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Relativity : the Special and General Theory by Albert Einstein

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A Smithsonian magazine special report

The Theory of Relativity, Then and Now

Albert Einstein’s breakthrough from a century ago was out of this world. Now it seems surprisingly down-to-earth

Brian Greene

Contributing Writer

"I am exhausted. But the success is glorious.”

It was a hundred years ago this November, and Albert Einstein was enjoying a rare moment of contentment. Days earlier, on November 25, 1915, he had taken to the stage at the Prussian Academy of Sciences in Berlin and declared that he had at last completed his agonizing, decade-long expedition to a new and deeper understanding of gravity. The general theory of relativity, Einstein asserted, was now complete.

The month leading up to the historic announcement had been the most intellectually intense and anxiety-ridden span of his life. It culminated with Einstein’s radically new vision of the interplay of space, time, matter, energy and gravity, a feat widely revered as one of humankind’s greatest intellectual achievements.

At the time, general relativity’s buzz was only heard by a coterie of thinkers on the outskirts of esoteric physics. But in the century since, Einstein’s brainchild has become the nexus for a wide range of foundational issues, including the origin of the universe, the structure of black holes and the unification of nature’s forces, and the theory has also been harnessed for more applied tasks such as searching for extrasolar planets, determining the mass of distant galaxies and even guiding the trajectories of wayward car drivers and ballistic missiles. General relativity, once an exotic description of gravity, is now a powerful research tool.

The quest to grasp gravity began long before Einstein. During the plague that ravaged Europe from 1665 to 1666, Isaac Newton retreated from his post at the University of Cambridge, took up refuge at his family’s home in Lincolnshire, and in his idle hours realized that every object, whether on Earth or in the heavens, pulls on every other with a force that depends solely on how big the objects are—their mass—and how far apart they are in space—their distance. School kids the world over have learned the mathematical version of Newton’s law, which has made such spectacularly accurate predictions for the motion of everything from hurled rocks to orbiting planets that it seemed Newton had written the final word on gravity. But he hadn’t. And Einstein was the first to become certain of this.

In 1905 Einstein discovered the special theory of relativity, establishing the famous dictum that nothing—no object or signal—can travel faster than the speed of light. And therein lies the rub. According to Newton’s law, if you shake the Sun like a cosmic maraca, gravity will cause the Earth to immediately shake too. That is, Newton’s formula implies that gravity exerts its influence from one location to another instantaneously. That’s not only faster than light, it’s infinite.

Relativity: The Special and the General Theory

Published on the hundredth anniversary of general relativity, this handsome edition of Einstein's famous book places the work in historical and intellectual context while providing invaluable insight into one of the greatest scientific minds of all time.

Einstein would have none of it. A more refined description of gravity must surely exist, one in which gravitational influences do not outrun light. Einstein dedicated himself to finding it. And to do so, he realized, he would need to answer a seemingly basic question: How does gravity work? How does the Sun reach out across 93 million miles and exert a gravitational pull on the Earth? For the more familiar pulls of everyday experience—opening a door, uncorking a wine bottle—the mechanism is manifest: There is direct contact between your hand and the object experiencing the pull. But when the Sun pulls on the Earth, that pull is exerted across space—empty space. There is no direct contact. So what invisible hand is at work executing gravity’s bidding?

Newton himself found this question deeply puzzling, and volunteered that his own failure to identify how gravity exerts its influence meant that his theory, however successful its predictions, was surely incomplete. Yet for over 200 years, Newton’s admission was nothing more than an overlooked footnote to a theory that otherwise agreed spot on with observations.

In 1907 Einstein began to work in earnest on answering this question; by 1912, it had become his full-time obsession. And within that handful of years, Einstein hit upon a key conceptual breakthrough, as simple to state as it is challenging to grasp: If there is nothing but empty space between the Sun and the Earth, then their mutual gravitational pull must be exerted by space itself. But how?

Einstein’s answer, at once beautiful and mysterious, is that matter, such as the Sun and the Earth, causes space around it to curve, and the resulting warped shape of space influences the motion of other bodies that pass by.

Here’s a way to think about it. Picture the straight trajectory followed by a marble you’ve rolled on a flat wooden floor. Now imagine rolling the marble on a wooden floor that has been warped and twisted by a flood. The marble won’t follow the same straight trajectory because it will be nudged this way and that by the floor’s curved contours. Much as with the floor, so with space. Einstein envisioned that the curved contours of space would nudge a batted baseball to follow its familiar parabolic path and coax the Earth to adhere to its usual elliptical orbit.

It was a breathtaking leap. Until then, space was an abstract concept, a kind of cosmic container, not a tangible entity that could effect change. In fact, the leap was greater still. Einstein realized that time could warp, too. Intuitively, we all envision that clocks, regardless of where they’re located, tick at the same rate. But Einstein proposed that the nearer clocks are to a massive body, like the Earth, the slower they will tick, reflecting a startling influence of gravity on the very passage of time. And much as a spatial warp can nudge an object’s trajectory, so too for a temporal one: Einstein’s math suggested that objects are drawn toward locations where time elapses more slowly.

Still, Einstein’s radical recasting of gravity in terms of the shape of space and time was not enough for him to claim victory. He needed to develop the ideas into a predictive mathematical framework that would precisely describe the choreography danced by space, time and matter. Even for Albert Einstein, that proved to be a monumental challenge. In 1912, struggling to fashion the equations, he wrote to a colleague that “Never before in my life have I tormented myself anything like this.” Yet, just a year later, while working in Zurich with his more mathematically attuned colleague Marcel Grossmann, Einstein came tantalizingly close to the answer. Leveraging results from the mid-1800s that provided the geometrical language for describing curved shapes, Einstein created a wholly novel yet fully rigorous reformulation of gravity in terms of the geometry of space and time.

But then it all seemed to collapse. While investigating his new equations Einstein committed a fateful technical error, leading him to think that his proposal failed to correctly describe all sorts of commonplace motion. For two long, frustrating years Einstein desperately tried to patch the problem, but nothing worked.

Einstein, tenacious as they come, remained undeterred, and in the fall of 1915 he finally saw the way forward. By then he was a professor in Berlin and had been inducted into the Prussian Academy of Sciences. Even so, he had time on his hands. His estranged wife, Mileva Maric, finally accepted that her life with Einstein was over, and had moved back to Zurich with their two sons. Though the increasingly strained family relations weighed heavily on Einstein, the arrangement also allowed him to freely follow his mathematical hunches, undisturbed day and night, in the quiet solitude of his barren Berlin apartment.

By November, this freedom bore fruit. Einstein corrected his earlier error and set out on the final climb toward the general theory of relativity. But as he worked intensely on the fine mathematical details, conditions turned unexpectedly treacherous. A few months earlier, Einstein had met with the renowned German mathematician David Hilbert, and had shared all his thinking about his new gravitational theory. Apparently, Einstein learned to his dismay, the meeting had so stoked Hilbert’s interest that he was now racing Einstein to the finish line.

A series of postcards and letters the two exchanged throughout November 1915 documents a cordial but intense rivalry as each closed in on general relativity’s equations. Hilbert considered it fair game to pursue an opening in a promising but as yet unfinished theory of gravity; Einstein considered it atrociously bad form for Hilbert to muscle in on his solo expedition so near the summit. Moreover, Einstein anxiously realized, Hilbert’s deeper mathematical reserves presented a serious threat. His years of hard work notwithstanding, Einstein might get scooped.

The worry was well-founded. On Saturday, November 13, Einstein received an invitation from Hilbert to join him in Göttingen on the following Tuesday to learn in “very complete detail” the “solution to your great problem.” Einstein demurred. “I must refrain from traveling to Göttingen for the moment and rather must wait patiently until I can study your system from the printed article; for I am tired out and plagued by stomach pains besides.”

But that Thursday, when Einstein opened his mail, he was confronted by Hilbert’s manuscript. Einstein immediately wrote back, hardly cloaking his irritation: “The system you furnish agrees—as far as I can see—exactly with what I found in the last few weeks and have presented to the Academy.” To his friend Heinrich Zangger, Einstein confided, “In my personal experience I have not learnt any better the wretchedness of the human species as on occasion of this theory....”

A week later, on November 25, lecturing to a hushed audience at the Prussian Academy, Einstein unveiled the final equations constituting the general theory of relativity.

No one knows what happened during that final week. Did Einstein come up with the final equations on his own or did Hilbert’s paper provide unbidden assistance? Did Hilbert’s draft contain the correct form of the equations, or did Hilbert subsequently insert those equations, inspired by Einstein’s work, into the version of the paper that Hilbert published months later? The intrigue only deepens when we learn that a key section of the page proofs for Hilbert’s paper, which might have settled the questions, was literally snipped away.

In the end, Hilbert did the right thing. He acknowledged that whatever his role in catalyzing the final equations might have been, the general theory of relativity should rightly be credited to Einstein. And so it has. Hilbert has gotten his due too, as a technical but particularly useful way of expressing the equations of general relativity bears the names of both men.

Of course, the credit would only be worth having if the general theory of relativity were confirmed through observations. Remarkably, Einstein could see how that might be done.

General relativity predicted that beams of light emitted by distant stars would travel along curved trajectories as they passed through the warped region near the Sun en route to Earth. Einstein used the new equations to make this precise—he calculated the mathematical shape of these curved trajectories. But to test the prediction astronomers would need to see distant stars while the Sun is in the foreground, and that’s only possible when the Moon blocks out the Sun’s light, during a solar eclipse.

The next solar eclipse, of May 29, 1919, would thus be general relativity’s proving ground. Teams of British astronomers, led by Sir Arthur Eddington, set up shop in two locations that would experience a total eclipse of the Sun—in Sobral, Brazil, and on Príncipe, off the west coast of Africa. Battling the challenges of weather, each team took a series of photographic plates of distant stars momentarily visible as the Moon drifted across the Sun.

During the subsequent months of careful analysis of the images, Einstein waited patiently for the results. Finally, on September 22, 1919, Einstein received a telegram announcing that the eclipse observations had confirmed his prediction.

Newspapers across the globe picked up the story, with breathless headlines proclaiming Einstein’s triumph and catapulting him virtually overnight into a worldwide sensation. In the midst of all the excitement, a young student, Ilse Rosenthal-Schneider, asked Einstein what he would have thought if the observations did not agree with general relativity’s prediction. Einstein famously answered with charming bravado, “I would have been sorry for the Dear Lord because the theory is correct.”

Indeed, in the decades since the eclipse measurements, there have been a great many other observations and experiments—some ongoing—that have led to rock-solid confidence in general relativity. One of the most impressive is an observational test that spanned nearly 50 years, among NASA’s longest-running projects. General relativity claims that as a body like the Earth spins on its axis, it should drag space around in a swirl somewhat like a spinning pebble in a bucket of molasses. In the early 1960s, Stanford physicists set out a scheme to test the prediction: Launch four ultra-precise gyroscopes into near-Earth orbit and look for tiny shifts in the orientation of the gyroscopes’ axes that, according to the theory, should be caused by the swirling space.

It took a generation of scientific effort to develop the necessary gyroscopic technology and then years of data analysis to, among other things, overcome an unfortunate wobble the gyroscopes acquired in space. But in 2011, the team behind Gravity Probe B, as the project is known, announced that the half-century-long experiment had reached a successful conclusion: The gyroscopes’ axes were turning by the amount Einstein’s math predicted.

There is one remaining experiment, currently more than 20 years in the making, that many consider the final test of the general theory of relativity. According to the theory, two colliding objects, be they stars or black holes, will create waves in the fabric of space, much as two colliding boats on an otherwise calm lake will create waves of water. And as such gravitational waves ripple outward, space will expand and contract in their wake, somewhat like a ball of dough being alternately stretched and compressed.

In the early 1990s, a team led by scientists at MIT and Caltech initiated a research program to detect gravitational waves. The challenge, and it’s a big one, is that if a tumultuous astrophysical encounter occurs far away, then by the time the resulting spatial undulations wash by Earth they will have spread so widely that they will be fantastically diluted, perhaps stretching and compressing space by only a fraction of an atomic nucleus.

Nevertheless, researchers have developed a technology that just might be able to see the tiny telltale signs of a ripple in the fabric of space as it rolls by Earth. In 2001, two four-kilometer-long L-shaped devices, collectively known as LIGO (Laser Interferometer Gravitational-Wave Observatory), were deployed in Livingston, Louisiana, and Hanford, Washington. The strategy is that a passing gravitational wave would alternately stretch and compress the two arms of each L, leaving an imprint on laser light racing up and down each arm.

In 2010, LIGO was decommissioned, before any gravitational wave signatures had been detected—the apparatus almost certainly lacked the sensitivity necessary to record the tiny twitches caused by a gravitational wave reaching Earth. But now an advanced version of LIGO, an upgrade expected to be ten times as sensitive, is being implemented, and researchers anticipate that within a few years the detection of ripples in space caused by distant cosmic disturbances will be commonplace.

Success would be exciting not because anyone really doubts general relativity, but because confirmed links between the theory and observation can yield powerful new applications. The eclipse measurements of 1919, for example, which established that gravity bends light’s trajectory, have inspired a successful technique now used for finding distant planets. When such planets pass in front of their host stars, they slightly focus the star’s light causing a pattern of brightening and dimming that astronomers can detect. A similar technique has also allowed astronomers to measure the mass of particular galaxies by observing how severely they distort the trajectory of light emitted by yet more distant sources. Another, more familiar example is the global positioning system, which relies on Einstein’s discovery that gravity affects the passage of time. A GPS device determines its location by measuring the travel time of signals received from various orbiting satellites.  Without taking account of gravity’s impact on how time elapses on the satellites, the GPS system would fail to correctly determine the location of an object, including your car or a guided missile.

Physicists believe that the detection of gravitational waves has the capacity to generate its own application of profound importance: a new approach to observational astronomy.

Since the time of Galileo, we have turned telescopes skyward to gather light waves emitted by distant objects. The next phase of astronomy may very well center on gathering gravitational waves produced by distant cosmic upheavals, allowing us to probe the universe in a wholly new way. This is particularly exciting because waves of light could not penetrate the plasma that filled space until a few hundred thousand years after the Big Bang—but waves of gravity could. One day we may thus use gravity, not light, as our most penetrating probe of the universe’s earliest moments.

Because waves of gravity ripple through space somewhat as waves of sound ripple through air, scientists speak of “listening” for gravitational signals. Adopting that metaphor, how wonderful to imagine that the second centennial of general relativity may be cause for physicists to celebrate having finally heard the sounds of creation.

Editors' Note, September 29, 2015: An earlier version of this article inaccurately described how GPS systems operate. The text has been changed accordingly.

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Brian Greene | | READ MORE

Science columnist Brian Greene is a mathematician and physicist at Columbia University, the author of bestselling cosmology books such as The Hidden Reality , co-founder of the World Science Festival and the prime mover behind the online education resource World Science U . Photo: Lark Elliott.

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Conventionality of Simultaneity

In his first paper on the special theory of relativity, Einstein indicated that the question of whether or not two spatially separated events were simultaneous did not necessarily have a definite answer, but instead depended on the adoption of a convention for its resolution. Some later writers have argued that Einstein’s choice of a convention is, in fact, the only possible choice within the framework of special relativistic physics, while others have maintained that alternative choices, although perhaps less convenient, are indeed possible.

1. The Conventionality Thesis

2. phenomenological counterarguments, 3. malament’s theorem, 4. other considerations, other internet resources, related entries.

The debate about the conventionality of simultaneity is usually carried on within the framework of the special theory of relativity. Even prior to the advent of that theory, however, questions had been raised (see, e.g., Poincaré 1898) as to whether simultaneity was absolute; i.e., whether there was a unique event at location A that was simultaneous with a given event at location B. In his first paper on relativity, Einstein (1905) asserted that it was necessary to make an assumption in order to be able to compare the times of occurrence of events at spatially separated locations (Einstein 1905, 38–40 of the Dover translation or 125–127 of the Princeton translation; but note Scribner 1963, for correction of an error in the Dover translation). His assumption, which defined what is usually called standard synchrony, can be described in terms of the following idealized thought experiment, where the spatial locations A and B are fixed locations in some particular, but arbitrary, inertial (i.e., unaccelerated) frame of reference: Let a light ray, traveling in vacuum, leave A at time t 1 (as measured by a clock at rest there), and arrive at B coincident with the event E at B . Let the ray be instantaneously reflected back to A , arriving at time t 2 . Then standard synchrony is defined by saying that E is simultaneous with the event at A that occurred at time ( t 1 + t 2 )/2. This definition is equivalent to the requirement that the one-way speeds of the ray be the same on the two segments of its round-trip journey between A and B .

It is interesting to note (as pointed out by Jammer (2006, 49), in his comprehensive survey of virtually all aspects of simultaneity) that something closely analogous to Einstein’s definition of standard simultaneity was used more than 1500 years earlier by St. Augustine in his Confessions (written in 397 CE). He was arguing against astrology by telling a story of two women, one rich and one poor, who gave birth simultaneously but whose children had quite different lives in spite of having identical horoscopes. His method of determining that the births, at different locations, were simultaneous was to have a messenger leave each birth site at the moment of birth and travel to the other, presumably with equal speeds. Since the messengers met at the midpoint, the births must have been simultaneous. Jammer comments that this “may well be regarded as probably the earliest recorded example of an operational definition of distant simultaneity.”

The thesis that the choice of standard synchrony is a convention, rather than one necessitated by facts about the physical universe (within the framework of the special theory of relativity), has been argued particularly by Reichenbach (see, for example, Reichenbach 1958, 123–135) and Grünbaum (see, for example, Grünbaum 1973, 342–368). They argue that the only nonconventional basis for claiming that two distinct events are not simultaneous would be the possibility of a causal influence connecting the events. In the pre-Einsteinian view of the universe, there was no reason to rule out the possibility of arbitrarily fast causal influences, which would then be able to single out a unique event at A that would be simultaneous with E . In an Einsteinian universe, however, no causal influence can travel faster than the speed of light in vacuum, so from the point of view of Reichenbach and Grünbaum, any event at A whose time of occurrence is in the open interval between t 1 and t 2 could be defined to be simultaneous with E . In terms of the ε-notation introduced by Reichenbach, any event at A occurring at a time t 1 + ε( t 2 − t 1 ), where 0 < ε < 1, could be simultaneous with E . That is, the conventionality thesis asserts that any particular choice of ε within its stated range is a matter of convention, including the choice ε=1/2 (which corresponds to standard synchrony). If ε differs from 1/2, the one-way speeds of a light ray would differ (in an ε-dependent fashion) on the two segments of its round-trip journey between A and B . If, more generally, we consider light traveling on an arbitrary closed path in three-dimensional space, then (as shown by Minguzzi 2002, 155–156) the freedom of choice in the one-way speeds of light amounts to the choice of an arbitrary scalar field (although two scalar fields that differ only by an additive constant would give the same assignment of one-way speeds).

It might be argued that the definition of standard synchrony makes use only of the relation of equality (of the one-way speeds of light in different directions), so that simplicity dictates its choice rather than a choice that requires the specification of a particular value for a parameter. Grünbaum (1973, 356) rejects this argument on the grounds that, since the equality of the one-way speeds of light is a convention, this choice does not simplify the postulational basis of the theory but only gives a symbolically simpler representation.

Many of the arguments against the conventionality thesis make use of particular physical phenomena, together with the laws of physics, to establish simultaneity (or, equivalently, to measure the one-way speed of light). Salmon (1977), for example, discusses a number of such schemes and argues that each makes use of a nontrivial convention. For instance, one such scheme uses the law of conservation of momentum to conclude that two particles of equal mass, initially located halfway between A and B and then separated by an explosion, must arrive at A and B simultaneously. Salmon (1977, 273) argues, however, that the standard formulation of the law of conservation of momentum makes use of the concept of one-way velocities, which cannot be measured without the use of (something equivalent to) synchronized clocks at the two ends of the spatial interval that is traversed; thus, it is a circular argument to use conservation of momentum to define simultaneity.

It has been argued (see, for example, Janis 1983, 103–105, and Norton 1986, 119) that all such schemes for establishing convention-free synchrony must fail. The argument can be summarized as follows: Suppose that clocks are set in standard synchrony, and consider the detailed space-time description of the proposed synchronization procedure that would be obtained with the use of such clocks. Next suppose that the clocks are reset in some nonstandard fashion (consistent with the causal order of events), and consider the description of the same sequence of events that would be obtained with the use of the reset clocks. In such a description, familiar laws may take unfamiliar forms, as in the case of the law of conservation of momentum in the example mentioned above. Indeed, all of special relativity has been reformulated (in an unfamiliar form) in terms of nonstandard synchronies (Winnie 1970a and 1970b). Since the proposed synchronization procedure can itself be described in terms of a nonstandard synchrony, the scheme cannot describe a sequence of events that is incompatible with nonstandard synchrony. A comparison of the two descriptions makes clear what hidden assumptions in the scheme are equivalent to standard synchrony. Nevertheless, editors of respected journals continue to accept, from time to time, papers purporting to measure one-way light speeds; see, for example, Greaves et al . (2009). Application of the procedure just described shows where their errors lie.

For a discussion of various proposals to establish synchrony, see the supplementary document:

Transport of Clocks

The only currently discussed proposal is based on a theorem of Malament (1977), who argues that standard synchrony is the only simultaneity relation that can be defined, relative to a given inertial frame, from the relation of (symmetric) causal connectibility. Let this relation be represented by κ, let the statement that events p and q are simultaneous be represented by S ( p , q ), and let the given inertial frame be specified by the world line, O , of some inertial observer. Then Malament’s uniqueness theorem shows that if S is definable from κ and O , if it is an equivalence relation, if points p on O and q not on O exist such that S ( p , q ) holds, and if S is not the universal relation (which holds for all points), then S is the relation of standard synchrony.

Some commentators have taken Malament’s theorem to have settled the debate on the side of nonconventionality. For example, Torretti (1983, 229) says, “Malament proved that simultaneity by standard synchronism in an inertial frame F is the only non-universal equivalence between events at different points of F that is definable (‘in any sense of “definable” no matter how weak’) in terms of causal connectibility alone, for a given F ”; and Norton (Salmon et al . 1992, 222) says, “Contrary to most expectations, [Malament] was able to prove that the central claim about simultaneity of the causal theorists of time was false. He showed that the standard simultaneity relation was the only nontrivial simultaneity relation definable in terms of the causal structure of a Minkowski spacetime of special relativity.”

Other commentators disagree with such arguments, however. Grünbaum (2010) has written a detailed critique of Malament’s paper. He first cites Malament’s need to postulate that S is an equivalence relation as a weakness in the argument, a view also endorsed by Redhead (1993, 114). Grünbaum’s main argument, however, is based on an earlier argument by Janis (1983, 107–109) that Malament’s theorem leads to a unique (but different) synchrony relative to any inertial observer, that this latitude is the same as that in introducing Reichenbach’s ε, and thus Malament’s theorem should carry neither more nor less weight against the conventionality thesis than the argument (mentioned above in the last paragraph of the first section of this article) that standard synchrony is the simplest choice. Grünbaum concludes “that Malament’s remarkable proof has not undermined my thesis that, in the STR, relative simultaneity is conventional, as contrasted with its non-conventionality in the Newtonian world, which I have articulated! Thus, I do not need to retract the actual claim I made in 1963…” Somewhat similar arguments are given by Redhead (1993, 114) and by Debs and Redhead (2007, 87–92).

For further discussion, see the supplement document:

Further Discussion of Malament’s Theorem

Since the conventionality thesis rests upon the existence of a fastest causal signal, the existence of arbitrarily fast causal signals would undermine the thesis. If we leave aside the question of causality, for the moment, the possibility of particles (called tachyons) moving with arbitrarily high velocities is consistent with the mathematical formalism of special relativity (see, for example, Feinberg 1967). Just as the speed of light in vacuum is an upper limit to the possible speeds of ordinary particles (sometimes called bradyons), it would be a lower limit to the speeds of tachyons. When a transformation is made to a different inertial frame of reference, the speeds of both bradyons and tachyons change (the speed of light in vacuum being the only invariant speed). At any instant, the speed of a bradyon can be transformed to zero and the speed of a tachyon can be transformed to an infinite value. The statement that a bradyon is moving forward in time remains true in every inertial frame (if it is true in one), but this is not so for tachyons. Feinberg (1967) argues that this does not lead to violations of causality through the exchange of tachyons between two uniformly moving observers because of ambiguities in the interpretation of the behavior of tachyon emitters and absorbers, whose roles can change from one to the other under the transformation between inertial frames. He claims to resolve putative causal anomalies by adopting the convention that each observer describes the motion of each tachyon interacting with that observer’s apparatus in such a way as to make the tachyon move forward in time. However, all of Feinberg’s examples involve motion in only one spatial dimension. Pirani (1970) has given an explicit two-dimensional example in which Feinberg’s convention is satisfied but a tachyon signal is emitted by an observer and returned to that observer at an earlier time, thus leading to possible causal anomalies.

A claim that no value of ε other than 1/2 is mathematically possible has been put forward by Zangari (1994). He argues that spin-1/2 particles (e.g., electrons) must be represented mathematically by what are known as complex spinors, and that the transformation properties of these spinors are not consistent with the introduction of nonstandard coordinates (corresponding to values of ε other than 1/2). Gunn and Vetharaniam (1995), however, present a derivation of the Dirac equation (the fundamental equation describing spin-1/2 particles) using coordinates that are consistent with arbitrary synchrony. They argue that Zangari mistakenly required a particular representation of space-time points as the only one consistent with the spinorial description of spin-1/2 particles.

Another argument for standard synchrony has been given by Ohanian (2004), who bases his considerations on the laws of dynamics. He argues that a nonstandard choice of synchrony introduces pseudoforces into Newton’s second law, which must hold in the low-velocity limit of special relativity; that is, it is only with standard synchrony that net force and acceleration will be proportional. Macdonald (2005) defends the conventionality thesis against this argument in a fashion analagous to the argument used by Salmon (mentioned above in the first paragraph of the second section of this article) against the use of the law of conservation of momentum to define simultaneity: Macdonald says, in effect, that it is a convention to require Newton’s laws to take their standard form.

Many of the arguments against conventionality involve viewing the preferred simultaneity relation as an equivalence relation that is invariant under an appropriate transformation group. Mamone Capria (2012) has examined the interpretation of simultaneity as an invariant equivalence relation in great detail, and argues that it does not have any bearing on the question of whether or not simultaneity is conventional in special relativity.

A vigorous defense of conventionality has been offered by Rynasiewicz (2012). He argues that his approach “has the merit of nailing the exact sense in which simultaneity is conventional. It is conventional in precisely the same sense in which the gauge freedom that arises in the general theory of relativity makes the choice between diffeomorphically related models conventional.” He begins by showing that any choice of a simultaneity relation is equivalent to a choice of a velocity in the equation for local time in H.A. Lorentz’s Versuch theory (Lorentz 1895). Then, beginning with Minkowski space with the standard Minkowski metric, he introduces a diffeomorphism in which each point is mapped to a point with the same spatial coordinates, but the temporal coordinate is that of a Lorentzian local time expressed in terms of the velocity as a parameter. This mapping is not an isometry, for the light cones are tilted, which corresponds to anisotropic light propagation. He proceeds to argue, using the hole argument (see, for example, Earman and Norton 1987) as an analogy, that this parametric freedom is just like the gauge freedom of general relativity. As the tilting of the light cones, if projected into a single spatial dimension, would be equivalent to a choice of Reichenbach’s ε, it seems that Rynasiewicz’s argument is a generalization and more completely argued version of the argument given by Janis that is mentioned above in the third paragraph of Section 3.

The debate about conventionality of simultaneity seems far from settled, although some proponents on both sides of the argument might disagree with that statement. The reader wishing to pursue the matter further should consult the sources listed below as well as additional references cited in those sources.

• Anderson, R., I. Vetharaniam, and G. Stedman, 1998. “Conventionality of Synchronisation, Gauge Dependence and Test Theories of Relativity,” Physics Reports , 295: 93–180.
• Augustine, St., Confessions , translated by E.J. Sheed, Indianapolis: Hackett Publishing Co., 2nd edition, 2006.
• Ben-Yami, H., 2006. “Causality and Temporal Order in Special Relativity,” British Journal for the Philosophy of Science , 57: 459–479.
• Brehme, R., 1985. “Response to ‘The Conventionality of Synchronization’,” American Journal of Physics , 53: 56–59.
• Brehme, R., 1988. “On the Physical Reality of the Isotropic Speed of Light,” American Journal of Physics , 56: 811–813.
• Bridgman, P., 1962. A Sophisticate’s Primer of Relativity . Middletown: Wesleyan University Press.
• Debs, T. and M. Redhead, 2007. Objectivity, Invariance, and Convention: Symmetry in Physical Science , Cambridge, MA and London: Harvard University Press.
• Earman, J. and J. Norton, 1987. “What Price Spacetime Substantivalism? The Hole Story,” British Journal for the Philosophy of Science , 38: 515–525.
• Eddington, A., 1924. The Mathematical Theory of Relativity , 2nd edition, Cambridge: Cambridge University Press.
• Einstein, A., 1905. “Zur Elektrodynamik bewegter Körper,” Annalen der Physik , 17: 891–921. English translations in The Principle of Relativity , New York: Dover, 1952, pp. 35–65; and in J. Stachel (ed.), Einstein’s Miraculous Year , Princeton: Princeton University Press, 1998, pp. 123–160.
• Ellis, B. and P. Bowman, 1967. “Conventionality in Distant Simultaneity,” Philosophy of Science , 34: 116–136.
• Feinberg, G., 1967. “Possibility of Faster-Than-Light Particles,” Physical Review , 159: 1089–1105.
• Giulini, D., 2001. “Uniqueness of Simultaneity,” British Journal for the Philosophy of Science , 52: 651–670.
• Greaves, E., A. Rodriguez, and J. Ruiz-Camaro, 2009. “A One-Way Speed of Light Experiment,” American Journal of Physics , 77: 894–896.
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Stephen Hawking: Everything you need to know about the thesis that 'broke the Internet'

Your cheat sheet into the mind of one of the world’s greatest physicists.

Scribbled in pencil on one of its early pages is "no copying without the author's consent". In October 2017, Stephen Hawking allowed his PhD thesis — Properties of Expanding Universes — to be made available online through the University of Cambridge's Apollo portal. The website crashed almost immediately under the sheer weight of traffic. It was downloaded almost 60,000 times in the first 24 hours alone.

Hawking was 24 years old when he received his PhD in 1966 and, despite being diagnosed with motor neurone disease at just 21, could still handwrite that "this dissertation is my original work." In a statement to accompany its release, the late physicist said: "By making my PhD thesis Open Access, I hope to inspire people around the world to look up at the stars and not down at their feet; to wonder about our place in the universe and to try and make sense of the cosmos." 

Here, we break it down, guiding you through the physics until we reach the conclusion that made Hawking a household name.

Step 1: What’s it about?

Hawking's PhD thesis relates to Albert Einstein's General Theory of Relativity — the more accurate theory of gravity that replaced Isaac Newton 's original ideas. Newton said gravity was a pull between two objects. Einstein said that gravity is the result of massive objects warping the fabric of space and time (space-time) around them. According to Einstein, Earth orbits the sun because we're caught in the depression our star makes in space-time.

Hawking applies the mathematics of general relativity to models of the birth of our universe ( cosmologies ). The earliest cosmologies had our universe as a static entity that had existed forever. This idea was so ingrained that when Einstein's original calculations suggested a static universe was unlikely, he added a "cosmological constant" to the math in order to keep the universe static. He would later reportedly call it his "greatest blunder".

Things began to change when Edwin Hubble made an important discovery. Hawking writes: "the discovery of the recession of the nebulae [galaxies] by Hubble led to the abandonment of static models in favor of ones in which we're expanding."

Step 2: Our expanding universe

Some astronomers seized the idea of an expanding universe to argue that the universe must have had a beginning — a moment of creation called the Big Bang . The name was coined by Fred Hoyle, an advocate of the alternative Steady State Model. This theory states that the universe has been around forever, and that new stars form in the gaps created as the universe expands. There was no initial creation event.

Hawking spends chapter one of his thesis taking down the premise, formally encapsulated in a model called Hoyle-Narlikar theory. Hawking laments that although the General Theory of Relativity is powerful, it allows for many different solutions to its equations. That means many different models can be consistent with it. He says that's "one of the weaknesses of the Einstein theory."

The famous physicist then shows that a requirement of Hoyle-Narlikar theory appears to "exclude those models that seem to correspond to the actual universe." In short, the Steady State Model fails to match observation.

Step 3: Space: It looks the same everywhere

Hawking says that the assumptions of the Hoyle-Narlikar theory are in direct contradiction of the Robertson-Walker metric, named after American physicist Howard P. Robertson and British mathematician Arthur Walker. Today it is more widely called the Friedmann–Lemaître–Robertson–Walker (FLRW) metric. A metric is an exact solution to the equations of Einstein's General Theory of Relativity. Devised in the 1920s and 1930s, FLRW forms the basis of our modern model of the universe. Its key feature is that it assumes matter is evenly distributed in an expanding (or contracting) universe — a premise backed up by astronomical observations. 

Interestingly, Hawking offers Hoyle and Narlikar a ray of hope. "A possible way to save the Hoyle-Narlikar theory would be to allow masses of both positive and negative sign," he writes, before adding: "There does not seem to be any matter having these properties in our region of space." Today, we know that the expansion of the universe is accelerating, perhaps due to dark energy — a shadowy entity with an anti-gravitational property perhaps akin to particles with a negative mass.

Step 4: The problem with galaxies

Even geniuses get it wrong sometimes. Hawking's second chapter covers perturbations — small variations in the local curvature of space-time — and how they evolve as the universe expands. He says that a small perturbation “will not contract to form a galaxy." Later in the chapter he goes on to say: "We see that galaxies cannot form as the result of the growth of small perturbations."

That couldn't be further from our modern-day picture of how galaxies form. The key ingredient Hawking was missing is dark matter , an invisible substance thought to be spread throughout the universe, which provides a gravitational glue that holds galaxies together. Dark matter gathered around small space-time perturbations, eventually drawing in more and more material until early galaxies formed.

Our modern working cosmological picture is known as the ΛCDM model (pronounced Lambda CDM). Lambda is the Greek letter cosmologists use to denote the cosmological constant that Einstein originally introduced (albeit for the wrong reasons). CDM stands for cold dark matter. These two factors have been added to the FLRW model since Hawking wrote his thesis.

Step 5: Gravitational waves don’t disappear

Where Hawking was wrong on galaxies, he was very right on gravitational waves —ripples in the fabric of space-time that move outwards through the universe. They were predicted by Einstein when he first devised his Theory of General Relativity back in 1915, and in Hawking's time were also known as gravitational radiation.

Hawking uses Einstein's equations to show that gravitational waves aren't absorbed by matter in the universe as they travel through it, assuming the universe is largely made of dust. In fact, Hawking says that "gravitational radiation behaves in much the same way as other radiation fields [such as light]."

The physicist does note how esoteric the topic is in the 1960s. "This is slightly academic since gravitational radiation has not yet been detected, let alone investigated."

It would take physicists until September 2015 to detect gravitational waves for the first time using the Laser Interferometer Gravitational-Wave Observatory (LIGO). They were produced by the collision of two black holes — one 36- and the other 29-times the mass of the sun — about 1.3-billion-light-years away.

Related: Lab-grown black hole analog behaves just like Stephen Hawking said it would

Step 6: Are we living in an open, closed, or flat cosmos?

Hawking is heading for a groundbreaking conclusion, but first he sets himself up by introducing the idea of the overall shape of space. There are three general forms the curvature of space can take: open, closed, or flat.

A closed universe resembles Earth's surface — it has no boundary. You can keep traveling around the planet without coming to an edge. An open universe is shaped more like a saddle. A flat universe, as the name suggests, is like a sheet of paper. 

Imagine a triangle drawn onto the surface. We all learn at school that the angles inside a triangle sum to 180 degrees. However, that's only the case for triangles on flat surfaces, not open or closed ones. Draw a line from the Earth's North Pole down to the equator, before taking a 90-degree turn to travel along it. Then make another 90-degree turn back towards the North Pole. The angle between your path away from and towards the North Pole cannot be zero, so the angles inside that triangle must add up to more than 180 degrees.

Step 7: The universe is flat!

Hawking then links the idea of open and closed universes to Cauchy surfaces, named after the French mathematician and physicist Augustin-Louis Cauchy (1789—1857). A Cauchy surface is a slice through space-time, the equivalent of an instant of time. All points on the surface are connected in time. A path along a Cauchy surface cannot see you revisit a previous moment. In Hawking's own words: “A Cauchy surface will be taken to mean a complete, connected space-like surface that intersects every time-like and null line once and once only.”

He then says that closed universes are known as "compact" Cauchy surfaces, and open universes as “non-compact” ones. The former example is said to have "positive" curvature, the latter "negative" curvature. 

A flat universe has zero curvature. He moves on to set up the landmark assertions he's about to make about singularities by saying they are “applicable to models... with surfaces... which have negative or zero curvature.” Modern astronomers believe the universe is flat, meaning its zero curvature satisfies Hawking's conditions. 

Step 8: Hawking drops a bombshell

Most of the early chapters of Hawking's thesis are unremarkable — they don't offer anything particularly revolutionary, and he even gets a few things wrong. However, in his final chapter the physicist drops a bombshell that will make his name and ignite a stellar career, during which he will become one of the most famous scientists on the planet.

He says that space-time can begin and end at a singularity , and what's more he can prove it. A singularity is an infinitely small and infinitely dense point. It literally has zero size, and space and time both end (or begin) at a singularity. They had been predicted for decades, particularly when physicists started to apply Einstein's General Theory of Relativity to the picture of an expanding universe. 

If the universe is expanding today then it was smaller yesterday. Keep working back, and you find all matter in the universe condensed into a tiny, hot point — the moment of creation, a Big Bang. But just how do you prove that you can indeed get singularities in space-time?

Step 9: Hawking’s proof that the Big Bang happened

Hawking's proof leans on a very old method for proving a mathematical theory: Proof by contradiction. First you assume the thing you are trying to prove isn't true, then show that the resulting conclusions are demonstrably false. In fact, Hawking's most important section begins with the words "assume that space-time is singularity-free." There then follows some very complicated maths to show that such a universe would be simultaneously both open and closed — compact and non-compact at the same time. "This is a contradiction," Hawking said. "Thus the assumption that space-time is non-singular must be false." 

In one swoop, Hawking had proven that it is possible for space-time to begin as a singularity — that space and time in our universe could have had an origin. The Big Bang theory had just received a significant shot in the arm. Hawking started to write his PhD in October 1965, just 17 months after the discovery of the Cosmic Microwave Background — the leftover energy from the Big Bang. Together, these discoveries buried the Steady State Model for good.

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The theory of relativity, and other essays

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https://www.nist.gov/blogs/taking-measure/nist-physicists-once-obscure-work-now-helping-researchers-learn-about-origins

Taking Measure

Just a Standard Blog

NIST Physicist’s Once Obscure Work Is Now Helping Researchers Learn About the Origins of the Universe

When physicist John “Ben” Mates completed his doctoral thesis in 2011, he figured few people would read it. 

It’s not that Mates, who conducted his Ph.D. research at NIST while a graduate student at the University of Colorado, thought his work unimportant. 

Mates was just being realistic. Most scientists don’t bother to wade through doctoral dissertations, which can run more than 100 pages. Dissertations tend to focus on highly specialized topics. 

And for several years of his career at NIST, Mates was right.

He had devised a novel method to read out the signals from an array of exquisitely sensitive sensors that measure tiny changes in the intensity of thermal radiation (heat), including the afterglow of the Big Bang, known as the cosmic microwave background (CMB).

Reading out data from the detectors, developed at NIST and known as transition edge sensor (TES) bolometers, had proved challenging. That’s because the bolometers can only operate at temperatures a fraction of a degree above  absolute zero , which is about minus 273 degrees Celsius or minus 459 degrees Fahrenheit. If too many wires link the ultracold detectors to room-temperature equipment, the sensors will heat up and stop functioning.

Mates’ dissertation described a way to minimize the number of these wires, enabling the sensors to maintain their chilly operating temperature. 

After completing his thesis, Mates pursued another research project at NIST. 

(Many) Bolometers Needed 

In late 2013, however, his NIST supervisor, Joel Ullom, asked him if he’d like to return to his original study. His dissertation, Ullom told Mates, had taken on added importance. 

Mates had previously demonstrated that signals from two of the TES bolometers could be read out using a single wire connected to a room-temperature device rather than using a separate wire for each sensor. 

Although he had designed the method to minimize the room-temperature connections for a much larger number of sensors, he had not actually shown it could work.

Now, that demonstration was urgently needed — and on a massive scale. 

Astronomers wanted to use not just two but thousands of the TES bolometers on a set of ground-based telescopes to examine the CMB with 10 times more sensitivity than ever before. Although researchers have studied the CMB for decades, the bolometers are able to capture details of the tiny temperature variations in the radiation that may put to the test the leading theory of how the universe was born.

With thousands of bolometers, however, it would be virtually impossible to attach a separate room-temperature wire to each one without heating the sensors beyond their operating temperature.

Over the next 10 years, Mates perfected his technique, showing how the signal from each TES — a change in a tiny current — could be converted to a unique frequency. Thousands of those frequencies, he showed, could be carried on a single room-temperature cable, dramatically reducing the flow of heat back to the detectors.

Using his method, known as microwave multiplexing, astronomers recently installed 67,080  bolometers on the  Simons Observatory , a suite of four telescopes in Chile devoted to studying the CMB. 

The NIST-designed sensors act like miniature thermometers and can discern tiny temperature variations — as small as ten-millionths of a degree — in the CMB over more than 40% of the sky. 

The minuscule hot and cold spots correspond to slight variations in the density of the universe in its infancy, 380,000 years after its violent birth. Studying those variations reveals how and where tiny clumps of matter, the seeds of the galaxies we see in the sky today, first formed in the cosmos.

The bolometers also record patterns of different polarizations in the CMB — wiggles in the electric field of the radiation. Those wiggles encode a wealth of information about the universe an instant after the Big Bang and could hold clues about its mysterious beginning.

Multiplexing Research Goes Mainstream 

Now Mates’ dissertation is a hot topic — required reading for many scientists interested in multiplexing. He’s gotten hundreds of requests for reprints and has traveled around the world, recently installing instrumentation at the Japan Proton Accelerator Research Complex in Tokai. 

“It’s sort of freakish how it all worked out,” Mates said. “I never imagined the work would have such an impact.”

His thesis is so popular that Mates said he’s considering publishing an updated version of his manuscript.

In the future, Mates hopes to keep refining the technique and reducing the cost, so there can be many more projects over the next decade or longer. 

While he appreciates the attention his work is currently receiving, for Mates, the measurement problems were motivation to keep researching. 

“I think I also find most of the problems of developing and improving the system to be interesting on their own,” he said. 

Measuring the Cosmos 

Many NIST technologies have found homes among the stars. Learn more about how this research is helping to better understand our world on our Measuring the Cosmos site . 

About the author

Ron Cowen has been a science writer and editor at NIST since 2016. When not working at NIST, he’s a freelance writer specializing in physics and astronomy. In 2019, he authored his first book, a popular-level account of the 100-year struggle to understand the general theory of relativity, Gravity’s Century: From Einstein’s Eclipse to Images of Black Holes . Cowen has written for Scientific American , The New York Times , U.S. News & World Report , The Washington Post , National Geographic and the news sections of Science and Nature . He was also a staff reporter for 21 years at Science News magazine. Cowen has twice won several awards: the American Institute of Physics' excellence in science writing award, the American Astronomical Society's science writing award in solar physics and the Society's David Schramm science writing award for feature articles on high-energy astrophysics. He has a master's in physics from the University of Maryland.

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Intriguing story of the CMB radiation, though I missed any reference to the presumed dark matter scaffold on which galaxies were formed.

Hi, Dan. Thanks for your comment. Here's some additional information from the author:

Details in the structure of the cosmic microwave background reveal, indirectly, the composition of the universe, including the amount of dark matter. This video from Fermilab explains it: https://www.youtube.com/watch?v=ri2LIEjXhmE .

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