Brit. ]. Phil. Sci. 40 (1989), 69-76 Printed in Great Britain

Simultaneity, Conventionality and
Existence*

VESSELIN PETKOV

ABSTRACT

The present paper pursues two aims. First to show that the experiment proposed by
Stolakis [1986] does not lead to absolute synchronization in a single frame of
reference and therefore also to the measurement of one-way velocity of light.
Second, by consecutively considering the problems of the conventionality of
simultaneity and of existence to show that the simultaneity of distant events can be
a matter of convention only in a four-dimensional world.

1 Introduction
2 Has the Conventionality of Simultaneity Been Refuted?
3 On the Essence of the Conventionality of Simultaneity

I INTRODUCTION

There have been three changes in the history of the idea of simultaneity. It has
changed for the first time in the 17th century after Romer had shown that light
propagated with extreme velocity. It was then clearly realized that the events
we observe simultaneously at a given moment of time have in actual fact taken
place at different previous moments. The idea of simultaneity was changed for
the second time when the theory of relativity showed that simultaneity was
not absolute, that there is meaning in speaking of the simultaneity of any
events only as regards a given reference frame (or observer). The third change
of view on simultaneity is connected with the elucidation of the fact that even
as regards a single reference system the definition of the simultaneity of events
is not absolute but is a matter of convention. The present paper is devoted
precisely to elucidating the essence of the conventionality of simultaneity.

The problem of the conventionality of simultaneity already stemmed from
Einstein and Reichenbach and chiefly boils down to the following. Two distant
clocks at the respective points A and B in a single frame of reference must be
synchronized. At moment t\ a light signal is emitted from point A, is reflected

* I am grateful to the anonymous referees for their useful suggestions on an earlier version of this
paper.

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70 Vesselin Petkov

on point B at moment £2 and returns to point A at moment (3. The times tA and
13 are measured by the clock at point A, while the time t\ is measured at point
B. The question is: is there an objective criterion on the basis of which it can be
determined which moment t\ of the interval (£A, t\) is simultaneous with the
moment of reflection of the light signal at point B, i.e. with the moment tA? The
different possibilities of this are expressed in the formula:

in which 0 < e < l . . H o w e v e r , it appears that in the attempt to establish this
moment, i.e. in the attempt to determine the 'real' value of e, a logical circle is
obtained (Reichenbach [1958]): if we try to synchronize the clocks at points A
and B by light signals we must know the one-way velocity of light; but in order
to measure it the clocks should have been synchronized beforehand. The
situation in the attempt to synchronize the clocks by slow transport of a third
clock is analogous.

This situation shows that it is not possible to determine the value of e by
means of a physical experiment, i.e. it is impossible to establish which events
are simultaneous at a given moment of time even as regards a single system of
reference. From this the conclusion is drawn that determining simultaneous
events is a matter of convention, from which it follows in particular that the
answer to the question: are the back and forth velocities of light one and the
same, also proves to be conventional. The latter conclusion appears to arouse
the strongest instinctive resistance, because it contradicts the statement
considered as an obvious one that in reality light travels at an exactly determined
velocity. If this is so, we should be able to measure its velocity in one way. This
dissatisfaction with the conventionality of simultaneity probably explains why
ideas on measuring one-way velocity of light are persistently suggested.

2 HAS THE CONVENTIONALITY OF SIMULTANEITY
BEEN REFUTED?

One of the most recent suggestions for an experimental solution of the problem
of the conventionality of simultaneity was made by Stolakis [1986]. The aim
pursued crossed the one factually achieved. The author's efforts were directed
to proposing an experimental way of measuring one-way velocity oflight, thus
showing the inconsistency of the conventionality thesis. Actually, however, he
proposes an experiment which had no relation to the conventionality of
simultaneity but could serve to discover a possible anisotropy of spacetime in a
space-like direction.

The essence of the experiment proposed by Stolakis would be revealed by a
more thorough analysis of the standard schema of synchronizing the clocks A
and B, distant in space, set forth in Part 1. Figure 1-a shows the world lines A
and B of the two clocks (at rest as regards one another) and the world line of the

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Simultaneity, Conventionality and Existence

t A tB { A

M

A B A B
(a) (b)

Figure 1

(c)

light ray propagating between them. In standard synchronization ( E = 1 / 2 )
event R (reflecting the light ray from clock B) is simultaneous with event M. In
this case space is orthogonal to the world lines of the clocks and the velocity of
the back and forth light signal is one and the same. In any other choice of e (in
the limits 0 < s < 1) space is not orthogonal to the world lines of clocks A and B
and the velocity of the back and forth light ray is different. Therefore velocity of
light in a straight direction is different from velocity in the opposite direction,
owing to the fact that space is not orthogonal to A and B. This conclusion is
only correct, however, if a premise (usually understood) is fulfilled, namely
that space-time is isotropic. In this case (depicted in Figure 1-a) the slant (in
relation to A and B) of the world line of the light ray propagating from clock A
to B is equal to the slant of the world line of the light ray from B to A, i.e. the
angle a,- is equal to the angle /?,-. But if space-time is anisotropic in a space-like
direction (see Figure 1-b) the slant (in relation to A and B) of the world line of
the light ray, propagating from clock A to clock B can be larger than the slant
of the world line of the light ray, propagating from B to A, i.e. the angle <xfl can
be larger than the angle /?„ (this anisotropy of space-time would appear as a
deformation of the light cone). In this case the orthogony of space in relation to
A and B does not ensure equality of light velocity in a backhand forth direction.
Therefore there is no simple connection between the equation of light velocity
in a back and forth direction and the orthogony of space in relation to the

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72 Vesselin Petkov

world lines of the clocks, i.e. in relation to the time axis of the system of
reference, in relation to which clocks A and B are at rest.

In both cases depicted in Figures 1-a and 1-b, we can arbitrarily choose the
value of e (of course in the limits of 0 < £ < 1), i.e. we can arbitrarily define
which moment of the interval (ft, t*) is simultaneous with t\. Thus, in both
cases we would obtain arbitrary values of back and forth velocity of light (as
Winnie [1970] showed and all relative velocities will also prove to be
arbitrary). In other words, we can arbitrarily (but nevertheless within definite
limits, ensuing from the restriction 0 < e < 1) choose the angle between space
and the world lines A and B. This arbitrariness shows that defining the
simultaneity of events M and R is a matter of convention and ensues from the
fact that not a single moment in the interval (ft, t*) is objectively priviledged so
that it can be considered simultaneous with the moment t§. The matter of the
essence of this objective privilege will be discussed in Part 3. But even if we
accept that any moment of the interval (ft, £3) is simultaneous with the
moment t\, the question of the orthogony of space in relation to the world lines
A and B (i.e. the matter of the isotropy of space-time in a space-like direction)
remains open. Let us suppose that event M is simultaneous with event R, i.e.
with the moment t\. Then in both cases depicted on Figures 1-a and 1-b it
appears that back and forth velocity of light is one and the same, but owing to
the isotropy of space-time, on Figure 1-a space is orthogonal to the world lines
A and B, while, owing to the anisotropy of space-time in a space-like direction,
on Figure 1-b space is not orthogonal to A and B. These two cases are
indiscernible experimentally. The matter of the isotropy of space-time cannot
be solved by a more complicated experiment (shown on Figure 1-c) in which
light signals are emitted from a clock in opposite directions to two other equally
distant clocks. After being reflected from the endmost clocks the light signals
return to the central clock. Regardless of whether the light signals propagate in
isotropic spacetime (in this case their world lines are depicted with dotted lines)
or in anisotropic space-time, they arrive simultaneously at the central clock.

Nevertheless, the question might be asked: is there any way of delimiting the
cases of isotropic and anisotropic space-time experimentally? The value of the
Stolakis experiment proposed consists namely in the fact that it attempts to
answer this question. The experiment is depicted in Figure 2. Figure 2-a shows
the case when space-time is isotropic, while on Figure 2-b we see the case of
anisotropic space-time. The smaller slant of the world lines of the light signals
to events M and N (lying on the world lines of the points at which the light rays
have been reflected) in Figure 2-a, compared with the slant of their world lines
after M and N, reflects the fact that up to the moment when the light signals are
reflected they have moved at a lesser velocity (owing to their propagation in a
medium with a refractive index of n > 1). In this case, owing to the isotrophy of
space-time, the light signals arrive simultaneously with event C2. However, this
is no proof that back and forth velocity of light is one and the same, because we

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Simultaneity, Conventionality and Existence 73

M N M

N

(b)

Figure 2

can again choose to have space not orthogonal to the time axis which will lead
to different values of back and forth velocity. We have no criterion (and the
Stolakis experiment does not give us one) which would force us to choose space
in such a way as to have events M and N occur simultaneously in it. Only if we
consider events M and N as simultaneous (i.e. as being privileged in comparison
with the remaining events on the world lines on which events M and N lie)
could it be asserted that the one-way velocity of light has been found. The
simultaneous arrival of light rays at the initial point from which they were
emitted is proof alone of the fact that space-time is isotropic in a space-like
direction.

Figure 2-b depicts the Stolakis experiment in a case of anisotropic space-
time in a space-like direction. In the case, the light signals emitted at event Ci
and respectively reflected in events M and N will not arrive simultaneously at
their point of emission. The reason for the non-simultaneous arrival of the light
signals (i.e. the appearance of the time difference tc2c3) is, however, due solely
to the anisotrophy of space-time. Even now the definition of back and forth
velocity of light is a matter of convention, because once again the criterion for
the simultaneity of events M and N has not been discovered. The only fact
which can be established by the Stolakis experiment is whether space-time is
anisotropic in a space-like direction.

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74 Vesselin Petkov

Yet even if the problem of the isotropy of space-time is solved, the problem of
the conventionality of simultaneity remains untouched. Thus the Stolakis
experiment has no relation to the problem of the conventionality of
simultaneity. The analysis of this experiment shows once again the impossibi-
lity in principle of measuring one-way velocity of light and therefore of rejecting
the thesis of the conventionality of simultaneity.

3 ON THE ESSENCE OF THE CONVENTIONALITY OF SIMULTANEITY

The discussion on the problem of the conventionality of simultaneity would
hardly have continued so long if it had been considered in close connection
with the problem of existence. The joint consideration of both problems makes
it possible (1) categorically to establish that the definition of simultaneous
events is a matter of convention, and (2) to elucidate the essence of this
convention. So that this joint consideration may be realized, from now
onwards simultaneously at the present moment of time will be implicit in the use
of the term simultaneity.

According to the classical (pre-relativistic) view of reality only the present
exists, i.e. only the constantly changing three-dimensional world at the
moment 'now'. But bearing in mind the fact that the present (the three-
dimensional world) is a set of simultaneous events at the present moment (i.e.
the set of material objects existing simultaneously at the moment 'now') it
follows that the simultaneity of events is an objective fact.

Therefore, the objectivity of simultaneity is expressed in the circumstance
that simultaneous events occur simultaneously, i.e. that the objects with which
these events take place exist simultaneously. It is precisely the existence at the
moment 'now' which objectively privileges one class of events. As is apparent a
conventionality of simultaneity in this case obviously leads to an unacceptable
conclusion regarding a conventionality also connected with what exists (R.
Weingard [1972] says that if two events are real, their simultaneity cannot be
a matter of convention). Therefore, according to the classical view of reality,
simultaneous events at the present moment of time (i.e. present events) are
objectively privileged in comparison with past and future events (since only
present events are considered as existing), which means that simultaneity is
not conventional.

However, things look quite different from the point of view of the theory of
relativity. The problem is additionally complicated by the fact that the problem
of the dimensionality of the world has not been convincingly solved to this day.
If we allow that even according to the special theory of relativity reality is a
three-dimensional world (a three-dimensional space-like slice of the Min-
kowski world—the present), the above-mentioned unacceptable conventiona-
lity as regards what exists follows from the conventionality of simultaneity.
Therefore reality cannot be a three-dimensional world if the definition of

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Simultaneity, Conventionality and Existence 75

simultaneity is a matter of convention, because if it could; it follows that it will
depend on our own will which three-dimensional space-like slice of the
Minkowski world we are to consider as reality. This unacceptable conclusion is
reached on the basis of the premises of the conventionality of simultaneity and
the three-dimensionality of reality. It can only be avoided if we give up one of
the premises. The impossibility of demonstrating by means of a physical
experiment the privileged state (existence) of only one set of events which are
to be considered as simultaneous at the present moment (i.e. the impossibility of
refuting the conventionality thesis of simultaneity), shows that the premise of
the three-dimensionality of the world should be abandoned. This conclusion
also follows directly from the relativity of simultaneity. The fact that the
observers in relative motion have different classes of simultaneous events
shows that not a single class is objectively privileged. Therefore all events are
equally real (Putnam [1967]), which means that it really is a question of
convention which events should be considered as simultaneous for a given
observer. In the contrary case—if any observer has objective grounds to
examine as simultaneous exactly defined events—this would mean that, being
objectively privileged, these events would be simultaneous for all observers in
relative motion. In such a case, however, the relativity of simultaneity proves
to be impossible. That is why it (the relativity of simultaneity) unambiguously
shows that the simultaneity of distant events is conventional (from which it
follows that the world cannot be three-dimensional). If we deny the view that
the world is three-dimensional and consider the Minkowski world as a
mathematical model of a real four-dimensional world, conventionality in the
choice of a three-dimensional space-like slice of this world becomes trivial,
because all slices are equally existent and the matter only concerns the
convenience of which slice is to be examined as the present. In other words,
since all events in a four-dimensional world exist in the same way the
conventionality of simultaneity does not also lead to conventionality of
existence, but is connected only with the description of the four-dimensional
world in our habitual 'three-dimensional language'. The elucidation of this
situation reveals the profound essence of the conventionality of simultaneity:
we cannot choose one class of events among the events of the Minkowski
world simply because there is no such a class of objectively priviledged events
(owing to the equal existence of all events), which we might examine as
simultaneous at the moment 'now'.

It can now be said that the logical circle obtained in an attempt to imagine
an experiment enabling us to define which events are simultaneous at a given
moment of time, or to establish the one-way velocity of light, convincingly
shows that we have tried, on the basis of an erroneous view of the
dimensionality of the world to discover the objective content of concepts (as
simultaneity and velocity), for which it appears that they have no such content
according to a more adequate view of reality. This explains why the

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76 Vesselin Petkov

conventionality of simultaneity does not presuppose some kind of 'agreements'
concerning physical magnitudes as to which we have been intuitively
convinced that they have an exactly defined objective content. And, indeed, the
concepts simultaneity and velocity have no adequate content, because among
the equally existing events of the four-dimensional world there is no set of events
which are privileged as being simultaneous (at the present moment). We can
speak of simultaneity and velocity only after the stratification of the four-
dimensional world, i.e. of space-time, into space and time. However, this
stratification has no objective basis (space-time is not really divided into space
and time precisely because of the equal existence of all events), but is the result
solely of a description of the four-dimensional world in 'three-dimensional
language'. Owing to this reason it is obvious that it is indeed a matter of
convention as to how to stratify space-time into space and time (of course,
within the framework of the requirement 0 < e < 1). Only in this case, when we
have chosen that space should be orthogonal to the time axis (i.e. with e = 1/2
and with isotropic space-time) the back and forth velocity of light is one and
the same. Depending on the manner in which we shall stratify space-time into
space and time (i.e. depending on the choice of e) different values of back and
forth velocity of light will be obtained and the experiment will always confirm
(in the supposed isotropy of space-time) the theoretically foretold result, simply
because in processing the experimental data the premise on the choice of e is
essentially made use of. That is why all attempts to measure one-way velocity
of light are doomed to failure beforehand.

Institute of Philosophy, Sofia

REFERENCES

PUTNAM, H. [1967]: 'Time and Physical Geometry', Journal of Philosophy, 64, pp. 2 4 0 -
7.

REICHENBACH, H. [1957]: The Philosophy of Space and Time. Dover.
STOLAKIS, G. [1986]: 'Against Conventionalism in Physics', British Journal for the

Philosophy of Science, 37, pp. 2 2 9 - 3 2 .
WEINGARD, R. [1972]: 'Relativity and the Reality of Past and Future Events'. British

Journal for the Philosophy of Science, 2 3 , pp. 1 1 9 - 2 1 .
WINNIE, J. [1970]: 'Special Relativity Without One-Way Velocity Assumptions',

Philosophy of Science, 37, pp. 81- 99; 2 2 3 - 3 8 .

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