What Becomes of a Causal Set?

Most causal sets sanctioned merely by the kinematic axiom do not stand in a relation of faithful approximation to spacetimes with low-dimensional manifolds.

“It turns out that most causal sets sanctioned merely by the kinematic axiom do not stand in a relation of faithful approximation to spacetimes with low-dimensional manifolds.”

• cites Smolin 2006 sourceSupported

The selection of events in a Poisson sprinkling must be sufficiently random and irregular.

“Since there can be no such observable di↵erences, the selection of events must be su ciently random and irregular (Dowker et al. 2004).”

• cites Dowker et al. 2004 sourceSupported

The absence of any empirical tests pointing to a violation of Lorentz symmetry indicates that the actual fundamental causal set cannot be highly regular.

“We take the absence of any empirical tests pointing to a violation of Lorentz symmetry (Mattingly 2005) to be an indication that the actual fundamental causal set-again, if any-cannot be highly regular.”

• cites Mattingly 2005 sourceSupported

If the set of events exhibits too much regularity, then Lorentz symmetry would be broken because a lattice could be distinguished from its Lorentz boosted analogue at an appropriately coarse-grained scale.

“If the set of events exhibits too much regularity, then Lorentz symmetry would be broken because we would be able to distinguish, at an appropriately coarse-grained scale, such a lattice from its Lorentz boosted analogue.”

• cites Dowker et al. 2004 sourceSupported

The relation R on the counterexample causet is invariant under automorphisms of the structure.

“Nonetheless R is invariant under automorphisms of the structure, as it only relies on the causal relations themselves, except for the last two pairs, which are however symmetrically included and obtain between points with identical 'relational profiles' and hence are invariant under structure-preserving maps.”

Any pair of distinct elements of a non-Hegelian subset cannot stand in the relation , i.e., they are spacelike to one another.

“It is clear that any pair of distinct elements of a non-Hegelian subset cannot stand in the relation , i.e., they are by necessity 'spacelike' to one another.”

On the counterexample causet, the relation R holds between the spacelike related events b and c.

“It is clear that this relation does not obtain only between events and events in their causal past, as it obtains between the spacelike related events b and c!”

The analogue of Stein's theorem from special relativity does not hold in causal set theory.

“The possibility of fundamental becoming is nourished by the fact that the analogue of Stein's theorem from special relativity does not hold in causal set theory.”

• Also stated in: § 3 Facing the same dilemma? · boosted

  “The analogue of this theorem, however, is straightforwardly false in CST.”

In CST there can be non-trivial and non-universal relations of 'being definite as of' that do not collapse to events in the causal past of the reference event.

“Thus, we see that in CST there can be non-trivial and non-universal relations of 'being definite as of' that do not collapse to events in the causal past of the reference event.”

A large subsector of general relativity can be described in a dynamical manner via its many 3+1 formulations.

“Against this, we note firstly that (a large subsector of) general relativity certainly can be described in a 'dynamical' manner via its many '3+1' formulations.12”

• cites [12] 12. Fine, K. [2005]: 'Tense and reality', in: K. Fine, 2005, Modality and Tense: Philosophical Papers, Oxford: Oxford University Press, pp. 259-320. sourceNot checked — no full text of the source

Earman's hidden variable addition to causal sets, where actuality takes one path or another, flies in the face of the normal interpretation of labels as pure gauge.

“But as Aristidis Arageorgis (2012) rightly points out, such a move really flies in the face of the normal interpretation of these labels as pure gauge.15”

• cites Arageorgis (2012)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

A consistent gauge invariant dynamics exists beneath the label-dependent dynamics.

“True, the dynamics is written in terms of a choice of label, but we know that a consistent gauge invariant dynamics exists 'beneath' this dynamics.”

Rewriting the theory in terms of a probability measure space, one can quotient out under relabellings to arrive at a label-invariant measure space.

“In fact, rewriting the theory in terms of a probability measure space, as indicated above, one can quotient out under relabellings to arrive at a label-invariant measure space (for construction and details, see Brightwell et al 2003).”

• cites Brightwell et al 2003Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

There is gauge-invariant growth: focusing on the number of elements and ignoring labeling, there are transitions from C to C' and so on.

“Focusing just on these and ignoring any labeling, we do have transitions from C to C 0 and so on.There is gauge-invariant growth.”

At a modest level, consistent with explicit pronouncements by advocates of causet becoming, one can articulate a localized, observer-dependent form of becoming.

“At a more modest level, and consistent with explicit pronouncements by advocates of causet becoming, we can articulate a localized, observer-dependent form of becoming.”

Large and many non-Hegelian subsets are few and far between in causal sets.

“Although we know of almost no pertinent analytical results or numerical estimates, we suspect that large and many non-Hegelian subsets are few and far between.”

• AssumptionRests on an assumption: There are many more irregular structures that satisfy the kinematic axiom than there are structures sufficiently regular to sustain large and many non-Hegelian subsets.

Only if the inverse problem is successfully solved can causal set theory become a viable approach to finding a quantum theory of gravity.

“Only if this di culty is successfully solved can causal set theory become a viable approach to finding a quantum theory of gravity.”

In general, some event does get determinate at some finite stage of becoming in a future-infinite causet.

“So does any event ever get determinate at any finite stage of becoming?In general, yes.”

The number of elements in any causet is gauge invariant.

“And one thing that we know is gauge invariant is the number of elements in any causet.”

There is nothing in the kinematics of CST that suggests any kind of ontological growth.

“There is nothing in the kinematics of CST that suggests any kind of ontological growth.”

We may have ontological indeterminacy without vagueness in the cardinality of the finite set of all existing objects.

“Thus, we may have ontological indeterminacy without vagueness in the cardinality of the (finite) set of all existing objects.”

Given the structure of causets, it is straightforward to define relativistically kosher forms of becoming that imitate past lightcone becoming compatible with Minkowski spacetime.

“Given the structure of causets, it is straightforward to define relativistically kosher forms of becoming that essentially imitate the past lightcone becoming already compatible with the geometry of Minkowski spacetime; for instance, let 'is definite as of' coincide with the causet order relation.”

The causal order of a sufficiently well-behaved relativistic spacetime determines its geometry up to a conformal factor.

“The theory finds its inspiration in a theorem (Malament 1977) that shows the precise sense in which the causal ordering of a su ciently well-behaved relativistic spacetime determines its geometry, up to a conformal factor.”

• cites Malament 1977 sourceNot checked — no full text of the source

The radical fragmentation of reality could be introduced in the ordinary theory.

“Not only would that be a high cost to introduce becoming, but it is also one that, again, could be introduced in the ordinary theory.”

Any dynamics distinguishing a particular label order will be non-relativistic.

“Any dynamics distinguishing a particular label order will be non-relativistic.”

We are generally prohibited from saying exactly what elements exist at any stage of growth.

“The problem is that we are generally prohibited from saying exactly what elements exist at any stage of growth.”

In rare circumstances where the necessary condition for evading the dilemma is satisfied, the causal set has regular 'lattice' structures.

“Yet observe that in these rare circumstances where the necessary condition is satisfied, we have regular 'lattice' structures.”

Discrete general covariance implies that the product of transition probabilities along any path from the singleton causet to a given n-element causet is the same.

“In particular, if ↵ is one path from the singleton causet to an n-element causet, and is another path to the same causet, then the product of the transition probabilities along the links of ↵ is the same as that for (and any other such path).”

In special relativity, any binary relation 'is definite as of' adapted to temporally oriented Minkowski spacetime must coincide with 'is in the causal past of' lest it degenerate into the trivial or universal relation, modulo a choice of temporal direction.

“In special relativity, Stein's theorem tells us, essentially, that any binary relation 'is definite as of' adapted to the structure of temporally oriented Minkowski spacetime must coincide with 'is in the causal past of' lest it degenerate into the trivial or the universal relation, modulo a choice of temporal direction.”

• cites Stein (1991) sourceNot checked — no full text of the source

To enforce consistency with relativity, there is no fact of the matter about which of two spacelike-related events happened right after the singleton element event.

“To enforce consistency with relativity, there is no fact of the matter about which one happened right after the singleton element event.”

Causets based on transitive percolation in general have many posts, where a post is an event comparable to every other event.

“As it turns out, causets based on transitive percolation in general have many 'posts', where a post is an event that is comparable to every other event, i.e., an event that either is causally preceded or causally precedes every other event in the causet.”

The causal set program, interpreted as here, supplies a possible model of a world that is ontologically vague.

“The causal set program, interpreted as we have here, supplies a possible model of a world that is ontologically vague.”

All sequential growth dynamics compatible with CST possess discrete general covariance.

“Used as a condition to derive the dynamics, all sequential growth dynamics compatible with CST possess this symmetry.”

A novel and exotic notion of becoming is compatible with causal sets.

“In particular, we show that a novel and exotic notion of becoming is compatible with causal sets.”

• Also stated in: § 1 Introduction · boosted

  “However, we show that if one is willing to pay them, a novel and exotic form of becoming is compatible with relativity.”

There is a weak sense in which a fully objective kind of becoming with relativistic credentials can be had.

“We will argue that it does not and that there is a weak sense in which a fully objective kind of becoming with relativistic credentials can be had.”

Sequential growth is compatible with a tenseless or block picture of time.

“Although we are interested in becoming, we should immediately remark that sequential growth is certainly compatible with a tenseless or block picture of time.”

Serious constraints threaten becoming if we take the time in CST's dynamics seriously.

“Serious constraints also threaten becoming if we take the time in CST's dynamics seriously too.”

As a causet grows, events that were once spacelike to the causet might acquire timelike links to future events.

“Second, as a causet grows, events that were once spacelike to the causet might acquire timelike links to future events.”

It is possible to modify the theory so that the world becomes in both directions, future and past, by choosing a here-now as the original point.

“Choose a here-now as the original point.Then it is possible to modify the theory so that the world becomes in both directions, future and past.”

• AssumptionThe authors state this as an assumption: Events can only be born to the future of existing events.

If the growth of a new timelike link to a spacelike event is regarded as making the spacelike event determinate, modulo the above vagueness, then future becoming can make events past.

“If we regard the growth of a new timelike link to a spacelike event as making the spacelike event determinate, modulo the above type of vagueness, then this is a way future becoming can make events past.”

If we relax the assumption that events can only be born to the future of existing events, then it is possible to have percolation-and hence becoming-going both to the future and past.

“If we relax the assumption that events can only be born to the future of existing events, then it is possible to have percolation-and hence becoming-going both to the future and past.”

• AssumptionThe authors state this as an assumption: Events can only be born to the future of existing events.

In the toy causal set, it is not true at C3 that C2 determinately is one way rather than the other.

“In our toy causal set, it is not true at C 3 that C 2 determinately is one way rather than the other.”

On the causal set picture, the past too can be indeterminate.

“We note that the above causal set vagueness is quite similar, but with one big di↵erence: on the causal set picture, the past too can be indeterminate!”