Higher-order corrections for the deflection of light around a massive object

0.65 MB·

by  Carlos Rodriguez

Higher-order corrections for the deflection of light
around a massive object
Carlos Rodriguez , Carlos A. Mar´ın 1
Universidad San Francisco de Quito
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1 1 [email protected]
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Abstract
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FromtheSchwarzschildmetricweobtainthehigher-orderterms(upto20-thor-
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8 der) for the deflection of light around a massive object using the Lindstedt-Poincar´e
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method to solve the equation of motion of a photon around the stellar object. Ad-

] ditionally, we obtain diagonal Pad´e approximants from the perturbation expansion,
c
q
and we show how these are a better fit for the numerical data. Furthermore, we use
-
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g these approximants in ray-tracing algorithms to model the bending of light around
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the massive object.

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v keywords: general relativity, light deflection, Einstein, black hole, Pad´e.
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3 Mathematics Subject Classification 2010: 83C10, 83C25, 83C57, 41A21
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. Introduction
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1 The General Theory of Relativity (GTR) is probably one of the most elegant theories ever
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v performed. It was put forth by Albert Einstein in its current form in a 1916 publication,
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which expanded on his previous work of 1915 [1, 2]. This is summarized in 14 equations
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a [3, 4]. The Einstein field equations (ten equations written in tensor notation)
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G ≡ R − Rg = kT +λg (1)
µν µν µν µν µν
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and the geodesic equations (4 equations)
d2xµ dxρ dxσ
+Γµ ( )( ) = 0 (2)
ds2 ρσ ds ds
In (1) G is the Einstein’ s Tensor, which describes the curvature of space-time, R is
µν µν
the Ricci tensor, and R is the Ricci scalar (the trace of the Ricci tensor), g is the metric
µν
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2
tensor that describes the deviation of the Pythagoras theorem in a curved space, T is
µν
the stress-energy tensor describing the content of matter and energy. k = 8πG, where
c4
c is the speed of light in vacuum and G is the gravitational constant. Finally, λ is the
cosmological constant introduced by Einstein in 1917 [5, 6, 7] that is a measure of the
contribution to the energy density of the universe due to vacuum fluctuations. In equation
(2) s is the arc length satisfying the relation ds2 = g dxµdxν and Γµ are the connection
µν ρσ
coefficients ( Christoffel symbols of the second kind). xµ is the position four-vector of the
particle. We use Greek letters as µ,ν,α,etc for 0,1,2,3. We have adopted the Einstein
summation convention in which we sum over repeated indices. Einstein’s equations ( 1)
tells us that the curvature of a region of space-time is determined by the distribution of
mass-energy of the same and they can be derived from the Einstein-Hilbert action [8, 9]:
(cid:90) √
S = d4x −g[R−2kL +2λ] (3)
F
(cid:60)
where (cid:60) represents a region of space-time, L is the Lagrangian density due to the fields
F
of matter and energy and g is the determinant of the metric tensor.
One of the most relevant predictions of General Relativity is the gravitational de-
flection of light. It was demonstrated during the solar eclipse of 1919 by two british
expeditions [10]. One of the expeditions was led by Arthur Eddington and was bound for
the island of Pr´ıncipe in East Africa. The other one was led by Andrew Crommelin in the
region of Sobral in Brazil. The light deflection can be measured taking a photograph of
a star near the limb of the Sun, and then comparing it with another picture of the same
star when the sun is not in the visual field. The observations are not easy. At present,
Very Long Baseline Interferometry (VLBI) is used to measure the gravitational deflection
of radio waves by the sun from observations of extragalactic radio sources [11]. The result
is very close to the value predicted by General Relativity [7] , which is Ω = 4GMΘ = 1.752
RΘc2
seconds of arc (M and R represent the solar mass and radius, respectively).
Θ Θ
In the literature we can find calculations to second order of the deflection of light by a
spherically symmetric body using Schwarzschild coordinates [12, 13, 14, 15] In this paper
using the Schwarzschild metric we obtain higher order corrections (up to 20-th order)
for the gravitational deflection of light around a massive object like a star or a black
hole using the Lindstedt-Poincar´e method to solve the equation of motion of a photon
around the stellar body. Additionally, we obtain diagonal Pad´e approximants from the
perturbation expansion, and we show how these are a better fit for the numerical data.
We also use these approximants in ray-tracing algorithms to model the bending of light
around the massive object.
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1 Schwarzschild metric
For a spherical symmetric space-time with a mass M in the center of the coordinate
system, the invariant interval is [16, 17]:
(ds)2 = γ(cdt)2 −γ−1(dr)2 −r2(dΩ)2 (4)
where (dΩ)2 = (dθ)2 +sin2θ(dφ)2 , with coordinates x0 = ct, x1 = r, x2 = θ and x3 = φ.
γ = 1− rs where r = 2GM is the Schwarzschild radius.
r s c2
The corresponding covariant metric tensor is given by
 
γ 0 0 0
 
 0 −γ−1 0 0 
 
gµν =   (5)
 0 0 −r2 0 
 
 
0 0 0 −r2sin2θ
Equation (4) has two singularities. The first one is when r = r (the Schwarzschild
s
radius) which defines the horizon event of a black hole. This is a mathematical singularity
that can be removed by a convenient coordinate transformation like the one introduced
by Eddington in 1924 or Finkelstein in1958 [17]:
r r
tˆ= t± sln| −1| (6)
c r
s
With this coordinate transformation the invariant interval reads:
(ds)2 = c2(cid:18)1− rs(cid:19)(cid:16)dtˆ(cid:17)2 −(cid:18)1+ rs(cid:19)(dr)2 ∓2c(cid:18)rs(cid:19)dtˆdr−r2(dΩ)2 (7)
r r r
The other singularity in r = 0 is a mathematical singularity. For a radius r < r ,
s
all massless and massive test particles eventually reach the singularity at r = 0. Thus,
neglecting quantum effects like Hawking radiation [10, 18], any particle (even photons)
that falls beyond this Schwarzschild radius will not escape the black hole.
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2 Geodesic equation for a photon in a Schwarzschild
metric
The geodesic equation can be written in an alternative form using the Lagrangian
(cid:32) dxµ(cid:33) dxαdxβ
L xµ, = −g (xµ) (8)
αβ
dσ dσ dσ
where σ is a parameter of the trajectory of the particle, which is usually taken to be the
proper time, τ, or an affine parameter for massless particles like a photon. The resulting
geodesic equation is:
du 1
µ = (∂ g )uαuβ (9)
µ αβ
dσ 2
where u = dxµ.
µ dσ
Consideraphotontravelingintheequatorialplane(θ = π/2)aroundamassiveobject.
For a photon, dτ = 0 and thus, we use an affine parameter, λ, to describe the trajectory
instead of the proper time, τ. For the coordinates ct (µ = 0) and φ (µ = 2) the geodesic
equation (9) give us, respectively :
(cid:34) (cid:35)
d dt
γc2 = 0 (10)
dλ dλ
and
(cid:34) (cid:35)
d dφ
r2 = 0 (11)
dλ dλ
Both of these equations define the following constants along the trajectory of the
photon around the massive object:
dt
γc2 = E(cid:48) (12)
dλ
and
dφ
r2 = J (13)
dλ
where E(cid:48) has units of energy per unit mass and J of angular momentum per unit mass
(when λ has units of time).
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The invariant interval for the Schwarzschild metric in the plane θ = π/2 is.
(ds)2 = c2(dτ)2 = γc2(dt)2 −γ−1(dr)2 −r2(dφ)2 = 0. (14)
Using dr = dr dφ the last equation can be written in the form
dλ dφdλ
(cid:32) (cid:33)2 (cid:32) (cid:33)2(cid:32) (cid:33)2 (cid:32) (cid:33)2
dt dr dφ dφ
γc2 −γ−1 −r2 = 0. (15)
dλ dφ dλ dλ
Multiplying (15) by γ, and inserting the definitions of E(cid:48) and J we obtain:
(E(cid:48))2 J2 (cid:32)dr(cid:33)2 γJ2
− − = 0 (16)
c2 r4 dφ r2
This equation can be turned into an equation for U(φ) = 1 , noting that
r(φ)
dU 1 dr
= − (17)
dφ r2dφ
so we arrive at the following equation for U(φ):
(E(cid:48))2 (cid:32)dU(cid:33)2
−J2 −J2U2(1−r U) = 0 (18)
c2 dφ s
By taking the derivative of equation (18) with respect to φ, we get the following
differential equation for U (φ)
(cid:32)dU(cid:33)(cid:32) d2U (cid:33)
2 +2U −3r U2 = 0 (19)
dφ dφ2 s
The differential equation in (19) can be separated into two differential equations for
U(φ). The first one is the equation for a photon that travels directly into or out from the
black hole:
dU
= 0 (20)
dφ
theotherdifferentialequation, applicablefortrajectoriesinwhichU(φ)isnotconstant
with respect to φ, is the following:
d2U 3
+U = r U2 (21)
dφ2 2 s
This equation can also be written in the following way, using the definition of the
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Schwarzschild radius:
d2U 3GMU2
+U = (22)
dφ2 c2
This is the equation for the trajectory of a massless particle that travels around a
black hole in the equatorial plane.
3 Differential equation for the trajectory of a photon
In the previous section, we obtained a differential equation for a photon traveling around
a masive object like a star or a black hole (see equation (22)). This equation has an exact
constant solution, for the unstable circular orbit of a photon around the black hole:
3GM
r = (23)
c c2
where r is the radius of the so-called photon sphere [16]. We note that the radius of
c
the photon sphere can be expressed in terms of the Schwarzschild radius:
3r
s
r = (24)
c
2
The orbit described by a photon in the photon sphere is actually an unstable orbit ,
and a small perturbation in the orbit can lead either to the photon escaping the black
hole or diving towards the event horizon [16].
Equation (22) is nonlinear, and is highly difficult to solve analytically. However, a
perturbative solution of this equation can be readily obtained. Let’s first rewrite Equation
(22) in terms of r :
c
d2U
+U = r U2 (25)
dφ2 c
ConsidertheinitialconditionsshowninFigure1. Thesmallestvalueofther−coordinate
in the trajectory, r = b, is taken such that the photon escapes the black hole, b > r . We
c
will rewrite Equation (25) in terms of (cid:15) = rc < 1, which we will use as a non-dimensional
b
small number for our following perturbative expansions. Note that by multiplying both
sides of Equation (25) by b, and defining the non-dimensional trajectory parameter
b
V(φ) = (26)
r(φ)
Equation (25), with the inclusion of the term (cid:15) = rc, then becomes a differential
b
equation in V(φ):
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Figure 1: Trajectory of a photon outside the photon sphere. The initial conditions are
takensuchthatr| = b, andiscalledtheimpactparameterofthetrajectory-theclosest
φ=0
distance from the trajectory to the center of the black hole. We thus have, dr| = 0
dφ φ=0
and dU| = 0. The photon experiences a total angular deflection of 2α.
dφ φ=0
d2V
+V = (cid:15)V2 (27)
dφ2
where 0 < (cid:15) = rc < 1, and with initial conditions given by
b
dV
V(φ = 0) = 1; (φ = 0) = 0 (28)
dφ
Under these conditions, V(φ) is bounded such that
|V(φ)| ≤ 1 (29)
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4 First-order solution for V (φ)
A first idea to obtain a solution of Equation (27) is to consider a V(φ) as a power series
in (cid:15):
V(φ;(cid:15)) = V (φ)+(cid:15)V (φ)+(cid:15)2V (φ)+... (30)
0 1 2
Plugging the expansion (30) into Equation (27) results in the following:
(cid:32)d2V0 +(cid:15)d2V1 +(cid:15)2d2V2 +...(cid:33)+(V +(cid:15)V +(cid:15)2V +...) = (cid:15)(cid:16)V +(cid:15)V +(cid:15)2V +...(cid:17)2 (31)
dφ2 dφ2 dφ2 0 1 2 0 1 2
We can group the powers of (cid:15) in Equation (31):
(cid:15)0 : d2V0 +V = 0 (32)
dφ2 0
(cid:15)1 : d2V1 +V = V 2 (33)
dφ2 1 0
(cid:15)2 : d2V2 +V = 2V V (34)
dφ2 2 0 1
(cid:15)3 : d2V3 +V = (V )2 +2V V (35)
dφ2 3 1 0 2
.
.
.
Note that the initial conditions of V(φ), applied to the asymptotic expansion in Equa-
tion (30), imply the following, by grouping powers of (cid:15):
(cid:15)0 : V (0) = 1; dV0(0) = 0 (36)
0 dφ
(cid:15)k : V (0) = 0; dVk(0) = 0; k ≥ 1 (37)
k dφ
From these differential equations and initial conditions, we can readily obtain V and
0
V iteratively 1:
1
1It is convenient to write the V (φ) in terms of polynomials in cos(φ)
k
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V (φ) = cos(φ) (38)
0
2 1 1
V (φ) = − cos(φ)− cos2(φ) (39)
1
3 3 3
Thus, we obtain an equation for V(φ), per Equation (30):
(cid:20)2 1 1 (cid:21)
V(φ) = cos(φ)+(cid:15) − cos(φ)− cos2(φ) +O((cid:15)2) (40)
3 3 3
According to the coordinate system shown in Figure 1, the photon goes through a
total angular deflection of 2α. This corresponds to setting V(φ) = 0 for both φ = π/2+α
and φ = −π/2 − α. From both of these conditions considering that α is very small, to
first order in (cid:15) we get:
2(cid:15)
α = 3 (41)
(cid:16) (cid:17)
1− (cid:15)
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The total deviation of the photon is then
4(cid:15) 4r 4GM
c
Ω = 2α ≈ = = (42)
3 3b bc2
For a light ray grazing the Sun’ s limb b = R = 695510km [19] and we get the very
Θ
well known value
4GM
Θ
Ω = 2α ≈ = 1.7516 arcseconds (43)
R c2
Θ
(cid:104) (cid:105)
where M = 1.9885 × 1030kg is the Sun’ s Mass, and c = 2.99792458 × 106 m is the
Θ s
value of the speed of light in vacuum [19].
5 Towards a second-order solution for Ω((cid:15))
We will now see how to obtain higher-order solutions for Ω. The differential equation in
(34) has the following solution:
4 41 2 1 5
V (φ) = − + cos(φ)+ cos2(φ)+ cos3(φ)+ φsin(φ) (44)
2
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However, the term in Equation (44) that goes as φsin(φ) grows without bound, and
occurs because the right-handed side of Equation (34) contains terms proportional to the
homogeneous solution of Equation (34): acos(φ) + b sin(φ). When this happens, the
solution contains terms that grow without bound, such as φsin(φ), called secular terms
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[20]. Thus, if we naively include Equation (44) in V(φ), our solution is no longer bounded.
Thus, we have to eliminate any and all secular term that arises to arrive at a well-behaved
solution for V(φ).
One method to do this, due to Lindstedt and Poincar´e, is by solving the differential
equation in the following strained coordinate [20]:
φ˜= φ(cid:16)1+ω (cid:15)+ω (cid:15)2 +...(cid:17) (45)
1 2
Where the ω are constants to be determined. In terms of this new strained coordinate
k
˜
φ, Equation (27) becomes
(cid:16)1+ω (cid:15)+ω (cid:15)2 +...(cid:17)2 d2V +V(φ˜) = (cid:15)V2(φ˜) (46)
1 2 dφ˜2
˜
We proceed in the previous way, and assume an asymptotic expansion on V(φ):
V(φ˜;(cid:15)) = V (φ˜)+(cid:15)V (φ˜)+(cid:15)2V (φ˜)+... (47)
0 1 2
Plugging the expansion(47) in Equation (46), we obtain:
(cid:16)1+ω (cid:15)+ω (cid:15)2 +...(cid:17)2(cid:32)d2V0 +(cid:15)d2V1 +(cid:15)2d2V2 +...(cid:33)+
1 2 dφ˜2 dφ˜2 dφ˜2
(cid:16) (cid:17)2
+(V +(cid:15)V +(cid:15)2V +...) = (cid:15) V +(cid:15)V +(cid:15)2V +... (48)
0 1 2 0 1 2
We can group the powers of (cid:15) in Equation (48):
(cid:15)0 : d2V0 +V = 0 (49)
dφ˜2 0
(cid:15)1 : d2V1 +V = V 2 −2ω d2V0 (50)
dφ˜2 1 0 1 dφ˜2
(cid:15)2 : d2V2 +V = 2V V −(ω 2 +2ω )d2V0 −2ω d2V1 (51)
dφ˜2 2 0 1 1 2 dφ˜2 1 dφ˜2
(cid:15)3 : d2V3 +V = V 2 +2V V −(2ω ω +2ω )d2V0 −(ω 2 +2ω )d2V1 −2ω d2V2 (52)
dφ˜2 3 1 0 2 1 2 3 dφ˜2 1 2 dφ˜2 1 dφ˜2