[(T ) (X)] (X)

(7.9)

u = m

) (T ) cosh[(T ) (X)] (T ) 0 (T )2 )

(7.10)

g =

1 sinh [(T ) (X)]2

( cosh2 [(T ) (X)] (X)2 0

(7.11)

(X, T ) (X , T )

dX = (dX v dT ) dT = (dT v dX ) X dX = 0

(7.12)

dX = v dT

(7.13)

dT =

1 1 v 2 dT = dT 1+T 2

(7.14)

7.3

71

T = sinh1 T T = 0 T = 0 7.13

(7.15)

X =

T dT = 1 + T 2 1+T

2

+ (X)

(7.16)

T = 0 X = X

T = sinh T X = cosh T 1 + X

(7.17)

T X T X

(X X)dX = T dT T dX /dT dX /dT

(X X)dT = T dX (X , T ) 7.5 X T 7.5 T

1 + T 2 dT = T dX T

1+T

2

sinh1

1 = X (T ) |T |

72

7

T 7.2 S T 1 1 1 + T 2 sinh1 = X sinh1 ( T ) (7.18) |T | sinh |T | 7.5 X 7.8 X T X

X =X+

1+T 2 ( )2 1 =X + 1+ 1 sinh[sinh (1/ sinh |T |) X] 1 T = sinh[sinh1 (1/ sinh |T |) X]

( X ) ( T )

(7.19)

T (X, T ) X T f (X)

T = sinh1 T = sinh1

1 sinh{X + sinh1 [1/ (X X)2 1]} 1 sinh[X + sinh1 (1/|T |)] 1+T

( X ) (7.20) ( T )

7.5 X

X=X

2

T = sinh1 = sinh umn = 1+ 2 1

1 sinh[X + sinh1 (1/|T |)] 1 sinh[X 1 + T 2 + sinh1 (1/|T |)]cosh(Xasinh (1/ sinh |T |))3

(7.21)

sinh(Xasinh (1/ sinh |T |))1 sinh 2 (Xasinh (1/ sinh |T |)) cosh(Xasinh (1/ sinh |T |)) sinh2 (Xasinh (1/ sinh |T |))

2 sinh |T |

cosh(Xasinh

sinh(Xasinh (1 sinh |T | cosh(Xasinh sinh |T | sinh2 (Xas

(7.22) gmn

1, 1

(g ) =

( 1 (gt /c)2 gt /c2

gt /c2 1/c2

)

7.4

73

x x = 0 x x t = x x = x t

7.4 ds2 = e2f (x ) c2 dt 2 + dx 2 + dy 2 + dz 2

xt t = e2f f t t x = f Rt t = e2f (f + f 2 ) Rx x = (f + f 2 )2

f +f

=02

f = f dx =

df f2

f =

1 x x0

f = log a(x x0 ) e2f = a2 (x x0 )2 xt t = e2f f = a2 (x x0 ) 1 t t x = f = x x0

74

7

2 2 dU x = xt t U t = a2 (x x0 )U t ds dU t 1 = 2t t x U t U x = Ut Ux ds x x0

d(F (x )U t ) dU t = F U ( x )U ( t ) + F =0 ds ds F /F = 1 d log F = x x0 dx a F = x x0

7.5 ds2 = e2f (r ) c2 dt 2 + e2g(r ) dr 2 + r 2 d 2 + r 2 sin2 2

xt t = e2(f g) f t t x = f xx x = g 1 x = r 1 x = r x = re2g = cot x = re2g sin2 = sin cos

7.5

75

Rt t = e2(f g) (f f g + f Rx x = (f f g + f R R = sin2 R 0 2

2

2 + f) r

2 g) r = 1 + e2g (rg rf 1)

g = f f + 2f2

+ 2f /r = 0

A K

anti-symmetricity . . . . . . . . . . . 43 B

Kronecker delta . . 6 L

Biannki equation . . 43, 45 C

length . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 lowering . . . . . . . . . . . . . . . . . . . . . . . . 20 M

Cartesian coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Christel symbol 27 commutation . . . . . . . . . . . . . 43 contravariant . . . . . . . . . . . . . . . . . 8, 12 contravariant component . . . . 12 contravariant vector . . . . 12 covariant . . . . . . . . . . . . . . . . . . . . . 8, 12 covariant component . . . . . . . . 19 covariant derivative . . . . . . . . . 30 covariant dierential . . . . . . 30 covariant vector . . . . . . . . . 19 curvature tensor . . . . . . . . 38 curvilinear coordinate system 3 D

metric tensor . . . . . . . . . . . . 5 Minkowski space . . . 6 N

natural basis . . . . . . . . . . . . . . . . . 4 O

oblique coordinate system . . 3 orthogonal coordinate system 3 orthogonal curvilinear coordinate system . . . . . . . . . . . . . . . . . . . . 3 orthonormal basis . . . . . . . 51 P

distance . . . . . . . . . . . . . . . . . . . . . . . . . 5 dual basis . . . . . . . . . . . . . . . . . . 17 E

physical component . . . . . . . . . 52 R

Einstein convention 4 Einstein tensor . 45 Euler equatio . . . . . . . . . 35 F forward transform . . . . . . . . . . . . . 7 forward transform coecient 8 G

raising . . . . . . . . . . . . . . . . . . . . . . . . . . 20 reverse transform . . . . . . . . . . . . . . 7 reverse transform coecient 8 Ricci formula . . . . . . . . . . . 42 Ricci tensor . . . . . . . . . . . 44 Riemanian geometry . . . 1 Riemann Christoel curvature tensor 38 S

geodesic line . . . . . . . . . . . . . . . . . . 34 I

inner product . . . . . . . . . . . . . . . . . . . . 5

scalar curvature . . . . . . . . . 45 scale factor . . . . . . . . . . . . . 51 spherical coordinate system . . 10, 59 square distance . . . . . . . . . . . . . . 5 square length . . . . . . . . . . . . . . . . . . 5

78 T

orthogonal coordinate system

tensor . . . . . . . . . . . . . . . . . . . . . . 13 transform . . . . . . . . . . . . . . . . . . . . . . . . 7 transform coecient . . . . . . . . . 7 translation . . . . . . . . . . . . . . . . . . 27 U

3Cartesian coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . 3 tensor . . . . . . . . . . . . . . . . . . . . . . 13 inner product . . . . . . . . . . . . . . . . . . . . length . . . . . . . . . . . . . . . . . . . . . . . . . . . square distance . . . . . . . . . . . . . . square length . . . . . . . . . . . . . . . . . .

unit vector . . . . . . . . . . . . . . . 3 Einstein convention

5 5 5 5

4Einstein tensor .

45Euler equatio . . . . . . . . . 35 reverse transform . . . . . . . . . . . . . . 7 reverse transform coecient 8 spherical coordinate system . .

10, 59covariant . . . . . . . . . . . . . . . . . . . . . 8, 12 covariant component . . . . . . . . 19 covariant derivative . . . . . . . . . 30 covariant dierential . . . . . . 30 covariant vector . . . . . . . . . 19 curvilinear coordinate system

anti-symmetricity . . . . . . . . . . . 43 contravariant . . . . . . . . . . . . . . . . . 8, 12 contravariant component . . . . 12 contravariant vector . . . . 12 Biannki equation . . 43, 45 physical component . . . . . . . . . 52 translation . . . . . . . . . . . . . . . . . . 27 transform . . . . . . . . . . . . . . . . . . . . . . . . 7 transform coecient . . . . . . . . . 7 Minkowski space . . . 6 Riemanian geometry . . . 1 Riemann Christoel curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . 38 Ricci tensor . . . . . . . . . . . 44 Ricci formula . . . . . . . . . . . 42

3curvature tensor . . . . . . . . 38 distance . . . . . . . . . . . . . . . . . . . . . . . . . 5 Christel symbol 27 Kronecker delta . . 6 metric tensor . . . . . . . . . . . . 5 lowering . . . . . . . . . . . . . . . . . . . . . . . . 20 commutation . . . . . . . . . . . . . 43 natural basis . . . . . . . . . . . . . . . . . 4 oblique coordinate system . . 3 forward transform . . . . . . . . . . . . . 7 forward transform coecient 8 raising . . . . . . . . . . . . . . . . . . . . . . . . . . 20 scalar curvature . . . . . . . . . 45 scale factor . . . . . . . . . . . . . 51 orthonormal basis . . . . . . . 51 dual basis . . . . . . . . . . . . . . . . . . 17 geodesic line . . . . . . . . . . . . . . . . . . 34 unit vector . . . . . . . . . . . . . . . 3 orthogonal curvilinear coordinate system . . . . . . . . . . . . . . 3