## Abstract

It has been argued that the standard formulation of the Special Theory of Relativity (hereafter STR) is not only incompatible with presentism, but also strongly indicates the truth of eternalism. We should, however, distinguish two claims concerning the ontological implications of STR: (1) STR is inconsistent with every ontology which requires an absolute relation of simultaneity; and (2) STR implies that eternalism is the only possible ontology of time. There have been a wide range of responses designed to reject these claims, both jointly and independently. For example, one way of rejecting claim (2) is by rejecting claim (1): thus, one would argue that STR can be revised or interpreted in such a way that it allows an absolute relation of simultaneity. Another way of rejecting claim (2) is by questioning the equivalency of the relation ‘being real as of’ (a relation known in the literature as relation R). The main purpose of this paper is to raise a new line of objection against concluding eternalism from the relativity of simultaneity. I argue that there is a way to deny claim (2) without denying claim (1) and also without denying the equivalency of relation R. The argument which I present rests on a metaphysical assumption concerning the relation of simultaneity (hereafter the SIM): the assumption that the SIM holds basically between space–time points as opposed to holding basically between events.

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Another deflationary approach can be found in Dieks (2006), in which he proposes a local notion of becoming.

As for what space–time points and events are, an adequate definition can be assumed for what will follow in this paper: space–time points are positions or places in space–time that can be referred to by coordinates. An event is what occurs in a space–time position or place (the occurrence understood either as temporal or timeless). Thus, space–time and events are understood here as having infinitesimal extension, both spatial and temporal. As Peterson and Silberstein noted, this definition allows us to bypass debates over identity and endurance-perdurance (2010, p. 211).

This paper limits itself to the framework of STR. It would be interesting to extend these ideas to the general theory. However, this calls for a more careful and detailed presentation of related materials and preliminaries than current space permits. However, in a footnote at the end of Sect. 4, I make some remarks about general relativity considered in the context of the ideas presented in this paper.

Some philosophers have questioned the transitivity of R (for instance, Sklar 1977, pp. 274–275; Hinchliff 1996, pp. 130–131). A natural response to this critique is to say that the non-transitivity of R would have odd consequences concerning the notion of

*being real*: it would become a relative instead of an absolute notion. As for a non-symmetric and/or non-reflexive R, it seems yet more problematic, and of no relevance to our purpose. From now on, I will assume R to be an equivalence relation.Putnam himself talks about “things” being simultaneous in a coordinate system or being real as of each other with no more specifications of their nature (Putnam 1967, pp. 241–242).

One might think that in addition to (a

_{i}), (a_{ii}), (b_{i}), (b_{ii}), (c_{ii}), and (d_{ii}), there could be other possibilities for connecting SIM to R, notably if we want to consider other temporal relations. For example, let BI(φ, ψ) be the relation ‘ψ is either identical to or in the absolute past of φ’ (assuming both the Newtonian theory of space and time and the Special Theory of Relativity can provide such a truth condition); then there could be other possibilities for connecting a fact about*this temporal relation*and R:(e

_{i}) BI(Y, X) → R(X, Y)(f

_{i}) BI(Y, X) ↔ R(X, Y)and

(e

_{ii}) (BI(Y, X) & C(*x*, X) & C(*y*, Y)) → R(X, Y)(f

_{ii}) (BI(Y, X) & C(*x*, X) & C(*y*, Y)) ↔ R(X, Y)Given that the relation BI is not symmetric, both (f

_{i}) and (f_{ii}) are permitted if R is also non-symmetric. One can even involve the simultaneity relation in these possibilities and so recognize new possibilities:(g

_{i}): (BI(Y, X) or SIM(X, Y)) → R(X, Y)(h

_{i}): (BI(Y, X) or SIM(X, Y)) ↔ R(X, Y)and.

(g

_{ii}): ((BI(*x*,*y*) or SIM(*x*,*y*)) & C(*x*, X) & C(*y*, Y)) → R(X, Y)(h

_{ii}): ((BI(*x*,*y*) or SIM(*x*,*y*)) & C(*x*, X) & C(*y*, Y)) ↔ R(X, Y)Possibility (e

_{i}) is very similar to a relation that Howard Stein introduced and defended (1968, 1991), although instead of the relation ‘being real as of’ he uses the non-symmetric relation ‘being definite or determined as of’. We can define an auxiliary relation DEF, instead of R, to determine the set of*events that have been determined*, rather than the set of*existing events*. So, by substituting R with the relation DEF in (e_{i}), the connection between temporal relation and the relation DEF would be:(e*

_{i}) BI(Y, X) ↔ DEF(X, Y)Moreover, similar to the implication β, we can assume:

γ: DEF(X, Y) → (D(X) → D(Y))

Where D(X) means that ‘“X is determined” is true.’ Thus, the set of events which

*are determined, providing that the event X is determined*, would be:D

_{X}: {Y | DEF(X, Y) & D(X)} .Similar to the points we discussed before, for determining the set D

_{X}, D(X) should be the case. D(X) is the case if X is a member of the set of events that have been determined or are definite. Stein thought that this event is the special event of me-now, which is happening at the space–time point of here-now. DEF is not symmetric, so in view of the non-equivalency of DEF, for every me-now, i.e., X_{i}, there would be a separate set of D_{Xi}, i.e., the set of determined events*providing*that X_{i}is determined. Hence, D_{Xi}is not necessarily the set of*all and only*the events which are determined. The set of*all and only*the events which are determined—a set that we might refer to as D—is a set of all members of all D_{Xi}s. All X_{i}s should be provided to determine the D.In other words, if SIM is absolute, then all the conditionals ((a

_{i}) and (a_{ii})) and biconditionals ((b_{i}) and (b_{ii}))—as possibilities to connect SIM to R—are permitted and the choice between them depends on what ontology is regarded as true. For example, if presentism is true, then only the biconditionals could be the case, and, if eternalism is true, then only the conditionals can be the case.They present more details about what R is (in terms of ‘definiteness’ and ‘distinctness’), in order to argue against possible reactions to their argument (including, notably, semantic considerations raised by Savitt (2006) and Dorato (2006)). So one could consider their argument to be a ‘more conclusive’ version of the old Putnam-Rietdijk argument in favor of eternalism, as indeed they themselves also claim (Peterson and Silberstein 2010, p. 209). However, we need not go into the details of their new definition of reality and how they respond to the objections, since, first, we do not restrict our notion of reality, and there is nothing in our notion of reality which is incompatible with theirs; and second, as stated in the introduction, the main purpose of this paper is to raise a

*new*line of objection against concluding eternalism from the relativity of simultaneity, whereas all those objections that Peterson and Silberstein list in their paper attack assumptions that we here*grant*the proponent of the argument from relativity in favor of eternalism (see Sect. 1).This paper has been insensitive to the important issue of ‘manifold substantivalism’ in the context of the general theory of relativity. In space–time theories before general relativity, in which it is possible for the structure of space–time to be independent of the distribution of matter and energy (i.e., independent of events), metrical properties of space and time (or space–time) were, just like manifold properties, independent from the distribution of matter and energy. However, in general relativity there is an interdependence between metric field and the stress-energy tensor (which represents the distribution of matter and energy). So one can distinguish between, on the one hand, manifold substantivalism, in which what is substantial is just the manifold, and, on the other hand, metrical substantivalism, which takes metrical properties as essential to the substantial space–time.

Manifold substantivalism must face the famous

*hole argument*developed by Earman and Norton (Earman and Norton 1987; Earman 1989). According to the hole argument, if space–time is substantial with respect to the manifold alone, then there would be an indeterminism with respect to what event occurs where-and-when (in the hole). This is because, given manifold substantivalism, there can be two different physically possible states (i.e., both solutions to Einstein’s Field Equations), related by a hole diffeomorphism (that is, the identity map outside a hole and diffeomorphism inside that hole), such that, in terms of the terminology utilized in this paper, the truth value of the sentence C(*q*, Q) is indeterminate in the hole. This indeterminism is, indeed, a consequence of a commitment to a kind of metaphysics (namely, manifold substantivalism) which is, as Earman and Norton put, akin to the idea of the container and the containment (1987, pp. 518–519), i.e., what we have referred to so far as dualistic substantivalism.How does all this matter to our discussion of the argument from the relativity of simultaneity to eternalism (both event eternalism and space–time point eternalism)? As long as one supposes that SIM is relative and R is an equivalence relation, and we also suppose relationalism to be the fundamental metaphysics of space–time (one could be motivated here by the hole argument), one would conclude event eternalism, just as we saw in DOi. On the other hand, if one is inclined to insist on manifold substantivalism, then, given the hole argument, the truth value of sentences like C(

*q*, Q), which are vital for an argument like DOii to proceed, are indeterminate in the hole region. To hold some substantivalist theories that are compatible with the hole argument (like metrical essentialism) would make the situation similar to the case of monistic substantivalism: event eternalism would directly follow from space–time point eternalism*and*monistic substantivalism.

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## Acknowledgements

I am grateful to Laleh Ghadakpour for her supports and invaluable helps in improving the earlier version of the paper. I am also thankful to Giuliano Torrengo, Aboutorab Yaghmaie and Ben Young for their helpful suggestions. Thanks to two anonymous referees for their constructive comments. This research was supported by funding from the Iranian Institute of Philosophy.

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Amiriara, H. On the possibility of non-eternalism without absolute simultaneity.
*Synthese* **199**, 5885–5898 (2021). https://doi.org/10.1007/s11229-021-03050-6

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DOI: https://doi.org/10.1007/s11229-021-03050-6