The deflection of star light in a gravitational field was first tested during solar eclipse in 1919, and was actually described in Sir Arthur Eddington’s book in detail[1]. A rough illustration of the effect is shown below:
Deflection of star light as predicted by
General Relativity
In the above diagram the light from the star at actual position S2 is seen to deflect by
some angle a, thereby
altering the image position to that seen at S1. This is a direct result of the
effect of the gravitational field of the Sun on the light rays. The true
direction is thus alone the ray ES2 while the deflected position is along the
ray ES1.
Theoretically and quantitatively, one can obtain an estimate of the magnitude of deflection by incorporating another parameter – call it b – as shown in the diagram below:
Determining the linear deflection (impact) parameter, b.
Then we
obtain for the deflection angle, a:
a = - 4 GM/ b or (in cgs units):
a = - 4 GM/ b c2
Einstein, in
his own paper: ‘On the Influence of Gravitation on the Propagation of Light’,
gives the result as:
a = 2k M/ c2 D
Diagram for
Einstein’s derivation of a (q).
Here k = G, the gravitational constant and D = the
distance of the ray from the center of the body. It is obtained from the
diagram of Fig. 12.
Einstein used the cosine of the angle and integrated from the
negative angle q = - p/ 2 to q = p/ 2:
a = 1/ c2
ò - p/ 2 p/ 2 k M/ r2 cos q ds
Enter now Neil Ashby who, in a world class presentation in May, 2008, presented an update based on second order corrections to gravitational deflection for a massive body.
i)The key initial quantity is the total elongation angle: f (b, rB , rA ) = f ( rA , b) + f ( rB , b) referenced to the diagram shown below (see element (ix) below):
Neil Ashby's dynamical diagram (May, 2008 paper)
ii)The parameters γ, β and e are equal to 1 in general relativity ( γ+ β + e = 1)
iii) Cassini’s 2002 experiment, in which the PPN (parameterized post-Newtonian formalism) parameter γ was measured with an
accuracy (standard deviation) σγ
= 2.3
× 10-5
iv) Light propagation in a static spacetime is equivalent to a problem in ordinary geometrical optics: Fermat’s action functional at its minimum is just the light-time between the two end points A and B. See e.g. Ashby Fig. 2 below and:
Proving Fermat's Principle Of Least Time For Reflection At A Plane Surface
v) For the case considered (Fig. 2) the best mathematical tool to deal with electromagnetic propagation is not null geodesics, but the theory of eikonal. It is known that in this problem Fermat’s Principle holds, corresponding to the refractive index:
N(r) = ÖB(r) /A(r)
The eikonal was developed ab initio and solved for by separation of variables. The radial part provides Fermat’s action as a radial integral containing N(r) and the parameter h; when computed at the true value true, such action is just the required light-time.
The solution can be obtained recursively, using appropriate expansions in powers of m ( bo is the Euclidean distance of the straight line AB - in Fig. 1- from the mass):
b = bo + m (b 1 / bo) + m2 b 2 / bo2
vi) In the framework of metric theories of gravity and the PPN ((parameterized post-Newtonian) formalism, the main violations of general relativity – those linear in the masses – are described by the single dimensionless parameter γ. The question is at what level and how general relativity is violated, in particular how much γ differs from unity. (Ashby brilliantly answered this in his paper.)
vii) To date the best measure of γ has been obtained in Cassini's experiment:
γ - 1 = (2.1 + 2.3) x 10 -5
The Orbit Determination Program (ODP) of NASA’s Jet Propulsion Laboratory, which was used in the data analysis, is based on an expression for the gravitational delay D t which differs from the standard formula; this difference is of second order in powers of m and the gravitational radius ( R⊙ ) of the Sun i.e.
m (m/ bo) = m (m /R⊙ ) (R⊙/ bo)= 0.3 (R⊙ / bo ) cm,
But in Cassini’s case it was much larger than the expected order of magnitude m2/b, where b is the distance of closest approach of the ray.
vii) For a typical velocity 10-4 c the correction is of order:
20 × 1.4 × 10 5 × 10-4 = 300 cm, and the a
priori accuracy in D t is
sufficient.
viii) It should also be noted that the spacetime coordinates of the end events are not directly provided in the experimental setup and depend on the gravitational delay D t, the very quantity one sets out to measure.
The trajectories rA (t) and rB (t) are given by the numerical code; the starting time t A is just a label of the ray, but the arrival time:
tB > t A + rAB.
The way out is to take for the end point:
rB (rB )
= rB
(t A +
rAB)
+ D t u
B(t A + rAB),
(where uB = drB/dt)
Ashby concluded with this solution using the triangles in Fig. 1 above. For reference the mass is at O and the end points (events) at A, B. The distance OH = bo and the angles a are taken to be positive. The internal angle fAB can be obtuse or acute. When: rB > rA >> bo we have a close superior conjunction for which the deflection is large. Asked for estimates of the key parameters from his data Ashby showed: b = 2R⊙ , rA = 1 AU, rB = 5 AU
Ashby then challenged us to show - using some algebra and trigonometry- that when R = O(bo ) order of bo the truncation error at order k is (m/ bo) k+1 with coefficient of order unity. Finally leading to the condition:
mR/ b o2 << 1
The above quantity then gives (by order of magnitude) the ratio between the gravitational deflection ( m/ bo ) and the angle ( bo /R ) which separates the central mass from the distant star as seen from distance R.
After less than ten minutes all attendees were able to show this and Ashby treated us to his derivation of the Asymmetric Eddington potential, accompanied by a round of applause.
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