Relativistic Universe
Relativistic stars
Metric
z
Locally any curved manifold is
characterized by its metric (interval), which
tells the distance between a point and
points in its neighborhood.
ds 2 = g µν dx µ dxν
µ
ν
ds = g µν dx dx
2
z
Tangent vector
xα(λ)
dxα (λ )
v ( x) =
dλ
α
vα(x)
z
We can rewrite the
geodesic equation as
d α
v ( x(λ )) + Γα µν ( x) v µ ( x) vν ( x) = 0
dλ
z
But
z
d
≡ vµ ∂µ
dλ
So that we can write the geodesic eqn. as
v µ ∂ µ vα ( x) + Γα µν ( x) v µ ( x) vν ( x) = 0,
(
)
(
0 = v µ ∂ µ vα ( x) + Γα µν ( x) vν ( x) = v µ ∇ µ vα ( x)
)
Covariant derivative
z
The covariant
derivative
z
Is the right derivative,
replacing partial
derivative of the flat
space and preserving
tensor character of
objects
∇ µ vα ( x) = ∂ µ vα ( x) + Γα µν ( x) vν ( x)
Covariant derivatives of tensors
α
α
α
∇ µ v ( x) = ∂ µ v ( x) + Γ
∇ µ vα ( x) = ∂ µ vα ( x) − Γ
∇ µ tαβ ( x) = ∂ µ tαβ ( x) − Γ
δ
µα
β
ν
µν
( x) v ( x)
µα
( x) vβ ( x)
( x) tδβ ( x) − Γ
δ
µβ
( x) tαδ ( x)
It can be easily checked
z
That the metric is covariantly constant
∇ µ gαβ ( x) ≡ 0
Geometrical meaning
In flat space, partial derivative ∂µ is
infinitesimal translation in direction µ.
z Similarly, in curved space, covariant
derivative ∇µ is infinitesimal parallel
translation in direction µ.
z        
     !!
z
Parallel transport
z
If there is a curve
with tangent vector
vα(x) and we want
to paralel transport
a vector uα(x) along
it we must solve the
equation
α
β
v ∇α u = 0
uα(x(1))
vα(x)
uα(x(0))
Curvature
If there is an
infinitesimal closed
curve (loop) and we
parallel transport a
vector along it
z Then the difference
is proportional to
curvature
z
Riemann tensor
z
R
Curvature is measured by Riemann
tensor, defined in terms of Christoffel
symbols
µ
ναβ
= ∂α Γ
µ
νβ
− ∂β Γ
µ
να
+Γ
µ
σα
Γ
σ
νβ
−Γ
µ
σβ
Γ
σ
να
Symmetries of Riemann tensor
Rµν ρσ = Rρσ µν
Rµν ρσ = − Rνµ ρσ
Rµν ρσ = − Rµν σρ
Contracting we get
z
Ricci tensor
Rνβ = R µνµβ
= ∂µΓ
z
µ
νβ
− ∂β Γ
µ
νµ
+Γ
µ
σµ
Curvature scalar
νβ
R = g Rνβ
σ
Γ
νβ
−Γ
µ
σβ
σ
Γ
νµ
Vacuum Einstein equations
Rµν = 0
Bianchi identity
1
⎛
⎞
∇ ⎜ Rµν − g µν R ⎟ ≡ 0
2
⎝
⎠
µ
z
This important identity is analogous to the
identity for the LHS of the second pair of
Maxwell equations
ε
µνρσ
∂ν Fρσ ≡ 0
Coupling to matter
Gravity tells matter how to move
z Matter’s energy and momentum tells
spacetime how to curve
z In empty space Einstein equations have
the form
z
Rµν = 0
z
In the presence of matter the equation
should be of the form
Rµν + … = 8π G Tµν
z
Tµν is called energy-momentum tensor.
Properties of e-m tensor
Tµν is symmetric in µ, ν;
z T00 is energy density, ρ;
z Tii are pressures in direction i , pi ;
z For perfect isotropic, relativistic fluid, in the
system, in which it is at rest
z
Tµν
⎛ρ
⎜
0
=⎜
⎜0
⎜
⎝0
0
0
p
0
0
0
0
p
0⎞
⎟
0⎟
0⎟
⎟
p⎠
Equations of motion
z
In flat space matter equations of motion
can be expressed with the help of Tµν as
follows
µ
∂ Tµν = 0
For perfect fluid
z
The general expression for energy
momentum tensor of the fluid (gas)
composed of particles is
Tµν = pg µν + ( ρ + p )uµ uν
Four velocity
In flat space
Componets of four velocity are uµ=(1,v i ),
where v is much smaller than 1;
z Pressure is much smaller from energy
density;
z
T
00
= ( ρ + p )u u + pg
0 0
00
≈ρ
T 0 j = T j 0 = ( ρ + p )u 0u j ≈ ρ v j
T
jk
= ( ρ + p )u u + pδ
j
k
jk
≈ ρ v v + pδ
j k
jk
Motion of the fluid
z
Continuity equation
0 = ∂ µT
z
µ0
( )
= ∂0 ρ + ∂i ρ v = ρ + ∇ ( ρ v )
i
Euler equation
0 = ∂ 0T
0j
+ ∂ iT = v + (v ⋅∇)v +
ij
1
ρ
∇p
Curved space generalization
z
To generalize our result to curved space
we must only replace partial derivative by
the covariant one
∇ µT
µν
=0
Einstein equations
z
We know that
1
⎛
⎞
∇ ⎜ Rµν − g µν R ⎟ ≡ 0
2
⎝
⎠
µ
z
Thus (c=1)
1
Rµν − g µν R = 8π G Tµν
2
Consistency with vacuum eqns
1
Rµν − g µν R = 0 ⇒
2
R − 2R = 0 ⇒
Rµν = 0
Metric of spherical relativistic star
z
It turns out that the most general
spherically symmetric metric is of the form
ds = −e
2
z
2Φ ( r )
dt + e
2
2Λ(r )
dr + r d Ω
2
2
2
Λ(r) and Φ(r) both go to zero at infinity (far
away from the source the metric is flat.)
Matter in star …
z
1.
2.
3.
4.
Is described by
Energy density ρ (r),
Isotropic pressure p (r),
Barion number density n (r), with EOS;
4-velocity of the fluid uµ = uµ (r) :
T
µν
µ ν
= ( p + ρ )u u + pg
µν
00 Einstein equation
1
R00 − g 00 R = 8π G T00
2
(
)
d ⎡
−2 Λ
2
⎤
r
1
e
8
π
Gr
ρ
−
=
⎦
dr ⎣
z
Denoting (m(r) is mass in radius r)
(
r 1− e
z
−2 Λ
) = 2Gm(r )
⇔e
2Λ
⎛ 2Gm(r ) ⎞
= ⎜1 −
⎟
r
⎝
⎠
And solving the equation
2G dm(r )
= 8π G ρ ⇒
2
r
dr
r
m(r ) = ∫ 4π ρ dr
0
−1
rr Einstein equation
1
Rrr − g rr R = 8π G Trr
2
(
)
−2 Λ
1
2e
−2 Λ
− 2 1− e
+
r
r
dΦ
= 8π G p
dr
dΦ
4π pr + m
=G
dr
r ( r − 2Gm )
3
z
With Newtonian limit
d Φ Gm
= 2
dr
r
Oppenheimer – Volkov eqn.
z
It follows from energy conservation that
dp
dΦ
= −( ρ + p )
dr
dr
z
So that we get
dp
( p + ρ )(4π pr 3 + m)
= −G
dr
r ( r − 2Gm )
Nonrelativistic vs relativistic stars
Solution of OV equation
Assume that density of star is constant:
ρ(r) =ρ0.
z Then
z
r
m(r ) = ∫ 4π ρ0 dr ⇒
0
4πρ0 3
m( r ) =
r
3
z Then
the OV equation takes the form
4π Gr ρ02 (1 + p / ρ0 c 2 )(1 + 3 p / ρ 0 c 2 )
dp
=−
dr
3 1 − 8π G ρ0 r 2 / 3c 2
(
z To
)
which we add the usual initial
condition p(R)=0
Solution
p(r ) = ρ0c 2
1 − 2GMr 2 / R 3c 2 − 1 − 2GM / Rc 2
3 1 − 2GM / Rc 2 − 1 − 2GMr 2 / R 3c 2
Central pressure
z
At r=0
p (0) = pc = ρ0 c 2
z
1 − 1 − 2GM / Rc 2
3 1 − 2GM / Rc 2 − 1
This is the pressure in the center of a
stable star of mass M and radius R
Condition of stability
z
The central pressure becomes infinite if
9 2GM 9
R=
= Rs
2
8 c
8
z
The star radius cannot be smaller if the
star is to be stable.
z
A more reasonable limit on the central
pressure would be to limit the speed of
sound to be less than or equal to the
speed of light. A sound speed in excess of
the speed of light would suggest
conditions where the gas would violate the
principle of causality. Namely, sound
waves could propagate signals faster than
the velocity of light. Since P/ρ0 is the
square of the local sound speed,
pc = ρ0 c
2
1 − 1 − 2GM / Rc 2
3 1 − 2GM / Rc − 1
2
= ρ0 c
2
4 2GM 4
R=
= Rs
2
3 c
3
Since causality will always dictate that
the sound speed be less than the speed
of light, we conclude that any stable star
must have a radius R such that R > 4/3
Rs.
z
From the NICMOS image we have seen that behind the
obscuring dust, inside the Centaurus A Galaxy, there are
a hot gas disk and a twisted jet and disk around a
feeding black hole.
1998-14-a-low_quicktime.mov
Matter falling on black hole
Black hole in globullar cluster