Deflection of Starlight

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Isaac Newton described gravity as a force exerted between two bodies and calculated the amount of the force as being proportional to the product of the bodies' masses. We know that light possesses zero mass, so we can ask Does gravity affect light? By Newton's formula any body, however massive, will exert zero force upon a ray of light. But because light has zero mass (which implies no resistance to acceleration) it might accelerate nonetheless and, then again, it might not. In classical physics we have no way of knowing the answer to that question.

Einstein described gravity as the warping of inertial frames by the presence of a massive body and claimed that anything in the space near that body would thus follow a curved trajectory due to that warping of space and time out of true. In accordance with General Relativity, then, and because we certainly include light in the term "anything", we should expect gravity to affect the motion of light. To test that proposition, to see whether it comes out true or false to Reality, Einstein suggested using starlight because, after traveling distances measured in lightyears, even lightcenturies, it comes into our solar system in the form of plane waves; that is, the crests of the electromagnetic swells, over spans of millions of kilometers, define almost perfectly flat planes oriented perpendicular to the direction of the waves' propagation. A telescope resolves a plane wave as a point of light and, though starlight retains some of its original curvature (it starts out as a spherical wave), even the most powerful telescopes cannot make the nearest stars appear as discs; thus, astronomers can use photographs of the stars to make exquisitely accurate measurements of the angular distances among the stars as they appear on the sky. By making such accuracy available to astronomers plane-wave starlight gave them the means to test Einstein's theory.

According to Einstein's theory, the sun's gravity will so bend starlight that the light will appear to observers on Earth to be emanating from a point displaced a small angle away from the point whence the light actually came. Because of light's great speed, even the stiff gravity of the sun will bend a ray of starlight through a very small angle, but one that astronomers can nonetheless measure with sufficient precision. Thus, if, from our perspective, the sun were to appear in the middle of some constellation, the stars in that constellation would appear to us to be farther apart from each other than they appear six months earlier or later, when the sun lies on the opposite side of the sky. Two effects bring about that bending of starlight - the direct acceleration of the light toward the sun and an effect analogous to refraction, the phenomenon that bends light when the light passes from one medium into another medium of greater or lesser density (such as passing from air into water or vice versa).

I stated previously that light cannot change its speed of flight and that statement remains true to Reality, but light can, nonetheless, accelerate. We define acceleration as the rate at which the velocity of some thing changes and we define velocity as a combination of speed and direction, so light can change its velocity without changing its speed if it changes only its direction of flight; that is, light can accelerate but only sideways. As a ray of starlight flies past the sun, skimming over roiling incandescent clouds and flashing through fountains of cosmic fire, in that brief elapse the sun's gravity, twenty-eight times as stiff as the field that clasps us to Mother Earth's bosom, nudges the ray downward, so altering its course that it would appear to us to come from a point on the sky 0.876 arcsecond from its actual point of origin if this were the only effect operating upon the light.

We now want to calculate that deflection. Imagine a pulse of light flying along the x-axis of a coordinate grid as it passes through its perihelion, a distance R from the center of the sun. We want to calculate the angle, represented by the Greek letter alpha, through which the sun's gravity deflects the pulse's path in the negative y-direction due solely to direct lateral acceleration of the pulse. Imagine, then, that only lateral acceleration deflects the pulse and that as it flies out of the solar system the pulse flies parallel to a straight line that, extended backward, crosses our x-axis at the angle alpha. Because we know that alpha will be a very small number, we can exploit several simplifying approximations.

In the first of those approximations we use the Pythagorean theorem to express the distance between the pulse and the sun's center as a function of distance along the x-axis. Because the angle of deflection is very small we can ignore the y-component of the pulse's location in that calculation; that is, we exploit the fact that at any given time the x-component of the pulse's location far exceeds the magnitude of the y-component, so we can ignore the contribution that the y-component makes to the overall distance. We thus have the distance as

(Eq'n 1)

We can now calculate the y-ward acceleration of the pulse by multiplying the magnitude of the gravitational acceleration by the ratio of the y-component of the pulse's distance from the sun's center to the overall distance given in Equation 1. Herein I make another approximation that I could not use if the pulse were to fly through a much stiffer gravitational field, such as one near the surface of a neutron star. I use the smallness of the angle alpha as an excuse to ignore the turning of the acceleration vector: that turning gives the acceleration of the pulse a very small component oriented in the negative x-direction and that component, in accordance with the Pythagorean theorem, makes the magnitude of the y-component of the acceleration ever-so-slightly smaller than the raw value that I intend to use. For that acceleration I have

(Eq'n 2)

To calculate the net y-ward velocity gained by the pulse as it goes from its perihelion to its exit from the solar system, we multiply Equation 2 by a small increment of time, dt = dx/c, and integrate the result. Again I have used my small-angle approximation to represent the pulse as if it were traveling only in the x-direction at the speed of light. We want to integrate the change in y-ward velocity between the limits x = 0 and x = x' > 150,000.000 kilometers, on the understanding that almost all of the deflection of the pulse's path occurs within one astronomical unit of the sun. We thus obtain

(Eq'n 3)

We can interpret that velocity as one side of a right triangle whose hypotenuse is the speed of light. Basic trigonometry then tells us that we can obtain the angle alpha from

(Eq'n 4)

the latter expression reflecting our assumption that x' is very much larger than R. But for our sun MG/c2 = 1.478 kilometers and R must exceed 696,000 kilometers, the radius of the photosphere, so the sine will be small enough that we can replace it with the angle alpha itself (expressed in radians). We must now make one final adjustment to this calculation.

Thus far we have only calculated the deflection of the pulse as it goes from its perihelion to its exit from the solar system. Now we must add to that result the deflection that the pulse's path accrues between the pulse's entry into the solar system and its arrival at its perihelion. Perfect mirror symmetry shapes the two parts of the pulse's path, so we need only double alpha to obtain the total deflection coming from direct lateral acceleration of the pulse. For a pulse whose path just skims the photosphere (for which R = 696,000 kilometers) we have, when we recall that one radian contains 206,264.8 arcseconds, a total deflection of 0.876 arcsecond, as noted above.

To that deflection we must now add the deflection caused by the gravitational refraction of light. In the previous calculation we represented light as a body subject to acceleration (albeit only laterally). Now we must envision starlight passing the sun as comprising trains of plane waves. In thus calculating the overall deflection that the sun's gravity imposes upon starlight we tacitly invoke the wave-particle duality of the quantum theory, seeing light in both its particle aspect (tacitly in the previous calculation) and in its wave aspect (in the calculation to come). In this invocation we may see a faint hint at the eventual union of the quantum theory and General Relativity, a union that physicists have striven for decades to produce in their theories, one that may come to us more easily than anyone currently guesses.

The gravitational analogue of refraction originates in the gravitational contraction of vertical distances. In order to understand what that fact means for the motion of light, image that we have available to us a perfectly massless, perfectly inelastic surveyor's chain one billion kilometers long, each link spanning one kilometer. If we apply a little tension, the chain will neither stretch nor sag in the sun's gravitational field. Now image that we have laid two parallel rails one billion kilometers apart, putting the rails equidistant from the sun with the sun lying just below the plane defined by the rails. If we make our surveyor's chain slide along the rails, it will just skim the photosphere before going back out into deep space. Although we make the rails perfectly straight, we also make them easily deformable so that they will follow any change in the length of the chain and record it. Having done that, we will find that the chain bends the rails, bowing them toward each other: the chain shrinks as it comes close to the sun and re-expands as it moves away; more precisely, the space in which we have embedded it shrinks and expands, carrying the chain with it. If we repeat the experiment, making the chain pass the sun farther from the previous experiment's perihelion, we will find that the chain bends the rails less. In that fact lies the basis for understanding the gravitational refraction of light.

We actually know of something that acts much like our magical surveyor's chain. Imagine a ray of light so propagating through the solar system that it just grazes the sun's photosphere. If we had a magic dye that we could apply to electromagnetic fields to mark them and if we used that dye to paint the parts of the ray where the electric field was strongest, then we would see a series of parallel planes moving in the direction absolutely perpendicular to their surfaces. If we were to slice a cross section through that painted array, we would see a series of parallel straight lines sliding sideways at 299,792.458 kilometers per second. But as that array approached the sun, we would see the distances between neighboring lines shrink, the distances passing closer to the sun shrinking more than those farther away. That differential shrinkage makes the painted lines tilt as they go through the space close to the sun and thus makes the direction of their propagation tilt as well: the ray seems to bend.

As another way to understand the bending of the ray consequent to the differential shrinkage of the distance between successive painted planes in the ray, we can note that the ray flies in the direction perpendicular to each plane, crossing each kilometer in 3.33564 microseconds, regardless of whether it's a shrunken kilometer or not. If you follow two such planes in your imagination, you will see that as a consequence of the differential shrinkage the part of the two planes that passes closer to the sun appears to a distant observer to move a trifle slower than the part farther from the sun. That means that one part of each plane will lag behind the other and will thus cause the plane to rotate in space, turning the direction of propagation. That fact, in turn, means that the ray turns through a small angle as it passes through the sun's gravitational field.

We now want to devise a mathematical description of that effect. We call the distance between to adjacent painted planes as described above half the wavelength of the electromagnetic radiation comprising the ray. We now identify the shrinkage of the wavelength with the gravitational blueshift of light falling in a gravitational field. At a radial distance R' from the center of the gravitating body the deep space wavelength λ0 shortens to

(Eq'n 5)

As before, I want to make the approximation that the ray is deflected through a small angle so that I can treat the mathematical description as if the ray were traveling entirely parallel to the x-axis of our coordinate grid. I then assert that the distance R, the distance of the ray's closest approach to the sun's center, lies on a line drawn perpendicular to the x-axis. Thus we obtain the distance R' by way of the Pythagorean Theorem as in Equation 1.

Now I want to calculate the distance along the ray from wave number zero to wave number N, in which I make N a very large number. If I make the wavelength short enough, then I can make λ0 = dx, the basic element of integration along the x-axis. I can then calculate the length the ray has as it flies through deep space toward the sun: L0 = dx. I imagine that when the ray reaches the stage where wave number N/2 lies on the ray's perihelion, that instant in time freezes and holds the ray motionless for my inspection via calculation. I now modify Equation 5 into a differential reflecting the gravitational blueshift and calculate L', the deformed length of the ray;

(Eq'n 6)

By comparing that length calculated along two lines a distance dR apart I can calculate the degree to which the wave tilts. But first I need to know the lengths of those lines expressed in terms of our coordinates R and x.

Substituting for R' from the equivalent of Equation 1 gives us

(Eq'n 7)

If the distance R at which the ray passes through its perihelion greatly exceeds the sun's Schwarzschild radius (which it does as 696,000 kilometers versus 2.956 kilometers), then we can replace the outer square root in Equation 7 with the first two terms of its Taylor series expansion to obtain

(Eq'n 8)

The Table of Integrals tells us that

(Eq'n 9)

so when we carry out the integration of Equation 8 and evaluate the integral for the situation in which one end of L' lies on the perihelion point and the other end is at the appropriately distant point x, we obtain

(Eq'n 10)

In that equation the logarithmic term represents the amount by which the sun's gravity shortens the wavetrain from its deep-space length of L0.

I want to use Equation 10 to calculate the length of the wavetrain from wave number zero to wave number N at two values of R that differ by the pseudo-infinitesimal value dR and use that calculation to derive the angle that has accumulated between wave number zero and wave number N. Imagine extending the lines marking the crests of wave number zero and wave number N until they cross each other. We can designate one of those lines the hypotenuse of an exceptionally shallow right triangle along whose base our ray of light propagates. We can calculate the sine of that triangle's vertex angle by dividing the length the wavetrain by the length of the hypotenuse or, equivalently, by dividing the amount by which the length of the wavetrain increases with an increase in the length of the hypotenuse by that change in the length of the hypotenuse. In that latter case we have

Sinb = dL/dR.

(Eq'n 11)

Because we have assumed, with good justification, that the angle is shallow, we can replace the sine with the angle itself measured in radians and suffer only negligible error. So we have

(Eq'n 12)

When we make x very large relative to R (e.g. one astronomical unit versus one solar radius for a ray of starlight just grazing the sun and then observed on Earth), the second term in that equation shrinks into the negligible realm and we can write our calculation as

(Eq'n 13)

But that just gives a deflection equal to that of Equation 4. And as we did with the deflection that we calculated from Equation 4, we must double the calculation of Equation 13 in order to account for the deflection that the sun's gravity imposes upon the ray as it comes to perihelion and as it recedes from perihelion.

To obtain the total deflection we must add the contributions due to lateral acceleration of the ray and due to gravitational refraction of the ray. We thus calculate the net deflection of the ray as

(Eq'n 14)

The total expected deflection of starlight grazing the sun's limb, the baseline deflection, then calculates out as 1.752 arcseconds. Does Reality fulfill that expectation? We need only a photograph of some constellation with the sun in it and one of the same constellation with the sun not in it to answer that question. Astronomers can obtain the latter picture easily by photographing the constellation when it's in the sky at night. Obtaining the other picture, the one that must be taken during the day, comes a little less easily. Aside from carrying a camera into space, the only way to obtain the required picture is to take it in the middle of a total solar eclipse, when the moon casts its shadow over a wide enough area that it suppresses the normal blue glow of the sky and lets the stars come out.

During the eclipse of 1919 May 29 a team led by Arthur Eddington went to the island of Principe, near Spanish Guinea, and a team led by Andrew Crommelin went to the city of Sobral in northern Brazil to obtain the first photographs intended to test Einstein's hypothesis. Once the glass plates bearing the photographs had been returned to England, Eddington and his colleagues analyzed the images. They first measured the distances among the stars near the edges of the plates in order to obtain the scaling data that they needed to compensate the fact that the images were not all the same size. Then they were able to measure the distances among the stars near the centers of the plates, compensating accordingly, to find out whether the presence of the sun in the picture makes a difference in those distances. It does: the baseline deflection calculated from the analysis of the plates from Principe was 1.60 arcsecond give or take 0.33 arcsecond and the baseline deflection calculated from the analysis of the plates from Sobral was 1.98 arcsecond give or take 0.12 arcsecond. Subsequent observations have enabled astronomers to refine those values to the value accepted in 1964 - 1.79 arcsecond give or take 0.06 arcsecond.

Another manifestation of gravitational bending of light deserves mention. There is a picture, now used to illustrate astronomy books, that shows a cluster of galaxies, glowing red-orange (from the high proportion of red stars in them), with a number of thin blue arcs appearing to float among them. The arcs, called Einstein arcs, are the images made on the sky by light from a young galaxy (one with a high proportion of hot, blue stars) on the opposite side of the cluster from us. The combined gravity of the galaxies in the cluster has so bent the light coming from that young galaxy that it appears to us to be coming from sets of points that trace out arcs on the sky for us. That picture and others like unto it are, so far, the only pictures that actually let us see directly a relativistic effect.

Why? To devise a possible answer to that question, it helps to remember that in 1919 the people who comprised what we call Western Civilization had only recently emerged from a thoroughly demoralizing experience - the world's first full-scale mechanized war. It was a war whose participants saw the first major use of machine guns and long-range artillery, of aerial bombing and strafing, of tanks and poison gas. It was the first war whose death tally went into the millions and the first in which the fighting killed more men than disease did. And it was the first major war that could be shown, via newsreels in local movie houses, to people who could not otherwise have imagined the intensity and scale of the destruction and matched it against the utter futility of the whole thing in a cost-benefit analysis guaranteed to send all but the most depraved of civilizations into the bluest of funks. Those people were ready for something - anything - to lift their spirits and restore at least some of their previous faith in the ultimate goodness of Humanity. I'm guessing that it was inevitable that they would choose to invest the tattered remnants of that faith in the funny little German fellow who seemed to have risen above all the horror, casually picked Nature's most complicated mathematical locks, and thereby revealed unsuspected secrets of the cosmos. By simple acclamation our whole civilization made Einstein the paragon of scientific genius.

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Appendix: Comment upon Space Warping

I noted above that the deflection of starlight by the sun comes from the warping of space and time by the sun's gravity. Thus saith Einstein. But Einstein also offered a more elegant proof and verification of the proposition that gravity deflects light, albeit a flawed one.

In accordance with the equivalence principle, Einstein stated that a person standing in an elevator could not be able to tell whether the elevator was standing still on Earth or being accelerated through deep space at one gee. No experiment that the person could perform would tell them whether they were experiencing a resistence to gravity or to acceleration.

But inside the accelerating elevator a ray of light would appear to follow a curved path. The photons within that ray would all be flying in straight lines, but to the observer their trajectories would appear bent. Therefore, Einstein reasoned, a ray of light passing horizontally through the elevator standing still in a gravitational field must also bend, though in this case the photons really and truly follow curved paths.

By such an elegant argument did Einstein illustrate part of his greatest discovery. But we now know that this imaginary experiment tells only half the story of the gravitational deflection of light, missing the truly wonderful effect of gravitational refraction of light. In the elevator experiment we must imagine the gravitational field as being uniform in order to conform it to the implied geometry of the acceleration field. In so doing we remove the possibility of using this particular imaginary experiment to demonstrate gravitational refraction, which only occurs in non-uniform gravitational fields. And non-uniform gravitational fields are the norm in our universe.

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