by Scott Soames
Nevertheless, one can trace relative spatial relations between distinct objects that persist through time. Changes in those relations can be visualized by imagining the slabs stacked vertically with lines connecting the occurrences of objects on lower slabs to the occurrences of those same objects on higher slabs (representing later times). The directions and distances between objects on a slab can be compared with those of the same objects on earlier and later slabs, indicating their relative movements over time. An object not subjected to external forces over a given period of time—i.e., a period in which the object, in Newton’s original system, would either be at rest or in a state of uniform motion in a straight line—is represented in the new system as following a straight-line trajectory from lower to upper slabs (i.e., no distinction is made between rest and inertial motion over time).
In this framework, Newton’s first law tells us that the trajectory (through time) of a body not acted on by external forces is a straight line. His second law says that when a force acts on a body, the trajectory of the body from lower to higher slabs is curved in the direction in which the force is applied, the amount of curvature being proportional to the amount of force applied and inversely proportional to the mass of the body. So, in a variant on the spinning bucket case, two globes connected by a cord and revolving around an axis running through the middle of the cord will remain at a constant distance from one another as they move from earlier to later simultaneity slabs, but their trajectories through space-time will be curved because of the constant force applied to them (as in the left side of the next figure). In this way, one accommodates the accelerated circular motions (rotations caused by the application of a constant force), without having to posit absolute space, thus avoiding the empirical conundrums to which it gives rise.
The next step toward the modern conception of space and time in physics was Albert Einstein’s theory of special relativity, presented in his 1905 paper “On the Electrodynamics of Moving Bodies.” The theory presented there describes a single inertial frame (where we don’t have to consider the motions of any objects other than those within a limited physical system). It targets the notion of temporal simultaneity, giving up absolute time along with absolute space, thereby modifying our understanding of simultaneity.5 One can get a sense of the change by considering how we normally establish the temporal simultaneity of two events occurring at a distance from one another. In daily life we judge two nearby events in our visual field to be simultaneous when we see them at the same time—when light emanating from one impacts our eyes at the same time as light emanating from the other. Since the distances are often short, this method works well for everyday purposes. But when we let the distances of the events from each other, and from the observer, vary, and become arbitrarily great, we need a method that takes into account the fact that the transmission of light from source to observer isn’t instantaneous.
Diagram of rotating (left) vs. stationary globes (right). From Maudlin (2012), p. 56.
The idea can be illustrated by imagining synchronized (ideal) clocks present at the sites of two events A and B located at arbitrary distances from an observer. Each clock starts the moment its paired event occurs. The clocks are then transported to the observer through different paths at different speeds. If the speed of their transmission didn’t affect their running, then an observer who knew the distance they traveled and their speeds could simply check their readings when they arrived. If one traveled twice as far but moved twice as fast, identical clock readings would register simultaneity of events.
According to relativity theory, however, the clocks’ behavior is affected by their transmission through space.6 If this sounds incoherent, it is probably because one is thinking of clocks as metaphysical know-not-what’s that, by definition, track the passage of time, which, by definition, exists independently of any physical phenomenon. But that thought is unfounded. It’s not true a priori that there must be such a thing as time conceived of in that way. Rather, the imagined clocks should be thought of as physical mechanisms, and so subject to physical laws. Because of this, it’s not obvious that their behavior will be unaffected by their movement through space. Relativity theory maintains that their behavior is affected, thereby questioning the pre-theoretic idea of simultaneity.
Suppose we try to replace this idea with a physically defined notion of simultaneity applying to events at a distance. Let us say that for events at a distance to be physically simultaneous, and so not separated in time, is (essentially) for there to be no possible causal connection (e.g., by light from one reaching the other) between them. The argument of Einstein’s 1905 paper shows that although physical simultaneity, so understood, is a symmetric relation (if x is simultaneous with y, then y is simultaneous with x), it’s not transitive (x and z may fail to be simultaneous, even though x is simultaneous with y and y is simultaneous with z).
This result is illustrated by a sequence of events—A, B, C, and D—all occurring in that temporal order at point 1 in space, and another event Δ occurring at a spatially distant point 2. Event A is the emission of a ray of light at point 1 that travels to point 2. Its arrival there is event Δ, which, since it took time for the light to make the journey, occurs later than A. The ray is instantaneously reflected back to point 1; its arrival there is event D, which occurs later than Δ. Because the transmission of light is not instantaneous, events B and C, which occur at point 1 after A but before D, can’t be connected by rays of light to the occurrence of Δ at point 2. (Since B follows A, light from B can reach point 2 only after Δ has occurred, and since C precedes D, light from Δ can’t reach point 1 at the moment prior to D at which C occurs.) So there are no physical relations capable of causally connecting event Δ at point 2 with any events occurring at point 1 after A but before D.7 This seems to suggest that events B and C at point 1, which occur after A but before D, are both physically simultaneous with Δ at point 2, even though B temporally precedes C.
But that seems impossible. Since we don’t want one event to be simultaneous with two temporally nonoverlapping events, one of which is later than the other, we need to adjust our understanding of these relations. One way to do so is to let the relations simultaneous with, before, and after be undefined for pairs one of which is Δ and the other of which is any event in the temporal interval from A to D at point 1. If we do this, then these temporal relations will be physically grounded, but only partially defined. A different way out is to choose a unique event in the range of indeterminacy at point 1 and simply stipulate that it is to count as the event at point 1 that is simultaneous with Δ at point 2. The adoption of such a rule means that the simultaneity relation embedded in the theory will be partially a matter of convention or convenience, rather than a fully objective physical relation.8 Einstein took the second option, offering a partially conventional synchronization rule for simultaneous events. This allowed him to assign a fixed numerical value to the speed of light, though different values could have been assigned had different conventions been stipulated.
Since, in special relativity, we give up absolute space and time, we must also modify our ordinary notions of motion, distance, and speed. When they are replaced with modified notions, the replacements don’t have all the properties of the originals. Instead of independent time and 3-D Euclidean space, special relativity posits a 4-D space-time continuum, made up of points represented by coordinates
One law of special relativity involved in generating these results is the experimentally validated hypothesis that the movement of light in a
vacuum is independent of the physical state of its source. In particular, light from two sources moving in opposite directions in a vacuum, each emitting light when passing one another, will arrive at any point in the universe at the same time. This would not be true of two physical objects initially moving in opposite directions, each, at the moment of passing the other, subjected to the same external force in the same direction (i.e., in the direction of movement of one of the two bodies).
Einstein’s realization that this is so was a crucial step in the development of special relativity. In “Fundamental Ideas and Methods in the Theory of Relativity, Presented in Their Development,” he says, “The phenomenon of magnetico-electric induction caused me to postulate the (special) principle of relativity.”9 According to Maxwell, a magnet at (absolute) rest is surrounded by a magnetic field, but when it moves, the magnetic field changes and an “induced” electrical field comes into being. Since the presence or absence of electric current should, in principle be detectable, the presence or absence of the current should tell us whether the magnet is moving in absolute space. However, Einstein knew that it couldn’t play this role, since whether the induced field could be detected depended on whether or not the observer was moving in absolute space in sync with the magnet—something that could not, in principle, be determined.
Einstein’s solution was to relativize space and time, thereby making the presence or absence of the electrical field an objective and observable effect of movement of the magnet relative to potential observers, each with their own space-time trajectories. But this posed the further problem of reconciling Maxwell’s electrodynamics of light, which Einstein accepted, with relativity. According to Maxwell, light consisted of waves in an electromagnetic field. How, then, should one conceptualize the velocity of light? In the Newtonian framework of the late nineteenth century, the velocity of light from a source moving in the same direction as the source should be the velocity of the source plus the constant velocity of light, which for Maxwell was 186,000 miles per second. (For light from a source moving in the opposite direction one would subtract the velocity of the source from the constant figure for light.) Call this “the emission theory of light.” When Einstein gave this up—positing that the state of motion of the light source doesn’t affect when light from a point will reach other points—he was able to incorporate Maxwell’s theory into special relativity.10
Another law of special relativity states that the path of a light ray emitted from a source in a vacuum is a straight line. One can represent this visually in two dimensions with a vertical temporal axis t, a horizontal spatial axis x, and a light-emitting event e at space-time point p. Ignoring the y and z spatial dimensions (and thinking in terms of a flat spatial plane), we may draw two lines from p at right angles to each other, each climbing vertically at a 45-degree angle (so the values of x and t always change by the same increment along the line). Everything between the lines is called e’s future light-cone.
The idea that nothing goes faster than light is given by a further law: that the path of a physical entity passing through the future light-cone of an event e never goes outside e’s light-cone (which would require the x-value to change more rapidly than the t-value). To this we may add the relativistic law of inertia: The trajectory of any physical entity (light or a body with mass) not acted on by external forces is always a straight line.11
With all this in place, one can get an idea of how time is measured in special relativity theory. Movements are represented by lines, straight or otherwise, connecting space-time points through which something—physical objects or light—passes. The points are represented by four-dimensional numerical coordinates. The interval between two points is given by applying a mathematical formula to the pair of 4-D coordinates assigned to the two points. Given this, we can illustrate the hypothesized relation between time, space, and movement postulated by special relativity.
From Maudlin (2012), p. 78.
Suppose a pair of objects A and B are at rest relative to one another at a space-time point o, which, for simplicity, we will assign the temporal coordinate 0 and each spatial coordinate x, y, and z also 0. As explained by Tim Maudlin in The Philosophy of Physics: Space and Time, A and B are identical twins each in her own rocket ship.12 B remains where she is throughout. A does not. By firing her rocket, she moves along the x spatial dimension only, arriving at point p, with temporal coordinate (t) 5, x coordinate 4, and y and z remaining at 0 (reflecting the fact that she is moving in just one of the three spatial dimensions). A then returns in the opposite x-direction, arriving at point q, where B is located, the temporal coordinate of which is 10, the x, y, and z coordinates 0. (B hasn’t moved in any spatial dimension). Visualizing this on a two-dimensional diagram, the line connecting space-time point o where A and B start out with the space-time point q where both end up (B by staying put and A by moving) is a straight vertical line.13 A’s movement from o to p is a straight diagonal line from one to the other; A’s movement from p to q is similar, resulting in an isosceles triangle.
The sides of the triangle with vertices o, p, q represent intervals through space-time. The special relativity formula for measuring them is a simple arithmetical computation on the
So special relativity tells us that A’s journey through space-time is shorter than B’s—in fact it is 3/5 of B’s. But, what, you may ask, is measured? Time is measured. The time A lived through by first moving away from B to p and the returning to B at q is less than the time B lived through by waiting for A to rejoin her.15 Thus, although A and B were (we may imagine) exactly the same age when they were together at o, and although both are older at q than they were at o, A is now younger than B. Whatever amount of time B experienced in going from point <0,0,0,0> to point <10,0,0,0>, say 100 days, A experienced 6/10 of that, or 60 days. If both had accurate clocks with them (whatever physical mechanisms might count as such), A’s would register 30 days when reaching p and 60 days when reaching q, while B’s would register 100 days while reaching q. In effect, A’s clock would tick more slowly than B’s. But it would not fail to keep correct time. According to special relativity, both clocks are correct. What the example shows is the essential interdependence, rather than independence, of space and time, and its consequences for our understanding of movement through space and time. This, special relativity tells us, is what space-time is really like, quite apart from how we are intuitively inclined to think of it.16
So far, we have simply assumed that clocks in our examples keep correct time. All that can be said at this point about what physical processes count as clocks is (i) that if they are on the same trajectory (moving neither nearer nor further from one another), they will, when synchronized, display the same time and “tick together,” (ii) that if they are moving apart (like the clocks of A and B on A’s journey away from B), each will seem to the other to be running slow, and (iii) that if they are moving toward one another (like the clocks of A and B on A’s journey back to B), each will seem to the other to be running fast (even though overall, A will have expended less time in her journey than B will have expended standing s
till).17 Ultimately, of course, these clocks have to be physical entities, which, one can demonstrate by experimental test, will perform as the theory predicts.
Although the experimental tests are difficult and complicated, there have been such demonstrations. The idea is often illustrated by imagining a pair of mirrors, between which a ray of light bounces—each round trip of the light between the mirrors counting as a “tick” of the clock.18 In order for the mechanism to count as a clock, the intervals measured by each tick must be the same. As Maudlin explains, this condition will be violated unless, as Einstein proposed, the mirrors are connected by a rigid rod as they move through space-time. Without it, special relativity predicts that changes in velocity resulting from a force acting on the two mirrors would lengthen the intervals after the force is applied, destroying the system’s ability to function as a clock accurately measuring time for the journey as a whole. According to relativity theory, the rigid rod connecting the mirrors prevents this by physically contracting as the force is applied, pulling the mirrors closer together in just the amount needed to keep the intervals traveled by the light back and forth between the mirrors the same.19 In other words, the spatial distance between the two mirrors, as measured from the perspective of the original reference frame (before the force was applied), contracts in a way that preserves the length of the intervals traveled by the light. This is the sense in which the speed of light is constant in special relativity, even though there is no absolute space or time to objectively measure speed independent of any frame of reference.
Subjecting special relativity to empirical test is complicated by the fact that the temporal intervals of round trips between mirrors in a laboratory are too short to be reliably measured by ordinary clocks. But the fact that the speed of light is constant in the way predicted by relativity theory can be tested empirically. One such test involves a light source emitted from a spatial location p in line behind a pair of rapidly spinning discs (one behind and at some distance from the other) connected by a rapidly spinning rod. Each disc has a small slit in it through which light might pass. The light source shines on the rear disc, resulting in some light passing through its slit as it spins. If the light passing through reaches the space occupied by the slit in the rotating second disc, the light will pass through it and illuminate the screen behind the apparatus; otherwise it will be blocked by the second disc. By adjusting the speed of the discs, as well as the placement and the angle of the slits, it is possible to put the system into a state in which the light from p always gets through. Relativity theory predicts that this will remain true whether the source emitting light at p is at rest, or is moving toward or away from the apparatus. This has been verified.
