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نظرية النسبية العامة

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جلال الحاج عبد

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  • 1

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    laretalirdauq irehccaS

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    .

    ( ) :

    hnis hsoc hnis: 2 2

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  • 23

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    CBA tcefeD k S

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  • 33

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    :

    CBA tcefeD k SCBA2

    r r r S KC B A CBA

    i1 Ri k k K12 .

    : hnis 42

    2k Srk

  • 43

    L QP L P

    : L PQ YPQ QPX

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    nat 2

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    nat k2

    . e PY PX .

    . L k

    2 .

    ( ) k L

    .

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    k

    k

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  • 53

    meroehT legnE 1

    2

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  • 63

    .

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    ypx PRQxPQ RPQ

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    09 QRP . 081 QRP

    ) ( 063

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  • 73

    L : 2 L

    :

    m L QP P - QP

    L t L R - t SP P -

    QRSP m S SP m 063 )

    ( 081

    ) L m SP :

    (

  • 83

    L L : 3 L L

    : L

    C ACA CC BB AA BB . L )

    (

    A CC C AAAB B CB B BC C CB B BB AAAB B BA A CC BB

    .L C B A

  • 93

    (ledoM imartleB nielK ) R O ( )

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  • 04

    ) (

    ( )

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    : : B A BA

    gol ) (12

    BA dQB PAQA PB

    . ledoM racnioP . ( shpromosI )

    -

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  • 14

    2 =

    _ 1

  • 24

  • 34

    .

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    : elpicnirP lacigolomsoC . :

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    . o o

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  • 44

    ( )

    . " ." " "

    .

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  • 54

    . x1 x4 x1 ( ) x1 . x3 x2

    4 1 1nis soc4 1 4soc nis

    2 2

    3 3

    x x x

    x x x

    x x

    x x

    S t ( zyx) i1 tci x4

    : x tci4 x z3 x y2 x x1

    : .

    nis socsoc nis

    tci x xtci x tciy yz z

    : t S d zc yb xa0

    O

  • 64

    : t S atci d zc yb x a0 nis ) soc (

    S O xy d b a0 yxO d b a0 Oyx z= 0

    : S tci xnat

    .Ox tcinat Oyx

    S S tcinat Ox

    .t

    S : u S

    ci unat :

    ui natc

    2

    soc1nat 1

    :

    nat tci-

  • 74

    2

    2

    2

    2

    soc11

    nis1

    uc

    uicuc

    nis soc

    soc nistci x xtci x tciy yz z

    :

    2

    2

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    2

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    1

    1

    xtu xuc

    y yz z

    tc txuuc

    : c u

    t

    tu x xy yz zt

    tt .

    .

  • 84

    : :

    .

    x x1 S x : S x x2

    x x L1 2 x x1 S t

    : x x2

    22 2

    12

    xtu xuc

    2 11 12

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    : x x L1 2 S

    2

    12L Luc

    L L. L . . L

    .

  • 94

    :

    v

    2 12 vc

    .

    , S S t2 t1 . t2 t1 v S

    : 1

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    12

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    c t t t tv 12 2 1 2 12

    2

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  • 05

    : u( ) u

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    ( c u u )

    .

  • 15

    :

    2

    12

    mmuc

    m S mo .u S m S

    : (ygrene citenik) : xd F K . u

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    : .

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    md u udum u mdu udm um d xd um xdF Kxd du uu u utd td

    : u m

    2

    12

    mmuc

    :

  • 25

    2 2 2 2 2 22

    12

    mc m u m c m m

    uc

    : m2

    md c md u udum md um udu m md cm0 2 2 22 2 2 2 2

    :

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    0

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    : cm E2

    K c m E2

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    u1 c :

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    c c

    mm

    c m K c m c K c m c K

    :

  • 35

    (2 4 22

    2 1

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    c

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  • 45

  • 55

    Universe d Earth = Human Being

    Atom

    Galaxy

    =d

  • 65

  • 75

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  • 85

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    nis ) (

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    (x-x) .

    :

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    R xR yR z

  • 59

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    2

    cosh sin2

    x R

    y R

    z

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    sin cossin sin

    (cos ln tan )2

    x Ry R

    z R

    Helicoid

    cossin

    xyz R

  • 06

    .

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    2 1

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  • 16

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    ( knar)

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    . .

  • 26

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    . .

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  • 36

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    z y x

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  • 64

    Kronecker Delta :

    j k ki j i jk kij i ii j ja a k i ikb b

    k kij ji

    1 ( )2

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    g gggu u u

    1 ( )2

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    g ggu u u

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    10

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    if

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    i j i j

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    3 3 3 311 12 13 144 4 4411 12 14131 1

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    ij

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    41

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  • 56

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  • 66

    :

    soc socnis soc ) (

    nis socsoc nis ) (

    nis

    R xR f R f

    R yR h R h

    R z

    2 2 22 2 2

    2 2 22 122

    112 2 2 2 2 2soc22 11

    socR h f g

    xd g xd g sdR f g

    d R d R sd

    sd2 .d xd2 d xd1 . R2 R2

    soc nis ) ( soc nisnis nis

    ) nat nl soc() nis ( ) nat nl soc( ) (12nis 2

    R f R f R xR y

    R zR h R h

    d R d R sdnis toc2 2 2 2 2 2 2

    ij

    ji ud ud b II

    .

    : soc ) (

    nis ) (

    ) (

    v u f x

    v u f y

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  • 67

    :

    ' '' ' ''

    11 '2 '2

    012 21'

    22 '2 '2

    f h h fb

    f hb b

    f hb

    f h

    :

    11 1221 22

    b bb

    b b

    :

    cos coscos sinsin

    x Ry Rz R

    2 2 2cosII Rd R d

    sin cossin sin

    (cos ln tan )2

    x Ry R

    z R

    2 2cot sin cosII R d R d

  • 86

    :

    21 22 11

    21 22 11

    2

    2

    Kbb b bg g g g

    12 21 22 11

    12 21 22 11

    Kb b b b bg g g g g

    2 2 2

    22 11 211 1 2 2 2 121 21 22 11

    ] ) ( [1 1 12 2

    s r s rsr

    g Kg g g bu u u u u u g g

    21 111 22 1 21 2 11

    s r s r s s 1u u gg Ks r r

    : 2 2

    2 2 4

    1 socsoc

    KRR R

    : 2 2

    2 2 4

    1 socsoc

    KRR R

    :

    2 2 2

    00 01

    yd xd sdII

    K

    : 2

    2 2 2

    soc ) (nis) (

    ) () (

    v u f xK v u f yh h f f hh f f

    u h z

  • 96

    ( ) : . ( )

    .

    k2 k1 (erutavruc lapicnirp) . ( erutavruc lamron)

    : k k K2 1

    .k2 k1

    : k1< 0 k2> 0

    K < 0

    erutavruc_lapicnirP/ikiw/gro.aidepikiw.ne//:ptth:

  • 70

    1 ( )2

    jk ijkiijk i j k

    g ggu u u

    1 ( )2

    jm ijk km miij i j m

    g gggu u u

    012 21g g

    11 11 1111111 1 1 11

    21 12 1111121 1 1 22

    21 22 2212221 2 1 12

    12 11 1121112 1 2 21

    22 1121122 1 2

    1 1( )2 2

    1 1( )2 2

    1 1( )2 2

    1 1( )2 2

    1 (2

    ggg gu u uu

    gg g gu u uu

    gg g gu u uu

    gg g gu u uu

    gg gu u

    222 1

    22 22 2222222 2 2 22

    1) 2

    1 1( )2 2

    g

    u ugg g g

    u u uu

    11 22 12 21g g g g g :

    11 22

    11

    22 11

    22

    12 21 12 211212 21

    1

    1

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    gg

    g g

    gg

    g g

    g gg

    g g g g gg

    11 2111 21

    12 2212 22

    1g gg gg gg g

  • 71

    1 ( )2

    sj jki i is sksjk kj jk

    gg gguuu

    1 1111 1

    11

    2 1111 2

    22

    1 1112 2

    11

    2 2212 1

    22

    1 2222 1

    11

    2 2222 2

    22

    12

    121

    21

    21

    21

    2

    gg x

    gg x

    gg x

    gg x

    gg x

    gg x

    1 2A gdu du

    : 1 2

    4 2

    22 2 220 0

    cos

    cos cos 2

    A gdu du

    g R

    A R d d R d d R

    t=b t=a ,

    b i j

    ijas g du du

    2 jiijds g du du : 2 22 1 1 2 211 12 222ds g du g du du g du

  • 27

    p

    . .

    kg

    2

    2

    i kjk

    jin gr ud ud u d

    sdsd sd k

    :

    2 1

    2 2 1 22 1

    2 2

    j i j i g

    ji ji

    ud udgsd sd

    ud ud u d ud ud u dsd sd sd sd sd sd

    k

    P S C P

    : n t

    g nk k n ktdsd

    kg kn

    . P .k

  • 37

    ( ) . : . ( )

    2

    0 2i kj

    kji

    ud ud u dsdsd sd

    : s

    1ij

    jigxd xdsd sd

    .

    : -0 2 2 :

    u dksd

    . .

    ( setanidrooc cisedoeg-imeS)

    u G g) ( 22 g g0 12 21 g1 11j i 2

    ud G ud sd) ( ) (2 2 2 1 2 ud ud g sdji

    2

    1 1

    KG 1Gu u

    ( setanidrooc cirtemosI) 2

    g g0 12 21 g g22 11 ud ud sd) ( ) (2 2 2 2 1 2 2 ud ud g sdjij i 2

    : 2 2

    2 2 1 1 2

    K) (nl nl 1u u u u

  • 47

    zd z yd y xd x) , , ( z y x) , , ( wd w vd v ud u) , , ( w v u) , , (

    : w v u z z) , , ( w v u y y) , , ( w v u x x) , , (

    :wd vd ud xdx x xw v u wd vd ud ydy y yw v u wd vd ud zdz z zw v u

    : sd2

    xd sdzd yd 22 2 2

    : zd yd xd

    udA sdvdudH udwdG wdvdF wdC vdB2 2 2 22 2 2

    : .w v u) , , ( C B A

    ij 2 xd xd g sdji

    . xj xi gji .

  • 57

    xi P xi :

    x x x x xn i i) ,..., , (2 1

    P : x xdi i x xdi i

    ij ij

    xxd xdx

    :(.tnairav artnoc) : ij iA j

    x Ax

    xi Ai

    .

    ( tnairavni) :

    A A . A

    Bi (. tnairavoc) :

    jij i

    xB Bx

  • 67

    (xedni) xi xdi .

    . xi xi .

    N2 Bj Ai : B Aij

    l k j il k

    x xB A B Aix x j

    B Aj i N2 Bj Ai :

    k l i

    j kl jx xB A B Aix x

    Akji :

    r it s ik j r kjts

    x x xA Ax x x

    : Ai : A A Ai j g iji

    . :

  • 77

    ( ) jixd xd g sdij 2 .

    jig xd xdij sd2

    gji . :

    jig g0

    jiA gj: Aj : iA

    ji iA g Aj . Ai Ai

    : :i k i k

    kjk j A A A g g A gji ji

    ri irjk kj

    A g A

    si

    jig g g g gsr ir irjs

    jiA A g Aij 2

  • 87

    xA Axi ik k

    : xj

    k l k2

    k ij i l j i jk

    x x x x x xA x A x x A

    xAji .

    xAji xdi

    xxd AdAj iij ( Aj i, )

    . .

    . P P Ai

    . Ad Ai i P A Ai i : A Adi i

    ;j

    xd A A Adj i i i xdj Aj i; Aj i,

    (.; , ) Aj i; .

    Bi P P Ai : Y

    j

    ij iB Ayx

    j

    ij iA Bxy

  • 97

    ( B ) B0 Bi Ai :

    ) ( ) (j j

    i ij i j iB A B Ay yx x

    j 2

    kxd k ij i

    B Ayx x

    yA Bxij ij :

    l j 2k

    xd j k il iA Ax y

    y x x

    :

    l j 2lj k i ki

    x yy x x

    . :

    k l A Axdl ki i

    .

    :

    i k kj Bxd B ji

  • 08

    C Ai

    . Ai

    . P C Ai

    p . i . U kji Aj : U kji U

    il ikj

    xl kj

    : U P Aj

    l r j j A Alr

    U xi i:

    U V d xi i i:

    U Ai V

    : k i ij

    d A Akj

    I

    II

    III

  • 18

    . P lrj Ar Aj : l I III II

    k l ) (ir i i ij j jkj

    xA A A Ad llr kj kj

    .C

    ) (ik l kr i i ij jkj

    c cxd A d A Al lj kr kj

    : C . r j dk0

    c

    d dl k k lc c

    : 1

    (2

    dd k ll k lk)c

    :

    ) (ilk r i ij kj

    xA Al lj kr

    : l k

    ) (ikl r i ij lj

    xA Ak kj lr

  • 28

    : kl lk

    1

    2()

    i ikl r i r i ij kj lj

    x xA Al k kj lr lj kr

    l k ( cirtemmys-weks) :

    ()i ir i r ij kj lj

    x xA l k kj lr lj kr

    : Aj

    i ir i r i ikj lj

    x x Bl k kj lr lj kr lkj

    .

    : Rkj l i Blkji

    i B Rikj kj

    m i m i i i iu u Rl kkj lm lj km kj lj lkj

    m

    R g Rlkj mi lkji

  • 38

    : :

    ij1 j i

    gg

    xd xd g sdjiij 2

    :

    sd sd gxd xd jiij1 : 2

    0 2k ij

    ikj

    xd xd x dsdsd sd

    , , ks ki k ji j ki kjiji js Rs s

    x xkks ki kjijji js Rkis s ji

    R Rkji jik

    ij

    j i R g R

    1

    ji jiji2 R g R G

    12

    R g R Gji ji ji

    : . - . - . - (. ) -

  • 48

    -

    .

    .

    r - x t

    : . t z y1

    2

    3

    4

    r xxxt x

    xd xd g sdjiij 2 jig - .

    0 0 0

    0 0 0

    0 0 0

    0 0 0

    1122

    3344

    gg

    gg

    . jik -

  • 58

    . - . .

    kjiRs - . R2121 :

    . jiR - .

    . .

    .

    R - .

    .

    jiG - .

  • 68

    . :1

    :

    td t z y x D zd t z y x C yd t z y x B xd t z y x A sd) , , , ( ) , , , ( ) , , , ( ) , , , (2 2 2 2 2

    ) , , , (1) , , , ( 2

    ) , , , (1) , , , ( 2

    ) , , , (1) , , , ( 2

    ) , , , (1) , , , ( 2

    t z y xt z y x A

    t z y xt z y x B

    t z y xt z y x C

    t z y xt z y x D

    : 2 2 2 2 2

    121

    21

    21

    2

    tdD zdC ydB xdA sd

    A

    B

    C

    D

    : , , ,

    ) ( 12

    g g g g

    ( DAC htaM) -1 .

    : . . ptth//:www.debajahlalalaj.moc/lareneg_ytivitaler.dcm

  • 78

    :

    2

    2 1 21

    AAx

    BBx x

    (. ) y x2 x x1

    DACTAMBALTAM . . ELPAM

    .

    . g g0 12 21

    1 C2 322 D4 444 A 11 0 32

  • 88

    x

    Add

    y

    Add

    z

    Add

    tAd

    d

    y

    Add

    x

    Bdd

    0

    0

    z

    Add

    0

    x

    Cdd

    0

    tAd

    d

    0

    0

    x

    Ddd

    y

    Add

    x

    Bdd

    0

    0

    x

    Bdd

    y

    Bdd

    z

    Bdd

    tBd

    d

    0

    z

    Bdd

    y

    Cdd

    0

    0

    tBd

    d

    0

    y

    Ddd

    z

    Add

    0

    x

    Cdd

    0

    0

    z

    Bdd

    y

    Cdd

    0

    x

    Cdd

    y

    Cdd

    z

    Cdd

    tCd

    d

    0

    0

    tCd

    d

    z

    Ddd

    tAd

    d

    0

    0

    x

    Ddd

    0

    tBd

    d

    0

    y

    Ddd

    0

    0

    tCd

    d

    z

    Ddd

    x

    Ddd

    y

    Ddd

    z

    Ddd

    tDd

    d

    1 2 3 411 1 1 1 1 1 11 2 3 41 2 1 2 1 2 1 21 2 3 41 3 1 3 1 3 1 31 2 3 41 4 1 4 1 4 1 41 2 3 42 1 2 1 2 1 2 1

    1 2 3 42 2 2 2 2 2 2 21 2 3 42 3 2 3 2 3 2 31 2 3 42 4 2 4 2 4 2 41 2 3 431 3 1 31 3 1

    1 2 3 43 2 3 2 3 2 3 21 2 3 43 3 3 3 3 3 3 31 2 3 43 4 3 4 3 4 3 41 24 1 4 1

    3 44 1 4 1

    1 2 3 44 2 4 2 4 2 4 21 2 3 44 3 4 3 4 3 4 31 2 3 44 4 4 4 4 4 4 4

  • 89

    : k

    i j i jkR R

    jii j

    R g R

    R R :

    R11 w11 w12 w13 w14 w15 w16 w17

    w11 2yAd

    d

    2 2z A

    dd

    2 x y z t( ) 2t A

    dd

    2

    w12 2xBd

    d

    2 2x C

    dd

    2 2x D

    dd

    2

    w13 x

    Bdd

    2

    x

    Cdd

    2

    x

    Ddd

    2

    w14 x

    Add

    xBdd

    xCdd

    xDdd

    w15 y

    Add

    yAdd

    yBdd

    yCdd

    yDdd

    w16 z

    Add

    zAdd

    zBdd

    zCdd

    zDdd

    w17 tAd

    d

    tAdd

    tBdd

    tCdd

    tDdd

  • 90

    : R12

    x yCd

    d

    dd

    x yDdd

    dd

    xCdd

    yCdd

    xDdd

    yDdd

    y

    Add

    xCdd

    yAdd

    xDdd

    xBdd

    yCdd

    xBdd

    yDdd

    10

    1915 . .

    : , ,

    s ssj ijijk ik j ij k ik skR

    R g R

    R R R R

    0R R R

    1234 0R

    R121312 y z

    Add

    dd

    yAd

    d

    zAdd

    yAdd

    zBdd

    zAdd

    yCdd

    R1212

    2yAd

    d

    2 2x B

    dd

    2 xA

    dd

    xBdd

    yAdd

    2

    yA

    dd

    yBdd

    xBdd

    2

    zA

    dd

    zBdd

    tAdd

    tAdd

  • 91

    : :

    jii j

    R g R

    w12yAd

    d

    2 2xB

    dd

    2 yA

    dd

    2

    yBd

    d

    2

    xAd

    d

    xBdd

    yAdd

    yBdd

    zAdd

    zBdd

    tAdd

    tBdd

    w22zAd

    d

    2 2xC

    dd

    2 zA

    dd

    2

    xCd

    d

    2

    xAd

    d

    xCdd

    yAdd

    yCdd

    zAdd

    zCdd

    tAdd

    tCdd

    w32zBd

    d

    2 2y C

    dd

    2 zB

    dd

    2

    yCd

    d

    2

    xBd

    d

    xCdd

    yBdd

    yCdd

    zBdd

    zCdd

    tBdd

    tCdd

    w42tAd

    d

    2 2xD

    dd

    2 tA

    dd

    2

    xDd

    d

    2

    xAd

    d

    xDdd

    yAdd

    yDdd

    zAdd

    zDdd

    tAdd

    tDdd

    w52tBd

    d

    2 2y D

    dd

    2 tB

    dd

    2

    yDd

    d

    2

    xBd

    d

    xDdd

    yBdd

    yDdd

    zBdd

    zDdd

    tBdd

    tDdd

    w62tCd

    d

    2 2z D

    dd

    2 tC

    dd

    2

    zDd

    d

    2

    xCd

    d

    xDdd

    yCdd

    yDdd

    zCdd

    zDdd

    tCdd

    tDdd

    Ricci Scalar

    R 4 w1 4 w2 4 w3 4 w4 4 w5 4 w6

  • 92

    : 1

    2G R Rg

    :

    g1 2zBd

    d

    2

    2yCd

    d

    2

    zBd

    d

    2

    yCd

    d

    2

    xBd

    d

    xCdd

    yBdd

    yCdd

    zBdd

    zCdd

    tBdd tC

    dd

    g2 2tBd

    d

    2

    2yDd

    d

    2

    tBd

    d

    2

    yDd

    d

    2

    xBd

    d

    xDdd

    yBdd

    yDdd

    zBdd

    zDdd

    tBdd tD

    dd

    g3 2tCd

    d

    2

    2zDd

    d

    2

    tCd

    d

    2

    zDd

    d

    2

    xCd

    d

    xDdd

    yCdd

    yDdd

    zCdd

    zDdd

    tCdd tD

    dd

    G11

    g1

    g2

    g3

    ) .(

    K ( )ijkl ik jl il jkR K g g g g

    1212 11 22 12 21( )R K g g g g 1212R Kg

  • 39

    elpicnirP lacigolomsoC ehT

    .

    .

    . :

    . .

    . .

    ( ) 8491 .

    ! . . 06

    .

    .

    .

  • 49

    . .

    ( ) .

    . .

    .

    .

    .

  • 59

    . .

    . .

    : 2 2 2

    xd xdH xdE td xdB tdA sd3 2 1 1

    . x2 x1 t B A :

    d tdA sd2 2 2

    xd xd xd0 3 2 1 3 2 1 =i xd di2 2 d2 :

    tdA sd2 2

    2

    e c Ac 22 :

    2 2 2 2

    2d td e c sdc2

    .

  • 69

    2 2

    c12 2c

    ec01 5.0 225

    td c2 2

    : 2 2 2 2 2 2

    td c sdxd xd xd3 2 1

    :

    T X xhsoc Y y Z z T X thnis

    T thnat: x xX t2 2 2

    : T T1 hsoc hsoc2 2

    yd xd Ad y x F AF F ) , (y x

    TdT X TdXdT T X T Xd xd tdT X XdT xdhnis hnis hsoc 2 hsoc hnis hsoc2 2 2 2 2 2 TdT X TdXdT T X T Xd td tdT X XdT tdhsoc hnis hsoc 2 hnis hsoc hnis2 2 2 2 2 2

    : Xd Td X xd td2 2 2 2 2

    Zd Yd Xd Td X sd zd yd xd td sd2 2 2 2 2 2 2 2 2 2 2 Zd Yd Xd Td X sd2 2 2 2 2 2

    . T Z Y X) , , , (

  • 79

    :

    rrtd c d d r sdrdmm 1) 1( nis 222 2 2 2 2 2 22 :

    1 T R t r G c14

    m

    : 2 2 2 2 2 1 2 2

    2 2

    nis ) 1( ) 1(1 1R R

    d d R Rd Td sd

    1 2

    R : X t x R1 22 2 2

    2 2 22

    Rd Xd XdX Rd X R2 2 1 21X

    : Zd Yd Xd Td X sdZd Yd Rd R Td R sd)1 2( )1 2(2 2 2 1 2 2 2 2 2 2 2 2

    1 2

    R

    ) t x laksurK

    ( v u :

  • 89

    d d R xd td sdRnis eR2) ( 12 2 2 22 2 2 2

    : t x2 2 R

    t x R eR)1 2(2 2 2

    R0 R0 . t .

    ( ) t snocnat t snocnat . 54

    : R T

    2)1 2(22 2 2 2 2

    1 2

    RRd Xd t x R ee R

    R

    Xd Td X xd td2 2 2 2 2 :

    d d R RR Rd Td sdR Rnis 2 2) ( ) (1 2 1 22 2 2 2 2 1 2 2

    ( ) 1

    2 . . R

  • 99

    R0 g R R0

    . :

    d d r rd e td e sdB Anis2 2 2 2 2 2 2 2

    111 1 14 4 2

    A B A A RBr

    B A r e RB21 11 22

    2 R Rnis 22 33

    244

    1 1 14 4 2

    A B A A e RA B Ar

    R0 R R0

    : g R B A B A

    er Ar A1 ) 1(2 g R22 22

    eA r r r12

    11 223

    r mr

    m g R :

  • 001

    d d r rr rd r td r sdm mnis 3 31 11 2 1 22 2 2 2 2 2 2 2 21

    12 6

    r mr

    : m0 m

    d d r rd r td r sdnis 3 31 11 12 2 2 2 2 2 2 2 21

    7191 (rettis ed) 1 .

    3

    m0 . r3 . ( )

    g R .

    . 2391 neliM ( )

    O .

    . t z y x S) , , , ( .

    . tnorfevaw .

  • 101

    M4

    d d r rd td c sdnis2 2 2 2 2 2 2 2 :

    ur ( ) t

    hnis thsoc c rhnis

    :2

    12tuc

    2

    2

    1

    1u c

    uc

    : 2

    2 2 2 2 2 2 2 2 21) nis ( 2

    sdd d c d cd

    : d0 2

    2 2 2 2 2 2 21) nis ( 2

    d d c dd

    d2 K1 2 2

    1 c

    A A ) (A21

  • 102

    12 2K c Robertson-Walker metric

    R k

    2 2 24 4 1 1 2 2 3 3 4 4 1 1 2 2 3 3( ) ( )x dx x dx x dx x dx x dx x dx x dx x dx

    2

    2 1 1 2 2 3 34

    2 2 2 21 2 3

    1

    ( )

    k

    x dx x dx x dxdxR k x x x

    : I

    22 2 2 2 1 1 2 2 3 3

    1 2 32 2 2 2

    1 2 31

    ( )

    k

    x dx x dx x dxdl dx dx dxR k x x x

    : 1

    2

    3

    cos sinsin sincos

    x Rrx Rrx Rr

    0k 1k 2 2 2 2 21 2 3 4 1k

    x x x x R k 2 2 2 2 2

    1 2 3 4dl dx dx dx dx

    2 2 2 2 2 2 2 2 2 21 2 3 4 4 1 2 3

    1 1

    k kx x x x R k x R k x x x

    2 2 2 2 21 2 3 4dl dx dx dx dx

  • 301

    1

    2

    3

    nis nis soc soc nis socnis soc soc nis nis nisnis soc

    d rR d rR rd R xdd rR d rR rd R xdd rR rd R xd

    : I

    22 2 2 2 2 2 2

    1) nis (2d r d r R ldrd

    rk

    ld2

    ( suonegomoh) ld td c sd2 2 2 2 : ( ciportosi)

    : t R R) ( :

    22 2 2 2 2 2 2 2 2

    1) nis () (2d r d r t R td c sdrd

    rk

    -

  • 401

    . .

    .

    :

    G 4 2

    2 2 2

    z y xG 4 2 2 2

    G

    G 4 2

    : .

    1ji ji ji2

    T R g R

    . Tji .

  • 501

    : T R

    :) (1ji ji ji2

    T T g R

    :

    22 44

    R1c

    :

    2 44

    g12c

    r . : M

    MGr

    :

    2 44

    gMG 12cr

  • 601

    td cb d d r rda sd) nis (2 2 2 2 2 2 2 2

    c r b a

    : = 1x r = 2x = 3x = 4x t

    . Rji0

    .

    112

    222 2

    332

    44

    nis

    a gr gr gcb g

    r cba gnis2 4 21 11

    2 212

    3 312 2

    ni s

    4 412

    ga

    gr

    gr

    gc b

  • 701

    j i

    :

    111

    2

    2 2112 21

    3 3113 31

    4 414 41

    2

    122

    23 32toc 3 3

    33nis2 1

    33soc nis 22

    144

    2

    a

    a

    r

    rb

    b

    r

    a

    r

    a

    b c

    a

    r b a

    :

    kj ikji

    R R

  • 801

    :2

    44 22 112 22 221 1

    22 33nis24 4 22 44) (22

    Ra b a b bbra ba bRar braa ba

    R R

    c Rb b a b bara ba a

    : ij

    R g Rj i2

    2 2 2 2 22 2 2 2

    2 2Ra b b a b br ra r a b a barba ba

    : 2

    44 2202 220 1 1

    24 4 220

    a b a b bbra ba b

    ar braa bab b a b bara ba a

    : b a ba0

    ( ) a. b =

  • 901

    1 mil

    1 mil

    1

    ar

    br

    ba

    2 220 1 1 b ar braa ba

    a a ar) 1(

    1

    12a

    mr

    m r bm 12

    :

    I

    . r) , , ( .

    . m

    rr mtd c d d r sdm rd12) 1( nis22 2 2 2 2 2 22

  • 011

    c mMG2 - G - M - c -

    :

    td cb d d r rda sd) nis (2 2 2 2 2 2 2 2 II

    : II I r c cbm) 1(2 2 2

    : c mMG2

    21

    2MG

    br c

    c m rMG 222 b = 0 : r

    gk M01 2479.542 sgk Gm 01 82476.62 113

    c 854 297 992m s

  • 111

    9 9 .

    c mMG2 2

    2 22 2 22 2 2 2

    2 2rba b a ba bar ra r a Ra b b a b b

    r Rm3 4 r = 0008736

    m 01 8.612 32

    2rMG2

    c

  • 211

    .

    .

    . b a 1

    12a

    mr

    r bm 12

    111

    221

    331

    441

    1223322 133

    233

    213 44

    ) 2 (1

    1

    ) 2 () 2 (

    toc

    nis) 2 (

    soc nis

    ) 2 (

    mm r r

    r

    rm

    m r rm r

    m r

    m r cmr

  • 113

    2

    2 0j kid x dx dxi

    jkds ds ds

    :

    2

    23

    22 2 2 2

    2 ( ) ( 2 ) ( ) sin ( ) ( ) 0( 2 )d r m dr d d mc dtr m

    r r m ds ds ds r dsds

    22

    2

    2 sin cos ( ) 0d dr d dds r ds ds ds

    2

    2

    2 2cot 0d dr d d dds r ds ds ds ds

    2

    2

    2 0( 2 )

    d t m dr dtds r r m ds ds

    2

    2 2 2 2 2 2 2 22sin (1 )2

    1

    dr mds r d r d c dtrm

    r

    2

    0dds :

    2

    2

    2

    2

    22 2 2 2

    2 0

    2 0( 2 )

    ( ) ( ) ( 2 )( ) 12

    d dr dds r ds dsd t m dr dtds r r m ds ds

    r dr d c dtr r mr m ds ds r ds

  • 411

    wd sdtd

    sd ( )

    w wd0 2r rd

    20

    ) 2 (

    m d

    m r r rd

    : 2

    2

    wdr sd

    vr tdm r sd

    d sdtd

    sd1 ) () 2 ( ) ( ) (2 2 2 22

    2m r rtd c d rd r

    sd r sd sd m r :

    22 2 2

    3

    c m rm rd1 ) 2 ( ) (2r r sd

    r sd wd2 sd :

    2 22 2 2

    3 2 2

    cm m rd1 ) (2 2r r r d r

    u1 r

    : 2 2

    3 2 22 2

    um u um c ud2 ) (2 1d

    : 2

    23 2 2

    um um u d d

    :

  • 511

    2

    2 2

    uMG u d h d

    h ( ) M :

    h rd 2td

    r sd wd2

    r sd wd2 tdci sd :

    ci rd 2td

    : dum um u d3 2 22 2 ci h 2 2

    23 2 2

    um ucm u d h d

    ( )

    um32 2

    mMGc

    .

    : hcm22 um32 2 2

    2 22 2

    r u h3 3c c

    r . 01 7.78 84

    : h duMG u d2 2 2

  • 611

    he u) (soc 1 2

    : fo edutignol ) )yticirtnecce( e cm MG2

    (noilehirep

    h dum ucm u d3 2 22 2 2 :

    he umm) (soc 1 334 22 2

    : h h de um u d) (soc 1 34 2 22 2 2

    :

    2 2

    e ue m) (nis ) (soc 13h h

    2 2

    m3h :

    he u) (soc 1 2

    he u) (soc 1 2

    .

  • 711

    : 2

    2 2 2 2

    m3 3 3l c h c h

    :

    3 302 03 11

    2 2

    2

    2

    2 2 22

    2 2

    01 723.1 01 99.1 01 76.6

    1

    26 3

    1

    61

    nusgk MGm m

    s sgk

    lh

    a e h

    MG MGca e c h

    MGa e c

    : m a01 97.501 ( )

    e602.0 c 854 297 992m

    s

    naidar 01 577910.57:

    06 06 06301 6260.25 naidar 2

    : naidar naidar45301.0 01 577910.501 6260.2 75

  • 811

    52.563 88 :

    7924.0 45301.0 52.56388

    . 34

    .

    0. 34 3. 8 8. 3

  • 911

    2) 1( nis22 2 2 2 2 2 2 2

    21

    td c d r d r sdm rdmr

    r

    sd0 :

    2 22

    3 2 2um ucm u d h d

    : ( sd0 ) sd0 h

    22

    3 2um u u d d

    :

    u) (soc 1R

    . R 0

    2 . 0

    : 2

    22 2

    um u dsoc 3R d

    : ( Rmsoc 2 22 )

  • 021

    R R umsoc 2 soc 122

    : u0 m m0 soc soc2 2

    R R

    m

    R :

    m soc2R

    22

    mR

    : m4R

    : c mMG2

    2

    MG4R c

    :

    gk M01 99.103 c 854 297 992m

    s

    sgk Gm 01 76.62 113

    m R01 59.68

  • 121

    303 11

    26

    28

    01 99.1 01 76.6 457.1 01 5.8

    01 59.6 854 297 992m

    gk mnaidar sgk

    ms

    . 57.1 :

    . .

  • 221

    :

    .

    .

    ( )

    .

    .

    (tfihsder lanoitativarG)

    .

  • 321

    . A B B A

    . A B

    . D C B A D C B A

    : ds sd xd g sdiii 22

    x3 x2 x1 xi

    x4 ci

    .

    : xd xd xd0 3 2 1 td g c xd g sd44 442 2 2 22

    (x3 x2 x1 ) t s x4 ict

    :

    td g ci s44

    ds s s

    ci : t T T

    td g T44

    : g44

  • 421

    2 44

    g12c

    :2

    td Td12c

    . td2 td1 2 1 Td

    ctd Td122 . :

    2 12 22 1

    td td Td1 12 2c c

    :

    22

    1

    1 22

    12

    12

    c tdtd

    c

    at ( ) bt at bt

    :

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  • 169

    Wormhole Gravity Scalar product Gravitational potential Potential Geodesic Event - Bounded Kinetic Tensor calculus Perihelion Conservation Gravitational field Einstein field Tensor field Invariant field Vector field Field due to Absurd statement Property World- line Osculation circle Kronecker delta Curl Ricci Aberation of light Velocity Umbilical Saddle surface Catenoid Helicoid Pseudoplane Pseudosphere Electrical intensity Magnetic intensity Radiation Sirius

  • 071

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  • 171

    Absolute past Principle of equivalence General principle of relativity Special principle of relativy Machs principle Curvature vector Coaxial Metric Isogonal Equidistance Isoareal Bianchi identity Orthogonal Variable Isometric Equivalent Manifold - Controversial paraboloid Conform Conformal Local Paradox Contravariant Light cone Conoid Orbit Observer Order Component Rectilinear Flat Postulate Trajectory Derivative Assertion Equation of continuity

  • 172

    Tensor equation Coexistence Fallacy Asymptotic Parabolic Discriminant Curve rectifiable Catenary Minkowski Covariant Axiom System Ideal point Model Incidence geometry Non-Euclidean geometry Elliptic geometry Affine geometry Intrinstic geometry Parabolic geometry Pan - geometry Spherical geometry Absolute geometry Hyperbolic geometry Inertial frame Existential Unit Parametric

  • 371

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  • 174

    Component Compton effect Conform Conformal Congruence Connection Conoid Conservation Consistency Contraction - Contradiction Contradiction Contravariant Controversial Coordinate Coordinate surface Cosmical constant Covariant Curl Curvature Curvature tensor Curvature vector Curve rectifiable Defect Density Derivative Differential Dilation of time Dimension Discriminant Divergence Doppler effect Einstein field Einsteins law of gravitation Einsteins tensor Electrical intensity Element -

  • 175

    Elliptic Elliptic geometry Energy Equation of continuity Equidistance Equivalence Equivalence of mass Equivalent Euclidean space Even horizon Event - Evidence Excess Existential Fallacy Field due to First fundamental form Fitzgerald contraction Flat Force Fundamentalist Galactic mass Galilean law of inertia General principle of relativity Geodesic Geodesic curvature Gradient Gravitational acceleration Gravitational constant Gravitational field Gravitational lens Gravitational potential Gravitational red shift Graviton Gravity Helicoid Helix

  • 176

    Homology Hyperbolic geometry Hypersphere Ideal point Incidence geometry Inertial frame Infinite Interval Intrinstic geometry Invariant Invariant field Isoareal Isogonal Isometric Kinetic Kronecker delta Levi-Civita tensor Light cone Local Lorentz transformation Machs principle Magnetic intensity Manifold - Mapping Mass Mass Energy equivalence Mass inertial - Mass less Massive mass Metric Metric tensor Minkowski Model Momentum Non- Euclidean space Non-Euclidean geometry Observer

  • 177

    Orbit Order Orthogonal Osculation circle Pan - geometry Parabolic Parabolic geometry paraboloid Paradox Parallel displacement Parallel transpotr Parametric Perihelion Phenomenon Physical space Postulate Potential Principle of equivalence Property Pseudoplane Pseudosphere Quadratic Quantum Radiation Rectification Rectilinear Representation Ricci Ricci curvature Riemannian space Saddle surface Scalar curvature Scalar product Schwarzschild Second fundamental form simultaneity Sirius

  • 178

    Skew- symmetric skew-symmetric Space Space time - Space warp Special principle of relativy Spherical geometry Spherically symmetric Summation convention Superposition Synchronization Synchronization of clocks System tensor Tensor calculus Tensor equation Tensor field Torsion Trajectory Ultra ideal Umbilical Unit Universe Variable Vector field Velocity World- line Wormhole

  • 179

    - 1973 )- - ( . . )- - ( . . 1979

    1999 . .

    Essential Relativity:special, General and Cosmological, Wolfang Rindler, 1977

    Introduction to Special Relativity, Wiley Fastern, Private Limited. 1972 Lecture Notes on General Relativity, Sean M. Carrroll, Institute for

    Theoretical Physics University of California, 1997

    Introduction to General Relativity, G. t Hooft, Institute for Theoretical Physics University Utrech University, version 8/4/2002

    Euclidean and Non-Euclidean Geometries, Second edition, Marvin Jay Greenberg, W. H. Freeman & Co. , 1979.

    An Introduction to Tensor Calculus, Relativity and Cosomology, Derek F. Lawden, 1982.

    Non-Euclidian Geometry, Harold E. Wolfe. Introduction to Differential Geometry, Abraham Goets.

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  • 181

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  • 281

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