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In an earlier paper I have discussed the effect of gravitation on the bending of the locus of a photon.(1)Even though the role played by the coordinate time associated with the gravitational motion in deflecting the path of light was correctly depicted therein, I believe that the mathematical implications were not correctly brought out. Especially, in the case of equation (6) there, though its existence was correctly recognized, its physical significance was misconstrued. The present paper, therefore, attempts to overcome these shortcomings and derives the mathematical expression for the angle through which the path of a light beam is deflected in the vicinity of a mass.

According to the Reciprocal System, an independent motion of speed v has associated with it an increase in the coordinate time amounting to (v/c sec/sec (c being the speed of light). In the case of a one-dimensional motion, like that of a planet orbitting the sun, or of a photon grazing the sun’s limb, I have pointed out(2) that the circumferential spatial effect arising out of the coordinate time amounts to 3(v/c sec/sec. It was further explained that, in the case of a photon this spatial effect manifests itself in the radially inward direction since no further circumferential effect is possible, as the photon is already moving at unit speed in the latter direction.(3)

The gravitational speed v at any radial distance r from a mass M is shown(4) to be

= GM/r

where G is the gravitational constant. Thus, we have the rate of coordinate time increase at a radial distance r outside a mass M as

dtc/dt = 3(v/c)² = 3 GM/rc²

where tc represents the coordinate time and t the clock time.

As shown in Fig. 1, let the straight line ABC represent the locus of a photon passing the sun situated at S.With SB perpendicular to AC, B is the point of closest approach to the sun. Let SB = ro.The equation of the line ABC in polar coordinates, with the origin at S, is given by

r0 = r cos

where r is the radial distance at any angle q measured counterclockwise from SB. Substituting r from eq. (2) into eq. (1), we have

dtc/dt = (3 GM/ro) cos

Now we note that the gravitational effect of any mass aggregate, according to the Reciprocal system, does not extend up to infinity, but becomes zero at a limiting distance, which Larson calls the “outer gravitational limit,” d1. As such, we need to compute the coordinate time increase in the case of the transitting photon, starting from the outer gravitational limit on one side (toward A, in Fig. 1), up to the outer gravitational limit on the other side (toward C, in Fig.1). Larson worked out the value of the outer gravitational limit for the sun to be nearly 13350 light years. As this will be very large compared to ro, we find that the limits on the two sides are given by q1 = –P/2 and q2 = +P/2.

Hence, using eq. (3), the average rate of coordinate time increase during this transit from q1 to q2 is given by

  ò q2
)dq / (q2 - q1)
  dt   q1

ò P/2
= (3 GM/ro)
cosq dq/P

= 6 GM/Pro sec/sec 

Since the total distance traveled is 2 d1, the total transit time is

t = 2 d1/c sec

Therefore the total coordinate time gained during this clock time t is

  tc,tot= t * = (6 GM/P ro)*(2d1/c)

  = 12 GM d1/P ro sec

In Fig. 2 the directions of approach and departure of the light beam are shown as ABC’ and BC respectively.CC’ represents the spatial shift in the radial direction arising out of the coordinate time component and is given by

x = c * tc,tot = 12 GM d1/ ro c² cm

Finally, from Fig. 2 we can see that the angular deflection, according to the Reciprocal System is given by

dRS = x/d1 = 12 GM/P ro c² radians

The corresponding expression from General Relativity is

dGR = 4 GM/ro

The discrepancy between the two formulae can be seen to be

dRS/dGR = 3/P

The value calculated from the Reciprocal System formula, for the sun, is 1.67 arcsec, whereas the General Relativity value is 1.75 arcsec. The reported values vary from 1.5 to 1.8 arcsec.


  1. K.V.K. Nehru, “Gravitational Deflection of a Light Beam in the Reciprocal System,” Reciprocity XI(1) (Spring, 1981), 28.
  2. Idem, “Precession of Planetary Perihelia due to Coordinate Time,” Reciprocity XIV(1) (Autumn, 1985), 11-13.
  3. Idem, “Gravitational Deflection. . .” op. cit., 29.
  4. Ibid., eq. (1), p. 28.
  5. D. B. Larson, The Universe of Motion (Portland, Ore.: North Pacific Publishers, 1984), p. 201.

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