Light Deflection on de-Sitter Space

Abstract

We study the light deflection on de-Sitter Space with negative cosmological
constant

Full text

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arXiv:0903.0143v1 [gr-qc] 1 Mar 2009
Light Deflection on de-Sitter Space
Luiz C.L. Botelho
Departamento de Matem´ atica Aplicada,
Instituto de Matem´ atica, Universidade Federal Fluminense,
Rua Mario Santos Braga
24220-140, Niter´ oi, Rio de Janeiro, Brazil
e-mail: botelho.luiz@superig.com.br
We study the light deflection on de-Sitter Space with negative cosmological constant.
Key-words: Cosmological deflection, anti-gravity, light deflection.
1The light deflection
The most general covariant second-order equation for the gravitation field generated by
a given (covariant) enegy-matter distribution on the space-time is given by the famous
Einstein field equation with a cosmological constant Λ [with dimension (length)−2([1]),
namely.
?
Rµν(g) −1
2gµνR + Λgµν
?
(x) = 8πGTµν(x)(1)
where x belongs to a space-time local chart.
It is well-known that studies on the light deflection by a gravitational field generated
by a massive point-particle with a pure time-like geodesic trajectory (a rest particle “sun”
for a three-dimensional spatial space-time section observer!) is a always carried out by
considering Λ ≡ 0([2],[3]).
Our purpose in this short note is to understand the light deflection phenomena in
the presence of a non vanishing cosmological term in Einstein equation (1), at least on a
formal mathematical level of solving trajectory motion equations.

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Let us, thus, look for a static spherically symmetric solution of eq. (1) in the standard
isotropic form ([2], [3])
(ds)2= B(r)(dt)2− A(r)(dr)2− r2[(dθ)2+ sen2θ(dφ)2] (2)
In the space-time region r = + | ? x |2> 0, where the matter-energy tensor vanishes
identically, we have that the Einstein equation takes the following form
Rµν(g)(x) = −Λ(gµν(x))(3)
In the above cited region, the Ricci tensor is given by
− Λgµν=








−ΛB(r)000
0ΛA(r)00
00Λr2
0
000Λr2sen2θ








=








Rtt
000
0Rrr
00
00Rθθ
0
000Rφφ








(4)
we have, thus, the following set of ordinary differential equations in place of Einstein
Partial Differential eq. (1)
Rtt= −B′′
2A+1
2B−1
4
?B′
?
?B′
A
??A′
−A′
??A′
A+B′
A+B′
B
A+B′
B
?
+1
?
−1
−1
r
?A′
?B′
A
?
?
= ΛA
= −ΛB(5)
Rrr≡B′′
4
r
2A
BBrA
(6)
Rθθ= −1 +
?
A= Λr2
(7)
Rφφ= sen2θRθθ= Λ(sen2θ)r2
(8)
At this point we note that
Rtt
B(r)+Rrr
A(r)= 0
(9)
or equivalently
A(r) =
α
B(r)
(10)
where α is an integration constant.
2

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Since Rθθ= −1+r
αB′+B
α= Λr2, we get the following expression for the B(r) function
B(r) =αΛr2
3
+ α +β
r
(11)
witch β denoting another integration constant.
In the literature situation, ([2],[3]), one always consider the case Λ ?= 0 in a pure
classical mathematical vaccum situation context, the so called de-Sitter vaccum pure
gravity. However in our case it becomes physical to consider that our solution depends
analytically on the cosmological constant. In other words, if the parameter Λ → 0 in our
solution, it must converges to the usual Schwarzschild solution with a mass singularity at
the origin r = 0. That is our boundary condition hypothesis imposed on our solution.
As a consequence, one gets our proposed Schwarzschild-de-Sitter solution
(ds)2=
?Λr2
?Λr2
3
+ 1 −2MG
r
?
(dt2)
−
3
+ 1 −2MG
r
?−1
(dr)2− r2[(dθ)2+ (sen2θ)(dφ)2](12)
At this point let us comment that for the space-time region exterior to the spatial
sphere r > (3mG
Λ)1/3, the field gravitation approximation leads to the anti-gravity (a
respulsion gravity force) if Λ < 0, so explain from this Einstein Gravitation theory of ours
the famous “Huble accelarating Universe expansion”.
In what follows we are going to consider a non-vanishing Λ < 0 and study the path of
a light ray on such negative cosmological constant Einstein manifold eq. (12).
We have the following null-geodesic equation for light propagating in θ = π/2 plane
(Einstein hypothesis) for light propagation on the presence of the sun (Section 2 for the
related formulae)
0 = B(r) −
1
B(r)
?dr
dt
?2
− r2
?dφ
dt
?2
(13)
At this point we note that
?dφ
dt
?
=
?B(r)J
r2
?
(14)
where J is a integration constant.
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After substituting eq. (14) into eq. (13) we hve the following differential equation for
the light trajectory as a function of the deflection angle φ
?dr
dφ
dφ
dt
?2
+J2B3(r)
r2
− B2(r) = 0(15)
which is exactly integrable
dφ =
dr
r2
?
1
J2−B(r)
r2
(16)
By supposing a deflection point rmwheredr
dt= 0 and, thus, J = rm/?B(rm), we get
the deflection angle.
∆1φ =
?rm
∞
dr
−B(r)
r2[B(rm)
r2
m
r2 ]1/2=
?
1
rm
0
dU
[(U2
m− U2) − 2MG(U3
m− U3)]1/2
(17)
which is exactly that one given in the pure (Λ = 0) Schwarzschild famous case. However,
if one supposes that there is no deflection (a continuous monotone trajectory r = r(φ)!),
the total deflection angle now depends on the cosmological constant and is given formally
by the expression below.
∆2φ =
?rm
0
dr











1
r2?
−1
r2+2mG
r3



1
?
1 +
?
r3(3−ΛJ2
3J2
)
2MG−r
?














?= ∆1φ(18)
As a general conclusion of our note we claim that the usual light-deflection experi-
mented test foes not make difference between the usual non-cosmological Schwarzschild
case and our case Eq.(12), and, thus, it should be not considered as a definitive physical
support for Einstein General Relativity without cosmological constant.
4

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2 The Trajectory Motion Equations
The body trajectory (t(p),r(p),θ(p),ϕ(p)) on the presence of the gravitational field gen-
erated by the metric eq.(2)–eq.(10) is described by the following geodesic equations
d2t
d2p+B′
d2r
d2p+A′
d2θ
d2p+2
d2φ
d2p+2
B
?dr
?dr
dθ
dr
dφ
dp
dp
??dt
?2
dp
?
= 0(19)
2Adp
−r
A
?dθ
dp
?2
?dφ
dθ
dp= 0
−rsen2θ
A
?dφ
dp
?2
+B′
2A
?dt
dp
?2
= 0(20)
r
dr
dp− senθ · cosθ
dr
dp+ 2cotg(θ)dφ
dp
?2
= 0(21)
r dp
(22)
At this point we remark that by multiplying eq.(19) by B(r(p)), it reduces to the
exactly integral form relating the Einstein proper-time (physical evolution parameter) p
with the geometrical dependent coordinate Newtonian time t:
dt
dp=
1
B(r)
(23)
We remark either that eq.(22) can be rewritten in the form
d
dp
?
ℓndφ
dp+ ℓnr2+ 2ℓn senθ
?
= 0(24)
which reduces to the following form
?dφ
dpr2(p)sen2(θ(p))
?
= J(25)
where J is a integration constant.
By substituting eq.(23) and eq.(25) into equations (20) and (21) we obtain the full set
of equations describing the body trajectory in relation to the (r,θ) variables
d2r
d2p−B′
2B
?dr
d2θ
d2p+2
dp
?2
− rB
?dθ
dp
?2
−J2B
r3
+B′
2B= 0(26)
r
dθ
dr
dr
dp−cosθ
sen3θ
J2
r4= 0(27)
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