Space and Time in Loop Quantum Gravity Carlo Rovelli CPT, Aix-Marseille Universit´e,Universit´ede Toulon, CNRS, F-13288 Marseille, France. (Dated: February 8, 2018) Quantum gravity is expected to require modifications of the notions of space and time. I discuss and clarify how this happens in Loop Quantum Gravity. [Written for the volume \Beyond Spacetime: The Philosophical Foundations of Quantum Gravity" edited by Baptiste Le Biha, Keizo Matsubara and Christian Wuthrich.] I. INTRODUCTION to in his Physics, Descartes founds on `contiguity', and so on. In mathematics it is studied by topol- Newton's success sharpened our understanding of the ogy. This is a very general notion of space, equally nature of space and time in the XVII century. Einstein's present in ancient, Cartesian, Newtonian, and rel- special and general relativity improved this understand- ativistic physics. ing in the XX century. Quantum gravity is expected to This notion of space is equally present in LQG. In take a step further, deepening our understanding of space LQG, in fact, we can say that something is in a cer- and time, by grasping of the implications for space and tain location with respect to something else. A par- time of the quantum nature of the physical world. ticle can be at the same location as a certain quan- The best way to see what happens to space and time tum of gravity. We can also say that two quanta are when their quantum traits cannot be disregarded is to adjacent. The network of adjacency of the elemen- look how this actually happens in a concrete theory of tary quanta of the gravitational field is captured by quantum gravity. Loop Quantum Gravity (LQG) [1{7] is the graph of a spin network (see Appendix). The among the few current theories sufficiently developed to links of the graph are the elementary adjacency re- provide a complete and clear-cut answer to this question. lations. Spin networks describe relative spacial ar- Here I discuss the role(s) that space and time play in rangements of dynamical entities: the elementary LQG and the version of these notions required to make quanta. sense of a quantum gravitational world. For a detailed discussion, see the first part of the book [1]. A brief sum- Newtonian space: In the XVII century, in the Prin- mary of the structure of LQG is given in the Appendix, cipia, Newton introduced a distinction between two for the reader unfamiliar with this theory. notions of space [9]. The first, which he called the \common" one, is the one illustrated in the previous item. The second, which he called the II. SPACE \true" one, is what has been later called Newto- nian space. Newtonian space is not a relation be- Confusion about the nature of space | even more so tween objects: it is assumed by Newton to exist for time| originates from failing to recognise that these also in the absence of objects. It is an entity with are stratified, multi-layered concepts. They are charged no dynamics, with a metric structure: that of a 3d with a multiplicity of attributes and there is no agree- Euclidean manifold. It is postulated by Newton on ment on a terminology to designate spacial or temporal the basis of suggestions from ancient Democritean physics, and is essential for his theoretical construc- notions lacking same of these attributes. When we say 1 `space' or `time' we indicate different things in different tion. Special relativity modifies this ontology only marginally, merging Newtonian space and time into arXiv:1802.02382v1 [gr-qc] 7 Feb 2018 contexts. The only route to clarify the role of space and time in Minkowski's spacetime. quantum gravity is to ask what we mean in general when In quantum gravity, Minkowski spacetime and we say `space' or `time' [8]. There are distinct answers to hence Newtonian space appear only as an approxi- this question; each defines a different notion of `space' or mations, as we shall see below. They have no role `time'. Let's disentangle them. I start with space, and at all in the foundation of the theory. move to time, which is more complex, later on. Relational space: `Space' is the relation we use when we locate things. We talk about space when we ask 1 During the XIX century, certain awkward aspects of this Newto- \Where is Andorra?" and answer \Between Spain nian hypostasis led to the development of the notion of `physical and France". Location is established in relation to reference system': the idea that Newtonian space captures the properties of preferred systems of bodies not subjected to forces. something else (Andorra is located by Spain and This is correct but already presupposes the essential ingredient: France). Used in this sense `space' is a relation a fixed metric space, permitting to locate things with respect to between things. It does not require metric conno- distant references bodies. Thus the notion of reference system tations. It is the notion of space Aristoteles refers does not add much to the novelty of the Newtonian ontology. 2 General relativistic space: Our understanding of the days". Location of events is given with respect to actual physical nature of Newtonian space (and something else. (We shall meet after three sun- Minkowski spacetime) underwent a radical sharp- rises.) Used in this sense time is a relation between ening with the discovery of General Relativity events. This is the notion Aristoteles refers to in (GR). The empirical success of GR |slowly cumu- his Physics2, and so on. It is a very general no- lated for a century and recently booming| adds tion of time, equally present in ancient, Cartesian, much credibility to the effectiveness of this step. Newtonian, and relativistic physics. What GR shows is that Newtonian space is indeed When used in this wide sense, `time' is definitely an entity as Newton postulated, but is not non- present in LQG. In LQG we can say that some- dynamical as Newton assumed. It is a dynamical thing happens when something else happens. For entity, very much akin to the electromagnetic field: instance, a particle is emitted when two quanta of a gravitational field. Therefore in GR there are gravity join. Also, we can say that two events are two distinct spacial notions. The first is the sim- temporally adjacent. A network of temporal adja- ple fact that dynamical entities (all entities in the cency of elementary processes of the gravitational theory are dynamical) are localized with respect to field is captured by the spinfoams (see Appendix). one another (\This black hole is inside this globu- lar cluster"). The second is a left-over habit from Newtonian time: In the Principia, Newton distin- Newtonian logic: the habit of calling `space' (or guished two notions of time. The first, which he `spacetime') one particular dynamical entity: the called the \common" one, is the one in the previous gravitational field. There is nothing wrong in do- item. The second, which he called the \true" one, ing so, provided that the substantial difference be- is what has been later called Newtonian time. New- tween these three notions of space (order of local- tonian time is assumed to be “flowing uniformly", ization, Newtonian non-dynamical space, gravita- even when nothing happens, with no influence from tional field) is clear. events, and to have a metric structure: we can say LQG treats space (in this sense) precisely as GR when two time intervals have equal duration. Spe- does: a dynamical entity that behaves as Newto- cial relativity modifies the Newtonian ontology only nian space in a certain approximation. However, in marginally, merging Newtonian space and time into LQG this dynamical entity has the usual additional Minkowski spacetime. properties of quantum entities. These are three: (i) In LQG (Minkowsky spacetime and hence) Newto- Granularity. The quantum electromagnetic field nian time appears only as an approximation. It has has granular properties: photons. For the same no role at all in the foundation of the theory. reason, the quantum gravitational field has granu- lar properties: the elementary quanta represented General relativistic time: What GR has shown is by the nodes of a spin network. Photon states form that Newtonian time is indeed (part of) an entity a basis in the Hilbert state of quantum electromag- as Newton postulated, but this entity is not non- netism like spin network states form a basis in the dynamical as Newton assumed. Rather, it is an Hilbert space of LQG. (ii) Indeterminism. The dy- aspect of a dynamical field, the gravitational field. namics of the `quanta of space' (like that of pho- What the reading T of a common clock tracks, for tons) is probabilistic. (iii) Relationalism. Quan- instance, is a function of the gravitational field gµν , tum gravity inherits all features of quantum me- Z p µ ν chanics including the weirdest. Quantum theory T = gµν dx dx : (1) (in its most common interpretation) describes in- teractions among systems where properties become In GR, therefore, there are two distinct kinds of actual. So happens in LQG to the gravitational temporal notions. The first is the simple fact that field: the theory describes how it interacts with all events are localized with respect to one another other systems (and with itself) and how its prop- (\This gravity wave has been emitted when the two erties become actual in interactions. More on this neutron stars have merged", \The binary pulsar after we discuss time. emits seven hundred pulses during an orbit"). The second is a left-over habit from Newtonian logic: the habit of calling `time' (in `spacetime') aspects III.