Gravitation:Schwarzschild Black Holes - Pablo...
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Gravitation: Schwarzschild Black HolesAn Introduction to General Relativity
Pablo Laguna
Center for Relativistic AstrophysicsSchool of Physics
Georgia Institute of Technology
Notes based on textbook: Spacetime and Geometry by S.M. CarrollSpring 2013
Pablo Laguna Gravitation: Schwarzschild Black Holes
Schwarzschild Black Holes
Schwarzschild Metric
Birkhoff’s Theorem
Singularities
Geodesics
Experimental Tests
Stars and Black Holes
Pablo Laguna Gravitation: Schwarzschild Black Holes
One of the most important solutions to the Einstein’s equations is that of spherically symmetric vacuumspacetimes.
The solution was discovered by Karl Schwarzschild 1915.
It represents the solution outside a spherical, static body.
Schwarzschild metric
ds2 = −(
1−2GM
r
)dt2 +
(1−
2GM
r
)−1dr2 + r2dΩ2
where dΩ2 = dθ2 + sin2 θ dφ2 and the parameter M is interpreted as the mass of the gravitating object.
Pablo Laguna Gravitation: Schwarzschild Black Holes
Derivation
Starting point, Einstein’s equation in vacuum, i.e. Rµν = 0
We require the spacetime to be static and spherically symmetric.
Static: i) The metric is time-independent and ii) the metric does not have time-space terms dt dx i
Example, of a static and spherically symmetric spacetime is Minkowski ds2 = −dt2 + dr2 + r2dΩ2
We will begin with the following ansatz:
ds2 = −e2α(r)dt2 + e2β(r)dr2 + e2γ(r)r2dΩ2
Introduce the following change of coordinates: r = eγ(r)r
Then
dr =
(1 + r
dγ
dr
)eγdr
so
ds2 = −e2αdt2 +
(1 + r
dγ
dr
)−2e2β−2γdr2 + r2dΩ2
and redefine
e2β =
(1 + r
dγ
dr
)−2e2β−2γ
to get
ds2 = −e2αdt2 + e2βdr2 + r2dΩ2
r is an areal coordinate.
Pablo Laguna Gravitation: Schwarzschild Black Holes
Derivation
Compute the Christoffel symbols (drop bars)
Γttr = ∂rα Γr
tt = e2(α−β)∂rα Γrrr = ∂rβ
Γθrθ = 1r Γr
θθ = −re−2β Γφrφ = 1
r
Γrφφ = −re−2β sin2 θ Γθφφ = − sin θ cos θ Γ
φθφ
= cos θsin θ .
Compute Riemann tensor
Rtrtr = ∂rα∂rβ − ∂2
r α− (∂rα)2
Rtθtθ = −re−2β
∂rα
Rtφtφ = −re−2β sin2
θ ∂rα
Rrθrθ = re−2β
∂rβ
Rrφrφ = re−2β sin2
θ ∂rβ
Rθφθφ = (1− e−2β ) sin2θ
Compute Ricci tensor
Rtt = e2(α−β)[∂2r α + (∂rα)2 − ∂rα∂rβ +
2
r∂rα]
Rrr = −[∂2r α + (∂rα)2 − ∂rα∂rβ −
2
r∂rβ]
Rθθ = e−2β [r(∂rβ − ∂rα)− 1] + 1Rφφ = Rθθ sin2
θ
Pablo Laguna Gravitation: Schwarzschild Black Holes
Derivation
Notice that
e2(β−α)Rtt + Rrr =2
r(∂rα + ∂rβ)
Since each component of Rµν vanishes independently, then α = −β.
Consider now Rθθ = 0,1 = e2α (2 r ∂rα + 1) = ∂r (r e2α)
which has the following solution
e2α = 1−RS
r
with RS a constant of integration.
Thus
ds2 = −(
1−RS
r
)dt2 +
(1−
RS
r
)−1dr2 + r2dΩ2
It is easy to check that e2α = e−2β = 1− RS/r satisfies Rtt = Rrr = 0.
Recall: in the weak field limit we saw that
gtt = −1 + htt = −(1 + 2 Φ) = −(
1−2 G M
r
)
thus if we take the weak field limit of our solution
gtt = −(
1−RS
r
)⇒ RS = 2 G M Schwarzschild Radius
Pablo Laguna Gravitation: Schwarzschild Black Holes
Birkhoff’s Theorem
Theorem: The Schwarzschild metric is the unique vacuum solution with spherical symmetry.Proof consist of showing that:
A spherically symmetric spacetime can be foliated by two spheres.
The spatial metric can always be recast as ds2 = dτ2(a, b) + r2(a, b)dΩ2 with (a, b) coordinatestransverse to the sphere.
the solution to Einstein’s equation is the Schwarzschild metric.
STEP 1
Our spacetime manifold M has the same symmetries as a S2 sphere, i.e. the ordinary rotations in 3-dimEuclidean space (special orthogonal group SO(3).
The Killing vectors associated with these symmetries are:
R = ∂φS = cosφ∂θ − cot θ sin θ∂φT = − sinφ∂θ − cot θ cos θ∂φ
The algebra of these Killing vectors is
[R, S] = T[S, T ] = R[T ,R] = S
Pablo Laguna Gravitation: Schwarzschild Black Holes
Then, M admits a foliation of S2 spheres if every point x ∈ M is on exactly one of these spheres.
An example of this type of foliation is the space R× S2 (a wormhole)
STEP 2:
Our submanifolds are 2-spheres with coordinates (θ, φ) and metric dΩ2 = dθ2 + sin2 θ dφ2
Our spacetime metric is then in general
ds2 = gaa(a, b)da2 + 2 gab(a, b) da db + gbb(a, b)db2 + r2(a, b)dΩ2
Without loss of generality, we can pick r = b, so
ds2 = gaa(a, r)da2 + 2 gar (a, r) da dr + grr (a, r)dr2 + r2dΩ2
Pablo Laguna Gravitation: Schwarzschild Black Holes
Next, find a function t(a, r) such that we eliminate the cross term dt dr in the metric and haveds2 = m dt2 + n dr2 + r2dΩ2.
Given t(a, r),
dt =∂t
∂ada +
∂t
∂rdr
so
dt2 =
(∂t
∂a
)2da2 + 2
(∂t
∂a
)(∂t
∂r
)da dr +
(∂t
∂r
)2dr2
Then,
ds2 = m dt2 + n dr2 + r2dΩ2
= m
[(∂t
∂a
)2da2 + 2
(∂t
∂a
)(∂t
∂r
)da dr +
(∂t
∂r
)2dr2]
+ n dr2 + r2dΩ2
= m(∂t
∂a
)2da2 + 2 m
(∂t
∂a
)(∂t
∂r
)da dr +
[m(∂t
∂r
)2+ n
]dr2 + r2dΩ2
Comparison withds2 = gaa(a, r)da2 + 2 gar (a, r) da dr + grr (a, r)dr2 + r2dΩ2
yields the following three equations for the three unknowns (t,m, n)
m(∂t
∂a
)2= gaa
n + m(∂t
∂r
)2= grr
m(∂t
∂a
)(∂t
∂r
)= gar
Pablo Laguna Gravitation: Schwarzschild Black Holes
After solving these equations our metric will have the form
ds2 = m(t, r)dt2 + n(t, r)dr2 + r2dΩ2
Since we are dealing with a Lorentzian metric, either m or n will have to be negative. We select t to be thetimelike coordinate and write the metric as
ds2 = −e2α(t,r)dt2 + e2β(t,r)dr2 + r2dΩ2
STEP 3: Solving the Einstein equations.
Calculate Christoffel symbols, using labels (0, 1, 2, 3) for (t, r, θ, φ).
Γ000 = ∂0α Γ0
01 = ∂1α Γ011 = e2(β−α)∂0β
Γ100 = e2(α−β)∂1α Γ1
01 = ∂0β Γ111 = ∂1β
Γ212 = 1
r Γ122 = −re−2β Γ3
13 = 1r
Γ133 = −re−2β sin2 θ Γ2
33 = − sin θ cos θ Γ323 = cos θ
sin θ .
Calculate the Reimann tensor
R0101 = e2(β−α)[∂2
0β + (∂0β)2 − ∂0α∂0β] + [∂1α∂1β − ∂21α− (∂1α)2]
R0202 = −re−2β
∂1α
R0303 = −re−2β sin2
θ ∂1α
R0212 = −re−2α
∂0β
R0313 = −re−2α sin2
θ ∂0β
R1212 = re−2β
∂1β
R1313 = re−2β sin2
θ ∂1β
R2323 = (1− e−2β ) sin2
θ .
Pablo Laguna Gravitation: Schwarzschild Black Holes
Calculate the Ricci tensor
R00 = [∂20β + (∂0β)2 − ∂0α∂0β] + e2(α−β)[∂2
1α + (∂1α)2 − ∂1α∂1β +2
r∂1α]
R11 = −[∂21α + (∂1α)2 − ∂1α∂1β −
2
r∂1β] + e2(β−α)[∂2
0β + (∂0β)2 − ∂0α∂0β]
R01 =2
r∂0β
R22 = e−2β [r(∂1β − ∂1α)− 1] + 1R33 = R22 sin2
θ .
From R01 = 0 we get ∂0β = 0
Taking the time derivative of R22 = 0 and using ∂0β = 0, we get ∂0∂1α = 0
Therefore, β = β(r) and α = f (r) + g(t)
The metric becomesds2 = −e2(f (r)+g(t))dt2 + e2β(r)dr2 + r2dΩ2
Pablo Laguna Gravitation: Schwarzschild Black Holes
Redefine the time coordinate by replacing dt → e−g(t)dt to finally get
ds2 = −e2α(r)dt2 + e2β(r)dr2 + r2dΩ2
That is, any spherically symmetric vacuum metric can be written in coordinates such that all its componentsare independent of the time coordinate.
Therefore, any spherically symmetric metric possesses a time-like Killing vector.
Recall, we have already shown that the metric above can be brought to have the form
Schwarzschild Metric
ds2 = −(
1−2GM
r
)dt2 +
(1−
2GM
r
)−1dr2 + r2dΩ2
Pablo Laguna Gravitation: Schwarzschild Black Holes
Stationary and Static
Stationary metric: one that possesses a time-like Killing vector near infinity.
For a stationary metric, one can choose coordinates (t,~x) such that the Killing vector is ∂t and the metrictakes the form
ds2 = g00(~x) dt2 + 2 g0i (~x) dt dx i + gij (~x) dx i dx j
Static metric: one that possesses a time-like Killing vector orthogonal to a family of hypersurfaces.
In the coordinates for which the metric is independent of the time coordinate, the Killing vector is orthogonalto the t = constant hypersurfaces.
In these coordinates, the metric reads
ds2 = g00(~x) dt2 + gij (~x) dx i dx j
Pablo Laguna Gravitation: Schwarzschild Black Holes
Singularities
ds2 = −(
1−2GM
r
)dt2 +
(1−
2GM
r
)−1dr2 + r2dΩ2
The Schwarzschild metric becomes singular at r = 0 and r = 2GM
Are these real singularities? After all the form of the metric depends on the coordinate system used.
Could they be coordinate singularities? Example: ds2 = dr2 + r2dθ2 becomes degenerate at r = 0 andthe component gθθ = r−2 of the inverse metric blows up.
We need a coordinate independent quantity to identify singularities.
Examples: R = gµνRµν , RµνRµν , RµνρσRµνρσ , RµνρσRρσλτRλτµν
In the case of the Schwarzschild metric,
RµνρσRµνρσ =12G2M2
r6.
thus r = 0 represents an honest singularity.
Notice that at r = 2 G M this quantity is not singular. We will show later that we can transform to a newcoordinate system for which the metric is regular at this point.
Form most realistic spherical object, their radius R is such that R RS = 2 G M. For example, the Sun:R = 106 G M
Pablo Laguna Gravitation: Schwarzschild Black Holes
Geodesics
Christoffel symbols for Schwarzschild:
Γ100 = GM
r3 (r − 2GM) Γ111 = −GM
r(r−2GM)Γ0
01 = GMr(r−2GM)
Γ212 = 1
r Γ122 = −(r − 2GM) Γ3
13 = 1r
Γ133 = −(r − 2GM) sin2 θ Γ2
33 = − sin θ cos θ Γ323 = cos θ
sin θ
Geodesic equations Uν∇νUµ = 0 with Uµ = dxµ/dλ
d2t
dλ2+
2GM
r(r − 2GM)
dr
dλ
dt
dλ= 0
d2r
dλ2+
GM
r3(r − 2GM)
( dt
dλ
)2−
GM
r(r − 2GM)
( dr
dλ
)2− (r − 2GM)
[( dθ
dλ
)2+ sin2
θ
( dφ
dλ
)2]
= 0
d2θ
dλ2+
2
r
dθ
dλ
dr
dλ− sin θ cos θ
( dφ
dλ
)2= 0
d2φ
dλ2+
2
r
dφ
dλ
dr
dλ+ 2
cos θ
sin θ
dθ
dλ
dφ
dλ= 0
with λ an affine parameter.
Pablo Laguna Gravitation: Schwarzschild Black Holes
Recall that for any Killing vector Kµ,
∇(µKν) = 0 ⇒ pµ∇µ(Kνpν ) = 0
with p = m Uµ the 4-momentum of a particle.
Since Schwarzschild space-time have 4 Killing vectors (3 rotations and time translation), there are then 4constants of motion:
Kµdxµ
dλ= constant
In addition, from the geodesic equation Uν∇νUµ = 0 and metric compatibility∇αgµν = 0
ε = −gµνdxµ
dλ
dxν
dλ= constant
For time-like curves we typically pick λ = τ thus ε = 1. For null geodesics, ε = 0.
Invariance under time translation implies conservation of energy
Invariance under spatial rotation implies conservation of angular momentum
Conservation of angular momentum consists of magnitude and direction
Conservation of the angular momentum direction implies particles move in a plane thus we can, without lossof generality, choose θ = π/2
Pablo Laguna Gravitation: Schwarzschild Black Holes
With θ = π/2, the two Killing vectors corresponding to conservation of energy and magnitude of theangular momentum are given, respectively, by K = ∂t and L = ∂φ or equivalently byKµ = (∂t )µ = (1, 0, 0, 0) and Lµ = (∂φ)µ = (0, 0, 0, 1).
Also,
Kµ =
(−(
1−2GM
r
), 0, 0, 0
)Lµ =
(0, 0, 0, r2 sin2
θ).
Therefore, the conserved quantities (energy and angular momentum) Kµ dxµdλ = constant are
(1−
2GM
r
) dt
dλ= E
r2 dφ
dλ= L
Pablo Laguna Gravitation: Schwarzschild Black Holes
From
ε = −gµνdxµ
dλ
dxν
dλ= constant
we have that
−(
1−2GM
r
)( dt
dλ
)2+
(1−
2GM
r
)−1 ( dr
dλ
)2+ r2
( dφ
dλ
)2= −ε
Using E and L
−E2 +
( dr
dλ
)2+
(1−
2GM
r
)( L2
r2+ ε
)= 0
Rewrite as1
2
( dr
dλ
)2+ V (r) = E
where
V (r) =1
2ε− ε
GM
r+
L2
2r2−
GML2
r3
E =1
2E2
Notice: The geodesic equation has the same structure as that for a classical particle of unit mass, energy E moving
in a 1-dimentional potential V
Pablo Laguna Gravitation: Schwarzschild Black Holes
We are interested on the solutions r(λ), t(λ), φ(λ)
There are different curves V (r) for different values of L
The type of orbit can be obtained by comparing the 12 E2 to V (r) and in particular looking for the turning
points where V (r) = 12 E2.
Circular orbits with radius rc = const happen if the potential is flat, i.e. dV/dr = 0. That is
εGMr2c − L2rc + 3GML2
γ = 0
where γ = 0 in Newtonian gravity and γ = 1 in general relativity.
Circular orbits will be stable if they correspond to a minimum of the potential, and unstable if theycorrespond to a maximum.
Bound orbits which are not circular will oscillate around the radius of the stable circular orbit.
Pablo Laguna Gravitation: Schwarzschild Black Holes
Orbits in Newtonian gravity
0 10 20 300
0.2
0.4
0.6
0.8
r
1
23
4L=5
Newtonian Gravity
massless particles
0 10 20 300
0.2
0.4
0.6
0.8
r
L=1
2
3
4
5
Newtonian Gravity
massive particles
Circular orbits in Newtonian gravity appear at
rc =L2
εGM.
For massless particles ε = 0. Thus, there are no circular orbits (no bound orbits in left Figure).
Massless particles move in a straight line since the Newtonian gravitational force on massless particles iszero.
For massive particles, we have the standard situation of bound orbits for E < 1 are either circles or ellipses,while unbound for E ≥ 1 ones are either parabolas or hyperbolas.
Pablo Laguna Gravitation: Schwarzschild Black Holes
Orbits in General Relativity
0 10 20 300
0.2
0.4
0.6
0.8
r
1
2
3
4L=5
General Relativity
massless particles
In general relativity the situation is different only for r → 0 when thepotential goes to−∞.
At r = 2GM the potential is always zero.
For massless particles there is always a barrier.
At the top of the barrier there are unstable circular orbits.
For ε = 0, γ = 1, the radius of this orbit is at rc = 3GM.
Pablo Laguna Gravitation: Schwarzschild Black Holes
Orbits in General Relativity
0 10 20 300
0.2
0.4
0.6
0.8
r
L=1
2
3
4
5
General Relativity
massive particles
For massive particles, circular orbits are at
rc =L2 ±
√L4 − 12G2M2L2
2GM.
For large L there will be two circular orbits, one stable and one unstable.For L→∞ their radii are
rc =L2 ± L2(1− 6G2M2/L2)
2GM=
(L2
GM, 3GM
).
As we decrease L the two circular orbits come closer together; theycoincide when
L =√
12GM ,
for whichrc = 6GM ,
and disappear entirely for smaller L.
6GM is the smallest possible radius of a stable circular orbit.
In summary, Schwarzschild spacetimes possesses stable circular orbitsfor r > 6GM and unstable circular orbits for 3GM < r < 6GM.
Pablo Laguna Gravitation: Schwarzschild Black Holes
Experimental Tests of General Relativity
Most the experimental tests involve motion of particles or photons (i.e. geodesics)Precession of periheliaGravitational redshiftDeflection of lightGravitational time or Shapiro delay
Early tests were confined to the Solar System
The deflection of light and the Shapiro delay arise in the weak-field limit; thus, they do not constitute astrong test of GR.
The ultimate test of GR is the detection of gravitational waves.
Pablo Laguna Gravitation: Schwarzschild Black Holes
Precession of the Perihelia
In GR, elliptical orbits are not closed, they are ellipses that precess.
The precession of Mercury was the first test of GR.
The observed precession of Mercury’s major axis is δφ ≈ 5601 arcsecs /100 yrs.
Most of the precession, δφ ≈ 5025 arcsecs /100 yrs, is due to the precession of equinoxes in ourgeocentric coordinate system.
The gravitational perturbations of the other planets contribute an additional δφ ≈ 532 arcsecs /100 yrs.
The remaining δφ ≈ 43 arcsecs /100 yrs are explained by GR.
Pablo Laguna Gravitation: Schwarzschild Black Holes
Precession of the PeriheliaDERIVATION:Recall from the geodesic equation:
( dr
dτ
)2= E2 −
(1−
2M
r
)(1 +
L2
r2
)dφ
dτ=
L
r2
Thus ( dr
dφ
)2=
E2 − (1− 2M/r)(
1 + L2/r2)
L2/r4r
Introduce u ≡ 1/r . ( du
dφ
)2=
E2
L2− (1− 2Mu)
( 1
L2+ u2
)In the Newtonian limit this equation reads
( du
dφ
)2=
E2
L2− (1− 2Mu)
1
L2− u2
A circular orbit in Newtonian theory has u = M/L2. Define y = u − M/L2 to represent the deviation from circularorbit. We then have for the GR case neglectingO(y3),
( dy
dφ
)2=
E2 + M2/L2 − 1
L2+
2M4
L6+
6M3
L2y −
(1−
6M2
L2
)y2
Pablo Laguna Gravitation: Schwarzschild Black Holes
The solution to this equation has the form
y = y + A cos (kφ + B)
where
k =
(1−
6M2
L2
)1/2
y =3M3
k2L2
A =1
k
[E2 + M2/L2 − 1
L2+
2M4
L6− y
]1/2
with B an arbitrary constant. For comparison, in the Newtonian case
y =
[E2 + M2/L2 − 1
L2
]cos (φ + B)
Notice that in this case after φ advances 2π, the orbit returns to the same radius (i.e. closed orbits). In the GR case,the situation is different because of the constant k . The orbit returns to the same radius after kφ goes through 2π.That is
∆φ =2π
k= 2π
(1−
6M2
L2
)−1/2
Pablo Laguna Gravitation: Schwarzschild Black Holes
For nearly Newtonian orbits
∆φ = 2π
(1−
6M2
L2
)−1/2
≈ 2π
(1 +
3M2
L2
)
Therefore, the perihelium advance is given by
δφ =6πM2
L2
For orbits about a non-relativistic star L2 ≈ Mr . Thus,
δφ =6πM
r
Consider the case of Mercury, r = 5.55× 107 km and M = 1 M = 1.47 km. Then
δφ = 4.99× 10−7 radians per orbit
Or equivalentlyδφ = 0.43′′/yr = 43′′/century
Pablo Laguna Gravitation: Schwarzschild Black Holes
Gravitational Redshift
The gravitational redshift is another effect which is present in the weak field limit.
It will be predicted by any theory of gravity which obeys the Principle of Equivalence.
Over large distances, the amount of redshift will depend on the metric, and thus on the theory underquestion.
DERIVATION:
Consider the Schwarzschild geometry.
Consider two observers who are not moving on geodesics at fixed spatial coordinate values (r1, θ1, φ1)and (r2, θ2, φ2).
From the line element with dθ = dφ = 0,
dτi
dt=
(1−
2M
ri
)1/2
Suppose that the observerO1 emits a light pulse which travels to the observerO2, such thatO1 measuresthe time between two successive crests of the light wave to be ∆τ1.
Each crest follows the same path toO2, except that they are separated by a coordinate time
∆t =
(1−
2M
r1
)−1/2
∆τ1 .
Pablo Laguna Gravitation: Schwarzschild Black Holes
The second observer measures a time between successive crests given by
∆τ2 =
(1−
2M
r2
)1/2
∆t =
(1− 2M/r21− 2M/r1
)1/2
∆τ1
∆τi measure the proper time between two crests of an electromagnetic wave, the observed frequenciesωi ∝ 1/∆τi are then related by
ω2
ω1=
∆τ1
∆τ2=
(1− 2M/r11− 2M/r2
)1/2
In the weak-field limit r >> 2M and
ω2
ω1= 1−
M
r1+
M
r2= 1 + Φ1 − Φ2
Consider an emitter at the location r = r1 near the object M and an observer at a location r2 >> r , then
λe
λ= 1−
M
ror equivalently
λ
λe= 1 +
M
r
Define the redshift z as
z ≡λ − λe
λe=
M
r
Pablo Laguna Gravitation: Schwarzschild Black Holes
Deflection of Light
Recall that for photons the orbits are given by
dt
dλ= E
(1−
2 G M
r
)−1
dφ
dλ=
L
r2( dr
dλ
)2= E2 − V (r)
where
V (r) =L2
r2
(1−
2 G M
r
)Therefore
dφ
dr= ±
L
r2
(E2 − V (r)
)−1/2
= ±1
r2
[E2
L2−
1
r2
(1−
2 G M
r
)]−1/2
dφ
dt=
1
r2
L
E
(1−
2 G M
r
)
Pablo Laguna Gravitation: Schwarzschild Black Holes
For r →∞
φ ≈b
rdr
dt≈ −1
dφ
dt=
dφ
dr≈
b
r2
butdφ
dt=
1
r2
L
E
(1−
2 G M
r
)≈
1
r2
L
E
thus the impact parameter b is given by
b =L
E
thereforedφ
dr= ±
1
r2
[ 1
b2−
1
r2
(1−
2 G M
r
)]−1/2
which yields a deflection angle
∆φ = 2∫ ∞
r1
dr
r2
[ 1
b2−
1
r2
(1−
2 G M
r
)]−1/2
with r1 the turning point radius, the radius where the square bracket vanishes.
Pablo Laguna Gravitation: Schwarzschild Black Holes
Define r = b/w . Thus
∆φ = 2∫ ∞
r1
dr
r2
[ 1
b2−
1
r2
(1−
2 G M
r
)]−1/2
becomes
∆φ = 2∫ w1
0dw[
1− w2(
1−2 G M w
b
)]−1/2
expanding in powers of 2 G M/b one gets
∆φ = 2∫ w1
0dw(
1 +G M w
b
)[1− w2 +
2 G M w
b
]−1/2
which yields after integration
∆φ ≈ π +4 G M
b
The deflection angle is then
δφ = ∆φ− π =4 G M
b
which for the Sun is 1.7”
Pablo Laguna Gravitation: Schwarzschild Black Holes
Time of Light
From
dφ
dr= ±
1
r2
[ 1
b2−
1
r2
(1−
2 G M
r
)]−1/2
dφ
dt=
1
r2
1
b
(1−
2 G M
r
)
we have thatdt
dr= ±
1
b
(1−
2 G M
r
)−1 [ 1
b2−
1
r2
(1−
2 G M
r
)]−1/2
Consider a pulse of light originating at the Earth r⊕ that grazes the Sun and is reflected back to Earth from a pointrR . Assume origin at the Sun.
The total travel time is(∆t)tot = 2 t(r⊕, r1) + 2 t(rR , r1)
where r1 is the distance of closest approach to the Sun and t is the travel time computed from
t(r, r1) =
∫ r
r1dr
1
b
(1−
2 G M
r
)−1 [ 1
b2−
1
r2
(1−
2 G M
r
)]−1/2
Pablo Laguna Gravitation: Schwarzschild Black Holes
To first order in M, the integral
t(r, r1) =
∫ r
r1dr
1
b
(1−
2 M
r
)−1 [ 1
b2−
1
r2
(1−
2 M
r
)]−1/2
becomes
t(r, r1) =√
r2 − r21 + 2 M log
r +√
r2 − r21
r1
+ M
(r − r1r + r1
)1/2
thus(∆t)excess ≡ (∆t)tot − 2
√r2⊕ − r2
1 − 2√
r2R − r2
1
For r1/rR << 1 and r1/r⊕ << 1
(∆t)excess ≈4 G M
c3
[log
(4 rR r⊕
r21
)+ 1
]
Pablo Laguna Gravitation: Schwarzschild Black Holes
Schwarzschild SpacetimeTo understanding the geometry of the Schwarzschild spacetime, we explore its causal structure as defined by thelight cones. Because of its spherical symmetric symmetry, let’s consider only radial null curves:
ds2 = 0 = −(
1−2GM
r
)dt2 +
(1−
2GM
r
)−1dr2
,
thusdt
dr= ±
(1−
2M
r
)−1.
Notice
dt
dr= ±1 for r →∞
dt
dr= ±∞ for r → 2M
The light that approaches 2m neverseems to get there!
We need to investigate if this is becauseof the choice of coordinates.
r
t
2GM
Pablo Laguna Gravitation: Schwarzschild Black Holes
QUESTION: Does a free-falling observer ever reaches or even crosses r = 2M in a finite amount of proper time?
Consider the following coordinate transformation:
t = ±r∗ + constant
where the tortoise coordinate r∗ is defined by
r∗ = r + 2M ln( r
2M− 1)
The metric in these coordinates becomes
ds2 =
(1−
2M
r
)(−dt2 + dr∗2
)+ r2dΩ2
with r = r(r∗)
Notice that the light-cones do not close up, i.e. dt/dr∗ = ±1 everywhere.
Pablo Laguna Gravitation: Schwarzschild Black Holes
ingoing Eddington-Finkelstein coordinate
Introduce the following ingoing coordinate u = t + r∗
The metric becomes
ds2 = −(
1−2M
r
)du2 + 2 du dr + r2dΩ2
At r = 2M, although guu = 0, there is no degeneracy since g = −r4 sin2 θ
For radial null curvesdu
dr=
0 , (infalling)
2(
1− 2Mr
)−1(outgoing)
Integration yiels
u = constant
u − 2(
r + 2M ln∣∣∣∣ r
2M− 1∣∣∣∣) = constant
Pablo Laguna Gravitation: Schwarzschild Black Holes
At r = 2M
Light-cones tilt inwards. It is impossible to seethe inside.
The metric is perfectly regular
The surface is null
This surface is defines as the event horizon
The event horizon is a global object, requiresthe knowledge of the entire spacetime todefine it.
u
r = 2GM
u =
r = 0
const
~
~
r
Pablo Laguna Gravitation: Schwarzschild Black Holes
Maximally Extended Schwarzschild SolutionRecall the metric in the form
ds2 =
(1−
2M
r
)(−dt2 + dr∗2
)+ r2dΩ2
Introduce both an ingoing coordinate u = t + r∗ and an outgoing coordinate v = t − r∗
The metric becomes
ds2 = −(
1−2GM
r
)du dv + r2dΩ2
with r = r(u, v) such that1
2(u − v) = r + 2M ln
( r
2M− 1)
Notice that in these coordinates r = 2M is infinitely far away (at either u = −∞ or v = +∞).
Introduce the following null coordinates: u′ = eu/4M and v′ = e−v/4M
The metric becomes
ds2 = −32M3
re−r/2GM du′ dv′ + r2dΩ2
Notice that none of the metric coefficients behave in any special way at the event horizon r = 2GM.
The transformation to the original coordinates reads:
u′ =
( r
2M− 1)1/2
e(r+t)/4M
v′ =
( r
2M− 1)1/2
e(r−t)/4M.
Pablo Laguna Gravitation: Schwarzschild Black Holes
Kruskal-Szekeres coordinates
Introduce time-like and space-like coordinates
T =1
2(u′ − v′) =
( r
2M− 1)1/2
er/4M cosh(t/4M)
R =1
2(u′ + v′) =
( r
2M− 1)1/2
er/4M sinh(t/4M)
The metric becomes
ds2 =32G3M3
re−r/2GM (−dT 2 + dR2) + r2dΩ2
,
where
T 2 − R2 =
( r
2M− 1)
er/2M
The coordinates (T ,R, θ, φ) are known as the Kruskal-Szekres coordinates.
Pablo Laguna Gravitation: Schwarzschild Black Holes
The event horizon (r = 2M) is defined byT = ±R
Surfaces of constant r = satisfy hyperbolas
T 2 − R2 = constant
Surfaces of constant t satisfy straight lines
T
R= tanh(t/4M)
Notice that t → ±∞ are the same surfacesas the horizon r = 2GM.
Pablo Laguna Gravitation: Schwarzschild Black Holes
Schwarzschild vs Kruskal-Szekeres coordinates
Pablo Laguna Gravitation: Schwarzschild Black Holes
Coordinates (T ,R) range over every value withouthitting the real singularity at r = 2GM; that is,−∞ ≤ u ≤ ∞ and T 2 < R2 + 1.
The T − R plane is known as the Kruskal diagramand has 4 regions.
Region I: The original region covered by theSchwarzschild coordinates.
Region II:
What we think of as the black hole.
Anything inside of this region can neverescape.
Every future-directed path in region II ends uphitting the singularity at r = 0.
The boundary of region II is called the futureevent horizon.
II
IV
III
I
Pablo Laguna Gravitation: Schwarzschild Black Holes
Region III:
The time-reverse of region II
Things can escape to us, while we can neverget there.
It is often called a white hole
There is a singularity in the past, out of whichthe universe appears to spring.
The boundary of region III is called the pastevent horizon
Region IV:
Cannot be reached from region I.
It is another asymptotically flat region ofspacetime, a mirror image of region I.
It is connected to region I via a wormhole
II
IV
III
I
Pablo Laguna Gravitation: Schwarzschild Black Holes
Wormholes
A
B
C
D
E
A B C D E
r = 2GM
v
Pablo Laguna Gravitation: Schwarzschild Black Holes
Stars and Black Holes
The spacetime outside a spherically symmetric star is that of Schwarzschild (Birkhoff theorem).
Stars eventually collapse under their own gravitational pull once hydrostatic equilibrium cannot be longermaintained.
For stars with mass M < 8 M, gravitational collapse is stopped by electron degeneracy pressure (i.e.Pauli exclusion principle: no two fermions can be in the same state). Hydrostatic equilibrium is once againachieved and the resulting object is a white dwarf. (Mwd ≈ 1.4 M and Rwd ≈ R⊕).
For stars with mass 8 M < M < 18 M , neutron degeneracy pressure stops the gravitational collapseand a neutron star is formed (1.2 M < Mns ≈ 2 M and Rns ≈ 10 km).
For stars with mass 18 M < M, the stars will shrink down to below r = 2GM and further into a singularity,resulting in a black hole.
r = 2GMr = 0
vacuum(Schwarzschild)
interiorof star 0.5
1.0
1.5
log R
white dwarfs
neutron stars
1 2 3 4
D
B
CA
10
M/M
(km)
Pablo Laguna Gravitation: Schwarzschild Black Holes
Stellar Models
Consider the general, static, spherically symmetric metric:
ds2 = −e2α(r)dt2 + e2 β(r)dr2 + r2 dΩ2
We need solutions to the Einstein equations Gµν = 8π Tµν such that
Tµν = (ρ + p)UµUν + p gµν
which ρ the energy density, p the pressure and Uµ the 4-velocity which for static solutions is given byUµ = (eα, 0, 0, 0).
The components of Tµν and Tµ ν are
Tµν = diag(e2αρ, e2βp, r2 p, r2 sin2
θ p)
Tµν = diag(−ρ, p, p, p)
Pablo Laguna Gravitation: Schwarzschild Black Holes
The components of Gµν and Gµ ν are
Gtt =1
r2e2(α−β)
(2 r ∂rβ − 1 + e2β
)Grr =
1
r2
(2 r ∂rα + 1− e2β
)Gθθ = r2 e−2β
[∂
2r α + (∂rα)2 − ∂rα∂rβ +
1
r(∂rα− ∂rβ)
]Gφφ = sin2
θ Gθθ
and
Gtt = −
1
r2e−2β
(2 r ∂rβ − 1 + e2β
)Gr
r =1
r2e−2β
(2 r ∂rα + 1− e2β
)Gθ
θ = e−2β[∂
2r α + (∂rα)2 − ∂rα∂rβ +
1
r(∂rα− ∂rβ)
]Gφ
φ = Gθθ
respectively
Pablo Laguna Gravitation: Schwarzschild Black Holes
The above yields the following three independent equations:
1
r2e−2β
(2 r ∂rβ − 1 + e2β
)= 8π ρ
1
r2e−2β
(2 r ∂rα + 1− e2β
)= 8π p
e−2β[∂
2r α + (∂rα)2 − ∂rα∂rβ +
1
r(∂rα− ∂rβ)
]= 8π p
Introduce the following new variable
m(r) =r
2(1− e−2β )
Φ = α
The metric takes the form
ds2 = −e2Φdt t +
(1−
2 m
r
)−1dr2 + r2 dΩ2
The first equation above becomesdm
dr= 4π r2
ρ
Pablo Laguna Gravitation: Schwarzschild Black Holes
Integrate this equation to get
m(r) = 4π∫ r
0ρ(r) r2 dr
If the star has a radius R, then let M = m(r = R).
Question: Is M the mass of the star?
Recall: The proper spatial volume element is
√γ d3x = eβ r2 sin θ dr dθ dφ
whereγij = diag(e2β
, r2, r2 sin2
θ)
In the integral for m(r), the spatial volume element used was
√η d3x = r2 sin θ dr dθ dφ
whereηij = diag(1, r2
, r2 sin2θ)
Pablo Laguna Gravitation: Schwarzschild Black Holes
It seems then that the “true” mass from the integrated energy density should be
M = 4π∫ R
0ρ(r) r2 eβdr
= 4π∫ R
0
ρ(r) r2[1− 2 m
r
]1/2dr
The difference between M and M is the binding energy; that is
Eb = M − M > 0
The binding energy is the energy needed to disperse the matter in the star to infinity.
Pablo Laguna Gravitation: Schwarzschild Black Holes
Consider now the rr component equation
1
r2e−2β
(2 r ∂rα + 1− e2β
)= 8π p
which in terms of m and Φ readsdΦ
dr=
m + 4π r3 p
r [r − 2m]
From the energy-momentum conservation equation∇µTµν = 0, one gets that
(ρ + p)dΦ
dr= −
dp
dr
thus one arrives to
Tolman-Oppenheimer-Volkoff equations
dp
dr= −
m ρ
r2
(1 +
p
ρ
)(1 +
4π r3 p
m
)(1−
2m
r
)−1
dm
dr= 4π r2
ρ
Pablo Laguna Gravitation: Schwarzschild Black Holes
Tolman-Oppenheimer-Volkoff equations
dp
dr= −
m ρ
r2
(1 +
p
ρ
)(1 +
4π r3 p
m
)(1−
2m
r
)−1
dm
dr= 4π r2
ρ
To close the above system, one need to provide an equation of state that relates pressure in terms ofenergy density and specific entropy, that is p = p(ρ, S)
In many situation the entropy is very small, so p = p(ρ)
Astrophysical systems can be often modeled with a polytropic equation of state: p = K ρΓo with K and Γ
constants and ρo the rest mass density.
The total energy density is then given by ρ = ρo(1 + ε) with ε the specific internal energy.
With the Gamma-law p = (Γ− 1)ρo ε one can show that
ρ =
( p
K
)1/Γ+
p
Γ− 1
Pablo Laguna Gravitation: Schwarzschild Black Holes
EXAMPLE: Constant density sphere
ρ(r) =
ρ∗ r < R0 r > R
Integration of the dm/dr equation yields
m(r) =
43π r3ρ∗ r < R
43π R3ρ∗ = M r > R
Integration of the dp/dr equation yields
p(r) = ρ∗
√
1− 2 MR −
√1− 2 M
Rr2
R2√1− 2 M
Rr2
R2 − 3√
1− 2 MR
Notice that p(r = R) = 0 and
p(r = 0) = ρ∗
√
1− 2 MR − 1
1− 3√
1− 2 MR
Notice also that the pressure becomes infinity if the mass of the star exceeds Mmax = 4 R/9.
Pablo Laguna Gravitation: Schwarzschild Black Holes
Chandrasekhar limit
WHITE DWARFS:
As stars run out of burnable fuel, they collapses.
A star burned-out star with mass M < 8 M , consists of a gas of electrons and ions.
When the electron separation becomes comparable to the de Broglie wavelength, λ = h/p, the starbecomes supported by electron degeneracy pressure because of the Pauli exclusion principle
The resulting object is a white dwarf with masses Mwd ≤ 1.4 M and radius Rwd ≈ R⊕ .
The maximum mass of a white dwarfs for which hydrostatic equilibrium can be maintained isMwd = 1.4 M . This limit is known as the Chandrasekhar limit
Pablo Laguna Gravitation: Schwarzschild Black Holes
Oppenheimer-Volkoff limit
NEUTRON STARS:
For stars with mass 8 M < M < 18 M , electron degeneracy pressure is not capable to stop thegravitational collapse.
Electrons combine with protons to make neutrons and neutrinos.
The results is an object supported by neutron degeneracy pressure called neutron star with masses1.2 M < Mns ≈ 2 M and radius Rns ≈ 10 km).
The maximum mass of a neutron star for which hydrostatic equilibrium can be maintained isMns = 3− 4,M . This limit is known as the Oppenheimer-Volkoff limit
Highly magnetized, rotating neutron stars that emit a beam of electromagnetic radiation are called pulsars.
Pablo Laguna Gravitation: Schwarzschild Black Holes