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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 :
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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