Time in Physics |

No one conception of time emerges from a study of physics. One’s understanding of physical time changes as science itself changes, either through the development of new theories or through new interpretations of a theory.

Each of these changes and resulting theories of time has been the subject of philosophical scrutiny, so there are many philosophical controversies internal to particular physical theories. For instance, the move to special relativity gave rise to debates about the nature of simultaneity within the theory itself, such as whether simultaneity is conventional.

Nevertheless, there are some philosophical puzzles that appear at every stage of the development of physics. Perhaps most generally, there is the perennial question, Is there a “gap” between the conception of time as found in physics and the conception of time as found in philosophy?

concept of time |

One can understand all of these changes and controversies as debates over what properties should be attributed to time. The history of the

**concept of time**in physics can then be understood as the history of addition and subtraction of these properties, and the philosophical controversies thus understood as debates about particular additions and subtractions.

Just as one may take a set of numbers and impose structure on this set to form the real number line, one may also take the set of moments or events (which will be used interchangeably) and impose various types of structure on this set.

Each property attributed to time corresponds to the imposition of a kind of structure upon this set of events, making sense of different claims about time. Let us begin with a bare set of events and successively add structure to this set. In particular, it helps to differentiate ordering properties, topological properties, and metrical properties of time.

**Time in Special Relativity**

Time in Special Relativity |

In classical physics, material processes take place on a background arena of space and time, described above. The move from classical physics to special relativity is usually taken as a change in the background arena from classical space and time to the “spacetime” of

**Hermann Minkowski**.

This new entity, spacetime, is fundamental, and space and time only exist in a derivative fashion. On this conception, there is not one metric for time and another for space; rather, there is one spacetime metric supplying spatiotemporal distances between four-dimensional events. These spacetime distances are invariant properties of the spacetime.

Time can be decoupled from space only in an observer-dependent way; each distinct possible inertial observer (one who feels no forces) carves up spacetime into space and time in a different way. In a sense, there is no such thing as time in Minkowski spacetime, if by “time” one conceives of something fundamental.

Hermann Minkowski |

There are, however, two “times” in Minkowski spacetime that correspond to different aspects of classical time, namely, “coordinate” time and “proper” time. Let us take coordinate time first. Think of an arrow in three-dimensional Euclidean space.

One can decompose this arrow relative to an arbitrary basis {x,y,z} by measuring how far the arrow extends in the x-direction, how far in the y-direction, and how far in the z-direction, where x, y, and z are perpendicular, and the arrow’s base lies at the origin. The same arrow would decompose differently in a different basis {x',y',z'}.

As one can decompose a vector in Euclidean space along indefinitely many different bases, so too can one decompose a four-dimensional spacetime vector along many different bases in Minkowski spacetime. Mathematically, coordinate time in special relativity is just one component of an invariant spacetime four-vector, just as y’ is one component of a Euclidean spatial vector.

Euclidean case |

In the

**Euclidean case**, the value of the arrow along the first component of the decomposition varies with basis; so too in spacetime, the value of the first component—here, coordinate time—varies with frame of reference.

The second bit of residue of the classical time is the so-called proper time. The proper time is a kind of parameter associated with individual trajectories in spacetime. It is often thought of as a kind of clock tied to an object through its motion. This time is a scalar—that is, just a number—and as such is an invariant of the spacetime.

All observers will agree on the value of proper time for A as he travels from e1 to e2; all will agree on the value of proper time for B as she travels from e1 to e2; and all will agree that these values will not be the same if they take different paths. Unlike with classical time, the temporal distance in Minkowski space is not independent of spatial distance.

path-dependent |

The amount of time between any two events is

**path-dependent**: if persons A and B leave an event e1 and then meet at a later event e2, the amount of time that has elapsed for A is in general not equal to the amount of time that has elapsed for B. Spatial distances can only be completely disentangled from temporal distance in a given inertial frame of reference.

Time in classical physics plays the role of coordinate time and the role of proper time. A little reflection reveals that it can accomplish this task because in classical physics the amount of time between any two events is path-independent.

Three consequences of the shift to special relativity ought to be highlighted. First, simultaneity is not absolute in Minkowski spacetime. Simultaneity is a temporal feature, yet the temporal does not disentangle from the spatial except within an inertial reference frame.

inertial observer |

What events are simultaneous with one another is observer-dependent. Given spacelike-related events e1 and e2, inertial observer A may (rightly) say they are simultaneous whereas

**inertial observer**B, traveling at a constant velocity with respect to A, may (rightly) say e1 is earlier than e2.

In Minkowski spacetime, they do not disagree over any observer-independent fact of the matter. In terms of the earlier discussion, it can then be said that the “simultaneous with” relation partitions Minkowski spacetime, but only within a frame of reference.

Second, the temporal ordering in Minkowski spacetime is partial, not total. The only temporal ordering that all observers agree on is the ordering among “timelike” events. Timelike related events are those that are in principle connectible by any particle going slower than the speed of light in a vacuum.

high school graduation |

Think of all the events that can be reached from any given event that way. Consider the event of your elementary school graduation (e1) and the event of your

**high school graduation**(e2). Obviously sub-luminal particles could make it from one to the other; for instance, you are a set of such particles.

Due to the finite speed of light, however, there are many events that such particles could not reach—for example, whatever was going on at Alpha Centuri simultaneous with (in your reference frame) e2. What happened on Alpha Centuri simultaneous with e2 is not an observer-independent fact. But that e2 follows e1 is an observer-independent fact. Only the timelike related events are invariantly ordered.

Third, and perhaps most famously, in a sense time passes more slowly for a moving observer than for one at rest. Consider two inertial observers, A and B, traveling at a constant velocity relative to one another, and let a clock be at rest in A’s frame.

time dilation |

Looking at the ticks of the clock, the special relativistic metric entails that B will conclude that the clock in A’s frame is running slow. This effect, known as

**time dilation**, is entirely symmetrical: A would find a clock at rest in B’s frame to be running slow, too.

Time dilation has many experimentally confirmed predictions, such as that atomic clocks on planes tick slowly relative to clocks on land and that mesons have longer lifetimes than they should from the earth’s frame of reference.

**Time in General Relativity**

General relativity, unlike special relativity, treats the phenomenon of gravitation. It famously does away with Newton’s gravitational force, understanding gravitational phenomena as instead a manifestation of spacetime curvature.

Time in General Relativity |

Loosely put, the idea is that matter curves spacetime and spacetime curvature explains the gravitational aspects of matter in motion. Hence the largest conceptual difference between special and general relativity is that Minkowski spacetime is flat whereas general relativistic spacetimes may be curved in an indefinite number of ways.

Otherwise, as regards time, again there is a division between coordinate time and proper time, no privileged foliation of spacetime, only a partial temporal ordering, and the possibility of time dilation.

OrderIn terms of the previous division, curvature is a metrical property, so the primary difference between special and general relativity is that the former’s metric is merely one of the many possible metrics allowed by the latter.

spacetime metric |

General relativity places various constraints between the

**spacetime metric**, or geometry, and the distribution of matter-energy. Thinking of these constraints as the laws of general relativity, general relativity claims a variety of spacetime geometries are physically possible.

Because these different metrics allow and sometimes demand different topologies and even orderings, time may have dramatically different ordering, topological, and metrical properties depending on the spacetime model. Some consequences of this fact are especially worthy of note.

First, there are spacetimes without a single global moment. In special relativity, simultaneity was observer-dependent. Minkowski spacetime could be carved up, or foliated, into a succession of three-dimensional spaces evolving along a one-dimensional time an indefinite number of ways—a distinct foliation for every possible inertial observer.

Gödel spacetime |

Though this may also be the case in general relativity, there are spacetime models that prohibit even one foliation of spacetime into space and time. The famous

**Gödel spacetime**, named after the great logician Kurt Gödel, is an example of such a spacetime.

Due to the effects of curvature, in such spacetimes it is impossible to find even a single global always-spatial three-dimensional surface. There is no global moment of time in such spacetimes. There is no way to conceive of world history, in such a spacetime, as the successive marching of three-dimensional surfaces through time.

Second, perhaps most famously, general relativity has models that permit interesting time travel. In these models a traveler can start off at event e, and by traveling always to the local future (that is, into e’s future lightcone), eventually come back to events that are to e’s past (that is, in e’s past lightcone).

proper time |

Indeed, these models will allow one to travel back to an earlier event: an observer’s worldline may intersect e, and then after some

**proper time**has elapsed, intersect e again. These “causal loops” are called closed timelike curves.

Of the many models that allow time travel, the Gödel model is again remarkable for it allows the time traveler the fullest menu of possibilities: in the model, it is possible (given enough time and energy) to get from any event e1 to any other event e2 on the entire spacetime, including the case where e1 = e2.

Third, whether time is infinite or finite can be an observer-dependent fact. When discussing Minkowski spacetime it was noted that there are different ways to decompose spacetime into space and time; alternatively, there are generally many ways to foliate a spacetime.

nontrivial topologies |

When

**nontrivial topologies**are considered, there are spacetimes consistent with general relativity that make whether time is infinite or finite a foliation-dependent matter. That is, there are foliations of one and the same spacetime that make time finite and foliations that make time infinite. In spacetimes admitting two such foliations, the age-old question of whether time is finite or infinite would be answered with a convention.

In such a world there is no coordinate-independent fact of the matter regarding how long time persists. The universe might last an infinite amount of time according to one coordinization, or language, and a finite amount of time according to another coordinization, or language.

It seems clear that different times are ordered to some extent. Intuitively, one can give a set an order by making sense of what times are between what other times. The time the cake baked is between the time of mixing the ingredients and the time of eating the cake; eating the cake is between the baking and the feeling full, and so on.

simultaneous events |

One can therefore impose an ordering on this set of events by adding a ternary “between-ness” relation of the form: “x is between y and z” defined for some or all moments in the set.

If betweenness is defined for some but not all distinct triples of moments, then it can be said that one has a partially ordered set; if betweenness is defined for every triple of the set, then it can be said that one has a totally ordered set. Newtonian physics, as will be shown, totally orders classes of

**simultaneous events**. Relativistic physics, by contrast, will only partially order the set of all events.

Between-ness as defined above is not always sufficiently powerful to order topologically nontrivial sets. To see this, consider a circle with four members of the set on it: “1” at twelve o’clock, “2” at three o’clock, “3” at six o’clock, and “4” at nine o’clock.

natural direction |

Because the set is closed, 2 is between 1 and 3, between 3 and 4, and between 1 and 4. Consequently, the between-ness relation is blind to the difference between this layout and the same but with “3” at three o’clock and “2” at six o’clock. For such sets more machinery is needed to order the set.

An ordering does not disclose much about the set of moments, {t1, t2, t3 ...}. It does not imply whether t2 is as far from t1 as from t3. Nor does it imply a direction, whether times goes from t1 to t3 or t3 to t1. Although the baking example suggests a

**natural direction**to the set of times, an ordering is strictly independent of a direction.

Nor does the ordering specify the dimensionality of the set or most other properties one normally attributes to time. The next level of structure, topology, will help make sense of some of these attributions to time.

**Topology**

Topology |

Topological properties are those that are invariant under “smooth” transformations. Technically, these transformations are one-to-one and bicontinuous; and what they leave invariant is the so-called neighborhood structure that is given by picking out a family of open subsets closed under the operations of union and finite intersection.

Intuitively, the transformations that leave this structure unchanged correspond to operations such as stretching or shrinking, as opposed to operations such as ripping and gluing. A coffee cup and a doughnut are, topologically speaking, the same shape; if made out of an infinitely pliable rubber, one could be smoothly transformed into the other.

Being closed like a circle, having an edge, and being one-dimensional are examples of topological properties. No amount of stretching and shrinking can (for instance) make the circle into a line, make an edge disappear, or make a one-dimensional set two-dimensional.

structure of time |

Many issues in the philosophy of time are in fact questions about the topology of time: is time closed or open? discrete? branching? two-dimensional? oriented (directed)? Formally, the answers to these questions are determined by the topological

**structure of time**.

**Metric**

Once topological structure is added to the set of times, most temporal properties are determined. However, there is still a major one remaining: duration. Of the set {t1, t2, t3 ...} it is still not known whether t2 is as far from t1 as it is from t3—even after all topological properties are specified. The temporal distance between two moments is not a topological invariant, for it can be smoothly stretched or shrunk.

To capture the idea of temporal distance, a metric must be put on the topological structure. The temporal metric is a function that gives one a number, the temporal distance or duration, between any pair of times. (In relativity what is imposed instead is a spacetime metric; see below.)

metric |

In principle, an infinite number of possible metrics are mathematically possible. One might choose a metric that makes the duration between 1980 and 1990 twice the duration between 1990 and 2000. However, such a choice would make a mess of almost all of science.

It would entail, for instance, that the earth went twice as fast around the sun in the 1990s as it did in the 1980s. One would then have to adjust the rest of physics so as to be compatible with this result. As Hans Reichenbach stresses, there are simpler and more complex choices of temporal metric.

**Time in Classical Physics**

Time in classical physics is normally assumed to have the ordering, topological, and metrical structure of the real number line. That is, it is one-dimensional, continuous, infinite in both directions, and so on. The temporal metric is just the one used for the real line: between any two times, a and b, the duration is b–a.

Time in Classical Physics |

Time in classical physics does have a number of remarkable properties, of which three will be mentioned here. The first two concern the metrical properties of time, whereas the third is more a property of the dynamics than of time itself.

First, the metric of time is independent of the metric of space. This feature implies that the amount of time between any two events is path-independent: if persons A and B leave an event e1 and then meet at a later event e2, the amount of time that has elapsed for A is equal to the amount of time that has elapsed for B. The distinct spatial distances traveled by A and B are irrelevant to how much time has passed between e1 and e2 .

Second, simultaneity is absolute. Before explaining “absolute,” consider the “simultaneous with” relation. For any event e, there is a whole class of events that are simultaneous with e. Indeed, the “simultaneous with” relation is an equivalence relation in classical physics.

totally ordered |

Equivalence relations are reflexive, symmetric, and transitive; for this example, what is important is that they partition a set into disjoint subsets. Hence the “simultaneous with” relation partitions the set of all events into proper subsets, all of whose members are simultaneous with one another. It is these classes of simultaneous events, rather than the events themselves, that are

**totally ordered**.

What is interesting about this partition in classical physics is that it is unique. Classical physics states that every observer, no matter their state of motion, in principle agrees on whether any two events are simultaneous.

This observation translates into only one partition (or foliation) being the right one. In this sense simultaneity is absolute—it does not depend on one’s frame of reference but is an observer-independent fact of the Newtonian world.

over time |

Third, classical physics is time reversal invariant. Consider a sequence of particle positions

**over time**, (x1,t1), (x2,t2), (x3,t3)...(xn,tn). The fundamental classical laws of evolution are such that if this sequence is a solution of the laws, then so is the time-reversed sequence (xn,tn)...(x3,t3), (x2,t2), (x1,t1). The classical laws are invariant under the transformation of –t for t.

This is true also of arbitrarily large

**multi-particle**systems and even of classical fields. If a bull entering a china shop and subsequently breaking vases is a lawful history, then so is a bunch of scattered vase shards spontaneously jumping from the ground and forming perfect vases while a bull backs out of a china shop.

multi-particle |