A Derivation of Space and Time Paul Bernard White1 March 4, 2019 (Updated: March 12, 2019)2 Abstract. Four simple postulates are presented, from which we derive a (3+1)-dimensional structure, interpreted as ordinary space and time. We then derive further properties of space: isotropy and homogeneity; a rapid expansion within the first instant of time (i.e. inflation); and a continual and uniform expansionary pressure, due to a continual influx of (non-zero-point) energy that is uniformly distributed (i.e. dark energy). In addition, the time dimension is shown to have an "arrow". These results suggest that the four postulates may be fundamental to the construction of the physical universe. Contents 1 Introduction..................................................................................................................................1 2 Levels, projections, and relations: the structure and basic properties of system P......................3 2.1 Some properties of system P................................................................................................4 3 Constructing space and time in system P.....................................................................................5 3.1 Constructing a (3+1)-dimensional structure at level 2 (and above).....................................5 3.2 Isotropy and homogeneity of space......................................................................................7 3.3 Rapid expansion of space within the first instant of time....................................................8 3.4 A continual influx of energy associated with p0, yielding a continual and uniform expansionary pressure on space...........................................................................................8 4 Conclusion.................................................................................................................................10 Acknowledgments......................................................................................................................10 References..................................................................................................................................10 1 Introduction Systems that are based on information typically contain a basic information element and a basic information structure. In Biological systems, for example, the basic information element is the nucleotide molecule, and the basic information structure is a sequence of nucleotides (e.g. a codon, or a gene). Likewise, for computer systems the basic information element is the bit, and the basic information structure is a sequence of bits (e.g. an 8-bit byte). And in natural language the basic information element is the letter or phoneme, and the basic information structure is a sequence of letters or phonemes (e.g. a word or a sentence). 1 pbwx@att.net / ORCID: 0000-0002-2681-3670 (BSc Physics) 2 There may be a newer/updated version of this paper at: http://www.paulbwhite.org/papers/ 1 Such systems must also have a way of translating or computing the information elements and structures into meaningful output. In biology this is accomplished by the operations of ribosomes, enzymes, etc., acting on the nucleotide strings. For computers, the operations of logic gates on the bit strings typically perform this function. And in natural language the operations of lexical analysis, parsing, and context translate a string of letters/phonemes into meaning. Similarly, if the physical universe is based on information (as many have speculated, e.g. [1], [2], [3]), then the following questions arise: (a) What is the basic information element for this system?; (b) what is the basic information structure for the system?; and (c) how are these elements and structures translated (or computed) into the meaningful output that we call the physical universe? In answer to questions (a) and (b) above, I propose the following two postulates: 1. For creation of the physical universe, the basic information element is a type of projection --more specifically, a projection from a prior level. 2. The basic information structure is a sequence of such projections. With respect to the first postulate, we may refer to both projections and levels as "elements" (or basic elements) of the system, but will reserve the term "basic information element" for the projections alone. We now add two more postulates: 3. Each such projection is a one-dimensional vector, constituting a different, but related, one-dimensional space. (The basic relations between these projections/vectors are stated in the next postulate.) 4. Prior things (e.g. projections, levels, and constructions from them) are independent of subsequent things; and, conversely, subsequent things are dependent on prior things. (The terms prior, subsequent, dependent, and independent denote here logical/ontological relations. See e.g. [4].) In [5], I use these four postulates (and two additional ones) to develop a model for the basic construction of the physical universe --including the construction of ordinary space and time themselves, the fundamental particles and interactions, etc. In the present paper, however, we will (for the sake of brevity) focus simply on constructing ordinary space and time, and their basic properties. That is, using the four postulates above, we will: • derive a (3+1)-dimensional structure, interpreted as ordinary space and time • show that the derived 3-dimensional space is isotropic and homogeneous, and that the time dimension has an "arrow" • show that space undergoes a rapid expansion within the first instant of time (i.e. inflation) • show that space undergoes a continual and uniform expansionary pressure, due to a continual influx of (non-zero-point) energy that is uniformly distributed (i.e. dark energy). With respect to question (c) above, it will be shown that a method for translating sequences of 2 projections into physical meaning is by taking into account the relations between projections --specifically , their dependence and independence relations (i.e. postulate 4). Once obtained, the above (bulleted) results can then be said to support the proposition that the four stated postulates are fundamental to the construction of the physical universe. From now on, we will often refer to the model for constructing the physical universe, developed herein, as system P. 2 Levels, projections, and relations: the structure and basic properties of system P To construct our model for the physical universe (i.e. system P), we must begin with a state at which the things of the universe do not exist (otherwise our construction would be circular), i.e. a state that is absent the energy, elementary particles, and even space and time, as we know them. We will call this state level 0 of system P, or just level 0. We do not, however, presume that level 0 is a state of nothingness, or that nothing exists at level 0. We merely claim that nothing that comes into being with the construction of the physical universe exists at level 0; for level 0 is by definition a state that is immediately prior to the construction of the physical universe. Recalling our first three postulates, we say that a projection from level 0, to be denoted as p0, generates a new state, which we call level 1. Likewise, a projection from level 1, denoted as p1, generates another new state, which we call level 2. And a projection from level 2, denoted as p2, yields level 3; and so on. So, in general, the projection pk represents a sort of displacement from level k that generates level k + 1 (for k = 0, 1, 2, ...); thus, relative to each other, level k is prior, and level k + 1 is subsequent; also, relative to each other, pk is prior, and pk+1 is subsequent. (Again, the terms "prior" and "subsequent" refer to logical/ontological priority and subsequence.) In Fig. 1, where levels are represented by horizontal lines, and projections are represented by vertical arrows from a prior level to the next subsequent level, we illustrate the construction of levels 1 through 3 via the projections p0, p1, and p2. To the right of each level in Fig. 1 is shown the sequence of projections that is required to construct that level (the round brackets indicate a sequence, as is common in mathematics). Thus, the sequences of projections that are required to create levels 0, 1, 2, and 3 are ( ), (p0), (p0, p1), and (p0, p1, p2), respectively; moreover, the latter sequence constructs all of the levels (above level 0) in Fig. 1. As just described, the order of construction in system P starts with level 0 at the bottom of Fig. 1 and proceeds in the upward direction. Thus, level 0 is prior to all other elements (levels or projections) in system P, and subsequent to none; p0 is subsequent to level 0, but prior to level 1, p1, level 2, etc.; and so on. So, in general, a given element x in system P is subsequent to everything below it in Fig. 1, but prior to everything above it. By postulate 4, this means that element x is dependent on everything below it in the Figure, but independent of everything above it. Thus, for example, level 0 is independent of all other elements in system P, and dependent on none. Since level 0 is our starting point (or starting state) for constructing system P, then we must say that it is a nonconstructed element of that system, whereas the subsequent projections and 3 levels (p0, level 1, p1, level 2, etc.) are constructed elements of system P. So anything subsequent to level 0 is a constructed entity of the system. Fig. 1 Construction of levels 1 through 3 of system P via the projection sequence (p0, p1, p2). The projection sequence that is required to construct a given level is shown to the right of that level. 2.1 Some properties of system P Let x be a thing of system P (e.g. x is a level, a set of one or more projections, or something constructed from them). By postulate 4, things that are subsequent to x are (logically/ontologically) dependent on x. Such dependence implies that x is in effect, effective, operative, or operant at those subsequent things; or, alternatively, we say that those subsequent/dependent things are within the scope of x. Conversely, since things that are prior to x are independent of it, we say that x is not in effect or operant at those prior things; or, alternatively, we say that those prior/independent things are not within the scope of x. All of this is summarized in what will be called the scope rule for system P, stated as follows: A given thing in system P is in effect/operant at (i.e. contains within its scope) those things which are subsequent, and is not in effect at (does not contain within its scope) those things which are prior. From this we may deduce the following corollary to the scope rule: A given element in system P (i.e. a projection or level) is in effect/operant at (contains within its scope) those elements that are above it in Fig. 1, and is not in effect at (does not contain within its scope) those elements that are below it in Fig. 1. Thus, for example, since all of the constructed elements of system P (i.e. p0, level 1, p1, level 2, etc.) are subsequent to level 0 (or, conversely, level 0 is prior to them), then level 0 is in effect/operant at all of those things; or, all of those things are within the scope of level 0. Likewise, p1, level 2, p2, and level 3 are within the scope of level 1; but level 0 is not within the scope of level 1. And so on. Since pk is not in effect at level k, but is in effect at level k + 1, then level k + 1 represents the state at which the projection pk first comes into effect; by the scope rule, pk then stays in effect for all subsequent levels. Thus, the projection p0 first comes into effect at level 1, and stays in 4 p 1 p 0 p 2 0 1 2 3 ( ) (p 0 ) (p 0 , p 1 ) (p 0 , p 1 , p 2 ) effect for levels 2 and 3; likewise, p1 first comes into effect at level 2, and stays in effect for level 3. Let us say that the level at which a projection first comes into effect is its native level. Thus, level 1 is the native level for p0; level 2 is the native level for p1; and so on. That is, the native level for pk is level k + 1. Moreover, the concept of native level can be extended to things that are constructed from projections; thus, for example, something that is constructed using p0 and p1 (and no other projections) is native to level 2, since those two projections are first jointly in effect at that level. We note also that the projections that are in effect/operant at a given level are the same as the ones that are required to construct that level (as described earlier, and as listed in the sequences to the right of each level in Fig. 1). In constructing the sequence of projections (p0, p1, p2), since any projections that are in effect at level k are also in effect at the subsequent level k + 1, then we can think of the latter level as inheriting all of the projections that are in effect at the former level. And since this is true of projections, then it is also true of anything that is associated with or constructed from them. This aspect of system P --whereby that which is in effect at one level (or, if you will, generation) is passed on to the next subsequent level (and thus, by extension, to all subsequent levels) --will be called the inheritance rule. 3 Constructing space and time in system P Following postulate 3, let us model each projection as a one-dimensional vector; i.e. we model each pk (k = 0, 1, 2) as a one-dimensional vector going from level k to level k + 1. Thus, p0 is a one-dimensional vector from level 0 to level 1; p1 is a one-dimensional vector from level 1 to level 2; and so on. These vectors are represented graphically by the vertical arrows in Fig. 1. Moreover, each pk constitutes a different one-dimensional space. Though they are different in this respect, the pk are nevertheless related by the dependence and independence relations that have been postulated and discussed. 3.1 Constructing a (3+1)-dimensional structure at level 2 (and above) Since p0 is the only projection in effect at level 1, and since (by postulate 3) it is one dimensional, then it is fair to say that system P is one dimensional at level 1. Since both p0 and p1 are in effect at level 2, and since (by postulate 3) each of these constitutes a different one-dimensional space, then it might seem --at first glance --that system P should be two dimensional at level 2. But this would be wrong. To get the correct dimensionality at level 2, we must take into account the relations between p0 and p1, as per postulate 4 --i.e. the fact that p0 is independent of p1, and that this relation is asymmetric (p1 is dependent on p0). Since p0 and p1 are vectors, we interpret that these relations imply a kind of (asymmetric) linear independence, with the following property: from the perspective of p1, the vector p0 may be collinear with p1, but is also free to be noncollinear with p1. With these considerations in mind, we ask the question: What is the direction of p0 with respect to p1? Or, in other words, how does p0 "look" relative to p1? 5 Since p0 may be both collinear and noncollinear with p1 (from the latter's perspective), then p0 may have a component parallel to p1, and may also have a component perpendicular/orthogonal (i.e. at 90 degrees) to p1. But, by symmetry, the perpendicular component can be anywhere in a two-dimensional plane orthogonal to p1. The two dimensions of this orthogonal plane, plus the one dimension parallel to p1, makes three dimensions. Thus, from the viewpoint of p1 (and from the perspective of level 2), p0 has three dimensions; i.e. p0 constitutes a three-dimensional space (whereas, recall that p0 has only one dimension at level 1). We might say, therefore, that the view of p0 from the perspective of p1 "bootstraps" the former from a one-dimensional vector into a three-dimensional space. In summary, to construct its interpretation of p0, we can think of p1 as applying postulates 3 and 4 in succession: first, by postulate 3, p0 is a one-dimensional vector; second, by postulate 4, p0 is independent of p1 --which allows the former to have a component that is orthogonal to p1, with the result that p1 sees p0 as three dimensional. Conversely, we can ask, how does p1 "look" relative to p0? Since p1 is dependent on p0, then the former is not free to have a component that is orthogonal to the latter, and so p0 sees p1 as being collinear; or, more simply, p0 sees p1 strictly as per postulate 3: as a one-dimensional vector. So, at level 2 we have the three dimensions of p0, plus the one dimension of p1, for a total of four dimensions. Since system P is a model for constructing the physical universe, we interpret that the three dimensions of p0 are just the three dimensions of ordinary space, and the one dimension of p1 is the dimension of time; thereby yielding at level 2 the signature 3+1 space and time dimensions of our experience. The dimension of time, therefore, being a consequence of p1 (and p0), does not exist at levels 0 and 1, but only comes into existence at level 2; likewise, since ordinary, three-dimensional space is a consequence of p0 and p1, it also does not exist at levels 0 and 1, but only comes into existence at level 2. Note that, although p0 itself is independent of p1, the triple dimensionality of p0 at level 2 is not independent of p1. That is, in the process described above, p0 only manifests as three dimensional when it is related to, or juxtaposed with, p1. Thus, the triple dimensionality of p0 at level 2 (i.e. the triple dimensionality of ordinary space) is in fact dependent on p1. Conversely, both p0 and p1 are prior to, and thus independent of, ordinary space. We have shown, among other things, that p0 manifests differently at levels 1 and 2. At level 1 it is one dimensional. But when juxtaposed with p1 at level 2 it manifests as a three-dimensional space. Note that p0 itself does not change from level to level: it represents a projection from level 0 to level 1 wherever it appears (i.e. wherever it is in effect). This is analogous to e.g. the G nucleotide in biology, which is always the same molecule wherever it appears, but yields a different output (i.e. amino acid) depending on what other nucleotides/letters it is juxtaposed with in a sequence. In other words, like the letter G in a DNA sequence, the meaning of p0 is context dependent; which is just what we might expect for an element of a language, thus supporting our earlier notion that the basis of the physical universe is, to some degree at least, informational in nature. We might say that level 2 has two dimensions as input (one dimension for p0, plus one for p1), 6 but has four dimensions as output --three for p0, and one for p1. Which brings us back to question (c) in the introduction: How are the basic information elements of the model (which at level 2 are the inputs p0 and p1) translated (or, if you will, computed) into the meaningful output that we call the physical universe? We now see that at least a partial answer is that the relations between prior and subsequent elements are what translate them into meaningful output. In the present case, the independence relation between p0 and p1 at level 2 translates/transforms the manifestation of the former from a one-dimensional entity into a three-dimensional space. We can thus say that the construction of each space at level 2 requires the participation of an observer, in the sense that p1 "observing" p0 constructs ordinary, three-dimensional space, and p0 "observing" p1 constructs one-dimensional time. With ordinary space itself constructed by an observation of sorts, it becomes more plausible that e.g. the position of an object within ordinary space might also be constructed by some type of observation, as seems to be the case in quantum mechanics (more about that in [5]). The projections p0 and p1 are also operant at level 3 (as per the scope rule), and the relations between them are the same as at level 2 (i.e. p0 is independent of p1, but not the converse). Thus, at level 3 --as at level 2 --p0 will appear to p1 as a three-dimensional space (i.e. ordinary space), and p1 will appear to p0 as a one-dimensional space (i.e. time). In other words, the spaces that exist at level 2 also exist at level 3. Indeed, as per the inheritance rule, we might say that level 3 inherits these spaces from level 2; or, more precisely, level 3 inherits p0, p1, and the relations between them from level 2, and uses them to construct ordinary space and time. 3.2 Isotropy and homogeneity of space Recall that ordinary, three-dimensional space is created when p0 is viewed from the perspective of p1. So it follows that (a) the creation/construction of ordinary space is dependent on p0 and p1; and (b) p0 and p1 are prior to, and thus (by postulate 4) independent of, ordinary space. Suppose now that an outcome of constructing ordinary space is that p0 (or p1) manifests with a particular orientation or direction within that space. Since this would make p0 (or p1) functionally dependent on ordinary space, and thus contradict (b) above, we conclude that the construction of ordinary space cannot result in p0 (or p1) having a particular direction/orientation within that space. Presumably, then, there is no way for the process that constructs ordinary space to establish a distinctive (i.e. special or preferred) direction within that space. We thus conclude that, as constructed above, ordinary space is perfectly isotropic. Now suppose that an outcome of constructing ordinary space is that p0 (or p1) manifests with a particular position within that space. This, again, would make p0 (or p1) functionally dependent on ordinary space and thereby contradict (b) above; and so we conclude that the construction of ordinary space cannot result in p0 (or p1) having a particular position within that space. Presumably, then, the process that constructs ordinary space cannot establish a distinctive (i.e. special or preferred) position within that space. We thus conclude that, as constructed above, ordinary space is perfectly homogeneous. In addition, the construction of ordinary space cannot result in either p0 or p1 manifesting as 7 vectors, or vector fields, within that space; for if they did, then these projections/vectors would be functionally dependent on ordinary space, which would again contradict (b). Given that vector fields have been ruled out, it seems we have little choice but to assume that p0 and p1 manifest within ordinary space as uniform scalar fields --uniform, because any nonuniformity would make the manifestations of p0 or p1 functionally dependent on ordinary space, which would, again, violate/contradict their independence from that space. Presumably, the uniform scalar field for p0 is just (raw, unstructured) ordinary space itself, and the uniform (onedimensional) scalar field for p1 is just proper time. Lastly, let us recall that p0 sees p1 as a one-dimensional vector. This, presumably, would impart some directionality to p1 --which, as we have concluded, could not manifest as a direction within ordinary space. Since p1 has been associated with time, we interpret that this directionality of p1 (with respect to p0) is just the "arrow" of time. 3.3 Rapid expansion of space within the first instant of time Recall that p0 at level 1 is one dimensional --having, let us say, a length of p0. The time dimension, being a result of p1, does not exist at this level/stage. Given that a one-dimensional object has zero volume, then the physical universe at this stage of development has a volume of zero. Since the time dimension comes into existence with the projection p1, then the advent of p1 defines the time point t = 0, at which point p0 has the value p0(t = 0), which may be denoted as p0,0. So, at exactly t = 0, or within the first instant after it, the existence/perspective of p1 causes p0 to manifest as three-dimensional ordinary space, with a volume on the order of p0,0 3 . Thus the volume of ordinary space goes from zero to around p0,0 3 within a time interval of zero, or nearzero, length --which constitutes a potentially very large, perhaps infinite, rate of spatial expansion. I propose, therefore, that this rapid spatial expansion, triggered by the advent of p1 at t = 0, is the process known as inflation [6]. Note that, under the above mechanism, inflation has a natural beginning: the advent of p1 at t = 0. And it also has a natural ending: it ends when the volume of ordinary space is around p0,0 3 . So inflation only lasts for the time (if any) that it takes (from the perspective of p1) for the onedimensional space of length p0,0 to become the three-dimensional space of approximate volume p0,0 3 . 3.4 A continual influx of energy associated with p0, yielding a continual and uniform expansionary pressure on space In constructing the sequence (p0, p1, p2) for system P, let us assume that energy is needed to create each of the projections pk (for k = 0, 1, 2). We can think of this energy as being stored along the length of pk, and/or as being stored in the level that is created by pk. So we can speak of "pk energy", and/or we can speak of the energy, Ek+1, that pk inputs into level k+1. Thus, p0 is a process through which energy E1 is input into level 1 of system P. Likewise, p1 is a process that 8 inputs energy E2 into level 2; and p2 is a process that inputs energy E3 into level 3. The total energy, Et, that is input into system P is therefore Et = E1 + E2 + E3 . We assume that all of these energies are nonzero and positive, so the energy of system P at level 1 and above, due to contributions from the sources mentioned, is positive. Now recall that the dimension of time is associated with p1. Since p1 does not exist at levels 0 and 1, then time also does not exist there; i.e. all time intervals are zero at those levels. Indeed, we can say that levels 0 and 1 are independent of time. But p1 does exist at level 2 and above; so time exists there, and all time intervals at those levels are nonzero (and presumably positive). Thus, at level 1, energy is nonzero, but time is zero. At level 2 (and above), however, both energy and time (intervals) are nonzero. Consequently, at level 2 and above, the product of energy and time --the quantity known as action --is nonzero, and thus has a positive lower bound; i.e. at level 2 (and above) the action is quantized. We thus have the derivation of an action quantum, which we interpret to be the basis for the empirically-known "quantum of action", commonly referred to as Planck's constant, and denoted as h. In the present model, therefore, the quantum of action, h, depends on both p0 and p1, and so does not exist at levels 0 and 1, but only comes into being at level 2. Thus, quantum mechanics, which is based on h, also comes into being at level 2 of system P. And therefore, due to the scope rule, both h and quantum mechanics are operant at level 2 and above; i.e. they are native to level 2. The presence of h at levels 2 and 3 can, and we assume does, partition the energies E2 and E3 into a multiplicity of smaller chunks, yielding many objects/particles at those levels. The absence of h at level 1, however, means that the energy E1 cannot be broken into chunks; and so the energy E1 at level 1 constitutes a single, continuous entity. In addition, given that time exists at levels 2 and 3, we assume (as per special relativity) that the particles at those levels possess mass; and, given that time does not exist at level 1, we assume that the single entity at level 1 is massless. Furthermore, in [5] it is shown that the objects at level 3 have internal structure, whereas the objects at level 2 are structureless. These results lead us to identify the level-3 objects as baryons, and the level-2 objects as leptons. Moreover, since time exists at levels 2 and 3, then the input of energies (E2 and E3) into those levels can be, and we assume is, time limited --yielding a finite number of baryons at level 3, and a finite number of leptons at level 2. Recall now that p0 is native to level 1, but time is native to level 2. Thus, p0 is prior to time. By postulate 4, this means that the p0 process, which pumps energy E1 into level 1, is independent of time, and is therefore a continual process --i.e. it never stops, and so it must be happening right now. Consequently, the quantity E1 is always increasing. Moreover, since E1 is the energy of p0 at level 1, and since p0 (as seen by p1) is ordinary space, then it is clear that E1 is just the energy of space itself. Hence, an always-increasing E1 should yield a continual expansionary pressure on space. Indeed, an increase in E1 may produce an increase in the length of p0 , and thus an increase in p0 3 (the size/volume of the physical universe). Suppose now that the p0 process distributes its energy E1 nonuniformly within space. This would make that process (and thus p0 itself) functionally dependent on space, and thereby contradict statement (b) in section 3.2. Consequently, the energy E1 must be distributed 9 uniformly throughout space. Since this process is also independent of time, then it is constant in time. So the continual influx of E1 energy into the system via the p0 process yields an input of energy per unit volume of space that is uniform throughout space, and constant in time; in other words, E1 yields a cosmological constant. Taken all together, the above results suggest that we interpret E1 to be the phenomenon known as dark energy [7]; i.e. dark energy = E1. Moreover, since the p0 process and E1 are level-1 phenomena, but h only becomes operant at level 2, then dark energy/E1 is prior to --and thus independent of --h and quantum mechanics, and so is not a zero-point energy. 4 Conclusion A truly fundamental model of the universe must derive space and time --not just take them as given. Firstly, such a model should derive the (3+1)-dimensionality of space and time, and the isotropy and homogeneity of space. Secondly, since inflation and dark energy are likely to be important factors in the construction of space, then the model should also derive them. As shown above, the present model meets these basic criteria, which indicates that the four stated postulates may be fundamental to the construction of the physical universe. Acknowledgments I would like to acknowledge the following people for enabling the research for this paper to take place: My late mother and late father, Margean A. White and Phillip B. White; my brother, Jeffrey C. White; John P. A. Higgins; and the late Louise L. Hay (The Hay Foundation). References [1] J. A. Wheeler, "It From Bit", in his At Home in the Universe (AIP Press, 1994). [2] J. A. Wheeler, "The Computer and the Universe", Int. J. Theor. Phys. 21, Nos. 6/7, (1982). [3] P. Davies, N. H. Gregersen (eds.), Information and the Nature of Reality (Cambridge University Press, 2010). [4] J. Van Cleve, "Dependence", in The Cambridge Dictionary of Philosophy, R. Audi (ed.), (Cambridge University Press, 1995), p. 191. [5] P. B. White, "A Model for Creation: Part I" (2019), http://www.paulbwhite.org/papers/AModel-for-Creation-part-1.pdf [6] A. Guth, The Inflationary Universe (Reading, MA: Perseus Books, 1997). [7] S. M. Carroll, "Dark Energy and the Preposterous Universe" (2001), arXiv:astroph/0107571v2