©Robert DiSalle 2000
1. Newton's definitions. Recall the classical distinction between analytic and synthetic propositions: an analytic proposition explains what is already contained in a given concept, while a synthetic proposition joins some new predicate to what a given concept already contains. Ordinarily a definition is thought to be analytic, since it only explains the meaning of a term. (E.g. "gravity is the tendency of bodies to fall toward the centre of the earth.") A law, in contrast, is thought to express a universal relation among several already-defined terms, and therefore to be synthetic as well as universal. (E.g., knowing what force, mass, and distance are, we learn from the law of universal gravitation that "the gravitational force between two masses is proportional to the product of the masses divided by the square of the distance between them.")
Against the background of this simple scheme, Newton's definitions have some interesting features. Each one is the definition of some quantity, and so it must explain how that quantity can be measured. E.g., "the quantity of matter is the measure of the same, arising from its density and bulk conjointly." This seems intuitively fairly obvious: the quantity of matter (i.e.mass) is "how much stuff" there is in a given volume. In the background of this definition, undoubtedly, is the Cartesian notion that matter is reducible to extension (cf. Meditation III, Principles II); Newton is making it plain from the beginning that identical extensions can differ in a fundamental physical property. We could think of this as a property determinable by counting: a given volume contains some definite number n of bits of matter, and a massive body differs from a less massive one by containing more units of "stuff". In practice, however, such counting isn't possible, and so we usually rely on weight to make the distinction. This is one reason why the distinction between mass and weight was not clear before Newton articulated it; apparently it was too obvious that weight provided a simple means of distinguishing among equally extended bodies, one that sufficed for most practical purposes. The fact that some bodies require more effort to lift seems to give a reasonably good measure of how much stuff there is in a given volume, e.g. of how much grain there is in a vessel.
At the same time, we can see that in many spheres of human activity, a rough distinction between weight and mass is implicit in our pre-systematic thought and practice. One need only consider the invention of the wheel: this reflects the implicit recognition that resistance to horizontal motion is somehow different from, and easier to overcome than, weight-that is, that one can practically separate the weight from the resistance one is trying to overcome, even if one can't quite conceptually separate the two. Of course, the fact that one is also removing a source of friction complicates the issue, and in general it would be a mistake to think that whoever invented the wheel ought to have isolated the concept of mass. Still, the fact that many practical problems are made easier to solve by some degree of isolation of resistance from weight- that we can move many bodies horizontally that we could never lift- suggests that Newton's conception of mass is not entirely remote from common sense. Newton's conception merely takes this separability and raises it to the status of a definition: mass is precisely the resistance to motion that bodies exhibit, even when we eliminate (to the extent possible) the effects of gravity. Understanding mass in this way gives us a more precise account of weight, as the effect of a given mass subjected to a given gravitational pull, and makes the proportionality of weight and mass perspicuous. For mass is the invariant quantity underlying weight- a body will have the same resistance to horizontal acceleration on the moon or the earth- while weight necessarily depends on the strength of the local gravitational field, so that the weight of a given mass will be different on the earth, at a great height above the earth, and on the moon. It follows from all of this that weighing an object and measuring its resistance to acceleration- its inertia- are two different things, even though in ordinary circumstances the difference might not seem to matter.
From the foregoing we can see, however, that the definition of mass already involves an appeal to the laws of motion: there is no way to measure mass except by assuming that it is proportional to force divided by acceleration, and then measuring how much acceleration is produced by a given force. Without knowing Newton's laws-i.e., without knowing that inertia is the tendency to resist acceleration (rather than, say, velocity)-we can't explain how to measure inertia by the amount of resistance to a given force. Therefore the laws themselves can't be said to express relations among already-defined quantities, because, as we've just seen, they themselves provide the definitions of mass and force. (The laws take for granted that acceleration can be independently defined; if we can measure distance and time, then in principle we can measure the rate of change of the rate of change of distance, i.e. acceleration.) In other words, we may have an "intuitive" notion of mass as "the measure of how hard it is to shove something out of the way," or "how much stuff is in a given volume," and an intuitive notion of force as "the effort it takes to shove something out of the way." But we don't have a precise concept of either one until we state that it is, specifically, acceleration that requires a force as its cause, and that mass is the measure of resistance to acceleration. That is, the laws of motion tell us what mass and force are by telling us how they are related to each other and to acceleration.
In the second half of the 19th century, it was not uncommon to criticise Newton's presentation because of this interdependence of the laws and the definitions; it seemed to be a flaw in the logical structure of the Principia. In the late 19th century, Henri Poincaré pointed out that this interdependence is characteristic of the fundamental principles of the sciences. In both mathematics (especially geometry) and physics, the fundamental axioms speak of concepts of which we have no clear definition independently of the axioms themselves. For example, we don't have a clear concept of "point," "line," "distance," etc. except through the axioms of geometry; therefore, the axioms do not relate already-defined concepts, but provide "implicit definitions" of those concepts. By the same token, the laws of physics are "implicit definitions" of force and inertia.
This seems to cast doubt on whether Newton's laws are really empirical
laws. If they are really definitions, how can they be tested? If we discover
a force that seems not to be proportional to mass times acceleration, we
can't say that Newton was wrong, but only that this phenomenon fails to
satisfy the definition of a force. To solve this problem, we have to understand
that the purpose of these definitions is to make certain empirical claims
possible, and to determine what are the empirical claims that can
be made with the help of the definitions.
2. Force, mass, and cause. By defining force and mass by acceleration, Newton's laws introduce a conception of cause: all, and only, accelerations require a causal explanation; any true cause manifests itself by producing accelerations. Therefore a causal account of the motions in the heavens requires measuring every acceleration, and determining its cause through the laws of motion; two bodies are in a causal relation if they interact according to those laws, i.e., if each determines an acceleration in the other, and their actions on each other are equal and opposite. For example, Jupiter can be recognized as causing the anomalous acceleration of Saturn, just in case that acceleration is proportional to the force exerted by Jupiter, and Saturn exerts an equal and opposite force. So we can see what empirical claim is being made by the laws: for every acceleration of every body in a system of bodies, some other body can be determined to be exerting the force that produces that acceleration. In this way we can understand the entire system as joined together in a set of causal relations.
The "mechanical philosophy," as represented by Descartes, Boyle, Huygens,
and Leibniz (though Leibniz felt that it was incomplete as a metaphysics)
implicitly assumed that causal agents produce accelerations. But they also
made the further assumption that every genuine causal explanation requires
the action by immediate contact of one body on another, so that
"action-at-a-distance" is impossible. Thus Newton's conception of his laws
is quite radical: even an apparent "action-at-a-distance" (such as that
between Jupiter and Saturn) can be regarded as a genuine cause as long
as it satisfies the laws of motion; in other words, Newton regards the
laws of motion as both necessary and sufficient conditions of causal
interaction. Evidently this view was crucial to the discovery of universal
gravitation.
3. Absolute motion. Implicit in the Newtonian conception of force and cause is that forces can provide a criterion of motion. While Galileo had shown that uniform motion is indistinguishable from rest, he also assumed that non-uniform motion would reveal itself, and the clearest example of this is centrifugal force: in a rotating system, bodies will appear to be accelerated away from the axis of rotation. This is a pseudo-force, because it is not really caused by an applied force; if it were, we could identify the cause by finding which body experiences an equal and opposite reaction. (E.g., if my car swerves to the right, I feel an apparent acceleration to the left, but this is not caused by an impressed force; in reality, my body is continuing to move in a straight line, while the car swerves. So the true motion belongs to the "frame of reference" [the car], not to me.) All the important 17th-century physicists assumed some version of this principle; so they assumed that the orbits of the planets around the sun must involve some balance between the planet's natural tendency to continue in a straight line (the source of the "centrifugal force") and some agent impelling them toward the sun.
Descartes, however, though he shared these assumptions about centrifugal force, also introduced a definition of "motion in the philosophical sense," which meant "motion of a body relative to those bodies immediately contiguous to it." Newton's "Scholium to the Definitions" therefore attempts to show that this definition is inconsistent with, and inferior to, the definition of motion provided by centrifugal forces. If we consider the rotation of water in a bucket, we see immediately that its motion in the Cartesian sense (i.e. relative to the sides of the bucket) does not correspond to the characteristic dynamical effect of rotation, centrifugal force (revealed by the water's climbing the sides of the bucket). So the absolute rotation of the water "may be measured by this endeavour" to recede from the axis of rotation. Of course these are two competing definitions, and so one can't decide between them on the basis of an empirical test. Nonetheless, as Newton argues, we can see that his criterion of motion corresponds directly to the physical conception of cause, because it is correlated with physical effects (the dynamical effects of rotation); while the Cartesian criterion has no relation to such causes and effects. Thus, the physical principles that provide the definition of causal interaction also provide a definition of the dynamical difference between rotation and non-rotation.
Newton's view remained controversial, however, because he also implied
that there is a real difference between absolute motion (change of absolute
place) and rest in space, even though he knew that he was unable to define
a dynamical difference (cf. Corollary 5 to the Laws of Motion). No one
could understand how it could be possible to determine physically the velocity
of rotation of a body, but impossible to determine the velocity of translation
through space. So philosophers tended to assume either that both are possible,
or that both are impossible. Only in the late 19th century was
this problem resolved, when physicists replaced absolute space with the
concept of "inertial frames." Instead of assuming that there is a single,
resting "space" relative to which every body's true velocity can be (in
principle) measured, we recognize that there is an infinity of equivalent
"spaces" (inertial frames) that may have any uniform velocity relative
to each other; if we measure the rotation or acceleration of a body in
one of those frames by applying Newton's laws, we know that these quantities
will be the same in every other such frame, though the velocity may differ.
E.g., if we measure the accelerations and rotations of all the bodies in
the solar system, we know that these will be the same in any other frame
in which the solar system is moving with any uniform velocity.
4. Relativity and invariance. The foregoing discussion provides a classic example of invariance. 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 a set of laws if those laws provide 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.
A more basic example is the invariance of distance in ordinary Euclidean
geometry. I can draw coordinates on the floor of a room in order to determine
precisely the distance between one chair and another; I assume that if
I draw the coordinate axes from a different origin, or in different directions,
I won't really change the distance that I measure. Any re-drawing of the
axes will be allowable by Euclidean geometry as long as it leaves the distance,
which is defined by the Pythagorean theorem, unchanged. So in this case
we can say, we know that position and orientation are relative because
we assume that distance is invariant, or, any change in position or orientation
is allowable as long as it leaves distance unchanged. If we didn't care
about distance, but only about angles, we might assume that a geometrical
figure (say, a triangle) and a scaled (up or down) version of it are equivalent.
In that case we would allow a larger class of changes, not only of position
and orientation but also of scale, and we would say that shape is invariant,
but not size.
5. Space-time. A wide variety of geometrical spaces
can be characterized by what particular quantities are invariant in those
spaces, just as Euclidean space is defined by the invariance of the Pythagorean
theorem. To represent the invariants of Newtonian physics geometrically,
we first have to consider that its geometrical invariant is acceleration,
a quantity that involves space and time together (the rate of change of
the rate of change of distance). Therefore the space characterized by such
an invariant would have to be a four-dimensional space, combining space
with time. As we will see, the difference between Newtonian physics and
special relativity can be understood as the difference between two models
of space-time.
Newtonian Assumptions about Time and Space
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.)
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 Rainbow Bridge can be made to coincide.
But we won't therefore admit that the bridge is one metre long, 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 electromagnetic
criterion of simultaneity might be the only one there is.
Spacetime.
1. The four-dimensional world. The concept of spacetime
arises first in physics, because physics has always been focussed on how
spatial configurations evolve through time. Thus the "law of inertia,"
that bodies not subject to forces travel equal spatial intervals in equal
time-intervals, makes a statement about the spatio-temporal behaviour
of bodies. But spacetime can be considered in a more familiar way, as arising
from the fact that "no one has ever observed a place except at a time."
We locate an object in space by its three spatial coordinates (usually
measured relative to the earth), or in time by its time coordinate. But
to locate an event, we simply need four coordinates: we locate it
by where and when it happened. If space is the set of all possible locations
(each represented by three coordinates), and time is the set of all possible
moments (each represented by a single time coordinate), spacetime is nothing
but the set of all possible locations of events in the history of the world:
each happens at a given time and place. If every object occupies a point
(or set of points) in three dimensional space, every event occupies a point
in four-dimensional spacetime.
2. Motion in spacetime. In spacetime, we are concerned
with how a particle moves through time: that is, we are comparing
its spatial coordinates with its temporal coordinate. The path of a particle
in spacetime, therefore, represents its "history" (and is therefore called
a "worldline"). What does the worldline of a particle look like? If a particle
isn't moving at all, its spatial coordinates don't change, but obviously
the temporal coordinate must change as time goes by; so the worldline of
a particle at rest must be a straight line parallel to the time axis. If
a particle is moving uniformly, its worldline will still be a straight
line, but no longer parallel to the time axis. This is because it crosses
equal intervals of space in equal intervals of time, i.e., its motion along
the spatial axes is always proportional to its motion along the time axis,
but because it is moving, it must uniformly increase its distance from
a particle at rest. And if two particles are moving uniformly, but at the
same velocity, their distance from one another won't change over time,
and so their worldlines will be parallel straight lines. But if a particle
is moving non-uniformly (accelerating), even if its direction in space
doesn't change, it cannot move in a straight line of spacetime.
And a if a stick is rotating around its midpoint, the two ends describe
a "helical" motion in spacetime. By these considerations we arrive at the
spatio-temporal version of Newton's first law: Particles not subject to
forces move in straight lines of spacetime.
Relativity of motion. We have said that a particle that
is at rest must not change its spatial coordinates over time, so that its
worldline remains parallel to the time axis. But we already know from Newton's
laws that we can't distinguish rest from uniform motion. That means that
we can't tell which straight worldline is parallel to the time axis after
all. It follows that we can represent any straight worldline as
parallel to the time axis. All such choices of how to draw the diagram
are equivalent, since the accelerations will be the same no matter which
particle we take to be at rest. The following two diagrams therefore represent
the same physical situation.
3. Newtonian time. We can already see how free particles
exemplify equal intervals of time. But we also said that the Newtonian
picture presupposes absolute simultaneity. This means that there is an
objective way to divide spacetime into three-dimensional "slices," each
of which represents "all of space at a given time" (and is therefore called
a "simultaneity slice," as in the above diagram). On a slice it is possible
to determine the lengths of bodies and the distances between them (since,
as we have seen, when we can determine simultaneity we can objectively
determine distance. Clearly if a particle could move infinitely fast, i.e.
pass through space without taking time, its worldline would lie entirely
in a simultaneity slice. Note that observers in different states of uniform
motion will agree on which events are simultaneous and therefore on spatial
distances measured at a given moment. Note also that a simultaneity slice
separates the future from the past, and thus defines the "causal structure"
of Newtonian spacetime, i.e. the limits of possible causal influence. An
event on a given simultaneity slice is, in principle, capable of having
a causal influence on any other event on the same slice, if the influence
can be propagated infinitely fast; if the influence travels at any finite
speed, it can (in principle) reach any events that lie in the future of
that slice; but no causal influence can reach events before that slice.
By the same token, no event in the future of a given slice can influence
an event on that slice.
This structure elaborates upon a view already articulated by Aristotle, that the future is distinct from the past because, at a given "present moment," statements about the past already have a truth-value (nothing can be done about them now), while statements about the future do not. As we will see, the Special Theory of Relativity denies that this distinction can be made objectively in a way that observers in relative motion can agree upon. Instead, it holds that which events one regards as simultaneous will depend on one's state of motion-that is, that simultaneity is relative.