Part I
Part II
8. McTaggart’s Argument against the Reality of Time
9. Reality of Time Distinguished from Discourse about Time
10. Gale’s Criticism of McTaggart: A-Theory and B-Theory
11. Philosophy of Temporal Series Applied to Special Relativity
12. Extension to General Relativity
Notes to Part II
While many philosophers, mostly idealists, have denied the reality of temporal succession, we are concerned only with whether there is anything in physics that demands this position. In recent years, more than a few physicists and philosophers of science have invoked the concepts of the idealist philosopher J. Ellis McTaggart in support of an all at once
interpretation of Minkowski spacetime. In fact, McTaggart made no appeal to relativity or to any natural science, relying on a purely abstract logical argument. Unlike many of those who invoke him, McTaggart recognized that the concepts of past, present and future are essential to time, so he thought that by refuting them he also refuted the reality of time altogether. A physicist, by contrast, cannot credibly make a complete denial of the reality of time, as this is an essential constituent of spacetime, not to mention a necessary presupposition of all measurements of motions. Further, it would be an unconscionable betrayal of scientific empiricism to hold that time is unreal because it is found only
in our observations and measurements.
In The Unreality of Time,
McTaggart offers various arguments against the objective reality of past, present and future as proofs that time itself is unreal.[1] McTaggart acknowledges, as no one can deny, that we always observe events in a temporal series, both as past, present and future, in what he calls an A series, and as earlier and later, in a B series.[2] Some might argue that only the latter series is real, while the former is an illusion of our minds. McTaggart finds this position untenable. One cannot logically have a B series without presupposing the existence of some A-series.
McTaggart offers a proof by contradiction against time being constituted solely by a B-series. For there to be time at all, there must be something that changes, but what changes if there is only a B-series of earlier and later events? An event can never cease to be an event, and its position in the B-series never changes. If N is ever earlier than O and later than M, it will always be, and has always been, earlier than O and later than M, since the relations of earlier and later are permanent.
Nor can it be said that an event merges into another event, becoming that new event, for an event never ceases to have a place as itself in the B series.
The argument that one event M cannot become or change into another event N is true enough, only because one has effectively confined events to spatiotemporal points (much as physicists are tempted to do with Minkowski diagrams, forgetting that this is only an approximation), thereby abstracting temporal process from eventhood and rendering events apparently static. This is only a formal abstraction, and it no more disproves the reality of change than does McTaggart’s remark that no moment in time can become another moment. As we shall see, McTaggart is not so much proving the unreality of time as identifying a problem with language about time when analyzed with formal logic.
Nonetheless, he does show that a B series per se is insufficient to establish the reality of change, so time cannot be constituted by a B series. Consequently, anyone who should interpret Minkowski diagrams as showing that time has only a B-series structure, and not that of past, present and future, is effectively arguing against the reality of change. Yet it is precisely the reality of measured change or motion upon which our construction of frame-dependent definitions of simultaneity depend. Without such constructions, Minkowski diagrams would be impossible, since their function is to allow us to convert between reference frames. If there is no change, it is unclear what the time variable in a given reference frame is even supposed to measure, much less what we mean by velocity and countless other rates of change. To solve one ontological problem, we would introduce countless others that would make physics, the science of sensible change or motion, without an object.
Since events do not change as such, nor do they change in their B-series position, the only thing left that may change is an event’s A-determination, i.e., whether it is past, present or future. What does it mean for a present event to become past, or a future event to become present? What sort of change is involved? Since an event cannot become what it is not, the change must be such that the events do not cease to be what they are even after the change (e.g., present becoming past). That is to say, a change in position in the A series leaves the event unchanged in itself. In fact, all so-called change can be nothing more than a change in position in A-series (i.e. going from future to present to past).
If we tried to remove A-series determinations from our notion of time, we would be left with what McTaggart calls a C series, which has only an order, but no direction. Thus the series MNOP is the same as PONM, but not MONP. McTaggart allows that events could be objectively ordered in a C series, but this would not imply any temporality. One must additionally postulate the reality of change and a direction of change to make the C series into a B series. These criteria are satisfied by an A series, so an A series and a C series suffice to constitute time. Again, the B series is less fundamental to time than an A series.
There is strong reason to believe that physical events do have an objective C-series order (though McTaggart is reluctant to commit to this), since this would enable us to parsimoniously situate events with extremely similar states of affairs (indeed nearly continuous) close to each other. It is hard to see, however, why McTaggart should allow a C series but deny time, since the only difference between a C series and a B series is that the latter specifies a distinction between forward and backward. Yet we also recognize a physical distinction between going forward and backward in time, as causes must precede effects. Or is it a bizarre accident of biology that we remember only the past and not the future? It is much more parsimonious to suppose that we remember only the past for the same reason that only the past can affect the present, or more properly that present affairs are the result of past affairs, and our mental state is just one particular present affair that is affected only by the past.
To confirm his thesis that the A series is essential to any notion of temporal change, McTaggart addresses a couple of objections, one of which is the hypothetical possibility that there is not a single A series, but many A series, each with its own past, present and future. This is much like what we have found in fact with special relativity. Note that McTaggart does not see this possibility as supporting his position, but as an objection that could potentially uphold the reality of time without an A series. As he understands it, the objection is that the time series (B series) would be real, but past, present and future would have meaning only within each series. Different nonsuccessive presents, in the view of our objector, cannot be real.
I cannot, however, regard this objection as valid. No doubt, in such a case, no present would be the present—it would only be the present of a certain aspect of the universe. But then no time would be the time—it would only be the time of a certain aspect of the universe. It would, no doubt, be a real time-series, but I do not see that the present would be less real than the time.[3]
Since the supposed unreality of the present due to its non-universality would equally apply to time, McTaggart’s assertion that the A-series is essential to time has not been contradicted. Consequently, anyone who uses relativity of simultaneity to argue against the reality of the A-series must thereby deny the reality of time. Those who make such arguments mistake the fact that there is no such thing as the time (or the present) implies that there is no time (or present) at all. That is to say, if time (or the present) is not absolute, it cannot be real, which is the opposite of a relativistic view. If time, and therefore motion, is unreal, what on earth are we measuring? Denying the reality of time would contradict both postulates of special relativity. Several of the laws of physics involve explicit time derivatives, as does the velocity of light. While special relativity means we cannot have a frame-invariant coordinate for time t, it does not deny the reality of change expressed as d/dt, but presupposes it.
McTaggart does not invoke relativity or any physics against the reality of the A-series, but instead relies on the purely formal logic used by analytic philosophers. He tries to show that an A-series, and therefore time, cannot exist, as this would involve either a contradiction or a vicious infinite regress. If the terms past, present and future are taken as qualities or relations of events, we have a contradiction. If they are relations, then only one term of that relation can be an event in the time series, for if both terms were events, we have seen that the relation between events in time is unchanging. The relations that generate the A-series must somehow relate events to something outside the time series. In any case, the A-series determinations are incompatible with each other, yet can all be predicated of the same event. The common explanation to avoid this contradiction, namely that an event is not past, present and future at once, but successively, involves a vicious circle or infinite regress.
To say that the same event is first in the future, then in the present, and later in the past, presupposes an A series. We cannot ask When does this event become past?
without identifying a time as present. That is to say, we cannot define an event’s A-determination without first specifying an A-determination. If we claim that event M is now present, has been future, and will be past, our tensed verbs indicate another A series that contains the first A series defined by our usage of past, present and future. This is a vicious regress, so it does not allow us to escape the contradiction of considering A-determinations as relations. The same problem of vicious regress if we treat the A-determinations as qualities that each event has successively. So it is unavoidable that the reality of an A series, and therefore the reality of time, involves a contradiction.
One might object that this only proves that time is a primitive concept that cannot be explained except in terms of itself. Yet McTaggart retorts that the notion of time is contradictory, because of the A series, and the attempt to remove the contradiction failed because it was a vicious circle. Thus time is rejected because it is contradictory, not because it is unexplainable. The B series of earlier and later is likewise rejected as resting on the A series. The reality of the C series is possible, but in any case is atemporal.
We may note, however, that the very notion of what constitutes a contradiction is informed in part by a notion of time. There is no contradiction in a substance having the same predication at different times. McTaggart has gotten around this by replacing substances with events, which by definition each have unique predication. (Such uniqueness is preserved only if we also stipulate the reality of the C series.) Is it any less contradictory to take slices (in his classical view) or points (in the relativistic view) of reality and call them substantive events
distinct from each other, while denying the reality of the time variable that distinguishes them?
Furthermore, despite his lip service to the possibility of time being primitive, he has effectively denied this possibility by insisting that changes in A-determination must correspond to some change in quality or relation in order to be real. This is to restrict ourselves to an object-property-relation ontology, with the further impoverishment that substances are not allowed to change, but each of their successive states is to be reified as an event-object. (More exactly, since McTaggart assumed classical physics, everything simultaneous was a single event.)
It could well be that time is a primitive ontological category, distinct from quality, substance, relation and space, much as Aristotle claimed, so any attempt to describe time in non-temporal terms is foolhardy. By making events
into objects, reality is artificially rendered static. The validity of such a formal abstraction is not in doubt, but to make such an abstraction into a description of physical reality is to assume the conclusion that time is unreal. When was the death of Socrates the present
? At the instant we call t1 It became past at some later moment t2 afterward. There is nothing illogical about such assertions, if we cast them as follows: If t1 is present, then the death of Socrates is present,
and If t2 is present, then the death of Socrates is past.
We run into problems only when we talk about something becoming
past as though becoming
meant coming into being
rather than a formal change in attribution. A-determinations are not ontological accidents, since a past
event no longer exists, so it can hardly have any accidents (even if we treated the event as a substance). When we say X is past,
we do not assert that X is an occurrent existent having some property called past.
Although A-determinations are not illogical, our way of speaking about them with tensed verbs can be problematic, and McTaggart is right to call this into question.
The problem is not much different in special relativity, except here we must also specify spatial coordinates to indicate the present
with respect to which we determine the A-status of other events. It should not be thought, however, that there is some fixed point in spacetime with the coordinates (x1, y1, z1,
t1). A coordinate system is only definable in a chosen frame of reference, and reference frames are definable only if you presuppose that signals and observers have velocities, which in turn supposes motion, and therefore time. You do not build up your reference frames coordinate system from the supposition that a single point has some 4D coordinates, but from the measured difference in spatial location and temporal duration between two events. Without clocks and rulers or some equivalent, it is impossible to make such a construction. Minkowski diagrams show the structure of the relationship between such differences in events, and how the relative proportion of distance and duration varies by reference frame. In particular, it lets us see the A-determinations of other events with respect to one we choose as present; not all other events; only those that could conceivably be in causal relation to our chosen present. This mapping is visually present to us at once, but this does not imply that events are not situated in time, for we have built temporal measurements into our construction.
McTaggart accomplishes his proof of the unreality of time only by speaking like a formal logician or mathematician, treating the existence of temporal events as something that can be abstracted from time. Yet precisely what is at issue is whether certain things, which we perceive as changeable, mobile, or perishable, exist in time. If there are such things that exist temporally, then it is invalid to abstract their existence from time. McTaggart suffers from a more general problem of the analytic school, namely that they treat formal logic as though it could be ontologically neutral, when in fact the validity of an argument often depends on the ontological category of its variables. Using a set theoretic logic to analyze events
as objects or elements of reality assumes what is supposedly proved, namely, that so-called time is really just an extension that divides reality into static parts. It creates pseudo-objects, whereby Object A at time t1
is a distinct substantive object from Object A at time t2.
By defining events at each time-point as distinct objects, we make time into a medium of pluralization, which is to mistake it for space.
Special relativity does show an interdependence between the magnitudes of measured space and time intervals, but it does not reduce time to space or space to time. So-called space-time
is not a four-dimensional extension, since the time variable in a given coordinate system actually subtracts from the pseudo-distance, so it does not add extension to a 4D volume. If we lose sight of the fact that Minkowski diagrams merely recapitulate what we gather from measurements of motion, which presupposes the reality of time, then we can easily fall into McTaggart’s error of abstracting events from time (though his critique of A-series discourse does not depend on this error). Each event becomes like Parmenides’ One. It never grows older than itself, so it cannot exist in time, or participate in past, present or future. Yet this is to forget that the notion of eventhood presupposes time.
We noted at the outset that propositions about physical facts include an implied specification of time, unlike timeless
mathematical relations. By adopting a formalistic treatment, McTaggart begs the question and supposes that the existence of physical objects can be abstracted from their situation in time. This conflates formal essence (what a thing is) with the ineffably simple act of existing now. (This is also why Russell’s logic, with its static treatment of existence, is not an adequate basis for a priori philosophizing.)
In general, McTaggart tries to show that time is incoherent by talking about existence as though it could be abstracted from time. He treats the existence of events as abstracted from their position in time, much like a formal logician, or an idealist. There is an implicit category error of treating events as substances or possible objects of change. Naturally they fail in this role, from which the false conclusion that change is logically impossible arises. Substances are primary existents, and the reality of time requires us to suppose nothing more than that substances can persist while changing in some properties. Once this is acknowledged, the reality of temporal succession must be upheld to avoid contradiction; otherwise the identical substance could both exist and not exist or both have and not have the same property.
While we have criticized McTaggart’s argument for the unreality of time on ontological grounds, it is far more difficult to find fault with his logical critique of discourse about A-determinations. Many philosophers in the analytic school did attempt such refutations in succeeding decades (or they all occur timelessly in directionless order), but often failed to grasp the subtlety of McTaggart’s position. Louis O. Mink summarized the counterarguments to date in 1960, finding them generally lacking. He considered McTaggart’s argument to be valid, yet not as proving the unreality of time, but as proving the logical incoherence of our discourse about time.[4]
Mink understands McTaggart to have forced a choice between self-contradiction or vicious infinite regress in interpreting the A-series. Since the A-determinations (past, present, future) are mutually contradictory, it would be a contradiction for the same event to have different A-determinations. The usual rejoinder, that events have different A-determinations not at once, but successively, forces us to consider what is meant by successive. If we mean successive only in a B-series sense, then we have eluded temporal change, since position in the B-series is not subject to temporal change. If it is successive in an A-series sense, then we seem to imply that the characteristic of each moment changes, in respect to some different time order. This higher order succession must be explained either as a B-series or an A-series. So we must either end in an unchanging B-series, denying that A-determinations involve change, or in an infinite regress of A-series. We can ignore this logical problem in ordinary speech only by unconsciously oscillating between A-series and B-series notions of succession.
Three classes of objectors are identified by Mink: (A) those who support a B-series-only conception of time; (B) those who admit the inadequacy of the B-series, but deny that there is a contradiction in the A-series; and (C) those who accuse McTaggart of misusing language.
Among those who think the B-series is sufficient to account for time, some have argued that it is things (substances), not events that change. Events do not change precisely because they are changes. Therefore McTaggart’s objection to the temporality of the B-series is unfounded. While we have noted that it is indeed a mistake to posit events (and their supposed qualities or relations) rather than substances as the subjects of change, Mink rightly notes that this error is immaterial to McTaggart’s argument against a B-series-only notion of time. Regardless of whether McTaggart is correct to call B-relations permanent
in the sense of persisting through time, the fact remains that there is no sense in which B-characteristics change. The B-series-only philosophers who would situate change in the terms of the B-series (i.e., the events themselves) rather than in relations between terms simply make change primitive and unanalyzable.
We may note, however, that McTaggart claimed to have refuted the primitivity of time, so situating change within events would be a valid counter to that claim. Nonetheless, it would be a mistake to consider change to be solely within events or solely between events. Neither conception does justice to the continuity of time. This is better understood with the use of derivatives. At any given point in time, a physical object is in process, undergoing some change in position or another variable as expressed by its instantaneous time derivative. On the other hand, for there to be any measurable change, we must compare between measurements at two time points. Thus change can be conceived as both occurrent at each event or time-point and between or through the continuous series of events. What McTaggart considered to be an objection to the A-series, namely the speciousness of the present since it is not experienced as absolutely infinitesimal, actually points to the resolution, which is the continuity of time. If we are not to fall prey to one of Zeno’s paradoxes, we must not render events static by making them unchanging point-monads, nor should we invoke the finite duration required for anything measurable to occur, including what we colloquially call the present, as implying that change is only an interval relation. The latter claim would make velocity nothing more than the ratio of distance to time elapsed, and render instantaneous velocity unintelligible. Differential calculus resolves these paradoxes with mathematical rigor.[5]
Nonetheless, Mink is correct to remark that the problem of describing changes in A-determination is not evaded by making things rather than events the subjects of change. Temporal process would still be a series of such changes rather than an assemblage of things.
[6] So we would still be left with the question of how to analyze a series of such changes, and this cannot be done purely in terms of a B-series. Perhaps Mink’s criticism could be averted by recognizing that temporal process requires perdurance as well as change, but this is not the position that was held by B-series-only philosophers.
The second class of critics admit that time is not reducible to a B-series, but deny that there is any contradiction in the notion of an A-series. Statements about A-characteristics have two possible interpretations, both of which McTaggart found inconsistent. First, the ‘is’ in X is present
can be considered a timeless copula. In that case, X is present
and X is past
cannot be distinguished as true at different times, since they are attributed timelessly. So it must be that the copula indicates time, so that X is present
means X is present now,
i.e., the copula indicates an implied moment (usually the time of the utterance) that is defined to be now or present. So we should say X is present at a moment M that is present,
or when M is the present.
In what sense should we interpret the second ‘is’? If it is taken as timeless, we will have the same problem as before. If we take it as indicating a time, we must refer to yet another present, and so on, in an infinite regress.
John Wisdom and Susan Stebbing both held that this regress is not vicious, for the higher order statements are just consequences, not prior conditions, of the statement X is present.
The present-tense of the copula points to the same now
as the predicate present,
so we do not lack any information about When is X present?
that would be supplied by the regress.
Mink, however, notes that such a criticism has no force when dealing with other A-relations. It clearly is meaningful to ask When was X present?
The tensed copula was
situates the predicate at a time in the past relative to the moment of speech, taken as present, so X is past
means X was present at a moment of time which is past.
This is equivalent to X is past at a moment of time which is present.
Again, we have an infinite regress, unless we are to deny the meaningfulness of saying X is past.
The only way to avoid this regress would be by denying that we can refer events to moments; i.e. we can say The battle of Jena is past in 2018,
or The battle of Jena was present in 1806,
but not that 1806 is past.
It is hardly deniable, however, that the latter statement is also meaningful.
Mink’s analysis here seems to assume that you can abstract a moment
(e.g., 1806) from its A-series status. If, in fact, moments exist only as they are present, then all other A-determinations (besides the simple present) have to do with discourse about moments, not changes in their quality.
The Wisdom-Stebbing criticism would make all three primitive tenses irreducible, so X is past at a present moment
does not add anything to X is present.
Then A-characteristics, Mink notes, cannot be analyzed in terms of B-characteristics, which would mean complex tenses are also unanalyzable. Yet this is contradicted by our practice in constructing and using such tenses.
It seems that a complicated conceptual structure (displayed by the A-series) is presupposed and used in our understanding of tensed statements in general. This structure is not merely extrapolated from tensed usage; it is rather logically prior to it (as Kant saw), and this indicated by the fact that we can construct unfamiliar but clear locutions such as ‘I had been about to have been…’, or can even construct a model of n-adic tenses for which no ordinary language suffices at all. Since this is a systematically constructed model, its elements cannot be irreducible. But then McTaggart’s argument has again triumphantly survived.[7]
A third class of critics say McTaggart was misusing language by combining tenseless and tensed language. If this were a valid criticism, Mink says, it should be possible to construct an A-language in which all verbs are temporal, and a B-language without any temporal verbs. He then shows that both of these endeavors are impossible even in principle. Regarding a B-language:
…even though we had a systematic model of the manifold of events and their B-relations, we should never know nor be able to say where we were in this manifold. Nor could the manifold be divided into events earlier than and events later than any present, since no ‘present’ can be denoted in B-language.[8]
We may note that mathematical representations of physical time are effectively a B-language, with a real variable t that has ordered values, with higher value meaning later and lower value meaning earlier. Classically, all events across space could have the same value of t regardless of reference frame, but in special relativity there is not a single B-series for all events, except as formally defined in a chosen reference frame. Still, there is a real B-series structure to every world line.
From Mink’s discussion of B-language, it should be clear that mathematical representation of physical time is not truly temporal unless we impose an A-series interpretation upon it. This is done by reference to empirical observation, without which our model of time would not be truly physical. The point at which an observation is made is defined to be present.
Once such an event is chosen, we can know the A-determinations of all other events that are least potentially causally related to the present event. It might be said that this A-series interpretation is superfluous, but this is to forget that it is only from many A-determinate observation points that we were able to construct our B-series in the first place. That is to say, we were able to arrange events into earlier and later only because we experienced them as successive presents in a certain order.
Nonetheless, as Mink remarks, a pure A-language is also impossible, i.e., you cannot denote A-determinations solely with tensed verbs. You could only say X is present, Y was past, Z will be future. So you could say when something is with respect to the present utterance, but otherwise, without timeless predication, you cannot compare events to each other, making a B-series structure impossible.
Any language that is to describe time must both account for an order of events and… the corruption of time
[9] (i.e., that things past pass out of existence). Otherwise we are not describing time at all.
Both McTaggart and his critics seem to ignore that there is a series of A-series. That is, we can define an A-series for when event L is the present, and another for when a later event M is the present, and so on. (To be rigorous, we should allow for a continuum of events and moments, but the relevant principles are captured even in the discrete case.) The series of A-series is itself a B-series, as Mink expounds, as each A-series is distinguishable from the others only by reference to the B-series of events (L, M, N, O…). Attempts to define how the A-relations change defy analysis, as …every attempt to know the A-relations of events is balked by the passage of time itself.
[10]
McTaggart and his critics, Mink says, have assumed that concept of time can be discursively exhausted. Can this be denied without taking Bergson’s position that time cannot be discursively understood at all?
A chronology of dates is effectively a stand-in for the B-series that spares us the tedium of having to think about temporal positions in explicitly relative terms (e.g., the French Revolution was after the American Revolution but before the battle of Waterloo). Basically a set of (astronomical) events is labeled by numbers, so an event’s position in the series compared to other events is easily gathered by comparing numeric values. In classical time with absolute simultaneity, we can define all events as unequivocally simultaneous to one of the events in our numbered chronology. In relativity, our numeric chronology must be defined by events at particular locations. If we wish to project these chronological values to distant locations, we can only do so for a chosen frame of reference, not unequivocally.
This chronological or historical approach to describing time focuses on the discursive aspect of time. Yet there is also a transient aspect to time, i.e. things or states passing away and coming into being, without which there would be no need for any notion of time in the first place. To describe this A-series aspect of time, Mink holds, we can never mean what we say, as McTaggart has proven.
Time has some discursive (B-series) characteristics and a non-discursive, transient (A-series) aspect. Discourse about the transient aspect, according to Mink, is necessarily Pickwickian, i.e., not to be taken literally. A picture of a classical timeline or relativistic world line does not really grasp the transience of time, since it displays everything at once. We can convey this transience only in the act of drawing the line, showing how events come into being, with the point of the pen indicating the ever-changing present. Likewise, if we try to describe transience with any fixed terms, in any language, infinite regress is unavoidable. Concepts as such are static and timeless. Treating present
past
and future
as fixed predicates leads to an analogous contradiction, not because the notion of transience is contradictory, but because it is contradictory to express this in static terms.
When we grasp this, we can see why Mink finds that saying that the terms present, past and future, and tensed verbs regarding them must point to another A-series is like saying that a sign shaped like a man’s hand must point to a mans hand.[11] Discourse about time is necessarily Pickwickian because discoursing takes time, or rather, discourse takes place in time.
In 1966, the analytic philosopher Richard Gale noted that many analytic criticisms of McTaggart unwittingly adopted some of his faulty premises.[12] In the process of analyzing this problem, he introduced the terms A-Theory
and B-Theory,
which have lately found currency in discussion of the philosophy of physics. Unfortunately, current usage of these terms is frequently confused with the poorly defined concepts of presentism
and eternalism.
As with McTaggart, Gale’s concepts bear little resemblance to the meanings later imputed to them.
Gale noted that McTaggart’s position had both a positive thesis and a negative thesis.
Positive: An A-series is essential to the concept of time, and a B-series is reducible to an A-series.
Negative: An A-series carries an internal contradiction regarding existence, therefore time is unreal.
Many analytic philosophers had already identified alleged flaws in McTaggart’s arguments, rejecting his negative thesis. Gale classifies these refutations into two approaches (akin to the first two classes of objectors identified by Mink):
He calls the first approach the B-Theory Answer
and the second the A-Theory Answer.
The A-Theory accepts McTaggart’s positive thesis, but denies the negative. The B-Theory denies the positive thesis, making any supposed internal contradiction of the A-series irrelevant to the question of the reality of time. Note that A-Theory and B-Theory do not correspond to the various notions called presentism
or eternalism.
Again, none of this has any dependence on special relativity. These are issues that can be raised even in a classical paradigm.
Gale finds fault with both the A-Theory and B-Theory answers, since they both ignore an obscurity in McTaggart’s definition of an A-series. McTaggart describes his A-series as not simply past, present and future, but also as running from far past to near past, and from near future to far future. (This inclusion of degree is followed by his interpreters, including Mink.) Thus it implicitly has a B-series aspect to it.
Gale remarks that no one denies the necessity of a B-series (earlier and later) for there to be time, but some have questioned the necessity of an A-series (past, present and future) as objectively real, arguing that this is only delusory or psychological.
Both A-theory and B-theory agree there cannot be time without change. McTaggart showed that, since each event is never other than what it is, the only possible change is in one’s position in the A-series. A B-theorist (or any sound ontologist) could retort that things rather than events change, and the changes of things can still be analyzed in terms of B-series.
Gale notes that McTaggart’s claim that an event always
is what it is equivocates in the use of ‘always.’ The poker is hot on Monday
is always true, but this does not mean that the state of affairs described by the statement is always occurrent. This is the logical fallacy of so-called eternalism.
Because statements describing events in a tenseless manner are always true it does not require that these events are sempiternal: to claim the opposite is an unwarranted addition of an eternalistic ontology to a tenseless (or token-reflexive free) logic.[13]
This is the only place in the paper where Gale even mentions eternalism, and here only dismissively. He claims that McTaggart upheld this eternalistic ontology of events against B-theorists. Obviously, those who identify B-theory with eternalism are confused.
Besides this appeal to the unchangeability of events, McTaggart had a second argument, namely that a B-series necessarily involves reference to an A-series, i.e., concepts of past, present and future. Gale finds this to be circular. McTaggart, for example, thinks P is earlier than Q
to be intelligible only in terms of notions such as P is present while Q is future,
or equivalently, Q is future at P.
Gale notes that the is
in the latter statement is timelessly true, so the statement is not an A-determination. Statements of A-determinations cannot be timelessly true. As we noted, the truth of The death of Socrates is past
depends on when it is uttered.
A third argument by McTaggart is definition in use.
That is to say, by declaring a B-series order P is earlier than Q,
you imply a conjunction of A-determinations: P is past and Q is present
OR P is present and Q is future
OR P is past and Q is future.
Gale observes that you have logical equivalence only if you also conjoin: P is further past and Q is nearer past
OR P is nearer future and Q is further future.
A B-series is only reducible to this impure
A-series that includes notions of near
and far
past and future. But this renders the reduction trivial, since such an expansive notion of an A-series clearly has the concept of a B-series built in.
An (impure) A-series is a specific determination of a B-series (i.e., choosing a particular moment in the B-series as present, much as Mink described), but a B-series can be defined without specifying an A-series. A B-series, with its notions of earlier and later, does presuppose that some A-series exists, but it does not require us to define exactly where the present is; i.e., to choose a particular A-series.
A pure A-series would not be a series at all, since past and future would each be a blob
with no specified internal order. The generating relation of an impure A-series (with ordered near-to-far future or past) is the same as that of a B-series: earlier (later) than.
Since a B-series must also be an A-series, though not any specific A-series (i.e., we can choose any value for the present), the generating relation of the B-series must be the generating relation of an A-series, though not any one specific A-series.
An A-series is a conjunction of two series with a common member being the terminus ad quem of one and the terminus ab quo of the other. One series has the generating relation more past than,
and the other has a generating relation more future than.
McTaggart thought he could derive the B-series from a conjunction of the A-series and a C series,
a non-temporal ordered set such that MNOP is the same as PONM, but not the same as MONP. Gale notes that this is either pointless or futile. If the A-series
is an impure A-series, then it already contains the notion of B-series, making any conjunction pointless. If the A-series is a pure A-series, then there is no way to impose the C-series ordering because the past and future are blobs with no distinction between more past
and less past.
Ultimately, Gale concludes that both the pure A-series (trichotomy of past, present, and future) and the B-series are essential to the notion of time. This contradicts both the A-Theory and B-Theory positions.
Admittedly, A-determinations are perspective-dependent; i.e., the truth of such statements depend on when (and under relativity, also where) they are said, or rather, on the time (or place) to which their use indicates. Anyone who wishes to deny the reality of A-series must deny the reality of time altogether. Apart from the illogic of invoking the reality of spacetime to deny one of its essential constituents, one must commit the fallacy of equating tenseless truths about events with their tenseless occurrence, or of equating tenselessness with eternalist ontology.
Successive presents are ordered in a B-series. It does make a physical difference which order we consider events in, so time a B-series is objectively real. Yet a B-series, as Gale has shown, logically implies some A-series, though not any specific A-series. This is what we have been saying all along, insisting on the reality of temporal succession (A-series), while admitting that values of A-determinations are perspective-dependent.
Since the reality of a B-series implies the reality of some indeterminate A-series, those who would deny the reality of temporal succession must deny that any event is really earlier or later than another. We have seen, however, that all timelike connected events, which is to say, all events that can be causally related to each other even in principle, have an unambiguous order of temporal priority, regardless of reference frame. This strongly indicates some objective reality to temporal succession, though the specific values of the A-determination of each event is dependent on perspective; i.e., on which event is experienced as the here-and-now present.
Some pairs of events, which are called spacelike or spatially separated, cannot be placed in an objective temporal order, unless we arbitrarily define a particular frame of reference to be privileged. Though each event belongs to some objective temporal sequence in a host of world lines passing through it, it does not follow that any two events can belong to the same temporal sequence. Attempts to define such a sequence objectively (without arbitrarily defining a privileged frame) leads to results that are contradictory. This impossibility, however, has no physical implications, since spatially separated events cannot send or receive signals from each other, much less be physical causes or origins of each other. Thus it leads to no ontological ambiguity except for those who are not content with an empiricist notion of physical reality.
Philosophers in classical and modern traditions alike can fall into this snare, when they insist on defining the present
univocally across space as an essential condition for the reality of temporal succession. Yet the arguments surrounding the interdependence of A-series and B-series have no logical dependence on this condition. We have seen that the B-series is not dependent on a particular A-series, but just on the reality of some indeterminate A-series. While the absence of a single A-series that can be defined everywhere at once may be intuitively unsatisfying, the compatibility of world lines, intersecting at mutually agreed presents,
suffices to maintain the lack of contradiction among A-determinations, even if different world lines cannot otherwise be brought into a common absolute chronology.
A lucid illustration of what happens when we try to insist on a spatially universal present can be found in a 2008 paper by V. Petkov.[14] After duly showing how presentism
in the sense of a uniquely definable 3D class of simultaneous events is impossible in special relativity, he proposes introducing a broader 4D concept of simultaneity across spacetime, rather than simply dispensing with the notion. He defines that all events outside the light cone of event O should be considered simultaneous with O. This is to take the conventional moniker for events outside the light cone as elsewhere
quite literally. Yet if we accept Petkov’s definition, we should consider all events in this elsewhere to be simultaneous with each other (since they are all simultaneous with O), even though many of them can be in each other’s light cones! This definition flounders precisely because it supposes that the light cone of a given point ought to impose A-determinations for all other events, ignoring the fact that the value of an A-determination is intrinsically perspective-dependent. (Again, this perspective-dependence refers to our choice of an event as the origin or point of observation, not to be confused with frames of reference defined by observers at the same event-point with differing relative velocity.)
The slab of spatiotemporal elsewhere
is a strange choice of definition for the physically real universe of O, since the class of events outside the light cone of O are, by definition, precisely those events that are unobservable from O even in principle, and cannot have any physical causal connection with O. This is a thoroughly anti-empirical, non-physical definition of reality. An empiricist instead ought to say that the rest of the universe,
from the perspective O, consists of the absolute past of O, since that is all that we can observe from that point. In fact, when astronomers make supposed cosmological maps of the universe, they are really just mapping a subset of our absolute past.
Naturally, once the dubious definition of simultaneity across spacetime as elsewhere
is accepted as the only one available, it is easy to infer that there can be no three-dimensional world
that is existentially simultaneous, since elsewhere
cannot be confined to a 3-dimensional slice
(or a 1-dimensional line in our planar Minkowski diagrams). Petkov infers from this that there can be no 3-dimensional world, but this is to demand that existence must be conceived statically, rather than in local flux. What is true is that we cannot contemplate anything larger than a point in relativity without also considering the fourth dimension
of time, since observing anything some distance away requires the elapse of time for some signal.
If we cannot define a univocal present
across space, does it follow that presentism
in a second sense (often confused with that of absolute simultaneity), namely that actual existence is confined to the temporal present,
is unsustainable? Petkov believes so, since the alternative to universal simultaneity would be to define a three-dimensional world containing different moments of time. Again, this assumes that reality must be considered statically rather than in local flux.
Petkov himself tacitly acknowledges the reality of time in the B-series sense when he points to the causal structure of spacetime. Yet we have seen that B-series structure is intelligible only on the supposition of the reality of some (indeterminate) A-series. It would seem that the relativity of simultaneity implies that there is a potentially infinite plurality of A-series, corresponding to each world line. These are all mutually compatible, intersecting at a present common to both objects in their respective world lines when meeting in a Minkowski diagram. This is no accident, for an event, considered as such, is actual or present, meaning that Minkowski diagrams give the spatiotemporal relations among all possible presents. The determinations of past and future are definable only relative to a given event as present. This is not fundamentally different in principle from the classical situation, except that present
cannot be abstracted from a determinate event to be some shared property held by various events across space at the same time.
Every event as such, considered in itself, is present, as should be obvious from the fact that the notion of eventhood entails specification of time. An event cannot be other than what it is, as McTaggart said. This does not prove the unreality of time or change, however, since an event is nothing more than substance(s) and properties abstracted from temporal flux and considered at a frozen instant.
Correct interpretations of relativity can easily elide into denials that temporal A-determinations have any bearing on the existential status of a fact (i.e., as potential, actual, or annihilated). In extreme cases, this can be taken in an Eleatic, monistic or eternalistic sense. Such interpretations, whether made by physicists or philosophers, generally equivocate with statements such as All spacetime events are real.
Is the copula to be taken in the present tense? If so, this is contradicted by the structure of spacetime, which makes it absolutely impossible for events in the same world line to be simultaneous, in any reference frame. If the copula is taken tenselessly or atemporally, then this does not contradict the notion that things may come into being and pass away in temporal sequences. All events are indeed real, in their turn.
Why should one insist that things that exist in time, coming into being or passing away, must have an unequivocal ontological status irrespective of where the observer inquiring about such status is in time? The truth of the statement Socrates is alive
depends on when the statement is made. All temporal being is relative in that sense, even in classical physics. The difference with special relativity is that specifying when is inseparable from specifying where. I can say that the death of Socrates is in the past with respect to now on Earth,
but the A-determination of that event may be unknowable even in principle at other spacetime points. Yet it would be impossible for anyone at such spacetime points to receive any news of the death of Socrates even in principle, so the question of its ontological status could not occur to them except hypothetically (i.e., guessing from earlier events that they can observe).
We may further consider that present
is the only A-determination properly predicable of an occurrent event, though that same event may be called future
or past
in an equivocal, B-series usage. For example, I may say that now (on Earth)
is in the future of the death of Socrates. Here I do not really mean that the A-determination of now is future,
but instead mean that now is later than the death of Socrates. Equivalently, it could mean that the A-determination of the event we call now
was future
when the death of Socrates was the present. In special relativity, when we speak of an event being in the future or past of another event, we usually intend this B-series usage, insofar as we are not concerned with which event is really the present. Yet even B-series discourse, to be intelligible, presupposes A-determinations, and we must treat an event as suppositionally present in order to define others as past or future. Note that the A-determination called present does not admit of B-series usage. We can speak of an event being in the past of
or future of
another event, but an event can only be in the present of
itself.
Petkov makes use of a physicist’s shorthand that treats observer
and frame of reference
as synonymous. This can give the misleading impression that it is really possible to map all of spacetime at once, when in fact, most of the map is inferred by an observer who, besides being in a reference frame (defined by relative velocity), is always localized in spacetime (i.e., confined to a world line) and moving at finite speed. He computes his map
of the rest of the universe from similarly localized signals he sends and receives in temporal succession. In short, the whole construction depends on the supposition of temporal succession. Since the observer is localized, there is no reason to expect his A-determination as present to be shared by other objects. All signals require finite duration, so by the time his signal reaches another object, he too has moved onward in time, though there is no uniform measure of how much time has elapsed for him compared to his signal, since the duration and path length of the signal depend on reference frame. No observer can travel infinite speeds to see everything at once. It is invalid to use Minkowski diagrams as a way of getting around this impossibility, since it contradicts a postulate of special relativity, of which the diagrams are just a shorthand expression.
The reality of temporal succession is further indicated by structural constraints on the order of events. Consider spatially separated events A and C from Figure 8. Neither is in the unambiguous past of the other, but there could be a third event D that is in the unambiguous future of both A and C. Further, any event in the future light cone of D cannot be in the past light cones of A or C. Such constraints on sequence indicate that time can only go forward,
and never reverse itself, though the precise direction of forward
varies by choice of reference frame. This notion of forward direction has physical significance, in that an effect can never be prior to its cause.
Special relativity does not abolish the notion of A-determinations, but forces us to dissociate the notion of present
from that of simultaneity. This can be seen diagrammatically:
In both the classical and relativistic models, A-determinations are definable only by a choice of zero-point; the line t=0 in the classical example, or the point (x,t) = (0,0) in the relativistic example. In the relativistic example, A-determinations are definable only for events in the light cone of the chosen origin. If you want to know the A-determinations of events elsewhere,
you would have to choose an origin there, but you cannot choose two different points simultaneously
to be the present any more than you could choose the lines t=0 and t=1 to both represent the present. A-determination or temporality is bound up with locality; this is a central insight of relativity.
It is true that, mathematically, once we have a spacetime diagram in front of us (be it relativistic or classical
), we can decline to define A-determinations and make calculations purely in terms of B-relations, treating the time coordinate values as if they were abstract quantities having nothing to do with time. If we are acting as physicists, however, we cannot allow ourselves to forget the empirical basis of our diagrams. Only an observer situated in some A-series is able to construct the B-relations for all other events, including those presently unobservable events in elsewhere.
This constructed B-series for elsewhere
is frame-dependent (i.e., the constructed sequence of presently unobservable events depends on the relative velocity of the observer). Thus even certain B-series are relative.
Shall we say therefore that all B-series are unreal?
Such a denial would ignore the constraints that causality imposes on the structure of spacetime. Furthermore, if we should deny any objective reality to A-series or B-series, we have clearly denied time altogether. What then does the variable t measure, and how can events with different values of t (in a given frame) be considered really distinct?
It is true that there is no single A-series or B-series for all events in the universe. Nonetheless, each event can be situated an A-series and B-series without contradiction of the A- and B-relations of those events with which it can interact. This suffices for consistency with the physical reality of A-series and B-series, even though it falls short of making them universal absolutes.
The speciousness of invoking relativity of simultaneity to deny time can be seen when we apply the same reasoning to deny space. For any pair of timelike separated events, there is a relativity of colocality; i.e., whether these events occur in the same place depends on choice of reference frame. Another way of putting this is that whether an object is at rest (remaining in the same place) or in motion (changing place over time) depends on reference frame. Since colocality is relative, then we should say that difference in place is unreal by the same reasoning used to deny the reality of temporal succession. Yet to deny the reality of difference in place is to deny the reality of space altogether, much as denying the A-series is a denial of time. Surely such reasoning proves too much.
All such arguments really prove is the interdependence of spatial and temporal measurement. If we try to isolate time from space or space from time, measuring one without the other, we get a contradiction. This is not because time and space are unreal, or because they are not really different from each other, but because they are interdependent, each measuring the same motion in a different aspect. How much of that motion is perceived as displacement versus duration depends on the observer’s motion relative to the observed motion.
Change is invisible if we pretend to abstract from all A-determinations and view the spacetime diagram as a pseudo-Cartesian space. One must be in the flux in order to experience change, which is why A-determinations are necessary for perceiving the reality of change, the object of study for physics.
Our perceptions of the magnitudes of duration and length are frame-dependent, but this does mean that motion is not physically real; only that its quantitative parameters are not absolute in value. In one frame it may seem that there is greater length and less time between two events than in another frame. Yet the order of succession of events in any physical process is always unambiguous and frame-independent.
Planes of simultaneity can be defined conventionally only after an observer situated in spatiotemporal flux, i.e., motion, has chosen a frame of reference, sent signals back and forth to distant locations, and made inferences about the chronology of distant events based on when those signals return. This constructed definition of simultaneity across space is frame-specific, so another observer moving at some non-zero relative velocity with respect to the first would get a different definition. Each would have his own coordinate system, in which all event-points with the same value of t are defined to be simultaneous,
constituting a 3D slice
of the 4D block
of spacetime.[15]
Minkowski diagrams make clear that observers in different frames of reference will have different planes of simultaneity, perpendicular to the time axis in each frame, at different angles in the diagram. Since these definitions are mutually inconsistent, some infer (with a big logical leap) that we must take the entire block of spacetime as prime reality, and that change or motion is not physically real, but just an artifact of perception.
This last inference is faulty because it tacitly assumes that a univocal notion of simultaneity across space is a necessary condition for the reality of motion. In fact, it suffices for events to belong to timelike intervals, occurring in a well-defined order along each world line, as long as they agree on the present
at points of intersection. This world in local flux admits of no static map, but that is hardly a condition for the reality of motion. If anything, relativity should point away from an Eleatic view of the world, since it makes all physical reality definable only in terms of relative motion, forbidding us from ignoring the motion of the observer. While physical motion has no absolute values for displacement, duration or velocity (except for the velocity of light), its unreality is not thereby inferred.
Failing to find a static map in three dimensions, those inclined to desire such a map must resort to four dimensions, the so-called block universe. Only here can one find a static depiction of reality. We must recall, however, that Minkowski spacetime has no mathematical or physical content regarding the relativity of simultaneity not already implicit in Einstein’s gedanken experiments. It is just a picture that makes the algebra easier to visualize. It does not authorize us to introduce new physical content not already present in the special theory of relativity. That theory, considered as a theory of empirical physics, i.e., grounded in observation, offers no justification for treating any slice or block of spacetime
as substantial stuff
that exists at once. True, there is an invariant pseudo-metric distance
s2, which gives us cause to believe that there is an objectivity to the relational structure of spacetime, but this is not sufficient reason to regard spacetime as a substantive thing. Such a notion loses sight of the fact that all constructions of spacetime
and its various coordinate systems rely in the first place on observations with suppositions about succession in time for physical processes and signals. Without such suppositions, e.g., if we could not know that a signal is received after it is sent, we would not have been able to construct a geometric model of spacetime in the first place. It is never valid to draw conclusions from a theory that contradict its own suppositions, unless one’s aim is to invalidate that theory.
Despite the considerable mathematical complexity introduced by general relativity, the basic notions of relativity of simultaneity and temporal succession along world lines remain fundamentally unchanged, so that what we have said regarding the reality of temporal succession is largely unaffected. There is nothing in general relativity that would make one deny time that would not already be present in interpretations of special relativity. If anything, general relativity, with its dynamic metric tensor with curvature, introduces further considerations that make the notion of a static block universe untenable, even unintelligible.
In general relativity, the spacetime metric tensor gμν is an indispensable construct with real physical content. We cannot avoid making use of this metric, unlike in special relativity, where Minkowski spacetime is just an optional convenience. General relativity suggests that spacetime is something more than a description of how matter relates to other matter, which is why Einstein ultimately abandoned a purely Machian, relational notion of space and time, in favor of a more substantive conception.
While the physical reality of the metric tensor seems to imply the reality of spacetime as a substantial manifold, this ignores two important considerations. First, the metric tensor does not uniquely specify a manifold, and such specification is anyway physically irrelevant; only the metric tensor matters. Second, the Lorentzian signature is not positive definite, so the metric tensor is only a pseudometric for a class of pseudo-Riemannian manifolds, just as Minkowski spacetime is pseudo-Euclidean. Thus we can refer to spacetime as a manifold only equivocally. It is not a manifold in the mathematical sense, which is locally similar to n-dimensional Euclidean space. Rather, it is locally similar to Minkowski spacetime, which is not Euclidean or even a metric space at all, since it has no distance function.
It is misleading to visualize the spacetime manifold
as Minkowski spacetime with some bending. Minkowski diagrams that map event-points in some definite structure are possible only when you have a path-independent distance. No static map of spacetime, not even a Minkowski-like pseudomap, is possible any more.[16] Spacetime is not globally representable as an array of coordinate points in any number of dimensions. The notion of a block universe, if it ever had any credibility, is inapt for a spacetime that cannot be represented as a vector space.
We cannot use anything analogous to a Minkowski diagram precisely because we are no longer working with inertial frames of reference, which Minkowski diagrams are able to superpose. Instead we can only locally define a comoving frame, i.e., a frame accelerating in the same way as a particular object is accelerating at point P. This does not, in general, give a global coordinate chart for the manifold, much less enable us to depict different coordinate systems as superposed in the same diagram, as in special relativity. This is a blessing in a way, for we might again mistake such a diagram for a frame-independent God’s-eye view of spacetime as it really is.
In general, we cannot simply say what is the spacetime metric
for the universe globally. The metric tensor components can only be evaluated at a point P in spacetime, using a chosen coordinate system for the (locally Minkowskian) tangent vector space at P. The metric tensor can then be represented as a 4 × 4 matrix, the components of which show the relative rate of displacement along each of the four coordinates of the local tangent vector space. Every other point in spacetime would, in general, have different values for the components of the metric tensor. They would not even necessarily be using the same vector space coordinates, since different points in curved spacetime have different tangent spaces. One may use a derivative operator to define parallel transport of a vector along a curve, so that its orientation with respect to the curve remains the same. Nonetheless, this is not a basis for making the vector space coordinates universal, since two different paths between the same two points might not preserve the same parallel transport. [See Basic Issues in Natural Philosophy, Sec. 14.2.1]
[If we were to attempt to make a pseudo-Minkowskian 2D diagram, with one space dimension and one time dimension, we should have to define a 2 × 2 array of values representing the metric tensor at each point (ignoring the problems of incompatible coordinate systems and parallel transport for the moment). This is already more than the three dimensions we can visually represent. Even if we could visualize more dimensions, the matrix components of the tensor do not represent four coordinate components of a vector, and plotting them as though they were would still not give us a map
of the spacetime manifold. Spacetime is not a vector space at all.]
Since a global coordinate system is not definable, the metric tensor is evaluated locally at a point P, with coordinate functions expressed in differential form. Unlike the Minkowski metric, there is no frame-invariant global pseudo-distance s2 to be gleaned from curvilinear metrics in general. The metric must be expressed in differential form ds2, representing an infinitesimal square distance
definable only in the neighborhood of P. Even this ds2 is not expressible as a single quantity (scalar), but an array of tensor components in a coordinate system of the local tangent space.
No matter what your choice of coordinate system for the tangent space, one of the dimensions will have a negative vector norm, so that is the time dimension for that coordinate system. To parallel transport that time unit coordinate vector along a world line is to make the length of that line the physically meaningful proper time.
[Minkowski’s artifice of defining the time variable as ict, so that we have a complex Euclidean space, only hides the non-definiteness by situating it in the vectors, which can have negative squared magnitudes. In general relativity, spacetime is no longer globally representable as a vector space, and making the time coordinate imaginary adds needless complication. Thus the time coordinate vectors in tangent space retain negative norms.]
Although there is no meaningful notion of 4D distance between two spacetime points, we can define the length
of a curve C in a (pseudo-)Riemannian manifold as the integral (continuous summation) of the norm of its tangent: l = ∫(gμνTμTν)1/2 dt, where t is not time, but an arbitrary parameter of the curve, which can be timelike, spacelike or null, or it can transition from one type to another. A curve that is everywhere timelike must have gμνTμTν < 0 everywhere along it. It is spacelike if gμνTμTν > 0 everywhere, and null if gμνTμTν = 0. If it transitions between spacelike and timelike, no length is defined. If the curve is timelike, we can regard its proper time τ as its length, after making a minus sign correction: τ = ∫ (-gμνTμTν)1/2 dt.[17]
Between any two points, you can always define curves of arbitrarily long length, but there is always a lower bound on length in a Riemannian metric. It does not follow, however, that this lower bound is realized in a particular curve. You may have a curve of minimum length between two points connected by spacelike curves, but this will never be the case for two points connected by timelike curves in a Lorentzian metric. You can always find curves in which the proper time between such points is arbitrarily small, by choosing a smooth curve that zigzags across a timelike curve.[18] In only some kinds of spacetime, the length of curves between two points can have an upper bound, in which case the curve of greatest proper time between two points will be a geodesic, though not every geodesic maximizes proper time.
Geodesics in general are the straightest possible paths
between two points. When two spacetime points in a pseudo-Riemannian manifold are spacelike separated, there can be a curve of minimum length l between them, which will also be a geodesic. Timelike geodesics correspond to the trajectories of bodies in freefall acceleration, though, as noted, there is no timelike curve of minimum proper time. It is mathematically impossible for a geodesic to transition from timelike to spacelike since its tangent vector is parallel propagated along itself.[19] Null curves are always geodesics, corresponding to the motion of light in a vacuum.
Analytic solution is possible only for some basic scenarios, such as uniform 4D curvature, or foliated 3D hypersurfaces that each have uniform 3D curvature. The latter enables us to speak of a given hypersurface having the present curvature of the universe,
but this is just a mathematically convenient frame (with time dimension orthogonal to spatial hypersurfaces), not a privileged frame, and the relativity of simultaneity holds the same significance.
The pseudometric changes dynamically, but there is no way to separate the dynamic and background aspects of the metric. This change is defined relative to the time dimension in a given frame, but there are cross terms in the tensor among all four dimensions. So you cannot just say that there is this space whose curvature changes over time (except in the particular case of spatial homogeneity, which conveniently is a good approximation of our universe in frames comoving with the cosmic microwave background). Additionally, Weyl curvature means that there are non-zero off-diagonal elements in the metric. This forces us to recognize that we cannot neatly isolate curvature as purely spatial
and independent of time.
It might be thought that the contribution of all mass in the universe to the curvature of spacetime implies that a point in spacetime can be instantly affected by a distant object, or even by an event in the future. In fact, we only measure curvature locally, as a differential. This is the whole point of having a stress-energy tensor model, which lets us preserve locality in our computation.
Despite all these differences from special relativity, physical motions are still represented as world lines, which remain timelike in all frames in general relativity, just as in the special theory. Although the theory of general relativity imposes no formal constraints on manifold curvature, all physically relevant models entail time orientability, so that it is possible to distinguish whether a local tangent vector points forward or backward in time. Even if we were to allow causal loops, this would not deny the directionality of temporal succession, though all events in the loop besides the present would be both past and future, and you could have a causal grandfather paradox.
Restricting ourselves to time-orientable manifolds, we can define future and past light cones locally, but not globally, since the precise orientation of the light cone boundaries (null curves) varies at different successive presents in a world line. We can, however, define a frame-independent chronological future
for each event P, which consists of all events that can be reached from P by a future-directed timelike curves. A curve is future-directed if its tangent vector at every point is future-directed. More broadly, we may define the causal future
of an event P as consisting of all future-directed timelike or future-directed null curves.[20] We exclude null curves from chronological
future since they involve no elapse of proper time. The timelessness of light presents no existential problem for the reality of time, since it is impossible to define a local frame that is a proper frame of light, just as it was impossible to have a rest frame of light in special relativity.
It is mathematically possible for there to be a spacetime manifold in which it is impossible to have any time slices,
i.e. global spacelike hypersurfaces defined to have the same time in a particular frame. Our universe, however, is not such a case, since such hypersurfaces can be defined in comoving coordinates.
A particular application of causal past
can be seen in an event’s coordinate-independent particle horizon,
within which is contained all the world lines since the beginning of the universe that can send signals to the event. The particle horizon can differ in scope from a Minkowskian past light cone, due to the metric expansion (or contraction) of space.
Talk of metric expansion or contraction seems to imply that spacetime is something substantive. This need not follow in the strict ontological sense of a being that is not predicated of another being, or in the materialistic sense of being an extensive body. It does require us, at the least, to consider spacetime as distinct from all other physical properties of matter.
Einstein clearly believed in his later years that the determinations past, present, and future were purely subjective, and that objective physical reality was the entire complex of events taken at once. Yet he was also a strenuous advocate of causal determinism, a definitely time-ordered notion. One might say he believed in a B-series but not an A-series, yet we have seen that this position is philosophically untenable. His denial of the reality of A-determinations was grounded in a generalization of his conviction that all perspectives are equally valid. The perspective of me being born is no less valid then that of me dying. However persuasive this may seem, especially as a consolation that helps us look away from the transience and destructiveness of time, as Einstein did upon the death of his friend Michele Besso, we can only regard this as a faith, not as something empirically validated, certainly not something we observe. Even those who mock the consolations of others are not immune to devising eternalistic consolations of their own.
I share nothing of that sadism, disguised as zeal for truth, which delights in denying people their consolations. Maybe there is a sense in which the past is still real, and everything that occurs is immortal, either in having once really occurred, or having been forseen by God or the Absolute. Such possibilities do not exclude or contradict the objective reality of change, which makes growth and life possible. Indeed, if there were no reality to growth and life, this would abolish precisely that aspect of the past one wistfully desires to preserve.
[1] J. Ellis McTaggart. The Unreality of Time.
Mind 17 (1908): 456-473.
[2] In the 1908 paper, the series names were unhyphenated, but commentators have used hyphens, following McTaggart’s convention in The Nature of Existence (Cambridge Univ. Press, 1921), where his arguments on time were restated.
[3] McTaggart, op. cit., p. 466.
[4] Louis O. Mink. Time, McTaggart and Pickwickian Language.
The Philosophical Quarterly 10(40) (1960): 252-263.
[5] For a rigorous definition of continuity of a function, see, e.g., Tom M. Apostol, Calculus, 2nd ed., Vol. I (New York: Wiley, 1967), pp.130-31; or Walter Rudin, Principles of Mathematical Analysis, 3rd ed. (New York: McGraw-Hill, 1976), pp.85-86.
In physics and other applications, it is common to treat infinitesimals as though they were finite quantities, but this is valid only as a first approximation where higher order derivatives can be ignored. It is not possible to define general norms for such usage. When in doubt, it is always valid to refer to the rigorous conception in terms of limits. [Francesco Severi. Lezioni di Analisi, 2nd ed. (Bologna: Nicola Zanichelli, 1946), pp.207-208.]
[6] Mink, op. cit., p.256.
[7] Ibid., p.258.
[8] Ibid., p.259.
[9] Loc. cit.
[10] Ibid., p.261.
[11] Ibid., p.263.
[12] Richard M. Gale. McTaggart’s Analysis of Time
American Philosophical Quarterly 3(2) (1966): 145-152.
[13] Ibid., p.148.
[14] Vesselin Petkov. Conventionality of Simultaneity and Reality.
Philosophy and Foundations of Physics, 4 (2008): 175-185.
[15] Since we cannot visualize 4-dimensional space and 3-dimensional slices, we can visually represent at most 2 space dimensions, x and y, with time t being the third axis. All points with the same value of t will be a plane parallel to the x-y plane.
[16] Penrose diagrams can represent the causal relations among events schematically, but not metrically like a map with scale. The schematic nature of such diagrams is evident, for example, with the representation of points at infinity as lines. They do not allow us to compare different coordinate representations of spacetime.
[17] Robert M. Wald. General Relativity (Chicago: Univ. of Chicago, 1984), pp.43-44.
[18] Ibid., p.234.
[19] Ibid., pp.44-45.
[20] Ibid., pp.189-90.
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