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Einstein’s Theory of Gravity
February 9, 2014
Einstein’s Theory of Gravity
Newtonian Gravity
Poisson equation
∇2 U(~x ) = 4πG ρ(~x ) → U(~x ) = −G
Z
d 3 x~0
ρ(~x )
|~x − x~0 |
For a spherically symmetric mass distribution of radius R
Z
1 R 02 0 0
U(r ) = −
r ρ(r )dr for r > R
r 0
Z
Z R
1 r 02 0 0
U(r ) = −
r ρ(r )dr −
r 0 ρ(r 0 )dr 0 for
r 0
r
Einstein’s Theory of Gravity
r <R
For a non-spherical distribution the term 1/|~x − x~0 | can be expanded as
1
|~x − x~0 |
=
xkxl
1 X x k x 0k
1 XX
+
+
3x 0k x 0l − ~r 02 δkl
+ ...
3
r
r
2
r5
k
U(~x ) = −
k
l
GM
G X k k G X kl x k x l
− 3
Q
x D −
+ ...
r
r
2
r5
kl
k
Gravitational Multipoles
Z
M
Dk
Q
kl
=
Z
=
=
ρ(x~0 )d 3 x 0
Mass
x 0k ρ(x~0 )d 3 x 0
Z Mass Dipole momenta
2
3x 0k x 0l − r~0 δkl ρ(x~0 )d 3 x 0
Mass Quadrupole tensorb
a
If the center of mas is chosen to coincide with the origin of the coordinates
then D k = 0 (no mass dipole).
b
If Q kl 6= 0 the potential will contain a term proportional to ∼ 1/r 3 and the
gravitational force will deviate from the inverse square law by a term ∼ 1/r 4 .
Einstein’s Theory of Gravity
The Earth’s polar and equatorial diameters differ by 3/1000 (Sun 10−5 ).
This deviation produces a quadrupole term in the gravitational potential,
which causes perturbations in the elliptical Kepler orbits of satellites.
Usually, we define the dimensionless parameter
J2 = −
Q 33
2
2M R
(1)
as a convenient measure of the oblateness of a nearly spherical body. For
the Sun the oblateness due to rotation gives J2 ≈ 10−7 .
The main perturbation is the precession of Kepler’s ellipse and this can
be used for precise determinations of the multipole moments and the
mass distribution in the Earth.
What about the Sun?
Einstein’s Theory of Gravity
Equivalence Principle
In GR gravitational phenomena arise not from forces and fields, but from
the curvature of the 4-dim spacetime. The starting point for this
consideration is the Equivalence Principle which states that the
Gravitational and the Inertial masses are equal. The equality mG = mI is
one of the most accurately tested principles in physics!
I
Now it is known experimentally that: γ = mGm−m
< 10−13
G
Experimental verification
Galileo (1610) ,
Newton (1680) γ < 10−3
Bessel (19th century) γ < 2 × 10−5
Eötvös (1890) & (1908) γ < 3 × 10−9
Dicke et al. (1964) γ < 3 × 10−11
Braginsky et al. (1971) γ < 9 × 10−13 ,
Kuroda and Mio (1989) γ < 8 × 10−10 ,
Adelberger et al. (1990) γ < 1 × 10−11
Su et al. (1994) γ < 1 × 10−12 (torsional)
Williams et al. (‘96), Anderson & Williams (‘01) γ < 1 × 10−13
Einstein’s Theory of Gravity
Equivalence Principle II
Weak Equivalence Principle :
The motion of a neutral test body released at a given point in
space-time is independent of its composition
Strong Equivalence Principle :
The results of all local experiments in a frame in free fall are
independent of the motion
The results are the same for all such frames at all places and
all times
The results of local experiments in free fall are consistent with
STR
Einstein’s Theory of Gravity
Equivalence Principle : Dicke’s Experiment
The experiment is based on measuring the effect of the gravitational field
on two masses of different material in a torsional pendulum.
(E )
(E )
(E )
GMmG
mI v 2
GM mG
2
=
⇒
v
=
R2
R
R
mI
The forces acting on both masses are:
"
(E ) (j) #
(j)
(j) 2
(j) GMm
m
v
GMm
m
m
G
G
I
G
= I
⇒ F (j) =
−
F (j) +
R2
R
R2
mI
mI
h
i
(1)
(2)
and the total torque applied is: L = F (1) − F (2) ` = GM
m
−
m
`
2
G
G
R
(1)
here we assumed that : mI
(2)
= mI
= m.
Einstein’s Theory of Gravity
Towards a New Theory for Gravity
Because of the success of Newton’s theory of gravity, our new theory
should obey two demands:
In an appropriate first (weak field) approximation the new theory
should reduce to the Newtonian one.
Beyond this approximation the new theory should predict small
deviations from newtonian theory which must be verified by
experiments/observations
Einstein’s equivalence principle:
Gravitational and Inertial forces/accelerations are equivalent and they
cannot be distinguished by any physical experiment.
This statement has 3 implications:
1
Gravitational forces/accelerations are described in the same way as
the inertial ones.
2
When gravitational accelerations are present the space cannot be
flat.
3
Consequence: if gravity is present there cannot exist inertial frames.
Einstein’s Theory of Gravity
1. Gravitational forces/accelerations are described in the same
way as the inertial ones.
• This means that the motion of a freely moving particle, observed
2 µ
from an inertial frame, will be described by ddtx2 = 0 while from a
non-inertial frame its movement will be described by the geodesic
equation
ρ
σ
d 2x µ
µ dx dx
+
Γ
= 0.
ρσ
ds 2
ds ds
• The 2nd term appeared due to the use of a non-inertial frame, i.e.
the inertial accelerations will be described by the Christoffel symbols.
• But according to Einstein the gravitational accelerations as well will
be described by the Christoffel symbols.
• This leads to the following conclusion: The metric tensor should play
the role of the gravitational potential since Γµρσ is a function of the metric
tensor and its derivatives.
Einstein’s Theory of Gravity
2. When gravitational accelerations are present the space cannot
be flat
since the Christoffel symbols are non-zero and the Riemann tensor is not
zero as well.
In other words, the presence of the gravitational field forces the space to
be curved, i.e. there is a direct link between the presence of a
gravitational field and the geometry of the space.
3. Consequence: if gravity is present there cannot exist inertial
frames.
If it was possible then one would have been able to discriminate among
inertial and gravitational accelerations which against the “generalized
equivalence principle” of Einstein.
The absence of “special” coordinate frames (like the inertial) and their
substitution from “general” (non-inertial) coordinate systems lead in
naming Einstein’s theory for gravity “General Theory of Relativity”.
Einstein’s Theory of Gravity
Einstein’s Equations
• Since the source of the gravitational field is a tensor (Tµν ) the field
should be also described by a 2nd order tensor e.g. Fµν .
• Since the role of the gravitational potential is played by the metric
tensor then Fµν should be a function of the metric tensor gµν and its 1st
and 2nd order derivatives.
• Moreover, the law of energy-momenum conservation implies that
T µν ;µ = 0 which suggests that F µν ;µ = 0.
• Then since F µν should be a linear function of the 2nd derivative of
gµν we come to the following form of the field equations (how & why?):
F µν = R µν + ag µν R + bg µν = κT µν
where κ =
8πG
c4 .
(2)
Then since F µν ;µ = 0 there should be
(R µν + ag µν R + bg µν );µ = 0
(3)
which is possible only for a = −1/2. Thus the final form of Einstein’s
equations is:
1
R µν − g µν R + Λg µν = κT µν .
(4)
2
where Λ = 8πG
c 2 ρv is the so called cosmological constant.
Einstein’s Theory of Gravity
Example: Another writing of Einstein Equations
We will show that Einstein’s equations can be written as
1
Rµν = −κ Tµν − gµν T
2
(5)
This is true because if we multiply
1
Rµν − gµν R = −κTµν
2
with g ρν they can be written as:
2R ρ ν − δ ρ ν R = −2κT ρ ν
and by contracting ρ and ν we get 2R − 4R = −2κT i.e. R = κT . Thus:
1
1
Rµν = −κTµν + gµν R = −κ Tµν − gµν T .
2
2
Einstein’s Theory of Gravity
Newtonian Limit
In the absence of strong gravitational fields and for small velocities both
Einstein & geodesic equations reduce to the Newtonian ones.
Geodesic equations:
d 2x µ
d 2 t/ds 2 dx µ
dx α dx β
=−
+ Γµαβ
2
dt
dt dt
(dt/ds)2 dt
⇒
If g00 ≈ η00 + h00 = 1 + h00 = 1 + 2 cU2 then
d 2xk
dt 2
d 2x j
≈ g00,j (6)
dt 2
∂U
≈ − ∂x
k
Einstein equations
Rµν
1
= −κ Tµν − gµν T
2
⇒ ∇2 U =
where κ = 8πG or κ = 8πG /c 4 .
Here we have used the following approximations:
Γj00 ≈
1
g00,j
2
and R00 ≈ Γj00,j ≈ ∇2 U
Einstein’s Theory of Gravity
1
κρ
2
(7)
Newtonian Limit: The geodesic equations
The geodesic equations are typically written with respect to the proper
time τ or the proper length s. In Newtonian theory the absolute time and
the proper time are identical thus the equations need to be written with
respect to the coordinate time t, which is not and affine parameter!
Thus we will use the form of the geodesic equations (for non affine
parameters) presented in Chapter 1 i.e. (??)
d 2 σ/ds 2 dx µ
d 2x µ
dx α dx β
=−
+ Γµαβ
2
dσ
dσ dσ
(dσ/ds)2 dσ
(8)
and if we select σ = x 0 = t the above relation will be written:
d 2x µ
dx α dx β
d 2 t/ds 2 dx µ
+ Γµαβ
=−
2
dt
dt dt
(dt/ds)2 dt
(9)
The 1st of the above equations (the one for x 0 ) simplified because
dx 0 /dt = dt/dt = 1 and d 2 x 0 /dt 2 = 0 and thus
Γ0αβ
dx α dx β
d 2 t/ds 2
=−
dt dt
(dt/ds)2
Einstein’s Theory of Gravity
(10)
This can be substituted in (9) and the remaining 3 equations will have
the form for the coordinates x k for (k = 1, 2, 3) become:
α β
k
dx dx
d 2x k
k
0 dx
= 0.
(11)
+ Γαβ − Γαβ
2
dt
dt
dt dt
Then we will use the approximation that all velocities are much smaller
than the speed of light i.e. u k = dx k /dt << 1 (in geometrical units we
assume that c = 1). Thus Γkαβ >> Γ0αβ dx k /dt and the previous
equation will be approximately:
d 2x k
≈ −Γk 00
(12)
dt 2
In other words the Christoffel symbol Γk00 corresponds to the Newtonian
force per unit of mass.
In a space which is “slightly curved” i.e. where
gµν ≈ ηµν + hµν
with ηµν >> hµν .
Under this assumption the Christoffel Γk 00 , gets the form
1
1
1
1
Γk 00 = g kλ (2gλ0,0 − g00,λ ) ≈ − η kλ g00,λ = δ kj g00,j = g00,k (13)
2
2
2
2
Einstein’s Theory of Gravity
Thus in the Newtonian limit the geodesic equations reduced to:
d 2x k
≈ g00,k .
dt 2
(14)
Which reminds the Newtonian relation
∂U
d 2x k
≈
dt 2
∂x k
(15)
which suggests that g00 has the form
g00 ≈ η00 + h00 = 1 + h00 = 1 + 2U
here we set U = 2h00 .
Einstein’s Theory of Gravity
(16)
Newtonian Limit: Einstein equations
We will show that Einstein’s equations in the Newtonian limit reduce to
the well known Poisson equation of Newtonian gravity.
For small concentrations of masses the dominant component of the
energy momentum tensor is the T00 . Thus the dominant component of
the Einstein’s equations is
1
1
1
R00 = −κ T00 − g00 T ≈ −κ T00 − η00 T ≈ − κT00
2
2
2
1
= − κρ
(17)
2
where we assumed that T = g µν Tµν ≈ η µν Tµν ≈ η 00 T00 = T00 .
The 00 component of the Ricci tensor is given by the relation
R00 = Γµ00,µ − Γµ0µ,0 + Γµ00 Γννµ − Γµ0ν Γν0µ ≈ Γµ00,µ ≈ Γj00,j
(18)
But as we have shown Γj00 ≈ g00,j /2 and thus:
R00 ≈ Γj00,j ≈
1
1 ∂ 2 g00
= ∇2 g00 ≈ ∇2 U
2 ∂x j ∂x j
2
leading to:
Einstein’s Theory of Gravity
(19)
1
κρ
(20)
2
From which by comparing with Poisson’s equation (1) we get the value of
the coupling constant κ that is:
∇2 U =
κ = 8πG .
Einstein’s Theory of Gravity
(21)
Solutions of Einstein’s Equations
• A static spacetime is one for which a timelike coordinate x 0 has the
following properties:
(I) all metric components gµν are independent of x 0
(II) the line element ds 2 is invariant under the transformation
x 0 → −x 0 .
Note that the 1st property does not imply the 2nd (e.g. the time reversal
on a rotating star changes the sense of rotation, but the metric
components are constant in time).
• A spacetime that satisfies (I) but not (II) is called stationary.
• The line element ds 2 of a static metric depends only on rotational
invariants of the spacelike coordinates x i and their differentials, i.e. the
metric is isotropic.
Einstein’s Theory of Gravity
The only rotational invariants of the spacelike coordinates x i and their
differentials are
~x · ~x ≡ r 2 ,
~x · d~x ≡ rdr ,
d~x · d~x ≡ dr 2 + r 2 dθ2 + r 2 sin2 θdφ2
Thus the more general form of a spatially isotropic metric is:
ds 2
2
= A(t, r )dt 2 − B(t, r )dt (~x · d~x ) − C (t, r ) (~x · d~x ) − D(t, r )d~x 2
= A(t, r )dt 2 − B(t, r )r dt dr − C (t, r )r 2 dr 2
−D(t, r ) dr 2 + r 2 dθ2 + r 2 sin2 θdφ2
(22)
= A(t, r )dt − B̃(t, r )dt dr − C̃ (t, r )dr − D̃(t, r ) dθ + sin θdφ2
= A0 (t, r̃ )dt 2 − B 0 (t, r̃ )dt dr̃ − C 0 (t, r̃ )dr̃ 2 − r̃ 2 dθ2 + sin2 θdφ2(23)
2
2
2
2
where we have set r̃ 2 = D(t, r ) and we redefined the A, B̃ and C̃ . The
next step will be to introduce a new timelike coordinate t̃ as
1 0
0
d t̃ = Φ(t, r̃ ) A (t, r̃ )dt − B (t, r̃ )dr̃
2
where Φ(t, r̃ ) is an integrating factor that makes the right-hand side an
exact differential.
Einstein’s Theory of Gravity
By squaring we obtain
2
2
d t̃ = Φ
1
A dt − A B dtdr̃ + B 02 dr̃ 2
4
02
2
0
0
from which we find
A0 dt 2 − B 0 dtdr̃ =
1
B0 2
2
d
t̃
−
dr̃
A0 Φ2
4A0
Thus by defining the new functions  = 1/(A0 Φ)2 and B̂ = C + B 0 /(4A0 )
the metric (23) becomes diagonal
ds 2 = Â(t̃, r̃ )d t̃ 2 − B̂(t̃, r̃ )dr̃ 2 − r̃ 2 dθ2 + sin2 θdφ2
(24)
or by dropping the ‘hats’ and ‘tildes’
ds 2 = A(t, r )dt 2 − B(t, r )dr 2 − r 2 dθ2 + sin2 θdφ2
(25)
• Thus the general isotropic metric is specified by two functions of t
and r , namely A(t, r ) and B(t, r ).
• Also, surfaces for t and r constant are 2-spheres (isotropy of the
metric).
• Since B(t, r ) is not unity we cannot assume that r is the radial
distance.
Einstein’s Theory of Gravity
Schwarzschild Solution
A typical solution of Einstein’s equations describing spherically symmetric
spacetimes has the form:
ds 2 = e ν(t,r ) dt 2 − e λ(t,r ) dr 2 − r 2 dθ2 + sin2 θdφ2
(26)
We next calculate the components of the Ricci tensor
Rµν = 0 .
(27)
Then the Christoffel symbols for the metric (26) are :
Γ000
Γ101
Γ112
Γ323
ν̇
ν0
ν0
,
Γ100 = e ν−λ ,
Γ001 =
2
2
2
0
λ̇
λ̇
λ
=
,
Γ011 = e λ−ν ,
Γ111 =
2
2
2
1
1
=
Γ313 = ,
Γ122 = −re −λ
r
r
= cot θ ,
Γ133 = −re −λ sin2 θ ,
Γ233 = sin θ cos θ
=
where ( · = ∂/∂t) and ( 0 = ∂/∂r )
Einstein’s Theory of Gravity
(28)
(29)
(30)
(31)
Then the components of the Ricci tensor will be:
R11
=
R00
=
R10
=
"
#
ν 00
ν 02
ν 0 λ0
λ0
λ̇2
λ̇ν̇
λ−ν λ̈
−
−
+
+
+e
+
−
2
4
4
r
2
4
4
00
ν
ν 02
ν 0 λ0
ν0
λ̈ λ̇2
λ̇ν̇
e ν−λ
+
−
+
− −
+
2
4
4
r
2
4
4
λ̇/r
(32)
(33)
(34)
−λ
R22
=
= −e
R33
=
sin2 θR22
h
i
r
1 + (ν 0 − λ0 ) + 1
2
(35)
(36)
in the absence of matter (outside the source) Rµν = 0 we can prove that
λ(r ) = −ν(r ), i.e. the solution is independent of time (how and why?).
Einstein’s Theory of Gravity
We will use the above relations to find the functional form of ν(t, r ) and
λ(t, r ) in an empty space i.e. when Rµν = 0.
Then from the sum: R11 + e λ−ν R00 = 0 we get:
ν 0 + λ0 = 0
(37)
while from
R10 =
λ̇
=0
r
(38)
2 λ
(e − 1)
r
(39)
we conclude that λ = λ(r ).
From R22 = 0 (eq. 35) we get:
ν 0 − λ0 =
Thus
ν0
=
λ0
=
1 λ
(e − 1)
r
1
− (e λ − 1)
r
Einstein’s Theory of Gravity
(40)
(41)
Since the right side of the above differential equation for ν is function
only of r the function ν = ν(t, r ) can be written as:
ν(t, r ) = α(t) + ν̃(r )
(42)
which leads to a redefinition of the time coordinate t̃,
d t̃ = e α/2 dt
(43)
i.e the time dependence of ν(t, r ) is ”absorbed” by the change of variable
from t to t̃.
Thus from (37) we lead to the relation:
λ(r ) = −ν̃(r )
(44)
i.e. both unknown components of the metric tensor are independent of
time and for simplicity we will write ν(r ) instead of ν̃(r ).
BIRKOFF’s THEOREM
If the geometry of a spacetime is spherically symmetric and is solution of
the Einstein’s equations, then it is described by the Schwarzschild
solution.
Einstein’s Theory of Gravity
The solution of equation (41) provides the function λ(r ). We can get it
by substituting f = e −λ then eq. (41) will be written as:
rf 0 + f = 1
(45)
with obvious solution:
f = e −λ = e ν = 1 −
k
r
(46)
where k and will be determined by the boundary conditions.
In the present case for r → ∞ our solutions should lead to Newton’s
solution that is
2
g00 = 1 + 2 U(r )
(47)
c
where U(r ) is the Newtonian potential and for the case of spherical
symmetry is
GM
(48)
U(r ) = −
r
which leads to
2GM
k=
(49)
c2
and Schwarzschild solutions gets the form:
Einstein’s Theory of Gravity
ds 2 =
1−
−1
2GM 2 2
2GM
c
dt
−
1
−
dr 2 − r 2 dθ2 + sin2 θdφ2
2
2
rc
rc
Sun: M ≈ 2 × 1033 gr and R = 696.000km
2GM
≈ 4 × 10−6
rc 2
Neutron star : M ≈ 1.4M and R ≈ 10 − 15km
2GM
≈ 0.3 − 0.5
rc 2
Neutonian limit:
g00 ≈ η00 + h00 = 1 +
2U
c2
⇒
U=
Einstein’s Theory of Gravity
GM
r
Schwarzschild Solution: Geodesics
1
1
r̈ − ν 0 ṙ 2 − re ν θ̇2 − re ν sin2 θφ̇2 + e 2ν ν 0 ṫ 2
2
2
2
θ̈ + ṙ θ̇ − sin θ cos θφ̇2
r
cos θ
2
θ̇φ̇
φ̈ + ṙ φ̇ + 2
r
sin θ
0
ẗ + ν ṙ ṫ
=
0
(50)
=
0
(51)
=
0
(52)
=
0
(53)
and from gλµ ẋ λ ẋ µ = 1 (here we assume a massive particle) we get :
−e ν ṫ 2 + e λ ṙ 2 + r 2 θ̇ + r 2 sin2 θφ̇2 = 1.
(54)
If a geodesic is passing through a point P in the equatorial plane
(θ = π/2) and has a tangent at P situated also in this plane (θ̇ = 0 at
P) then from (51) we get θ̈ = 0 at P and all higher derivatives are also
vanishing at P. That is, the geodesic lies entirely in the plane defined by
P, the tangent at P and the center of symmetry of the space.
Since the symmetry planes are equivalent to each other, it will be
sufficient to discuss the geodesics lying on one of these planes e.g. the
equatorial plane θ = π/2.
Einstein’s Theory of Gravity
The geodesics on the equatorial plane are:
1
1
r̈ − ν 0 ṙ 2 − re ν φ̇2 + e 2ν ν 0 ṫ 2 =
2
2
2
φ̈ + ṙ φ̇ =
r
ẗ + ν 0 ṙ ṫ =
ν 2
λ 2
2 2
−e ṫ + e ṙ + r φ̇
=
0
(55)
0
(56)
0
(57)
1
(58)
Then from equations (56) and (57) we can easily prove (how?) that:
d 2 r φ̇ = 0
dτ
d
e ν ṫ = 0
dτ
⇒
r 2 φ̇ = L = const : (Angular Momentum)(59)
⇒
e ν ṫ = E = const : (Energy)
(60)
• For a massive particle with unit rest mass, by assuming that τ is the
affine parameter for its motion we get p µ = ẋ µ . Thus
p0 = g00 ṫ = e ν ṫ = E
and pφ = gφφ φ̇ = −r 2 φ̇ = −L
Einstein’s Theory of Gravity
(61)
• An observer with 4-velocity U µ will find that the energy of a particle
with 4-momentum p µ is:
E = pµ U µ
An observer at infinity, U µ = (1, 0, 0, 0) will find E = p0 = Ec 2 .
Actually, for a particle with rest mass m0 we get E = E/(m0 c 2 ) i.e. it is
the energy per unit rest-mass.
For a particle at infinity we assume E = m0 c 2 .
————
Finally, by substituting eqns (60) and (59) into (58) we get the “energy
equation” for the r -coordinate:
eν 2
L + eν = E 2
r2
which suggests that at r → ∞ we get that E = 1.
ṙ 2 +
Einstein’s Theory of Gravity
(62)
By combining the 2 integrals of motion and eqn (58) we can eliminate the
proper time τ to derive an equation for a 3D path of the particle (how?)
d
u
dφ
2
+ u2 =
E 2 − 1 2Mu
+ 2 + 2Mu 3
L2
L
(63)
where we have used u = 1/r and
ṙ =
dr dφ
L dr
dr
=
= 2
dτ
dφ dτ
r dφ
The previous equation can be written in a form similar to Kepler’s
equation of Newtonian mechanics (how?) i.e.
d2
M
u + u = 2 + 3Mu 2
dφ2
L
(64)
The term 3Mu 2 is the relativistic correction to the Newtonian equation.
This term for the trajectories of planets in the solar system, where
M = M = 1.47664 km, and for orbital radius r ≈ 4.6 × 107 km
(Mercury) and r ≈ 1.5 × 108 km (Earth) gets very small values
≈ 3 × 10−8 and ≈ 10−8 correspondingly.
Einstein’s Theory of Gravity
Radial motion of massive particles
For the radial motion φ is constant, which implies that L = 0 and eqn
(62) reduces to
ṙ 2 = E 2 − e ν
(65)
and by differentiation we get an equation which reminds the equivalent
one of Newtonian gravity i.e.
r̈ =
M
.
r2
(66)
• If a particle is dropped from the rest at r = R (ṙ = 0) we get that
E 2 = e ν(R) = 1 − 2M/R and (65) will be written
1
1
2
−
(67)
ṙ = 2M
r
R
which is again similar to the Newtonian formula for the gain of kinetic
energy due to the loss in gravitational potential energy for a particle (of
unit mass) falling from rest at r = R.
Einstein’s Theory of Gravity
• For a particle dropped from the rest at infinity E = 1 the geodesic
equations are simplified (how?):
r
dr
dt
2M
−ν
=e
and
=−
(68)
dτ
dτ
r
The component of the 4-velocity will be:
dx µ
u =
=
dτ
µ
r
e
−ν
,−
!
2M
,0,0
r
(69)
Then by integrating the second of (68) and by assuming that at
τ = τ0 = 0 that r = r0 we get
r
r
2
r03
2
r3
τ=
−
(70)
3 2M
3 2M
q
r03
which suggests that for r = 0 we get τ → 32 2M
i.e. the particle takes
finite proper time to reach r = 0.
Einstein’s Theory of Gravity
If we want to map the trajectory of the particle in the (r , t) coordinates
we need to solve the equation
r
dr dτ
2M
dr
−ν
=
= −e
(71)
dt
dτ dt
r
The integration leads to the relation
!
r
r
r
r
2
r03
r3
r0
r
t =
−
+ 4M
−
3
2M
2M
2M
2M
p
! p
!
r /2M + 1
r0 /2M − 1 p
+ 2M ln p
r /2M − 1
r0 /2M + 1 (72)
where we have chosen that for t = 0 to set r = r0 .
Notice that when r → 2M then t → ∞. In other words, for an observer
at infinity, it takes infinite time for a particle to reach r = 2M.
———QUESTION : Can you find what will be the velocity of the radially
falling particle for a stationary observer at coordinate radius r̃ ?
Einstein’s Theory of Gravity
8.0
6.0
r/M
t : coordinate time
4.0
2.0
r=2M
τ : proper time
0.0
0
5
10
15
20
25
Time/M
Figure: Radial fall from rest towards a Schwarschild BH as described by a
comoving observer (proper time τ ) and by a distant observer
(Schwarschild coordinate time t)
2
τ=
3
t =
2
3
s

r03
2M
s
−
r3
2M

 + 4M
r
r
r0
2M
r03
2
−
2M
3
r
−
r
2M
r
r3
2M
+ 2M ln p
p
r /2M + 1
!
r /2M − 1
Einstein’s Theory of Gravity
!
p
r0 /2M − 1 p
r0 /2M + 1 Circular motion of massive particles
The motion of massive particles in the equatorial plane is described by
eqn (64)
M
d2
u + u = 2 + 3Mu 2
(73)
dφ2
L
For circular motions r = 1/u=const and ṙ = r̈ = 0. Thus we get
L2 =
r 2M
r − 3M
(74)
if we also put ṙ = 0 in eqn (62) we get :
1 − 2M/r
E=p
1 − 3M/r
(75)
The energy of a particle with rest-mass m0 in a circular radius r is then
given by E = E · (m0 c 2 ).
For the circular orbits to be bound we require E < m0 c 2 , so the limit on
r for an orbit to be bound is given by E = 1 which leads to
2
(1 − 2M/r ) = 1 − 3M/r
true when
r = 4M or r = ∞
Thus over the range 4M < r < ∞ circular orbits are bound.
Einstein’s Theory of Gravity
(76)
Figure: The variation of E = E/(m0 c 2 ) as a function of r /M for a
circular orbit of a massive particle in the Schwarzchild geometry
Einstein’s Theory of Gravity
Figure: The shape of a bound orbit outside a spherical star or a
black-hole
Einstein’s Theory of Gravity
From the integral of motion r 2 φ̇ = L and eqn (74) we get
dφ
dτ
2
=
M
r 2 (r − 3M)
(77)
NOTICE : This equation cannot be satisfied for circular orbits with
r < 3M. Such orbits cannot be geodesics and cannot be followed by
freely falling particles.
We can also calculate an expression for Ω = dφ/dt
2
Ω =
dφ
dt
2
=
dφ dτ
dτ dt
2
e 2ν
= 2
E
dφ
dτ
2
=
M
r3
which is equivalent to Kepler’s law in Newtonian gravity.
Einstein’s Theory of Gravity
(78)
Stability of massive particle orbits
According to the previous discussion the
closest bound orbit around a massive
body is at r = 4M, however we cannot
yet determine whether this orbit is
stable.
In Newtonian theory the particle motion
in a central potential is described by:
1
2
dr
dt
2
+ Veff (r ) = E 2
(79)
M
L2
+ 2
r
2r
(80)
where
Veff (r ) = −
The bound orbits have two turning points while the circular orbit
corresponds to the special case where the particle sits in the minimum of
the of the effective potential.
Einstein’s Theory of Gravity
In GR the ‘energy’ equation (62) is
ṙ 2 +
eν 2
L + eν = E 2
r2
(81)
which leads to an effective potential of the form
Veff (r ) =
L2
M
ML2
eν 2
ν
+
L
+
e
=
−
−
r2
r
2r 2
r3
(82)
Circular orbits occur where dVeff /dr = 0 that is:
dVeff
M
L2
3ML2
= 2 − 3 +
dr
r
r
r4
so the extrema are located at the solutions of the eqn
Mr 2 − Lr + 3ML2 = 0 which occur at
p
L r=
L ± L2 − 12M 2
2M
Einstein’s Theory of Gravity
(83)
√
√
Note that if L = 12M = 2 3M
then there is only one extremum and
no turning points in the orbit for
lower values of L. Thus the
innermost stable circular orbit
(ISCO) has
√
rISCO = 6M and L = 2 3M ≈ 3.46M
and it is unique in satisfying both
dVeff /dr = 0 and d 2 Veff /dr 2 = 0,
the latter is the condition for
marginal stability of the orbit.
Figure: The dots indicate the
locations of stable circular orbits
which occur at the local minimum of
the potential. The local maxima in
the potential are the locations of the
unstable circular orbits
Einstein’s Theory of Gravity
Figure: Orbits for L = 4.3 and different values of E .
The upper shows circular orbits. A stable (outer) and an unstable
(inner) one.
The lower shows bound orbits , the particle moves between two
turning points marked by dotted circles.
Einstein’s Theory of Gravity
Figure: Orbits for L = 4.3 and different values of E .
The upper shows a scattering orbit the particle comes in from
infinity passes around the center of attraction and moves out to
infinity again.
The lower shows plunge orbit , in which the particle comes in from
infinity.
Einstein’s Theory of Gravity
Trajectories of photons
Photons, as any zero rest mass particle, move on null geodesics. In this
case we cannot use the proper time τ as the parameter to characterize
the motion and thus we will use some affine parameter σ.
We will study photon orbits on the equatorial plane and the equation of
motion will be in this case
ν 2
e ṫ − e
−ν 2
e ν ṫ
=
E
(84)
2 2
=
0
(85)
r φ̇ =
L
(86)
ṙ − r φ̇
2
The equivalent to eqn (62) for photons is
ṙ 2 +
eν 2
L = E2
r2
(87)
while the equivalent of eqn (64) is
d2
u + u = 3Mu 2
dφ2
Einstein’s Theory of Gravity
(88)
Radial motion of photons
For radial motion φ̇ = 0 and we get
e ν ṫ 2 − e −ν ṙ 2 = 0
from which we obtain
2M
dr
=± 1−
dt
r
(89)
integration leads to:
Outgoing Photon
r
t = r + 2M ln − 1 + const
2M
Incoming Photon
r
t = −r − 2M ln − 1 + const
2M
Figure: Radially infalling particle
emitting a radially outgoing photon.
Einstein’s Theory of Gravity
Circular motion of photons
For circular orbits we have r =constant and thus from eqn (88) we see
that the only possible radius for a circular photon orbit is:
3GM
r = 3M
or r =
(90)
c2
There are no such orbits around typical stars because their radius is much
larger than 3M (in geometrical units). But outside the black hole there
can be such an orbit.
Einstein’s Theory of Gravity
Stability of photon orbits
We can rewrite the “energy” equation (87) as
where D = L/E and Veff
ṙ 2
eν
1
+
= 2
L2
r2
D
= e ν /r
(91)
Figure: The effective potential for photon orbits.
We can see that Veff has a single maximum at r = 3M where the value
of the potential is 1/(27M 2 ). Thus the circular orbit r = 3M is unstable.
There are no stable circular photon orbits in the Schwarzschild
geometry.
Einstein’s Theory of Gravity
The slow-rotation limit : Dragging of inertial frames
4Ma 2
sin θdtdφ
(92)
r
where J = Ma is the angular momentum.
The contravariant components of the particles 4-momentum will be
2
ds 2 = dsSchw
+
p φ = g φµ pµ = g φt pt + g φφ pφ
and p t = g tµ pµ = g tt pt + g tφ pφ
If we assume a particle with zero angular momentum, i.e. pφ = 0 along
the geodesic then the particle’s trajectory is such that
pφ
g tφ
2Ma
dφ
= t = tt = 3 = ω(r )
dt
p
g
r
which is the coordinate angular velocity of a zero-angular-momentum
particle.
A particle dropped “straight in” from infinity ( pφ = 0) is dragged just
by the influence of gravity so that acquires an angular velocity in the
same sense as that of the source of the metric.
The effect weakens with the distance and makes the angular
momentum of the source measurable in practice.
Einstein’s Theory of Gravity
Useful constants
Useful Constants in geometrical units
Speed of light
Planck’s constant
Gravitation constant
Energy
Distance
Time
Light year
Astronomical unit (AU)
Earth’s mass
Earth’s radius (equator)
Solar Mass
Solar Radius
c =299,792.458 km/s = 1
~ = 1.05 × 10−27 erg ·s = 2.612 × 10−66 cm2
G = 6.67 × 10−8 cm3 /g ·s 2 =1
eV=1.602×10−12 erg = 1.16 × 104 K
=1.7823×10−33 g =1.324×10−56 km
1 pc=3.09×1013 km=3.26 ly
1 yr = 3.156×107 sec
1 ly = 9.46×1012 km
1AU = 1.5 × 108 km
M⊕ = 5.97 × 1027 g
R⊕ = 6378 km
M = 1.99 × 1033 g =1.47664 km
R = 6.96 × 105 km
Einstein’s Theory of Gravity
The Classical Tests: Perihelion Advance
For a given value of angular momentum Kepler’s equation (64)
d2
M
u + u = 2 + 3Mu 2
2
dφ
L
(64)
(without the GR term 3Mu 2 ) admits a solution given by
u=
M
[1 + e cos(φ + φ0 )]
L2
(93)
where e and φ0 are integration constants.
• We can set φ0 = 0 by rotating the coordinate system by φ0 .
• The other constant e is the eccentricity of the orbit which is an
ellipse if e < 1.
Now since the term 3Mu 2 is small we can use perturbation theory to get
a solution of equation (64).
d2
M
3M 3
M
3M 3
6eM 3
2
u + u ≈ 2 + 4 [1 + e cos(φ)] ≈ 2 + 4 +
cos(φ)
2
dφ
L
L
L
L
L4
Because, 3M 3 /L4 M/L2 and can be omitted and its corrections will be
small periodic elongations of the semiaxis of the ellipse.
Einstein’s Theory of Gravity
The term 6eM 3 /L4 cos(φ) is also small but has an accumulative effect
which can be measured. Thus the solution of the relativistic form of
Kepler’s equation (94) becomes (k = 3M 2 /L2 )
M
M
3eM 2
u = 2 1 + e cos φ +
φ sin φ ≈ 2 {1 + e cos[φ(1 − k)]} (94)
L
L2
L
The perihelion of the orbit can be found be maximizing u that is when
cos[φ(1 − k)] = 1 or better if φ(1 − k) = 2nπ. This mean that after each
rotation around the Sun the angle of the perihelion will increase by
φn =
2nπ
1−k
⇒ φn+1 − φn =
2π
6πM 2
≈ 2π(1 + k) = 2π +
(95)
1−k
L2
Einstein’s Theory of Gravity
i.e. in the relativistic orbit the perihelion is no longer a fixed point, as it
was in the Newtonian elliptic orbit but it moves in the direction of the
motion of the planet, advancing by the angle
δφ ≈
6πM 2
L2
(96)
By using the well know relation from Newtonian Celestial Mechanics
L2 = Mr0 (1 − e 2 ) connecting the perihelion distance r0 with L we derive
a more convenient form for the perihelion advance
6πM
(97)
δφ ≈
r0 (1 − e 2 )
Solar system measurements
Mercury
Venus
Earth
(43.11 ± 0.45)00
(8.4 ± 4.8)00
(5.0 ± 1.2)00
43.0300
8.600
3.800
In binary pulsars separated by 106 km the perihelion advance is extremely
important and it can be up to ∼ 2o /year (about 1000 orbital rotations)
Einstein’s Theory of Gravity
Sources of the precession of perihelion for Mercury
Amount (arcsec/century)
5025.6
531.4
0.0254
42 ± 0.04
5600.0
5599.7
Gause
Coordinate ( precession of the equinoxes)a
Gravitational tugs of the other planets
Oblateness of the Sun
General relativity
Total
Observed
a
Precession refers to a change in the direction of the axis of a rotating
object. There are two types of precession, torque-free and torque-induced
Einstein’s Theory of Gravity
The Classical Tests : Deflection of Light Rays
Deflection of light rays due to presence of a gravitational field is a
prediction of Einstein dated even before the GR and has been verified in
1919.
Photons follow null geodesics, this means that ds = 0 and the integrals
of motion (E and L) are divergent but not their ratio L/E . Thus the
photon’s equation of motion on the equatorial plane of Schwarszchild
spacetime will be:
d 2u
+ u = 3Mu 2 .
(98)
dφ2
with an approximate solution (we omit at the moment the term 3Mu 2 )
1
cos(φ + φ0 )
(99)
b
which describes a straight line, where b and φ0 are integration
constants. Actually, with an appropriate rotation φ0 = 0 and the length b
is the distance of the line from the origin.
u=
Einstein’s Theory of Gravity
Since the term 3Mu 2 is very small we can substitute u with the
Newtonian solution (99) and we need to solve the non-homogeneous ODE
d 2u
3M
+ u = 2 cos2 φ .
2
dφ
b
(100)
admitting a solution of the form
cos φ M
+ 2 1 + sin2 φ
(101)
b
b
which for distant observers (r → ∞) i.e. u → 0, and we get a relation
between φ, M and the parameter b
cos φ M
+ 2 1 + sin2 φ = 0
(102)
b
b
and since r → ∞, this means that φ → π/2 + and cos φ → 0 + and
sin φ → 1 − we get:
2M
≈−
(103)
b
Since, φ → π/2 + 2M/b for r → ∞ on the one side & φ → 3π/2 − 2M/b
on the other side the total deviation will be the sum of the two i.e.
4M
δφ =
.
(104)
b
u=
Einstein’s Theory of Gravity
For a light ray tracing the surface of the Sun gives a deflection of
∼ 1.7500 .
The deflection of light rays is a quite common phenomenon in Astronomy
and has many applications. We typically observe “crosses” or “rings”
Figure:
Einstein Cross (G2237+030) is the most characteristic case of gravitational lens where a galaxy at a
distance 5 × 108 lys focuses the light from a quasar who is behind it in a distance of 8 × 109 lys. The focusing
creates 4 symmetric images of the same quasar. The system has been discovered by John Huchra.
Figure:
“Einstein rings” are observed when the source, the focusing body and Earth are on the same line of
sight. This ring has been discovered by Hubble space telescope.
Einstein’s Theory of Gravity
The Classical Tests : Gravitational Redshift
Figure:
Let’s assume 3 static observers on a Schwarszchild spacetime, one very close to the source of the field
the other in a medium distance from the source and the third at infinity.
The clocks of the 3 observers ticking with different rates. The clocks of
the two closer to the source are ticking slower than the clock of the
observer at infinity who measures the so called ‘ coordinate time” i.e.
1/2
1/2
2M
dτ1 = 1 − 2M
dt
and
dτ
=
1
−
dt this means that
2
r1
r2
dτ2
=
dτ1
1−
1−
2M
r2
1/2
M
M
1/2 ≈ 1 + r − r .
1
2
2M
r1
Einstein’s Theory of Gravity
(105)
If the 1st observer sends light signals on a specific wavelegth λ1 from
c = λ/τ we get a relation between the wavelength of the emitted and
received signals
M
M
λ2
≈1+
−
λ1
r1
r2
⇒
λ2 − λ1
∆λ
M
M
=
=
−
.
λ1
λ1
r1
r2
(106)
A similar relation can be found for the frequency of the emitted signal:
ν1
=
ν2
1−
2M
r2
1−
2M
r1
1/2
1/2 .
(107)
While the photon redshift z is defined by
1+z =
ν1
ν2
(108)
QUESTION : What will be the redshift for signals emitted from the
surface of the Sun, a neutron star and a black hole?
Einstein’s Theory of Gravity
The Classical Tests : Radar Delay
A more recent test (late 60s) where the delay of the radar signals caused
by the gravitational field of Sun was measured. This experiment
suggested and performed by I.I. Shapiro and his collaborators.
The line element ds 2 = gµν dx µ dx ν for the light rays i.e. for ds = 0 on
the equatorial plane has the form
−1 2
2
2M
dr
dφ
2M
0= 1−
− 1−
− r2
(109)
r
r
dt
dt
For the study of the radial motion we should substitute the term dφ/dt
from the integrals of motion (59) and (60) i.e. we can create the
quantity:
−1
L
2M
dφ
2
=r 1−
(110)
D=
E
r
dt
Einstein’s Theory of Gravity
Then eqn (109) becomes
−1 2
2
2M
dr
D2
2M
2M
− 1−
− 2 1−
= 0.
1−
r
r
dt
r
r
(111)
At the point of the closest approach to the Sun, r0 , there should be
r02
dr /dt = 0 and thus we get the value of D 2 = 1−2M/r
. Leading to an
0
equation for the radial motion:
r 2 1 − 2M/r 1/2
2M
dr
0
= 1−
1−
(112)
dt
r
r
1 − 2M/r0
leading to
Z
t1
r1
dr
=
q
2 1−2M/r
1 − rr0 1−2M/r
0
" p
#
r
q
2
2
r1 r1 − r0
r1 − r0
2
2
=
r1 − r0 + 2M ln
+M
r0
r1 + r0
2r1
≈ r1 + 2M ln
+M
r0
r0
1−
2M
r
Einstein’s Theory of Gravity
For flat space we have t1 = r1 i.e. the term t̃1 = 2M ln(2r1 /r0 ) + M is
the relativistic correction for the first part of the orbit in same way we get
a similar contribution as the signal returns to Earth. Thus the total
“extra time” is:
4r1 r2
∆T = 2 (t̃1 + t̃2 ) = 4M 1 + ln
.
(113)
r02
Einstein’s Theory of Gravity
Figure: Comparison of the experimental results with the prediction of the
theory. The results are from I.I. Shapiro’s experiment (1970) using Venus
as reflector.
Einstein’s Theory of Gravity
EXAMPLE: weighting a pulsar
Figure: As a pulsar passes behind its heavy companion star, its pulses are
delayed by the mass of the companion. (Credit B. Saxton/NRAO/AUI)
• If a pulsar is in orbit around a companion WD, its pulses of light will
follow the space curve caused by that star. When the companion WD
star is in front of the pulsar, the pulses take a little longer to reach us
than when the WD is clear of the pulsar. The amount of delay tells you
the amount of mass of the star causing the delay. 1
• Astronomers using the NSF’s Green Bank Telescope (GBT) have
discovered the most massive neutron star (2M ) yet found, a discovery
with strong and wide-ranging impacts across several fields of physics and
astrophysics.
1
NRAO:http://www.nrao.edu/index.php/learn/science/weighing-pulsars
Einstein’s Theory of Gravity