Space-time structure may be topological and not geometrical
aa r X i v : . [ phy s i c s . g e n - ph ] J un Space-time structure may be topological and notgeometrical
Gabriele Carcassi, Christine A. Aidala
Physics Department, University of Michigan, 450 Church StreetAnn Arbor, MI 48109-1040, United StatesE-mail: [email protected]
February 2020
Abstract.
In a previous effort we have created a framework that explains whytopological structures naturally arise within a scientific theory; namely, they capturethe requirements of experimental verification. This is particularly interesting becausetopological structures are at the foundation of geometrical structures, which play afundamental role within modern mathematical physics. In this paper we will showa set of necessary and sufficient conditions under which those topological structureslead to real quantities and manifolds, which are a typical requirement for geometry.These conditions will provide a physically meaningful procedure that is the physicalcounter-part of the use of Dedekind cuts in mathematics. We then show that thoseconditions are unlikely to be met at Planck scale, leading to a breakdown of the conceptof ordering. This would indicate that the mathematical structures required to describespace-time at that scale, while still topological, may not be geometrical.
Keywords : Foundations of physics, space-time structure, topology
Submitted to:
Phys. Scr.
1. Introduction
In our ongoing project, Assumptions of Physics, the idea is to find a minimal set ofassumptions from which the different basic theories can be derived. The idea is that,by doing this as formally as possible, we are forced to specify all our implicit startingpoints, thus clarifying what could possibly be done differently. While other approachesstart with a similar goal, see for example Refs. [1, 2, 3, 4], there are a number ofkey differences. One difference is that in our work both classical and quantum casesare derived on equal footing, identifying the key point of divergence between the twotheories.[5] Another difference is that our approach aims to start with primitives that arenecessary to do physics. Our basic building block is the notion of a verifiable statement:an assertion for which an experimental test is available that would confirm, in finite time,that the assertion is true. As one can imagine, without such a notion, we would have no pace-time structure may be topological and not geometrical U is an open set if and only if “ x is in U ” is experimentally verifiable). Forreal valued quantities, the verifiable statements correspond to open intervals to signifythat our measurements always have finite precision. The fact that a continuous quantityhas precisely a given real value (e.g. the length of the side is exactly 1 meter and thediagonal is exactly √ must be well behaved in physics. It is not a matter of convenience: if they are not theywould break experimental verifiability.This idea also clarifies the division between topology and geometry. Topologicalconstructs are more primitive than geometrical ones because they have no notion ofsize. Physically, it means that before we can assign distances between two points, wemust be able to distinguish them, to tell them apart. The topology, then, comes first,as it tells us if and how the points are distinguished experimentally. This insight alsotells us why different spaces, though topologically equivalent, have different geometry.In everyday space, the distance along any direction can be measured in the same unit,say meters. The geometry is Riemannian. On phase space, within a degree of freedom,units of position cannot be compared to units of momentum, therefore we cannot defineangles. Yet, phase-space areas for a degree of freedom are proportional to the number ofpossible configurations. The fact that we can quantify areas for each degree of freedomis what gives us symplectic geometry. If we imagine for example the space of all possibleblood work results, however, we have neither notion, and therefore we simply have amanifold that is not geometric.For those interested in the ultimate structure of space-time these insights naturallylead to the following line of inquiry. Before we want to understand whether space-timeis geometrically truly a Riemannian manifold, we need to understand whether space-time is topologically truly a manifold. A manifold, simply put, is a set of possible cases X that can be experimentally identified by a set of real values. So, the question is:when can a set of possible cases, or possibilities for short, be experimentally identifiedby a real value? To be clear, the question is not when the results of our measurementsare real numbers. The answer to this is simple: never. Our measurements are always pace-time structure may be topological and not geometrical x i = x i ( x ).As we mentioned before, functions need to be, at the very least, topologically continuousor they would break the notion of experimental verifiability (i.e. verifiable statementswould not be mapped to other verifiable statements). This means that in any theorythat includes time evolution, the topology one gives to time will severely constrain thetopology of the space within which time evolution takes place. Moreover, in any theoryof space-time we have the additional constraint that time and spatial coordinates canbe mixed. Therefore the argument will work in reverse as well: the topology one givesto space has to be the same as the one of time because one may use spatial distance asa time parameter. For example, two spaceships drifting at constant velocity may usetheir spatial separation as a clock.Even though this work is mathematically very technical and abstract, we strive tohave a well understood map to more tangible and physical concepts; that is one of themain goals in our work. Therefore let us first give a summary of the results in physicsterms, leaving the details to subsequent sections. Mathematics offers many constructions of the real numbers from more primitive notions.Common techniques include completion through Cauchy sequences, using Dedekind cutsor adding constraints over an algebraic field. In fact, Faltin et al. state[7]:Few mathematical structures have undergone as many revisions or have beenpresented in as many guises as the real numbers. Every generation re-examinesthe reals in the light of its values and mathematical objectives.Our present purpose is one such re-examination in the light of how these quantities aredefined through experiments.If we think about how quantities are measured, the general idea is that we areable to define references with preset amounts, and then compare our object to severalreferences to find the ones that bound the value from above and below. For example, aruler is a series of marks and measuring the position with it means finding the closestone. A clock is a series of ticks and measuring time means noting the tick right beforethe event. A balance scale compares the weight of an object to a few known ones.Intuitively, we can understand that a quantity is continuous if, in line of principle, wecan prepare references ever closer to each other. Our aim is to make this notion precise,so we have a practical/operational definition of what we mean experimentally by acontinuous quantity. pace-time structure may be topological and not geometrical
4A single reference is something that allows us to distinguish a before and an after, asmaller and a greater, a higher and a lower. We can formalize a reference with our morebasic notion of a verifiable statement: a reference gives us two verifiable statements, “theobject is before this reference” and “the object is after this reference”. This allows us torecast ordering relationship in terms of logical relationship. For example, if reference A isafter reference B, then if “the object is before reference B” is true, “the object is beforereference A” must also be true. And these logical relationships remain conceptuallythe same regardless of whether “before” means “before in time” or “before along aspatial direction”, etc. These are the sort of basic ideas and language we use within ourframework, and from these we work out a set of necessary and sufficient conditions thata set of references needs to satisfy so that they define a set of possible distinguishablecases, or possibilities for short, that is ordered like the real line.The idea is that the quantities themselves are not a priori objects, but a constructionbuilt upon a set of references that form our system of measurement. A reference framein space-time, then, is also a construction consisting of fixed elements (e.g. the stars, theborders of the experiment table, the elements of a timing system in a particle accelerator,...) and signals exchanged among them. The topology of space-time is then an idealizedcharacterization of the set of all possible such constructions, that is, all possible referencesystems. Our goal is to understand the extent and the limitation of such idealization.The biggest obstacle in this undertaking, and possibly in fully appreciating it, isthat our standard intuition traps us in circular arguments. For example, one issue isthat references have an extent: they occupy some space and so do objects. If we are tomeasure the position of an object, then, it may extend before, on and after the markof a ruler. In these cases we typically note the position of the beginning of the objectand of the end of the object. This implicitly relies on the fact that the resolution of oureyes is higher than that of the ruler, and that we can independently recognize parts ofthe object, namely the beginning and end. Identifying parts using a higher resolutioncorresponds, in the end, to new references. So, assuming this can always happen meansimplicitly assuming what we are tasked to derive.Unfortunately, thinking visually and drawing pictures is misleading becauseeverything we draw is inherently ordered. It confuses more than it helps, so muchso that we ourselves progressed only when we stopped trying to reason with picturesand concentrated on just the boolean logic of our statements in truth tables. In thatsetting, the only elements we are allowed to define are those simple logical relationshipswe described before: if the object is before A, then it is not after B and it is beforeC. All properties and relationships between references, then, must be defined in thatmanner.Using that strategy, we reached two major findings. The first major finding is that most of the conditions are not required by the real numbers specifically, but they arerequired by ordered quantities in general . That is, the set of references has to satisfy afew conditions just to be sure that the cases they distinguish experimentally are ordered(i.e. can be thought to be all being one after the other). The conditions are the following. pace-time structure may be topological and not geometrical • The references must be strict, meaning that the cases before, on and after aremutually exclusive. In practice, if objects have an extent, this must be muchsmaller than the extent of the reference, such that it will always be consideredwholly before, on or after the reference. • The references must be aligned, meaning their before and after statements identifyincremental regions. In practice, this means we can always distinguish betweenthem and they remain with constant before/after relationships (e.g. they do notfluctuate). • The references must be refinable, meaning we can always resolve overlaps and fillin the whole space. For example, if an object can be between two references, thenwe must be able to put a new reference between those two references as well. Ifwe have two references that overlap, then we must be able to find other referencesthat fit within that overlap, to tell what is before and after.These are the conditions one must require to have any linear order, either discrete (whichcan be labeled by integers), continuous (which can be labeled by real numbers) or anyother. The divergence between real and integer quantities lies in how many referencesone can put in between two references. In the integer case, we can only fit finitely manyreferences. In the real case, we can fit infinitely many.The second finding is that the inability to distinguish below a certain scale most likelyinvalidates those conditions, leading to a breakdown in ordering.
In other words, it isnot just that the points of our space cannot be labeled by real numbers: they cannot beordered and therefore cannot be labeled by any ordered quantity. To obtain that result,one must simply argue that at least one of the three conditions fails. For example, onecould argue that if at the finest level the references are particles, once they start beingvery close to each other it may become impossible to keep them distinguishable, whichmakes it impossible to keep well established before/after relationships and thereforesatisfy the alignment condition. One could argue that if both references and the objectbeing measured are particles, their extent (i.e. the support of the wavefunction or ofthe field excitation) is comparable, which breaks the strictness condition. Alternatively,one can also argue that if we approach Planck scale, we cannot refine our references anylonger, which breaks the refinement condition. The gist is that, in the case of quantitiesover which objects have an extent, the requirements for ordering can only be understoodin terms of simplifying assumptions.The inability to create an ordering at the finest scale means we are not able to defineother geometrical quantities, such as distances and angles. In this light, the ultimatestructure of space-time may not be geometrical but it will still be topological, as wewill still need to describe what is experimentally accessible. At large scale, geometricalstructure will need to emerge to recover the established theories. The point is thatthese large scale geometric structures cannot emerge from other geometrical structures,as these would necessarily suffer from the same fundamental problems. pace-time structure may be topological and not geometrical The full mathematical account, which is available at [8], uses tools and constructionsthat, in our experience, are not widespread among physicists. Moreover, that level ofdetail may not be of interest to most. Therefore, we have recast the definitions and themain arguments to the most basic elements of set theory and order theory, which webelieve are more accessible and cover the most important points. This should allow usto give a more intelligible account of the physics without hiding it in the math.We will first review the link between point-set topology and experimentalverifiability. We will show how a reference can be defined in terms of sets. We willgive the basic notions of order theory we need. We will then formalize the requirementsa set of references needs to satisfy to identify a continuous quantity. Finally, we will seehow these requirements describe idealized conditions.
2. Elements of topology and its link to experimental verifiability
The first thing we need to establish is the link between topology and experimentalverifiability. Point-set topology (or general topology) is, knowingly or unknowingly,widely used in physics as it is the foundation of many other tools, such as differentialgeometry and Lie groups. Most mathematical structures used in physics are topologicalspaces. Therefore it is crucial to understand what physical content they capture. ‡ The formal definition of a topology is the following: given a set X , a topology T is a collection of subsets of X that is closed under finite intersection and arbitrary(infinite) union. § This is similar to other algebraic structures, like groups, where youhave elements and operations that return other elements of the same structure. Here theelements are subsets of X and the operations are set operations. The formal definitionis very abstract and does not have an apparent connection to physics.Seemingly unrelated, consider a “verifiable statement”: an assertion that can beverified to be true experimentally. For example, “the mass of the neutrino is 0 . ± . X is the set of all possible cases, each verifiable statement is associated with a set U ⊆ X and the statement can be re-expressed as “ x is in U ”. In our first example, x isthe mass of the neutrino and U is the set of possible values between 0 .
045 and 0 . X correspond to a verifiable statement. Forexample, “the mass of the neutrino is exactly 0 .
05 eV” is not verifiable because wecannot perform a measurement with infinite precision. We call U ⊆ X a verifiable set ‡ In computer science, the link between topology and computability is already accepted, see e.g. [9],though not widely known. There is a link between those concepts and the ones presented here, but, forbrevity, we are not going to expand on it. The gist is that any computation device is also a physicalsystem, and the output of a computation can be experimentally verified (i.e. we can read it). § Technically, it also must contain the full set X and the empty set ∅ . This does not play an importantrole. pace-time structure may be topological and not geometrical x is in U ” is, at least in line of principle, something we can verifyexperimentally. Let T be the collection of all verifiable sets. The idea is that T is atopology.To convince ourselves of that, consider the following. If we have a finite collectionof verifiable sets { U i } ni =1 , then the intersection corresponds to the logical AND (i.e. theconjunction) of all the statements “ x is in U i ”. We can test the intersection simply bytesting each assertion one at a time, and therefore { U i } ni =1 is a verifiable statement. Ifthe collection were infinite, though, we would have to go through infinitely many tests toverify the logical AND, so we would never terminate. Therefore, in general, the infiniteintersection of verifiable sets is not verifiable.On the other hand, the union of verifiable sets would correspond to the logicalOR (i.e. disjunction) of all the associated statements. In this case, as long as onestatement is verified, the OR is verified and we can terminate. Therefore we do not carehow many statements there are after the one that was verified, so the infinite union ofverifiable sets is verifiable. Note that topologies are closed under arbitrary unions, notjust countable. Yet, because we cannot test more than countably many statements, theset of all verifiable statements must be constructable for a countable set of statements, orwe would not be able to fully explore the space even with unlimited time. Therefore thetopology T must have a countable basis, which means all arbitrary unions are countableunions.We can make this more concrete by looking at the standard topology on the realnumbers R , which contains all the open intervals ( a, b ) where a, b ∈ R ∪ {−∞ , + ∞} .These, in fact, correspond to all the finite precision measurements. It also contains theirunions. For example “the absolute value of the charge of the electron is 1 . ± .
005 10 − C” would correspond to “ x is in ( − .
65 10 − , − .
55 10 − ) ∪ (1 .
55 10 − , .
65 10 − )”.Singletons, sets with a single value, are not part of the topology and in fact we cannotverify them experimentally. Note that the topology can be generated by the set of allrational intervals, which is countable, since the infinite unions will allow us to constructlimits, which correspond to the intervals of the reals.A topology, then, is not just some mathematically abstract construction thathappens to be useful in physics. It captures the way that we can experimentallydistinguish a set of possible cases. The main point is that statements have a binarylogic in terms of TRUE/FALSE, while experimental tests have a ternary logic in terms ofSUCCESS/FAILURE/UNDEFINED, where undefined corresponds to non-termination.The topology is generally used to keep track of those differences. It makes sense, then,that topologies are so pervasive among the mathematical structures used in physics:experimental verifiability is at the heart of science.Note that we will call verifiable sets, instead of open sets, the sets within thetopology. We will also call falsifiable sets, instead of closed sets, the complement ofverifiable sets. It will make our work more connected to this physical notion andtherefore a little bit more intuitive, especially to those who are not deeply familiarwith point-set topology. pace-time structure may be topological and not geometrical X , which we will call possibilities , and a wayto distinguish them experimentally, which means we have a topology T that captures allour verifiable sets. What are the requirements on ( X, T ) such that it is homeomorphic(i.e. topologically equivalent) to the set of the real numbers with standard topology?
3. References as the starting point for measurement scales
The first thing we need to develop is a conceptual model to represent how quantitiesare measured experimentally. Let us first gather some requirements from the physics.We start from the notion of a reference, such as a mark on a ruler or the tick ofa clock. A reference, then, is a physical object that allows us to distinguish betweenthree cases: a before, an on and an after the reference. That is, a point can be before,on or after the mark on a ruler; an event can happen before, on or after the tick of aclock. We take references to be the basic conceptual element upon which quantitativemeasuring devices are constructed in practice.In general, the three cases a reference defines are not mutually exclusive: an objectcan extend before, on and after the mark; an event can start before and end after thetick of a clock. In fact, if the object is both before and after, it will be on the referenceas well. In all cases, the object should be found at least in one of the three cases.If a reference is at an end of the range, then it will either not have an after or abefore. Yet, a reference is a physical object and needs to take some space, thereforethere will always be an on case.We should also note that the before and after cases are easier to test experimentally.When comparing two weights on a scale, for example, it is easy to tell when one is greaterthan the other. If they are very close, we can only typically say that they are closerthan a certain threshold. This is also something we will have to take into account.If ( X, T ) is the topological space of the physically distinguishable cases, the numberof ways an object can be found to be, we can formalize a reference as a triplet r = ( B, O, A ) of three subsets
B, O, A ⊆ X . Respectively, they will represent thecases in which the object is before, on or after the reference. We can capture the restof the requirements with the following: • B ∩ A ⊆ O (all possibilities in which the object is found before and after, the objectis also found on) • B ∪ O ∪ A = X (before/on/after cover all possibilities) • O = ∅ (the reference itself must be found somewhere) pace-time structure may be topological and not geometrical • B, A ∈ T (before and after are verifiable)Note that we require only before and after to be experimentally verifiable, leaving theon case unspecified. It turns out this is sufficient and avoids the problem of testing forperfect equality.A measuring device will be composed of several references. Intuitively, forcontinuous quantities we are assuming that, though we always have a finite set ofreferences with finite precision, we could in principle refine our instruments with evergreater precision. Our work is to make this intuitive notion precise.
4. Elements of order theory
There are many ways to mathematically characterize the real numbers. The one weare interested in is in terms of their order, instead of operations like addition ormultiplication, as it is the one that imposes the least requirements and it is enoughto identify the topology. k Let us review, then, a few key concepts of order theory, whichis the branch of mathematics that studies ordered sets and their properties.A partial order ≤ is a binary relationship that is reflexive (i.e. a ≤ a ), antisymmetric(i.e. if a ≤ b and b ≤ a then a = b ) and transitive (i.e. if a ≤ b and b ≤ c then a ≤ c ).In a partial order, two distinct elements are not necessarily one before the other. On aplane, we can order points by their horizontal position, yet that would not order pointsthat lie on a vertical line. A total order, or linear order, is a partial order such that anytwo elements are comparable (i.e. at least a ≤ b or b ≤ a ). If not explicitly stated, wewill assume orders to be linear.Given an ordered set ( X, ≤ ) we can construct the order topology in the followingway. Take all sets of the form ( a, ∞ ) = { x ∈ X | x > a } , ( −∞ , b ) = { x ∈ X | x < b } .This will be a basis for the order topology. We then take all the sets that can begenerated from the basis through finite intersection and arbitrary union. This will bethe order topology. Both the integers and the reals are totally ordered sets with theirstandard ≤ relationship. Their respective order topologies correspond to their standardtopologies. Note how each set within the basis corresponds to a statement like “ x isafter a ” or “ x is before b ”. Note how these, already, are very similar to the before andafter cases we introduced in the previous section.Like groups or topological spaces, two ordered sets are isomorphic if they haveequivalent structure. That is, two ordered sets are isomorphic if there is a bijection thatpreserves the ordering (i.e. an invertible monotonic function). If two ordered sets areisomorphic, then they will have the same order topology and vice-versa. So the orderingidentifies the topology of an ordered set and vice-versa. k Note how all transformations that preserve addition or multiplication must preserve the ordering,while the converse is not true. For example, consider x → x , which is a non-linear monotonictransformation. It preserves the ordering (i.e. x ≤ x if and only if x ≤ x ) while it does notpreserve addition (i.e. ( x + x ) = x + x ). pace-time structure may be topological and not geometrical X is linearlyordered and its topology is the order topology. Then understand what the additionalrequirements are to be ordered like the real numbers or the integers. It turns out thatmost of the work lies in the first part, in recovering the linear order.
5. Experimental requirements for constructing real-valued quantities
As we saw, in general a reference defines three cases (before/on/after) that may notbe mutually exclusive. We say a reference is strict if they are. That is, a reference r = ( B, O, A ) is strict if B ∩ O = ∅ , O ∩ A = ∅ and B ∩ A = ∅ . Technically, we wouldjust need that before and after are disjoint ( B ∩ A = ∅ ) and then redefine on as whatremains ( O = X \ ( A ∪ B )). That is, as long as before and after are separate cases, wecan redefine on as “not before and not after”.The order topology always allows us to construct strict references. If we take a, b ∈ R such that b < a , we have (( −∞ , b ) , [ b, a ] , ( a, + ∞ )) that represents a referencethat physically extends from b to a . Mathematically, we can see that the extent ofthe reference is represented by a falsifiable (i.e. closed) set [ b, a ], while the othersare verifiable (i.e. open) sets. Therefore if we want to rederive the order topologystarting from references alone, those will need to be strict. Intuitively, defining an orderexperimentally on all possibilities means that, given two distinct ones, we are able tofind references that will confirm unambiguously that one possibility is before the other.Therefore we must be able to tell before and after apart.What does this requirement entail in practice? References for quantities that wecount (i.e. discrete quantities) can always be made strict: either you have n elements,less than n or more than n . The reason is that the quantity does not really have anextent and the possible values are well separated. Quantities over which we have anextent, like space and time, are another matter. When measuring the position of anobject, this can extend before, on and after the mark. If the extent of the object beingmeasured is much smaller than the extent of the reference, then we can assume theobject to be wholly found either before, on or after. In those regimes we can treat thereferences as strict. But this is an idealization that will break down if the extent of thereferences is comparable to the extent of what is being measured. As we put references together, we must be sure they are related to each other in thecorrect way if we want to end up with a linear ordering. Intuitively, if we mix referencesfor horizontal position with references for vertical position we will not end up with alinear ordering. pace-time structure may be topological and not geometrical b , b ∈ R .Suppose that b ≤ b . Then if we find that our object is before b then it will also bebefore b . In terms of sets, ( −∞ , b ) ⊆ ( −∞ , b ). In a linear order, the idea that onepoint is before the other can be translated in terms of set inclusion. Alignment betweenreferences, then, can be defined by requiring that the before and after statements havean inclusion relationship.We also note that the negation of “ x is greater than a ” is “ x is less than or equalto a ”. In terms of sets, ( a, + ∞ ) C = ( −∞ , a ]. For a strict reference extending from b to a , we will have ( −∞ , b ) ⊆ ( −∞ , a ].We will say that two references are aligned if their edges can be ordered.Mathematically two references r = ( B , O , A ) and r = ( B , O , A ) are aligned if thesets B , B , A C , A C can be ordered by inclusion. For example, if B ⊂ A C ⊂ B ⊂ A C then we will be in the case where r is completely before r . If B ⊂ B ⊂ A C ⊂ A C then the two references are overlapping. Alignment of references is another necessarycondition if we want to reconstruct an ordering that is linear.What does this requirement entail in practice? It means that there are clear fixedordering relationships between the references. For example, every time something isbefore one reference it will also be after the other. It means we are perfectly able toprepare, control and identify our references.To give a better understanding of how these definitions work in a multidimensionalsetting, let us see how they apply to a Cartesian frame with coordinates ( x, y, z ). Areference for the x coordinate would be something like r =(“ x < ≤ x ≤ x > y and z . The reference r =(“ y < ≤ y ≤ y > r as something can be before r and still be before, on orafter r . On the other hand, r =(“ x < − − ≤ x ≤ x > r even though it is not either before or after r because the on regions overlap. Note thatthese references would not have such a simple expression in, say, spherical coordinates.Yet the regions themselves are coordinate independent and, therefore, the relationshipsbetween them are too.What happens is that each reference frame would have its own set of references thatcan be nicely expressed using its coordinates. The before/after relationships expressedusing one set of references may not be simply expressible using another, precisely becausethey are not aligned with each other. Relativistically, the references used for time byone frame will not be aligned with the ones of a boosted frame: the coordinate surfacesdo not partition (i.e. do not foliate) space-time in the same way. Reference alignmentessentially captures these various requirements with one simple formal definition. Strict and aligned references allow us to define ordering between sets of possibilities. Weneed an additional condition to make sure we have enough references at an appropriate pace-time structure may be topological and not geometrical r = ( B , O , A )and r = ( B , O , A ) are such that A ∩ B = ∅ , then we can find a reference r = ( B , O , A ) such that O ⊆ A ∩ B . On the other hand, if the extent of onereference is within the extent of the other, then we need to be able to find a referencethat covers another part. That is, if O ⊂ O then we can find another reference suchthat O ⊂ O and O ∩ O = ∅ . If a set of references has this property, then we say it isrefinable: the references can be refined to non-overlapping references that cover all thepossibilities.What does this requirement entail in practice? It means that we can place areference wherever we want and can shrink its extent such that it occupies only onepossibility. If we take the traditional manifold structure of space-time literally, thiswould mean having at our disposal ever shrinking references that can be placed anywherein space and time. For example, it would require being able to create a timing systemwith as many well separated pulses as desired distributed to as many places as desired.All of this, without changing the nature of the process we are studying. It can be demonstrated that a set of experimentally distinguishable possibilities islinearly ordered if and only if its topology can be generated by a set of refinable,aligned strict references. That is, if we have an order topology, we can construct aset of references that have those properties, and if we have a set of references that havethose properties, the order topology can be generated by finite intersection and arbitraryunion from the before and after sets of the references.The idea is that, under those conditions, we can find references that are so fine thatthey extend over only one possibility. Therefore we have a one-to-one correspondencebetween experimentally distinguishable cases and the finest references. Since allreferences are aligned and strict, two different references must be sequential, one beforethe other. This order corresponds to the one that defines the order topology.
The order of the integers can be characterized as the one that does not have a minimum,does not have a maximum, and given any two elements, there are finitely many elementsbetween them.If we have an ordered set with the same characteristics, we can put it in a bijectivecorrespondence with the integers that preserves the order. Roughly, one can proceedthis way. Take one element and arbitrarily label it zero. Since between two elementsthere are only finitely many, each element will have a successor and a predecessor. Label pace-time structure may be topological and not geometrical
The order of the reals can be characterized as one that is dense (i.e. between twoelements there is always another one), complete (i.e. it contains all the limit points), hasa countable dense subset (i.e. between two real numbers we can always find a rational,which is a countable set), and has no minimum or maximum.If we have an ordered set X with the same characteristics, we can put it in a bijectivecorrespondence with the reals that preserves the order. Roughly, one can proceed thisway. We start by noticing that we have a countable dense subset Q and, as Cantorshowed, it can be put into correspondence with the rationals. The set Q is dense inthe original set X and the original set X is complete. Therefore X is the completion of Q . Another theorem in order theory states that the completion of any order is unique.Since the real numbers are the completion of the rationals, and Q has the same orderof the rationals, then X has the same order of the reals.If we have a set of refinable aligned strict references, then the only additionalrequirement needed to recover the reals is that between two references there is alwaysanother reference. That is, we are only imposing that the references are dense. Thecompletion and the countable dense subset are a consequence of the more generaltopological requirements. The inclusion of all the limit points will come from the infiniteunion allowed by the topology while the dense subset is associated with the countablebase, which is a requirement for our topology to be physically meaningful.
6. Breakdown of ordering and of geometry
In the previous sections we have identified the experimental requirements needed to beable to operationally define continuous quantities. These are:(i) the ability to compare the quantity (i.e. position, time, mass, ...) with referencessuch that we can say whether the object is before or after each one(ii) the three different cases, before/on/after, must be mutually exclusive(iii) the references must be aligned, we must be able to place them in a line(iv) we must be able to find enough references such that overlaps can be avoided andall the possible values can be covered(v) between two references we can always place another pace-time structure may be topological and not geometrical pace-time structure may be topological and not geometrical
7. Conclusion
We have seen what the requirements are to give rise to the real numbers through a setof experimentally verifiable statements, which captures the most basic elements of howcontinuous quantities are measured in practice. We have also seen that those can only beconsidered idealized conditions, and that, when those requirements fail, ordering itselffails and no geometrical structure can be constructed. Naturally, this opens the questionof what topological spaces would be appropriate in those regimes. This is something wedo not have an answer for, though we can mention two of the possibly many scenarios.One can imagine to try and construct a topological space that is not ordered at afine scale but is ordered at a large scale. A difficulty here is that typically the notion ofdistance is absent from topological spaces. Though it is not clear to us whether this ispossible, at least it is a clear venue to explore.There is, however, a much more drastic scenario. General relativity tells us that thegeometry depends on the content of the space (i.e. the energy-momentum tensor). Whatis in the space affects the notion of distance and angle. It would not be far-fetched toassume that what is in the space also affects what can be distinguished experimentally.As the energy-matter distribution changes the geometry by curving space-time, it would,at a more fundamental level, change the topology as well. It is not clear to us at thistime how one would even start exploring such a scenario.
Acknowledgments
We would like to thank Mark J. Greenfield for review and help on the mathematicaldetails. Funding for this work was provided in part by the MCubed program of theUniversity of Michigan. This article is part of a larger project, Assumptions of Physics,that aims to identify a handful of physical principles from which the basic laws can berigorously derived [8].
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