On 11 February 2016 a paper was published in Physical Review Letters, arguably the most important physics journal in the world, announcing the discovery of gravitational waves. [Reference 1]  On the same day the New York Times made the same eannouncement. [Reference 2]

Clearly this was a BIG DEAL.  How big?  Almost certainly (>99% probability, in this author’s estimation) it will mean a Nobel Prize for three men:  Ronald Drever, Kip S. Thorne, and Rainer Weiss. They are listed, alphabetically with over 100 authors, in the Letters article.  The editor of the Letters, according to the Times article, said that he got goose bumps when he read the paper.  People are saying it is the most important discovery in physics in years.  (And to top it off, two cartoons in the 15 February 2016 Deseret News featured the discovery in a satirical commentary on the 2016 Republican campaign for the presidential nomination!)

There is a caveat.  The results must be confirmed by independent experiments in order to qualify as a real discovery.  The research was very carefully done, and I think that it will indeed be verified, but we still have to wait and see.

Incidentally, some people have asked me if I had any connection to the gravitational wave theoretical and experimental teams.  The answer is no.  I merely have had the good fortune to have had John Archibald Wheeler (see below) as my Ph. D. advisor, who was also Kip Thorne’s advisor, and to have had a continuing friendship with Kip. Some of the work we did on stars at the endpoint of their lives is discussed below in the section on white dwarfs, neutron stars, and black holes. 


So what are gravitational waves?  And what is the theory of Einstein’s that is confirmed, as mentioned in the NYT headline?  And why should we care?

Sir Isaac Newton, in his 1687 work Mathematical Principles of Natural Philosophy, presented laws of nature that we now call Newton’s laws of motion and Newton’s law of gravitation.  This work is considered by most people to mark the beginning of modern science.  The laws of motion introduced the idea of a force, a push or a pull.  Also, Newton realized that the agent that caused objects at the surface of the Earth to fall downwards, gravity, was the same agent that caused the Moon to orbit the Earth, and he formulated his law of gravitation, to wit, that gravity is an attractive force between any two objects with the properties of inverse square distance behavior; e.g., if the distance between them was doubled, the force would be cut in half, and if the distance was tripled, the force would be cut by three, and so on.  The law can be expressed succinctly in mathematical terms as F = G M m/d2, where F is the force, M and m are the masses involved, d is the distance between them, and G is a proportionality constant, now known as Newton’s gravitational constant. 

Newton had the audacity to suggest that his law applied throughout the universe, that any two objects with mass would experience a gravitational force between them.  Yet it has turned out to be correct so far as man can see into space.  It applies to rockets sent into space, to planets as they move around the Sun, to satellites of planets, to space probes sent beyond the solar system, to meteorites and comets, to stars and their structure, and to galaxies and their interactions.

Almost correct, that is.

Optics, the Science of Light

Newton also studied optics and proposed that light might be considered as a stream of particles.  As such, they would be acted on by gravity.  This led an unsung astronomer named John Michell, in 1783, to suggest that under certain conditions, if a star were large enough, its gravitational escape speed would be greater than the speed of light  (which was known to good accuracy at the time), so that light could not escape.  In modern terms, it would be a black hole!

Alas for the idea of the particle stream, the physicist Thomas Young, in 1800, showed that light was actually a wave.  Waves of course are familiar to us from water motions.  A wave typically is considered to have a crest, the top, and a trough, the bottom.  If two waves come together, they will interfere. (They should be of one color, or wavelength, so they will have a simple form.)  If two crests or two troughs in a light wave come together, they add in constructive interference and there will be a bright spot; if a crest and a trough come together, they cancel out in destructive interference in a dark spot. Suppose two light waves (of the same color, or wavelength) arrive together on a movie screen such that their crests align at one point and constructively interfere. Now consider a point slightly to one side or the other.  One of the waves will have to move slightly further to reach that point than the other, and consequently will have to traverse a slightly different distance.  Now the crests no longer come together; a crest will align with a trough and we have destructive interference, a dark spot.  As we shift a little further, two crests will come together again and there will be a bright spot.  As we continue to move to one side or the other, there will be an alternating succession of bright and dark spots.  This is what is called an interference pattern.  The alternating spots are called fringes.

If a light is shone through a single narrow opening, called a slit, the light from various parts of the slit, arriving at different places on a screen, will cause an interference pattern.  The overall brightness will decrease to the left or right.  But if one shines a light through several slits side by side, the pattern exhibits a uniform brightness throughout its length.

Young showed that light behaved in this manner.  It is a proof that light is a wave, since a series of particles, bullets, for example, fired together at the screen will not cancel out and produce darkness.  And Michell’s idea was quietly shelved; where it was mentioned in pre-1800 publications, it does not show up at all in publications after 1800.

Young worked with a series of slits in one direction.  If you have an object with openings or slits in two directions, the pattern will have two dimensions.  One can see such a pattern by looking toward a distant light source through such objects, such as curtains, an umbrella, or a pocket handkerchief, all of which have a fine weave.  The interference appears as a regular rectangular pattern of light and dark regions. This kind of pattern can appear through a the regular openings in a microwave oven screen, if there is something dark in the background so the pattern shows up. (Note:  there are two different meanings of the word screen in common use; a movie screen, and an object with regular holes in it like the microwave screen or a window screen.  I have used both here.)  Oil on water shows various colors in various patterns; these are interference patterns, caused by varying thickness in the oil layer.

Electricity and Magnetism

In the late 1700s and early 1800s, the laws of electricity and magnetism were being discovered.  Coulomb’s law gave the force between two electric charges, q and Q.  It will look familiar:  F = k Q q/d2, very much like the law of gravitation.  There are two differences:  charges may be positive or negative while mass is only positive, and the proportionality constants G and k are different.  The sign of the charge will not concern us here, but the difference between the sizes of G and k is profound, as we shall see.

Further discoveries in electricity and magnetism led to such facts as: 1) electric current causes magnetic fields, the principle at work in electromagnets, and 2) changing magnetic fields cause electric fields and current, this fact being the basis for electric power generation.  Finally, James Clerk Maxwell, in the 1860s, by introducing an additional term in the equations to insure mathematical consistency, showed 3) that waves could exist in electromagnetism.  The speed of electromagnetic waves, derived from Maxwell’s equations, turned out to be exactly the same as the speed of light, which had been previously determined astronomically, so that it could now be inferred that 4) light itself is an electromagnetic wave.  Heinrich Hertz later demonstrated the existence of electromagnetic waves in laboratory experiments.

In modifying and unifying the equations, Maxwell combined the sciences of electricity, magnetism, and optics, showed that electromagnetic waves exist, and changed the world.  We now know various types of electromagnetic waves, besides light: radio, TV, infrared, ultraviolet, X-rays, and gamma rays.  I used to ask my students who had heard of Thomas Edison.  Of course everyone had.  Then I would ask them who had heard of James Clerk Maxwell.  No one had.  Yet he changed the world far more than Edison.  (I should also credit Maxwell’s predecessors Charles Augustin de Coulomb, Hans Christian Oersted, André-Marie Ampère, Michael Faraday, and Joseph Henry, who laid the groundwork leading up to Maxwell’s unified theory.)

Enter Albert Einstein; Special Relativity

Einstein was fascinated with light and with electromagnetic theory.  As he studied these matters he came in contact with certain paradoxes that had been known and puzzled over for much of the 19th century, having to do with the possible existence of the aether, a substance that was supposed to permeate all of space and to be at rest.  This substance was presumably the stuff that wiggled as light and other electromagnetic waves passed through it.  There were intense discussions in the scientific community about its properties and whether the earth carried it along in its orbit.  One could hopefully find the speed of light relative to this substance, and show that this speed was enhanced by the earth’s forward motion, or decreased if light shone in a direction opposite to the earth’s motion. 

It was shown in the 1880s by Albert Michelson and Edward Morley that light speed was unaffected by the Earth’s motion.  It traveled at the same speed in all directions.  This caused great confusion about the existence and nature of the aether.  Some prominent physicists announced defensively and vociferously that the aether did indeed exist.

Michelson and Morley used an interferometer, in which a light beam was split into two beams and sent in two directions at right angles.  The separate beams were reflected and returned to meet again.  The apparatus was adjusted so that the beams met in constructive interference.  When the apparatus was rotated, the presumably different speeds of light in the two directions would cause a shift in the fringes in the interference pattern.  But no shift was detected.  There seemed to be, as noted above, no difference in light speed in any direction.  In a separate matter, It became clear that Newtonian mechanics and electromagnetism had mutual mathematical inconsistencies.

The puzzles were all resolved in 1905. In one of a quartet of enormously significant German publications, Einstein introduced his theory of special relativity, which redefined our understanding of space and time.  He used only two simple postulates, one of which stated that the speed of light was independent of its source (and thus the Earth’s motion would not affect it.)  The aether was abandoned.  He introduced certain formulas (actually anticipated by H. A. Lorentz and G. F. Fitzgerald) relating measurements of time and space in systems moving constantly relative to each other (from which came the name, the theory of relativity.)  These formulas were consistent with Maxwell’s equations, but were not consistent with Newton’s equations of motion.  However, the difference was insignificant in systems moving at speeds small compared to the speed of light, which is large, and so we do not see relativistic effects in everyday life.  Nothing moves at such speeds except atomic particles in some atomic and nuclear reactions.

Several consequences resulted from this theory.  One was that moving clocks run slower than stationary clocks and another was that moving rods are shorter than rods at rest. Again, these differences are not observable at ordinary speeds, although the time change effect has been observed using precise clocks on airlines. 

These consequences caused many to question the theory, on the grounds that it had paradoxes.  If system B moves relative to system A, one infers that B’s clocks will run slow compared to A.  But one could also say that system A moves relative to B, so A’s clocks should run slow.  Which is right?  The answer lies in careful definition of experiments.  Considering system B to be moving past A, one can only detect the slowdown of a clock of B’s by comparing it with two clocks in A.  That introduces a nonsymmetrical situation.  In order to see if one of A’s clocks runs slow, one would have to compare it with two clocks in B.  There is no paradox.  Similarly, to measure the shortening of a moving rod in B, one needs two cameras in A to take simultaneous pictures of the ends of B’s rod and then one measures the distance between these two clocks.  Other paradoxes can be dealt with similarly by careful definition.

Another consequence was that the speed of light is a limiting speed; no signal can move faster than light.  (This has obvious implications for space travel.)  A fourth was that simultaneous events in one system may not be simultaneous in another.  Finally, there is the famous equation E = m c2, where c is the speed of light.  Energy and mass are interchangeable,

As noted above, special relativity required slight changes in Newtonian mechanics in the laws of motion.  They were almost correct.  Thus did Einstein deal the first of two blows to Newton.  The second follows below.

The Beginnings of General Relativity

The differences of measurements spoken of in the last paragraphs result from relative speeds between systems, which speeds are unchanging, or uniform.  We will now see the amazing consequences that followed from Einstein’s attempt to generalize the theory to nonuniform, or accelerated motion.

Consider a box in which you are confined.  You while away the boredom by doing experiments.  In one experiment, in which the box sits at rest on the surface of the earth, you drop a brick.  The brick accelerates or speeds up as it falls, due to gravity.  In a second experiment, your jailer pulls the box upwards with a rope.  The pull, which is a force, accelerates the box (force causes acceleration, Newton’s second law of motion.)  Now if you, still inside, let a brick go, the floor of the box accelerates up to meet it.  From your observation inside, the behavior of the brick is the same in the two experiments. You cannot tell which experiment is being done. The effect of the acceleration is indistinguishable from the effect of the gravity.

Now drop two objects, a brick and a book.  They fall together, as Galileo is supposed to have demonstrated.  (Assume there is no air. You are wearing an oxygen mask!  David Scott, a member of the Apollo 15 team to the moon, wearing an oxygen mask, demonstrated this in the absence of air by dropping a hammer and a feather.  They fell together.)  Alternatively, if the box is accelerated by the rope pull, the floor accelerates upward and the same thing happens.  It looks inside as if both objects accelerate downwards relative to the box.  The nature of the objects does not affect this.  (This is in contrast with other forces, like electricity, in which a proton and an electron, say, with the same charge but different masses, have different motions.)  If one writes out an equation in which the force in Newton’s laws of motion (which equals mass m times acceleration a), equals the force in the law of gravitation, F = ma = G M m/d2, the masses on the two sides cancel, leaving a = G M/d2.  The physical significance of the masses is very different—m in Newton’s laws refers to “inertia”, the tendency of an object to move uniformly, in a straight line, while m in the law of gravitation is a source of gravitational pull—but experimentally they are equal.  We call the two types of masses inertial mass and gravitational mass.  Their equality—which has been tested to high accuracy in numerous experiments—is called the principle of equivalence.  It plays a fundamental role in Einstein’s general relativity, which we are about to discuss.   

Incidentally, in the equation a = G M/d2 one may take M to be the mass of the Earth, d to be the Earth’s radius R, and a to be the acceleration of gravity g, yielding g = G M/R2.  The value of G has been measured carefully in a laboratory measurement, enabling the determination of the mass of the Earth from this equation, since g and R are well known.

So the acceleration of gravity on an object is not due to any property of that object.  It is determined solely by the mass and radius of the object on which it rests—in other words, by what is around it—or, to use another word, by its environment—or to use still another word, the space around it.  This is a bit of a stretch; we will flesh this out shortly.

Let us now consider a new experiment.  We go to Lagoon (Utah) and ride the turning wheel in the Fun House to see if we can stay on.  (If you have not had that experience, you haven’t lived!)  We assume that the wheel is built very sturdily so that it can be sped up to a point where the edge travels at near the speed of light.  We leave it to you to devise a way to stay on. 

Again to combat boredom you do experiments.  You have with you a meter stick and you measure the radius of the wheel.  We assume at the beginning that the wheel is not rotating.  You go out to the edge and measure the circumference of a circle painted on the floor just below the edge of the wheel.  Not surprisingly, you find that the circumference is two pi times the radius.  But now let the operator turn the wheel on.  You measure the circumference again.  You and your meter stick are being carried around on the wheel, and your meter stick is therefore shortened compared to a meter on the stationary floor below.  It now takes more applications of the meter stick to determine the circumference than before, and so you conclude that the circumference is larger than it was before.  The circumference is now greater than two pi times the radius.

How can this be?  Euclidean geometry, or plane geometry, seems no longer to apply.  We have actually emerged into the realm of non-Euclidean geometry.  To use other words:  Euclidean space is flat space; non-Euclidean space is curved space.

The rotating wheel is similar to our box, but the motion is circular.  Circular motion is accelerated motion; the motion is not in a straight line.  On the wheel you seem to feel a pull outward; it is rather like your experience in the box when the box is pulled upward.  In both cases, you think there is a force, seemingly gravity, acting on you, when that sensation is actually due to the motion of your system.  (This idea is used in science fiction stories in which a spaceship rotates in order to simulate gravity.)

Let us now summarize.  Acceleration seems interchangeable with gravity, in some sense; and in the rotating wheel, it seems also to indicate that the space is curved.

We mention another example of curved space:  the surface of the Earth.  Consider the circle of the equator.  Draw a radius for that circle, not lying inside the Earth, but on its surface.  This radius goes from the equator to the north pole.  You will see that it is one-fourth of the Earth’s circumference.  Or, the circumference is four times the radius, again not equal to two pi times the radius.  (In the previous case, it was greater than two pi times the radius; in this case it is less.  These represent two kinds of curvature, negative and positive, respectively.)

There is another curvature effect.  Draw a triangle on the Earth’s surface.  Start at the equator at 0o longitude.  Move along the equator to 90o west longitude (or east if you prefer), then north to the pole, then south along the 0o longitude line.  Your triangle has three 90o angles, adding to 270o, very different from the 180o you get in Euclidean geometry.  If you have negative curvature, the total is less than 180o, not greater.  Mariners have to learn spherical trigonometry, the mathematics of the Earth’s surface.

As Einstein generalized his special relativity, to general relativity, to include accelerated systems, he realized that he was getting a theory of gravity—and furthermore, that he would have to generalize space (and time) to a curved spacetime.  (We will not attempt to illustrate the time part of this here.)

In Newton’s theory, gravity is caused by the presence of matter (mass), like the mass of the Sun.  So Einstein incorporated that idea in his theory too.  We are now ready to present a nice illustration of how this works.

Go to the fabric store and buy a yard of Spandex©, which is elastic.  Build yourself a frame of wood, in the form of a circle or a square, it doesn’t matter.  Tack the fabric to the wood, stretching it out so it makes a tight, level surface.  (The author has done this for demonstration purposes.  If you don’t want to make your own, you can come to my house and I will show it to you.)  Set it on a table, maybe resting on books, so it is level.

The surface that you have is flat.  It represents Euclidean space.  If you roll a small ball across it, the ball moves in a straight line, actually showing Newton’s first law of motion (which was originally demonstrated by Galileo.)  (Friction will be present and will slow down the ball.  Never mind.)

Now plunk a big ball down on the fabric.  The fabric dips into a sort of funnel.  Roll the little ball again.  It no longer moves in a straight line, but in an orbit around the big ball.  (It will spiral into the big ball because of friction.  Never mind.)  The big ball represents the Sun; the little ball represents the Earth or any one of the planets. The nature of the little ball does not affect its motion, assuming it is small; if it is big, it will cause its own depression in the fabric, and then you have a double star.  The little ball orbits the big ball, not because there is a force between them, but because of the curvature of the fabric.  (Actually, in this model, there is a force between them due to the elasticity of the fabric, but in space, there is no fabric; there is just, well, space.)  You may choose to say that space exerts a force on the planet, but it is simpler just to say that space is curved and the planet moves in that space.  The Sun is what causes the curved space. Note that the nature of the motion does not depend on what kind of object moves, like the simple experiments in the box spoken of above.

John Archibald Wheeler (who was, incidentally the author of the term “black hole”) coined the phrase: “Matter tells space how to curve; space tells matter how to move.” [Reference 3]   This concept, due to Einstein, that gravitation is an effect of curved space, is simple and beautiful; some have called it the most elegant theory ever devised by man.

Einstein gradually came to the realization that the technical mathematics of curved space was necessary for his theory.  In this he was aided by a mathematician named Marcel Grossmann, who introduced him to the mathematics that he needed, Riemannian geometry, worked out by Bernhard Riemann in the 19th century.  (Grossmann was instrumental in helping Einstein in other ways; he helped Einstein with the math he missed when he skipped class, and his father-in-law helped Einstein get the job in the Swiss patent office where Einstein devised his famous papers of 1905.  There is now a triennial international series of professional meetings named after Grossmann to honor his behind-the-scenes contribution to relativity.)

Einstein, with Grossmann’s help, was able eventually to formulate his theory mathematically, using Riemannian geometry.  He constructed it so that at base it was consistent with Newton’s theory of gravitation.  Here are the final Einstein’s equations, the foundation of general relativity, as formulated in the year 1915:

Rik – (1/2) R gik  = (8πG/c2) Tik ,

where the left-hand side represents the curvature of space-time and the right-hand side represents the matter that causes the curvature.  There are no fudge factors, no adjustable constants; the equations are simply elegant and elegantly simple.  Another set of equations, not shown here, shows how curvature causes matter to move.  They represent geodesics, the shortest line between two points in space-time.  In other words, objects move in as straight a path as possible in four dimensional curved space-time.

Unfortunately, while the concept is simple, the mathematics of the equations above is horrible. The apparently simple symbols shown conceal ten very nonlinear, four-dimensional, second order partial differential equations—which means, as I said, that the mathematics is horrible.  Maxwell’s equations also represent partial differential equations, but they are linear—which means that you can add any two solutions and get a third—and the theory of linear differential equations is well known.  But there is no general theory of nonlinear partial differential equations; solutions have to be found one at a time, by assuming simplifications and by hard work.  By 1963, nearly fifty years after Einstein proposed his equations, only a few exact solutions were known.  But they were important ones:  the gravitational field of a single stationary object such as the Earth or Sun, which, when carried to the extreme, gives the field of a stationary black hole (the (Karl) Schwarzschild solution); the generalization to a rotating object (the (Roy) Kerr solution); and cosmological solutions which represent the Big Bang, found by Alexander Friedmann.  There were also approximate solutions.  These included the generalization of Newton’s gravitation theory, a very small correction.  Again, Newton was almost correct but not quite. The additional terms in the equations resulting from Einstein’s corrections have been explored in the last fifty or sixty years, mostly in astronomy, and in all cases Einstein’s theory has been verified without question, at least to the limits of measurement.

The fact that the correction is small means that general relativistic effects will not be important in practical affairs at the surface of the Earth—with one exception.  GPS satellites provide detailed information about one’s position on Earth, to great accuracy.  General relativistic (and special relativistic) corrections to the details of their orbits are necessary, otherwise the coordinates of positions would shortly become compromised and meaningless.

Shortly after the publication of the theory, by putting the equations into an approximate linearized form, Einstein found an approximate solution that represented waves—gravitational waves. (By the way, they travel at the speed of light, like electromagnetic waves.)  At first, people weren’t sure that the equations really represented physical reality, but after while it was accepted that they did.  Now to look for them.

Ah, there’s the rub.  Electromagnetic waves have been known for a century and a half.  They are strong.  Gravitational waves are very weak.

But, you say, we know gravity.  It is very familiar to us.  How come it is weak? 

Gravity is familiar to us only because we stand on a very large Earth.  Let us compare the gravitational force between two objects with the electrical force between the same objects, say two protons.  If one looks at the gravitational force equation and the electrical force equation, we remember that there are two proportionality constants:  G and kk is large; G is small.  How do the forces compare?

The electrical force between two protons is 1036 times as large as the gravitational force between them.  That is ten followed by thirty-six zeroes, a difference far beyond our experience.  For example, consider the largest distance we normally experience—say 100 kilometers (10,000 if we travel the Earth.) The smallest is maybe the width of a human hair, maybe a tenth of a millimeter.  The difference in scale is nine powers of ten.  But thirty-six?

With your apparatus you can illustrate a gravitational wave.  Put a small object (I used a jack) on one side of it, on the fabric.  Hit the fabric on the other side.  A wave will travel quickly to the jack and cause it to jump.  That is a gravitational wave. 

This illustrates an important point.  One does not look for a gravitational wave with a telescope, as one would look for light or radio waves.  One needs an apparatus that detects slight displacements of objects due to the passage of a gravitational wave.

In the 1950s and 1960s, Joseph Weber of the University of Maryland built the first such apparatus:  a large aluminum cylinder with detectors around it to detect any slight distortion due to a passing gravitational wave.  He built several, and in 1969 he claimed that he had detected gravitational waves.  The announcement caused great excitement, but after a few years no one had been able to verify his claim.  No one ever did, though to the end of his life he claimed he had discovered gravitational waves.  He is a tragic figure; but he began the field of searching for gravitational waves and for that he deserves an enormous amount of credit.

In 1974, two astronomers, Russell Hulse and Joseph Taylor, made careful observations on a distant pair of orbiting neutron stars.  They compared the results on the observed decreasing period of rotation with theoretical predictions from general relativity, finding excellent agreement.  The decreasing period results from radiation by gravitational waves.  Thus Hulse and Taylor had demonstrated the existence of gravitational waves, although indirectly.  For that they received the Nobel Prize.

In the 1980s Kip Thorne, Ronald Drever, and Rainer Weiss began investigating a new idea for gravitational waves, using an interferometer, the same kind of instrument used by Michelson and Morley to look at the speed of light.  But the idea was quite different.  In the gravitational wave interferometer, called LIGO, Laser Interferometer Gravitational-wave Observatory, two arms are mounted at right angles.  A laser beam is split and the two parts travel to two mirrors, attached to two pendulums, and are reflected back to interfere when they meet.  They are adjusted so that fringes align when at rest, in constructive interference.  If a gravitational wave comes by, the pendulums will be disturbed and the interference pattern will show a fringe shift.  The time between the disturbances of the two pendulums will tell something about the direction from which the wave comes; if for example, the wave hits pendulum A before it hits pendulum B. you know it came from the A side of the apparatus.  Careful determination of the time difference gives good information about the direction.   There are two such systems, in Washington and Louisiana, to help provide even more accurate information about direction.  Detection of a signal at both systems is excellent verification that a wave has indeed been detected.

The arms are long, four kilometers in length, and provide multiple interior back-and-forth reflections of the beams to give a long travel distance.  That is needed to improve accuracy.  The apparatus is shielded from outside disturbances such as traffic and earthquakes.  Extensive, very sensitive processing equipment is used to analyze the results.  The expected deviation of a mirror is extremely tiny. One four-thousandth of the size of a proton was the deviation for the reported experiment.  The only sources that would produce detectable waves are systems with strong gravitational, changing, fields, such as orbiting or merging neutron stars or black holes, and maybe the Big Bang itself. (Stationary systems will not send out waves; there must be change in order to disturb the surrounding space.) The source of the waves detected last September was a pair of merging black holes. 

The experimental work of detection must be coupled with theoretical computer calculations of what might be observed from the various types of wave sources.  These calculations are large and arduous because the fields are strong, making approximations useless; one must work with the full set of nonlinear equations.  Nevertheless, such calculations have been made by several research groups.  The results are typically a wave form whose size and frequency increase rapidly in the case of black hole merger, as the black holes spiral together faster and faster, cutting off at the end.  This is the “chirp” alluded to in the original article and press release.   (The frequencies involved are actually in the human auditory range, so that with the right equipment you could actually hear the chirp.)  The calculated results are so good that the masses of the merging black holes could be determined to high accuracy by fitting the theoretical calculations to the observed wave form.  The strength of the signal relates to the distance of the black holes, and as already noted, the differing arrival times of the wave at the various detectors provided information about the location.  The result was a rather astonishing list of details about the observed black holes.

White Dwarfs, Neutron Stars, and Black Holes

Most people are familiar with the terms white dwarf and black holeNeutron star is perhaps less familiar.  A few remarks on these are in order.

Stars go through a fairly definite life cycle.  Details depend on the mass and on the composition of the star.  As stars travel through this cycle, they will blow off outer layers and will end up as some sort of a dense core.  Most stars, like our sun, will become a nova and end life as a white dwarf, which is a dim, white, but very dense object, with a density about a million times the density of water, and made of carbon.  The definitive theoretical work on white dwarfs was done in the 1930s by Subrahmanyan Chandrasekhar (it was rather controversial; his own Ph. D. advisor publically ridiculed it.)  But white dwarfs had been observed, the first being the so-called companion of Sirius.  A star with somewhat greater mass may become a supernova, finishing as a neutron star, an object made mainly of neutrons with density 1014 times that of water.  Fritz Zwicky predicted, also in the 1930s, the existence of neutron stars. (Pulsars, discovered in 1967 by Jocelyn Bell—for which her Ph.D. advisor Anthony Hewish got the Nobel Prize and she did not (!)—turned out to be rotating neutron stars.)  Theoretical work in the late 1930s by Robert Oppenheimer and his student G. M. Volkoff, using equations of stellar structure with general relativity, showed theoretically that white dwarfs and neutron stars should exist.  Chandrasekhar’s work also predicted an upper limit to the mass of endpoint stars of 1.4 times the mass of the Sun, which value now bears his name.  After there was good observational evidence that that limit did indeed exist, he received the Nobel Prize for the prediction. 

So what happens to stars that end up with a mass greater than the Chandrasekhar limit?  They are not stable; they go into a never-ending collapse, into what we now call a black hole.  Such an object has gravity so strong that it will not allow anything, not even light, to escape.

A 1939 calculation by Oppenheimer and his student H. Snyder considered an ideal preliminary situation, a shell of dust, and showed that it would collapse.  Work by David Finkelstein in the late 1950s led to a theoretical mathematical description of a black hole by Martin Kruskal in 1959, using the Schwarzschild solution of Einstein’s equations.  This, however, also is ideal, since it considers just a point object surrounded by vacuum. (Only by using the equations for the structure of a star formulated in general relativistic terms could one get a good physical model.) The central point to which the mass collapses is called the singularity.  It has infinite density and curvature.  Quantum effects will prevent the collapse from going to infinite values, although we have no theory that combines quantum theory with general relativity.

In the late 1950s, Wheeler undertook to explore stars at the endpoint of their lives in terms of physical models.  Masami Wakano and I, graduate students of his, were recruited to consider the nature of matter in such stars (Harrison) and to work out the structure of them from simple stellar theory with general relativistic corrections (Wakano).  Later, after Wakano and I had finished our graduate degrees and left, Kip Thorne came to work with Wheeler and finished the work. The results were eventually published in 1965 in a book by the four of us. [Reference 4] The calculations in the book unified the previous work of Chandrasekhar, Oppenheimer and Volkoff, mentioned above, also some by Lev Landau.  It appeared from those calculations that only the two families of stars, white dwarfs and neutron stars, were possible in a stable form in nature.  The Chandrasekhar limit was clear from those calculations; no stars beyond 1.4 solar masses were stable.  Further separate investigations by Wheeler and myself extended the results to stars of higher and higher density, proving definitely that there were no other families of stable stars beyond the two (although Wheeler and a later student reopened the question, finding the same conclusion.)  So black holes should exist in nature. 

The black holes considered above result from evolution of stars to their endpoints.  Two other types of black holes need to be mentioned.  Amalgamation of many stars produces large black holes, which are now known to be present at the centers of many galaxies, including our own Milky Way. Also, Stephen Hawking proposed the idea of very small black holes, which may arise in the Big Bang.  Such mini--black holes should actually radiate light in various wavelengths and various atomic particles.  This effect arises from quantum considerations and will not be discussed here.

The idea of black holes proved rather controversial.  It was not accepted for some time, even after the theoretical work of the 1950s and 1960s.  I remember seeing a photo of a well-known astronomer standing by a blackboard on which were the words “black hole—dirty words.”  Various researchers tried to prove their nonexistence.  (I was once asked to referee a paper which attempted such a proof by manipulation of the Schwarzschild solution mathematics.  I read it with interest until I came to the words, “Black holes tear the delicate tissue of spacetime.”  At that point I knew the author was emotionally involved!  I rejected the paper.  After further discussion between the author, the journal editor, and me, the paper was published with much more modest claims.) 

Wheeler gave black holes their name in about 1968, and finally in 1972 observation of a double star, Cygnus X-1, in the constellation Cygnus showed that one of its components was a black hole, the first black hole observation.  (John Michell was right after all!)   Discovery of the large black holes at the centers of galaxies has taken place in the years since.  Radiating mini-black holes, predicted by Hawking, have not yet been observed.


As noted above, general relativistic effects have essentially no practical importance to us on Earth except for the GPS coordinate determinations.  So what is the use of gravitational waves?

The reader will have noted the use of the term observatory in the name LIGO of the apparatus.  The title of this article indicates that use:  a new window on the Universe.  Up to now, our knowledge of the Universe has depended on electromagnetic waves—light, radio waves, ultraviolet waves, x-rays, and gamma rays.  Now we will be able to gain new information from gravitational waves, to excellent accuracy as noted above.  We will be able to gather information about black holes, their numbers, size, position, and such, perhaps neutron stars, and maybe even information about the Big Bang.  This promising possibility enabled NSF funding for LIGO to be continued when there was some danger of its being cut off.  I hasten to add that the sources may be few, although a few more events have been detected since the first.  But any information is better than none.

The general relativity explanation of the Big Bang provided initial ideas about the creation of the Universe.  Later theories have become much more sophisticated.  One theory, admittedly very far-fetched, has suggested the possibility that multiple universes (bubble universes) could come into being.  One is reminded of D. & C. 88:37.  Needless to say, it will be a very long time before we have any knowledge of such matters.

The curved space idea of general relativity has also given rise to various theories of space and time travel.  Wheeler suggested the idea of wormholes, tunnels, so to speak, from one point of space to another.  Imagine a sheet of paper folded in half.  If one is confined to the two-dimensional surface of the paper, one can travel from one corner to an opposite corner only by motion on that surface; but if one is allowed to move through the third dimension by jumping from a corner to the corner right next to it, one can travel through a much shorter distance than by going the long way on the surface.  I have often wondered whether some such sort of travel is possible, suggesting to the mind a way in which Moroni might have appeared in Joseph Smith’s bedroom.

Kip Thorne, with students, looked at the question whether one could travel through a black hole to another space.  They concluded that only very strange matter, not known to us, could make that possible.  However, the properties of the Kerr solution for a rotating black hole indicate that travel near it, not through the singularity, might be possible.  The travel must be carefully designed so one doesn’t fall in.  Such a trip might lead to a different space, even a different time.  In a sense, this could be viewed as travel through a wormhole.  Hypothetical travel through a wormhole was originally depicted in the movie “Contact”; the later, more sophisticated movie “Interstellar”, considers a more detailed model of travel involving a rotating neutron star.  (Kip Thorne was the scientific advisor for Interstellar.  He insisted that the situations proposed, while unlikely, must be consistent with science, including general relativity.  For example, the complicated equations on the blackboard shown in the film, derived and written there by Kip, are mathematically and physically correct!  They were actually used to calculate the details of the space travel in the movie.)

Perhaps of most interest to Latter-day Saints is the fact that a carefully worked out theory, general relativity, has had many confirmations of previous predictions, culminating in the spectacular discovery of gravitational waves.  That supports the concept that the universe is a rational place, not a random or chaotic one.  There is randomness in the universe, to be sure.  That is not all bad, as my biologist friend Riley Nelson, has pointed out.  There is randomness in things like the Big Bang.  Yet underlying it all seems to be a basic order.  While order may not be the first law of heaven, it is surely up there close to being first.  The demonstration of rationality in the universe, is, in some way, a demonstration of the existence of God.  BYU president Merrill Bateman once quoted Einstein to the effect that an idea for his theories in physics just came to him out of the blue.  I do not know which theory was referred to, but I suspect it was relativity.  I have been unable to find the quote.  Einstein was not a believer in an anthropomorphic god, and I do not know what he made of that experience, but I like to think that the Holy Spirit influenced him in his work and that the Spirit has influenced scientists over the world, and continues to do so, in work beneficial to the world and to God’s kingdom.

Another, admittedly tenuous connection to the heavens was kindly provided to me by my son Paul Harrison.  A Wall Street Journal letter to the editor (22 February 2016), citing Michio Kaku’s description of gravity waves as a fabric that bends and stretches, makes a connection to Psalms 104:2, which says that God “spread out the heavens like a curtain.”  Some LDS scriptures use similar language.

At some time in the future, we will understand more of the harmony of all things, as created by God.  In the meantime, we are still in kindergarten, as expressed in Isaiah 55:8:  “For my thoughts are not your thoughts, neither are my ways your ways, saith the Lord.”


  1. B. P. Abbott, et al, “Observation of Gravitational Waves from a Binary Black Hole,” Physical Review Letters, vol. 116, 061102 (2016), pp. 061102-1-061102-16.

  2. Dennis Overbye, “Gravitational Waves Detected, Confirming Einstein’s Theory,” New York Times, 11 February 2016, http://nyti.ms/1Xlr2hB.

  3. John Archibald Wheeler and Kenneth William Ford, Geons, Black Holes, and Quantum Foam: A Life in Physics, (W. W. Norton, New York and London),p. 235.

  4. B. Kent Harrison, Kip S. Thorne, Masami Wakano, and John Archibald Wheeler, Gravitational Theory and Gravitational Collapse (University of Chicago Press, Chicago, 1965.)

Readers may find the following books and articles, on Einstein, general relativity, and gravitational waves in particular, of interest.  They are just a sample of many available.

  1. Abraham Pais, Subtle is the Lord…: the Science and the Life of Albert Einstein,(Oxford University Press, Oxford and New York, 1982.)

  2. Kenji Sugimoto, Albert Einstein; A Photographic Biography, (Schocken Books, New York, 1989.)

  3. John Archibald Wheeler, A Journey into Gravity and Spacetime, (Freeman, New York, 1990.)

  4. John Archibald Wheeler, At Home in the Universe, (American Institute of Physics Press, American Institute of Physics, Woodbury, N. Y., 1994.)

  5. Kip S. Thorne, Black Holes and Time Warps: Einstein’s Outrageous Legacy, (W. W. Norton, New York and London, 1994.)

  6. Kip Thorne, The Science of Interstellar, (W. W. Norton, New York and London, 2014.)

  7. Barry C. Barish and Rainer Weiss, “LIGO and the Detection of Gravitational Waves”, Physics Today, October 1999, pp. 44-50.


Full Citation for this Article: Harrison, B. Kent (2016) "The Discovery of Gravitational Waves: A New Window on the Universe," SquareTwo, Vol. 9 No. 1 (Spring 2016), http://squaretwo.org/Sq2ArticleHarrison Gravity.html, accessed <give access date>.

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