Is Einstein Still Right?

by Clifford M. Will

  At the end of the study Cliff was required to make a final report to a meeting of the AFSB, headed by a four-star general who was at the time in charge of the Air Force Systems Command. Confronted with the challenge of explaining general relativity and its importance for GPS to an audience more familiar with military matters than with Einstein’s theories, Cliff prepared the most simple and colorful viewgraphs possible (this was in the pre-PowerPoint era), and tried to make his presentation lively and engaging. Unfortunately, within thirty seconds the general’s eyelids started closing…closing…until eventually he fell asleep! Luckily his aide, a second lieutenant, took careful notes, and Cliff presumes that she briefed the General afterward on what he said. But still, for a general relativist to be briefing a military leader on Einstein’s theory because it was important for US national security, it didn’t matter whether he was awake or asleep, this was a moment to be remembered!

  Without the proper application of relativity GPS would fail in its navigational functions within about two minutes. So the next time your plane approaches an airport in bad weather, and you just happen to be wondering “What good is Einstein’s theory?”, think about the GPS tracker in the cockpit, helping the pilots guide you to a safe landing.

  Let us return to using clocks to test Einstein’s equivalence principle. Over the 40 years since Vessot’s rocket experiment using hydrogen maser clocks, atomic timepieces have become much better. During the 1980s a new technique was discovered that allowed researchers to use crossed laser beams tuned to specific wavelengths to trap and slow down clouds of atoms. One of the leading enemies of time precision is the Doppler effect caused by the random motions of atoms, which smears the fundamental frequency of the light emitted by the atom, making it slightly less precise as a standard for measuring time. But the new method of “laser cooling” could slow down atoms to such a degree that the only way to characterize how fast they were moving was to express the speed in terms of the apparent temperature of a gas, with absolute zero (0 kelvin or − 459.67°F) representing absolute stillness. Temperatures measured in millionths of a degree above absolute zero (microkelvins) are now achieved routinely. In Vessot’s clocks the hydrogen atoms were at room temperature, about 300°K warmer.

  New ideas for ever more accurate and stable clocks have been developed, based on such concepts as “atom fountains,” “Bose–Einstein condensates” and “atom interferometry.” To describe these would take us too far afield, but we cannot resist mentioning one experiment that demonstrates just how “everyday” Einstein’s gravitational redshift has become.

  This was an experiment done in 2010 in the laboratory of David Wineland at the National Institute of Standards and Technology in Boulder, Colorado. The experimenters set up two clocks based on ultracold trapped aluminum ions, separated by a height of only 33 centimeters, or about a foot. They were able to measure that the higher clock was ticking a bit faster than the lower clock. If you have ever wondered why it seems that your brain is aging faster than your feet, now you have the answer. But cheer up, the difference is only about 7 nanoseconds per year.

  The effects we have been discussing, whether in the laboratory, on GPS satellites or on white dwarfs, are extremely tiny differences in the rate at which time moves forward. So what were the crew of the Endurance talking about: one hour equaling seven years back on Earth? How can that be possible? The answer is that the warpage of time becomes that extreme near the event horizon of a black hole. In Interstellar, Miller’s planet was orbiting very close to Gargantua’s horizon. As a result, during their stay on Miller’s planet their clocks would be slower than Earth’s clocks by a whopping factor of 60,000. In fact, the closer you get to the horizon of a black hole, the larger this factor becomes. Upon Cooper’s return to Earth (spoiler alert!) the effect of this warpage of time leads to a tear-inspiring scene as he visits his daughter Murph, now a very old woman.

  These numbers were not invented out of thin air by the movie’s director Christopher Nolan and the screenwriter, his brother Jonathan. In fact, many of the details related to the black hole Gargantua and the planets revolving around it were worked out carefully using Einstein’s theory by Caltech astrophysicist Kip Thorne, who developed the original concept on which the movie was based and was an executive producer of the film. In addition to his talents as a moviemaker, Christopher Nolan is a self-confessed “science geek,” and so he wanted Thorne to help make the movie as scientifically accurate as possible, within the confines of the science fiction genre. For example, the plot of the movie requires that the crew pass through a “wormhole,” which is a popular science fiction tool, but which no relativist, including Thorne, believes is possible in nature according to our current understanding of the laws of physics.

  But, is the warpage of time near a black hole a purely theoretical thing, useful for nothing more than movie plot twists? Not at all, because on 19 May 2018 a star designated by the prosaic name S2 passed its point of closest approach to the supermassive black hole known as Sgr A* that resides in the center of our Milky Way galaxy, a black hole with the mass of 4.3 million Suns. During that close encounter, reaching just 120 times the Earth–Sun distance away from the horizon, astronomers using advanced infrared telescopes in Chile and Hawaii were able to measure Einstein’s gravitational redshift of the spectrum of S2. But you’ll have to wait until Chapter 6 to read more about this, when we discuss how to test Einstein’s theory near black holes.

  CHAPTER 3

  How Light Sheds Light on Gravity

  After over a year of painstaking preparation and numerous rehearsals, Don Bruns felt he was ready. The skies over Casper, Wyoming that morning were clear and blue, with only a few thin clouds. Winds were calm. His TeleVue Optics NP101is telescope and its attached CCD camera were ready. The computer programs he had written to send commands to the telescope during the crucial two and a half minutes had been tested and rehearsed. All that remained was to sit back and wait for what was being called the Great American Eclipse.

  Bruns was one of an estimated 215 million people who viewed the eclipse of the Sun either in person or electronically that 21 August 2017. But Bruns, a retired physicist and amateur astronomer, was not in Wyoming to ooh and ahh over the sight of the eclipse, he was there hoping to repeat one of the most famous measurements of the twentieth century, an experiment that made Albert Einstein famous to the world at large.

  That earlier experiment generated over-the-top headlines in the autumn of 1919. “Revolution in Science / New Theory of the Universe / Newtonian Ideas Overthrown,” proclaimed The Times of London on 7 November. “Lights all Askew in the Heavens / Men of Science More or Less Agog over Results of Eclipse Observations,” declared the New York Times three days later (notice only “men” of science, a typical attitude of that era). It heralded a brave new world in which the old values of absolute space and absolute time were lost forever. To a world emerging from the devastation of World War I, it meant the overthrow of all absolute standards, whether in morality or philosophy, music or art. In a 1983 survey of twentieth-century history, the British historian Paul Johnson argued that the “modern era” began not in 1900 or in August 1914, but with the event that spawned these headlines in 1919.

  This event made Einstein a celebrity. Set aside for a moment his genius, the triumph of his theories, and the new scientific order he created almost singlehandedly. That alone might have been enough, but Einstein was also, in today’s terminology, a very “media-friendly” fellow. His absentmindedness, his wit, his willingness to expound upon politics, religion and philosophy in addition to science, his violin playing—all these characteristics sparked an intense curiosity on the part of the public. The press, tired of printing battle reports and casualty lists from the war, was only too eager to satisfy its readers’ curiosity.

  The event that caused such a commotion was the successful measurement of the bending of starlight by the Sun. The amount of bending agreed with the prediction of Einstein’s general theory of
relativity, but disagreed with the prediction of Newton’s gravitational theory. This was the experiment that Don Bruns planned to repeat, and if all went well, to beat in precision.

  The story of how gravity affects the trajectory of light is one of the most fascinating in all science. It actually has its roots in the eighteenth century, yet the story continues to evolve to this day. It journeys from the heights of theoretical and experimental accomplishments to the depths of racist propaganda, from our solar system to the most distant galaxies.

  It is believed that the first person to consider seriously the possible effect of gravity on light was a British theologian, geophysicist and astronomer, Reverend John Michell (1724–1793). Ever since the time of Newton, who had himself speculated vaguely that gravity might affect light, it had been assumed that light consisted of particles or “corpuscles.” In 1783, Michell reasoned that light would be attracted by gravity in the same way that ordinary matter is attracted. He noted that light emitted outward from the surface of a body such as the Earth or the Sun would slow down after traveling great distances (Michell, of course, did not know the theory of special relativity, which requires the speed of light to be the same from the viewpoint of any inertial observer). He then asked how large would a body of the same density (same number of grams per cubic centimeter) as the Sun have to be in order that light emitted from it would be stopped by gravity and pulled back before escaping? The answer he obtained was 500 times the diameter of the Sun. Light could never escape from such a body.

  This remarkable idea describes what we now refer to as a black hole. In today’s language, Michell’s object would be 100 million times more massive than our Sun. Fifteen years later, the great French mathematician Pierre Simon Laplace performed a similar calculation. Although Michell and Laplace were wrong in the fundamental theory, their basic premise is right: gravity affects light.

  But Michell didn’t stop there. He then asked, how would one ever detect such a body if light could not escape from it? His remarkable answer was that if such a dark body were to be in a double star or binary orbit with a normal star, one could infer its existence by measuring the wobble in the normal star’s position as the two bodies revolved around each other. What made this remarkable was that in 1783 there was no evidence that such binary star systems existed. It appears that Michell’s writings and speculations on such possibilities, along with his groundbreaking statistical analyses of close associations of stars on the sky, played a role in getting astronomers to start looking for evidence of binaries. The first solid discovery of two regular stars in a mutual orbit was announced by Wilhelm Herschel in 1803.

  Michell’s friend and colleague, Henry Cavendish (1731–1810), shared his interest in gravity. Already famous for his discovery of hydrogen, Cavendish inherited from his recently deceased friend an instrument that Michell had built to measure gravity, and after some modifications of it, he used it to measure what we now call Newton’s constant of gravitation (called “big G” by physicists), which relates the gravitational force between two bodies to their masses and separation.

  But around 1784, Cavendish also asked the question: If gravity affects light as Michell suggested, would it not also bend it? According to Newtonian gravity, the orbit of one body about another is a “conic section,” the figure formed by the intersection of a cone with a plane tilted at various angles: an ellipse or a circle if the orbit is bound, so that the body never escapes, or a hyperbola if it is unbound (Figure 3.1). If light is a corpuscle undergoing the same gravitational attraction as a material particle, then because its speed is so large, its orbit will be a hyperbola that is very close to being a straight line [the bottom panel of Figure 3.1(b)]. However, the deviation, while small, is calculable, and apparently Cavendish did the calculation.

  Figure 3.1 Newtonian orbits. Orbits (a) are bound, either circular or elliptical (also called eccentric). Orbits (b) are unbound hyperbolae of ever greater speed, moving from top to bottom.

  Why apparently? Cavendish was notorious for not bothering to publish his work, or even to discuss it with colleagues (the neurologist Oliver Sacks has speculated that Cavendish may have had Asperger syndrome). Around 1920, during a project to complete the publication of Cavendish’s work in physics (his work on chemistry having been compiled and published earlier), researchers discovered a scrap of paper among his documents which stated “To find the bending of a ray of light which passes near the surface of any body by the attraction of the body …,” followed by a formula. No calculational details, just the correct answer to the problem posed.

  Some fifteen years after Michell and Cavendish, a similar story played out on the other side of the European continent, but with a somewhat different outcome. Prompted by Laplace’s speculations, a Bavarian astronomer named Johann Georg von Soldner (1776–1833) asked the same question: Would gravity bend light? Von Soldner was a largely self-taught man who became a highly respected astronomer. He made fundamental contributions to the field of precision astronomical measurements known as astrometry, and eventually rose to the position of director of the observatory of the Munich Academy of Sciences. But in 1801, he was still an assistant to the astronomer Johann Bode in the Berlin Observatory. Von Soldner calculated the bending (his and Cavendish’s answers agree), and determined that, for a path that skims the surface of the Sun, the bending would be 0.875 seconds of arc. An arcsecond is the angle subtended by a human finger at a distance of about 4 kilometers or 2.5 miles (3,600 arcseconds equals 1 degree of arc).

  Von Soldner’s work was published in 1804 in one of the German astronomical journals. It was then immediately forgotten, partly because the effect was beyond the current limits of telescope precision, and partly because of the rise during most of the nineteenth century of the wave theory of light, according to which light moves as a wave through an imponderable “aether,” and presumably suffers no deflection. Einstein was certainly not aware of either von Soldner’s paper or Cavendish’s calculation. It was not until 1921 that von Soldner’s work was rediscovered and resurrected, but then it was for a different, more unsavory purpose.

  Like Cavendish and von Soldner over a century before, Einstein in 1907 was also interested in the effect of gravity on light. He recognized that if the principle of equivalence led to an effect on the frequency of light, the gravitational redshift (Chapter 2), it should also result in an effect on its trajectory. In 1911 he determined that the deflection of a ray grazing the Sun should be 0.875 arcseconds. He proposed that astronomers should look for the effect during a total solar eclipse, when stars near the Sun would be visible and any bending of their rays could be detected through the displacement of the stars from their normal positions (Figure 3.2). Several teams, including one headed by Erwin Finlay-Freundlich of the Berlin Observatory, one headed by William Campbell of the Lick Observatory in the USA, and one headed by Charles Perrine of the National Argentinian Observatory, traveled to the Crimea to observe the eclipse of 21 August 1914. But World War I intervened, and Russia sent many of the astronomers home, interned others, and temporarily confiscated much of the equipment; in any case, the weather at the site on eclipse day would have been too bad to permit useful observations.

  Figure 3.2 Deflection of light by the Sun. Top: Deflection of light causes the apparent position of a star to be displaced away from the Sun. Bottom left: A field of stars viewed at night. Bottom right: The same field with the Sun in the middle, obscured by the Moon. Stars whose light passes close to the Sun (the six stars in black) have their locations displaced more than stars farther from the Sun (the stars in gray). The amount of bending is greatly exaggerated, of course.

  It is quite easy to see how Einstein’s equivalence principle leads to a deflection of light. Imagine a laboratory with glass sides containing an observer well versed in the equivalence principle (see Figure 3.3). The laboratory is moving with constant speed far from any star or galaxy; it is therefore an inertial reference frame in which special relativity is valid. Because there
is no gravity, the observer inside floats freely. The following sequence of events is shown in the upper panel of Figure 3.3: (a) A light ray enters the laboratory from the left at a spot right at the middle of the lab. (b) As the ray crosses the lab in a straight line, the lab moves forward (upwards in the figure), so the ray is still at the midpoint. (c) The ray exits the lab also at the midpoint. The ray crosses the lab in a straight line.

  Figure 3.3 Einstein’s principle of equivalence and the bending of light. Top panel: A light ray (dashed lines) enters a laboratory moving at constant speed in empty space. It crosses the laboratory in a straight line, with the angle changed because of aberration. Bottom panel: A light ray enters a laboratory accelerating via the thrust of a rocket. Because the laboratory is moving faster in each panel, the light ray leaves the laboratory slightly below the level where it entered, as if it were deflected downward.

  But the angle of the ray as seen by the observer in the lab is different from the angle seen by external observers at rest! This is the well-known phenomenon of aberration, discovered by James Bradley in 1725. It manifests itself in an annual back and forth motion of stellar images by about 40 arcseconds as the Earth moves around the Sun. On a more mundane level, it is the same phenomenon that occurs when you carry an umbrella quickly through a vertical rainfall. You observe that the drops are tilted toward you and get your feet wet. In the top panel of Figure 3.3, the difference in angle between what we outside observers see for the light ray entering the lab and what the observer in the moving lab sees is substantial because we have made our lab move at around half the speed of light.