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The Deflection ( Bending ) of Light

Einstein's Razor: "Everything should be made as simple as possible, but not simpler" - an approximate quote attributed to Albert Einstein ( 1879 - 1955 )

NASA: The Incandescent Sun

"The Treatise on Opticks", by Sir Issac Newton

Sir Isaac Newton ( 1643 – 1727 )

Query 1. Do not Bodies act upon Light at a distance, and by their action bend its Rays; and is not this action ( coeteris paribus ) strongest at the least distance?

Qu. 4. Do not the Rays of Light which fall upon Bodies, and are reflected or refracted, begin to bend before they arrive at the Bodies; and are they not reflected, refracted, and inflected, by one and the same Principle, acting variously in various Circumstances?

Qu. 5. Do not Bodies and Light act mutually upon one another; that is to say, Bodies upon Light in emitting, reflecting, refracting and inflecting it, and Light upon Bodies for heating them, and putting their parts into a vibrating motion wherein heat consists?
source:   "Treatise on Opticks - The Third Book", The Third Edition, corrected, 1730, by Sir Issac Newton, The Gutenberg Project publisher, page 339 - 340 ( 128 of the pdf )
. . .

Qu. 20. Doth not this AEthereal Medium in passing out of Water, Glass, Crystal, and other compact and dense Bodies into empty Spaces, grow denser and denser by degrees, and by that means refract the Rays of Light not in a point, but by bending them gradually in curve Lines? And doth not the gradual condensation of this Medium extend to some distance from the Bodies, and thereby cause the Inflexions of the Rays of Light, which pass by the edges of dense Bodies, at some distance from the Bodies?
source:   "Treatise on Opticks - The Third Book", The Third Edition, corrected, 1730, by Sir Issac Newton, The Gutenberg Project publisher, page 350 ( 132 of the pdf )
. . .

But Light is never known to follow crooked Passages nor to bend into the Shadow. For the fix'd Stars by the Interposition of any of the Planets cease to be seen. And so the Parts of the Sun by the Interposition of the Moon, Mercury or Venus. The Rays which pass very near to the edges of any Body, are bent a little by the action of the Body, as we shew'd above; but this bending is not towards but from the Shadow ( see: pg. 336 of Opticks, pg. 127 of the pdf, for Newton's meaning of 'Shadow' ), and perform'd only in the passage of the Ray by the Body, and at a very small distance from it. So soon as the Ray is past the Body, it goes right on.
source:   "Treatise on Opticks - The Third Book", The Third Edition, corrected, 1730, by Sir Issac Newton, The Gutenberg Project publisher, page 363 ( 136 of the pdf )

. . .

note: Newton is basically speaking of the refractive nature of the AEthereal Medium ( the all - pervasive luminescent "Theory of Aether" ) which emanates from each and every Body of matter spreading out into deep, empty space by which Rays of Light are thereby refracted as by a "prism of aether", and hence bent, as these rays approach the Body. This sort of physics thinking is totally different from that of Riemannian curved geometry of space and time [ Bernhard Riemann, doctoral thesis 1854, published 1868 ], and certainly different from that of Einstein's General Relativity analysis for the deflection ( bending ) of light at the very nearest coronal edges of the sun for starlight entering into the sun's coronal region coming from an infinite distance away.

A Simple Experiment

In all of Einstein's other writings, it is spacetime geometry itself which is pulling and pushing gravitational masses [ not mysterious "beings"! ] which in turn are deforming spacetime's malleable fabric; in this way, according to Einstein, gravitational masses are best understood in terms of non - Euclidian metric geometries for deformable spacetime fabrics and therefore General Relativity mathematics employs 4 - dimensional tensors, a specialized matrix algebra and calculus for curvilinear Riemannian surface geometries of which Euclid's cartesian geometry is a unique case.

The Geometry of Light Bending

The Relativistic Light Bending Term

From the null geodesic nature of starlight, over vast expanses of cosmic space and time, we know simply that

and, therefore, Schwarszchild's equation

However we might also have arrived at the above equation much quicker by noting that in deriving "Mercury's Perihelion Precession Advance", there are the following equations:

The Family of Light Rays in the Equatorial Plane

this equation represents the family of light rays in the equatorial plane but will obviate the null geodesic characteristic of normal light transits; therefore, light deflection ( bending ) occurs!

As a first approximation, we shall assume that the family of light rays transit parallel to the y - axis and - i.e., no light bending. This is the same as saying

So let's submit

as a solution to

to get

no light bending which is what we'd expect.

However, as we're still interested in an overall solution to the generalized light bending equation as shown above, but still maintaining the special case of a straight line null geodesic light path, we continue to submit

as a solution into the "light bending term" as follows

but reserving

as an overall 2nd - degree quadratic equation.

The Derivation of the Light Deflection Angle

But this is only true when

This derivation, thus, means that

Now at the two directional ends of the light path which goes towards infinity as the light ray again assumes a null geodesic ( Newton - Euclid ) straight line, the "angular distance" between the two zeros of , and hence between the two asymptotic lines, becomes . That is,

which arises from the Newton - Euclid straight line equation for the null geodesic light path

parallel to the y - axis ( ) where .

Finally,

That is, for a pure null geodesic light path of a Euclidean straight line of no light bending, we will have

but the total deviation of the "angular distance" between two successive must include the "bending effect" of the sun's massive gravity field which is accounted for by the factor amount of

Calculating Deflection of Starlight by the Sun

Confirmation! Tentative, but still Confirmation!

Sir Arthur Eddington's experimental observation for "The Bending of Starlight" by a total solar eclipse on May 29, 1919 off the west coast of Africa on the island of Isle Principe in the Gulf of Guinea provides to the world of human knowledge its first positive, but still tentative, proof for Einstein's General Relativity Theory, the most towering intellectual achievement since the time of Isaac Newton:

Philosophic Musing on the Human Condition for Understanding

We humans in many respects are still simple creatures and for us humans "naïve realism" in terms of basic numbers appearing "on a dial" becomes our most compelling "reality".

In other words, in both for Special Relativity and now , starlight deflection in General Relativity, it all comes down to plugging in simple numbers to arrive at other simple numbers to explain "reality"!!

"When you can measure what you are speaking about and express it in numbers, you know something about it" - Lord Kelvin ( 1824 - 1907 )

The Moral of this Story

Gravity is a gossamer tissue cloth spread unevenly throughout space and time. The contours of this cloth of gravity is outlined in the geodesic paths that parallel transported gyroscopes and beams of light naturally follow.

§ References:

1.   "On the influence of Gravitation on the Propagation of Light" ( "Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes" ), by Albert Einstein, Annalen der Physik 35(10): 898 - 908, 1911, in the original German.

2.   "On the influence of Gravitation on the Propagation of Light", by Albert Einstein, Annalen der Physik 35(10): 898 - 908, 1911, English version, translator Michael D. Godfrey, Information Systems Lab, Stanford University. Click here for a 2nd English translation version.

3.   "A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations made at the Total Eclipse of May 29, 1919", by Sir F.W. Dyson, F.R.S, Astronomer Royal, Prof. A.S. Eddington, F.R.S. and Mr. C. Davidson

4.   "The Foundations of Einstein's Theory of Gravitation", 1919, by Erwin Freundlich, Director of The Einstein Tower, Preface by Albert Einstein, The Library of the University of California, Berkeley

5. "The Theory of Relativity", 2nd edition, by R.K. Pathria, Distinguished Professor Emeritus University of Waterloo, Dover Publications, Inc.

6. "Relativity: Modern Large - Scale Spacetime Structure of the Cosmos", Editor Moshe Carmeli, formerely of Ben Gurion University, Israel, World Scientific Publishing Co. Pte. Ltd.

7. "Introduction to Tensor Calculus, Relativity and Cosmology", by D. F. Lawden, Emeritus Professor, University of Aston in Birmingham, U.K., Dover Publications, Inc.

8. "Introduction to the Theory of Relativity, with Foreword by Albert Einstein", by Peter Gabriel Bergmann, Dover Publications, Inc.

9. "Gravitation and Inertia", by Ignazio Ciufolini and John Archibald Wheeler, Princeton Series in Physics, pg. 118 for providing simplified light deflecting diagram

10. "Mathematical Physics", 2nd edition, by Donald H. Menzel ( 1901 - 1976 ), Professor of Astrophysics, Harvard University, Dover Publications, Inc., first published in 1947 under the title "Theoretical Physics", Prentice - Hall, Inc., first published in 1947 under the title "Theoretical Physics", Prentice - Hall, Inc.

11. "General Relativity and Gravitational Waves", by J. ( Joseph ) Weber, Professor of Physics, University of Maryland, 2004 Dover Edition of an unabridged republication in 1961 by Interscience Publishers, Inc., New York

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