GENERAL RELATIVITY

A BRIEF INTRODUCTION

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    So far I have presented what people commonly call the theory of Special Relativity. They call it special because it applies to a very special circumstance in which the observers and reference frames under consideration all move with uniform, unchanging velocities. But some of the more interesting phenomena of nature involve events observed from frames that do not move with unchanging speeds or do not move in straight lines. We need, therefore, to consider a more general theory of Relativity, one in which the reference frames move with nonuniform velocities, one in which our observers move under acceleration.

    Can we have a theory of Relativity that involves accelerating frames of reference? Recall the theorems that I deduced as equivalent to Einstein=s postulates of Special Relativity. I deduced those theorems from the conservation laws by way of the finite-value theorem, so they have a strong logical foundation. Those theorems express the proposition which tells us that Existence so structures the Universe that it always presents its phenomena to any observers in such a way as to convince those observers that they each occupy a point at the center of the Universe. Does the fact of that proposition standing true to Reality on a strong foundation of logic thus in turn necessitate that the postulates of Relativity must remain unaffected by acceleration; that is, must observers in accelerating frames apply the postulates as do observers occupying inertial frames?

    We start to answer that question by noticing that acceleration does not refer merely to the shifting of a body through a series of inertial frames. If two observers measure some velocity between them, no experiment will reveal which observer is moving and which remains at rest. But if the velocity between the observers changes, they can readily attribute the acceleration to one or the other of themselves. Unlike velocity, acceleration appears to us as an absolute state of motion. Newton=s first law of motion necessitates that fact and, thus, that law immediately calls into question the validity of the first postulate of Relativity with respect to accelerations.

    Recall that Newton=s first law states that a body cannot change its motion unless forced to do so by some interaction with another body or bodies. In order to change the velocity of a body floating in space, we must apply a force to that body and that body must transmit that force throughout itself to each and every one of its components and contents. If you and I occupy two rocketships floating in space and we notice that the velocity between our ships is changing, we can determine readily which ship is accelerating and which is not: we need only ask AIn which ship do we see free-floating objects migrating abaft en masse?@ We will both answer that question by indicating either your ship or mine. The asymmetry of our answers to that question tells us that Existence mandates a fundamental difference between non-accelerating (inertial) frames of reference and accelerating (non-inertial) frames of reference. That difference may mitigate the applicability of the postulates of Relativity in non-inertial frames. Does it and if so, how?

    I need to be clear from the beginning about what I mean by an accelerating frame of reference. Like an inertial frame, a non-inertial frame comprises points that all remain motionless relative to each other, the points all accelerating together as if they comprise a perfectly rigid solid body. We define the acceleration of the frame by way of the fact that an observer occupying the frame must exert a force (F0) upon a body of a certain rest mass (m0)in order to keep that body motionless in the frame; we then calculate the acceleration as the ratio of those two numbers (a0=F0/m0). We need that particular definition, involving a body=s rest mass, because not all observers in other frames will agree on the frame=s rate of acceleration; indeed, only observers occupying a frame, either inertial or accelerating, that comes temporarily to rest relative to the accelerating frame will measure the same acceleration between the respective frames.

    We must certainly modify the first postulate of Relativity, if only to accommodate the fact that acceleration gives the non-inertial frame an anisotropy not found in inertial frames. But, as we so often do in theoretical physics, we look for similarities before we seek out the differences. We can expect that Existence so structures accelerating frames that they maintain continuity with inertial frames; that is, an accelerating frame will not display any dilation or contraction relative to any inertial frame relative to which it stands temporarily at rest. If that proposition did not stand true to Reality, a system that begins accelerating might shrink or expand relative to objects in the inertial frame it just left and someone could exploit this fact to create or destroy energy in violation of the conservation law pertaining to energy.

    We have made the second postulate of Relativity central to our current understanding of the structure of space and time. We use the uniform propagation of light in all inertial frames as the touchstone against which we test the laws of Relativity and the laws of physics constrained by them. We rightly regard the invariance of the speed of light in vacuum as the beating heart of Special Relativity: does it also beat for General Relativity? It turns out that it does and that fact allowed Einstein to modify his Special Theory of Relativity more or less directly into General Relativity.

    According to his own recollection, Albert Einstein gained the inspiration to conceive the basis for General Relativity one morning in 1907 while he sat at his desk in the Swiss Patent Office reading the newspaper. He read a report of a carpenter who had fallen off a roof and, fortunately, landed in a pile of sand and something in that report engaged his imagination. The carpenter had told the reporter something incredible, something not only astonishing to the average person but all the more so to the trained physicist: he claimed that in the brief time of his fall he saw his tools floating before him as if they had become weightless. How can that happen? Einstein wondered. Weightlessness denotes a property of matter occupying inertial frames of reference, frames that do not accelerate, and yet the carpenter and his tools had clearly accelerated, under Earth= s gravity, toward Earth=s surface at 9.81 meters (32.2 feet) per second per second. Einstein resolved the apparent contradiction in the carpenter=s claim by conceiving the Equivalence Principle, which states that

We may conceive a system falling freely in a gravitational field as perfectly equivalent to an identical system occupying an inertial frame of reference and conceive a system constrained to move at any unaccelerated speed in a gravitational field as perfectly equivalent to an identical system accelerating through a series of inertial frames in empty space.

In that principle Einstein found the means to extend his theory of Special Relativity, which applies only to unaccelerated motions, and develop a more general theory of Relativity, one that would apply to all possible motions and would thus comprise a description of the relationship between time and space applicable everywhere in the Universe.

    Einstein=s next major inspiration toward General Relativity came, in 1908, from Hermann Minkowski, one of his math teachers. Minkowski had shown that we may regard Special Relativity as describing a Euclidean geometry in four dimensions and he had gone so far as to suggest that the concepts of space and time as separate entities would fade away and that we would replace them with a kind of fusion of the two (more properly, a con-fusion, with all the connotations that word carries), which four-dimensional continuum he called A spacetime@ (actually A raumzeit@ : he was speaking German at the time). (I have not used the word spacetime in this monograph largely because, in my experience, people generally use it more to cultivate mystical mindfuzz and less to promote scientific clarity.) Einstein took Minkowski=s description as a cue to reconceive the Equivalence Principle, to envision a reference frame marked by a gravitationally accelerated object (that is, a frame moving with the object and thereby sharing its acceleration) as actually being an inertial frame, but one that the presence of the gravitating body has warped out of true. And then Einstein knew that he would have to describe that deformed Euclidean entity with a non-Euclidean geometry.

    In the geometry of Euclid, the geometry created by the classical Greeks, we have the science of the description and deduction of relationships among figures drawn upon a perfectly flat plane (whence the name plane geometry). Euclid gave us the geometry upon which Rene Descartes drew his coordinate grid in the early part of the Seventeenth Century and, through it, worked out the means to translate algebraic equations into geometric figures and vice versa. Mathematicians call the resulting fusion of basic algebra and plane geometry analytic geometry. Physicists have used its extension into three dimensions to work out the laws of classical Newtonian physics and Einstein used it to work out the laws of Special Relativity. In the Nineteenth Century mathematicians began exploring non-Euclidean geometries; that is, geometries whose figures geometers must draw upon curved surfaces, such as the surface of a sphere (with positive curvature) or a saddle (with negative curvature). Of the results of those explorations, Einstein chose that of Georg Friedrich Bernhard Riemann (1826 Sep 17 - 1866 Jul 20) as the analogue of analytic geometry in which he would work out General Relativity. Because Riemannian geometry can describe space of any curvature (including zero, which describes the Aflat@ realm of Special Relativity), it seemed ideally suited to describing a truly general theory of Relativity.

    Unfortunately, when Riemann created his analogue of analytic geometry, the warped-space manifestations of non-Euclidean geometry obliged him to use a number-manipulating mathematics that makes the algebraic formulae of Special Relativity seem like toddlers= play. We call that part of mathematics tensor calculus and it makes General Relativity the most difficult theory in physics both to learn and to apply to an understanding of natural phenomena. Where basic algebra takes formulae made of letters representing numbers as its fundamental objects, tensor calculus takes multi-dimensional arrays of algebraic formulae and operators (entities that transform algebraic formulae into other algebraic formulae, much as algebraic formulae transform numbers into other numbers) as its fundamental objects. But with that mathematics, however difficult we may find it to use, Einstein completed his theory of General Relativity in November of 1915. As the centerpiece of that theory Einstein presented the equation that relates the tensor describing the four-dimensional curvature of space and time to the tensor describing the distribution of matter and energy in the region of space and time under study.

    How much difficulty can tensor calculus give us? When I was attending UCLA my classmates and I used to refer to difficult problems (more properly, the equations that represented those problems) as bears, seeing ourselves somewhat as Davy Crockett going into the bushes with a knife and a grin. In that metaphor Einstein=s Equation appears as a steel-plated, nuclear-powered, fire-breathing Tyrannosaurus rex. Nobody goes after Godzilla with a knife and a grin and nobody has ever solved Einstein=s Equation. What physicists and mathematicians have presented as solutions of that equation are actually well-educated guesses.

    The first of those guesses came early in 1916 from Karl Schwarzschild (1873 Oct 09 - 1916 May 11), an astronomer who was serving in the German Army on the Russian front at the time. As his solution Schwarzschild presented a metric equation; that is, he gave us an equation like Minkowski=s equation, which gives us a four-dimensional analogue of the Pythagorean theorem. Schwarzschild=s metric equation describes how the presence of a spherical body of uniformly distributed mass deforms Minkowski=s equation (in a manner analogous to the way in which we would have to modify the Asum of the squares of the sides@ rule to accommodate a right triangle drawn upon the surface of a sphere). The second of those guesses, devised in 1963 by Roy P. Kerr (1934 May 16 - ?), a New Zealander mathematician, describes how the presence of a spinning spherical body of uniformly distributed mass warps space and time. Mathematicians and physicists have also devised solutions involving bodies bearing electric charge or displaying magnetic fields, but the Schwarzschild and Kerr solutions stand as the main ones, the ones of primary interest in the study of General Relativity.

    In metric equations we do not have the kinds of equations that we can use readily to solve problems pertaining to possible experiments or observations. We can=t use metric equations in the same way, for example, that Heinrich Hertz used Maxwell=s Equations to design the electromagnetic equipment that he used to conduct his experiments with radio waves in the early 1890's. No, in Relativity we need to use transformation equations, like those of the Lorentz Transformation, but to this date no one had published transformation equations that correspond to the Schwarzschild and Kerr metric equations as the Lorentz Transformation corresponds to Minkowski=s metric equation. Nonetheless, physicists have solved some fundamental problems in this realm by using a bit of mathematical finesse. Even before he had the complete theory worked out, Einstein solved three problems that have become famous B I) that of the precession of the orbit of the planet Mercury, II) that of the gravitational Doppler shift in light, and III) that of the deflection of starlight by the gravitation of the sun.

I: Precession of Mercury= s Orbit

    We have become so accustomed to conceiving the magnifying power of an astronomical telescope as a means of giving us better pictures of planets, moons, and nebulae that few of us remember, if we ever knew, that one of the most important uses to which astronomers put that magnifying power measures the positions of the planets on the sky (i.e. relative to certain distant stars). By 1838 telescopic makers had made their instruments accurate enough to enable Friedrich Wilhelm Bessel (1784 Jul 22 - 1846 Mar 17) to measure the first parallax of a nearby star (61 Cygni, 11.2 lightyears away), accurate enough that over the remainder of the century astronomers had the ability to track the motions of the planets on their orbits to within one second of arc (1/3600 of a degree) per century. With that kind of accuracy they could determine, in particular, that the orbit of Mercury, the little planet closest to the sun, precesses at a rate of 5600 arcseconds per century; that is, that the point of the planet=s closest approach to the sun (the perihelion) as seen from the sun shifts across the sky by 5600 seconds of arc (a bit over 1.5 degrees) every hundred years.

    According to the celestial mechanics based on Newton=s law of gravity, the orbits of planets in a multi-planet system with a rotating sun should display apsidal precession (also called the advance of the perihelion). Because of the perturbing forces that the other planets and the sun=s equatorial bulge exert upon it, a planet moves in such a way that its line of apsides (the long axis of the ellipse that describes the orbit and passes through both the planet=s aphelion and perihelion) shifts in the direction in which the planet moves. Add to that the fact that Earth precesses on its axis, which shifts the reference point on the sky to which astronomer refer all other celestial motions, and the net amount by which the classical theory tells us that Mercury=s orbit must shift comes out to 5557 arcseconds per century, noticeably less than the amount inferred from observation and measurement.

    The 43 arcsecond per century discrepancy between theory and measurement vexed astronomers mightily for years. Some suggested that the extra precession came from perturbations caused by another, undiscovered planet, one flying closer to the sun than Mercury does. That suggestion actually gave astronomers a very reasonable hypothesis, even though no astronomer ever found observational evidence for such a planet: a similar discrepancy between the theoretical orbit of Uranus and the orbit inferred from observations led to the discovery of Neptune in 1846. Other astronomers suggested that Newton=s law of gravity does not give us a perfectly accurate description of the actual phenomenon of gravity and that the exponent in the inverse-square law, instead of coming out exactly equal to two, would equal two plus some small fraction. Nobody liked that idea: it felt too much like an assertion that God had failed to sand down the rough edges on the Universe after taking it out of the mold in which It had cast it.

    Fortunately, Einstein found a way to show that General Relativity accounts for the discrepancy. As a consequence (largely in light of the suggestion to modify Newton=s law) physicists often call General Relativity Einstein=s theory of gravity: more properly, though, we should call it Einstein=s theory of relativistic motion in a Newtonian gravitational field. Einstein discovered, indirectly, that we must take four additional effects into account when we calculate a description of the orbit of Mercury. The sun=s gravity produces analogues of time dilation and of the Lorentz-Fitzgerald contraction and Mercury=s motion brings the time dilation and the Lorentz-Fitzgerald contraction from the Lorentz Transformation into play. When we bring the algebraic descriptions of those effects together and work out the required numbers, we obtain a description that tells us that if those effects alone affected Mercury=s orbit, then Mercury would reach each perihelion (the point on its orbit at which it comes closest to the sun) a little over one tenth of an arcsecond past the previous perihelion; that is, starting at one perihelion, Mercury goes a full 360 degrees around the sun and then goes one tenth of an arcsecond further before reaching the next perihelion. Doing that 415 times in a century, Mercury thus accumulates the required 43 arcseconds that we must add to the other theoretical contributions to the full description of the orbit to obtain a result that matches the number obtained from observations.

II: Gravitational Doppler Shift

    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. Light possesses zero mass. Does gravity affect light? By Newton= s formula any body 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 the 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. In accordance with General Relativity, then, we should expect gravity to affect the motion of light. But, on the other hand, Relativity tells us that the speed of light cannot be changed. So what happens to a ray of light that flies straight up or straight down in a gravitational field?

    Keeping Einstein= s falling carpenter in mind, let=s imagine that two observers have set up their experiments in an elevator shaft, one observer on the ground floor and the other 74 feet (22.55 meters) above him. If the carpenter occupied a warped inertial frame during his fall, as Einstein believed, then we know that any point of an inertial frame that comes to rest at our upper observer=s position will do so for only an instant and then it will accelerate downward, passing through the lower observer= s position at 69 feet per second (21 meters per second). We also know, then, that such a point must have previously passed upward through the lower observer=s position at that same speed. But if we want that warped inertial frame to keep the full meaning of inertial frame, then we must in some way regard our two observers as moving relative to each other, with 69 feet per second between them.

    If two observers have no actual motion between them but nonetheless occupy different inertial frames, then we must use some modified version of the Lorentz Transformation to convert one observer=s measurements of two events into the measurements that the other observer would make of the same two events. Because no actual motion comes between the observers, the displacement and temporal offset terms in the transformation equations will drop out. That leaves only time dilation and the Lorentz-Fitzgerald contraction to stand between warped inertial frames. But without the temporal offset to resolve the twin paradox, how can time dilation come real in a gravitationally warped inertial frame?

    Our observers conduct a simple experiment: the upper observer drops a body made of equal parts of matter and antimatter, the body falls into a machine that the lower observer has set up to convert the matter and antimatter into radiation, and the lower observer beams the radiation to a device that the upper observer has set up to convert radiation into matter and antimatter. As the body falls it gains kinetic energy, which Relativity manifests as an increase in the body=s mass. That increased mass-energy gets converted into radiation and the lower observer=s machine beams it to the upper observer. The law of conservation of energy requires that when the radiation reaches the upper observer=s apparatus, it must turn into the same amount of matter and antimatter that the body possessed before the upper observer dropped it. Thus, we must infer that as the radiation rose in the gravitational field the amount of energy carried by the radiation diminished by the amount the body gained as it fell. The amount of energy in a beam of radiation stands in direct proportion to the number of photons that comprise the beam and to the frequency assigned to the photons. Because photons cannot simply vanish into thin air, we infer that the diminution of the energy in the beam of radiation appears as a decrease in the frequency assigned to the photons. If we had conducted the experiment in the reverse order (tossing a body upward and converting it to radiation beamed downward), we would also infer that the frequency of radiation descending in a gravitational field increases.

    On the basis of that result and of the knowledge that whatever affects the frequency of photons will affect the frequency of pulses in a Feynman clock in the same way, our observers infer that the upper observer will see the lower observer=s clocks running slower than his do and that the lower observer will see the upper observer=s clocks running faster than his do. That asymmetry, time dilation for one observer and time contraction for the other, differs from the symmetry of Special Relativity and it does so because acceleration, unlike uniform velocity, appears to us as an absolute state of motion. That asymmetry also eliminates the twin paradox from General Relativity and contributes to the resolution of the twin paradox in Special Relativity.

    Can anyone possibly test that result? The Lorentz factor between two frames moving at 21 meters per second relative to each other works out to one plus 4.907 quadrillionths. Two clocks whose rates differ by that factor would take a little over 6,457,731 years to accumulate a difference of one second in their readings. And yet in 1960 Robert V. Pound and Glen A. Rebka set up an apparatus in an elevator shaft 74 feet high and measured the effect with an accuracy of ten percent. To achieve that measurement they exploited the Mössbauer effect, in which atoms of iron-57 embedded in certain crystals emit and absorb gamma-ray photons with energies of 14.4 thousand electron-volts over a very narrow frequency range. They set up the emitting crystal at one end of the shaft, set up the absorbing crystal and its associated detector at the other end of the shaft, and used a motor to move the absorbing crystal up or down to make the classical Doppler shift compensate the expected gravitational shift of the gamma photons= frequency. They found that, in order for both effects to cancel each other, the crystal had to move at a speed of 5.3 millimeters per hour, a pace that we can=t even call glacial (glaciers move faster). For all its sloth, that pace matched the theory perfectly.

    That warping of time doesn=t occur only for observers that remain stationary in gravitational fields. The Equivalence Principle tells us that it also occurs in any frame that accelerates in space free of gravitational fields. That set includes frames that we have constrained to move horizontally across Earth=s surface, so it would include the frames that we used to explore the various distortions of time caused by relative motion in our imaginary experiments involving trains moving between Fresno and Modesto. Recall those experiments to mind and look at them again in this new light. We inferred that the motion of the train leaving Fresno somehow causes the clocks in the Modesto station to advance their counting of time over that of the clocks in the Fresno station. We made that inference in order to avoid paradox, but now we can see how it came about. When the train accelerates out of the Fresno station, it and everyone in it occupy a non-inertial frame, in which any given clock ticks faster than the clocks behind it do and slower than the clocks ahead of it do. In that frame and for the observers in it the clocks in Modesto tick faster than do the clocks in Fresno in just the right way to create the required temporal offset between those clocks in the inertial frame that the train occupies after it stops accelerating and travels to Modesto at a uniform 86.6 miles per hour. Of course, the train=s deceleration into the Modesto station reverses the effect, thereby erasing the temporal offset between the Fresno and Modesto clocks, as it should.

III: Deflection of Starlight

    Where time dilates, Relativity tells us, distances contract. That theorem prepares us to deduce the proposition that an object lowered into a gravitational field will shrink in the vertical direction as it descends and will expand in the vertical direction as it ascends. How would you test that proposition to see whether it stands true or false to Reality?

    Einstein suggested using starlight. We can see the feature that makes starlight ideal for such a test of General Relativity in the fact that, after traveling distances measured in lightyears, even lightcenturies, starlight 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. In a telescope a plane wave appears 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. That accuracy gave Einstein what he wanted to exploit.

    According to Einstein=s theory, the sun=s gravity will so bend starlight that the light will appear to observers on Earth to emanate from a point displaced a small angle away from the point whence the light actually came. Thus, if, from our perspective, the sun appeared in the middle of some constellation, the stars in that constellation would appear to us to lie farther apart from each other than they appear six months earlier or later, when the sun lies on the opposite side of the sky. That bending of starlight comes about from two effects B 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. Acceleration denotes the rate at which the velocity of some thing changes and velocity denotes the 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 0.875 arcsecond from its actual point of origin if this direct acceleration provided the only effect acting upon the light.

    The gravitational analogue of refraction originates in the gravitational contraction of vertical distances. Any line that you draw past the sun will have components oriented partly in the vertical direction, relative to the sun, except for one very short component at the point where the line comes closest to the sun. That line will participate in some small degree in the shrinkage of space near the sun. In order to envision how that shrinkage makes light bend you might imagine drawing two straight lines parallel to each other, one just grazing the sun=s photosphere on its way to Earth and the other passing somewhat further from the sun. If you had marked those lines off into kilometers, you could see that the kilometers on the line grazing the sun have shrunk slightly more than have the kilometers on the farther line. But a ray of light crosses every kilometer, shrunk or not, in 3.33564 microseconds.

    Imagine a ray of light so propagating through the solar system toward a telescope on Earth that a portion of it just grazes the sun. Now imagine tracking the wavefronts of the ray, the flat planes defined by the crests of the electromagnetic wave comprising the ray, as they cross the kilometer markings on the lines defined above. Because of the relative shrinkage of the lines, the part of a wavefront closer to the sun will lag behind the part on the line farther from the sun. That lag will make the wavefront tilt, turning through a small angle as it passes the sun. Because light always propagates in the direction perpendicular to the plane of its wavefronts, that tilting of the wavefronts makes the ray itself turn through the same small angle. Thus we have a description of the refractive effect of the sun=s gravity. The mathematical description of that effect matches identically the mathematical description of the sideways acceleration of light, so we get the same result: a ray just grazing the sun bends by 0.875 arcseconds and rays passing progressively farther from the sun bend progressively less.

    We thus calculate the total expected deflection of starlight grazing the sun= s limb, the baseline deflection, as 1.75 arcseconds. Does Reality fulfill that expectation? All that we need to answer that question will appear on a photograph of some constellation with the sun in it and on a photograph of the same constellation with the sun not in it. Astronomers can obtain the latter picture easily by photographing the constellation when it appears in the sky at night. Obtaining the other picture, the one that we must take during the day, offers a bit more of a challenge. Aside from carrying a camera into space, the only way to obtain the required picture goes right through 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 they brought the glass plates bearing the photographs back 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 did not all have the same size. Then they measured 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 came out to 1.60 arcsecond give or take 0.33 arcsecond and the baseline deflection calculated from the analysis of the plates from Sobral came out to 1.98 arcsecond give or take 0.12 arcsecond. Subsequent observations have enabled astronomers to refine those values to the value accepted in 1964 B 1.79 arcsecond give or take 0.06 arcsecond.

    Another manifestation of gravitational bending of light deserves mention. The astrophysicists Fritz Zwicky (1898 Feb 14 - 1974 Feb 08) first suggested the idea as it applies to clusters of galaxies in 1937, though nobody actually observed the effect until 1979. A picture, now used to illustrate astronomy books, 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, which astronomers call Einstein arcs, display on the sky images made by gravitational bending of 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 distorted the light coming from that young galaxy that it appears to us to come from sets of points that trace out arcs on the sky for us. That picture and others like it give us the only pictures taken so far that actually let us see directly a relativistic effect.

    Let me also point out that Eddington and Crommelin=s observational verification of Einstein=s hypothesis made Einstein famous. The two team leaders presented their results to a joint meeting of England=s Royal Society of Sciences and Royal Astronomical Society on 1919 Nov 06 and the next day=s London Times proclaimed in its headlines AREVOLUTION IN SCIENCE@ and ANEW THEORY OF THE UNIVERSE@. The New York Times of 1919 Nov 08 came out with somewhat less restraint: ALIGHTS ALL ASKEW IN THE HEAVENS,@ the headline blared and then added, AMen of Science More or Less Agog Over Results of Eclipse Observations@ . (Please note that Aall askew@ refers to stars displaced from their normal positions on the sky by angles narrower than the angle that you would measure between two straight lines drawn from the tip of your nose to opposite sides of a baseball 46 miles away.) Some newspapers even took up the practice of publishing equations from Einstein=s latest work, even knowing that none of their readers could come any closer to understanding then than they could come to reading Egyptian hieroglyphs: people simply wanted some connection, however tenuous, to the previously obscure physicist.

    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 B the world=s first full-scale mechanized war. In that war the participants saw the first major use of machine guns and long-range artillery lobbing explosive shells, of aerial bombing and strafing, of tanks and poison gas, of submarines and dreadnought battleships. It went into the history books as the first war whose death tally went into the millions and the first in which the fighting killed more men than disease did. And as the first major war that could appear, via newsreels in local movie houses, to people who could not otherwise have imagined the intensity and scale of the destruction, it provided a horror show that displayed the utter futility of the whole thing and triggered a cost-benefit analysis guaranteed to send all but the most depraved of civilizations into the bluest of funks. Those people needed something B anything B to lift their spirits and restore at least some of their previous faith in the ultimate goodness of Humanity. I can only guess at the inevitability 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. But inevitable or not, by simple acclamation our whole civilization made Einstein the paragon of scientific genius.

IV: Shapiro= s Delay

    In 1964 Irwin Shapiro published a paper with the elegantly descriptive title AFourth Test of General Relativity@ . While attending a lecture in 1961 he had discerned the fact that I pointed out in the previous section B that a line drawn between two established points spans more kilometers if it passes close to the sun than it spans if it passes far from the sun. Shapiro gained from that fact the idea that radio signals passing the sun as they travel between planets would suffer a short delay beyond the time that astronomers would normally calculate from the positions of the two planets and an assumption that space remains unwarped. To avoid confusion, I=ll point out here that Einstein did not discover this effect because he worked out the bending of light by means of a highly abstract mathematical treatment that did not involve acknowledging, even tacitly, that the shrinkage of space by gravity increases the optical path length of the light passing the sun.

    Using essentially the same idea that I presented in the previous section, Shapiro calculated that a line drawn from Earth to Mars would, if it just grazed the sun, have 37.5 more kilometers in it than we would expect from a calculation based on the Newtonian description of the planets= orbits. That extra distance would add about 250 microseconds to the flight time of a radio signal sent from Earth to Mars and back. In July of 1976 Shapiro gained the means to test his hypothesis when Project Viking landed a space probe on Mars. Used as a radio transponder, the Viking lander provided a pinpoint location for the far end of the line from Earth to Mars. Thus Shapiro and his collaborators verified the expected delay. Measuring the reception of the radio signal to an accuracy of one part in one thousand, Shapiro and his team made this observation the best confirmation of General Relativity available so far.

    As you would expect, all four of the effects described above conform to Schwarzschild=s solution of Einstein=s spacewarping equation, though Einstein described three of them before Schwarzschild devised his metric equation and he could have described the fourth one. But one phenomenon, whose description emanates directly and only from Schwarzschild=s solution, looms large in the popular imagination. One of the coefficients in the equation has the form of a fraction and the denominator contains a minus sign. When the indicated subtraction yields a zero, the coefficient Ablows up@; that is, its value becomes infinite. At the place described by that infinite-valued coefficient vertical distance flattens to nothing and the river of time freezes up solid. Physicists and astronomers call the collection of all such points, forming a spherical shell around the body whose mass generates them, an event horizon (because we cannot detect any events beyond it) and it and they call the volume that it encloses a black hole.

    In order for a body to meet the requirements of a black hole all of the body=s mass must exist within the body=s Schwarzschild radius, the radius of the event horizon (as measured from a great distance, of course). Reality cannot easily satisfy that condition, as you can discern in the fact that the sun has a Schwarzschild radius of 2.956 kilometers (the sun=s mass actually spreads throughout a volume with a radius of 690,000 kilometers) and that Earth has a Schwarzshild radius of 8.87 millimeters, about the size of a standard seedless grape. Astrophysicists believe that only the collapse of a very massive star=s core when that star explodes as a supernova has the power to create a black hole. Once a black hole comes into existence it may then grow by drawing more matter into itself. This, astronomers believe, provides the driving force behind quasars, the exceptionally bright objects found in galaxies far out in deep space. In those objects, according to the theory, the light that we see comes from matter heated up by compression as it gets drawn into a giant black hole at the center of a young galaxy. For a short time such a central black hole can outshine its own galaxy, devouring whole stars when it has grown large enough. Eventually, though, the black hole uses up all of the matter available to it and it goes quiescent. Astronomers believe that just such a fasting black hole floats at the center of every spiral galaxy, ours in particular. Do we have any evidence to support such a belief? Professor Andrea Ghez and her coworkers in the Physics and Astronomy Department at UCLA have used an infrared telescope over the past several years to measure the motions of a small number of stars at the center of the Milky Way, a feat only made possible at a range of 26,000 lightyears by the fact that those stars move rapidly. Doctor Ghez interprets those speeds as representing the motions of those stars in orbits about an object five million times as massive as our sun, an object that does not show up in any of the photographs, even though ordinary stars in that region show up clearly in the pictures.

    Finally, General Relativity contains other topics beyond these few introductory pieces. I have shown you here only a brief introduction and in subsequent essays I intend to expand the scope of this presentation. I will revisit topics that I presented above, but in more detail. And I will cover topics that I did not include in this brief introduction. I hope that I will eventually provide a good overview of the complete Theory of Relativity. But for now what I have shown you here should suffice to give you a good sense of what General Relativity means.

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