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(x, y, z) ∆s (∆x, ∆y, ∆z)
(∆s)
2
= (∆x)
2
+ (∆y)
2
+ (∆z)
2
,
ds
2
= dx
2
+ dy
2
+ dz
2
.
(z, r, ϕ) (r, θ, ϕ)
ds
2
= dz
2
+ dr
2
+ r
2
dϕ
2
,
ds
2
= dr
2
+ r
2
dθ
2
+ sen
2
θdϕ
2
.
(x
1
, x
2
, x
3
)
ds
2
=
3
i=1
3
j=1
g
ij
dx
i
dx
j
g
ij
diag(1, 1, 1) (x
1
, x
2
, x
3
) = (x, y, z) diag(1, 1, r
2
) (x
1
, x
2
, x
3
) = (z, r, ϕ)
diag(1, r
2
, r
2
sen
2
θ) (x
1
, x
2
, x
3
) = (r, θ, ϕ) g
ij
ds
2
= −c
2
dt
2
+ dx
2
+ dy
2
+ dz
2
η
ij
= diag(−1, 1, 1, 1) (ct, x, y, z) =
(x
0
, x
1
, x
2
, x
3
) 0 3
ds
2
= −e
2φ(l)
c
2
dt
2
+ dl
2
+ r(l)
2
dθ
2
+ sen
2
θdϕ
2
,
l ∈ (−∞, ∞) l
r(l) l = 0
l l
l(r) r(l)
l
+
(r)
l
−
(r) φ
+
(r)
φ
−
(r)
ds
2
= −e
2φ
±
(r)
c
2
dt
2
+
∂l
±
∂r
2
dr
2
+ r
2
dθ
2
+ sen
2
θdϕ
2
.
ds
2
= −e
2φ
±
(r)
c
2
dt
2
+
dr
2
1 −
b
±
(r)
r
+ r
2
dθ
2
+ sen
2
θdϕ
2
.
l r
dl
±
dr
= ±
1
1 −
b
±
(r)
r
⇒ l
±
(r) = ±
r
r
0
dr
1 −
b
±
(r
)
r
,
r
0
l
±
(r
0
) = 0
lim
r→∞
b
±
(r)/r = 0
φ
±
(r) = φ
±
r → ∞
r(l) dr/dl = 0
dl /dr → ∞
b
±
(r
0
) = r
0
r
0
b(r) r
∗
r ∈ (r
0
, r
∗
)
d
2
r
dl
2
=
d
dl
dr
dl
=
dr
dl
d
dr
dr
dl
=
1
2
d
dr
dr
dl
2
> 0.
dr
dl
= ±
1 −
b
±
r
d
2
r
dl
2
=
1
2r
b
±
r
− b
±
,
f(r) f
r b
±
(r) <
b
±
(r)
r
r ∈ (r
0
, r
∗
)
l = l(r)
∂l
±
∂r
=
dl
±
dr
b
±
(r
0
) = r
0
d
2
r
dl
2
r
0
=
1
2r
0
1 − b
+
(r
0
)
=
1
2r
0
1 − b
−
(r
0
)
⇒ b
+
(r
0
) = b
−
(r
0
).
b
±
(r
0
) ≤ 1
r
0
b(r
0
) = r
0
b
(r
0
) ≤ 1
r
0
r
0
[0, r
0
) r > r
0
r > r
0
[0, r
0
)
b(r) ≤ r r ≥ r
0
t
θ =
π/2 t
ds
2
=
dr
2
1 −
b
±
(r)
r
+ r
2
dϕ
2
.
ϕ
z r ϕ
ds
2
= dz
2
+ dr
2
+ r
2
dϕ
2
.
z
r
ds
2
=
1 +
dz
dr
2
dr
2
+ r
2
dϕ
2
.
(r, ϕ) z(r)
dz
dr
= ±
1
r
b(r)
− 1
.
b(r)
ds
2
= g
αβ
dx
α
dx
β
x
0
= ct x
1
= r x
2
= θ x
3
= ϕ
Γ
α
βγ
R
α
βγδ
Γ
α
βγ
=
1
2
g
αλ
(∂
γ
g
λβ
+ ∂
β
g
λγ
− ∂
λ
g
βγ
) ;
R
α
βγδ
= ∂
γ
Γ
α
βδ
− ∂
δ
Γ
α
βγ
+ Γ
α
λγ
Γ
λ
βδ
− Γ
α
λδ
Γ
λ
βγ
,
∂
α
≡
∂
∂x
α
0 3
R
0
101
=
e
−2φ
(
1−
b
r
)
R
1
001
= −φ

+
(b
r−b)
2r(r−b)
φ
− (φ
)
2
;
R
0
202
= r
2
e
−2φ
R
2
002
= −rφ
1 −
b
r
;
R
0
303
= r
2
e
−2φ
sen
2
θR
3
003
= −rφ
1 −
b
r
sen
2
θ;
R
1
212
= −r
2
1 −
b
r
R
2
112
=
(b
r−b)
2r
;
R
1
313
= −r
2
1 −
b
r
sen
2
θR
3
113
=
(b
r−b)sen
2
θ
2r
;
R
2
323
= sen
2
θR
3
232
=
b
r
sen
2
θ.
R
α
βγδ
= −R
α
βδγ
(e
0
, e
1
, e
2
, e
3
)
e
t
= e
−φ
e
0
, e
r
=
1 −
b
r
e
1
,
e
θ
=
1
r
e
2
, e
ϕ
=
1
rsenθ
e
3
.
e
a
α
e
α
b
= δ
a
b
a b
e
t
= e
φ
e
0
, e
r
=
1
√
1−
b
r
e
1
,
e
θ
= re
2
, e
ϕ
= rsenθe
3
.
T
T
α
T = T
α
e
α
T = T
a
e
a
T
a
= T e
a
= T
b
e
b
e
a
= T
b
δ
a
b
T
a
= T e
a
= T
α
e
α
e
a
= T
α
e
a
α
T
abc...
= T
αβγ...
e
a
α
e
b
β
e
c
γ
...
g
ab
= η
ab
≡
−1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
,
R
t
rtr
= R
r
ttr
=
1 −
b
r
−φ

+
(b
r−b)
2r(r−b)
φ
− (φ
)
2
,
R
t
θtθ
= R
θ
ttθ
= −
1 −
b
r
φ
r
,
R
t
ϕtϕ
= R
ϕ
ttϕ
= −
1 −
b
r
φ
r
,
R
r
θrθ
= R
θ
rθr
=
(b
r−b)
2r
3
,
R
r
ϕrϕ
= R
ϕ
rϕr
=
(b
r−b)
2r
3
,
R
θ
ϕθϕ
= R
ϕ
θϕθ
=
b
r
3
,
R
ab
R
R
ab
= R
c
acb
,
R = g
ab
R
ab
;
G
ab
= R
ab
−
1
2
Rg
ab
.
G
tt
=
b
r
2
;
G
rr
= −
b
r
3
+ 2
1 −
b
r
φ
r
;
G
θθ
= G
ϕϕ
=
1 −
b
r
φ

−
(b
r−b)
2r(r−b)
φ
+ (φ
)
2
+
φ
r
−
(b
r−b)
2r
2
(r−b)
.
10m/s
2
γ
s
(t) s γ
s
s
s t γ
s
(t)
T
a
= (∂/∂t)
a
X
a
= (∂/∂s)
a
X
a
T
a
v
a
= T
b
∇
b
X
a
a
a
= T
c
∇
c
v
a
= T
c
∇
c
T
b
∇
b
X
a
.
T
a
T
a
∇
a
T
b
= 0,
T
b
∇
b
X
a
= T
b
(∂
b
X
a
+ Γ
a
bc
X
c
) = T
b
∂
b
X
a
+ T
b
Γ
a
bc
X
c
.
∂
b
= ∂/∂x
b
T
b
∂
b
X
a
=
∂
∂t
b
∂
∂x
b
∂
∂s
a
=
∂
∂t
∂
∂s
a
=
∂
2
∂t∂s
a
=
∂
∂s
∂
∂t
a
=
∂
∂s
b
∂
∂x
b
∂
∂t
a
= X
b
∂
b
T
a
,
Γ
a
bc
= Γ
a
cb
T
b
Γ
a
bc
X
c
= X
c
Γ
a
cb
T
b
T
b
∇
b
X
a
= X
b
∂
b
T
a
+ X
b
Γ
a
bc
T
c
= X
b
(∂
b
T
a
+ Γ
a
bc
T
c
)
= X
b
∇
b
T
a
.
∇
a
∇
b
T
c
− ∇
b
∇
a
T
c
= −R
c
dba
T
d
.
a
a
= T
c
∇
c
X
b
∇
b
T
a
=
T
c
∇
c
X
b
(∇
b
T
a
) + X
b
T
c
∇
c
∇
b
T
a
=
T
c
∇
c
X
b
(∇
b
T
a
) + X
b
T
c
∇
b
∇
c
T
a
− R
a
dbc
X
b
T
c
T
d
= X
c
∇
c
T
b
∇
b
T
a
− R
a
dbc
X
b
T
c
T
d
= −R
a
dbc
X
b
T
c
T
d
X
a
T
a
b(r) φ(r)
X
a
X
a
X
0
= 0 T
a
T
a
= cδ
a
0
a
a
= −c
2
η
a
b
R
b
0
k
0
X
k
R
abcd
a
0
= 0 η
ab
a = 0
δ
b
a
a
a
= −c
2
R
a
0
k
0
X
k
.
e
0
= γe
t
± γ
v
c
e
r
, e
2
= e
θ
,
e
1
= γ
v
c
e
t
± γe
r
, e
3
= e
ϕ
,
v = v(r) r
γ ≡ [1 −v/c]
−1/2
+ −
e
1
R
a
0
k
0
R
1
0
1
0
= −
1 −
b
r
−φ

+
(b
r − b)
2r (r − b)
φ
− (φ
)
2
,
R
2
0
2
0
= R
3
0
3
0
=
γ
2
2r
2
v
c
2
b
−
b
r
+ 2 (r − b) φ
,
a
1
= −c
2
R
1
0
1
0
X
1
, a
2
= −c
2
R
2
0
2
0
X
2
a
3
= −c
2
R
3
0
3
0
X
3
.
|a| g ∼ 10 m/s
2
|X| ∼ 2 m
|R
1
0
1
0
| =
1 −
b
r
−φ

+
(b
r − b)
2r (r − b)
φ
− (φ
)
2
g
c
2
· 2 m
,
|R
2
0
2
0
| =
γ
2
2r
2
v
c
2
b
−
b
r
+ 2 (r − b) φ
g
c
2
· 2 m
,
10
−16
m
−2
l = −l
1
l = l
2
τ
ds
2
= −e
2φ
c
2
dt
2
+ dl
2
+ r(l)
2
dθ
2
+ sen
2
θdϕ
2
= −e
2φ
c
2
dt
2
+ dl
2
= −e
2φ
c
2
dt
2
1 −
dl
2
e
2φ
c
2
dt
2
= −e
2φ
c
2
dt
2
1 −
v
2
c
2
.
ds
2
= −c
2
dτ
2
−c
2
dτ
2
=
−e
2φ
c
2
dt
2
γ
2
⇒ γdτ = e
φ
dt,
v =
dl
e
φ
dt
=
dl
γdτ
.
∆t =
l
2
−l
1
dl
ve
φ
∆τ =
l
2
−l
1
dl
vγ
1
a
g
|a| g
u = u
a
e
a
= T
a
e
a
= ce
0
a
a
= u
β
∇
β
u
a
a · u = ca · e
0
= ca
0
= 0 a
0
= −a
0
= 0 a
2
= a
3
= 0
a = ae
1
a u
α
r a
0
= u
α
∇
α
u
0
= u
1
∂
1
u
0
− u
α
Γ
β
α0
u
β
u = ce
0
= c
γe
t
± γ
v
c
e
r
= c
γe
−φ
e
0
± γ
v
c
1 −
b
r
e
1
,
u
0
= cγe
−φ
, u
1
= ±cγ
v
c
1 −
b
r
, u
2
= u
3
= 0,
u
0
= −cγe
φ
, u
1
= ±cγ
v
c
1 −
b
r
−
1
2
, u
2
= u
3
= 0.
a
0
= ∓c
2
γ
v
c
1 −
b
r
γe
φ
,
a
0
= g
00
a
0
= g
00
ae
0
1
= (−e
2φ
)aγ (v/c) e
−φ
= −ae
φ
γ (v/c)
−ae
φ
γ
v
c
= a
0
= ∓c
2
γ
v
c
1 −
b
r
γe
φ
⇒ a = ±
1 −
b
r
γe
φ
c
2
e
−φ
= c
2
e
−φ
d
dl
γe
φ
,
|a| g
a = 0
a = c
2
e
−φ
d
dl
γe
φ
= 0
γe
φ
r
0
G
ab
=
8πG
c
4
T
ab
T
tt
= ˜ρ(r)c
2
T
rr
= ˜p(r) T
θθ
= T
ϕϕ
= ˜τ(r) ˜ρ(r)
˜p(r) ˜τ(r) θ
ϕ ρ(r) =
8πG
c
4
T
tt
p(r) =
8πG
c
4
T
rr
τ(r) =
8πG
c
4
T
θθ
ρ =
b
r
2
p = −
b
r
3
+ 2
1 −
b
r
φ
r
τ =
1 −
b
r
φ

+ φ
φ
+
1
r

−
(b
r − b)
2r
2
φ
+
1
r
b
= ρr
2
,
φ
=
b + pr
3
2r
2
1 −
b
r
p
= −(ρ + p) φ
+
2 (τ − p)
r
.
φ

b
ρ + p =
1
r
2
b
−
b
r
+ 2
1 −
b
r
φ
r
= −
1
r

1 −
b
r
− 2φ
1 −
b
r

=
= −
e
2φ
r
e
−2φ
1 −
b
r

.
e
−2φ(r
0
)
1 −
b(r
0
)
r
0

=0
= 0,
b(r) = 2GM/c
2
r
0
= 2GM/c
2
b
(r
0
) = 0 1
r (r
0
, r
∗
)
˜r
∗
(r
0
, ˜r
∗
)
∃˜r
∗
| ∀r ∈ (r
0
, ˜r
∗
) ,
e
−2φ
1 −
b
r

> 0.
ρ + p < 0.
b(r
0
) = r
0
b
0
= b
(r
0
) 1,
ρ
0
= ρ(r
0
)
1
r
2
0
.
b
r
(2φ
r + 1) = 2φ
r − pr
2
∴ b(r
0
) = r
0
⇒ p
0
= p(r
0
) = −
1
r
2
0
.
ρ
0
+ p
0
0.
ρ + 3p 0
ξφ
2
R
φ R
ξ = 0
ξ = 0
r = 0
τ
ξ
a
ξ
a
ξ
a
= −ξ < 0
B
ab
B
ab
= ∇
b
ξ
a
.
ξ
a
B
ab
ξ
a
= ξ
a
∇
b
ξ
a
=
1
2
∇(ξ
a
ξ
a
) = 0,
B
ab
ξ
b
= ξ
b
∇
b
ξ
a
= 0,
γ
s
(τ) η
a
γ
0
ξ
b
∇
b
η
a
= η
b
∇
b
ξ
a
= B
a
b
η
b
.
B
a
b
η
a
γ
0
B
a
b
h
ab
h
ab
= g
ab
+ ξ
a
ξ
b
,
h
a
b
ξ
a
θ = B
ab
h
ab
,
σ
ab
= B
(ab)
−
1
3
θh
ab
,
ω
ab
= B
[ab]
,
B
(ab)
B
[ab]
B
ab
B
ab
= B
(ab)
+ B
[ab]
B
(ab)
=
1
2
(B
ab
+ B
ba
) ,
B
[ab]
=
1
2
(B
ab
− B
ba
) .
σ
ab
ω
ab
ξ
b
σ
ab
ξ
b
= ω
ab
ξ
b
= 0
B
ab
B
ab
=
1
3
θh
ab
+ σ
ab
+ ω
ab
B
a
b
η
b
=
1
3
θη
a
+ σ
a
b
η
b
+ ω
a
b
η
b
.
θ
ω
ab
B
a
b
σ
ab
ξ
c
∇
c
B
ab
= ξ
c
∇
c
∇
b
ξ
a
= ξ
c
∇
b
∇
c
ξ
a
+ R
d
abc
ξ
c
ξ
d
= ∇
b
(ξ
c
∇
c
ξ
a
) − (∇
b
ξ
c
) (∇
c
ξ
a
) + R
d
abc
ξ
c
ξ
d
= −B
c
b
B
ac
+ R
d
abc
ξ
c
ξ
d
,
ξ
a
∇
a
θ =
dθ
dτ
= −
1
3
θ
2
− σ
ab
σ
ab
+ ω
ab
ω
ab
− R
ab
ξ
a
ξ
b
,
T k = 8πG/c
4
G
ab
= R
ab
−
1
2
Rg
ab
= kT
ab
⇒ R = −kT ∴ R
ab
= k
T
ab
−
1
2
g
ab
T
,
R
ab
ξ
a
ξ
b
= k
T
ab
−
1
2
g
ab
T
ξ
a
ξ
b
= k
T
ab
ξ
a
ξ
b
+
1
2
T ξ
.
T
ab
ξ
a
ξ
b
ξ
a
T
ab
ξ
a
ξ
b
0
ξ
a
T
ab
ξ
a
ξ
b
+
1
2
T ξ 0.
T
ab
= ρe
t
a
e
t
b
+ p
1
e
x
a
e
x
b
+ p
2
e
y
a
e
y
b
+ p
3
e
z
a
e
z
b
,
{e
t
, e
x
, e
y
, e
z
} e
t
ρ
p
1
p
2
p
3
T
ab
T
ab
ξ
a
ξ
b
= ρ
e
t
a
ξ
a
2
+ p
1
(e
x
a
ξ
a
)
2
+ p
2
(e
y
a
ξ
a
)
2
+ p
3
(e
z
a
ξ
a
)
2
= ρ
ξ
t
2
+ p
1
(ξ
x
)
2
+ p
2
(ξ
y
)
2
+ p
3
(ξ
z
)
2
0.
ξ
t
ξ
t
+ ξ
x
ξ
x
+ ξ
y
ξ
y
+ ξ
z
ξ
z
= −ξ
η
ab
−
ξ
t
2
+ (ξ
x
)
2
+ (ξ
y
)
2
+ (ξ
z
)
2
= −ξ
∴
ξ
t
2
= ξ + (ξ
x
)
2
+ (ξ
y
)
2
+ (ξ
z
)
2
.
ρξ + (ρ + p
1
) (ξ
x
)
2
+ (ρ + p
2
) (ξ
y
)
2
+ (ρ + p
3
) (ξ
z
)
2
0 ∀ (ξ
x
, ξ
y
, ξ
z
) ∈ R
3
.
(ξ
x
, ξ
y
, ξ
z
) = (0, 0, 0) ρ 0
(ρ + p
i
) < 0 (i = 1, 2, 3)
(ρ + p
i
) 0 (i = 1, 2, 3)
(ξ
y
, ξ
z
) = (0, 0)
ρξ + (ρ + p
1
) (ξ
x
)
2
0 ⇒ (ρ + p
1
) −
ρξ
(ξ
x
)
2
ξ
x
(ρ + p
1
) 0
ρ 0 ρ + p
i
0 (i = 1, 2, 3).
T = −ρ + p
1
+ p
2
+ p
3
ξ
2
ρ +
3
i=1
p
i
+ (ρ + p
1
) (ξ
x
)
2
+ (ρ + p
2
) (ξ
y
)
2
+ (ρ + p
3
) (ξ
z
)
2
0 ∀ (ξ
x
, ξ
y
, ξ
z
) ∈ R
3
.
ρ +
3
i=1
p
i
0 ρ + p
i
0 (i = 1, 2, 3).
ω
ab
= 0
−σ
ab
σ
ab
σ
ab
ξ
a
dθ
dτ
+
1
3
θ
2
0 ∴ −
d
dτ
θ
−1
+
1
3
0
d
dτ
θ
−1
1
3
⇒ θ
−1
(τ) θ
−1
0
+
1
3
τ.
θ
0
< 0
θ
−1
τ 3/ |θ
0
|
θ → −∞
ω
ab
= 0
θ
dθ
dτ
−
1
3
θ
2
< 0,
dθ/dτ > 0
ds
2
=
dτ
2
− a
2
(τ) [dψ
2
+ sen
2
ψ (dθ
2
+ sen
2
θdϕ
2
)] ,
dτ
2
− a
2
(τ) [dψ
2
+ ψ
2
(dθ
2
+ sen
2
θdϕ
2
)] ,
dτ
2
− a
2
(τ) [dψ
2
+ senh
2
ψ (dθ
2
+ sen
2
θdϕ
2
)] ,
a(τ)
a(τ)
η
ab
G
ττ
= 3
(˙a
2
+ K)
a
2
,
G
rr
= G
θθ
= G
ϕϕ
= −
˙a
2
+ 2a¨a + K
a
2
,
K = 0 1 −1
τ
T
ab
= ˜ρu
a
u
b
− ˜p(g
ab
− u
a
u
b
),
u
u
a
= cδ
0
a
3
(˙a
2
+ K)
a
2
= ρ,
−
˙a
2
+ K
a
2
−
2¨a
a
= p.
¨a
a
= −
1
6
(ρ + 3p) .
ρ + 3p < 0 a(τ)
ρ + p =
2
a
2
K + ˙a
2
− ¨aa
=
2
a
2
B,
B < 0
a(τ) ¨a 0
B = K − ¨aa a(τ)
K = −1
˙a → 0 ¨a > 0
K = 0
a
0
a(τ) = a
0
+ bτ
2n
+ dτ
2n+1
+ eτ
2n+2
,
n 1 B(τ)
B(τ) = K −2n(2n−1)a
0
bτ
2n−2
−2n(2n+1)a
0
dτ
2n−1
−2(n+1)(2n+1)a
0
eτ
2n
+. . .
τ → 0 B(τ ) ∼ K − 2n(2n − 1)a
0
bτ
2n−2
B
K = 1
L = −
1
4
F + mF
2
+ nG
2
,
F := F
µν
F
µν
,
G :=
1
2
η
αβµν
F
αβ
F
µν
.
m n η
αβµν
L g
µν
T
µν
= −4
∂L
∂F
F
α
µ
F
αν
+
G
∂L
∂G
− L
g
µν
.
E
H
E
i
= 0, H
i
= 0, E
i
H
j
= 0,
E
i
E
j
= −
1
3
E
2
g
ij
,
H
i
H
j
= −
1
3
H
2
g
ij
.
A ≡ lim
V →V
0
1
V
A
√
−gd
3
x
i
;
A V =
√
−gd
3
x
i
V
0
H
2
E
2
= 0
ρ = k
H
2
2
1 − 8mH
2
,
p = k
H
2
6
1 − 40mH
2
.
m > 0
m
H
H = H(r)
p(r) = τ(r)
b
= ρr
2
;
φ
=
b + pr
3
2r
2
1 −
b
r
;
p
= −(ρ + p) φ
.
φ
b
= ρr
2
;
p
= −(ρ + p)
(
b+pr
3
)
2r
2
(
1−
b
r
)
.
p
r
0
p
0
= p
(r
0
)
p
= −
(ρ + p)
2r
·
(b + pr
3
)
(r − b)
⇒ p
0
= −
(ρ
0
+ p
0
)
2r
0
·
(b + pr
3
)
r
0
(r − b)
r
0
=
= −
(ρ
0
+ p
0
)
2r
0
·
(b
0
+ 3p
0
r
2
0
+ p
0
r
3
0
)
(1 − b
0
)
.
p
0
=
(1 − b
0
)
2r
3
0
·
(b
0
− 3 + p
0
r
3
0
)
(1 − b
0
)
=
b
0
− 3
2r
3
0
+
p
0
2
⇒ p
0
=
b
0
− 3
r
3
0
.
b
0
1 p
0
< 0
X(r) = kH
2
(r) α = m/k X
0
= X(r
0
)
X
0
= X
(r
0
)
X
0
6
(1 − 40αX
0
) = −
1
r
2
o
⇒ X
0
=
1 +
√
1 + γ
80α
, γ =
960α
r
2
0
,
X(r)
2X
0
3
(1 − 16αX
0
) 0 ⇒ X
0
1
16α
.
p
p
=
X
6
(1 − 80αX) .
p
0
< 0
X
0
6
(1 − 80αX
0
) < 0.
X
0
1 − 80αX
0
= −
√
1 + γ < 0
γ > 0
X
0
> 0.
X(r) ∞ b(r) p(r) ρ(r)
−∞
X
=
X (16αX − 1)
40αX
2
r − Xr − 6
b
r
2
1 −
b
r
(80αX − 1)
.
X
0
1/16α > 0 X
0
> 0 X(r)
X(r
m
) > X
0
X
(r
m
) = 0 r
m
> r
0
X
m
= X(r
m
)
X
b(r)/r
X
m
= 0;
16αX
m
− 1 = 0;
40αX
2
m
r
m
− X
m
r
m
− 6
b
m
r
2
m
= 0.
X
m
= 0 X
0
X
0
X
m
=
1
16α
X
0
,
X
m
=
1 +
1 + 960α
b
m
r
3
m
80α
=
1 +
1 + γ
r
2
0
b
m
r
3
m
80α
.
r
m
> r
0
b
m
= b(r
m
) r
m
r
0
< r
m
⇒
r
2
0
r
2
m
< 1, b
m
r
m
⇒
b
m
r
m
1 ⇒
r
2
0
r
2
m
b
m
r
m
r
2
0
r
2
m
< 1 ∴
∴
r
2
0
b
m
r
3
m
< 1.
X
m
X
0
1 + γ
r
2
0
b
m
r
3
m
< 1 + γ ⇔
1 + γ
r
2
0
b
m
r
3
m
<
1 + γ
∴ X
m
=
1 +
1 + γ
r
2
0
b
m
r
3
m
80α
1 +
√
1 + γ
80α
= X
0
X
X X
0
X
(r
m
) = 0 r
m
→ ∞
lim
r→∞
X(r) X
0
X
b(r) −∞
b(r) = 2GM/c
2
M
b(r)
φ
(r) −∞ φ(r)
|b(r)|/r → ∞ X(r)
p(r) ρ(r)
z(r)
0 1 2 3 4
5 6
7
8
r
-5
0
5
z(r)
z(r) × r r
0
= 4 m z r
7, 5 m
∇
λ
g
µν
= ω
λ
g
µν
,
ω
λ
= ω
λ
(x)
ω
λ
= ∂
λ
ω(x)
Γ
α
µν
∇
ν
g
λµ
= ∂
ν
g
λµ
− Γ
β
νλ
g
βµ
− Γ
β
νµ
g
λβ
= ω
ν
g
λµ
,
∇
µ
g
νλ
= ∂
µ
g
νλ
− Γ
β
µν
g
βλ
− Γ
β
µλ
g
νβ
= ω
µ
g
νλ
,
∇
λ
g
µν
= ∂
λ
g
µν
− Γ
β
λµ
g
βν
− Γ
β
λν
g
µβ
= ω
λ
g
µν
.
Γ
β
µν
g
βλ
=
1
2
(∂
µ
g
λν
+ ∂
ν
g
λµ
− ∂
λ
g
µν
) −
1
2
(ω
µ
g
λν
+ ω
ν
g
λµ
− ω
λ
g
µν
)
g
λα
Γ
α
µν
=
1
2
g
αλ
(∂
µ
g
λν
+ ∂
ν
g
λµ
− ∂
λ
g
µν
) −
1
2
ω
µ
δ
α
ν
+ ω
ν
δ
α
µ
− g
µν
ω
α
=
ˆ
Γ
α
µν
−
1
2
ω
µ
δ
α
ν
+
ω
ν
δ
α
µ
−
g
µν
ω
α
.
R
µν
=
ˆ
R
µν
+
3
2
ˆ
∇
ν
ω
µ
−
1
2
ˆ
∇
µ
ω
ν
+
1
2
ω
µ
ω
ν
+
1
2
g
µν
ˆ
∇
α
ω
α
− ω
α
ω
α
.
ω
µ
= ∂
µ
ω =
ˆ
∇
µ
ω,
ˆ
∇
ν
ˆ
∇
µ
ω =
ˆ
∇
ν
(∂
µ
ω) = ∂
ν
∂
µ
ω −
ˆ
Γ
α
νµ
∂
α
ω
= ∂
µ
∂
ν
ω −
ˆ
Γ
α
µν
∂
α
ω =
ˆ
∇
µ
(∂
ν
ω)
=
ˆ
∇
µ
ˆ
∇
ν
ω,
ˆ
∇
α
ω
α
=
ˆ
∇
α
ˆ
∇
α
ω =
ˆ
ω,
R
µν
=
ˆ
R
µν
+
ˆ
∇
ν
ω
µ
+
1
2
ω
µ
ω
ν
+
1
2
g
µν
ˆ
ω − ω
α
ω
α
.
R =
ˆ
R +
ˆ
ω +
1
2
ω
α
ω
α
+ 2
ˆ
ω − ω
α
ω
α
=
ˆ
R + 3
ˆ
ω −
3
2
ω
α
ω
α
.
S =
√
−g (R + ξ∇
α
ω
α
) ,
ξ
(g
µν
, ω)
R
µν
−
1
2
Rg
µν
−
ˆ
∇
ν
ω
µ
+ 2 (ξ − 1) ω
µ
ω
ν
−
ξ −
1
2
ω
α
ω
α
g
µν
= 0,
ˆ
ω =
1
√
−g
∂
β
√
−gg
αβ
∂
α
ω
= 0,
g R
µν
R
ˆ
G
µν
+ λ
2
ω
µ
ω
ν
−
λ
2
2
ω
α
ω
α
g
µν
= 0,
ˆ
ω = 0,
λ
2
= (4ξ − 3)/2
ω(x)
T
µν
=
λ
2
k
1
2
ω
α
ω
α
g
µν
− ω
µ
ω
ν
,
k
ω ω = ω(r )
ω
α
= ∂
α
ω(r) = ω
δ
1
α
, ω
α
= g
αβ
ω
β
= g
α1
ω
= g
11
ω
δ
α
1
,
ω
α
ω
α
= (ω
)
2
g
11
δ
1
α
δ
α
1
=
1 −
b
r
(ω
)
2
.
T
µν
=
λ
2
k
1
2
1 −
b
r
(ω
)
2
g
µν
− (ω
)
2
δ
1
µ
δ
1
ν
,
T
00
= −
λ
2
2k
1 −
b
r
(ω
)
2
e
2φ
, T
11
= −
λ
2
2k
(ω
)
2
,
T
22
=
λ
2
2k
1 −
b
r
(ω
)
2
r
2
, T
33
=
λ
2
2k
1 −
b
r
(ω
)
2
r
2
sen
2
θ.
T
tt
= T
rr
= −T
θθ
= −T
ϕϕ
= −
λ
2
2k
1 −
b
r
(ω
)
2
,
ρ = p = −τ = −
λ
2
2
1 −
b
r
(ω
)
2
.
b
= −
λ
2
2
1 −
b
r
(ω
)
2
r
2
,
φ
=
b −
r
3
λ
2
2
1 −
b
r
(ω
)
2
2r (r − b)
,
1
2
1 −
b
r
(ω
)
2
= −
1 −
b
r
(ω
)
2
φ
−
2
r
1 −
b
r
(ω
)
2
.

1 −
b
r
(ω
)
2
= 2
1 −
b
r
(ω
)
2
−φ
−
2
r
= 2r
2
e
φ
1 −
b
r
(ω
)
2
−
φ
r
2
e
φ
−
2
r
3
e
φ
= 2r
2
e
φ
1 −
b
r
(ω
)
2
1
r
2
e
φ
= −
2r
2
e
φ
r
2
e
φ
(r
2
e
φ
)
2
1 −
b
r
(ω
)
2
= −
r
2
e
φ
2
(r
2
e
φ
)
2
1 −
b
r
(ω
)
2
,
r
2
e
φ
2

1 −
b
r
(ω
)
2
+
r
2
e
φ
2
1 −
b
r
(ω
)
2
=
r
2
e
φ
2
1 −
b
r
(ω
)
2
= 0
∴ r
4
e
2φ
1 −
b
r
(ω
)
2
=
A
2
λ
2
,
b
= −
A
2
2r
2
e
2φ
,
φ
=
1
2r (r − b)
b −
A
2
2re
2φ
.
φ
= 0 ⇒ φ(r) = φ
0
b(r) =
A
2
2re
2φ
0
,
b(r
0
) = r
0
A = r
0
e
φ
0
√
2
b
(r
0
) = −1
ω
=
r
0
√
2
r
r
2
− r
2
0
⇒ ω(r) =
√
2 cot
−1
r
0
r
2
− r
2
0
+ ω(r
0
).
ρ(r) = p(r) = −τ(r) = −
r
2
0
r
4
,
b(r) =
r
2
0
r
φ(r) = φ
0
.
ds
2
= −e
2φ
0
dt
2
+
dr
2
1 −
r
2
0
r
2
+ r
2
dΩ
2
,
dΩ
2
= dθ
2
+ sen
2
θdϕ
2
l(r) = ±
r
2
− r
2
0
⇒ r
2
(l) = l
2
+ r
2
0
,
ds
2
= −e
2φ
0
dt
2
+ dl
2
+
l
2
+ r
2
0
dΩ
2
.
φ
b(r)
g
11
(1 − b/r)
l
1
= l
+
(r
1
) = l
2
= −l
−
(r
2
) = l r
1
= r
2
= r
d =
b(r)
r
=
r
2
0
r
2
10
−2
⇒
√
d 10
−1
l ≈ r.
γ
φ v(r)
γ
2
2r
2
v
c
2
b
−
b
r
= γ
2
v
c
2
r
2
0
r
4
1
10
16
m
−2
.
r
0
γ ≈ 1
v
c
2
1
r
2
0
1
10
16
m
−2
∴ v 3r
0
s
−1
.
v = 3r
0
α s
−1
α 1
∆τ ≈ e
φ
0
∆t ≈
1
v
l
2
−l
1
dl ≈
2r
v
T,
T
r
vT
2
,
r v
10
4
m/s
r 1, 6 ·10
11
m
r
3r
0
αT
2
s
−1
⇒ α
2r
3r
0
T
s =
2
3T
√
d
s,
α 1
2
3T
√
d
s 1 ⇒
√
d
2
3T
s ≈ 2, 1 · 10
−8
∴ 2, 1 · 10
−8
√
d 10
−1
r
0
= r
√
d
2, 1 · 10
−8
r r
0
10
−1
r.
10
4
m/s
φ
0
0
φ
0
r
0
v/3
r
0
= 7 ·10
8
m
r = 1, 5 ·10
11
m d = 2, 5 ·10
−5
v
2r
T
≈ 10
4
m/s,
∆τ
φ
0
r
4
10
9
10
9
1
10
−1
90 8765
z(r)
2
3
0
−2
21
z(r) ×r r
0
= 7 ·10
8
m
z(r) z(r
0
) = 0
z(r) = ±r
0
ln
r
r
0
+
r
2
r
2
0
− 1
r
0
RW
µ
W
ν
g
µν
R
µν
W
µ
W
ν
W
µ
−12
−10
−8
−6
−4
−2
−6
−5
−4
−3
−2
−6
−4
−1
−2
10
9
0
00
2
1
4
6
2
8
10
3
4
5
6
2
4
6
8
10
12
10
9
r
0
= 7 ·10
8
m
L
acop.
=
√
−gRW
µ
W
ν
g
µν
,
L = L
A
+ L
B
+ L
C
,
L
A
=
1
k
√
−g (1 + λW
µ
W
µ
) R,
L
B
= −
1
2
√
−gF
µν
F
µν
L
C
F
µν
= ∇
ν
W
µ
− ∇
µ
W
ν
λ
k R
g
µν
1 + λW
2
G
µν
+ λW
2
g
µν
− λ∇
ν
∂
µ
W
2
+ λRW
µ
W
ν
= kE
µν
+ kT
µν
,
T
µν
L
C
W
2
= W
µ
W
µ
E
µν
E
µν
= F
µα
F
α
ν
+
1
4
g
µν
F
αβ
F
αβ
.
R = −kT + 3 λW
2
,
T T
µν
W
µ
∇
ν
F
µν
= −
λ
k
RW
µ
+ J
µ
,
J
µ
∇
µ
J
µ
−
λ
k
∇
µ
(RW
µ
) = 0.
J
µ
= 0
R = 0,
W
2
= 0.
Ω ≡ 1 + λW
2
G
µν
=
∇
ν
∂
µ
Ω
Ω
Ω = 0.
Ω = Ω(r) ∂
µ
Ω =
Ω
δ
1
µ
∇
ν
∂
µ
Ω = ∂
ν
∂
µ
Ω − Γ
α
νµ
∂
α
Ω
= Ω

δ
1
µ
δ
1
ν
− Γ
1
µν
Ω
g
µν
= g
µν
(r)
Γ
1
µν
=
1
2
g
11
g
1µ
δ
1
ν
+ g
1ν
δ
1
µ
− g
µν
,
∇
0
∂
0
Ω = −e
2φ
1 −
b
r
φ
Ω
, ∇
1
∂
1
Ω = Ω

+
(b − rb
)
2r
2
1 −
b
r
Ω
,
∇
2
∂
2
Ω = r
1 −
b
r
Ω
, ∇
3
∂
3
Ω = rsen
2
θ
1 −
b
r
Ω
.
∇
t
∂
t
Ω = −
1 −
b
r
φ
Ω
, ∇
r
∂
r
Ω =
1 −
b
r
Ω

+
(b − rb
)
2r
2
Ω
,
∇
θ
∂
θ
Ω = ∇
ϕ
∂
ϕ
Ω =
1 −
b
r
Ω
r
.
Ω
Ω = ∇
A
∂
A
Ω =
1 −
b
r
φ
Ω
+
1 −
b
r
Ω

+
(b − rb
)
2r
2
Ω
+ 2
1 −
b
r
Ω
r
=
1 −
b
r
r
2
e
φ
1 −
b
r
r
2
e
φ
φ
Ω
+ r
2
e
φ
Ω

+ 2re
φ
Ω
+
r
2
e
φ
2
1 −
b
r
b
r
2
−
b
r
Ω
=
1 −
b
r
r
2
e
φ
1 −
b
r
r
2
e
φ
Ω
+ r
2
e
φ
(Ω
)
+
r
2
e
φ
Ω
+ r
2
e
φ
1 −
b
r
Ω
=
1 −
b
r
r
2
e
φ
r
2
e
φ
1 −
b
r
Ω
= 0,
r
2
e
φ
1 −
b
r
Ω
= A,
∇
r
∂
r
Ω =
1 −
b
r
1 −
b
r
Ω

+
b
r
2
−
b
r
Ω
2
1 −
b
r
=
1 −
b
r
1 −
b
r
Ω
= A
1 −
b
r
1
r
2
e
φ
,
T
ab
=
1
k
∇
b
∂
a
Ω
Ω
ρ = −
1 −
b
r
φ
Ω
Ω
,
p =
A
Ω
1 −
b
r
1
r
2
e
φ
,
τ =
1 −
b
r
Ω
rΩ
,
−
1 −
b
r
φ
Ω
Ω
=
b
r
2
,
A
Ω
1 −
b
r
1
r
2
e
φ
= −
b
r
3
+ 2
1 −
b
r
φ
r
,
1 −
b
r
Ω
rΩ
=
1 −
b
r
φ

+ φ
φ
+
1
r

−
(b
r − b)
2r
2
φ
+
1
r
.
b(r) = r
0
,
φ(r) = φ
0
,
Ω(r) =
2A
r
0
e
φ
0
1 −
r
0
r
.
ρ = 0,
p = −
r
0
r
3
,
τ =
r
0
2r
3
.
ds
2
= −e
2φ
0
c
2
dt
2
+
dr
2
1 −
r
0
r
+ r
2
dθ
2
+ sen
2
θdϕ
2
.
l(r) = ±r
1 −
r
0
r
+
r
0
2r
ln
2r
r
0
1 +
1 −
r
0
r
− 1

,
r(l)
b(r) φ(r)
l
1
= l
+
(r
1
) = l
2
= −l
−
(r
2
) = l
r
1
= r
2
= r
d =
b(r)
r
=
r
0
r
10
−2
l ≈ r.
γe
φ
γ φ
γ ≈ 1
r
0
2r
3
v
c
2
1
10
16
m
−2
,
1
2r
2
0
v
c
2
1
10
16
m
−2
∴ v 3
√
2r
0
s
−1
.
v = 3
√
2r
0
α s
−1
, α 1 v c.
∆τ ≈ e
φ
0
∆t ≈
1
v
l
2
−l
1
dl ≈
2r
v
T
r 1, 6 · 10
11
m
10
4
m/s
α
√
2
3T d
s ∴ d
√
2
3T
s
α 1
1, 5 · 10
−8
d 10
−2
∴ 1, 5 · 10
−8
r r
0
10
−2
r.
10
4
m/s φ
0
0
r
0
v/(3
√
2)
r
0
= 7 · 10
8
m
r = 1 , 5 · 10
11
m d = 0, 5 · 10
−2
10
4
m/s
φ
0
0
z(r) = ±2r
0
r
r
0
− 1
25
1
r
10
9
1251007550
z(r)
2
−1
−2
0
0
z(r) × r r
0
= 7 · 10
8
m
−6
−5
−4
−3
−2
−1
10
10
0 1412108642−2
0
1
2
3
4
5
r
0
= 7 · 10
8
m
z(r)
z(r) = ±
r
r
0
1
r
b(r
)
− 1
dr
.
b(r)
b
=
r
2
X
2
(1 − 8αX) ,
X
=
X (16αX − 1) (40αX
2
r
3
− Xr
3
− 6b)
r (r − b) (80αX − 1)
,
dy
dx
= y
= f (x, y(x)) .
y(x
0
+ h) = y(x
0
) + y
(x
0
)h +
1
2
y

(x
0
)h
2
+ O(h
3
),
y(x
0
+ h) = y(x
0
) + f (x
0
, y(x
0
)) h +
1
2
f
(x
0
, y(x
0
)) h
2
+ O(h
3
)
y x
0
+ h h
3
f
f
x
0
x
0
+ h
y(x
0
+ h) = y(x
0
) + h (a
1
k
1
+ a
2
k
2
) + O(h
3
),
k
1
≡ f (x
0
, y(x
0
)) ≡ f
0
,
k
2
≡ f (x
0
+ b
2
h, y(x
0
) + c
2
k
1
h) .
k
2
= f
0
+ b
2
h
∂f
0
∂x
+ c
2
f
0
h
∂f
0
∂y
+ O(h
2
),
k
1
c
2
f
0
h h
h
2
y(x
0
+ h) = y(x
0
) + h (a
1
+ a
2
) f
0
+ h
2
a
2
b
2
∂f
0
∂x
+ c
2
f
0
∂f
0
∂y
+ O(h
3
).
df
dx
=
∂f
∂x
+
dy
dx
∂f
∂y
=
∂f
∂x
+ f
∂f
∂y
,
y(x
0
+ h) = y(x
0
) + hf
0
+
h
2
2
∂f
0
∂x
+ f
0
∂f
0
∂y
+ O(h
3
).
a
1
+ a
2
= 1,
a
2
b
2
= a
2
c
2
=
1
2
.
a
1
= 0 a
2
= 1 b
2
= c
2
=
1
2
y(x
0
+ h) = y(x
0
) + hk
2
+ O(h
3
),
k
1
= f (x
0
, y(x
0
)) ,
k
2
= f
x
0
+
h
2
, y(x
0
) +
h
2
k
1
.
y x
0
+ h x
0
h
3
y(x
0
+ 2h)
n
y(x
0
+ nh).
h
4
x
0
x
0
+ h
y(x
0
+ h) = y(x
0
) +
h
6
(k
1
+ 2k
2
+ 2k
3
+ k
4
) + O(h
5
),
k
1
= f (x
0
, y(x
0
)) ,
k
2
= f
x
0
+
h
2
, y(x
0
) +
h
2
k
1
,
k
3
= f
x
0
+
h
2
, y(x
0
) +
h
2
k
2
,
k
4
= f (x
0
+ h, y(x
0
) + hk
3
) .
k
1
r
0
z(r) r r
0
z(r) =
r
0
+
r
0
1
r
b(r
)
− 1
dr
+
r
r
0
+
1
r
b(r
)
− 1
dr
= δ +
r
r
0
+
1
r
b(r
)
− 1
dr
z
−
(r) = z(r) − δ
z(r) z(r) × r
z
−
(r
0
+) = 0
z(r)
b
=
r
2
X
2
(1 − 8αX) ,
r − r
0
< d ⇒X
= X
0
+ X

0
(r − r
0
),
r − r
0
d ⇒X
=
X (16αX − 1) (40αX
2
r
3
− Xr
3
− 6b)
r (r − b) (80αX − 1)
,
r − r
0
< ⇒z
−
= 0,
r − r
0
⇒z
−
=
1
r
b(r)
− 1
,
d [r − b(r)]
−1
z
−
α r
0
α = 1 m
2
r
0
= 4 m
X
0
= 0.1042565109,
X

0
= 0.399292709.
z
−
(r) ≈ z(r)
z
−
(r
0
+ )