Newtonian Assumptions about Time and Space
(© Robert DiSalle 2001)
 

Newton's Laws of Motion (from The Mathematical Principles of Natural Philosophy, 1687)

1. Every body, left to itself, maintains its state of uniform motion or rest until acted upon by a force.

2. Acceleration is in the direction in which a forced is impressed, and is proportional to the magnitude of the force and the mass of the body.

3. To every action there is an equal and opposite reaction.
 

It might seem as if these statements presuppose that you know what mass and force are, and then tell you how they relate to one another. But it is impossible to say what the Newtonian notions of force and mass without actually stating the laws of motion. Therefore the three laws together define force and mass, by telling you how they are to be measured. The measure of mass is the measure of inertia, i.e. of a body's degree of resistance to being disturbed from its state of uniform motion.
 

We can see from this why mass is distinct from weight. Mass is the measure of a body's resistance to acceleration by any force whatever. Weight is specifically the acceleration of a mass by gravity. So the mass of the body is an absolute quantity: the same force will always produce the same acceleration on the same mass. But the weight of a body depends on the strength of the gravitational attraction acting on the body. Therefore if we weigh an object on the earth and the moon, we are measuring the same mass being acted upon by two different forces. The same mass is easier to lift off the surface of the moon than off the surface of the earth, and so it weighs more on the earth.
 

Recall that force, mass, and acceleration are the fundamental "absolute" or "invariant" quantities of Newtonian mechanics--the physical quantities that can be measured in mechanical experiments, according to Newton's laws, and that have the same magnitudes for all observers who are moving uniformly . Since we are using acceleration to measure mass and force, we are presupposing that we know how to measure acceleration; since acceleration is change of velocity over time, and velocity is distance travelled per unit of time, we are also presupposing that we know how to measure distance and time.

1. Time. Measurement of time usually invokes some repetitive process supposed to mark "equal intervals of time". Any natural process that is repetitive is, in some sense, a "natural clock," but some clocks seem to be better than others; for example, my heartbeat is not as good a clock as the rotation of the earth, and someone designing a clock would want it to be more like the earth's rotation than like my heartbeat. But why? In principle it would seem that we can only judge a clock by another clock, and the other by a third, and so on to infinity-in other words, it seems meaningless to say of a clock that the intervals it measures are "really" equal. Can I say that my watch is "really" slow? Perhaps I can only say, "it is slow relative to some other clock that I have chosen as a standard." In that case, "equality of time intervals" would be a matter of convention.

But Newton defined "absolute time" as time that "flows equably without regard to anything external"-he thought that there is a notion of "equality of time intervals" that is meaningful without regard to any particular clock. This notion is actually defined by the laws of motion: Equal intervals of time are those in which a body not subject to forces moves equal distances.

 

It follows from the laws of motion that a sphere set in rotation will continue rotating at the same rate as long as it is not disturbed by friction or other external forces ("conservation of angular momentum"); so we have another definition: Equal intervals of time are those in which a freely rotating sphere turns through equal angles. The earth is a better clock than my heartbeat because it is a better approximation to the "ideal" clocks defined by the laws of motion. And since the earth is known to be subject to forces that slow it down, an atomic oscillation, isolated in a laboratory, is (in a sense) a better approximation to a freely-rotating sphere than the earth is (according to the laws of atomic physics). So absolute time doesn't depend on any particular clock, but it does depend on what the laws of physics tell us about what fundamental processes are uniform. If we don't assume some physical laws, it's not clear how we could define absolute time.

Yet an ambiguity remains. Suppose I compare the rotation of the earth with that of some other sphere, which appears to be rotating non-uniformly relative to the earth. Which one should I use to measure time, that is, which one is really turning through equal angles in equal times? Obviously the one that is not subject to external forces. But which one is it? Obviously the one that is turning through equal angles in equal times...and so on. (This is like the situation of comparing the motion of a buzzing fly to that of a freely-moving particle.) If I can pick any motion and say that it is uniform, what content does the definition of time really have?

Suppose I have two uniform motions (say, a rotating sphere and a freely-moving particle). There is no way to refute the claim that one of them is or is not moving uniformly. But I can say of the two of them, that their motions are uniform and mutually proportional: in intervals of time in which the sphere rotates through equal angles, the particle moves equal distances. Or of two particles with different velocities: in intervals of time in which one moves a given distance d, the other moves a proportional distance d' = kd (where k is some constant factor; i.e. d'/d = k). 
 
 

The empirical content of the definition of equal times turns out to be: Of all "natural clocks" that we can observe, or artificial clocks that we can construct, there is a special subset, all of which mark approximately proportional intervals of time, i.e., each moves equal distances (or, more generally, performs approximately equal physical operations, as in the case of atomic oscillations) in time intervals in which the others move approximately equal distances. And the more closely clocks approximate the ideal clocks defined by the laws of motion, the more exactly proportional their time intervals will be.
 
 

2. Space. Space was always thought to be described by Euclidean geometry, the "science of space," whose postulates are:

1. That a unique line can be drawn connecting any two points;

2. That a line can be extended arbitrarily far in either direction;

3. That a circle can be drawn around a given point with any radius;

4. That all right angles are congruent;

5. That, if two lines L₁ and L₂ both pass through a third line L₃, L₁ and L₂ will intersect only on that side of L₃ where their internal angles with L₃ are less than right angles. (Or: Given a line L and a point P not on L, there is only line through P that does not intersect L.)
 

Postulate 5 is known as the parallel postulate.

 

Obviously these postulates enable us to prove arbitrarily many propositions about plane figures, such as the Pythagorean theorem, the theorem that the internal angles of a triangle sum to two right angles, etc. But are the postulates true? It is tempting to say that Postulate 5 is true for the Euclidean plane, but false on a sphere: the "straightest lines" of a sphere are great circles (circles that cut the sphere in halves), so that two straight lines can both be perpendicular to a third straight line, and yet intersect at one of the poles. (Alternatively, take one straight line L on a sphere and a point P not on L: there is no straight line through P that does not intersect L at the poles.)
 
 

But Postulate 5 is not an empirical claim about the Euclidean plane; rather (analogously to other cases we have seen), it is part of the definition of the Euclidean plane. In saying that these axioms are "true," we must be thinking that they are true of "real space". For example, two real straight lines in space that are perpendicular to a third line will never meet-even if two straight lines on a sphere might do so. And we can see that the straight lines of the sphere are not "really" straight: the latter follow the curvature of the earth, whereas I can see in a straight line tangent to the horizon (which is why objects following the curvature of the earth disappear from sight over the horizon). In saying this, however, we are presupposing that light travels in a straight line. That is, we can first apply the Euclidean concept of "straight line" to the world (as opposed to the spherical concept, or some other one) when we stipulate that geometrical straight lines correspond to physical paths of light rays. Only if we say what real objects correspond to straight lines can we ask the question, "which of these geometries is true of real space?"

A similar point can be made about the measurement of distance. Measurement is based on coincidence: the claim that two objects are the same size means that they can be made to coincide. So two sticks S₁ and S₂ have the same length if their ends can be made to coincide. But if I take S₁ to another place and compare it to a third, S₃, I assume that I can now compare S₂ with S₃. And in assuming this, I am assuming that S₁ did not change its dimensions in moving from S₂ to S₃. In other words, measurement of congruence depends on the assumption that there are rigid bodies, i.e. bodies that can be displaced through space without change of form. To summarize: The measurement of space, like the measurement of time, depends on physical assumptions. Geometry is not a description of space until we specify which physical objects or processes correspond to geometrical concepts.

3. Simultaneity. When we say that congruence depends on coincidence, we are omitting a crucial condition. Of course the two ends of a metre stick and the two ends of the Leaning Tower of Pisa can be made to coincide. We won't therefore admit that the tower is one metre high, because we demand that the two pairs of ends coincide at the same time.

    This seems utterly obvious. Yet it reveals that a further assumption underlies our idea of spatial measurement: measurement of spatial congruence is possible only if it is possible to determine objectively which events happen simultaneously; if two observers can't agree on which events are simultaneous, they can't agree on which lengths are the same. Moreover, if they can't agree on lengths, they can't agree on time intervals. And if they can't agree on time intervals, they can't agree on the invariant quantities of Newtonian physics-for, as we have seen, force and mass are defined by acceleration, which is defined by length and time intervals. In sum: all of Newtonian mechanics depends on the assumption of absolute simultaneity.

By itself this doesn't pose a problem. Usually we use light signals, or other electromagnetic transmissions (radio waves, etc) as a criterion of simultaneity; if we see things at the same time, we think of them as having happened at the same time. We know that light travels at a finite velocity, so that what we see as simultaneous need not have been really simultaneous, unless the events occur at equal distances from us (so that the light reaches us in precisely the same amount of time). But the velocity of light is so great, compared to the pace of everyday life, that these differences are immeasurably small. A precise determination of simultaneity would require something that propagates infinitely fast, moving through space without taking any time. And Newton's law of gravitation proposes such a thing: gravity acts immediately at a distance between one mass and another. (The force depends only on the masses and the distance, so if the distribution of mass changes, the force must change instantaneously.) But gravity has always been useless as a signal: the objects that would be large enough to create a detectable signal are too large for humans to manipulate. Newtonian mechanics implies that we could, in principle, create an infinitely fast signal; since the acceleration produced by a force is independent of the initial velocity, there is in principle no limit to how much an object could be accelerated. All of this means that, in Newtonian physics, infinitely fast signals are possible in principle, and therefore absolute simultaneity can be determined in principle. But for practical purposes, electromagnetic signals provide the only useful criterion of simultaneity. As we will see, Einstein raised the question whether the practical problem of determining absolute simultaneity might be a problem in principle, and whether the light-signaling method of determining simultaneity might be the only one there is.
 

The Newtonian theory of relativity.
 

According to the laws of motion, force, mass, and acceleration are the "invariant quantities" of classical mechanics. This means that we can allow any change of "frame of reference," or any re-orientation of the system of coordinates in which we measure-in general, any re-description of the physical situation we are interested in-as long as it leaves the magnitudes of force, mass, and acceleration unchanged. This is just another way of saying that, in Newtonian mechanics, position and velocity are "relative," while acceleration and rotation are "absolute": switching from one frame, or coordinate system, to another that is moving uniformly relative to the first is a kind of "transformation" that will obviously change position and velocity, but leave force, mass, and acceleration unchanged. Simply, a quantity is "invariant" according to some theory if that theory provides an unambiguous way of measuring that quantity; it is "relative" if it is changed by a transformation that leaves the "invariants" unchanged. Every relativity principle in physics is based on an invariance principle; i.e., we can say that some quantities are relative just because physical laws tell us which quantities are invariant, and thus they tell us what changes in coordinates, or frame of reference, are possible without changing the physical quantities in which we are interested.

Thus, while it is common to distinguish "Newtonian physics" from "relativity," it is more precise to distinguish "Newtonian relativity" (also called "Galilean relativity") from "special relativity." Both theories claim that uniform motion is indisinguishable from rest, because the fundamental physical quantities will be the same in all uniformly-moving frames of reference, no matter what their velocities relative to one another might be. The theories disagree on what the fundamental physical quantities are. For Newton, they are force, mass and acceleration- and whatever is presupposed by these, namely, time-measurement, distance-measurement, and simultaneity. For Einstein, the fundamental quantity is the speed of light. Why that is such a bizarre idea, how Einstein arrived at it, and why it is after all quite natural, we will see later.

Newtonian relativity is normally called "Galilean relativity" because it was Galileo who first made precise the idea of physically equivalent frames of reference, in his explanation of how it is that the earth could be moving without our noticing it. The rotation of the earth is so close to being a uniform motion that our own frame of reference, as we stand on the earth and perform ordinary experiments (e.g. throwing projectiles in the air, or dropping objects to the earth), is very close to being a uniformly-moving frame of reference. The fact that we don't notice this uniform motion is explained by the principle of inertia: since bodies naturally persist in uniform motion, the stone that we that we drop from a tower fall vertically, because its vertical falling motion is simply combined with its uniform motion in a straight line. Defenders of the idea that the earth is at rest argued that if the earth were moving, the stone would be left behind by the tower as it falls to the ground; Galileo showed that, because the stone preserves the uniform forward motion of the tower, it keeps up with the tower while simultaneously falling to the earth. Therefore a stone falling from a tower to the rotating earth behaves like a stone falling from the mast to the deck of a moving ship.

Newton was the first to make Galileo's idea precise, and to show its precise relation to the laws of motion; in fact, he deduced the Galilean principle of relativity from the laws of motion as Corollary V: "The motions of the bodies in a given space are the same among themselves whether that space is at rest or moving uniformly in a straight line." That is, because force is determined by acceleration, no experiment can measure the velocity of the system in which it takes place. Acceleration is "absolute," but velocity is relative.