## 1. Introduction



Around 500 B. C. Heraclitus wrote the following:



> 
> Everything flows and nothing abides; everything gives way and nothing
> stays fixed.
> 
> 
> You cannot step twice into the same river, for other waters and yet
> others, go flowing on.
> 
> 
> 
> Time is a child, moving counters in a game; the royal power is a
>  child's.[1]
> 
> 
> 



Transience is basic, and the present is primary. Those things which
exist now do not abide. They slip into the past and non-existence,
devoured by time, as all experience attests.



A generation or so later we have a classic statement of the opposing
view by Parmenides:



> 
> There remains, then, but one word by which to express the [true] road:
> Is. And on this road there are many signs that What Is has no
> beginning and never will be destroyed: it is whole, still, and without
> end. It neither was nor will be, it simply is--now, altogether,
> one, continuous...
> 



Permanence is basic. No things come to be or, slipping into the past,
cease to be. Past, present, and future are distinctions not marked in
the static Is. Time and becoming are at best secondary, at worst
illusory, as our understanding of the world confirms.



Turn now to modern times and to a paragraph in Rudolf Carnap's
intellectual autobiography (Carnap 1963, pp. 37-38):



> 
> Once Einstein said that the problem of the Now worried him seriously.
> He explained that the experience of the Now means something special
> for man, something essentially different from the past and the future,
> but that this important difference does not and cannot occur within
> physics. That this experience cannot be grasped by science seemed to
> him a matter of painful but inevitable resignation. I remarked that
> all that occurs objectively can be described in science; on the one
> hand the temporal sequence of events is described in physics; and, on
> the other hand, the peculiarities of man's experiences with
> respect to time, including his different attitude towards past,
> present, and future, can be described and (in principle) explained in
> psychology. But Einstein thought that these scientific descriptions
> cannot possibly satisfy our human needs; that there is something
> essential about the Now which is just outside the realm of science. We
> both agreed that this was not a question of a defect for which science
> could be blamed, as Bergson thought. I did not wish to press the
> point, because I wanted primarily to understand his personal attitude
> to the problem rather than to clarify the theoretical situation. But I
> definitely had the impression that Einstein's thinking on this
> point involved a lack of distinction between experience and knowledge.
> Since science in principle can say all that can be said, there is no
> unanswerable question left. But though there is no theoretical
> question left, there is still the common human emotional experience,
> which is sometimes disturbing for special psychological reasons.
> 



This difference as expressed here between Einstein and Carnap (that
is, between the Heraclitean and Parmenidean attitude towards time and
change) is the subject of this article, which will use modern
physics--especially modern spacetime theories--as a set of
lenses through which it is hoped the riddles of time will come into
sharper focus. There are many ways, however, to approach these
questions. Early in the twentieth century, Anglo-American philosophy
turned to consideration of language as a way to clarify philosophical
disputes. Philosophers of time debated the relations between and the
relative primacy of tensed language (concerning the notions of
present, past, and future) and tenseless language (concerning the
relations of simultaneity and temporal precedence). Our considerations
of physics will generally, though not completely, skirt linguistic
disputes. The reader interested in following these debates can find a
useful introduction in the entry on
 time,
 a classic presentation in Gale (1968), a review and discussion in
Tooley (1999), and Dyke (2007 chapter 3, 2013).



Other philosophers have been influenced by analogies between time and
modality. The reader interested in this way of thinking about time
should consult the article on
 temporal logic.
 The present article will focus on time in physics and the relations
between time and space. Other philosophical approaches focus on the
primacy of experience in our understanding of time. The reader
interested in these approaches may wish to consult the entry
 the experience and perception of time.


## 2. Newtonian Spacetime



Modern physical theories are often formulated in a language that
permits one to express a variety of different views with respect to
time and its relation to space. One can, for example, formulate the
basic ideas of classical (that is, Newtonian) physics, the special
theory of relativity, and the general theory of relativity in this
language. For a brief introduction to manifolds and the spacetime
view, see the section on
 modern spacetime theories
 in the entry on
 the hole argument
 in this Encyclopedia. For more detail with minimal technical demands
the reader should see the first four chapters of Geroch (1978) or the
more demanding chapter 2 of Friedman (1983).



For our purposes, the defining feature of a manifold that is a
Newtonian spacetime is that the temporal interval between any two
points or events in the spacetime, \(p\) and \(q\),
is a well-defined quantity. This quantity is
well-defined in that it does not depend upon point of view, reference
frame, coordinate system or "observer". This quantity,
then, is *absolute* in the sense of being frame- or
observer-independent. (In the special theory of relativity the
temporal interval between two distinct spacetime points fails to be
absolute in this sense.)



If the temporal interval between two events is 0, then we say that the
two events are *simultaneous*. This relation of (absolute)
simultaneity is an equivalence relation (That is, it is reflexive,
symmetric, and transitive.) that slices (partitions or foliates) the
spacetime or manifold into mutually exclusive and exhaustive planes of
simultaneity. These planes of simultaneity can then be completely
ordered by the relation 'is earlier than' or its converse
'is later than'.


### 2.1 Presentism, Possibilism, Eternalism



The geometrical structure of Newtonian spacetime reflects the way we
ordinarily think about time and is the proper backdrop for introducing
the three major rival metaphysical views of time, as illustrated
below:



![Figure 1](3metaphysics.jpg)

Figure 1. Three Metaphysics of Time




The first view, represented on the left, is the ontologically austere
view called *presentism*, the view that only the present
exists. The past has been but is no longer, while the future will come
to be but is not yet. Note that it is the convention of these diagrams
that one spatial dimension is suppressed. The present is actually a
three dimensional global slice of the spacetime. Moreover, the
illustration necessarily represents the spatial extent of the present
as finite and may suggest that time also has a beginning and/or end.
These views are, however, merely artifacts of the representation and
not integral to presentism, possibilism, or eternalism. The diagram
illustrating presentism also has four arrows pointing towards the top
of the page (conventionally taken to represent the future direction)
attached to the plane representing the present. These arrows are meant
to indicate something that is integral to presentism, the idea that
the present (and hence the existent) constantly shifts or changes.
These arrows represent, then, the dynamic aspect of time called
*temporal becoming* or *passage.* It is widely thought
that the deepest problems in the metaphysics of time concern the
understanding of passage or temporal becoming and its relation to
existence.



In contrast to the radical Heraclitean view of presentism, the
Parmenidean eternalist picture on the far right lacks these arrows and
indicates that there is nothing more special about the temporal
present (the *now*) than the spatial present (the
*here*). Future and past events at a place, on this view, are
no more or less real than distant events at a time. The *now*
like the *here* is a function of one's perspective,
one's position in the spacetime, and these positions are
indicated by the line in the spacetime representing the history of
spacetime locations of a particular object or person. Such a line is
often called a *world line.*



The middle view, possibilism, is indeed an intermediate view. It is a
passage view, but it is less ontologically sparse than presentism.
While on this view the future is still merely possible rather than
actual (hence its name), the past has become and is fully actual. If
one thinks of the future as a branching structure of alternative
possibilities (as the result, perhaps, of free human choices or
indeterministic quantum measurements), then one can think of the past
and present as the trunk of that tree, growing as possibilities become
actual in the present. This view is also known as *the growing
block* view.



Possibilism seems to capture much of the way we think about time and
being. While the sparse symmetry of presentism is attractive, there
are many deep asymmetries concerning past and future that it fails to
reflect. I can easily ascertain, for instance, yesterday's
closing number for the Dow Jones Industrial Average, but by no
efforts, however great, can I now ascertain tomorrow's close.
And it seems as if my future actions (or certain sorts of quantum
measurements) can actualize some future possibilities as opposed to
others, whereas past actions (or the results of past quantum
measurements) seem no longer to admit of past alternatives. Even if
one allows for the possibility of retrocausation, that is, for the
possibility of an effect preceding its cause in time, it is generally
held that a present cause can not change or alter the past. It would
merely make the past what it was. (See the entry on
 backwards causation
 for further consideration of this topic.)



Eternalism too, *prima facie*, would seem to have trouble
accounting for the asymmetries built into possibilism, while also
apparently bearing the burden of an implausible denial of passage. But
the first topic to which we shall turn is an argument, prominent in
twentieth century philosophy of time, that passage or becoming is a
self-contradictory idea. If the argument is correct, then neither
presentism nor possibilism can be correct metaphysical views of time
and being.


### 2.2 McTaggart's Argument



At the beginning of the 20th century, J. M. E. McTaggart (1908)
presented an argument which purported to prove that time is unreal.
According to McTaggart (1927, pp. 9-10):



> 
> Positions in time, as time appears to us prima facie, are
> distinguished in two ways. Each position is Earlier than some and
> Later than some of the other positions... . In the second place,
> each position is either Past, Present, or Future. The distinctions of
> the former class are permanent, while those of the latter are not. If
> \(M\) is ever earlier than \(N\), it is
> always earlier, But an event, which is now present, was future, and
> will be past.
> 



The first structure of "positions in time," McTaggart
called *the B-series.* I will assume that McTaggart intended
the B-series to coincide with the Newtonian spacetime structure
described above. McTaggart noted that there was something static or
"permanent" about the B-series. If, for example, event
\(e\_1\) is earlier than event \(e\_2\) at some time or other, then one
may truly assert at all times that \(e\_1\) is earlier than \(e\_2\)
.



The dynamic element of time must be represented, in McTaggart's
view, by the series of properties of pastness, presentness, and
futurity, which (in contrast to the static B-series) are constantly
changing. A given event becomes less future, becomes present, and then
becomes increasingly past. This latter ever-shifting series McTaggart
called *the A-series.*



While there are many obscurities in McTaggart's writing, it
seems clear that his argument that time is unreal runs along the
following lines:



(1)
 there can be no time unless it has a dynamic element (that is, on
his view, unless there is an A-series),
(2)
 there can be no A-series, because the supposition that there is an
A-series leads to contradiction.


The contradiction alleged by McTaggart is that:



(A1)
 every event must have many, if not all, the A-properties (or
A-determinations, as they are sometimes called) whereas,
(A2)
 since the A-properties are mutually exclusive, no event can have
more than one of them.


Near the end of a career in which he spent much time and effort in
thinking about McTaggart's argument, C. D. Broad (1959, p. 765)
wrote:



> 
> I felt from the first, and still feel, that the difficulty which
> arises is (a) embarrassing enough prima facie to demand the serious
> attention of anyone who philosophises about time, and (b) almost
> certainly due to some purely linguistic source (common, and perhaps
> peculiar, to the Indo-European verb-system), which it ought to be
> possible to indicate and make harmless.
> 



Broad's claim (a) was vindicated by the fact that
McTaggart's argument has received serious attention from most
subsequent philosophers who pondered the metaphysics of time. Much of
this debate concerns the relative relations of the two series. Is the
A-series fundamental and the B-series derived from it, or vice versa;
or does, perhaps, one series supervene upon the other? In the formal
mode, the questions become whether the B-series may somehow be reduced
to, may be defined in terms of, the A-series (or vice versa). These
debates concern mainly language rather than physics and will not be
considered
 here.[2]



What emerges from the McTaggart literature is, first of all, a
tendency to identify the existence of passage or temporal becoming
with the existence of the A-series (that is, to think of becoming as
events changing their properties of pastness, presentness or nowness,
and futurity) and hence the tendency for debates about the existence
of passage to focus on the merits or incoherence of the A-series
rather than examining alternative accounts of becoming. (But Cf.
Fitzgerald, 1985)



There is a contrary tendency amongst those philosophers who take
modern physics seriously to be sceptical of entities like constantly
shifting temporal properties of events, since such properties play no
role in modern physical theory. One view, defended by Paul Horwich
(1987, Chapter 2) and Huw Mellor (1981, 1998), is that even though
McTaggart showed that the A-series is impossible, the B-series (that
is, static classical spacetime structure) suffices for time.



Before we expand on this theme, though, first a few words about
Broad's (b), his suspicion that there is some peculiarity of our
language(s) that creates or at least reinforces the credibility of
McTaggart's anti-passage argument. Broad suspected that there
was a subtle ambiguity in the copula 'is' between tensed
and tenseless uses, between the uses in, for instance:



It is raining



and



Seven is prime,



the former sentence containing a tensed and the latter sentence a
non-tensed or tenseless copula. It has been further suggested (Sellars
1962) that one might understand a non-tensed copula (indicated by
'be' rather than 'is') after the following
fashion



\(S\) be \(F\) at \(t\)
iff \((S\) was \(F\) at \(t\) or \(S\)
is \(F\) at \(t\) or
\(S\) will be \(F\) at \(t)\),



where the verbs to the right of the 'iff' (a
logician's abbreviation for 'if and only if') are
usual tensed verbs.



Alternatively, one might think of a tenseless copula as the usual
copula stripped of temporal information (Quine, 1960, p. 170, Mellor
1981, 1998, Chapter 7), just as the usual copula carries no spatial
information. If we indicate this tenseless copula by writing
'BE' instead of 'is,' we can say that
'It BE windy in Chicago' carries information about the
place but not the time of the wind, just as 'It BE windy at
t' tells us about its time but not its place.



These distinctions will prove helpful in the subsequent discussion of
becoming in modern physics. For the moment, one might note that Broad
could argue that McTaggart's (A1) seems plausible if
the copula is understood in some tenseless fashion, whereas
(A2) is plausible if the copula is tensed. If, however, the
copula is not univocal in (A1) and (A2), then
there is no contradiction involved in accepting both (Savitt,
2001).



The claim that the copula is ambiguous (or more precisely, polysemous)
has also been thought to have some implications for the metaphysical
dispute between presentism and the two non-presentist views sketched
above. A number of authors (Meyer (2005), Savitt (2006), Dorato
(2006), Lombard (2010), and Arthur (2019, section 2.3). See Dolev
(2007) for a different anti-ontological argument.) have argued that in
order for there to be a genuine difference between these allegedly
distinct metaphysical views, in addition to the suggestive pictures
displayed above there must be some univocal statement or proposition
concerning temporal ontology that one side asserts and the other side
denies. When the polysemy of the copula (or of the verb 'to
exist') is taken into account, it has proven very difficult to
find a clear example of such a statement or proposition. What many
philosophers take to be a profound dispute in the metaphysics of time
may well turn out to be a mere verbal tangle. (Putnam (2008), by the
way, has recently joined these critics: "[T]he question whether
the past and future are 'real' is a
pseudo-question.")



According to one author (Sider, 2006), there seems to be disagreement
concerning the following assertion: dinosaurs exist. Presentists are
understood to deny this claim, while eternalists are supposed to
affirm it. If those who endorse the claim understand the verb
'exist' to be tenseless and those who deny it take
'exist' to be a present tense verb, then both sides seem
to endorse correct claims. There is in this instance, however, no
univocal statement that is being disputed, since 'exist'
is understood differently by the two sides. This observation has led
many to try to state a version of presentism that is neither trivial
nor obviously false and to locate precisely the point at which it
differs from non-presentist views. (See Crisp (2004a,b), Ludlow
(2005), Wuthrich (2012), Golosz (2013). Tallant (2013) reviews
several such efforts negatively and adds one of his own.)



The few authors that have approached this debate from a linguistic
perspective differ on the behaviour of the verb 'exist'.
Ludlow (2005) considers both a tensed and a tenseless verb; Stoneham
(2009) is sceptical with regard to tenseless predication; and Moltmann
(2013) considers 'exist' only as a present tense verb. It
is worth noting that he claims that 'exist' functions
quite differently from 'there are' in that the latter can
be used to quantify over past, merely possible, and intensional
entities whereas the former can not. If his claim is correct, then a
common test sentence for the difference between eternalists and
presentists introduced in Sider (2006)--There exist dinosaurs--seems
almost purpose-built to breed confusion.


### 2.3 How (and How Not) To Think About Passage



If McTaggart's argument that passage is conceptually absurd or
self-contradictory fails, philosophers mindful of modern physics are
still left with Einstein's concern that passage and the now,
while deeply embedded in human experience, seem to find no place in
physics. One may agree with Carnap above that "all that occurs
objectively can be described in science" and then argue that
passage reflects something perspectival or subjective and so is
rightly omitted by physics.



The most popular version of this view holds that *now* is a
token-reflexive or indexical term, like *here* (Smart 1963,
chapter VII; Mellor 1981, 1998). Physics is not felt to be incomplete
because it fails to treat *hereness.* Why should its
indifference to *nowness* be of any greater concern?



Early proponents of this view often claimed that '\(S\)
is now \(F\)' meant '\(S\)'s
being \(F\) is simultaneous
with this utterance,' a quite implausible claim. A more
sophisticated version of the view is that the truth-conditions of
sentences like '\(S\) is now \(F\)'
can be given solely in terms of the
(tenseless) facts that exist or events that occur at the time of the
utterance or inscription of the given sentence. One can treat
*past* and *future* in a similar fashion. (Dyke,
2013)



Smart claimed that excessive attention to the tensed notions of
*now, past,* and *future* serve to project a "sort
of anthropocentric idea on the universe at large."(1963, 132)
But even if the tensed temporal locutions are anthropocentric and do
locate *us* in the universe, it may still be asked whether
these temporal locations are in a static structure, "a
four-dimensional continuum of spacetime entities," (132) or in
an unfolding or dynamic universe. Smart dismisses this latter view
because, in his view, it involves the obscure or mistaken idea that
events "become" or "come into existence."
Becoming and passage are mistakes, in his view, and harmful ones at
that. Smart writes: "Our notion of time as flowing, the
transitory aspect of time as Broad has called it, is an illusion which
prevents us seeing the world as it really is" (132).



It will be useful to untangle a couple of ideas that are confounded in
these quotes from Smart, with the help of some arguments of (mostly)
Broad's (1938, section 1.22 of Chapter 35). First is the idea
that time "flows" or, more generally, that passage is
somehow to be thought of as like motion. Perhaps time itself somehow
moves. Or perhaps, as Broad wrote in a famous passage (1923, 59):



> 
> We are naturally tempted to regard the history of the world as
> existing eternally in a certain order of events. Along this, and in a
> fixed direction, we imagine the characteristic of presentness moving,
> somewhat like the spot of light from a policeman's
> bull's-eye traversing the fronts of the houses in a street. What
> is illuminated is the present, what has been illuminated is the past,
> and what has not yet been illuminated is the future.
> 



This picture of passage has come to be called *the moving
spotlight* view, or just *the moving
 NOW*.[3]



Motion is one sort of change, change of spatial position with respect
to time. The motion of time, then, must be change of time with respect
to ... What? If the answer, by analogy with motion, is
"time", one might be rightly puzzled as to how time (or
anything else, for that matter) can change *with respect to
itself.* Furthermore, if it is just time again, then the ratio of
these two quantities expressing the supposed rate of change is a pure
or dimensionless number if the dimensions of the quantities in this
ratio cancel. (See Price 1996, p. 13.) A pure number is not a rate of
change, although it may represent various rates of change (for
instance, 30 meters/second or 30 miles/hour). As Price remarks,
"We might just as well say that the ratio of the circumference
of a circle to its diameter flows at \(\pi\) seconds per
second!"



Tim Maudlin (2007) has replied to Price that the rate of one second
per second is a perfectly fine rate. Consider, he says, a fair
exchange rate (based, say, on purchasing power parity) for currencies.
The fair exchange rate for the British pound might, at some time, be
1.52 US dollars per pound. Similarly, Maudlin observes that the fair
exchange rate for the US dollar with respect to itself must be 1 US
dollar per US dollar. "If you think that this answer is
meaningless," Maudlin writes, "imagine your reaction to an
offer of exchange at any other rate" (2007, 112).



Price (2011) has replied to Maudlin by abandoning the claim that the
units in "one second per second" cancel, leaving a pure
number. He now insists, rather, that a ratio of a quantity to itself
can not be a rate at all. Maudlin's example only seems to evade
this point by supposing a ratio of *dollars offered* to
*dollars returned*. That ratio differs, as he sees it, in an
essential way from the ratio of one second to one second or, for that
matter, from the ratio of one mile to one mile, which are both ratios
of quantities to themselves. Neither in his view can be a rate, and
the former ratio is no more characteristic of time than the latter
ratio is of space. (See also Markosian (1993), Olson (2009), Raven
(2010), Skow (2012), and Arthur (2019, 134-35).)



If (in order to avoid the apparent collapse of a ratio or rate into a
pure number) the time in the denominator of the ratio expressing the
rate of time's motion is held to be a different temporal
dimension from the one in the numerator, then for it to be a genuine
time there will have to be passage in it, requiring yet a third
temporal dimension. One can see that we are at the beginning of an
infinite regress, unless the third temporal dimension is identified
with the first (as in Schlesinger 1980, Chapter II), leaving us still
in the uncomfortable position of having two temporal dimensions. It
seems at best heroic, at worst hopeless, to try to understand passage
as a kind of motion.



Broad thought that trying to explain or represent passage in terms of
qualitative change was "doomed to failure." A thing or
substance, \(S\), can change in terms of a quality or
property if property \(P\_1\) and property \(P\_2\) are determinates
under a given determinable and \(S\) is \(P\_1\) at
\(t\_1\) but \(P\_2\) at \(t\_2\). The passage of time, then, is to be
thought of as an event's having (say) the property of
presentness and then immediately losing that property but gaining (and
losing in turn) a long and possibly endless series of properties of
the increasing degrees of pastness.



In order for a thing to change from having \(P\_1\) at \(t\_1\) to
having \(P\_2\) at \(t\_2\) it must evidently persist at least from
\(t\_1\) to \(t\_2\), but the events usually supposed in discussions of
passage are instantaneous events, which have no duration at all. They
can not undergo qualitative change. It is sometimes argued that the
properties that make up the A-series (and so change of which
represents passage) are special properties, which even instantaneous
events can gain and lose, but this is special pleading. As noted
above, physics has so far no need of such special properties and such
special change and so is unlikely to be sympathetic to this special
pleading.



Finally, Broad notes that (assuming one wishes to think of passage as
like qualitative change) the acquisition and loss of (say) presentness
by an event would itself be an event, a second-order event, in the
history of a first-order event. Since the first-order events are, by
hypothesis, durationless, it is tempting to suppose that this history
takes place in a second temporal dimension. We find ourselves again
launched on what looks to be an infinite regress of temporal
dimensions.



These are strong arguments against two perennially tempting ways to
construe temporal becoming--as like motion or qualitative change.
They are strong arguments against the existence of temporal becoming
if there is no other way to understand it. Broad thought, however,
that he had a third way. Having pointed out the superficial
grammatical similarity between '\(E\) became
louder' and '\(E\) became present',
Broad said that our understanding of these two kinds of assertions
need not be dictated by it. He wrote (1938, p. 280-1):



> 
> Again, any subject of which we can significantly say that it
> "became louder" must be a more or less prolonged
> noise-process, which divides into an earlier phase of less loudness
> adjoined to a later phase of greater loudness. But a literally
> *instantaneous* event-particle can significantly be said to
> "become present"; and, indeed, in the strict sense of
> "present" *only* instantaneous event-particles can
> be said to "become present". To "become
> present" is, in fact, just to "become", in an
> absolute sense; i.e., to "come to pass" in the Biblical
> phraseology, or, most simply, to "happen". Sentences like
> "This water became hot" or "This noise became
> louder" record facts of *qualitative change.* Sentences
> like "This event became present" record facts of
> *absolute becoming.*
> 



The terminology may be pretentious, but the idea is simple. Absolute
becoming is just the happening of events. The *raison
d'etre,* the very being or existence of events, is in
their happening (at some place and time). If one is willing to embrace
this category of entity at all, then one has the tools for a
minimalist understanding of passage. Given the geometric richness of
Newtonian spacetime, we can say that some events occur at the same
time and so form a class of simultaneous events. These classes can be
ordered by their times, so we can say that some events occur before or
after others. The passage of time is then just the successive
happening of (simultaneity sets of)
 events.[4]
 It may be this picture of passage that the great logician Kurt
Godel had in mind when he wrote (1949, p. 558): "The
existence of an objective lapse of time ... means (or, at least,
is equivalent to the fact) that reality consists of an infinity of
layers of 'now' which come into existence
successively."



There is an ambiguity in this last quote, however, that we must note.
Did Godel think that the layers of now come into existence (as
what is to be becomes what is now) and then immediately cease to exist
(as what is now becomes what once was), which is the presentist
metaphysics of time? Or did he think that the layers of now come into
existence and forever stay in existence, as the possibilist picture
maintains? If one's basic ontology consists of the sort of
events characterized above and often invoked in discussions of time,
(idealized) instantaneous happenings, then the presentist picture
seems inevitable.



The metaphysics of time is, however, one of the cross-roads of
philosophy where issues intersect. If one thinks of a basic ontology
consisting not of events but of substances or continuants, then one is
apt to wonder what it is that makes sentences marking episodes in the
histories of such substances--sentences like '\(S\)
is \(\Phi\) at \(t\)'--true.
One frequent suggestion is that
the "truth-makers" of such sentences are facts, the fact
that at \(t\), \(S\) *is*
\(\Phi\). Then one might further note that in the current year, 2021,
we can say:


1. It is a fact that Mount St. Helens erupted in Washington in
1980.
2. It is a fact that Justin Trudeau is now Prime Minister of
Canada.
3. It is a fact that there will be a total eclipse of the sun in the
United States in 2045.



These facts, when compared to evanescent events, seem to have great
stability, the first one lasting (since it *is* a fact
...) at least from 1980 till the present. The third one is,
however, a special sort of fact, clearly not dependent on human will
or choice and almost certainly not dependent upon any quantum
measurements either. Future facts that do depend upon human choice or
quantum measurement, should they be facts now, would seem to constrain
human choice or quantum measurement in ways that many philosophers
find undesirable. It is easy to convince oneself, then, that future
facts of those two sorts can not really be part of the existing.
Perhaps, then, facts like fact 3 above can be argued away as well. The
result of this (lightly sketched) train of thought is, of course, the
possibilist picture of time.



It seems unlikely that a simple argument will decide between these two
metaphysical pictures of time, presentism and possiblism. Showing that
McTaggart's argument is flawed, because it relies on an
ambiguity in the copula 'is', and that there is a way to
construe passage that side-steps the traditional objections, moreover,
does not show that eternalism is false but only that it is optional.
In Newtonian spacetime it may appear implausible, but it may fare
better when we turn to Minkowski spacetime.


## 3. The Special Theory of Relativity



The Special Theory of Relativity (Einstein, 1905) was presented as a
geometric theory of spacetime in Minkowski
 (1908).[5]
 For our purposes, the key change from Newtonian spacetime to
Minkowski spacetime is that in the latter it is no longer the case
that the temporal interval between any two points or events in the
spacetime, \(p\) and \(q\), is a
well-defined quantity. In fact, the temporal interval between two
points in the spacetime (and hence the simultaneity of two points in
the spacetime) is not defined at all until a coordinate system or
frame of reference (with some arbitrarily chosen spacetime point as
origin of the frame) is chosen. A peculiar feature of special
relativity (as opposed Newtonian physics) is that each inertial frame
defined by an "observer" passing through the chosen origin
and moving with some constant non-zero velocity that is less than the
speed of light (the only coordinate systems or frames of reference
that will be considered in our discussion of the special theory) picks
out a *distinct* set of points as simultaneous with the origin.
This feature of special relativity is called *the relativity of
simultaneity.*



The relativity of simultaneity is a consequence of two startling
assumptions. First, each of these "observers", no matter
at what (subluminal) speed or in which direction they or the source of
the light are moving (as long as neither the speed nor the directions
change), must come to the same result (conventionally indicated as
\(c)\) when they measure the speed of light. We will not attempt to
justify the assumption of the constancy of the speed of light here,
though many standard texts present the empirical and theoretical
background that led to it. Nor is it obvious that this assumption
leads to the relativity of simultaneity, though one of the joys of
even elementary presentations of the subject is that this *prima
facie* astonishing connection can be convincingly demonstrated to
persistent non-specialists.



The second assumption typically made in presentations of the special
theory is *the Principle of Relativity:* All inertial frames
are completely equivalent for the formulation of the laws of
 physics.[6]



A glance back at Figure 1 reminds us that presentism and possibilism
suppose that one plane of simultaneity is uniquely metaphysically
important. In the former view, it represents all that exists. In the
latter view, it is the locus of becoming, the dividing line between
the merely possible future and the actual past-plus-present. The
special theory of relativity tells us that there is an infinity of
planes of simultaneity passing through any given spacetime point and
that no physical test can distinguish one from amongst the lot. What
was supposedly metaphysically distinguished is physically
indistinguishable. Assuming that we humans are complex physical
systems, then we have no way to distinguish *the* present from
amongst the multitude of putative presents.



An enthusiast could make much of this fact. For instance, the
mathematician (and science fiction writer) Rudy Rucker wrote (1984, p.
149):



> 
> As it turns out, it is actually *impossible* to find any
> objective and universally acceptable definition of "all of
> space, taken at this instant." This follows ... from
> Einstein's special theory of relativity. The idea of the block
> universe is, thus, more than an attractive metaphysical theory. It is
> a well-established scientific fact.
> 



On the other hand, the distinguished philosopher and logician Arthur
Prior thought that the above conclusion showed that special relativity
is an incomplete view of reality (Prior,
 1970):[7]



> 
> One possible reaction to this situation, which to my mind is perfectly
> respectable though it isn't very fashionable, is to insist that
> all that physics has shown to be true or likely is that in some cases
> we can never *know,* we can never *physically find out,*
> whether something is actually happening or merely has happened or will
> happen.
> 



We shall look at more nuanced reactions to the relativity of
simultaneity below, but first it will be useful to introduce an
argument that plays somewhat the same role in Minkowski spacetime as
McTaggart's argument did in Newtonian spacetime. Versions of the
argument are endorsed in papers by the physicist Wim Rietdijk (1966,
1976) and the philosopher Hilary Putnam (1967), but the presentation
here will be based on an example found in Roger Penrose's book,
*The Emperor's New Mind.*



Imagine that the Andromeda galaxy, which is about two million light
years or \(2\times 10^{19}\) kilometers from Earth, is at rest with
respect to Earth. On Earth two friends walk past each other, Alice
walking along the Earth-Andromeda line towards Andromeda, Bob walking
along that line but away from Andromeda. Each is walking at a
comfortable pace, say 4 km/hour. One can calculate that their planes
(or spaces) of simultaneity at the instant at which they pass each
other on Earth (Call the event of their meeting \(\mathbf{O})\)
intersect the history or *world line* of Andromeda about 5
3/4 days apart. (Call these two events \(\mathbf{A}\) and
\(\mathbf{B}\), respectively. We are idealizing Andromeda as a point,
for the purpose of this example.) Imagine, finally, that during this 5
3/4 day period between \(\mathbf{B}\) and \(\mathbf{A}\) a
momentous thing happens. The Andromedeans launch a space fleet aimed
at invading Earth.



![image](andromedeans.jpg)

Figure 2. The Andromedean Invasion




The launch of the invading fleet is prior to \(\mathbf{A}\) and so
*in some sense* in Alice's past. But since the launch is
after \(\mathbf{B}\), it is in that same sense in Bob's future.
Penrose comments:



> 
> Two people pass each other on the street; and according to one of the
> two people, an Andromedean space fleet has already set off on its
> journey, while to the other, the decision as to whether or not the
> journey will actually take place has not yet been made. How can there
> still be some uncertainty as to the outcome of that decision? If to
> *either* person the decision has already been made, then surely
> there *cannot* be any uncertainty. The launching of the space
> fleet is an inevitability. (p. 303)
> 



This is an odd situation indeed. An event in Bob's future seems
in some way to become fixed or inevitable by being in Alice's
past. But that is not the end of the oddness here. Imagine that at
point \(\mathbf{A}\) (where Alice's plane of simultaneity
intersects the world line of Andromeda) there is an Andromedean,
Carol, who is walking directly towards Earth at about 4 km/hour. Then
Carol's plane of simultaneity intersects Earth at some point
\(\mathbf{C}\) which is about 11 1/2 days after \(\mathbf{O}\),
the meeting of Alice and Bob. *If* all events (like
\(\mathbf{A})\) in Alice's past or present at \(\mathbf{O}\)
have happened, are fixed, or are real, then the principle of
relativity (the second assumption of the special theory introduced
above) suggests that we must also extend the same courtesy to Carol;
and so simultaneous with the fixed and real event \(\mathbf{A}\)
(Carol's walking towards Earth at exactly the point at which
Alice's plane of simultaneity intersects the history of
Andromeda) is the event \(\mathbf{C}\) (and so fixed and real), the
intersection of Carol's plane of simultaneity with Earth, which
is in the future of *both* Alice and Bob. It is easy to see
that, by adjusting the speeds of Alice and Carol, any event to the
future of \(\mathbf{O}\) can be shown to be fixed or real or
inevitable. But \(\mathbf{O}\) itself was just an arbitrarily chosen
point in the spacetime. "It begins to seem that if anything is
definite at all," we might echo Penrose, "then the entire
space-time must indeed be definite! There can be no
'uncertain' future." (p. 304)



Roberto Torretti (1983, p. 249) calls the resulting view of the
definiteness or fixity of all events in the spacetime
*chronogeometrical determinism.* A slightly better name might
be *chronogeometrical fatalism,* as we will see below. In order
to see more clearly, however, what has gone wrong in the argument
above, it will be useful first to look more closely at the problems
attendant upon trying to import our commonsense or classical
intuitions about time into the understanding of Minkowski spacetime
and then to describe briefly the structures peculiar to that spacetime
itself. To begin with the first task, one of the most notable attempts
to bring our time into Minkowski spacetime is to be found in Sellars
(1962), a determined attempt by one of the most profound systematic
metaphysicians of the latter half of the 20th century.


### 3.1 Relativizing the Present



Wilfrid Sellars
 believed that the various invariant or observer-*independent*
elements of Minkowski spacetime (like the light cone structure to be
described below) that are typically given primary consideration in
treatments of relativity from a spacetime perspective are abstractions
from and secondary to the 'perspectival' pictures or
*perspectives*, the myriad of inertial systems or reference
frames. When it comes to time, however, he believed that there is
something even more fundamental than these perspectives:



> 
> ...we must distinguish between a moment, t, and the event of the
> moment's being present with respect to a given perspective and,
> above all, between the event of the moment's being present with
> respect to a given perspective and the event of the moment's
> being *present*. The latter, of course, is the essential
> feature of a temporal picture of the world. (577)
> 



While there is in Sellars' paper a lengthy and illuminating
series of reflections on the relation between events, facts, and
substances, there is no guidance offered on the relation between a
moment's being present with respect to a given perspective and a
moment's simply being present, a concept which is ill-defined
from a relativistic point of view. If this latter is indeed an
essential feature of a temporal picture of the world, then it would
seem that special relativity does not provide us with a temporal
picture of the world. If on the other hand the world *is*
fundamentally temporal in the way that Sellars insists it must be,
then (at least as far a special relativity as a representation of that
world is concerned), Sellars' famous scientific realism is
compromised because what he deems for metaphysical reasons to be an
'essential feature' of a temporal picture of the world
does not appear in a fundamental theory of spacetime.



There are some useful lessons to learn from the failure of this
attempt to import pre-relativistic temporal categories (like simply
being present as opposed to present with respect to an inertial frame)
into Minkowski spacetime. First, Sellars is careful to distinguish
between events as things that happen or occur or take place and the
'events' (the use of single quotes is Sellars') that
are basic in relativity. The latter are just spacetime points. They do
not take place or occur, and they are not the relata in causal
relations, whereas events are. (But cf. Tooley (1997, Chapter 9))



Sellars also presents a distinction between what he calls (p. 586)
*categorial existence statements* and what, for lack of a
better term, I will call *non-categorial existence statements.*
The former invoke frameworks, like the framework of substances or the
framework of 'events', the frameworks that Sellars takes
great pains to compare in his essay. He is inclined towards a view he
attributes to Carnap (1950) that to say that, for instance,
'Things exist' is to make the categorial, metalinguistic
claim that there are thing words in our language L now. This use of
'exist,' claims Sellars, has no (future or past) tensed
contrast.



Non-categorial existence statements, on the other hand, assert the
existence of individuals or less general kinds in a fully tensed
fashion. Sellars would construe them in the following way (p.
592):



> 
> \(a\) be existent \(\{\) before *now*,
> *now*, after *now* \(\} \equiv\)
>   
> 
> \(\exists x(x\) be \(\{\) before *now*, *now*, after
> *now* \(\}\) *and* \(x\) be \(\Phi\_1
> ,\ldots ,\Phi\_n\) *and*
>   
> 
> '\(\Phi\_1\)', ..., '\(\Phi\_n\)' be our
> criteria *now* for [being] '\(a\)')
> 



Leaving aside Sellars' idiosyncratic way of construing existence
statements, if a distinction between categorial and non-categorial
existence claims can be made, then it would be perfectly coherent to
indicate that one is adopting or working in the framework of
'events' by asserting that 'events exist'
(tenselessly, in the categorial sense) without being committed to the
"tenseless existence" of particular events which may be
past, present, or future (in the non-categorial sense).



It has sometimes been thought that commitment to a spacetime
framework, as is often explicit in treatments of special (and general)
relativity, is tantamount to commitment to eternalism or the
'block universe', since to say that spacetime points exist
seems inconsistent with saying that some spacetime points are future
and so do not exist yet or are past and so exist no longer.
*If* some distinction of the type just sketched can be made
between categorial and non-categorial existence statements, then
eternalism is not a straightforward consequence of adopting the
spacetime
 viewpoint.[8]
 One can consistently affirm the category of events without also
affirming that particular events exist (occur, happen) now or at any
other specified time. (See SS3.4 of Arthur (2019) for more
detail.)



Granting Sellars all the distinctions that he wishes, however, does
not give him the tools to avoid the central problem sketched above.
Since the problem is, in one form or another, the problem that any
view that tries to define a notion of temporal becoming in Minkowski
spacetime must address, it is worth examining it a bit more closely.
Sellars wrote (p. 591):



> 
> ... in the case of an 'event' framework, a primary
> temporal picture is a picture with a *now.* And even if one
> observer's *now* is another observer's
> *then,* or one observer's simultaneous cross sections of
> the world are another observer's sets of differently dated
> 'events',... each of their *now*-pictures is a
> primary picture, and the purely topological picture (which includes
> the measurements performed by \(S\) and \(S'\) as
> topological facts) which is common to them is not *the primary
> picture* of the world construed as a system of
> 'events,' but merely a topological abstraction common to
> the various primary pictures; and the topologically formulated
> location of individual events in the topological picture is merely the
> topologically invariant features of the criteria which identify these
> 'events' in a primary picture.
> 



In this quote Sellars is using the term 'topological'
where one would now normally use the term 'geometric', and
he is forcefully reiterating his view that the spacetime manifold of
'events' is merely an abstraction from the infinity of
distinct *primary* now-pictures of individual observers.



The first question one will surely want to ask about this view is: how
can an infinity of distinct "now-pictures" *each*
be primary? No answer is forthcoming. The second, and more troubling
question, is: how can this infinity of distinct
"now-pictures" be related to the traditional metaphysical
views under discussion? What, in short, is the connection (if any)
between the temporal notions implicit in each of the pictures and
existence of the past, present, and future? The striking fact about
Sellars' schema above for 'a be existent
*now*' is that it is *not* relativized to a
reference frame, inertial system or "perspective" and so
is not meaningful relativistically. The definition gives us no
guidance as to how to parcel out existence to elements in the infinity
of inertial frames that are admissible at a spacetime point.



If the definition or schema above were relativized to inertial frames
\(F, F'\), etc., so as to connect existence to relativistically
acceptable "primary now-pictures", its interpretation
would be either unhelpful or mysterious. Consider the following
modification of Sellars' schema above:



\(a\) be existent *now* in \(F \equiv\)
  

\( \exists x(x\) be *now* in \(F\) *and*
\(x\) be \(\Phi\_1 ,\ldots \Phi\_n\) *and*
  

'\(\Phi\_1\)',...,'\(\Phi\_n\)' be our
criteria *now* for '\(a\)')



Suppose it is not the case that \(a\) be existent
*now* in some other frame \(F'\). It seems as if this
difference must result from \(a\)'s being
simultaneous with some spacetime point \(\mathbf{O}\), say, in \(F\)
while not being simultaneous with the same point
\(\mathbf{O}\) in \(F'\). But on this reading Sellars' schema is
just a round-about way to indicate that simultaneity is
relative--the point of departure for our metaphysical questions
rather than the answer to any.



The schema looks as if it is meant to do something more, to connect
temporal notions to existence. But if so, how is existence relative to
a frame to be understood? Classical presentism, for instance, wishes
to identify existence with present existence or existence
*now.* Since the present is relativized to frames in special
relativity, may not existence be relativized to frames as well? This
is a difficult notion to understand or accept. Kurt Godel (1949,
p. 558) said flatly, "The concept of existence ... cannot
be relativized without destroying its meaning completely." Is
the concept of existence, then, like the concept of truth, which, when
relativized (as true-for-me, true-for-you), comes to something more
like belief than truth? Or is it like simultaneity, about which
thoughtful persons a century or so ago might have made pronouncements
much like Godel's? This difficult and fundamental question
has by no means been resolved.



Were this question resolved in favor of the relativization of
existence, what would be the import of a relativized version of
presentism? It would have to hold that what existed changed radically
with one's state of motion. Certain events (say on Mars or a
planet orbiting a distant star) may be existent for you now, sitting
at your computer screen or reading a printout, but other events will
replace those as existent should you decide to walk one way or
another. This seems (once again) less like an interesting metaphysical
insight than a restatement of the relativity of simultaneity.
Possibilism is not better off in this regard, for it relies on a
metaphysically distinguished present to separate the real from the
potential. (See the symposium "The Prospects for the Present in
Spacetime Theories" in Howard (2000) for further arguments and
references.)



To sum up, then, Sellars' attempt to tie existence to temporal
notions, when properly relativized, is either a bland re-statement of
what special relativity tells us already about simultaneity or an
opaque statement about relativized existence. This dilemma confronts
any attempt to import pre-relativistic temporal notions into Minkowski
spacetime, such as that in SS5.7 of Dolev (2007), though Dolev
explicitly eschews ontologizing temporal relations. Let us, then, turn
to efforts to understand Minkowski spacetime in a different way,
efforts that will help clarify the puzzling argument about the
Andromedean invasion presented above.


### 3.2 Chronogeometrical Fatalism Again



We have said much about the relativity of simultaneity but little
about the invariance of the speed of light. We now rectify that
situation.



Imagine that at some point \(\mathbf{O}\) of the spacetime an
idealized point-sized flashbulb flashes for (literally) an instant. It
follows from the invariance of the speed of light in all directions
that Alice, passing through \(\mathbf{O}\) as above, will find herself
at the center of an expanding sphere of photons. The radius of the
sphere expands with speed \(c\). (It follows that Bob,
also passing through \(\mathbf{O}\) but moving with some constant
velocity with respect to Alice, must find himself also at the center
of such a sphere, even though he and Alice are moving away from each
other. Such is relativistic life!) If we try to diagram this
situation, it is helpful to suppress one spatial dimension, as we have
in all the figures above, and so the two-dimensional cut through the
expanding sphere looks like an expanding circle, which becomes a cone
when that growth is plotted vertically up the diagram (and so is
called *the light cone.*) More precisely, this figure is just
half the light cone. If two photons (restricting ourselves to two
dimensions now) converged on point \(\mathbf{O}\) from opposite
directions, the lines indicating their histories would mark the other
half, the past lobe, of the light
 cone.[9]



Light cones exist at each point of the spacetime and are invariant
structures. Since the speed of light is an invariant quantity, all
"observers" agree as to which points of the spacetime are
illuminated by the popping of the flashbulb at \(\mathbf{O}\).
Furthermore, as special relativity is standardly understood, the speed
of light is a limiting speed. No material particle can be accelerated
from a speed less than \(c\) to a speed equal to or
greater than \(c\). Electromagnetic radiation
(including light) always propagates in a vacuum at speed \(c\).
(To see why \(c\) is held to be
limiting speed, see Mermin (1968, Chapter 15) and Nahin (1999, pp.
342-353 and Tech Note 7.) Given these suppositions, the light
cone structure divides all spacetime into three distinct sorts of
regions relative to each spacetime point \(\mathbf{O}\). (See chapters
5 and 6 of Geroch (1978) for a thorough discussion.)



![lightcone](lightcone.gif)

Figure 3. The Light Cone




First, there are the points from which a photon (travelling at speed
\(c\), of course) may reach \(\mathbf{O}\) or which
(like \(\mathbf{A})\) may be reached by a photon from \(\mathbf{O}\).
We say that these points are *lightlike separated* from
\(\mathbf{O}\).



Second, there are the points inside (rather than on) the future or
past light cone of \(\mathbf{O}\). We say that these points are
*timelike separated* from \(\mathbf{O}\). If \(\mathbf{B}\) is
a point in the spacetime timelike separated from \(\mathbf{O}\) and
future to it (that is, inside \(\mathbf{O}\)'s future light
cone), then a material particle travelling at some relativistically
acceptable speed (that is, less than \(c)\) can travel from
\(\mathbf{O}\) to \(\mathbf{B}\). Similarly, a material particle at a
point like \(\mathbf{C}\) inside the past light cone of
\(\mathbf{O}\), can travel at some speed less than \(c\)
from \(\mathbf{C}\) to \(\mathbf{O}\).



Finally, there are the points of the spacetime that are neither in nor
on the light cone of \(\mathbf{O}\). We say that such points are
*spacelike separated* from \(\mathbf{O}\). If \(\mathbf{D}\) is
spacelike separated from \(\mathbf{O}\), then neither a light signal
nor a material body can travel from \(\mathbf{O}\) to \(\mathbf{D}\)
or *vice versa*, because such travel would require superluminal
speed. If one makes the natural assumption that information and causal
influence are propagated by electromagnetic signals and material
particles, then if \(\mathbf{D}\) is spacelike separated from
\(\mathbf{O}\), events or occurrences at \(\mathbf{O}\) can have no
causal influence at all on events at \(\mathbf{D}\), and *vice
versa*.



We have reached this last conclusion by means of quite straightforward
reasoning from the invariance of the speed of light. But consider the
following observation of Torretti (1983, p. 247):



> 
> Before Einstein ... nobody appears to have seriously disputed
> that any two events might be causally related to each other,
> regardless of their spatial and temporal distance. The denial of this
> seemingly modest statement is perhaps the deepest innovation in
> natural philosophy brought about by Relativity. It has completely
> upset our traditional views of time, space, and causality ...
> 



As one illustration of how our traditional views of time and causality
are upset by restricting the propagation of causal influence to the
light cone structure, let us revisit the reasoning of the example of
the Andromedean invasion that we used to illustrate and motivate
chronogeometrical fatalism. We may be able to see now that this
reasoning is not so compelling as it first seemed, and we may be able
to see why some philosophers have proposed that we look at becoming in
Minkowski spacetime in a way quite different from the traditional
way.



To make the exposition easier, let us add to the story of the
Andromedean invasion a fourth "observer", Ted, who is at
rest with respect to Earth (and so also Andromeda) at the spot where
Alice and Bob meet. Ted too defines an inertial frame, and there is a
point at Andromeda (We can call it \(\mathbf{D})\) that (in
Ted's frame) is simultaneous with the meeting of Alice and Bob
and Ted. To make our exposition easier still, let us suppose that
Alice and Bob and Ted set their clocks to read 0 at the instant at
which they all
 meet.[10]
 Let us focus on \(\mathbf{D}\).



Ted (at the meeting of Alice and Bob) assigns to \(\mathbf{D}\) the
time 0, since it is simultaneous (in his frame) with his time 0. Alice
assigns \(\mathbf{D}\) (roughly) the time \(-3\) days, whereas Bob
assigns it time (roughly) \(+3\) days. \(\mathbf{D}\) is, of course,
spacelike separated from \(\mathbf{O}\), and we have been at pains to
explain that from a special relativistic standpoint this spacelike
separation precludes the (physical) possibility that there is any
causal influence upon \(\mathbf{D}\) of the events at \(\mathbf{O}\),
or *vice versa*. Once the labelling of spacetime points like
\(\mathbf{D}\) with coordinates is complete, what further content is
there, what further could be meant, by adding that for Alice and Ted
\(\mathbf{D}\) is real or fixed? Or that for Alice it is past whereas
for Bob it is future? If there is indeed no further content, then what
possible implications with regard to 'reality' or
'fixity' or 'determinateness' or
'tense' can be drawn from the fact that Bob labels this
point with a positive number, Alice labels it with a negative number,
and Ted labels it with
 0?[11]



Any good text in special relativity will sooner or later prove that
for *any* pair of spacelike separated points (but let us
continue to call them \(\mathbf{O}\) and \(\mathbf{D})\) there is
precisely one admissible inertial frame (with \(\mathbf{O}\) as
origin) in which \(\mathbf{O}\) and \(\mathbf{D}\) are simultaneous,
an infinity of admissible inertial frames in which \(\mathbf{D}\) is
assigned a positive number (that is, in which \(\mathbf{O}\) occurs
before \(\mathbf{D})\), and an infinity of other admissible inertial
frames in which \(\mathbf{D}\) is assigned a negative number (that is,
in which \(\mathbf{D}\) occurs before \(\mathbf{O})\). What
metaphysical significance could be gleaned from the fact that some
"observers" (the usual anthropomorphized way to refer to
admissible inertial frames) at \(\mathbf{O}\) must assign positive
times, some negative times, and one time 0 to the distant event
\(\mathbf{D}\), which, again, can not be influenced by and can not
itself influence the events at \(\mathbf{O}\), according to special
relativity at least?



Inability to provide any positive answer to this question can motivate
a different approach to conceptualizing becoming in Minkowski
spacetime, an approach presented by the philosopher Howard Stein
(1968, 1991). The basic idea of this approach is to begin from or to
define concepts in terms of the geometric structure intrinsic to the
spacetime rather than in terms of coordinates. In the present case,
this approach leads one to try to define 'becoming' in
terms of spacetime points and light cones. Pre-relativistically,
'has become' is defined relative to a plane of
simultaneity. We have seen the limitations of the notion of a plane of
simultaneity in special relativity in our discussion of the relativity
of simultaneity. Stein begins, then, by proposing that one define the
relation of 'having become' or 'already
definite' with respect to spacetime points. A two-place relation
schematically written as \(Rxy\) will be intended to capture the idea
that point \(y\) has already become or is definite with
respect to point \(x\).



There are two other formal features that this relation \(R\)
should possess. It should be
*transitive*--that is, if \(z\) has already
become with respect to \(y\) and \(y\)
has already become with respect to \(x\), then it seems
reasonable to require that \(z\) has already become
with respect to \(x\). It should also be
*reflexive*--that is, is seems reasonable to require that
\(x\) has become with respect to \(x\)
itself. (We can indicate these conditions briefly as (1) \(Rzy\) and
\(Rxz\) entail \(Rxy\), for all \(x, y, z\) and (2) \(Rxx\), for all
\(x\).)



Finally, Stein proposes that the relation \(R\) not
hold between every two points in the spacetime. That is, he proposes
that given some choice of spacetime point \(x\), there
is at least one distinct point \(y\) that has not
become, that is not already definite, with respect to \(x\).
But is there a relation that has all these
intuitively desirable characteristics? The answer is *yes*. The
relation is that between a point \(x\) and each point
in or on its past light
 cone.[12]
 If one can accept that the relation \(Rxy\) represents in special
relativity the notion of becoming (or, having become), then the
existence of the relation specified and found by Stein is a formal
refutation of the Rietdijk-Putnam-Penrose argument for chronogeometric
fatalism.



That chronogeometric fatalism (or, the block universe) is refuted is,
of course, controversial. Is the relation that Stein *calls*
becoming, or having become, a genuine notion of becoming in Minkowski
spacetime? Stein points out that if its true for his relation R that
there is even one point or event x such that there is another point or
event y such that both Rxy and Ryx are true, then R must be the
universal
 relation.[13]
 That is, it would follow that Rxy for all x and y. If, then, one
thinks of the present as a set of events that have become for each
other but that not all events have become, then it follows that
"in Einstein-Minkowski space-time an event's present is
constituted by itself alone." (Stein, 1968, p. 15)



Is this an acceptable view? Callender (2000, S592) remarks that
requiring that an event's present must contain at least one
event distinct from it, which he calls the *non-uniqueness
condition,* "seems the thinnest requirement one might put on
becoming." He would then not accept Stein's relation \(R\)
as representing a genuine relation of becoming since
it fails to meet this condition, but then he also must accept the
conclusion of the Rietdijk-Putnam-Penrose argument, since the only
alternative to \(R\) is the universal relation. If one
wishes to evade chronogeometric fatalism, as far as the special theory
of relativity is concerned, then it seems there is no alternative to
accepting Stein's relation \(R\) as representing
a genuine relation of becoming and to considering that an
event's present is constituted by itself alone. It is a truism
that the relativistic revolution in physics has profound implications
for our concepts of space and time. This last dilemma supports that
truism.



There may seem to be another obstacle to accepting Stein's
relation \(R\) as representing a genuine relation of
becoming. \(R\) is supposed to represent becoming, but
the light cone structure of Minkowski spacetime, in terms of which it
is defined, is inert. This reaction was voiced, for instance, by Palle
Yourgrau, who wrote that "Stein's mistake is to adduce a
*structural* property as what 'justifies the use of our
notion of *"becoming"* in relativistic
spacetime."' (1999, p. 77) If Yourgrau has put his finger
on a "mistake", then it is a "mistake" at the
very heart of Stein's effort. There are, however, a few remarks
to be made on this score.



First, there have been attempts to articulate positions like
Stein's that try to account for passage in terms of geometric
structure and that seem to incorporate more dynamic elements,
exploiting the fact that persistent objects or substances (including
"observers") are represented by timelike world lines,
rather than by points. The mathematician G. J. Whitrow (1980, p. 348)
wrote:



> 
> At a given instant \(E\) on the world line of an
> observer \(A\) (who need not be regarded as anything
> more than a recording instrument), all the events from which \(A\)
> can have received signals lie within the
> backwards-directed light cone with its vertex at \(E\ldots\) . Signals
> from events [outside the light cone at \(E\)] can only
> reach \(A\) after the event \(E\), and
> when they do reach \(A\) they will then lie within \(A\)'s
> backward-directed light cone at that instant.
> The passage of time corresponds to the continual advance of this light
> cone.
> 



The physicist-philosopher Abner Shimony, in responding to the claim
that special relativity shows that becoming is subjective or
"mind-dependent," wrote (1993, p. 284):



> 
> Something fleeting does indeed traverse the world line, but that
> something is not subjective; it is the transient *now,* which
> as a matter of objective fact is momentarily present and thereafter is
> past.
> 



In the felicitous phrase of Park (1971), we have here two different
sorts of *animated Minkowski diagrams.* Each seems to involve a
kind of motion, of the light cone or of the transient *now*
advancing along a world line. Our initial restrictions on accounts of
transience inspired by Broad's arguments above, however, should
make us wary of invoking motion to account for passage. Park,
moreover, sees no benefit to adding the animation.



> 
> I want now to make the vital point that the animated diagram may be
> more intuitive, or more picturesque, or make better cinema than the
> atemporal one, but that it contains no more specific, verifiable
> information. All of the science of dynamics, that is, all we know
> about how complex systems (including ourselves) behave and interact,
> is already represented on the atemporal Minkowski diagram.
> 



The non-animated Minkowski diagram may be "static", but,
as Park points out, the static diagram *represents* the
evolution in (proper) time of systems along their world lines. The
diagram, if Park is correct, need not itself be animated to represent
dynamical phenomena. If Park is correct, then what Yourgrau called a
"mistake" is in fact a virtue of Stein's account, a
feature and not a bug. Stein makes no attempt to animate his geometric
*picture* but leaves whatever transience there may be to reside
in that which it depicts.


### 3.3 Localizing Time



Katherine Brading claimed that with proper attention to the history of
physics one can find a satisfying version of presentism that does not
require a global present. If one grounds the unity of objects in
dynamical laws, as she, invoking DiSalle (2006), claims that Newton
did, then she says the present can be viewed as "a
spatiotemporal region of whatever size is necessary to sustain the
dynamical system in question." (2013, 1109) Presents construed
this way will typically be larger than a single point but will not
necessarily exhaust all of Minkowski spacetime or fall afoul of the
relativity of simultaneity.



But this is a characterization of a relativistic present, not
relativistic presentism. Presumably every dynamical system that has
existed, does exist or will exist had, has or will have a region of
whatever size is necessary to sustain it. Furthermore, it is unclear
under what conditions two dynamical systems share a present. Even if
they are intuitively spatially close, what sustains one may, or may
not, sustain the other. While Brading's suggestion might be
developed into a way to define a relativistic present, it may not help
to define presentism in a relativistic context, as she intended.



Let us now turn to two variants of Stein's approach--two
closely related ways to understand the present and temporal becoming
in terms of structure intrinsic to Minkowski spacetime. One might see
these two views as attempts to capture Shimony's transience
without animating the Minkowski diagram.



Two sources of the view are Arthur (2006, and, especially, 2019,
chapter 5) and Dieks (2006). The latter paper begins by presenting a
series of arguments that hyperplanes of simultaneity or global
*nows* are not the proper relativistic successors to the common
sense *now*.



In SS1 Dieks presents the following argument:



(P1)
 The experiences of observers are of such short duration and
occupy such a small amount of space that they can, without loss, be
idealized as point-like. 
(P2)
 Amongst these experiences are those that convince observers that
time flows or passes. 
(P3)
 Given the upper limit of speed of propagation of causal signals,
no event that is spacelike separated from a given event can influence
it causally. Therefore, 
(C)
 The human experiences that suggest at any given event \(e\) in
the history of an observer that time flows or passes are invariant
under different choices of global hypersurface containing \(e\).



Even if it has no relevance to human experience, can we nevertheless
choose *our* hyperplane of simultaneity (the one orthogonal to
our world line) as *the* hyperplane of simultaneity that marks
the passage of time? No, Dieks argues in SS2 of his paper, for two
reasons.



First, there are too many. Were we inertial observers, then there
would be a hyperplane of simultaneity orthogonal to our world line
(and uniquely definable from our world line using the Minkowski
metric). But every inertial world line defines such a hyperplane.
Choosing a particular one, according to Dieks, "is tantamount to
augmenting the structure of Minkowski spacetime" (Dieks, p.
5).



But, second, there really aren't any. We are not inertial
observers. We are (very nearly) rotating observers, and, as Dieks
points out, "local Einstein synchrony \((\varepsilon =1/2)\) in
a rotating system does not extend to a consistent global definition of
simultaneity" (Dieks, p. 6).



Partisans of a global now in Minkowski spacetime face, according to
Dieks, a dilemma. "If we are not going to refer to the actual
material worldlines in the universe, but only to the spacetime
structure itself, we have insufficient resources to fix a unique set
of global nows. If we do attempt to rely on the actual material
worldlines, however, we will not succeed in defining global nows at
all" (Dieks, p. 7).



Although we are jumping ahead of our story, it is worth remarking here
that the situation does not improve in the general theory of
relativity, as Dieks notes in SS3. It has been proposed that the
mean motion of matter could be used to define a preferred frame that,
in turn, could be foliated (or sliced) into spacelike hypersurfaces
that define a preferred global time. But this procedure would work
only at a large scale, wherein there would be an arbitrary element in
both determining the scale and the averaging procedure. Should one
retreat to the use of actual world lines, the rotation problem
reasserts itself.



Time, in the special theory of relativity, appears in two
guises--coordinate time, which has held center stage thus far,
and proper time. The histories of material objects, always moving with
speeds less than that of light, are represented in Minkowski spacetime
by timelike world lines (curves in Minkowski spacetime such that the
tangent vector at each point is timelike). Timelike world lines can be
parameterized by a quantity, *proper time,* which is measured
by ideal clocks following such world lines.



Proper time is *path-dependent*. If two distinct timelike
worldlines connect point a to point b, they will in general indicate
different proper times between these points. On the other hand, proper
time is not frame-dependent. It is an invariant quantity (given a
worldline and the specified points on it). (See pages 73-80 of
Geroch (1978) for an illuminating discussion of the path-dependence of
proper time.)



The potential philosophical importance of proper time was first
pointed out, like so much else, in a footnote by Stein:



Proper time is not a quantity attached to space-time points or to
pairs of space-time points; it is in this respect a notion utterly
different from the quantity "time" or "time
interval" of pre-relativistic theory ... The fundamental
physical role of proper time comes from the principle (here stated
roughly) that whenever a process takes place along a well-defined line
of space-time ("world-line"), the time rates in the
dynamical principles that govern that process are to be understood in
terms of the proper time along that line (and *not* in terms of
a "time coordinate"...) (Stein, 1968, fn 6)



This idea was adopted by Richard Arthur (1982, 107): "It is ...
proper time which is understood to measure *the* rate of
becoming for the possible process following this timelike line (or
*worldline*)." Arthur (2019) is an extended development
of this view. The idea was also adopted by Dennis Dieks (1988, 456):
"Only time *along worldlines* (proper time) has an
immediate and absolute significance as an ordering parameter of
physical processes."



In the absence (or, rather, the superabundance) of global
hypersurfaces, which are defined in terms of coordinate time, becoming
in Minkowski spacetime may be thought of locally, as the advance of
proper time along a timelike worldline or, even more basically, as the
successive happening of events along such a world line. The passage of
time or temporal becoming (along a given timelike world line, of
course) will be directly indicated by a clock, albeit an ideal
clock.



What about the present? Dieks proposes that the present should be
thought of as corresponding to a point (a now-point) and the present
for that point is just that point itself (as did Robb (1914, 1921,
1936) and Stein (1968)).



One could also allow a present that corresponds to some small length
of time that humans typically think of as the present--an
interval that may range from about .5 seconds to 3 or perhaps 4
seconds. This is the time it takes to read a sentence or hear a
musical phrase and is sometimes referred to as *the specious
present*. If we let *T* indicate the length of a specious
present and we try to follow Stein's injunction to define
notions in terms of intrinsic structure, then Arthur (2006) and Savitt
(2009) have suggested that the present for a stretch of worldline or
process of length *T* would be the region of spacetime
comprised of the intersection of the future light cone of the point at
the beginning of that process and the past light cone of the point at
the end of it (assuming the process is or occupies a closed interval).
This region is often called a *causal diamond*. (See Winnie
(1977) for more on causal diamonds.)



Taken literally, Robb-Stein-Dieks point presents cannot be shared,
which some find an unattractive feature of that view. Causal diamonds,
on the other hand, are surprisingly large. For a specious present of
length three seconds, the corresponding causal diamond includes the
moon (that is, a contemporaneous bit of the moon's history). As
the length of a specious present shrinks to zero as a limit, the
corresponding causal diamond shrinks to a point, regaining the
Robb-Stein-Dieks view as a limiting case.



Dorato and Wittman (2015) take as their starting point "the
indisputable fact that we *share* the 'now; despite the
fact we don't share a 'here'" (2015, 200), but
they propose a different structure as a relativistic present at a
point, a sphere around that point of at least 40 ms length. (According
to them, 40 ms is *the temporal order threshhold*. That is, one
cannot reliably determine the temporal order of event occurring less
than 40 ms apart in time in one's immediate vicinity. In their
view, this sets a minimum bound on the length of the perceived or
psychological present.) They add that "... passage
physically correspond[s] to a succession of spheres..."
(202)



Rovelli (2019, 1330) contrasts "finite bubble presents"
with "diamond presents":



Each one of these definitions matches our common sense use of
"present" and "simultaneous" in the everyday
contexts. They show that the common usage of "present" and
"simultaneous" is not in contradiction with relativity,
provided that it is used within the appropriate approximation and
within the appropriate context. None of them deserves to be charged
with ontological weight. (1330-31)



It is also possible to have less geometrically well-defined shapes as
relativistic presents, as with Brading's presents above and
Callender's "*now* patches." (2017, 2016)



In pre-relativistic spacetime (and common sense) it is tempting to
identify the passing of time with a succession of presents, as did
Kurt Godel in the quote from him in SS 2.3 above. So, if one
thinks of presents in Minkowski spacetime as spheres or causal
diamonds, one might try to identify passage as a succession of such
diamonds or spheres along each timelike path, as in Dorato and Wittman
(2015) or Savitt (2009, 2020). It is not easy, however, to order a
succession of such regions of spacetime when their size is variable.
Perhaps, then, one should not succumb to this temptation. The lesson
to learn is that temporal concepts that coincide in common sense may
more clearly be seen to be distinct when re-considered in more
mathematically sophisticated spacetime settings.