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Explicit expressions for optical scalars in gravitational
lensing from general matter sources
talk by: Emanuel Gallo
in collaboration with Osvaldo Moreschi
FaMAF, Universidad Nacional de Córdoba,
Instituto de Fı́sica Enrique Gaviola (IFEG), CONICET,
Ciudad Universitaria, (5000) Córdoba, Argentina
I CosmoSul, Rio De Janeiro; Agosto 1, 2011
(FaMAF, UNC, CONICET)
I CosmoSul, Rio De Janeiro; Agosto 1, 2011
Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Content
1
Introduction
Standard gravitational lensing notation
2
Some observations on the standard gravitational lensing expressions
3
Integrated expansion and shear
General equations: The geodesic deviation equation
Approximation method for solving the geodesic deviation equation
Expressions for the lens optical scalar in terms of Weyl and Ricci curvature from
geodesic deviation equation
The thin lens approximation
Spherically symmetric lenses
Spherically symmetric lenses
4
Two simple examples
5
Summary
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I CosmoSul, Rio De Janeiro; Agosto 1, 2011
Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Content
1
Introduction
Standard gravitational lensing notation
2
Some observations on the standard gravitational lensing expressions
3
Integrated expansion and shear
General equations: The geodesic deviation equation
Approximation method for solving the geodesic deviation equation
Expressions for the lens optical scalar in terms of Weyl and Ricci curvature from
geodesic deviation equation
The thin lens approximation
Spherically symmetric lenses
Spherically symmetric lenses
4
Two simple examples
5
Summary
(FaMAF, UNC, CONICET)
I CosmoSul, Rio De Janeiro; Agosto 1, 2011
Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Introduction:
Standard gravitational lensing notation
I
Gravitational lensing in astrophysics
Gravitational lensing has became a significant tool to make progress in our
knowledge on the matter content of our Universe. In particular, there is a large
number of works that use gravitational lensing techniques in order to know how
much mass are in galaxies or clusters of galaxies.
One of the most exiting results was to reafirm the need for some kind of dark
matter, that appears to interact with the barionic matter only through gravitation.
The question in which there is yet not general agreement is on the nature of this
dark matter. The most common conception is that it is based on collisionless
particles[Wei08], and where the pressures are negligible. However in the context of
cosmological studies, one often recurs to models of dark matter in terms of scalar
fields or spinors fields.
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Introduction:
Standard gravitational lensing notation
II
Standard gravitational lensing notation
In many astrophysical situations, the gravitational efects on light rays is weak, and the
source and observer are far away from the lens, therefore they are studied under the
formalism of weak field and thin gravitational lenses.
In the framework of weak field gravitational lensing the lens equation reads
β a = θa −
dls a
α .
ds
s1
(1)
s2
d ls
l1
l2
β
ds
dl
θ
^α
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Introduction:
Standard gravitational lensing notation
III
The differential of this equation can be written as
δβ a = Aab δθb ,
where the matric Aab is in turn expressed by
1 − κ − γ1
Aab =
−γ2
−γ2
1 − κ + γ1
(2)
;
(3)
where the optical scalars κ, γ1 and γ2 , are known as convergence κ and shear
components {γ1 , γ2 }, or γ = γ1 + iγ2 and have the information of distortion of the
image of the source due to the lens effects.
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Content
1
Introduction
Standard gravitational lensing notation
2
Some observations on the standard gravitational lensing expressions
3
Integrated expansion and shear
General equations: The geodesic deviation equation
Approximation method for solving the geodesic deviation equation
Expressions for the lens optical scalar in terms of Weyl and Ricci curvature from
geodesic deviation equation
The thin lens approximation
Spherically symmetric lenses
Spherically symmetric lenses
4
Two simple examples
5
Summary
(FaMAF, UNC, CONICET)
I CosmoSul, Rio De Janeiro; Agosto 1, 2011
Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Introduction:
Standard gravitational lensing notation
I
It is somehow striking that in most astronomical works on gravitational lensing, it is
assumed that the lens scalars and deflection angle, can be obtained from a
Newtonian-like potential function. These expressions although are easy to use, have
some limitations:
They neglect more general distribution of energy-momentum tensor Tab , in
particular they only take into account the timelike component of this tensor. In
this way they severely restrict the possible candidates to dark matter that can be
studied with these expressions.
They are not expressed in terms of gauge invariant quantities.
Since these expressions are written in terms of a potential function, it is not easily
seen how different components of Tab contribute in the generation of these images.
Most of them assume from the beginning that thin lens is a good approximation.
4 We extend the work appearing in standard references on gravitational lensing and
present new expressions that do not suffer from the limitations mentioned above.
4 We present gauge invariant expressions for the optical scalars and deflection angle
for some general class of matter distributions.
4 In this work we study gravitational lensing over a flat background.
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Content
1
Introduction
Standard gravitational lensing notation
2
Some observations on the standard gravitational lensing expressions
3
Integrated expansion and shear
General equations: The geodesic deviation equation
Approximation method for solving the geodesic deviation equation
Expressions for the lens optical scalar in terms of Weyl and Ricci curvature from
geodesic deviation equation
The thin lens approximation
Spherically symmetric lenses
Spherically symmetric lenses
4
Two simple examples
5
Summary
(FaMAF, UNC, CONICET)
I CosmoSul, Rio De Janeiro; Agosto 1, 2011
Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
General equations: The geodesic deviation equation I
General equations: The geodesic deviation equation
Let us consider the general case of a null geodesic starting from the position ps (source)
∂
and ending at po (observer). Let us characterize the tangent vector as ` = ∂λ
; so that
`b ∇b `a = 0;
(4)
that is, λ is an affine parameter.
We can now consider also a continuous set of nearby null geodesics. This congruence of
null geodesics can be constructed in the following way. Let S be a two dimensional
spacelike surface (the source image) such that the null vector ` is orthogonal to S. Next
we can generalize ` to be a vector field in the vicinity of the initial geodesic in the
following way: let the function u be defined so that it is constant along the congruence
of null geodesics emanating orthogonally to S and reaching the observing point po .
Then, without loss of generality we can assume that
`a = ∇a u;
(5)
which implies that the congruence has zero twist.
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
General equations: The geodesic deviation equation II
We can complete to a set of null tetrad, so that ma and m̄a are tangent to S. At other
points ma is chosen so that it is tangent to the surfaces u =constant and λ =constant.
Then a deviation vector at the source image can be expressed by
ς a = ς m̄a + ς¯ma .
(6)
In order to propagate this deviation vector along the null congruence one requires, that
its Lie derivate vanishes along the congruence; that is
L` ς a = 0.
(7)
Using the GHP notation one can it can be shown thta the geodesic deviation equation
can be written as
0 = Þ(ς) + ςρ + ς¯σ;
where Þ is the well behaved derivation of type {1, 1} in the direction of `(the null
geodesic vector of the congruence).
Defining X by
ς
X =
;
ς¯
(FaMAF, UNC, CONICET)
(8)
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
General equations: The geodesic deviation equation III
the equation for ς can be written as
`(`(X )) = −QX ;
(9)
where Q is given by
Q=
with
Φ00
Ψ̄0
Ψ0
Φ00
;
(10)
1
Φ00 = − Rab `a `b ,
2
(11)
Ψ0 = Cabcd `a mb `c md .
(12)
and
Therefore, this form of the equation only involves curvature quantities.
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Approximation method for solving the geodesic deviation
equation I
Approximation method for solving the geodesic deviation equation
Let us first transform to a first order differential equation. Defining V to be
V≡
and
X≡
one obtains
`(X) =
dX
=
dλ
with
A≡
dX
;
dλ
X
V
;
V
−Q X
0
−Q
(13)
I
0
(14)
= A X;
(15)
.
Equation (15) can be re-expressed in integral form, which gives
Z λ
X(λ) = X0 +
A(λ0 ) X(λ0 ) dλ0 .
(16)
(17)
λ0
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Approximation method for solving the geodesic deviation
equation II
The complete linear iteration is
!
R λ R λ0
(λ − λ0 )I − λ0 λ0 (λ00 − λ0 )Q 00 dλ00 dλ0
X0 .
X3 (λ) =
Rλ
I − λ0 (λ0 − λ0 )Q 0 dλ0
(18)
Now in order to integrate the geodesic deviation equation, we must choose the correct
initial conditions. In the case of light rays belonging to the past null cone of the
observer and intersecting S at the source, this initial conditions are X = 0 and V =
6 0;
since one can think the beam, starts backwards in time from the observer position, and
so initially has vanishing departure, but with nonzero expansion and shear.
Therefore in the linear approximation one has
Z λ
X (λ) = (λ − λ0 )I −
(λ − λ0 )(λ0 − λ0 )Q 0 dλ0 V(λ0 );
(19)
I−
R λ R λ0 00 00 0
Q dλ dλ
λ0 Rλ0
λ
− λ0 Q 0 dλ0
λ0
and
V(λ) =
Z
I−
λ
(λ0 − λ0 )Q 0 dλ0 V(λ0 ).
(20)
λ0
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Approximation method for solving the geodesic deviation
equation III
If the metric were flat (Q = 0), in order to get a deviation vector constructed from X1 ,
defined as X evaluated at λs = λ0 + ds , one must choose as initial condition
V(λ0 ) =
1
1
X (λs = λ0 +ds ) = X1 .
(λs − λ0 )
ds
(21)
Using a complex displacement ς of unit modulus; namely ς = e iϕ , to represent the
deviation vector, one can express the equation in the form
Z ds
Z ds
1
1
ςs (ϕ) = 1 −
λ0 (ds − λ0 )Φ00 (λ0 ) dλ0 −
λ0 (ds − λ0 )Ψ0 (λ0 ) dλ0 e −2iϕ e iϕ .
ds 0
ds 0
(22)
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Optical scalar in terms of the curvature I
Optical scalar in terms of the curvature
In order to compare with the standard representation of the lens scalar we note that the
original deviation vector in the source will be given by the previous equationi.e.
Z ds 0
λ (ds − λ0 ) 0 0
ςs
ςo
= I−
Q dλ
;
(23)
ς¯s
ς¯o
ds
0
if we make the following decomposition into real and imaginary part,
ςo
=
ςoR + iςoI ,
(24)
ςs
=
ςsR + iςsI ,
(25)
Ψ0
=
Ψ0R + iΨ0I ;
(26)
we obtain from eq.(23) that
Z ds 0
Z ds 0
λ (ds − λ0 )
λ (ds − λ0 ) 0
Φ000 + Ψ00R dλ0 ςoR −
Ψ0I dλ0 ςoI ,
ςsR = 1 −
ds
ds
0
0
Z ds 0
Z ds 0
0
λ (ds − λ )
λ (ds − λ0 ) 0
ςsI = 1 −
Φ000 − Ψ00R dλ0 ςoI −
Ψ0I dλ0 ςoR .
d
d
s
s
0
0
(27)
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Optical scalar in terms of the curvature II
Note also that in principle the integration must be made through the actual geodesic
followed by a photon in its path from the source to observer. However the last
expressions are valid only in the limit where the linear approximation is valid. If one
considers a linear perturbation from flat spacetime, then the curvature components Φ00
and Ψ0 would be already of linear order. Then, in the context of weak field gravitational
lensing, it is consistent to consider a null geodesic in flat spacetime; since the actual null
geodesic can be thought as a null geodesic in flat spacetime plus some corrections of
higher orders.
Now, in order to compare with the usual expressions for the lens scalars κ, γ1 and γ2 , let
us recall that they are defined via the relation eq.(2); but since it is a linear relation, one
can relate the deviation vectors by the same matrix, namely
ςsi = Aij ςoj ;
(28)
where {ςsi , ςoi } are the spatial vector associated with {ςs , ςo } respectively. Therefore, by
replacing into eq.(28), we obtain
(FaMAF, UNC, CONICET)
ςsR
=
(1 − κ − γ1 )ςoR − γ2 ςoI ,
(29)
ςsI
=
−γ2 ςoR + (1 − κ + γ1 )ςoI ;
(30)
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Optical scalar in terms of the curvature III
By taking real and imaginary part of previous equation one obtains
Z ds
1
κ =
λ0 (ds − λ0 )Φ000 dλ0 ,
ds 0
Z ds
1
λ0 (ds − λ0 )Ψ00R dλ0 ,
γ1 =
ds 0
Z ds
1
γ2 =
λ0 (ds − λ0 )Ψ00I dλ0 .
ds 0
(31)
(32)
(33)
These expressions for the weak field lens quantities are explicitly gauge invariant,
since they are given in terms of the curvature components,
Note that these expressions are valid for any weak field gravitational lens on a
perturbed flat spacetime, without restriction on the size of the lens compared with
the other distances.
Note that the equations for the shears can be written as
Z ds
1
γ1 + iγ2 =
λ0 (ds − λ0 )Ψ00 dλ0 .
ds 0
(FaMAF, UNC, CONICET)
(34)
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
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The thin lens approximation I
The general case
If we assume a thin lens, then Φ00 and Ψ0 will be sharply peaked around λ = dl where is
located the lens. Then, the expressions for the lens scalars are reduced to
κ=
dl dls
Φ̂00 ,
ds
γ1 + iγ2 =
dl dls
Ψ̂0 ,
ds
(35)
(36)
where
Z
ds
Φ00 dλ,
Φ̂00 =
0
Z
Ψ̂0 =
(37)
ds
Ψ0 dλ,
0
are the projected curvature scalars along the line of sight.
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
The thin lens approximation II
The axially symmetric case
Let us define
Ψ̂0 (J)
=
−e 2iϑ ψ̂0 (J),
(38)
and
γ1 + iγ2 = −γe 2iϑ ;
(39)
then one has
κ
=
γ=
(FaMAF, UNC, CONICET)
dls dl
Φ̂00 (J),
ds
dls dl
ψ̂0 (J).
ds
(40)
(41)
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
The thin lens approximation III
Deflection angle in terms of projected Ricci and Weyl Scalars
We wish now to express the deflection angle in terms of the curvature scalars.
If we define the components of αi = (α1 , α2 ) as
(αi ) = α(J)(
z0 x0
, );
J J
(42)
Then, it can be shown then, that the optical scalars can be written in terms of α(J) as
α(J)
1 dls dl dα
+
,
(43)
κ =
2 ds
dJ
J
α(J)
1 dls dl
dα
γ1 =
cos(2ϑ)
−
,
(44)
2 ds
dJ
J
α(J)
1 dls dl
dα
sin(2ϑ)
−
.
(45)
γ2 =
2 ds
dJ
J
It is interesting to note that
κ − γ1 cos(2ϑ) − γ2 sin(2ϑ) =
(FaMAF, UNC, CONICET)
dl dls α(J)
;
ds
J
(46)
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
The thin lens approximation IV
Finally from the last expression it is deduced that the deflection angle for axially
symmetric spacetimes is given by
α(J) = J(Φ̂00 (J) + ψ̂0 (J)) .
(47)
This constitutes a very simple equation for the bending angle expressed in terms of the
gauge invariant curvature components in compact form.
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Spherically symmetric lenses I
Expressions for the bending angle in terms of energy-momentum
components and M(r )
A stationary spherically symmetric spacetime can be expressed in terms of the standard
line element
ds 2 = a(r )dt 2 − b(r )dr 2 − r 2 (dθ2 + sin2 θdϕ2 );
(48)
where it is convenient to define Φ(r ) and m(r ) from
a(r ) = e 2Φ(r ) ,
and
b(r ) =
(FaMAF, UNC, CONICET)
1
1−
2M(r )
r
(49)
.
(50)
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Spherically symmetric lenses I
For our purpose, it is more convenient to use a null coordinate system to describe the
spherically symmetric geometry. Let us introduce then, a function
u = t − r ∗;
(51)
where r ∗ is chosen so that u is null. Then by inspection of equation (48) one can see
that
r
b
dr ∗
du = dt −
dr = dt −
dr ;
(52)
dr
a
since then one has
√
(53)
ds 2 = a du 2 + 2 abdudr − r 2 (dθ2 + sin2 θdϕ2 ).
It is natural to define the principal null direction `˜P from
`˜P = du;
(54)
which implies that the vector is
1
`˜aP = g ab dub = √
ab
(FaMAF, UNC, CONICET)
∂
∂r
a
.
(55)
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Spherically symmetric lenses II
We complete to a null tetrad with
ñP =
∂
∂
+ UA ,
∂u
∂r
with the complex null vector
(56)
√
2P0 ∂
;
r ∂ζ
in terms of the stereographic coordinate ζ. In these expressions
m̃P =
1
A= √ ,
ab
(57)
(58)
and
1
a
=− .
2bA2
2
The energy momentum tensor can be given by
U=−
Ttt = %e 2Φ(r ) ;
Trr = (FaMAF, UNC, CONICET)
Pr
1−
2M(r )
r
(59)
(60)
;
(61)
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
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Spherically symmetric lenses III
Tθθ = Pt r 2 ;
2
(62)
2
Tϕϕ = Pt r sin(θ) ;
(63)
where we have introduced the notion of radial component Pr and tangential component
Pt , due to our general anisotropic asumption.
The (t, t) component of the field equations implies
dM
= 4πr 2 %.
dr
(64)
One can show that the Ricci scalars are given by
Φ̃00 =
4π
(% + Pr ) ,
a
(65)
Φ̃11 = π (% − Pr + 2Pt ) ,
(66)
Φ̃22 = aπ (% + Pr ) ,
(67)
and the Weyl scalar
Ψ̃2 =
(FaMAF, UNC, CONICET)
4π
M
(% − Pr + Pt ) − 3 .
3
r
(68)
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Spherically symmetric lenses IV
When one made a tetrad transformation between the spherically symmetric tetrad and
the tetrad adapted to the photon geodesic, one find that the function α(J) expressed in
terms of the spherically symmetric null tetrad reads,
Z dls 3J 2
1
2J 2
α(J) = J
− 2 Ψ̃2 + 2 (Φ̃11 − Φ̃00 ) + Φ̃00 dy .
(69)
r
r
4
−dl
Note that in this case, the integrationpis on the coordinate y , instead of using arbitrary
affine parameter. Also note that r = J 2 + y 2 .
This constitutes an important explicit relation for the bending angle in terms of the
curvature as seen in an spherically symmetric frame; which is the natural frame for the
sources of the gravitational lens.
In terms of the physical energy-momentum tensor it is obtained
dls
Z
α(J) = J
−dl
(FaMAF, UNC, CONICET)
3J 2
r2
M(r )
4π
−
%(r
)
+
4π
(%(r
)
+
P
(r
))
dy ;
r
r3
3
(70)
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
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Spherically symmetric lenses V
Expressions for the lens scalars in terms of energy-momentum
components and M(r )
κ=
γ=
4πdl dls
ds
dl dls
ds
Z
Z
dls
−dl
dls
ρ + Pr +
−dl
J2
r2
J2
(P
−
P
)
dy .
t
r
r2
3M
−
4π(ρ
+
P
−
P
)
dy .
t
r
r3
(71)
(72)
These new expressions let us see explicitly the contributions of different components of
the energy-momentum tensor on the optical scalars. One can see that a couple of terms
disappear in the isotropic case in which Pr = Pt .
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Content
1
Introduction
Standard gravitational lensing notation
2
Some observations on the standard gravitational lensing expressions
3
Integrated expansion and shear
General equations: The geodesic deviation equation
Approximation method for solving the geodesic deviation equation
Expressions for the lens optical scalar in terms of Weyl and Ricci curvature from
geodesic deviation equation
The thin lens approximation
Spherically symmetric lenses
Spherically symmetric lenses
4
Two simple examples
5
Summary
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Two simple examples I
In order to show the application of our treatment of gravitational lens, we will consider
next two standard models that are often used in representing the source of gravitational
lenses.
A monopole mass (Schwarzschild)
As a simple example let there be a monopole distribution characterized by a mass M,
, then by considering that
therefore a simple computation gives Φ̃00 = 0, and Ψ̃2 = − M
r3
the observer and the source are far away, one can replace in the extremes of the
integration (as is usually made) ds → ∞ and dl → ∞, then
Φ̂00
=
0,
(73)
Z
ψ̂0
=
∞
−2
0
2
3J
4M
Ψ̃2 dy = 2 ,
r2
J
(74)
and by replacing into eqs.(40), (41) and (47), we readily obtain the well known results
(FaMAF, UNC, CONICET)
α(J)
=
κ
=
γ
=
4M
,
J
0,
dl dls 4M
.
ds J 2
(75)
(76)
(77)
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
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Two simple examples II
The isothermal profile
One simple model of dark matter that is used to explain the rotation curves of galaxies
is the isothermal profile, which is defined by the density function
ρ=
vc2
,
4πr 2
where vc is the circular velocity.
Since vc c, the pressures in this model are negligible. Then we obtain,
Z ∞ 2
vc
v 2π
Φ̂00 =
dy = c ,
2
J
−∞ r
Z ∞
2 2
2J vc
v 2π
ψ̂0 =
dy = c .
4
r
J
−∞
(FaMAF, UNC, CONICET)
(78)
(79)
(80)
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Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Two simple examples III
From these relations follow the well know results,
α
(FaMAF, UNC, CONICET)
=
κ
=
γ
=
2πvc2 ,
dl dls vc2 π
,
ds J
dl dls πvc2
.
ds J
(81)
(82)
(83)
I CosmoSul, Rio De Janeiro; Agosto 1, 2011
Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Content
1
Introduction
Standard gravitational lensing notation
2
Some observations on the standard gravitational lensing expressions
3
Integrated expansion and shear
General equations: The geodesic deviation equation
Approximation method for solving the geodesic deviation equation
Expressions for the lens optical scalar in terms of Weyl and Ricci curvature from
geodesic deviation equation
The thin lens approximation
Spherically symmetric lenses
Spherically symmetric lenses
4
Two simple examples
5
Summary
(FaMAF, UNC, CONICET)
I CosmoSul, Rio De Janeiro; Agosto 1, 2011
Explicit expressions for optical scalars in gravitational lensing from general matter /source
34
Summary I
We have presented explicit expressions for the bending angle and optical scalars in
terms of all the components of the energy-momentum tensor for a variety of cases.
The incidence of the spacelike components of the energy-momentum tensor is not
trivial.
This work can be extended of the study of sources with different structure and in a
cosmological background.
It should be very interesting to study differents models of dark matters in this
context.
N
It might be that the description of dark matter needs for the consideration of the
spacelike components of the energy-momentum tensor.
(FaMAF, UNC, CONICET)
I CosmoSul, Rio De Janeiro; Agosto 1, 2011
Explicit expressions for optical scalars in gravitational lensing from general matter /source
34