arXiv:gr-qc/0610049v2  23 Apr 2007
Absolute Being vs Relative Becoming∗
Joy Christian1,2
1 Perimeter Institute, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada
jchristian@perimeterinstitute.ca
2 Department of Physics, Oxford University, Oxford OX1 3PU, United Kingdom
joy.christian@wolfson.ox.ac.uk
Summary. Contrary to our immediate and vivid sensation of past, present, and
future as continually shifting non-relational modalities, time remains as tenseless and
relational as space in all of the established theories of fundamental physics. Here an
empirically adequate generalized theory of the inertial structure is discussed in which
proper time is causally compelled to be tensed within both spacetime and dynamics.
This is accomplished by introducing the inverse of the Planck time at the conjunction
of special relativity and Hamiltonian mechanics, which necessitates energies and
momenta to be invariantly bounded from above, and lengths and durations similarly
bounded from below, by their respective Planck scale values. The resulting theory
abhors any form of preferred structure, and yet captures the transience of now along
timelike worldlines by causally necessitating a genuinely becoming universe. This is
quite unlike the scenario in Minkowski spacetime, which is prone to a block universe
interpretation. The minute deviations from the special relativistic eﬀects such as
dispersion relations and Doppler shifts predicted by the generalized theory remain
quadratically suppressed by the Planck energy, but may nevertheless be testable in
the near future, for example via observations of oscillating ﬂavor ratios of ultrahigh
energy cosmic neutrinos, or of altering pulse rates of extreme energy binary pulsars.
1 Introduction
From the very ﬁrst imprints of awareness, “change” and “becoming” appear
to us to be two indispensable norms of the world. Indeed, prima facie it
seems impossible to make sense of the world other than in terms of changing
things and happening events through the incessant passage of time. And yet,
the Eleatics, led by Parmenides, forcefully argued that change is nothing but
an illusion, thereby rejecting the prevalent view, expounded by Heraclitus,
that becoming is all there is. The great polemic that has ensued over these
two diametrically opposing views of the world has ever since both dominated
and shaped the course of western philosophy [1]. In modern times, inﬂuential
∗To appear in Relativity and the Dimensionality of the World (within the series
Fundamental Theories of Physics), edited by Vesselin Petkov (Springer, NY 2007).

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Joy Christian
neo-Eleatics such as McTaggart have sharpened the choice between the being
and the becoming universe by distinguishing two diﬀerent possible modes of
temporal discourse, one with and the other without a clear reference to the
distinctions of past, present, and future; and it is the former mode with explicit
reference to the tenses that is deemed essential for capturing the notions
of change and becoming [2]. Conversely, the latter mode—which relies on a
tenseless linear ordering of temporal moments by a transitive, asymmetric,
and irreﬂexive relation precedes—is deemed incapable of describing a genuine
change or becoming. Such a sharpening of the temporal discourse, in turn, has
inspired two rival philosophies of time, each catering to one of the two possible
modes of the discourse [3]. One tenseless philosophy of time holds that time is
relational, much like space, which clearly does not seem to “ﬂow”, and hence
what we perceive as the ﬂow or passage of time must be an illusion. The other
tensed philosophy of time holds, on the other hand, that there is more to time
than mere relational ordering of moments. It maintains that time is rather
a dynamic or evolving entity unlike space, and does indeed “ﬂow”—like a
refreshing river—much in line with our immediate experience of it. That is to
say, far from being an illusion, our sensation of that sumptuous moment now,
ceaselessly streaming-in from nowhere and slipping away into the unchanging
past, happens to reﬂect a truly objective feature of the world.
In terms of these two rival philosophies of time, a genuinely becoming
universe must then correspond to a notion of time that is more than a mere
set of “static” moments, linearly ordered by the relation precedes. In addition,
it must at least allow a genuine partition of this ordered set into the moments
of past, present, and future. From the perspective of physics, the choice of a
becoming universe must then necessitate a theory of space and time that not
only distinguishes the future events from the past ones intrinsically, but also
thereby accounts for the continual passage of the ﬂeeting present, from a non-
existing future into the unalterable past, as a bona ﬁde structural attribute
of the world. Such a theory of space and time, which would account for the
gradual coming-into-being of the non-existent future events—or a continual
accumulation of the unalterable past ones—giving rise to a truly becoming
universe, may be referred to as a Heraclitean theory of space-time, as opposed
to a Parmenidean one, devoid of any such explicit dictate to becoming.
One such Heraclitean theory of space-time was, of course, that of Newton,
for whom “[a]bsolute, true, and mathematical time, of itself, and from its
own nature, ﬂow[ed] equably without relation to anything external...” [4]. To
be sure, Newton well appreciated the relational attributes of time, and in
particular their remarkable similarities with those of space:
Just as the parts of duration are individuated by their order, so that
(for example) if yesterday could change places with today and become
the later of the two, it would lose its individuality and would no longer
be yesterday, but today; so the parts of space are individuated by their
positions, so that if any two could exchange their positions, they would

Absolute Being vs Relative Becoming
3
also exchange their identities, and would be converted into each other
qua individuals. It is only through their reciprocal order and positions
that the parts of duration and space are understood to be the very
ones that they truly are; and they do not have any other principle of
individuation besides this order and position [5].
And yet, Newton did not fail to recognize the non-relational, or absolute,
attributes of time that go beyond the mere relational ordering of moments.
He clearly distinguished his neo-platonic notion of “equably” ﬂowing absolute
time, existing independently of changing things, from the Aristotelian notion
of “unequably” ﬂowing relative times, determined by their less than perfect
empirical measures (such as clocks) [4]. What is more, he well appreciated
the closely related need of a temporally founded theory of calculus within
mathematics, formulated in terms of his notion of ﬂuxions (i.e., continuously
generated temporally ﬂowing quantities [6]), and defended this theory vigor-
ously against the challenges that arose from the quiescent theory of calculus
put forward by Leibniz [6]. Thus, the notions of ﬂowing time and becoming
universe were central to Newton not only for his mechanics, but also for his
mathematics [6]. More relevantly for our purposes, according to him the rate
of ﬂow of time—i.e., the rate at which the relationally ordered events succeed
each other in the world—is determined by the respective moments of his ab-
solute time, which ﬂows by itself, continuously, uniformly, and unstoppably,
without relation to anything external [7]. Alas, as we now well know, such a
Newtonian theory of externally ﬂowing absolute time, giving rise to an objec-
tively becoming universe, is no longer physically viable. But is our celebration
of Einstein’s relativistic revolution complete only through an unconditional
renunciation of Newton’s non-relationally becoming universe?
The purpose of this essay, ﬁrst, is to disentangle the notion of a becoming
universe from that of an absolute time, and then to diﬀerentiate two physically
viable and empirically distinguishable theories of spacetime: namely, special
relativity—which is prone to a Parmenidean interpretation—and a generalized
theory [8]—which is intrinsically Heraclitean by construction. The purpose of
this essay may also be taken as a case study in experimental metaphysics,
since it evaluates conceivable experiments that can adjudicate between the
two rival philosophies of time under discussion. Experimental metaphysics is
a term suggested by Shimony [9] to describe the enterprise of sharpening of
the disputes traditionally classiﬁed as metaphysical, to the extent that they
can be subjected to controlled experimental investigations. A prime example
of such an enterprise is the sharpening of a dispute over the novel conceptual
implications of quantum mechanics, which eventually led to a point where
empirical evidence was brought to bear on the traditionally metaphysical con-
cerns of scientiﬁc realism [9]. Historically, recall how resistance to accept the
novel implications of quantum mechanics had led to suggestions of alterna-
tive theories—namely, hidden variable theories. Subsequently, the eﬀorts by
de Broglie, von Neumann, Einstein, Bohr, Bohm and others led to theoretical

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Joy Christian
sharpening of the central concepts of quantum mechanics, which eventually
culminated into Bell’s incisive derivation of his inequalities. The latter, of
course, was a breakthrough that made it possible to experimentally test the
rival metaphysical positions on quantum mechanics [10]. As this well known
example indicates, however, experimental investigations alone cannot be ex-
pected to resolve profound metaphysical questions once and for all, without
careful conceptual analyses. Indeed, Shimony [9] warns us against overplay-
ing the signiﬁcance of experimental metaphysics. He points out that without
careful conceptual analyses even those questions that are traditionally clas-
siﬁed as scientiﬁc cannot be resolved by experimental tests alone. Hence, it
should not be surprising that questions as slippery as those concerning time
and becoming would require more than a mere experimental input. On the
other hand, as the above example proves, a judicious experimental input can,
indeed, facilitate greatly towards a possible resolution of these questions.
Bearing these cautionary remarks in mind, the question answered, aﬃr-
matively, in the present essay is: Can the debate over the being vs becoming
universe—which is usually also viewed as metaphysical [11]—be sharpened
enough to bear empirical input? Of course, as the above example of hidden
variable theories suggests, the ﬁrst step towards any empirical eﬀort in this
direction should be to construct a physically viable Heraclitean alternative
to special relativity. As alluded to above, this step has already been taken in
Ref. [8], with motivations for it stemming largely from the temporal concerns
in quantum gravity. What is followed up here is a comparison of these two
alternative theories of causal structure with regard to the status of becoming.
Accordingly, in the next section we begin by reviewing the status of becoming
within special relativity. Then, in Sec. 3, we review the alternative to special
relativity proposed in Ref. [8], with an emphasis in Subsec. 3.3 on the causal
inevitability of the strictly Heraclitean character of this alternative. Finally,
before concluding, in Sec. 4 we discuss the experimental distinguishability of
the two alternatives, and its implications for the status of becoming.
2 The status of becoming within special relativity
The prevalent theory of the local inertial structure at the heart of modern
physics—classical or quantal, non-gravitational or gravitational—is, of course,
Einstein’s special theory of relativity. This theory, however, happens to be
oblivious to any structural distinction between the past and the future [12].
To be sure, one frequently comes across references within its formalism to the
notions of “absolute past” and “absolute future” of a given event. But these
are mere conventional choices, corresponding to assignment of tenseless linear
ordering to “static” moments mentioned above, with the ordering now being
along the timelike worldline of an ideal observer tracing through that event
(see Fig. 1). There is, of course, no doubt about the objectivity of this ordering.
It is preserved under Lorentz transformations, and hence remains unaltered

Absolute Being vs Relative Becoming
5
b
c
b
c
b
c
From eternity
Conventional
Past
To eternity
Conventional
Future
A
B
C
Fig. 1. Timelike worldline of an observer tracing through an event B in a Minkowski
spacetime. Events A and C in the conventional past and conventional future of
the event B are related to B by the transitive, asymmetric, and irreﬂexive relation
precedes. Such a linear ordering of events is preserved under Lorentz transformations.
for all inertial observers. But such a sequence of moments has little to do with
becoming per se, as both physically and mathematically well appreciated by
Newton [5][6], and conceptually much clariﬁed by McTaggart [2]. Worse still,
there is no such thing as a world-wide moment “now” in special relativity,
let alone the notion of a passage of that moment. Due to the relativity of
simultaneity, what is a “now-slice” cutting through a given event for one
observer would be a “then-slice” for another one moving relative to the ﬁrst,
and vise versa. In other words, what is past (or has “already happened”) for
one observer could be the future (or has “not yet happened”) for the other,
and vise versa [13]. This indeterminacy in temporal order cannot lead to any
causal inconsistency however, for it can only occur for spacelike separated
events—i.e., for pairs of events lying outside the light-cones of each other.
Nevertheless, these facts suggest two rival interpretations for the continuum of
events presupposed by special relativity: (1) an absolute being interpretation
and (2) a relative becoming interpretation. According to the ﬁrst of these
interpretations, events in the past, present, and future exist all at once, with
equal ontological status, across the whole span of time; whereas according to
the second, events can be partitioned, causally, consistently, and ontologically,
into the sets of deﬁnite past and indeﬁnite future events, mediated by a ﬂeeting
present, albeit only in a relative and observer-dependent manner.
The ﬁrst of these two interpretations of special relativity is sometimes also
referred to as the “block universe” interpretation, because of its resemblance

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Joy Christian
to a 4-dimensional block of “already laid out” events. The moments of time
in this block are supposed to be no less actual than the locations in space are.
Just as London and New York are supposed to be there even if you may not be
at either of these locations, the moments of your birth and death are “there”
on your time-line, even if you are presently far from being “at” either of these
two moments of your life. More precisely, along your timelike worldline all
events of your life are ﬁxed once and for all, beyond your control, and in
apparent conﬂict with your freedom of choice. In fact, in special relativity, a
congruence of such non-intersecting timelike worldlines—sometimes referred
to as a ﬁbration of spacetime—represents a 3-dimensional relative space (or
an inertial frame). The 4-dimensional spacetime is then simply ﬁlled by these
“lifeless” ﬁbers, with the proper time along any one of them representing the
local time associated with the ordered series of events laid out along that
ﬁber. Informally, such a ﬁber is a track in spacetime of an observer moving
subluminally for all eternity. In particular, for a given moment, all the future
instants of time along this track—in exactly the same sense as all the past
instants—are supposed to be ﬁxed, once and for all, till eternity.
Such an interpretation of time in special relativity, of course, sharply diﬀers
from our everyday conception of time, where we expect the nonexistent future
instants to spring into existence from nowhere, streaming-in one after another,
and then slipping away into the unalterable past, thus gradually materializing
the past track of our worldline, as depicted in Fig. 2. In other words, in our
everyday life we normally do not think of the future segment of our worldline
to be preexisting for all eternity; instead, we perceive the events in our lives
to be occurring non-fatalistically, one after another, rendering our worldline
to “grow”, like a tendril on a wall. But such a “dynamic” conception of time
appears to be completely alien to the universe purported by special relativity.
Within the Minkowski universe, as Einstein himself has been quoted as saying,
“the becoming in three-dimensional space is transformed into a being in the
world of four dimensions” [14]. More famously, Weyl has gone one step further
in endorsing such a static view of the world: “The objective world simply is, it
does not happen” [15]. Accordingly, the appearances of change and becoming
are construed to be mere ﬁgments of our conscious experience, as Weyl goes
on to explain: “Only to the gaze of my consciousness, crawling upward along
the life line of my body, does a section of this world come to life as a ﬂeeting
image in space which continuously changes in time.” Not surprisingly, some
commentators have reacted strongly against such a grim view of reality:
But this picture of a “block universe”, composed of a timeless web of
“world-lines” in a four-dimensional space, however strongly suggested
by the theory of relativity, is a piece of gratuitous metaphysics. Since
the concept of change, of something happening, is an inseparable com-
ponent of the common-sense concept of time and a necessary component
of the scientist’s view of reality, it is quite out of the question that the-
oretical physics should require us to hold the Eleatic view that nothing

Absolute Being vs Relative Becoming
7
b
c
Unchangeable Past
Non-existent Future
Becoming Events
Moving “Now”
Eternal Present
Growing Worldline
Fig. 2. The tensed time of the proverbial man in the street, with a degree in special
relativity. His sensation of time is much richer than a mere tenseless linear ordering
of events. Future events beyond the moving present are non-existent to him, whereas
he, at least, has a memory of the past events that have occurred along his worldline.
happens in “the objective world.” Here, as so often in the philosophy
of science, a useful limitation in the form of representation is mistaken
for a deﬁciency of the universe [16].
The frustration behind these sentiments is, of course, quite understandable.
It turns out not to be impossible, however, to appease the sentiments to some
extent. It turns out that a formal “becoming relation” of a limited kind can
indeed be deﬁned along a timelike worldline, uniquely and invariantly, with-
out in any way compromising the principles of special relativity. The essential
idea of such a relation goes back to Putnam [17], who tried to demonstrate
that no meaningful binary relation between two events can exist within the
framework of special relativity that can ontologically partition a worldline into
distinct parts of already settled past and not yet settled future. Provoked by
this and related arguments by Rietdijk [18] and Maxwell [19], Stein [20][21]
set out to expose the inconsistencies within such arguments (without unduly
leaning on either side of the debate), and proved that a transitive, reﬂexive,
and asymmetric “becoming relation” of a formal nature can indeed be deﬁned
consistently between causally connected pairs of events, on a time-orientable
Minkowski spacetime. Stein’s analysis has been endorsed by Shimony [22] in
an approach that is diﬀerent in emphasis but complementary in philosophy,
and extended by Clifton and Hogarth [23] to a more natural setting for the
becoming along timelike worldlines. This coherent set of arguments, taken
individually or collectively, amounts to formally proving the permissibility of

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Joy Christian
objective becoming within the framework of special relativity, but only rela-
tive to a given timelike worldline. And since a timelike worldline in Minkowski
spacetime is simply the integral curve of a never vanishing, future-directed,
timelike vector ﬁeld representing the direction of a moving observer, the be-
coming defended here is meaningful only relative to such an observer. There
is, of course, no inconsistency in this relativization of becoming, since—thanks
to the absoluteness of simultaneity for coincident events—diﬀerent observers
would always agree on which events have already “become”, and which have
not, when their worldlines happen to intersect. Consequently, this body of
works make it abundantly plain that special relativity does not compel us
to adopt an interpretation as radical as the block universe interpretation, but
leaves room for a rather sophisticated version of our common-sense conception
of becoming. To be sure, this counterintuitive notion of a worldline-dependent
becoming permitted within special relativity is a far cry from our everyday
experience, where a world-wide present seems to perpetually stream-in from a
non-existent future, and then slip away into the unchanging past. But such a
pre-relativistic notion of absolute, world-wide becoming, occurring simultane-
ously for each and every one of us regardless of our motion, has no place in the
post-relativistic physics. Moreover, this apparent absolute becoming can be
easily accounted for as a gross collective of “local” or “individual” becomings
along timelike worldlines, emerging cohesively in the nonrelativistic limit. Just
as Newtonian mechanics can be viewed as an excellent approximation to the
relativistic mechanics for small velocities, our commonly shared “world-wide”
becoming can be shown to be an excellent approximation to these relativistic
becomings for small distances, thanks to the enormity of the speed of light in
everyday units. Consequently, the true choice within special relativity should
be taken not as between absolute being and absolute becoming, but between
the former (i.e., block universe) and the relativity of distant becoming.
There has been rather surprising reluctance to accept this relativization
of becoming, largely by the proponents of the block universe interpretation
of special relativity. As brought out by Stein [21], some of this reluctance
stems from elementary misconceptions regarding the true physical import of
the theory, even by philosophers with considerable scientiﬁc prowess. There
seems to remain a genuine concern, however, because the notion of worldline-
dependent becoming tends to go against our pre-relativistic ideas of existence.
This concern can be traced back to G¨odel, who ﬂatly refused to accept the
relativity of distant becoming on such grounds: “A relative lapse of time, ... if
any meaning at all can be given to this phrase, would certainly be something
entirely diﬀerent from the lapse of time in the ordinary sense, which means
a change in the existing. The concept of existence, however, cannot be rela-
tivized without destroying its meaning completely” [24]. In the similar vein,
in a certain book-review Callender remarks: “... the relativity of simultaneity
poses a problem: existence itself must be relativized to frame. This may not
be a contradiction, but it is certainly a queer position to hold” [25]. Perhaps.
But nature cannot be held hostage to what our pre-relativistic prejudices ﬁnd

Absolute Being vs Relative Becoming
9
queer. Whether we like it or not, the Newtonian notion of absolute world-
wide existence has no causal meaning in the post-relativistic physics. Within
special relativity, discernibility of events existing at a distance is constrained
by the absolute upper-bound on the speeds of causal propagation, and hence
the Newtonian notion of absolute distant existence becomes causally mean-
ingless. To be sure, when we regress back to our everyday Euclidean intuitions
concerning the causal structure of the world, the idea of relativized existence
seems strange. However, according to special relativity the topology of this
causal structure—i.e., the neighborhood relations between causally admissible
events—happens not to be Euclidean but pseudo-Euclidean. Once this aspect
of the theory is accepted, it is quite anomalous to hang on to the Euclidean
notion of existence, or equivalently to the absoluteness of distant becoming.
It is of course logically possible to accept the relativity of distant simultaneity
but reject the relativity of distant becoming, as G¨odel seems to have done,
but conceptually that would be quite inconsistent, since the former relativity
appears to us no less queer than the latter. In fact, perhaps unwittingly, some
textbook descriptions of the relativity of simultaneity explicitly end up using
the language of becoming. Witness for example Feynman’s description of a
typical scenario [26]: “ ... events that occur at two separated places at the
same time, as seen by Moe in S′, do not happen at the same time as viewed
by Joe in S [emphasis rearranged].” Indeed, keeping the geometrical formal-
ism intact, every statement involving the relativity of distant simultaneity
in special relativity can be replaced by an identical statement involving the
relativity of distant becoming, without aﬀecting either the theoretical or the
empirical content of the theory. In other words, Einstein could have written
his theory using the latter relativity rather than the former, and that would
have made no diﬀerence to the relativistic physics—classical or quantal—of
the past hundred years. The former would have been then seen as a useful but
trivial corollary of the latter. Thus, as Callender so rightly suspects, there is
indeed no contradiction in taking the relativity of distant becoming seriously,
since any evidence of our perceived co-becoming of objectively existing distant
events (i.e., of our perceived absoluteness of becoming) is quite indirect and
causal [22]. Therefore, the alleged queerness of the relativity of distant be-
coming by itself cannot be taken as a good reason to opt for an interpretation
of special relativity as outrageous as the block universe interpretation.
There do exist other good reasons, however, that, on balance, land the
block interpretation the popularity it enjoys. Einstein-Minkowski spacetime
is pretty “lifeless” on its own, as evident from comparisons of Figs. 1 and 2
above. If becoming is a truly ontological feature of the world, however, then
we expect the sum total of reality to grow incessantly, by objective accretion
of entirely newborn events. We expect this to happen as non-existent future
events momentarily come to be the present event, and then slip away into the
unchanging past, as we saw in Fig. 2. No such objective growth of reality can
be found within the Einstein-Minkowski framework for the causal structure.
It is all very well for Stein to prove the deﬁnability of a two-place “becoming

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Joy Christian
relation” within Minkowski spacetime, but in a genuinely becoming universe
no such relation between future events and a present event can be meaningful.
Indeed, as recognize by Broad long ago, “...the essence of a present event is,
not that it precedes future events, but that there is quite literally nothing
to which it has the relation of precedence” [27]. Even more tellingly, in the
Einstein-Minkowski framework there is no causal compulsion for becoming. In
a genuinely becoming universe we would expect the accretion of new events to
be necessitated causally, not left at the mercy of our interpretive preferences.
In other words, we would expect the entire spatio-temporal structure to not
only grow, but this growth to be also necessitated by causality itself. No such
causal dictate to becoming is there in the Einstein-Minkowski framework of
causality. A theory of local inertial structure with just such a causal necessity
for objective temporal becoming is the subject matter of our next section.
3 A purely Heraclitean generalization of relativity
Despite the fact that temporal transience is one of the most immediate and
constantly encountered aspects of the world [11], Newton appears to be the
last person to have actively sought to capture it, at the most fundamental level,
within a successful physical theory. Equipped with his hypothetico-deductive
methodology, he was not afraid to introduce metaphysical notions into his
theories as long as they gave rise to testable experimental consequences. After
the advent of excessively operationalistic trends within physics since the dawn
of the last century, however, questions of metaphysical ﬂavor—questions even
as important as those concerning time—have tended to remain on the fringe
of serious physical considerations.3 Perhaps this explains why most of the
popular approaches to the supposed quantum gravity are entirely oblivious
to the profound controversies concerning the status of temporal becoming.4
If, however, temporal becoming is indeed a genuinely ontological attribute of
the world, then no approach to quantum gravity can aﬀord to ignore it. After
all, by quantum gravity one usually means a complete theory of nature. How
can a complete theory of nature be oblivious to one of the most immediate
and ubiquitous features of the world? Worse still: if temporal becoming is a
genuine feature of the world, then how can any approach to quantum gravity
possibly hope to succeed while remaining in total denial of its reality?
3 There are, of course, a few brave-hearts, such as Shimony [28] and Elitzur [29],
who have time and again urged the physics community to take temporal becoming
seriously. However, there are also those who have preferred to explain it away as a
counterfeit, resulting from some sort of “macroscopic irreversibility” [30][31][32].
4 A welcome exception is the causal set approach initiated by Sorkin [33]. However,
the stochasticity of “growth dynamics” discovered a posteriori within this discrete
approach is a far cry from the inevitable continuity of becoming recognized by
Newton [6]. Such a deﬁciency seems unavoidable within any discrete approach to
quantum gravity, due to the “inverse problem” of recovering the continuum [34].

Absolute Being vs Relative Becoming
11
GN
h
GTR
STR
+
CHM
CTN
QTF
t−1
P
Fig. 3. Introducing the inverse of the Planck time at the conjunction of Special
Theory of Relativity (STR) and Classical Hamiltonian Mechanics (CHM), with a
bottom-up view to a Complete Theory of Nature (CTN). Both General Theory of
Relativity (GTR) and Quantum Theory of Fields (QTF) are viewed as limiting cases,
corresponding to negligible quantum eﬀects (represented by Planck’s constant h) and
negligible gravitational eﬀects (represented by Newton’s constant GN ), respectively.
Partly in response to such ontological and methodological questions, an
intrinsically Heraclitean generalization of special relativity was constructed in
Ref. [8]. The strategy behind this approach was to judiciously introduce the
inverse of the Planck time, namely t−1
P , at the conjunction of special relativity
and Hamiltonian mechanics, with a bottom-up view to a complete theory of
nature, in a manner similar to how general relativity was erected by Einstein
on special relativity (see Fig. 3). The resulting theory of the causal structure
has already exhibited some remarkable physical consequences. In particular,
such a judicious introduction of t−1
P
necessitates energies and momenta to be
invariantly bounded from above, and lengths and durations similarly bounded
from below, by their respective Planck scale values. By contrast, within special
relativity nothing prevents physical quantities such as energies and momenta
to become unphysically large—i.e., inﬁnite—in a rapidly moving frame. In
view of the primary purpose of the present essay, however, we shall refrain
form dwelling too much into these physical consequences (details of which
may be found in Ref. [8]). Instead, we shall focus here on those features of the
generalized theory that accentuate its purely Heraclitean character.

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Joy Christian
M
4
e1
e2
×
s1
s2
2n
N
Fig. 4. (a) The motion of a clock from event e1 to event e2 in a Minkowski Spacetime
M. (b) As the clock moves from e1 to e2, it also inevitably evolves, as a result of its
external motion, from state s1 to state s2 in its own 2n-dimensional phase space N.
3.1 Fresh look at the proper duration in special relativity
To this end, let us reassess the notion of proper duration residing at the very
heart of special relativity. Suppose an object system, equipped with an ideal
classical clock of unlimited accuracy, is moving with a uniform velocity v in a
Minkowski spacetime M, from an event e1 at the origin of a reference frame
to a nearby event e2 in the future light cone of e1, as shown in Fig. 4a. For our
purposes, it would suﬃce to refer to this system, say of n degrees of freedom,
simply as “the clock.” As it moves, the clock will also necessarily evolve, as
a result of its external motion, say at a uniform rate ω, from one state, say
s1, to another state, say s2, within its own relativistic phase space, say N.
In other words, the inevitable evolution of the clock from s1 to s2—or rather
that of its state—will trace out a unique trajectory in the phase space N, as
shown in Fig. 4b. For simplicity, we shall assume that this phase space of the
clock is ﬁnite dimensional; apart from possible mathematical encumbrances,
the reasoning that follows would go through unabated for the case of inﬁnite
dimensional phase spaces (e.g. for clocks made out of relativistic ﬁelds).
Now, nothing prevents us from thinking of this motion and evolution of the
clock conjointly, as taking place in a combined 4 + 2n−dimensional space, say
E, the elements of which may be called event-states and represented by pairs
(ei, si), as depicted in Fig. 5. Undoubtedly, it is this combined space that truly
captures the complete speciﬁcation of all possible physical attributes of our
classical clock. Therefore, we may ask: What will be the time interval actually
registered by the clock as it moves and evolves from the event-state (e1, s1)
to the event-state (e2, s2) in this combined space E? It is only by answering
such a physical question can one determine the correct topology and geometry
of the combined space in the form of a metric, analogous to the Minkowski

Absolute Being vs Relative Becoming
13
(e1, s1)
(e2, s2)
4 + 2n
E
?= M × N
Fig. 5. What is the correct metric-topology of the combined space E—made up of
the external Minkowski spacetime M and the internal space of states N—in which
our clock moves as well as evolves from event-state (e1, s1) to event-state (e2, s2)?
metric corresponding to the line element
dτ 2
E = dt2 −c−2dx2 ≥0 ,
(1)
where the inequality asserts the causality condition. Of course, after Einstein
the traditional answer to the above question, in accordance with the line
element (1), is simply
∆τE =
Z (e2, s2)
(e1, s1)
dτE =
Z t2
t1
1
γ(v) dt ,
(2)
with the usual Lorentz factor
γ(v) :=
1
p
1 −v2/c2 > 1.
(3)
In other words, the traditional answer is that the metrical topology of the
space E is of a product form, E = M × N, and—more to the point—the clock
that records the duration ∆τE in question remains insensitive to the passage
of time that marks the evolution of variables within its own phase space N.
But from the above perspective—i.e., from the perspective of Fig. 5—
it is evident that Einstein made an implicit assumption while proposing the
proper duration (2). He tacitly assumed that the rate at which a given physical
state can evolve remains unbounded. Of course, he had no particular reason
to question the limitlessness of how fast a physical state can evolve. However,
for us—from what we have learned from our eﬀorts to construct a theory of
quantum gravity—it is not unreasonable to suspect that the possible rate at
which a physical state can evolve is invariantly bounded from above. Indeed,
it is generally believed that the Planck scale marks a threshold beyond which
our theories of space and time, and possibly also of quantum phenomena,

14
Joy Christian
(t1, s1)
(t2, s2)
1 + 2n
O
?= R × N
Fig. 6. The evolution of our clock from instant-sate (t1, s1) to instant-state (t2, s2)
in the odd dimensional extended phase space O. What is the correct topology of O?
are unlikely to survive [8][35][36]. In particular, the Planck time tP is widely
thought to be the minimum possible duration. It is then only natural to suspect
that the inverse of the Planck time—namely t−1
P , with its approximate value of
10+43 Hertz in ordinary units—must correspond to the absolute upper bound
on how fast a physical state can possibly evolve. In this context, it is also
worth noting that the speed of light is simply a ratio of the Planck length
over the Planck time, c := lP /tP , which suggests that perhaps the assumption
of absolute upper bound t−1
P
on possible rates of evolution should be taken to
be more primitive in physical theories than the usual assumption of absolute
upper bound c on possible speeds of motion. In fact, as we shall see, the
assumption of upper bound c on speeds of motion can indeed be viewed as a
special case of our assumption of upper bound t−1
P
on rates of evolution.
To this end, let us then systematize the above thoughts by incorporating
t−1
P
into a physically viable and empirically adequate theory of the local causal
structure. One way to accomplish this task is to ﬁrst consider a simpliﬁed pic-
ture, represented by what is known as the extended phase space, constructed
within a global inertial frame in which the clock is at rest (see Fig. 6). Now,
in such a frame the proper time interval the clock would register is simply
the Newtonian time interval ∆t. Using this time t ∈R as an external parame-
ter, within this frame one can determine the extended phase space O = R × N
for the dynamical evolution of the clock using the usual Hamiltonian prescrip-
tion. Suppose next we consider time-dependent canonical transformations of
the dimensionless phase space coordinates yµ(t) (µ = 1, . . . , 2n), expressed in
Planck units, into coordinates y′µ(t) of the following general linear form:
y′µ(yµ(0), t) = yµ(0) + ωµ
r (y(0)) t + bµ ,
(4)

Absolute Being vs Relative Becoming
15
where ωµ
r and bµ do not have explicit time dependence, and the reason for
the subscript r in ωµ
r , which stands for “relative”, will become clear soon.
Interpreted actively, these are simply the linearized solutions of the familiar
Hamiltonian ﬂow equations,
dyµ
dt
= ωµ
r (y(t)) := Ωµν ∂H
∂yν ,
(5)
where ωr is the Hamiltonian vector ﬁeld generating the ﬂow, y(t) is a 2n-
dimensional local Darboux vector in the phase space N, Ωis the symplectic
2-form on N, and H is a Hamiltonian function governing the evolution of the
clock. If we now denote by ωµ the uniform time rate of change of the canonical
coordinates yµ, then the linear transformations (4) imply the composition law
ω′µ = ωµ + ωµ
r
(6)
for the evolution rates of the two sets of coordinates, with −ωµ
r interpreted
as the rate of evolution of the transformed coordinates with respect to the
original ones. Crucially for our purposes, what is implicit in the law (6) is
the assumption that there is no upper bound on the rates of evolution of
physical states. Indeed, successive transformations of the type (4) can be used,
along with (6), to generate arbitrarily high rates of evolution for the state of
the clock. More pertinently, the assumed validity of the composition law (6)
turns out to be equivalent to assuming the absolute simultaneity of “instant-
states” (ti, si) within the 1 + 2n−dimensional extended phase space O. In
other words, within the 1 + 2n−dimensional manifold O, the 2n−dimensional
phase spaces simply constitute strata of hypersurfaces of simultaneity, much
like the strata of spatial hypersurfaces within a Newtonian spacetime. Indeed,
the extended phase spaces such as O are usually taken to be contact manifolds,
with topology presumed to be a product of the form R × N .
Thus, not surprisingly, the assumption of absolute time in contact spaces
is equivalent to the assumption of “no upper bound” on the possible rates of
evolution of physical states. Now, in accordance with our discussion above,
suppose we impose the following upper bound on the evolution rates5:

dy
dt
 =: ω ≤t−1
P
.
(7)
If this upper bound is to have any physical signiﬁcance, however, then it
must hold for all possible evolving phase space coordinates yµ(t), and that is
amenable if and only if the composition law (6) is replaced by
ω′µ =
ωµ + ωµ
r
1 + t2
P ωµ ωµ
r ,
(8)
5 The “ﬂat” Euclidean metric on the phase space that is being used here is the
“quantum shadow metric”, viewed as a classical limit of the Fubini-Study metric
of the quantum state space (namely, the projective Hilbert space), in accordance
with our bottom-up philosophy depicted in Fig. 3. See Ref. [8] for further details.

16
Joy Christian
which implies that as long as neither ωµ nor ωµ
r exceeds the causal upper
bound t−1
P , ω′µ also remains within t−1
P . Of course, this generalized law of
composition has been inspired by Einstein’s own such law for velocities, which
states that the velocity, say vk (k = 1, 2, or 3), of a material body in a given
direction in one inertial frame is related to its velocity, say v′k, in another
frame, moving with a velocity −vk
r with respect to the ﬁrst, by the relation
v′k =
vk + vk
r
1 + c−2 vk vkr
.
(9)
Thus, as long as neither vk nor vk
r exceeds the upper bound c, v′k also remains
within c. It is this absoluteness of c that lends credence to the view that it is
merely a conversion factor between the dimensions of time and space. This fact
is captured most conspicuously by the quadratic invariant (1) of spacetime.
In exact analogy, if we require the causal relationships among the possible
instant-states (ti, si) in O to respect the upper bound t−1
P
in accordance with
the law (8), then the usual product metric of the space O would have to be
replaced by the pseudo-Euclidean metric corresponding to the line element
dτ 2
H = dt2 −t2
P dy2 ≥0,
(10)
where the phase space line element dy was discussed in the footnote 5 above.
But then, in the resulting picture, diﬀerent canonical coordinates evolving
with nonzero relative rates would diﬀer in general over which instant-states
are simultaneous with a given instant-state. As unorthodox as this new picture
may appear to be, it is an inevitable consequence of the upper bound (7).
Let us now raise a question analogous to the one raised earlier: In its rest
frame, what will be the time interval registered by the clock as it evolves from
an instant-state (t1, s1) to an instant-state (t2, s2) within the space O? The
answer, according to the pseudo-Euclidean line element (10), is clearly
∆τH =
Z (t2, s2)
(t1, s1)
dτH =
Z t2
t1
1
γ(ω) dt = | t2 −t1|
γ(ω)
=
∆t
γ(ω) ,
(11)
where
γ(ω) :=
1
p1 −t2
P ω2 > 1.
(12)
Thus, if the state of the clock is evolving, then we will have the phenomenon of
“time dilation” even in the rest frame. Similarly, we will have a phenomenon of
“state contraction” in analogy with the phenomenon of “length contraction”:
∆y′ = ω ∆τH = ω ∆t
γ(ω) = ∆y
γ(ω) .
(13)
It is worth emphasizing here, however, that, as in ordinary special relativity,
nothing is actually “dilating” or “contracting”. All that is being exhibited by

Absolute Being vs Relative Becoming
17
these phenomena is that the two sets of mutually evolving canonical coordi-
nates happen to diﬀer over which instant-states are simultaneous.
So far, to arrive at the expression (10) for the proper duration, we have
used a speciﬁc Lorentz frame, namely the rest frame of the clock. In a frame
with respect to which the same clock is uniformly moving, the expression for
the actual proper duration can be obtained at once from (10), by simply using
the Minkowski line element (1), yielding
dτ 2 = dt2 −c−2dx2 −t2
P dy2 ≥0.
(14)
This, then, is the 4 + 2n−dimensional quadratic invariant of our combined
space E of Fig. 5. We may now return to our original question and ask: What,
according to this generalized theory of relativity, will be the proper duration
registered by a given clock as it moves and evolves from an event-state (e1, s1)
to an event-state (e2, s2) in the combined space E? Evidently, according to
the quadratic invariant (14), the answer is simply:
∆τ =
Z (e2, s2)
(e1, s1)
dτ =
Z t2
t1
1
γ(v, ω) dt ,
(15)
with
γ(v, ω) :=
1
p
1 −c−2 v2 −t2
P ω2 > 1.
(16)
We are now in a position to isolate the two basic postulates on which the
generalized theory of relativity developed above can be erected in the manner
analogous to the usual special relativity. In fact, the ﬁrst of the two postulates
can be taken to be Einstein’s very own ﬁrst postulate, except that we must
now revise the meaning of inertial coordinate system. In the present theory
it is taken to be a system of 4 + 2n dimensions, “moving” uniformly in the
combined space E, with 4 being the external spacetime dimensions, and 2n
being the internal phase space dimensions of the system. Again, the internal
dimensions of the object system can be either ﬁnite or inﬁnite in number.
Next, note that by eliminating the speed of light in favor of pure Planck scale
quantities the quadratic invariant (14) can be expressed in the form
dτ 2 = dt2 −t2
P

l−2
P
dx2 + dy2	
≥0 ,
(17)
where lP is the Planck length of the value ∼10−33 cm in ordinary units. The
two postulates of generalized relativity may now be stated as follows:
(1)
The laws governing the states of a physical system are insensitive to “the
state of motion” of the 4 + 2n−dimensional reference coordinate system
in the pseudo-Euclidean space E, as long as it remains “inertial”.
(2)
No time rate of change of a dimensionless physical quantity, expressed
in Planck units, can exceed the inverse of the Planck time.

18
Joy Christian
Clearly, the generalized invariance embedded within this new causal theory
of local inertial structure is much broader in its scope—both physically and
conceptually—than the invariance embedded within special relativity. For ex-
ample, in the present theory even the four dimensional continuum of spacetime
no longer enjoys the absolute status it does in Einstein’s theories of relativity.
Einstein dislodged Newtonian concepts of absolute time and absolute space,
only to replace them by an analogous concept of absolute spacetime—namely, a
continuum of in principle observable events, idealized as a connected pseudo-
Riemannian manifold, with observer-independent spacetime intervals. Since
it is impossible to directly observe this remaining absolute structure without
recourse to the behavior of material objects, perhaps it is best viewed as the
“ether” of the modern times, as Einstein himself occasionally did [37]. By
contrast, it is evident that in the present theory even this four-dimensional
spacetime continuum has no absolute, observer-independent meaning. In fact,
apart from the laws of nature, there is very little absolute structure left in the
present theory, for now even the quadratic invariant (14) is dependent on the
phase space structure of the material system being employed.
3.2 Physical implications of the generalized theory of relativity
Although not our main concern here6, it is worth noting that the generalized
theory of relativity described above is both a physically viable and empirically
adequate theory. In fact, in several respects the present theory happens to
be physically better behaved than Einstein’s special theory of relativity. For
instance, unlike in special relativity, in the present theory physical quantities
such as lengths, durations, energies, and momenta remain bounded by their
respective Planck scale values. This physically sensible behavior is due to the
fact that present theory assumes even less preferred structure than special
relativity, by positing democracy among the internal phase space coordinates
in addition to that among the external spacetime coordinates.
Mathematically, this demand of combined democracy among spacetime
and phase space coordinates can be captured by requiring invariance of the
physical laws under the 4 + 2n−dimensional coordinate transformations [8]
zA = ΛA
B z′B + bA
(18)
analogous to the Poincar´e transformations, with the index A = 0, . . . , 3 + 2n
now running along the 4 + 2n dimensions of the manifold E of Fig. 5. These
transformations would preserve the quadratic invariant (17) iﬀthe constraints
ΛA
C ΛB
D ξAB = ξCD
(19)
are satisﬁed, where ξAB are the components of the metric on the manifold E.
At least for simple ﬁnite dimensional phase spaces, the coeﬃcients ΛA
B are
6 In this subsection we shall only brieﬂy highlight the physical implications of the
generalized theory of relativity. For a complete discussion see Sec. VI of Ref. [8].

Absolute Being vs Relative Becoming
19
easily determinable. For example, consider a massive relativistic particle at
rest (and hence also not evolving) with respect to a primed coordinate system
in the external spacetime, which is moving with a uniform velocity v with
respect to another unprimed coordinate system. Since, as it moves, the state
of the particle will also be evolving in its six dimensional phase space, say at
a uniform rate ω, we can view its motion and evolution together with respect
to a 4 + 6−dimensional unprimed coordinate system in the space E.
Restricting now to the external spatio-temporal sector where we actually
perform our measurements, it is easy to show [8] that the coeﬃcients ΛA
B
are functions of the generalized gamma factor (16), with the corresponding
expression for the length contraction being
∆x′ =
∆x
γ(v, ω) ,
(20)
which can be further evaluated to yield
∆x′ = ∆x
s
1 −v2
c2 −l2
P
∆x −∆x′
∆x′∆x
2
.
(21)
Although nonlinear, this expression evidently reduces to the special relativistic
expression for length contraction in the limit of vanishing Planck length. For
the physically interesting case of ∆x′ ≪∆x, it can be simpliﬁed and solved
exactly, yielding the “linearized” expression for the “contracted” length,
∆x′ = ∆x
v
u
u
t1
2

1 −v2
c2

+
s
1
4

1 −v2
c2
2
−
l2
P
(∆x)2 ,
(22)
provided the reality condition
1
4

1 −v2
c2
2
≥
l2
P
(∆x)2
(23)
is satisﬁed. Substituting this condition back into the solution (22) then gives
∆x′ ≥
p
lP ∆x ,
(24)
which implies that as long as ∆x remains greater than lP the “contracted”
length ∆x′ also remains greater than lP , in close analogy with the invariant
bound c on speeds in special relativity. That is to say, in addition to the
upper bound ∆x on lengths implied by the condition γ(v, ω) > 1 above, the
“contracted” length ∆x′ also remains invariantly bounded from below, by lP :
∆x > ∆x′ > lP .
(25)
Starting again from the expression for time dilation analogous to that for
the length contraction,

20
Joy Christian
∆τ =
∆t
γ(v, ω) ,
(26)
and using almost identical line of arguments as above, one analogously arrives
at a generalized expression for the time dilation,
∆τ = ∆t
v
u
u
t1
2

1 −v2
c2

+
s
1
4

1 −v2
c2
2
−
t2
P
(∆t)2 ,
(27)
together with the corresponding invariant bounds on the “dilated” time:
∆t > ∆τ > tP .
(28)
Thus, in addition to being bounded from above by the time ∆t, the “dilated”
time ∆τ remains invariantly bounded also from below, by the Planck time tP .
So far we have not assumed or proved explicitly that the constant “c”
is an upper bound on possible speeds. As alluded to above, in the present
theory the observer-independence of the upper bound c turns out to be a
derivative notion. This can be easily appreciated by considering the ratio of the
“contracted” length (22) and “dilated” time (27), along with the deﬁnitions
u := ∆x
∆t
and
u′ := ∆x′
∆τ
(29)
for velocities, leading to the upper bound on velocities in the moving frame:
u′ ≤u
q
1 +
p
1 −c2 u−2 .
(30)
Hence, as long as u does not exceed c, u′ also remains within c. In other words,
in the present theory c retains its usual status of the observer-independent
upper bound on causally admissible speeds, but in a rather derivative manner.
In addition to the above kinematical implications, the basic elements of the
particle physics are also modiﬁed within our generalized theory, the central
among which being the Planck scale ameliorated dispersion relation
p2 c2 + m2 c4 = E2
"
1 −
 E −m c22
E2
P
#
,
(31)
where EP is the Planck energy. It is worth emphasizing here that this is an
exact relation between energies and momenta, which in the rest frame of the
massive particle reproduces Einstein’s famous mass-energy equivalence:
E = m c2.
(32)
Moreover, in analogy with the invariant lower bounds on lengths and durations
we discussed above, in the present theory energies and momenta can also be
shown to remain invariantly bounded from above by their Planck values:

Absolute Being vs Relative Becoming
21
E′ ≤
p
EP E
and
p′ ≤
p
kP p ×
rv
c ,
(33)
where kP is the Planck momentum. Thus, as long as the unprimed energy E
does not exceed EP , the primed energy E′ also remains within EP . That is to
say, in addition to the lower bound E on energies implied by the condition
γ(v, ω) > 1, the energies E′ remain invariantly bounded also from above, by
the Planck energy EP :
E < E′ < EP .
(34)
Similarly, as long as the relative velocity v does not exceed c and the unprimed
momentum p does not exceed kP , the primed momentum p′ also remains within
kP . Hence, in addition to the lower bound p on momenta set by the condition
γ(v, ω) > 1, the momenta p′ remain invariantly bounded also from above, by
the Planck momentum kP :
p < p′ < kP .
(35)
Thus, unlike in special relativity, in the present theory all physical quantities
remain invariantly bounded by their respective Planck scale values.
Next, consider an isolated system of mass msys composed of a number
of constituents undergoing an internal reaction. It follows from the quadratic
invariant (17) that the 4 + 2n−vector Psys, deﬁned as the abstract momentum
of the system as a whole, would be conserved in such a reaction (cf. [8]),
∆Psys = 0 ,
(36)
where ∆denotes the diﬀerence between initial and ﬁnal states of the reaction,
and Psys is deﬁned by
msys
dzA
dτ
=: PA
sys :=
 Esys/c , pk
sys , P µ
sys

,
(37)
with k = 1, 2, 3 denoting the external three dimensions and µ = 4, 5, . . ., 3 + 2n
denoting the phase space dimensions of the system as a whole. It is clear from
this deﬁnition that, since dzA is a 4 + 2n−vector whereas msys and dτ are
invariants, PA
sys is also a 4 + 2n−vector, and hence transforms under (18) as
P′A
sys = ΛA
B PB
sys .
(38)
Moreover, since Λ depends only on the overall coordinate transformations
being performed within the space E, the diﬀerence on the left hand side of
(36) is also a 4 + 2n−vector, and therefore transforms as
∆P′A
sys = ΛA
B ∆PB
sys .
(39)
Thus, if the conservation law (36) holds for one set of coordinates within the
space E, then, according to (39), it does so for all coordinates related by the

22
Joy Christian
transformations (18). Consequently, the conservation law (36), once unpacked
into its external, internal, and constituent parts as
0 = ∆Psys =
 ∆Esys/c , ∆psys , ∆Pint
sys

,
(40)
leads to the familiar conservation laws for energies and momenta:
0 = ∆Esys :=
X
f
Ef −
X
i
Ei
(41)
and
0 = ∆psys :=
X
f
pf −
X
i
pi ,
(42)
where the indices f and i stand for the ﬁnal and initial number of constituents
of the system. Thus in the present theory the energies and momenta remain as
additive as in special relativity. In other words, in the present theory not only
are there no preferred class of observers, but also the usual conservation laws
of special relativity remain essentially unchanged, contrary to expectation.
3.3 The raison d’ˆetre of time: causal inevitability of becoming
With the physical structure of the generalized relativity in place, we are now
in a position to address the central concern of the present essay: namely, the
raison d’ˆetre of the tensed time, as depicted in Fig. 2. To this end, let us ﬁrst
note that the causal structure embedded within our generalized relativity is
profoundly unorthodox. One way to appreciate this unorthodoxy is to recall
the blurb for spacetime put forward by Minkowski in his seminal address at
Cologne, in 1908. “Nobody has ever noticed a place except at a time, or a
time except at a place”, he ventured [38]. But, surely, this famous quip of
Minkowski hardly captures the complete picture. Perhaps it is more accurate
to say something like: Nobody has ever noticed a place except at a certain time
while being in a certain state, or noticed a time except at a certain place while
being in a certain state, or been in some state except at a certain time, and a
certain place. At any rate, this revised statement is what better captures the
notion of time aﬀorded by our generalized theory of relativity. For, as evident
from the quadratic invariant (14), in addition to space, time in our generalized
theory is as much a state-dependent attribute as states are time-dependent
attributes, and as states of the world do happen and become, so does time.
Intuitively, this dynamic state of aﬀairs can be summarized as follows:
x = x(t, y)
t = t(x, y)
y = y(t, x),
(43)
where y is the phase space coordinate as before. In other words, place in the
present theory is regarded as a function of time and state; time is regarded as

Absolute Being vs Relative Becoming
23
Phase
space
Space
Time
The moving now
b
c
b
c
b
c
b
c
b
c
(e1, s1)
(e5, s5)
Fig. 7. Space-time-state diagram depicting the ﬂow of time. The solid blue curves
represent growing timelike worldlines at ﬁve successive stages of growth, from s1 to
s5, whereas the dashed green curve represents the growing overall worldline from
(e1, s1) to (e5, s5). The red dot represents the necessarily moving present. In fact,
the entire space-time-state structure is causally necessitated to expand continuously.
a function of place and state; and state is regarded as a function of time and
place. As we shall see, it is this state-dependence of time that is essentially
what mandates the causal necessity for becoming in the present theory.
To appreciate this dynamic or tensed nature of time in the present theory,
let us return once again to our clock that is moving and evolving, say, from
an event-state (e1, s1) to an event-state (e5, s5) in the combined space E, as
depicted in the space-time-state7 diagram of Fig. 7. According to the line
element (14), the proper duration recorded by the clock would be given by
∆τ =
Z t5
t1
1
γ(v, ω) dt ,
(44)
where γ(v, ω) is deﬁned by (16). Now, assuming for simplicity that the clock
is not massless, we can represent its journey by the integral curve of a timelike
4 + 2n−velocity vector ﬁeld V A on the space E, deﬁned by
V A := lP
dzA
dτ
,
(45)
such that its external components V a (a = 0, 1, 2, 3) would trace out, for each
possible state si of the clock, the familiar four dimensional timelike worldlines
7 Here perhaps “space-time-phase space diagram” would be a much more accurate
neologism, but it would be even more mouthful than “space-time-state diagram.”

24
Joy Christian
in the corresponding Minkowski spacetime. In other words, the overall velocity
vector ﬁeld V A would give rise to the familiar timelike, future-directed, never
vanishing, 4-velocity vector ﬁeld V a, tangent to each of the external timelike
worldlines. As a result, the “length” of the overall enveloping worldline would
be given by the proper duration (44), whereas the “length” of the external
worldline, for each si, would be given by the Einsteinian proper duration
∆τi
E =
Z ti
t1
1
γ(v) dt ,
(46)
where γ(v) is the usual Lorentz factor given by (3). In Fig. 7, ﬁve of such
external timelike worldlines—one for each si (i=1,2,3,4,5)—are depicted by
the blue curves with arrowheads going “upwards”, and the overall enveloping
worldline traced out by V A is depicted by the dashed green curve going from
the “initial” event-state (e1, s1) to the “ﬁnal” event-state (e5, s5).
It is perhaps already clear from this picture that the external worldline
of our clock is not given all at once, stretched out till eternity, but grows
continuously, along with each temporally successive stage of the evolution of
the clock, like a tendril on a wall. That is to say, as anticipated in Fig. 2, the
future events along the external worldline of the clock simply do not exist.
Hence the “now” of the clock cannot even be said to be preceding the future
events, since, quite literally, there exists nothing to which it has the relation of
precedence [27]. Moreover, since the external Minkowski spacetime is simply
a congruence of non-intersecting timelike worldlines of idealized observers,
according to the present theory the entire sum total of existence must increase
continuously [27]. In fact, this continuous growth of existence turns out to be
causally necessitated in the present theory, and can be represented by a Growth
Vector Field quantifying the instantaneous directional rate of this growth:
U a := ˆV a dτE
dy ,
(47)
where ˆV a is a unit vector ﬁeld in the direction of the 4-velocity vector ﬁeld V a,
dy := |dy| is the inﬁnitesimal dimensionless phase space distance between the
two successive states of the clock discussed before, and dτE is the inﬁnitesimal
Einsteinian proper duration deﬁned by (1). It is crucial to note here that in
special relativity this Growth Vector Field would vanish identically everywhere,
whereas in our generalized theory it cannot possibly vanish anywhere. This is
essentially because of the mutual dependence of place, time, and state in the
present theory we discussed earlier (cf. Eq. (43)). More technically, this is
because the 4-velocity vector V a of an observer in Minkowski spacetime, such
as the one in Eq. (47), can never vanish, whereas the causality constraint (14)
of the present theory imposes the lower bound tP on the rate of change of
Einsteinian proper duration with respect to the phase space coordinates,
dτE
dy ≥tP ,
(48)

Absolute Being vs Relative Becoming
25
which, taken together, causally necessitates the never-vanishing of the Growth
Vector Filed U a. Consequently, the “now” of the clock (the red dot in Fig. 7)
moves in the future direction along its external worldline, at the rate of no less
than one Planck unit of time per Planck unit of change in its physical state.
And, along with the non-vanishing of the 4-velocity vector ﬁeld V a, the lower
bound tP on the growth rate of any external worldline implies that not only
do all such “nows” move, but they cannot not move—i.e., not only does the
sum total of existence increase, but it cannot not increase. To parody Weyl
quoted above, the objective world cannot simply be, it can only happen.
This conclusion can be further consolidated by realizing that in the present
theory even the overall enveloping worldline (the dashed green curve in Fig. 7)
cannot help but grow non-relationally and continuously. This can be conﬁrmed
by ﬁrst parallelling the above analysis for the 1 + 2n−dimensional internal
space O instead of the external spacetime M, which amounts to slicing up
the combined space E of Fig. 7 along the spatial axis instead of the phase
space axis, and then observing that even the “internal worldline” (not shown
in the ﬁgure) must necessarily grow progressively further as time passes, at
the rate given by the internal growth vector ﬁeld
U α = lP ˆV α dtH
dx .
(49)
Here ˆV α is a unit vector ﬁeld in the direction of the 1 + 2n−velocity vector
ﬁled V α corresponding to the internal part of the overall velocity vector ﬁeld
V A, dx := |dx| is the inﬁnitesimal spatial distance between two slices, and dtH
is the inﬁnitesimal internal proper duration deﬁned by Eq. (10). Once again,
it is easy to see that the causality condition (14) gives rise to the lower bound
lP
dtH
dx ≥tP .
(50)
Thus, “now” of the clock necessarily moves in the future direction also along
its internal worldline within the internal space O. As a result, even the overall
worldline—namely, the dashed green curve in Fig. 7—can be easily shown
to be growing non-relationally and continuously. Indeed, using Eqs. (47) to
(50), an elementary geometrical analysis [8] shows that the instantaneous
directional rate of this growth is given by the overall growth vector ﬁeld
U A =

ˆV a dtE
dy , lP ˆV µ dtH
dx

,
(51)
whose magnitude also remains bounded from below by the Planck time tP :
p
−ξABU AU B ≥tP .
(52)
Thus, in the present theory, not only are the external events in E not all
laid out once and for all, for all eternity, but there does not remain even an

26
Joy Christian
overall 4 + 2n−dimensional “block” that could be used to support a “block”
view of the universe. In fact, the causal necessity of the lower bound (52) on
the magnitude of the overall growth vector ﬁeld U A—which follows from the
causality constraint (14)—exhibits that in the present theory the sum total
of existence itself is causally necessitated to increase continuously. That is to
say, the very structure of the present theory causally necessitates the universe
to be purely Heraclitean, in the sense discussed in the Introduction.
4 Prospects for the experimental metaphysics of time
As alluded to in the Introduction, any empirical confrontation of the above
generalized relativity with special relativity would amount to a step towards
what may be called the experimental metaphysics of time. However, since
the generalized theory is deeply rooted in the Planck regime, any attempt to
experimentally discriminate it from special relativity immediately encounters
a formidable practical diﬃculty. To appreciate this diﬃculty, consider the
following series expansion of expression (27) for the generalized proper time,
up to second order in the Planck time:
∆τ = ∆t
r
1 −v2
c2 −1
2
t2
P
∆t

1 −v2
c2
−3
2
+ . . .
(53)
The ﬁrst term on the right hand side of this expansion is, of course, the familiar
special relativistic term. The diﬃculty arises in the second term, i.e. in the
ﬁrst largest correction term to the special relativistic time dilation eﬀect, since
this term is modulated by the square of the Planck time, which in ordinary
units amounts to some 10−87 sec2. Clearly, the precision required to directly
verify such a miniscule correction to the special relativistic prediction is well
beyond the scope of any foreseeable precision technology.
Fortunately, in recent years an observational possibility has emerged that
might save the day for the experimental metaphysics of time. The central
idea that has emerged during the past decade within the context of quantum
gravity is to counter the possible Planck scale suppression of physical eﬀects by
appealing to ultrahigh energy particles cascading the earth that are produced
at cosmological distances. One strategy along this line is to observe oscillating
ﬂavor ratios of ultrahigh energy cosmic neutrinos to detect possible deviations
in the energy-momentum relations predicted by special relativity [39]. Let
us brieﬂy look at this strategy, as it is applied to our generalized theory of
relativity (further details can be found in Refs. [36] and [39]; as in these
references, from now on we shall be using the Planck units: ℏ= c = G = 1).
4.1 Testing Heraclitean relativity using cosmic neutrinos
The remarkable phenomena of neutrino oscillations are due to the fact that
neutrinos of deﬁnite ﬂavor states |να⟩, α = e, µ, or τ, are not particles of

Absolute Being vs Relative Becoming
27
deﬁnite mass states |νj⟩, j = 1, 2, or 3, but are superpositions of the deﬁnite
mass states. As a neutrino of deﬁnite ﬂavor state propagates through vacuum
for a long enough laboratory time, its heavier mass states lag behind the
lighter ones, and the neutrino transforms itself into an altogether diﬀerent
ﬂavor state. The probability for this “oscillation” from a given ﬂavor state,
say |να(0)⟩, to another ﬂavor state, say |νβ(t)⟩, is famously given by
Pαβ(E, L) = δαβ −
X
j̸=k
U ∗
αjUαkUβjU ∗
βk
h
1 −e−i(∆m2
jk/2E)Li
.
(54)
Here ∆m2
jk ≡m2
k −m2
j > 0 is the diﬀerence in the squares of the two neutrino
masses, U is the time-independent leptonic mixing matrix, and E and L are,
respectively, the energy and distance of propagation of the neutrinos. It is
clear from this transition probability that the experimental observability of
the ﬂavor oscillations is dependent on the quantum phase
Φ := 2π L
LO
,
(55)
where
LO := 2 π
∆p =
4πE
∆m2
jk
(56)
is the energy-dependent oscillation length. Thus, changes in neutrino ﬂavors
would be observable whenever the propagation distance L is of the order of the
oscillation length LO. However, in deﬁnition (56) the diﬀerence in momenta,
∆p ≡pj −pk, was obtained by using the special relativistic relation
pj =
q
E2 −m2
j ≈E −m2
j
2E .
(57)
In the present theory this relation between energies and momenta is, of course,
generalized, and given by (31), replacing the above approximation by
pj ≈E −m2
j
2E + E2
m2
P
mj
(58)
up to the second order, with mP being the Planck mass. The corresponding
modiﬁed oscillation length analogous to (56) is then given by
L
′
O := 2 π
∆p =
2 π
1
2E ∆m2
jk −
E2
m2
P ∆mjk
,
(59)
where ∆m2
jk ≡m2
k −m2
j as before, and ∆mjk ≡mk −mj > 0. Consequently,
according to our generalized relativity the transition probability (54) would
be quite diﬀerent in general, as a function of E and L, from how it is according
to special relativity. And despite the quadratic Planck energy suppression of

28
Joy Christian
the correction to the oscillation length, this diﬀerence would be observable for
neutrinos of suﬃciently high energies and long propagation distances. Indeed,
it can be easily shown [39] that the relation
L ∼π m4
P
E5
(60)
is the necessary constraint between the neutrino energy E and the propagation
distance L for the observability of possible deviations from the standard ﬂavor
oscillations. For instance, it can be readily calculated from this constraint
that the Planck scale deviations in the oscillation length predicted by our
generalized relativity would be either observable, or can be ruled out, for
neutrinos of energy E ∼1017 eV, provided that they have originated from a
cosmic source located at some 105 light-years away from a terrestrial detector.
The practical means by which this can be achieved in the foreseeable future
have been discussed in some detail in the Refs. [36] and [39] cited above.
4.2 Testing Heraclitean relativity using γ-ray binary pulsars
The previous method of confronting the generalized theory of relativity with
special relativity is clearly phenomenological. Fortunately, a much more direct
test of the generalized theory may be possible, thanks to the precise deviations
it predicts from the special relativistic Doppler shifts [8]:
E′
E
= ε′ 
ε′ −v
c cos φ

q
(ε′)2 −v2
c2
,
(61)
with
ε′ :=
s
1 −E2
E2
P

1 −E′
E
2
.
(62)
Here v is the relative speed of a receiver receding from a photon source, E
and E′, respectively, are the energy of the photon and that observed by the
receiver, and φ is the angle between the velocity of the receiver and the photon
momentum. Note that ε′ here clearly reduces to unity for E′ −E ≪EP , thus
reducing the generalized expression (61) to the familiar linear relation for
Doppler shifts predicted by special relativity.
Even without solving the relation (61) for E′ in terms of E, it is not
diﬃcult to see that, since ε′ < 1, at suﬃciently high energies any red-shifted
photons would be somewhat more red-shifted according to (61) than predicted
by special relativity. But one can do better than that. A Maclaurin expansion
of the right hand side of (61) around the value E/EP = 0, after keeping terms
only up to the second order in the ratio E/EP , gives
E′
E ≈1 −v
c cos φ
q
1 −v2
c2
+ 1
2
E2
E2
P

1 −v
c cos φ
 1 −v2
c2
3/2 −2 −v
c cos φ
 1 −v2
c2
1/2



1 −E′
E
2
+ . . . (63)

Absolute Being vs Relative Becoming
29
This truncation is an excellent approximation to (61). The quadratic equation
(63) can now be solved for the desired ratio E′/E, and then the physical root
once again expanded, now in the powers of v/c. In what results if we again
keep terms only up to the second order in the ratios E/EP and v/c, then,
after some tedious but straightforward algebra, we arrive at
E′
E ≈1 −v
c cos φ + 1
2

1 −E2
E2
P
cos2φ
 v2
c2 ± . . . ,
(64)
which, in the limit E ≪EP , reduces to the special relativistic result
E′
E ≈1 −v
c cos φ + 1
2
v2
c2 ± . . . .
(65)
Comparing (64) and (65) we see that up to the ﬁrst order in v/c there
is no diﬀerence between the special relativistic result and that of the present
theory. The ﬁrst deviation between the two theories occur in the second-
order coeﬃcient, precisely where special relativity diﬀers also from the classical
theory. What is more, this second-order deviation depends non-trivially on the
angle between the relative velocity and photon momentum. For instance, up to
the second order both red-shifts (φ = 0) and blue-shifts (φ = π) predicted by
(64) signiﬁcantly diﬀer from those predicted by special relativity. In particular,
the red-shifts are now somewhat more red-shifted, whereas the blue-shifts
are somewhat less blue-shifted. On the other hand, the transverse red-shifts
(φ = π/2 or φ = 3π/2) remain identical to those predicted by special relativity.
As a result, even for the photon energy approaching the Planck energy an
Ives-Stilwell type classic experiment [40] would not be able to distinguish
the predictions of the present theory from those of special relativity. The
complete angular distribution of the second-order coeﬃcient predicted by the
two theories, along with its energy dependence, is displayed in Fig. 8.
In spite of this rather non-trivial angular dependence of Doppler shifts, in
practice, due to the quadratic suppression by Planck energy, distinguishing
the expansion (64) from its special relativistic counterpart (65) would be a
formidable task. The maximum laboratory energy available to us is of the
order of 1012 eV, yielding E2/E2
P ∼10−32. This represents a correction of
one part in 1032 from (65), demanding a phenomenal sensitivity of detection
well beyond the means of any foreseeable precision technology. However, an
extraterrestrial source, such as an extreme energy γ-ray binary pulsar, may
turn out to be accessible for distinguishing the second order Doppler shifts
predicted by the two theories. It is well known that binary pulsars not only
exhibit Doppler shifts, but the second-order shifts resulting from the periodic
motion of such a pulsar about its companion can be isolated, say, from the
ﬁrst order shifts, because they depend on the square of the relative velocity,
which varies as the pulsar moves along its two-body elliptical orbit [41]. Due
to these Doppler shifts, the rate at which the pulses are observed on Earth
reduces slightly when the pulsar is receding away from the Earth, compared to

30
Joy Christian
0
90
180
270
360
0
0.5
Angle between relative velocity and photon momentum
Second order coeﬃcient
Fig. 8. The energy-dependent signatures of Heraclitean relativity. The green curves
are based on the predictions of the present theory, for E/EP = 0.3 to 0.9 in the
descending order, whereas the dashed black line is the prediction of special relativity.
when it is approaching towards it. As a result, the period, its variations, and
other orbital characteristics of the pulsar, as they are determined on Earth,
crucially depend on these Doppler shifts. In practice, the parameter relevant
in the arrival-time analysis of the pulses received on Earth turns out to be a
non-trivial function of the gravitational red-shift, the masses of the two binary
stars, and other Keplerian parameters of their orbits, and is variously referred
to as the red-shift-Doppler parameter or the time dilation parameter [41]. For
a pulsar that is also following a periastron precession similar to the perihelion
advance of Mercury, it can be determined with excellent precision.
The arrival-time analysis of the pulses begins by considering the time of
emission of the N th pulse, which is given by
N = NO + ντ + 1
2 ˙ντ 2 + 1
6 ¨ντ 3 + . . . ,
(66)
where NO is an arbitrary integer, τ is the proper time measured by a clock in
an inertial frame on the surface of the pulsar, and ν is the rotation frequency
of the pulsar, with ˙ν ≡dν/dτ|τ=0 and ¨ν ≡d2ν/dτ 2|τ=0. The proper time τ is
related to the coordinate time t by
dτ = dt

1 −α∗
2m2
r
−1
2
v2
1
c2 + ...

,
(67)
where the ﬁrst correction term represents the gravitational red-shift due to the
ﬁeld of the companion, and the second correction term represents the above

Absolute Being vs Relative Becoming
31
mentioned second-order Doppler shift due to the orbital motion of the pulsar
itself. The time of arrival of the pulses on Earth diﬀers from the coordinate
time t taken by the signal to travel from the pulsar to the barycenter of the
solar system, due to the geometrical intricacies of the pulsar binary and the
solar system [41]. More relevantly for our purposes, the time of arrival of the
pulses is directly aﬀected by the second-order Doppler shift appearing in Eq.
(67), which thereby aﬀects the observed orbital parameters of the pulsar.
Now, returning to our Heraclitean generalization of relativity, it is not
diﬃcult to see that the generalized Doppler shift expression (64) immediately
gives the following generalization of the inﬁnitesimal proper time (67):
dτ = dt

1 −α∗
2m2
r
−1
2

1 −E2
E2
P
cos2 φ
 v2
1
c2 + ...

.
(68)
Thus, in our generalized theory the second-order Doppler shift acquires an
energy-dependent modiﬁcation. The question then is: At what radiation energy
this nontrivial modiﬁcation will begin to aﬀect the observable parameters of
the pulsar? The most famous pulsar, namely PSR B1913+16, which has been
monitored for three decades with exquisite accumulation of timing data, is a
radio pulsar, and hence for it the energy-dependent modiﬁcation predicted in
(68) is utterly negligible, thanks to the quadratic suppression by the Planck
energy. However, for a γ-ray pulsar with suﬃciently high radiation energy
the modiﬁcation predicted in (68) should have an impact on its observable
parameters, such as the orbital period and its temporal variations.
The overall precision in the timing of the pulses from PSR B1913+16, and
consequently in the determination of its orbital period, is famously better than
one part in 1014 [42]. Indeed, the monitoring of the decaying orbit of PSR
B1913+16 constitutes one of the most stringent tests of general relativity to
date. It is therefore not inconceivable that similar careful observations of a
suitable γ-ray pulsar may be able to distinguish the predictions of the present
theory from those of special relativity. Unfortunately, the highest energy of
radiation from a pulsar known to date happens to be no greater than 1013 eV,
giving the discriminating ratio E2/E2
P to be of the order of 10−30, which is
only two orders of magnitude improvement over a possible terrestrial scenario.
On the other hand, the γ-rays emitted by a binary pulsar would have to be of
energies exceeding 1021 eV for them to have desired observable consequences,
comparable to those of PSR B1913+16. Moreover, the desired pulsar have to
be located suﬃciently nearby, since above the 1013 eV threshold γ-rays are
expected to attenuate severely through pair-production if they are forced to
pass through the cosmic infrared background before reaching the Earth. It is
not inconceivable, however, that a suitable binary pulsar emitting radiation of
energies exceeding 1021 eV is found in the near future, allowing experimental
discrimination of our generalized relativity from special relativity.

32
Joy Christian
5 Concluding remarks
One of the perennial problems in natural philosophy is the problem of change;
namely, How is change possible? Over the centuries, this problem has fostered
two diametrically opposing views of time and becoming. While these two views
tend to agree that time presupposes change, and that genuine change requires
becoming, one of them actually denies the reality of change and time, by
rejecting becoming as a “stubbornly persistent illusion” [43]. The other view,
by contrast, accepts the reality of change and time, by embracing becoming as
a bona ﬁde attribute of the world. Since the days of Aristotle within physics we
have been rather successful in explaining how the changes occur in the world,
but seem to remain oblivious to the deeper question of why do they occur at all.
The situation has been aggravated by the advent of Einstein’s theories of space
and time, since in these theories there is no room to structurally accommodate
the distinction between the past and the future—a prerequisite for the genuine
onset of change. By contrast, the causal structure of the Heraclitean relativity
discussed above not only naturally distinguishes the past form the future by
causally necessitating becoming, but also forbids inaction altogether, thereby
providing an answer to the deeper question of change. Moreover, since it is
not impossible to experimentally distinguish the Heraclitean relativity from
special relativity, and since the ontology underlying only the latter of these
two relativities is prone to a block universe interpretation, the enterprise of
experimental metaphysics of time becomes feasible now, for the ﬁrst time,
within a relativistic context. At the very least, such an enterprise should help
us decide whether time is best understood relationally, or non-relationally.
Acknowledgments
I would like to thank Huw Price and Abner Shimony for their comments on
Ref. [8], of which this essay is an apologia. I would also like to thank Lucien
Hardy, Lee Smolin, Antony Valentini, and other members of the Foundations
of Physics group at the Perimeter Institute for their hospitality and support.
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