De Felice F., Bini D. Classical Measurements in Curved Space-Times
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9.7 Ray-tracing in Kerr space-time 223
Direct image
As the photon reaches the observer, on the photon orbit we have dθ/dr > 0 (i.e.
dμ/dr < 0) if β>0, and dθ/dr < 0 (i.e. dμ/dr > 0) if β<0. Therefore, when
β>0 the photon must encounter a turning point at μ = μ
+
: μ starts from 0,
goes up to μ
+
, then goes down to μ
obs
(which is ≤μ
+
). When β<0, the photon
must not encounter a turning point at μ = μ
+
: μ starts from 0 and monotonically
increases to μ
obs
.
The total integration over μ along the path of the photon from the emitting
source to the observer is thus given by
I
μ
=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
3
μ
+
0
+
3
μ
+
μ
obs
dμ
√
Θ
μ
(μ)
(β>0),
3
μ
obs
0
dμ
√
Θ
μ
(μ)
=
3
μ
+
0
+
3
μ
+
μ
obs
dμ
√
Θ
μ
(μ)
(β<0).
(9.126)
By using (9.116)weget
I
μ
=
1
a
2
(μ
2
+
+ μ
2
−
)
K(m
μ
)+
β
cn
−1
μ
obs
μ
+
"
"
"
"
m
μ
, (9.127)
where
β
=1ifβ>0,0ifβ =0,and−1ifβ<0, and K(m) is the complete
elliptic integral of the first kind.
Now let us consider the integration over r. Since the observer is at infinity, the
photon reaching him must have been moving in the allowed region defined by
r ≥ r
1
when R(r) = 0 has four real roots (case A), or the allowed region defined
by r ≥ r
3
when R(r) = 0 has two complex roots and two real roots (case B).
There are then two possibilities for the photon during its trip: it has encountered
a turning point at r = r
t
(r
t
= r
1
in case A, r
t
= r
3
in case B), or it has not
encountered any turning point in r. Define
I
∞
≡
!
∞
r
t
dr
R(r)
,I
r
em
≡
!
r
em
r
t
dr
R(r)
. (9.128)
Obviously, according to (9.113), a necessary and sufficient condition for the occur-
rence of a turning point in r on the path of the photon is that I
∞
<I
μ
. Therefore,
the total integration over r along the path of the photon from the emitting source
to the observer is
I
r
=
⎧
⎪
⎨
⎪
⎩
I
∞
+ I
r
em
(I
∞
<I
μ
),
3
∞
r
em
dr
√
R(r)
(I
∞
≥ I
μ
).
(9.129)
By definition, I
∞
, I
r
em
,andI
r
are all positive. According to (9.113), we must
have I
r
= I
μ
for the orbit of a photon. The relevant cases to be considered are
the following.
224 Measurements in physically relevant space-times
Case A: R(r) = 0 has four real roots.
When r
1
= r
2
,byusingEq.(9.118) to evaluate integrals in (9.128) with
r
t
= r
1
,weget
I
∞
=
2
(r
1
− r
3
)(r
2
− r
4
)
sn
−1
r
2
− r
4
r
1
− r
4
"
"
"
"
"
m
4
,
I
r
em
=
2
(r
1
− r
3
)(r
2
− r
4
)
×sn
−1
(r
2
− r
4
)(r
em
− r
1
)
(r
1
− r
4
)(r
em
− r
2
)
"
"
"
"
"
m
4
. (9.130)
Substitute these expressions into (9.129), then let I
r
= I
μ
, and finally
solve for r
em
:
r
em
=
r
1
(r
2
− r
4
) −r
2
(r
1
− r
4
)sn
2
(ξ
4
|m
4
)
(r
2
− r
4
) −(r
1
− r
4
)sn
2
(ξ
4
|m
4
)
, (9.131)
where
ξ
4
=
1
2
(I
μ
− I
∞
)
(r
1
− r
3
)(r
2
− r
4
). (9.132)
Since sn
2
(−ξ
4
|m
4
)=sn
2
(ξ
4
|m
4
), the solution given by (9.131) applies
whether I
μ
− I
∞
is positive or negative, i.e. whether or not there is a
turning point in r along the path of the photon.
Case B: R(r) = 0 has two complex roots and two real roots.
By using (9.121) to evaluate integrals in (9.128) with r
t
= r
3
,weget
I
r
em
=
1
√
AB
cn
−1
(A −B)r
em
+ r
3
B − r
4
A
(A + B)r
em
− r
3
B − r
4
A
"
"
"
"
"
m
2
,
I
∞
=
1
√
AB
cn
−1
A −B
A + B
"
"
"
"
"
m
2
, (9.133)
where A and B are given by (9.122). Substitute these expressions into
(9.129), then let I
r
= I
μ
, and finally solve for r
em
:
r
em
=
r
4
A −r
3
B − (r
4
A + r
3
B)cn(ξ
2
|m
2
)
(A −B) − (A + B)cn(ξ
2
|m
2
)
, (9.134)
where
ξ
2
=(I
μ
− I
∞
)
√
AB. (9.135)
Since cn(−ξ
2
|m
2
) = cn(ξ
2
|m
2
), the solution given by Eq. (9.134) applies
whether I
μ
− I
∞
is positive or negative, i.e. whether or not there is a
turning point in r along the path of the photon.
9.8 High-precision astrometry 225
First-order image
In this case the photon trajectory crosses the equatorial plane once.
The integration over μ along the path of the photon from the emitting source
to the observer is given by
I
μ
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
−
3
−μ
+
0
+
3
μ
+
−μ
+
−
3
μ
obs
μ
+
dμ
√
Θ
μ
(μ)
=
−
3
μ
+
0
+2
3
μ
+
−μ
+
+
3
μ
+
μ
obs
dμ
√
Θ
μ
(μ)
(β>0),
−
3
−μ
+
0
+
3
μ
obs
−μ
+
dμ
√
Θ
μ
(μ)
=
−
3
μ
+
0
+2
3
μ
+
−μ
+
−
3
μ
+
μ
obs
dμ
√
Θ
μ
(μ)
(β<0).
(9.136)
By using (9.116)weget
I
μ
=
1
a
2
(μ
2
+
+ μ
2
−
)
3K(m
μ
)+
β
cn
−1
μ
obs
μ
+
"
"
"
"
m
μ
. (9.137)
Now let us consider the integration over r. The discussion holding for the case of
the direct image applies also in this case. The solution is thus given by (9.131)
and (9.134), with I
μ
given by (9.137).
9.8 High-precision astrometry
Modern space technology allows one to measure with high accuracy the gen-
eral relativistic corrections to light trajectories due to the background curvature.
Astrometric satellites like GAIA and SIM, for example, are expected to provide
measurements of the position and motion of a star in our galaxy with an accuracy
of the order of O(3), recalling that we set O(n) ≡ O(1/c
n
). With such accuracy,
we must model and interpret the satellite observations in a general relativistic
context. In particular the measurement conditions as well as the determination
of the satellite rest frame must be modeled with equal accuracy. This frame con-
sists of a clock and a triad of orthonormal axes which is adapted to the satellite
attitude. We shall give here two examples of frames which best describe the
satellite’s measuring conditions. First we construct a Fermi frame which can be
operationally fixed by a set of three mutually orthogonal gyroscopes; then we
find a frame which mathematically models a given satellite attitude.
Since we have in mind applications to satellite missions, we fix the background
geometry as that of the Solar System, assuming that it is the only source of
gravity. Moreover we assume that it generates a weak gravitational field, so we
shall retain only terms of first order in the gravitational constant G and consider
these terms only up to order O(3).
226 Measurements in physically relevant space-times
The rest frame of a satellite consists of a clock which measures the satellite
proper time and a triad of orthonormal axes. The latter are described by 4-vectors
whose components are referred to a coordinate system which in general is not
connected to the satellite itself. Moreover, they are defined up to spatial rotations;
there are infinitely many possible orientations of the spatial triad which can be
fixed to a satellite, so our task is to identify those which correspond to actual
attitudes.
The space-time metric is given by
ds
2
≡ g
αβ
dx
α
dx
β
=[η
αβ
+ h
αβ
+ O(h
2
)]dx
α
dx
β
, (9.138)
where O(h
2
) denotes non-linear terms in h, the coordinates are x
0
= t, x
1
= x,
x
2
= y,x
3
= z, the origin being fixed at the center of mass of the Solar System,
and η
αβ
is the Minkowski metric, so that the metric components are
g
00
= −1+h
2
00
+ O(4),g
0a
= h
3
0a
+ O(5),g
ab
=1+h
2
00
δ
ab
+ O(4). (9.139)
Here h
2
00
=2U , where U is the gravitational potential generated by the bodies of
the Solar System, and the subscripts indicate the order of 1/c (e.g. h
3
0a
∼ O(3)).
Following Bini and de Felice (2003) and Bini, Crosta, and de Felice (2003), we
shall not specify the metric coefficients, in order to ensure generality.
Let us fix the satellite’s trajectory by the time-like, unitary 4-vector u
(u
α
u
α
= −1), given by
u
= T
s
(∂
t
+ β
1
∂
x
+ β
2
∂
y
+ β
3
∂
z
), (9.140)
where the ∂
α
are the coordinate basis vectors relative to the coordinate system,
and β
i
are the coordinate components of the satellite 3-velocity with respect to a
local static observer, recalling that we use subscripts here to refer to contravariant
components, so as not to confuse them with power indices. Finally we define
T
s
=1+(U +
1
2
β
2
)andβ
2
= β
2
1
+ β
2
2
+ β
2
3
.
Fermi frame
A Fermi frame adapted to a given observer can be obtained from any frame
adapted to that observer, provided we know its Fermi coefficients. The latter tell
how much the spatial axes of the given triad must rotate in order to be reduced
to a Fermi frame.
We shall apply this procedure to a tetrad adapted to u
. Although we use the
simplest possible frame, we find that an algebraic solution for a Fermi frame is
possible only if we confine ourselves to terms of the order O(2) and set β
3
=0.
To this order a tetrad solution is given by Bini and de Felice (2003) as
9.8 High-precision astrometry 227
λ
ˆ
0
=
1+U +
1
2
β
2
∂
t
+ β[cos ζ(t)∂
x
+sinζ(t)∂
y
],
λ
ˆ
1
=(1− U)[sin ζ(t)∂
x
− cos ζ(t)∂
y
], (9.141)
λ
ˆ
2
= β∂
t
+
1
2
β
2
− U +1
(cos ζ(t)∂
x
+sinζ(t)∂
y
),
λ
ˆ
3
=(1− U)∂
z
,
where we have set β
1
= β cos ζ(t), β
2
= β sin ζ(t), and β
3
=0,ζ being the
angular velocity of orbital revolution of the given observer. This tetrad is not a
Fermi tetrad because one Fermi coefficient is different from zero, namely
C
(fw)
ˆ
2
ˆ
1
= −
˙
ζ, (9.142)
a dot meaning derivative with respect to the coordinate time. Subtracting the
Fermi rotation at each time, the triad {λ
ˆa
} reduces to a Fermi triad:
R
ˆ
1
= −β sin ζ(t)∂
t
−
1
4
β
2
sin 2ζ(t)∂
x
−
β
2
2
sin
2
ζ(t) − U +1
∂
y
,
R
ˆ
2
= β cos ζ(t)∂
t
+
1 −U +
β
2
2
cos
2
ζ(t)
∂
x
(9.143)
+
1
4
sin 2ζ(t)β
2
∂
y
,
R
ˆ
3
=(1− U)∂
z
.
It is easy to verify that all the Fermi coefficients of the triad (9.143) vanish
identically. The operational setting of a Fermi triad by means of three mutually
orthogonal gyroscopes has a degree of arbitrariness which arises from the freedom
one has in fixing a spatial triad; a Fermi triad, in fact, is defined up to a constant
rotation. In most cases it is more convenient to define a frame which is fixed
to the satellite and is constrained according to criteria of best efficiency for the
mission goal.
Attitude frame
Let us find now a frame adapted to the satellite’s attitude, keeping the approxima-
tion to the order O(3). We first fix a coordinate system centered at the baricenter
of the Solar System, with the spatial axes pointing to distant sources; the latter
identify a global Cartesian-like spatial coordinate representation (x, y, z) with
respect to which the space-time metric takes the form (9.138). The world line of
an observer at rest with respect to the chosen coordinate grid is given by the unit
4-vector
u =(g
tt
)
−1/2
∂
t
≈ (1 + U)∂
t
, (9.144)
228 Measurements in physically relevant space-times
where t is a coordinate time. The observer u, together with the spatial axes as
specified, is a static observer, termed baricentric, and the parameter on its world
line is the baricentric proper time. Obviously, at each point in space-time there
exists a static observer u who carries a triad of spatial and mutually orthogonal
unitary vectors which point to the same distant sources as for the baricentric
frame.
As shown in Bini, Crosta, and de Felice (2003), the spatial triad of a static
observer at each space-time point is given to O(3) by the following vectors:
λ
ˆ
1
= h
01
∂
t
+(1− U)∂
x
,
λ
ˆ
2
= h
02
∂
t
+(1− U)∂
y
, (9.145)
λ
ˆ
3
= h
03
∂
t
+(1− U)∂
z
.
We need to identify the spatial direction to the geometrical center of the Sun as
seen from within the satellite. To this end we first identify this direction with
respect to the local static observer which is defined at each point of the satellite’s
trajectory, then we boost the corresponding triad to adapt it to the motion of
the satellite.
Let x
0
(t),y
0
(t),z
0
(t) be the coordinates of the satellite’s center of mass with
respect to the baricenter of the Solar System, and x
(t),y
(t),z
(t)thoseof
the Sun at the same coordinate time t. Here the time dependence is assumed to
be known. The relative spatial position of the Sun with respect to the satellite
at the time t is then
x
= x
− x
0
,
y
= y
− y
0
, (9.146)
z
= z
− z
0
.
With respect to a local static observer, the Sun direction is fixed, rotating the
triad (9.145) by an angle φ
s
around λ
ˆ
3
and then by an angle θ
s
around the vector
image of λ
ˆ
2
under the above φ
s
-rotation, where
φ
s
= tan
−1
y
x
,θ
s
= tan
−1
z
x
2
+ y
2
. (9.147)
Thus we have the new triad adapted to the observer u,
λ
s
ˆa
= R
2
(θ
s
)R
3
(φ
s
)λ
ˆa
, (9.148)
where
R
2
(θ
s
)=
⎛
⎝
cos θ
s
0sinθ
s
010
−sin θ
s
0cosθ
s
⎞
⎠
(9.149)
9.8 High-precision astrometry 229
and
R
3
(φ
s
)=
⎛
⎝
cos φ
s
sin φ
s
0
−sin φ
s
cos φ
s
0
001
⎞
⎠
. (9.150)
It should be noted here that, since the Sun is an extended body, its geometrical
center may be difficult to determine with great precision. The uncertainty in this
measurement may affect the precision in fixing the angles φ
s
and θ
s
from on
board the satellite, contributing to the determination of the error box.
From Eqs. (9.148)–(9.150), the explicit expressions for the coordinate compo-
nents of the vectors of the new triad are
λ
s
ˆ
1
= [cos θ
s
(cos φ
s
h
01
+sinφ
s
h
02
)+sinθ
s
h
03
]∂
t
+cosφ
s
cos θ
s
(1 −U)∂
x
+sinφ
s
cos θ
s
(1 −U)∂
y
+sinθ
s
(1 −U)∂
z
, (9.151)
λ
s
ˆ
2
= −(sin φ
s
h
01
+cosφ
s
h
02
)∂
t
−sin φ
s
(1 −U)∂
x
−cos φ
s
(1 −U)∂
y
, (9.152)
λ
s
ˆ
3
= −[sin θ
s
(cos φ
s
h
01
+sinφ
s
h
02
) −cos θ
s
h
03
]∂
t
−cos θ
s
sin θ
s
(1 −U)∂
x
−sin φ
s
sin θ
s
(1 −U)∂
y
+cosθ
s
(1 −U)∂
z
. (9.153)
It is easy to verify that the set {u, λ
s
ˆa
} forms an orthonormal tetrad; moreover,
it is equally straightforward to see that
cos θ
s
λ
s
ˆ
2
=
d
dφ
s
λ
s
ˆ
1
,λ
s
ˆ
3
=
d
dθ
s
λ
s
ˆ
1
. (9.154)
All quantities in equations (9.151)–(9.153) are defined at the position (x
0
,y
0
,z
0
)
of the satellite at time t.
Let us remember that our aim here is to identify a tetrad frame which is
adapted to the satellite and whose spatial triad mirrors its attitude. Recalling
that the satellite 4-velocity is given by u
as in (9.140), we boost the vectors of the
triad {λ
s
ˆa
} along the satellite’s relative motion to obtain the following boosted
triad (see (3.143)):
λ
bs
α
ˆa
= P (u
)
α
σ
λ
s
σ
ˆa
−
γ
γ +1
ν
σ
(ν
ρ
λ
s
ρˆa
)
ˆa=1,2,3
, (9.155)
where
P (u
)
α
σ
= δ
α
σ
+ u
α
u
σ
(9.156)
230 Measurements in physically relevant space-times
is the operator which projects onto the rest-space of u
, ν
α
≡ ν(u
,u)
α
is the
relative spatial velocity of u
with respect to the local static observer u, defined as
ν
α
=
1
γ
(u
α
− γu
α
), (9.157)
and γ ≡ γ(u
,u)=−u
α
u
α
is the relative Lorentz factor. The vector λ
bs
ˆ
1
identifies
the direction to the Sun as seen from within the satellite. The other vectors of
the boosted triad are related to λ
bs
ˆ
1
by the simple relations
λ
bs
ˆ
2
=
d
dφ
s
λ
bs
ˆ
1
,λ
bs
ˆ
3
=
d
dθ
s
λ
bs
ˆ
1
. (9.158)
The tetrad {λ
bs
ˆ
0
≡ u
,λ
bs
ˆa
} will be referred to as the Sun-locked frame. The relation
between the components ν(u
,u)
α
of the spatial velocity ν(u
,u) and the compo-
nents β
i
appearing in (9.140) is easily established from (9.140) itself and (9.157),
and is given by
ν(u
,u)
α
=
1
γ
T
s
β
i
δ
iα
+ δ
0α
− u
α
γ
. (9.159)
The explicit expressions for the components of the vectors λ
bs
ˆa
can be found in
the cited literature (Bini, Crosta, and de Felice, 2003; de Felice and Preti, 2008).
10
Measurements of spinning bodies
A test gyroscope is a point-like massive particle having an additional “structure”
termed spin. A spinning body is not, strictly speaking, point-like because its
average size cannot be less than the ratio between its spin and its mass, in
geometrized units. This guarantees that no point of the spinning body moves
with respect to any observer at a velocity larger than c.
Rotation is a common feature in the universe so knowing the dynamics of
rotating bodies is almost essential in modern physics. Although in most cases
the assumption of no rotation is necessary to make the equations tractable, the
existence of a strictly non-rotating system should be considered a rare event, and
a measurement which revealed one would be of great interest. This is the case
for the massive black holes which appear to exist in the nuclei of most galaxies.
Detailed measurements are aimed at detecting their intrinsic angular momentum
from the behavior of the surrounding medium. A black hole, however, could also
be seen directly if we were able to detect the gravitational radiation it would emit
after being perturbed by an external field. A direct measurement of gravitational
waves is still out of reach for earthbound detectors; nevertheless they are still
extensively searched for since they are the ultimate resource to investigate the
nature of space-time. Because of the vast literature available on this topic, in
what follows we shall limit ourselves to the interaction of gravitational waves
with a spinning body, with the aim of searching for new ways to detect them.
10.1 Behavior of spin in general space-times
As already stated, we shall distinguish between a point-like gyroscope and an
extended spinning body. In the first case, let us consider the world line γ of an
observer U who carries the gyroscope. We denote the spin of the gyroscope by
a vector S defined along γ, everywhere orthogonal to its tangent vector U and
undergoing Fermi-Walker transport:
232 Measurements of spinning bodies
D
(fw,U)
S
dτ
U
= ∇
U
S − (a(U) · S)U =0,S· U =0. (10.1)
In the second case, the behavior of an extended spinning body is described by
the Mathisson-Papapetrou equations (Mathisson, 1937; Papapetrou, 1951), later
generalized by Dixon (1970a; 1970b; 1979):
D
dτ
U
P
μ
= −
1
2
R
μ
αβγ
U
α
S
βγ
≡ F
(spin)μ
, (10.2)
D
dτ
U
S
μν
=2P
[μ
U
ν]
, (10.3)
where U is the unit time-like vector tangent to the line representative of the body
(hereafter, the center of mass line), and P
μ
and S
μν
are its momentum 4-vector
and antisymmetric spin tensor, satisfying the additional conditions
S
μν
P
ν
=0. (10.4)
Because of the spin–curvature coupling appearing in (10.2), U is in general accel-
erated, with a(U )=f(U); therefore, following the notation of Section 6.7 (see
(6.75)), we set
P (u, U)f(U)=γF(U, u). (10.5)
In the limit of small spin relative to the length scale of the background curvature,
Eqs. (10.2)–(10.4) imply (10.1), so the two descriptions agree.
In what follows we shall consider first the motion of a test gyroscope.
Test gyroscope: the projected spin vector
Let u be a vector field tangent to a congruence C
u
of curves which cross the world
line of U. With respect to C
u
the spin vector S admits the representation
S =Σ(U, u)+[ν(U, u) · Σ(U, u)]u, Σ(U, u)=P (u, U)S. (10.6)
Since S is Fermi-Walker transported along U , its magnitude is constant, whereas
the projected vector
Σ(U, u)=||Σ(U, u)||
ˆ
Σ(U, u) (10.7)
varies. The evolution of Σ(U, u) along U , as measured by the observer u,canbe
obtained as follows. Starting from (10.1) and using (10.6), we find
DS
dτ
U
=
D
dτ
U
Σ(U, u)+(ν(U, u) · Σ(U, u))
Du
dτ
U
+ u
D
dτ
U
ν(U, u) · Σ(U, u)+ν(U, u) ·
D
dτ
U
Σ(U, u)
, (10.8)