Spacetime

• 1. Introduction

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  Spacetime arrived in public with a warning that sounded almost like an obituary. In September 1908, the mathematician Hermann Minkowski told a gathering of German scientists and physicians: "Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow, and only some sort of union of the two shall preserve independence." That union is what physicists now call spacetime, a mathematical model that fuses three dimensions of space and one of time into a single four-dimensional continuum.

  Until the turn of the 20th century, almost nobody would have thought to ask for such a thing. The geometry of the universe, meaning its locations, shapes, distances and directions, was one subject. Time, the measurement of when events occur inside that universe, was another.

  So why did two ideas that seem so obviously separate have to be welded together? Who actually saw the weld first, and who got the credit? What does it mean that a single point in this model takes four numbers to describe? And how can observers who disagree about both time and distance still agree on something real?

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• 2. Cracks in the Aether

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  By the middle of the 1800s, experiments such as the observation of the Arago spot had persuaded physicists that light behaves as a wave. Measurements comparing the speed of light in air with its speed in water pointed the same way. Waves were assumed to need something to wave in, so light was given a hypothetical medium called the luminiferous aether.

  In 1851, the French physicist Hippolyte Fizeau measured light moving through flowing water. The speed came out lower than the speed of light in air plus the speed of the water, by an amount that depended on the water's index of refraction. That index varies with wavelength. Taken literally, the result meant the aether would flow at different speeds for different colors of light, a conclusion physicists found hard to swallow.

  The Michelson-Morley experiment of 1887 made matters worse. It found no differential influence of the Earth's motion through the aether on the speed of light. The most likely explanation, that the aether is dragged along completely, collided with the observation of stellar aberration.

  George Francis FitzGerald in 1889 and Hendrik Lorentz in 1892 each proposed a way out, working independently. Material bodies moving through a fixed aether, they said, are physically altered by the passage. They contract along the direction of travel by exactly the amount needed to cancel the null result. Nothing changes in directions across the motion.

  By 1904, Lorentz had pushed his theory until it produced equations formally identical to those Einstein would later derive, now known as the Lorentz transformation. His version was a theory of dynamics, the study of forces and their effect on motion, and it assumed real physical deformations of the matter itself. It also contained a quantity Lorentz named local time, which let him explain the aberration of light and the Fizeau experiment. Nobody yet had a clear answer to what local time physically was.

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• 3. A Crowded Discovery

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  Henri Poincaré argued in 1898 that whether two events are simultaneous is a matter of convention. In 1900 he realized that Lorentz's local time is simply what moving clocks indicate, once you synchronize clocks by assuming light travels at a constant speed. In 1900 and again in 1904, he stressed that the aether could never be detected, and he called the underlying idea the principle of relativity.

  In 1905 and 1906, Poincaré mathematically perfected Lorentz's theory of electrons so that it matched that principle. While exploring gravitation, he introduced a four-dimensional spacetime by defining four-position, four-velocity and four-force. Then he stepped back from it. He said this line of research seemed to "entail great pain for limited profit" and concluded "that three-dimensional language seems the best suited to the description of our world." As late as 1909, he was still describing the Lorentz transformation in terms of dynamics.

  Albert Einstein took a different road in 1905. He analyzed special relativity through kinematics, the study of moving bodies without reference to forces. His results were mathematically equivalent to those of Lorentz and Poincaré, but he built the whole theory on just two postulates: the principle of relativity and the constancy of the speed of light. His papers were full of concrete images, such as light signals exchanged between moving clocks and careful measurements of moving rods. That same year he also introduced the general equivalence of mass and energy, which fed into his equivalence principle of 1907.

  Hermann Minkowski, who had once been Einstein's mathematics professor, was racing the same finish line. Max Born recalled that Minkowski told him Einstein's paper came as a great shock, because Minkowski had reached the same conclusions independently. He had held back from publishing because he wanted to work out the mathematical structure first. According to Born, Minkowski never made a priority claim and always gave Einstein his full share.

  Minkowski had been troubled by the state of electrodynamics since at least the summer of 1905. That summer he and David Hilbert led an advanced seminar where notable physicists studied the papers of Lorentz, Poincaré and others. Minkowski regarded Einstein's work as an extension of Lorentz's, and he was most directly influenced by Poincaré. On the 5th of November 1907, he gave the Göttingen Mathematical Society a lecture titled The Relativity Principle. His 1908 talk, Space and Time, then offered the first public spacetime diagrams. It also showed that the invariant interval, combined with the finite speed of light, is enough to derive all of special relativity.

  Einstein was initially unimpressed. He dismissed Minkowski's geometry as überflüssige Gelehrsamkeit, which means superfluous learnedness. But his search for general relativity, begun in 1907, turned out to depend on that geometry. In 1916 he fully acknowledged his debt to Minkowski. Because curved spacetime also exists in general relativity, the flat spacetime of special relativity now carries Minkowski's name.

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• 4. Events and World Lines

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  Four numbers pin down a point in spacetime. In ordinary space a position takes three, which in the Cartesian system are often called x, y and z. A point in spacetime is called an event, and it adds the time coordinate t to those three. That is why spacetime is described as four-dimensional.

  Popular writing often explains events with firecrackers or sparks, but the mathematical version is stricter. A mathematical event has zero duration and is a single point. You can be in motion relative to a firecracker popping, but you cannot be in motion relative to an event. String events together and you get the path of a particle through spacetime, called its world line.

  Mathematically, spacetime is a manifold. That means it looks flat near any one point, the way a small enough patch of a globe's surface looks flat. A scale factor, c, conventionally called the speed of light, links distances in space to distances in time. Nearly 300000 kilometers of space count as equivalent to one second of time. At everyday speeds and human-scale distances, this makes the world look almost exactly like the Euclidean geometry of common sense.

  The word observer has a special meaning here. In special relativity, an observer usually means an entire frame of reference, pictured as a dense lattice of synchronized clocks stretching through all of space. Every point has its own clock, and each event is recorded instantly. For that reason, it makes no sense to say such an observer has a location.

  A real observer, of course, sees a delay between a signal being emitted and detected, because light takes time to travel. Physicists correct for that delay, and the corrected result is what they call a measurement. They separate what is measured from what is visually seen. Confusing the two is a common source of trouble for students of relativity. Older books on the subject often use the word observer in its everyday sense instead, so the context has to settle which meaning is intended.

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• 5. The Interval and the Light Cone

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  Two viewers using different coordinate systems to measure the distance between two points in three dimensions will agree on the answer, as long as they use the same units. The Pythagorean theorem gives that distance, and it is called invariant. In relativity, the distance between two points stops being invariant once one observer is moving, because of Lorentz contraction. Add a separation in time, and the situation gets still messier.

  The replacement is the spacetime interval. Different observers disagree about elapsed time because of time dilation, and about distance because of length contraction. Yet all of them compute the same interval between any two events. In words, the squared interval is the square of the speed of light times the time separation, minus the squares of the separations in each spatial direction.

  That minus sign matters. Unlike distances in Euclidean geometry, intervals in Minkowski spacetime can be negative, so physicists treat the squared interval as a symbol in its own right rather than taking a square root. Two sign conventions are in use, and both are widely accepted. A positive squared interval is called timelike, a negative one is spacelike, and zero is lightlike, also called null. A photon reaching your eye from a distant star has not aged at all, even though it spent years crossing the gap from our perspective.

  On a spacetime diagram, the world lines of photons have a slope of plus or minus 1. Every meter a photon travels to the left or right takes approximately 3.3 nanoseconds. Draw the zero-interval lines through any event and you get its light cone. With a second spatial dimension added, the picture becomes two cones meeting at a point, one opening toward the future and one toward the past.

  Inside the future cone lie all the events the origin event could influence, at or below light speed. Inside the past cone lie all the events that could have influenced it. The outside region is labeled Elsewhere on the source's diagram. Nothing there can affect the event, or be affected by it, if no signal outruns light. Every observer draws the identical cone for a given event, so all of them agree on this division.

  They do not all agree on order, however. Events inside each other's cones keep the same before-and-after sequence for every observer. Spacelike-separated events can swap order depending on how the observer moves. Two events that are simultaneous in one frame must be spacelike separated, which means the relativity of simultaneity is built into the geometry. A spacelike interval gives proper distance, while a timelike interval gives proper time, the cumulative ticking of a clock traveling along its world line.

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• 6. Moving Clocks, Shrinking Rods

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  Consider a clock whose own ticking adds up to 5 meters of proper time. A clock at rest relative to the observer has a vertical world line, and the observer measures the same 5 meters. For a clock traveling at 0.3 c, the observer measures 5.24 meters. At 0.7 c, the observer measures 7.00 meters. Faster clocks take longer, in the observer's frame, to tick off the same proper time, and this is time dilation.

  Length shrinks by a related rule. Picture a 1 meter rod traveling at 0.5 c along the x axis. Measuring its length means marking both endpoints at the same moment. But events simultaneous in the rod's frame are not simultaneous in the observer's frame, so the measured length comes out foreshortened. Length contraction, like time dilation, is a consequence of the relativity of simultaneity.

  The effect is mutual. If observer O measures the clocks of O-prime as running slow, then O-prime measures the clocks of O as running slow. Each also finds the other's rulers contracted. The two frames are rotated relative to each other by a hyperbolic angle, which is the analog of an ordinary angle in Euclidean geometry. Because of that rotation, a meter stick from either frame, projected onto the other frame's axis, appears shortened.

  Beginners often hear mutual time dilation as nonsense, since two clocks cannot each run slower than the other. The source notes that this question goes to the heart of understanding special relativity. The resolution is that the two measurements use different equipment. To time a moving clock, an observer needs two synchronized clocks at rest, a fixed distance d apart, plus the moving clock itself. That makes three clocks. The reverse comparison requires a different set of three, so nothing forces the two results to match.

  The word measure carries a lot of weight in that explanation. In classical physics an observer cannot affect the object observed, though the object's motion can still shape what the observer sees. That reciprocity, in which each party sees the other as slower, is exactly what sets the trap in the most famous puzzle of the subject.

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• 7. The Twin Question

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  One twin boards a high-speed rocket, travels into space and returns to find that the twin who stayed home has aged more. This is the twin paradox. It seems puzzling because each twin sees the other as the one in motion, so each should find the other younger.

  The source insists it is not a true paradox. Introductions to special relativity often present a series of paradoxes, but these are ill-posed problems that come from our unfamiliarity with speeds near that of light. The remedy is to solve many such problems, and the geometrical approach is considered one of the best ways to build modern intuition.

  Here the asymmetry is real. Special relativity does not declare all frames equivalent, only inertial ones, and the traveling twin's frame is not inertial while she accelerates. The difference between the twins can also be detected: she must fire her rockets to come home, and her sibling does not. On a spacetime diagram of a twin who goes straight out and immediately turns back, the stay-at-home twin's proper time from start to finish exceeds the sum of the traveler's two legs.

  The source names the twins Terence, who stays home, and Stella, who travels, using Weiss's nomenclature. From Stella's side the analysis is harder. On both the outbound and inbound legs she measures Terence's clocks as running slower than hers. During the turnaround, however, the angle of her lines of simultaneity shifts, and she skips over a stretch of events on Terence's world line. By the end, she concludes that Terence has aged more.

  It is sometimes claimed that this puzzle requires general relativity to resolve. The source calls that incorrect, since a pure special relativity analysis works. Still, the equivalence principle adds insight. When Stella fires her rockets, she feels a force as if she were in a gravitational field, with Terence high up in it. Gravitational time dilation makes his clock appear to run fast, enough that he ends up older. Any theory of gravity that respects the equivalence principle predicts that effect, including Newton's.

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• 8. Adding Speeds

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  A train moving at 0.4 c passes a platform, and a passenger fires a bullet at 0.4 c in the train's frame. Someone standing by the tracks measures the bullet at 0.8 c. That matches naive expectation and is the Galilean law of adding velocities. Relativity replaces it with Einstein's formula: the combined speed equals the sum of the two speeds divided by 1 plus their product over c squared.

  The formula has two key features. For speeds tiny compared with light, the correction term nearly vanishes and Newton's rule returns, so the Galilean formula is a low-speed special case. And if one of the speeds is c, the result is c, whatever the other speed is. Light moves at the same speed for everyone, regardless of the motion of its source.

  One expression keeps recurring in the algebra. One divided by the square root of 1 minus the squared ratio of speed to c is called the Lorentz factor, written with the Greek letter gamma. For any speed above zero it exceeds 1, though at low speeds it sits extremely close to 1. If the speed reaches or exceeds c, the expression becomes physically meaningless, which implies c is the maximum possible speed in nature.

  The Lorentz factor sits inside the Lorentz transformations, which convert an event's coordinates from one frame to another. A worked example sends Stella racing from Earth to Mars, with Terence as an official at the starting line. The distance is 300 light-seconds. Her ship accelerates instantly to 0.5 c. Stella calculates the distance between the starting and finish lines, in her own frame, as 259.81 light-seconds.

  The transformations are linear, so two applied in sequence give another Lorentz transformation. In the source's example, Terence sees Stella recede at 0.500 c, and Stella sees Ursula recede at 0.250 c. Terence can relate Ursula's measurements directly to his own.

  Light also shifts frequency when source and receiver move relative to each other. Sound needs a medium, so its Doppler shift depends on whether the source or the listener moves. Light has no medium, so no such distinction exists. The transverse case is stranger. When source and receiver are at closest approach, the receiver can see light blueshifted by a factor of gamma in one setup and redshifted by that factor in another. The source calls the transverse Doppler shift a relativistic effect with no classical analog.

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