Another idea was to resurrect what Einstein called his “greatest blunder.” Already in 1916 he was thinking about applying his new theory to the universe as a whole. But to his horror, he realized that the equations demanded that the universe must either expand or contract. It could not be static. At the time, conventional wisdom held that the universe was actually static, perfectly unchanging. In fact, it was not even known that there were galaxies outside the Milky Way. To get around this, he added what he called a “cosmological term” to his original equations. This term would have the effect of introducing a repulsion that would counteract the natural tendency of the universe to contract under its own gravitational attraction, leading to a nicely balanced, static universe. The size of the term was governed by a “cosmological constant,” which came to be denoted by the Greek letter Λ (in homage to Einstein, dark energy aficionados have adopted the same symbol). So by picking the “right” value for his cosmological constant, Einstein could make everything right with the consensus view of the universe.
But along came data to mess with Einstein’s world view. First was the discovery of galaxies external to our own, along with evidence that many of them were moving away from us. And then, Edwin Hubble, an astronomer at the Mount Wilson Observatory in California, announced in 1929 that the data on the motions of galaxies implied that the universe was not static but was expanding. Einstein was now forced to drop his cosmological term, since its sole purpose was to make the universe static. With the new knowledge that the universal expansion is accelerating, it is a simple matter to bring back Einstein’s cosmological term, since it naturally provides the repulsive effect needed to counteract the gravitational deceleration. The value of the cosmological constant needed to explain the acceleration of the universe is much smaller than what Einstein needed for a static universe, and so the effect of adding this term to his equations is utterly negligible for everything but cosmology itself. You might call this the “minimal” modification to Einstein’s theory.
A third idea is to modify general relativity more drastically, but still with enough fine tuning so as not to violate the agreement with the many experiments that we will be describing in this book. This turns out to be not so easy. In developing general relativity, Einstein was driven by a desire for elegance and simplicity in the structure of the theory, and he was remarkably successful. For all the talk about how complicated his theory appeared at the time of Eddington’s announcement, from a more modern perspective, general relativity is the simplest theory of gravity you could possibly imagine. And it turns out that efforts to modify it for cosmological purposes generally lead to very ugly, complicated theories. Of course, it is not clear that the true theory of nature needs to be elegant or beautiful, since after all, these are human concepts that may have nothing to do with the physical world. In other words, the universe is a messy and dirty place, so it may be that the correct theory to describe it is equally messy and dirty.
None of these “clouds,” quantum gravity, dark matter or dark energy (and others we don’t have space to discuss), directly invalidates general relativity. Still, they leave us with the disquieting feeling that it might be necessary to take gravity “beyond Einstein”; to develop a theory that agrees with general relativity in all the realms where it has been tested precisely, but that might deviate from it, either in the realm of ultra-short scales and ultra-high energies, as in quantum gravity, or in the realm of ultra-large scales as in cosmology.
This book will focus on the many precision experiments and observations that have been carried out to test Einstein’s theory in different realms (laboratory, solar system and astrophysical). But after reading about how general relativity has passed test after test, you might be tempted to say “Einstein is still right,” so let’s be done. However, in science in general, and in physics specifically, the acceptance of any theory is always provisional, because no theory can be fully tested in all possible realms of its applicability to perfect or infinite precision. The best we can do is to extend our experimentation into wider and wider realms and to higher and higher precision, in the hope of either building further confidence in the theory, or finding a deviation that might lead us to a new, more fundamental, and more complete theory. The history of science is full of examples of both outcomes.
For general relativity, the arena for experimental tests began with the solar system, notably with the famous measurement of light bending in 1919. By the 1970s, the arena had been extended to astrophysical scales, with the discovery of the binary pulsar. But since the 1970s, the precision to which tests can be performed in both of these realms has greatly improved, and completely new arenas have opened up, to include gravitational waves and black holes, for example. The events of Einstein’s wonderful summer of 2017 illustrate all these arenas for putting general relativity to the test.
CHAPTER 2
Wrinkles in Time
Midway through the movie Interstellar, the crew of the Endurance discuss how to explore Miller’s planet, which orbits just outside the event horizon of the supermassive black hole Gargantua:
COOPER (Matthew McConaughey) Look, I can swing around that neutron star to decelerate …
BRAND (Anne Hathaway) It’s not that, it’s time. That gravity will slow our clock compared to Earth’s. Drastically.
COOPER How bad?
ROMILLY (David Gyasi) Every hour we spend on that planet will be maybe … seven years back on Earth.
COOPER Jesus …
ROMILLY That’s relativity, folks.
Until the twentieth century, it was accepted by everybody that time was the universal time of Newton, flowing at the same rate everywhere and forever, the same for everybody. Einstein upended that comforting absolutism with his 1905 special theory of relativity, pointing out that time could flow at different rates for people who are moving relative to each other. And this is not just a funny effect of clocks. It is actual physical time that flows at different rates, making people age differently. And in 1911 he argued that gravity could also affect time in a similar way.
This striking prediction was based on what Einstein later called his “happiest thought.” It was 1907, eight years before the general theory of relativity, and Einstein was doing well. The five papers he had published in 1905 on the photoelectric effect, the quantum nature of light, Brownian motion, special relativity and the mass–energy equivalence were generating a lot of buzz, and he would soon leave the patent office in Bern, Switzerland, where he had been working for the last six years, for a faculty position at the nearby University of Bern. He had also been invited to write a review article on special relativity for the scientific journal Jahrbuch der Radioaktivität und Elektronik. Part of the article would be a standard review of his special relativity theory, plus work that he and others had done on it since 1905, but part of it would be devoted to his latest preoccupation: gravity.
For all its successes, special relativity had a weakness. It was based on the premise of the “inertial reference frame,” a laboratory that moves with constant speed in the same direction. Inside this laboratory, free particles (those not acted upon by forces such as electric or magnetic forces) move on straight lines at a constant speed. By analyzing the laws of physics within such frames of reference, Einstein was able to account in a natural way for the experimental fact that the speed of light was independent of the speed of the emitter or of the observer. He could also make sense of the interrelations between electric and magnetic fields as seen by observers in relative motion. His theory of special relativity had led to some surprising, and at that time untested, predictions, such as the idea that a moving clock would tick more slowly than clocks at rest, and that energy and mass were really the same thing, related by his famous E = mc2 equation, destined for T-shirts and coffee mugs everywhere.
It was not too difficult to imagine an inertial reference frame, such as a spaceship with its engines turned off, in outer space far from any stars or galaxies, but what about on or near the
Earth? Any freely moving particle would experience an acceleration, a change of its speed and of the direction of its motion, because of Earth’s gravity. No reference frame could be truly inertial in the presence of gravity. Already in 1907, Einstein realized that he would have to find a way to make special relativity compatible with gravity.
It was here that Einstein demonstrated his special genius for taking a simple experimental observation, combining it with an idealized imaginary experiment (called a gedanken or “thought” experiment) that incorporates the essence of the original experiment, and pushing the result to its logical limit. In this case, the observational result was the commonplace one that bodies fall with the same acceleration, regardless of their internal makeup, in the absence of air resistance. This brings to mind the image of Galileo Galilei dropping objects from the top of the Leaning Tower of Pisa, although there is no actual contemporary account of his ever doing such a thing. But by the turn of the twentieth century this observational result had been verified to a few parts in a billion by a Hungarian physicist, Lorand Eötvös. Einstein took this simple observation and imagined what it would imply for an observer inside an enclosed, freely falling laboratory.
Of course, in 1907, when Einstein first began to ponder this question, it had to be a pure thought experiment, for the dawn of the space age and of astronauts floating around their space capsules was still fifty years into the future. There is also a story that he once observed a worker falling from a roof, and began to imagine what it would be like to be weightless (forgetting about what would happen when the worker encountered the ground, which from the worker’s point of view would be rising ever more rapidly toward him!). Nevertheless, the weightlessness, or vanishing of gravity, that such an observer would experience seemed so significant to Einstein that he elevated it to the status of a principle. He called it the principle of equivalence.
“Equivalence” came from the idea that life in a freely falling laboratory should be equivalent to life with no gravity. It also came from the converse idea that, if you were in an enclosed rocket with no windows, far away from any star or galaxy, and accelerating with the right amount (called “one g”) of constant rocket thrust, then you would not be able to tell that you are inside a rocket, as opposed to being safely inside a building on Earth. To Einstein, the acceleration due to a rocket and an acceleration due to gravity were the same thing! From this principle of equivalence, Einstein was able to conclude that time at the top of a tower at rest in a gravitational field ticks a little more quickly than time at the bottom.
The easiest way to understand this is to consider a simple thought experiment involving the Earth, an emitter and receiver of light, and a freely falling laboratory (Figure 2.1). This thought experiment is a somewhat modernized version of what Einstein wrote in a 1911 paper, but the idea is the same. Imagine a device that emits light at a well-defined frequency or wavelength, placed at the top of a high tower with its beam directed downward. A tunable receiver is placed on the ground and is tuned to receive the incoming signal from the emitter on the tower. Relative to the emitted frequency, is the received frequency larger, smaller, or the same?
Figure 2.1 Gravitational redshift thought experiment. A laboratory is released at the moment the emitter sends the pulse of light (left panel). The observer inside can use the gravity-free laws of special relativity to analyze the emission, propagation and reception of the pulse. Second panel: the laboratory has started to fall downward, but because the observer inside senses no gravity, she sees the packet of light propagating with the same frequency as before. Third panel: the laboratory is falling faster, and the observer sees the receiver coming up toward her. Because the light packet still has the same frequency as seen by her, the ascending receiver will see a higher frequency (a blueshift), because of the Doppler shift (fourth panel). The velocity that determines the amount of the shift is just the speed that the laboratory picked up in the time it took for the light packet to go from the emitter to the receiver.
To answer this question, let’s imagine a laboratory that is suspended next to our tower by a mechanism that can release the laboratory in an instant, letting it fall freely toward the ground. For a more visceral idea, picture being at the top of the 200-foot vertical drop of a roller coaster ride, such as “SheiKra” at Busch Gardens in Tampa, Florida, where the cars stop momentarily before being released, followed by a few seconds of breathtaking vertical free fall (and screaming). In this ride, the roller coaster obviously follows the curved tracks before hitting the ground, while in our thought experiment, the lab does crash against the ground, although this is irrelevant to our physics problem.
Let us also imagine that the laboratory is released at the precise moment that a short pulse or “packet” of light is also released by the emitter. Inside the laboratory, an observer prepares to measure the frequency of the emitted packet of light as she falls freely toward the ground. Clearly, the wave packet will travel toward the receiver at the speed of light, therefore faster than the laboratory and the observer are falling. The packet will thus overshoot the observer and reach the receiver before the laboratory hits the ground, as depicted in the four panels of Figure 2.1.
Now, let us think about what the observer measures at each stage of her descent. At the very start of the drop (the first panel in Figure 2.1), the laboratory is initially at rest with respect to the emitter, if only for a moment, and so the emitted frequency is the “rest” frequency, unaffected by any slowing down of moving clocks predicted by special relativity. The measured frequency would therefore be the standard value for that emitter, and could be looked up, say, in standard tables of physical constants, or calculated using the standard laws of atomic or nuclear physics.
At the next instant (the second panel of Figure 2.1), the wave packet travels down while the laboratory and observer begin to fall because of the Earth’s gravitational pull, so what does the observer measure now? She realizes that throughout her descent she is in free fall, and because she is well versed in the principle of equivalence, she also realizes that, from her point of view, gravity is absent! The wave packet thus obeys the laws of special relativity, which state that light moves at a constant speed with an unchanging frequency. She thus measures that the frequency of the wave packet remains unchanged, as seen by her, during every stage of her fall.
A little while later, however, the observer notices that from her viewpoint the ground, and in particular the receiver, are coming up toward her! She is falling, so of course this is what she will experience, even though from the viewpoint of a person standing on the ground the receiver is clearly not moving (the third panel of Figure 2.1). Thus, from our observer’s viewpoint, when the onrushing receiver absorbs the packet of light (the fourth panel of Figure 2.1), it will measure a higher frequency than our observer measured in the freely falling laboratory because of the Doppler effect, the effect that causes the frequency or pitch of an ambulance siren to be higher when the ambulance is approaching you and lower when it is moving away from you. And that is the answer to our question! Relative to the emitted frequency, the received frequency is higher.
The emitter and receiver, of course, are still at rest with respect to each other, but this is not the point. The important point is that from the point of view of the observer in the freely falling laboratory, in which the frequency has its standard value, the receiver is moving toward her. The velocity of the laboratory relative to the receiver is the same as the velocity that the freely falling laboratory has picked up in the time taken for the wave packet to travel the distance between the emitter and the receiver, and from this one can calculate the shift in frequency of the light. For example, for a difference in height of 100 meters, the shift would be only ten parts in a million billion, or one trillionth of a percent! If the emitter and receiver are at the same height, but separated in the horizontal direction, there is no frequency shift at all.
In this thought experiment the observed shift was toward higher frequenci
es—the blue end of the visible spectrum—because the freely falling frame was heading toward the receiver. If the emitter had been at the bottom and the receiver at the top, the shift would have been toward lower frequencies—the red end—because by the time the wave packet reached the top, the freely falling frame would be falling away from the receiver. Even though the result can be either a redshift or a blueshift, depending on the experiment, the generic name for this effect is the gravitational redshift. It is called a “gravitational” shift because it occurs only in the presence of a mass (the Earth in our case) that exerts a gravitational force on the lab (forcing it to accelerate down in our example).
It should be apparent from our thought experiment that the gravitational redshift is a truly universal phenomenon. It was the behavior of the freely falling laboratory that was the crucial element in the analysis. The nature of the emitter and receiver did not play a significant role, nor did our treatment of the nature of light. The light could have been in the visible spectrum, or it could have been in the radio or X-ray wavelengths. The signal could have been a continuous beam, or it could have been in the form of packets, such as might be emitted by a strobe light set to flash once per second. In the latter example, the observer at the bottom of the tower would observe not only that the intrinsic frequency of the light emitted by the strobe was shifted toward the blue, but also that the flashes arrived more quickly than once per second. Thus, all frequencies appear to be shifted. If the strobe’s flashes were timed by some sort of clock, then the observer on the ground would argue that the clock at the top of the tower was ticking faster than his ground clock; in other words, that the clock rate was “blueshifted.”
Is Einstein Still Right? Page 3