Is Einstein Still Right?

by Clifford M. Will

  Soon after the pulsar in a triple was discovered, the team set about looking for the Nordtvedt effect. If, for example, the neutron star were to fall with larger acceleration than its companion, then its orbit would be displaced slightly toward the distant white dwarf, while the orbit of the inner white dwarf would be displaced slightly in the opposite direction, and the displacement would rotate in time to track the distant companion (as in Figure 5.7). In July 2018, Anne Archibald, who led the data analysis, announced that they had found no evidence in the pulsar signal for such orbital dispacement of the pulsar. The data showed that the neutron star and the white dwarf fall with the same acceleration to about three parts in a million, showing no Nordtvedt effect, and in complete agreement with general relativity.

  Because the strong internal gravity of the neutron star could have produced an anomalous effect (and does, in many alternative theories), this constitutes a beautiful test of general relativity in the strong-field regime. Even more extreme tests of Einstein’s theory are possible, but for these, we must first discuss black holes, as we do in the next Chapter.

  1 If you are uncomfortable with the notion of negative energy, think of it in terms of bookkeeping. If you have a total amount of money, but you also owe something (say an automobile loan), that debt counts as a negative, reducing your total assets. But if you get an infusion of money (say by working overtime) and pay off the loan, you can own the car free and clear. In the same way, if you inject a lot of energy into the neutron star, enough to “pay off” the negative binding energy, you can free all the protons and neutrons. For all bound systems, from nuclei to neutron stars, physicists think of binding energy as negative because it simplifies the mass–energy bookkeeping.

  CHAPTER 6

  How to Use a Black Hole to Test General Relativity

  The black hole is probably the most bizarre, exotic and fascinating prediction of Einstein’s general relativity. This object, composed purely of warped spacetime, endowed with the ability to trap anything, from light rays to Marvel superheroes, that crosses its famous “event horizon,” has lodged itself in the public imagination like no other aspect of physics. Ask the average person on the street to name the four fundamental forces of nature or the basic particles that make up the atom, and more often than not you will get a blank stare. But ask that person to say something about black holes, and you may well get a fairly coherent summary of their basic properties. You will almost certainly hear the name Stephen Hawking, the person universally attached to the black hole idea, and you may also get a list of movies that have featured black holes.

  But even more bizarre is that we know, almost as certainly as one knows anything in science, that black holes exist. In Chapters 7 and 8 we will learn about the spectacular evidence for merging black holes gleaned from the gravitational waves they emitted. But long before those detections, the evidence for black holes obtained by astronomers using light, ranging from gamma rays to radio waves, was so solid that there was very little doubt that these things are really out there. Our goal here is to describe black holes and the evidence (apart from gravitational waves) for them. More importantly, we will explain to you how we may be able to carry out some remarkable new tests of general relativity using black holes.

  We have already seen in Chapter 3 how John Michell and Pierre Simon Laplace in the late 1700s speculated on the possibility of a body so dense that it could prevent light from escaping its surface and reaching great distances. While it is tempting to say that they “predicted” black holes, the fact is that the physics they used—Newtonian gravity and the corpuscular theory of light—was not correct in the end. We now know that gravity is governed by general relativity (at least all the evidence supports that so far), and that light is governed by Maxwell’s equations, even if you choose to describe it using the quantum concept of the photon. Nevertheless, it is amusing to learn about the ideas that these great Enlightenment thinkers came up with using their imaginations and the accepted physics of their day.

  The modern, relativistic story of black holes began on the battlefields of World War I, with a soldier-scientist named Karl Schwarzschild. He was born in Frankfurt, Germany, on 9 October 1873, the son of a banker. The family traced its roots to medieval Frankfurt, when Jews were confined to a ghetto, but under the protection of the king, or “Kaiser.” Even then, the Schwarzschild family was well-to-do. The family name evidently originates in a tradition of that period of identifying a family by a plaque (“schild” in German) that it would affix to the front of its house. For Karl’s ancestral family the plaque was black (“schwarz”), hence the name. One of their neighbors in the ghetto had a plaque that was red (“rot”). The descendants of that family would become the famous Rothschild family of financiers. By 1811 the Frankfurt ghetto was abolished and some measure of civil rights was granted to Jews.

  Schwarzschild showed an early interest in science, and by the age of 23 had received a Ph.D. from the Ludwig Maximillian University in Munich. His career advanced rapidly, with posts in Vienna and Göttingen, culminating in 1909 in the directorship of the Astrophysical Observatory in Postdam, one of the premier observatories of Europe at the time. He contributed seminal work to a wide range of topics, including solar physics, the statistics of stellar motions, the optics of astronomical lenses, the determination of the orbits of comets and asteroids, and the classification of stars by their spectra. He even traveled to Algiers to study the solar eclipse of 1905. This was long before the bending of light was an issue, of course; in those days eclipses were used mainly to study the corona of the Sun, and to search for the hypothetical planet Vulcan between the Sun and Mercury.

  When World War I broke out in August 1914 he was 41 years old, and therefore not subject to the draft, nor required to enlist. But, like many Jews in Germany at the time he volunteered, desiring to demonstrate the loyalty of the Jewish community to Germany (other Jewish scientists who served this way included Fritz Haber, James Franck and Gustav Hertz). He never fought on the front lines, but because of his technical background he was assigned to head a meteorological station at Namur, Belgium, and then was attached to the artillery staff in France and later on the Eastern front. He even wrote a paper in 1915 on the effects of air resistance on the path of a projectile, but publication was held up for security reasons.

  But in late 1915, at the Russian front, he contracted a rare and painful autoimmune skin disease called pemphigus, that even today is difficult to treat. While undergoing treatment he received copies of the papers that Einstein had delivered a few weeks earlier at the Prussian Academy of Sciences, describing his new theory of general relativity. Schwarzschild recognized that the theory was very complicated mathematically, but he set out to see if, by making some simplifying assumptions, he could find a solution to the Einstein equations of general relativity.

  For his first attempt, he assumed that the sought-after solution would be static, or unchanging in time. He then assumed that the solution would be spherically symmetric, or the same no matter how you rotated the system about a central point. He imagined that the solution might apply to a perfectly spherical body, such as a star, sitting at rest, and isolated or far away from any other perturbing bodies. For his first attempt at a solution, he further assumed that the body was so small in size that you could ignore its extent or its internal structure. He called it a “Massenpunkt,” or point mass, an object whose mass was entirely concentrated at a point. This kind of assumption is routinely used in physics because it allows you to get an idea of what the solution outside the body looks like, without having to worry about the messy details of the body’s internal structure. To his surprise, Schwarzschild found a simple exact solution to Einstein’s complex equations.

  For his second attempt, he assumed that the body had a finite size, but that its interior density (the amount of mass per unit volume) was constant throughout the body. This is not a totally realistic assumption, because we know that bodies bound by their own gravity, such as the
Earth or the Sun, are more dense at the center than at the surface. But once again, such an assumption might give useful insights without having to sweat a lot of messy details such as whether the body is solid, liquid or gas (or a combination), or whether it is hot or cold. Here again, he found an exact solution to the Einstein equations.

  He wrote up his solutions in two papers and sent them to Einstein, who communicated the Massenpunkt paper to the Prussian Academy of Sciences on 13 January 1916, and the finite-body paper on 24 February. Einstein was amazed that Schwarzschild had managed to obtain these exact solutions, and pronounced himself very pleased with the second paper. The Massenpunkt paper, not so much.

  The Massenpunkt paper had a feature that troubled Einstein greatly. Schwarzschild’s solutions gave formulae for the warped geometry of spacetime that varied as a function of the distance from the center. Very far from the body, the geometry became that of ordinary flat space and time, as you would expect when the gravity of the body becomes negligible at great distances. As you approach the body, the geometry becomes more and more warped, again as expected when gravity gets stronger. But in the Massenpunkt solution, when the distance from the center reaches a value given by twice the mass of the point (multiplied by Newton’s constant of gravitation and divided by the square of the speed of light), things got crazy. One function became infinite (one divided by zero), while another became zero. This behavior is what physicists call a “singularity,” and for years afterward this was called the “Schwarzschild singularity.” Since it occurs wherever the distance from the center has this special value, it is actually a spherical surface surrounding the point mass. It’s not a surface made of matter that you could bounce off or crash into, but is rather an imaginary surface or boundary that we define mathematically because odd mathematical things happen to the solution on that surface. The question was, what?

  Einstein believed that such singularities were unacceptable (we will learn in Chapter 7 that he thought he had disproved gravitational waves in 1938 because of the apparent existence of singularities in the solution), and that such a Massenpunkt would therefore not occur in nature. On the other hand, Schwarzschild’s second solution for an extended body was completely acceptable. The geometry of spacetime became more warped as one approached the body (in fact, for the same mass, the external solution was identical to the solution for the point mass), but once inside the body, the solution changed and was perfectly finite all the way to the center. Einstein and others who examined Schwarzschild’s two solutions believed that there must be something in the laws of physics to prevent a body from ever being so small in relation to its mass that it would reside completely inside this special “Schwarzschild” radius. To get a sense of scale, for an object with the mass of the Earth, the Schwarzschild radius is about a centimeter, roughly the size of the tip of your baby finger. For an object with the mass of the Sun, it is about three kilometers. For what we today would call a fifty million solar mass black hole, it is the radius of the Earth’s orbit around the Sun. A body would have to be compressed to an incredibly small size and an enormous density to fit inside its Schwarzschild radius. To Einstein and his contemporaries, the world seemed safe from Schwarzschild’s horrible singularity.

  Unfortunately, nothing could save Schwarzschild from the ravages of pemphigus; he died on 16 May 1916. Even though he was German, he was such a famous astronomer that The Observatory in England published an obituary that August. It described his many contributions to astronomy and astrophysics, but made no mention of his solution to the Einstein equations of general relativity. Einstein’s theory was still too new and obscure.

  And that pretty much was it for black hole research for the next forty years. A few papers related to Schwarzschild’s solution were published here and there, but they were largely ignored. Ironically, a young student named Johannes Droste, working under the tutelage of the great Dutch physicist Hendrik Lorentz, found the Massenpunkt solution in the spring of 1916, totally independently of Schwarzschild. But his paper was published in the relatively obscure Proceedings of the Royal Netherlands Academy of Sciences (in Dutch), and wasn’t “discovered” until many decades later. During the 1920s and 1930s, a few researchers, including Eddington and Howard P. Robertson, tried to explore what was really going on at the Schwarzschild singularity. Was this a place where bizarre physics occurred, or was the singular behavior merely an artifact of the coordinates that Schwarzschild used, in the same way that the coordinates of latitude and longitude on Earth are singular at the north and south poles where all the lines of longitude meet at one point. Some of these papers, seen in retrospect, held important clues, but they were barely noticed.

  In 1939, the American theoretical physicist J. Robert Oppenheimer and his student Hartland Snyder published a remarkable paper entitled “On continued gravitational contraction.” In it they showed that a massive enough star that runs out of thermonuclear sources of energy will no longer be able to support itself against the crushing force of its own gravity, and will contract. Using Einstein’s equations, they showed that the decreasing radius of the star would reach the Schwarzschild radius, but the star would continue to shrink. An observer riding on the surface of the star as it collapses inward would not observe any “singular” or bizarre behavior during the contraction, while observers at great distances would observe light emitted from the star’s surface becoming progressively redder (the gravitational redshift effect) and fainter until, after a long time, the star’s light would be essentially undetectable. For all practical purposes it would be “black,” and the final object would be described by Schwarzschild’s Massenpunkt solution. Eighty years later, this paper reads like a modern paper on black hole physics, with many of the insights into the nature of these objects that we have come to understand. But at the time, it also had almost no impact. Oppenheimer never followed up on the paper, and within three years would turn all of his attention to leading the Manhattan Project to develop an atomic bomb.

  We must remember that this was a period when general relativity was considered a backwater of physics. Very few people worked in the field. The noted general relativity theorist Peter Bergmann, who had been an assistant of Einstein during the late 1930s, once joked that if he ever needed to find out everything that was going on in general relativity, he only needed to call up his five best friends. Physicists were much more concerned with quantum mechanics, atomic and nuclear physics, field theory and elementary particles, and, after World War II, with developing the new technologies that had emerged from wartime research, such as radar, nuclear power, transistors, semiconductors, masers and lasers.

  In 1956, Martin Kruskal, a mathematical physicist working in the plasma physics laboratory at Princeton University, realized that he could find a new system of coordinates in which Schwarzschild’s “singularity” would disappear. He described his discovery to his Princeton colleague John Wheeler, who had begun to take an interest in general relativity. Wheeler thought it was nice, but otherwise paid little attention to it. But by 1959, Wheeler suddenly realized the significance of the discovery and wrote a paper with Kruskal’s name as the author and submitted it to Physical Review. He somehow neglected to tell Kruskal what he was up to. A few months later, while on sabbatical in Germany, Kruskal received out of the blue the galley proofs of a paper that he didn’t know he had written. But he recognized the figures in the paper as being in a style typical of his friend Wheeler, and he urged him to be a co-author. Wheeler declined, and Kruskal’s paper became one of the foundations of a new understanding of the nature of black holes.

  At the same time, George Szekeres, a Hungarian mathematician working at the University of Adelaide, Australia, and David Finkelstein, an American physicist at the Stevens Institute of Technology in New Jersey, were also working on similar approaches to resolving Schwarzschild’s singularity.

  These researchers showed conclusively that nothing “singular” or infinite happens at the Schwarzschild radius. A light ray, an a
tom or a graduate student can head toward the object and cross the Schwarzschild radius without experiencing anything infinite. To be sure, there will be the inevitable stretching and squeezing of a body as it approaches the object. This is nothing more than the same kind of tidal effects that the Moon induces upon the Earth, for example, stretching it along the line directed toward the Moon, and squeezing it in the perpendicular directions. These forces can be large enough to squeeze and stretch the poor graduate student into a long thin noodle, but no matter what, the forces remain finite as the student crosses the “magic” radius.

  The true significance of the Schwarzschild radius turned out to be rather different, and quite astonishing. The sphere defined by this special radius turns out to be the boundary between two realms. Outside the sphere is the normal external universe, where people can travel freely, subject to the speed of light limitation, and can communicate with each other using light signals. You can even safely go into orbit around the object.