The birth of (an intuition about) at(t)oms of spacetime geometry

Thinking for some time now on how to conceptually grasp and empirically catch attoms of spacetime geometry I could not fail to share this interesting and accessible lecture, taking the opportunity to prove today the progress made by physics community to put women contributions in the forefront!

Fay Dowker Public Lecture - 

Spacetime Atoms and the Unity of Physics (Perimeter Public Lecture)

One will find below a reading proposal to experience again the very pedagogical quality of Fay Dowker (I am not competent to emit judgment on her research) talking about the dynamics of spacetime in general relativity:

... I wish to suggest, our inability to reach consensus on the passage of time could be a consequence of our not yet having made the necessary scientific progress. We do not have a successful theory of spacetime that coherently incorporates the quantum nature of the physical world, so we do not yet know the nature of the deep structure of spacetime. Some of the observational facts on which the new theory will be built may, therefore, now be only roughly communicable and our sense-experience of the passage of time may be an example of such a fact. In the last decades, however, progress on one approach to finding a theory of quantum spacetime – or quantum gravity as it is usually called – affords us a forward look at how the passage of time may eventually find a place in science. The approach, causal set theory, is based on the hypothesis that spacetime is fundamentally granular or atomic at the Planck scale and this atomicity opens the door to new dynamical possibilities for spacetime and, hence, to a new perspective on the dichotomy of Being and Becoming. In this article I will describe this progress and will expand upon R. D. Sorkin’s claim that it gives us scientific purchase on the concept of passage [2].

 Spacetime is a continuous, smooth, four-dimensional material that bends, warps and ripples according to dynamical law as specified by the Einstein equations. Even when there is no matter present, “empty” spacetime itself can carry energy in the form of gravitational waves. Indeed this is the explanation for the variation in the rotation rate of the Hulse-Taylor binary pulsar system, which can be accurately modelled as a system losing energy via this gravitational radiation. The spacetime material is, however, very different from those substances that populated pre-relativistic physical theory in that it is intrinsically four-dimensional. It cannot be understood as a three-dimensional entity – “space” – evolving in time because that would imply a global time in which the evolution occurs and there is no such global, physical time in General Relativity (GR). The notion that at one moment of time there is space, a 3-d geometry, and at the next moment space has evolved to another 3-d geometry is wrong in GR. There is no such physically meaningful entity as 3-d space, no physically meaningful slicing of spacetime into space-changing-in-time. 

Having focussed on what spacetime in GR is not, we can ask what it is. The structure of spacetime that takes centre stage in understanding the physics of GR is its causal structure. This causal structure is a partial order on the points of spacetime.1 Given two points of spacetime, call them A and B, they will either be ordered or unordered. If they are ordered then one, let’s say A without loss of generality, precedes – or, is to the past of – the other, B. This ordering is transitive: if A precedes B and B precedes C then A precedes C. The order is partial because there are pairs of spacetime points such that there is no physical sense in which one of the pair precedes the other, they are simply unordered. This lack of ordering does not mean the points of the pair are simultaneous because that would imply they occur at the same “time” and require the existence of a global time for them to be simultaneous in. Again: global time does not exist in GR. 

This partial ordering of the points of spacetime is referred to as the causal structure of spacetime because it coincides with the potential for physical effects to propagate. Physical effects can propagate from A to B in spacetime if and only if A precedes B in the causal structure. If two spacetime points are unordered then no physical effect can propagate from one to the other because to do so would require something physical to travel faster than light. Causal structure plays a central role in GR and indeed the epitome of the theory, a black hole, is defined in terms of the causal structure: it is a region of spacetime such that nothing that happens in that region can affect anything outside the region. It is only by thinking of a black hole in terms of causal structure that its physics can be understood. [As an example of this, it is very difficult to answer the question, “Does someone falling feet first into a black hole lose sight of their feet as their feet cross the horizon?” without drawing the conformal “Penrose” diagram that depicts the causal structure.] 

If there is no global, universal time, where do we find within GR the concept of physical time at all? Physical time in GR is associated, locally and “personally,” with every localised physical system, in a manner that more closely reflects our intimate experience of time than the global time of pre-relativistic Newtonian mechanics. Each person or object traces out a trajectory or worldline through spacetime, a continuous line in spacetime that is totally ordered by the causal order: for any 2 points on the worldline, one precedes the other. GR also provides a quantitatively precise concept of proper time that elapses along each timelike worldline. A clock carried by a person following a worldline through spacetime will measure this proper time as it elapses, locally along the worldline. Viewed from this perspective, the famous “twin paradox” is no longer a paradox: two people who meet once, then follow different worldlines in spacetime and meet a second time in the future will in general have experienced different amounts of proper time – real, physical time – elapsing along their different worldlines between the meetings. Clocks are “odometers for time” along worldlines through spacetime. The remarkable thing, from this perspective, is that we get by in everyday life quite well under the assumption that there is a global Now, a universal global time, and that we can synchronise our watches when we meet, then do different things and when we meet again our watches will still be synchronised. GR explains this because it predicts that as long as the radius of curvature of spacetime is large compared to the physical scale of the system and the relative velocities of the subsystems involved are small compared with the speed of light, the differences in proper time that elapse along our different worldlines will be negligible. We can behave as if there is a global time, a global moment of Now, because for practical everyday purposes our clocks will remain synchronised to very high precision.    

In addition to being the key to understanding GR, the causal structure of spacetime is a unifying concept. Theorems by Kronheimer and Penrose [4], Hawking [5] and Malament [6] establish that the causal order unifies within itself topology, including dimension, differentiable structure (smoothness) and almost all the spacetime geometry. The only geometrical information that the causal structure lacks is local physical scale.[Technically, the result states that if two distinguishing spacetimes are causally isomorphic then they are conformally isometric. In 4 dimensions this implies that the causal structure provides 9/10 of the metrical information as the metric is given by a symmetric 4 × 4 matrix field of 10 spacetime functions, 9 of which can be fixed in terms of the 10th.]  This local scale information can be furnished by providing the spacetime volume of every region of spacetime or, alternatively, the amount of proper time that elapses – the duration – along every timelike worldline. In the continuum, the causal structure and local scale information complement each other to give the full geometry of spacetime, the complete spacetime fabric... 

There are strong, physical arguments that the smooth manifold structure of spacetime must break down at the Planck scale where quantum fluctuations in the structure of spacetime cannot be ignored. The most convincing evidence that spacetime cannot remain smooth and continuous at arbitrarily small scales and that the scale at which the continuum breaks down is the Planck scale is the finite value of the entropy of a Black Hole [7]. Fundamental spacetime discreteness is a simple proposal that realises the widely held expectation that there must be a physical, Planck scale cutoff in nature. According to this proposal, spacetime is comprised of discrete “spacetime atoms” at the Planck scale. The causal set programme for quantum gravity [8, 9, 10] is based on the observation that such atomicity is exactly what is needed in order to conceive of spacetime as pure causal order since in a discrete spacetime, physical scale – missing in the continuum – can be provided by counting. For example, a worldline in a discrete spacetime would consist of a chain of ordered spacetime atoms and its proper time duration, in fundamental Planckian scale units of time of roughly 10 ^-43 seconds, would be simply the number of spacetime atoms that comprise the worldline.

Causal set theory thus postulates that underlying our seemingly smooth continuous spacetime there is an atomic spacetime taking the form of a discrete, partially ordered set or causal set, whose elements are the spacetime atoms. The order relation on the set gives rise to the spacetime causal order in the approximating continuum spacetime and the number of causal set elements comprising a region gives the spacetime volume of that region in fundamental units. The Planckian scale of the atomicity means that there would be roughly 10²⁴⁰spacetime atoms comprising our observable universe. 

According to causal set theory, spacetime is a material comprised of spacetime atoms which are, in themselves, featureless, with no internal structure and are therefore identical. Each atom acquires individuality as an element of a discrete spacetime, a causal set, in view of its order relations with the other elements of the set. Let me stress here a crucial point: the elements of the causal set, the discrete spacetime, are atoms of 4-d spacetime, not atoms of 3-d space. An atomic theory of space would run counter to the physics of GR in which 3-d space is not a physically meaningful concept. An atom of spacetime is an idealisation of a click of the fingers, an explosion of a firecracker, a here-and-now.

The birth of spacetime atoms as the passage of time

Fay Dowker

(Submitted on 14 May 2014)

More about causal set theory and quantum gravity in the former post...

I remind the reader that my choice of the words attoms of spacetime geometry has several motives, the most natural one can be formulated in the following way:

The Higgs boson discovery which is a new fundamental piece of local information at the attometer scale, if it is understood in a spectral noncommutative geometric perspective, is conceptually linked to the global or rather spectral piece of information provided by dark matter and possibly dark energy interpreted as mimetic gravity aspects of the "quanta of geometry" uncovered through the higher Heisenberg equation proposed by Chamseddine, Connes and Mukhanov to model spacetime.

//the last paragraph has been slightly edited on March 12th 2018.
//this post has been edited on March 28th 2019