Different theories use different models of time. In preparation for the next sub-section we decompose the standard notion of time, the real numbers R, into a hierarchy of substructures, cf. [Rov95, Kro85]:
(A) LetT be a set with the cardinality of the continuum,|T | =|R|.
(B) LetT be equipped with a topology T.
(C) LetT be a differentiable manifold with local chartsϕT atT∈ T such that the topology induced by open sets of R via ϕ−1T coincides with T.
(D) LetTmoreover be isomorphic to the topology of R; this implies that a (global) chart ϕ: T →R (bijective and C∞) exists.
(E) Let an Euclidean metric d on T be defined (fixing one global chart ϕ),
d(T1,T2) :=|ϕ(T1)−ϕ(T2)| (T1,T2 ∈ T).
(F) Let a linear order relation ≤ onT be defined by T1 ≤T2 :⇔ϕ(T1)≤ϕ(T2) (T1,T2 ∈ T).
(G) LetT be equipped with the field structure inherited via ϕ−1 fromR, i.e. define addition inT by
T1+T2 :=ϕ−1(ϕ(T1) +ϕ(T2)) (T1,T2 ∈ T),
let the neutral element beϕ−1(0), and similarly for multiplication:
T1·T2 :=ϕ−1(ϕ(T1)·ϕ(T2)) (T1,T2 ∈ T), with the neutral element ϕ−1(1).
Conditions (A) to (G) imply that T is isomorphic to the real numbers R with Euclidean metric. We callT∈ T aninstant (of time) ormoment of time, T the set of instants, d the duration (of time) or temporal distance or time lapse and
≤the time order. By just time we mean T together with a specification of some of the structures (A) to (G), or possibly others.
ad G:The multiplicative group structure among instants has no physical mean-ing; neither does addition of instants. What is used in the description of dynam-ical flows is the sum of durations: Let φt be a flow with “time t” on a set M, i.e. a mappingφ :R×M →M, (t, x)7→φt(x) with the flow propertiesφ0 =1M
and φs ◦φt = φt+s (s, t ∈ R). The meaning of s, t and t+s is the following:
Given any instant T0 ∈ T the expression φt means that the flow φ has to be evaluated at that instant Tt ∈ T which is uniquely defined by d∗(T0,Tt) = t, where we have introduced a signed duration by d∗(T,T0) := ±d(T,T0), with
“+” applying in case T ≤ T0 and “−” otherwise. I.e., given any fixed instant T0, metric and time order provide a bijection between instants and real num-bers. The expressiont+s signifies that instantTt+s which is uniquely defined by d∗(T0,Tt+s) = d∗(T0,Tt) +d∗(T0,Ts). The flow φ can thus be characterized in terms of instants, duration and time order, while the field structure on T has no physical meaning. Note that the mapping T0 7→ Tt (T0 ∈ T, t ∈ R) is a time translation; time translations build a group reflecting the additive group of reals5. (Similarly, the time that enters into Galilei boosts has the meaning of a duration.)
Note also that independence of the choice ofT0 is implicitly contained in the flow properties. T0 can be understood as “now” and allows an observer to separate all other instants into two sets, called “past” ({T∈ T : T≤T0, T6=T0}) and
“future” ({T∈ T : T0 ≤T, T6=T0}). Two unsolved problems are connected with this: 1) Is it possible to distinguish the “past” and the “future” on a physical basis? - This is known as the problem of the arrow of time, see /2.1/ below. 2) Can the meaning of T0 as “now” or “present” be established on a physical basis?
- Because of time translation invariance (sometimes also called homogeneity of
5The set of reals involved here is notT, but the range of the metric.
time) it is hard to single out exactly one instant (called “now”) in an objective way6. This might be possible however in a theory describing the subjectiveness of an observer on a physical basis, cf. the discussion in [Kro85, Epilogue]. A combi-nation of both problems amounts to the problem of explaining theflow of time7: There is a change of the “now” (T0 7→T00) which causes future instants to become
“nows” and past instants later on, “during the flow of time”. Related questions are why the past is determined, cannot be influenced and can be remembered, while the future is not determined, can be influenced and not remembered. These are as yet open questions.
If we omit (G), we are left with an affine line.
ad F: Given any two instants T1,T2 ∈ T we have either T1 = T2 (“T1 and T2 are simultaneous”), or T1 < T2 (:⇔ T1 ≤ T2 ∧T1 6= T2, “T1 lies in the conventional past of T2” or, equivalently, “T2 lies in the conventional future of T1”), or T2 < T1 (:⇔ T2 ≤ T1 ∧ T1 6= T2). The question of whether (F) represents some physically observable relation is known as the debate on the arrow of time, see /2.1/. In relativity weaker versions of (F) may appear, see below.
ad E: The topology in (D) is induced by d. As will be seen later, the choice of a specific metric has limited physical meaning. E.g. in relativity duration is dependent on the observer’s path in space-time (proper time), see /1.2.4/.
ad D: This allows one to speak of time intervals, i.e. sets of instants, which ϕ maps to intervals of R. And it fixes the one-dimensional character of time and as well its linearity; other possibilities would include a cyclic time (T isomorphic to the topology of S1), a many fingered time (“curves with bifurcations”), a time with endpoints (T isomorphic to a closed interval), or even more dimensional times. (See also [Wic03], where a topology generated by half-open intervals is used in order to render time itself asymmetric.)
Time structure (D) together with (F) is known as topological time [Mit89,Mit95].
ad C: This condition rules out e.g. a discrete time, which is being considered in quantum gravity /2.3/.
ad B: Some sense of neighborhood seems to be necessary for any notion of con-tinuous time.
ad A:The cardinality ofT could even be lower, cf. our treatment of simultaneity within the relational concept of time in /1.3.3/.
Generalizations of the standard notion of time can be obtained by successively dropping the more specialized items (G,F,E,...), but there is no need for this to be done in the given order. One could e.g. define a metric, linear order and preferred point directly on T without having a topological or differential structure. (The
6While the instant of the big bang in cosmological models, i.e. in certain solutions of Einstein’s equations, is certainly a preferred point, it is not with regard to Einstein’s equations.
7This is not to be confused with the concept of a dynamical flow.
chosen hierarchy will prove useful in the next section, since it is similar to the hierarchy of physical theories.)
For further discussion of mathematical time structures cf. also [Pim95].