Discrete spacetime: Things happen, they just happen in a partial - - PowerPoint PPT Presentation

0.5 setgray0 0.5 setgray1

Discrete spacetime: Things happen, they just happen in a partial order

Fay Dowker Blackett Laboratory Imperial College London

Discrete spacetime: Things happen, they just happen in a partial order – p. 1/2

Plan of Talk

Timeless Spacetime: The Block Universe of General Relativity (GR) Discrete Spacetime: Introducing the Causal Set ’tHooft; Myrheim; Bombelli, Lee, Meyer and Sorkin Causal sets could be the inner basis for spacetime Classical Sequential Growth: A Law of Motion for a Discrete Universe Rideout, Sorkin; Brightwell “Becoming” is compatible with General Covariance. Discussion Points

Discrete spacetime: Things happen, they just happen in a partial order – p. 2/2

Past, Present and Future

When Einstein’s close and longtime friend Michele Besso died in March 1955, he wrote in a letter of condolence to Besso’s widow: “In quitting this strange world he has once again preceded me by just a little. That doesn’t mean anything. For we convinced physicists the distinction between past, present, and future is only an illusion, however persistent” Einstein died in April that year. One can find many instances of physicists making similar comments (e.g. Paul Davies in Scientific American, “That Mysterious Flow” and Brian Greene in the New York Times, “The Time We Thought We Knew.”) What makes them believe this when our experience speaks otherwise so insistently?

Discrete spacetime: Things happen, they just happen in a partial order – p. 3/2

General Relativity

Our current best theory of gravity, GR, explains that what we observe as the force of gravity due to some massive body is the consequence of the gravitational field caused by that mass and not of any action at a distance. This gravitational field takes the form of the curvature of spacetime itself and the analogy that is often used to give a flavour of this is of a bowling ball placed on a horizontal stretched rubber sheet creating a dimple in the sheet so that marbles rolled on the sheet behave as if they are being attracted by the ball. As useful as this analogy is, there is one crucial aspect of GR that it cannot capture. The problem with the rubber sheet is that it is purely an analogy for the curvature of space whereas in GR, space and time are unified into a single indivisible fabric, spacetime, and it is this whole spacetime that is curved.

Discrete spacetime: Things happen, they just happen in a partial order – p. 4/2

Minkowski

This view was eloquently expressed by Hermann Minkowski who began a talk in Cologne in 1908, “The conceptions of space and time which I would like to develop before you arise from the soil of experimental physics. Therein lies their strength. Their tendency is radical. From this hour on, space by itself and time by itself are to sink fully into the shadows and only a kind of union of the two should yet preserve autonomy.” According to this modern scientific view, spacetime, and not space, is the proper arena for considering everything that exists, everything that happens. We live out our lives in spacetime, the history of the universe itself is a spacetime.

Discrete spacetime: Things happen, they just happen in a partial order – p. 5/2

Spacetime

Spacetime is the collection of all idealised events where an event is something that happens at a point in space at a moment in time, something like the popping of a champagne cork. Even when there is no actual cork there to mark it, the position of a potential pop is included in spacetime. Thus empty, or vacuum, spacetime is a physical entity in the theory. In GR spacetime is a 4 dimensional differentiable manifold endowed with a metric, a two-index symmetric tensor field, gµν, with “Lorentzian” signature (− + ++). This metric is the geometry of the spacetime. ds2 = gµνdxµdxν

Discrete spacetime: Things happen, they just happen in a partial order – p. 6/2

Can’t sit still

A crucial difference between space and spacetime is that in spacetime one cannot sit still. Staying at the same point in spacetime means staying at the same place and the same time, “Here-and-Now”, and that can’t be done: time must pass! So to position myself in spacetime I have to give my coordinates as time passes and thus I trace out a one dimensional path with time a parameter along it, my worldline. Let’s have a picture.

Discrete spacetime: Things happen, they just happen in a partial order – p. 7/2

Sketch of 2+1 dimensional spacetime

‘‘time’’ ‘‘space2’’ ‘‘space1’’ Q R P γ

A chunk of a 3-d spacetime. P, Q and R are events. The mini-axes represent a three dimensional coordinate grid to be imagined as drawn throughout the chunk. The axis labels are written in inverted commas to emphasise that they are just coordinates.

Discrete spacetime: Things happen, they just happen in a partial order – p. 8/2

Block Universe: Past, Present, Future

GR is a deterministic theory. If the spacetime to the past of some 3-d surface is fixed then the future spacetime is determined by the Einstein equations. What is real in GR is the whole spacetime, all 4-dimensions of it, from its beginning (if it had a beginning) to its end (if it had an end), laid out once and for all in a timeless way.

‘‘time’’ ‘‘space2’’ ‘‘space1’’ Q R P γ

γ might be my worldline from my birth, P, to my death, Q, which also exists in the theory in a timeless way. All existing things similarly have worldlines and nowhere in the theory is there any such thing as a Marvellous Moment, the Now of our conscious experience. [How is this different from Classical Mechanics?]

Discrete spacetime: Things happen, they just happen in a partial order – p. 9/2

GR not the final story

So there is no “becoming” in GR. But we already know that GR cannot be correct. In particular, spacetime responds dynamically to the matter in it and our best theory of matter is quantum mechanical and quantum mechanics is a stochastic theory. At the very least, this must result in a stocasticity in the behaviour of spacetime. But what theory should replace GR? Could the new theory alleviate the tension between our experience of time passing and

• ur scientific understanding of spacetime?

We’re searching for a theory of quantum gravity and there are several different approaches, including Causal Set Theory which postulates that spacetime is fundamentally discrete on very small scales and which will be the focus of the rest of the talk.

Discrete spacetime: Things happen, they just happen in a partial order – p. 10/2

Why discreteness

The Three Infinities caused by the continuum assumption: Infinite curvatures in General Relativity Infinite amplitudes in Quantum Field Theory Infinite black hole entropy In the last case, we know the physical answer: it is one quarter of the area of the black hole horizon in units in which G = = c = 1, the so-called Planck units. This suggests a discreteness scale of roughly 10−33cm and 10−43s. There are also more philosophical motivations for discreteness e.g. Zeno’s paradoxes, and also the rise of digital computing has made discreteness part of the Zeitgeist.

Discrete spacetime: Things happen, they just happen in a partial order – p. 11/2

Introducing the Causal Set

A causal set (or causet) is a set of elements, C, with a binary relation, ≺, which we can call “precedes” which satisfies the following axioms: Transitivity: if x ≺ y and y ≺ z then x ≺ z, ∀x, y, z ∈ C; Non-circularity: if x ≺ y and y ≺ x then x = y ∀x, y ∈ C; Local finiteness: for any pair of fixed elements x and z of C, the set {y|x ≺ y ≺ z} of elements lying between x and z is finite. Of these axioms, the first two say that C is a partially ordered set or poset and the third makes the set discrete.

Discrete spacetime: Things happen, they just happen in a partial order – p. 12/2

The Causal Set Hypothesis

The deep structure of spacetime is a causal set The discrete elements of the causal set are related to each other only by the partial ordering that corresponds to a microscopic notion of before and after, and the continuum notions of length, space and time arise only as approximations at large scales. Just as ordinary matter appears smooth and continuous on large scales but is really made of atoms, so it is proposed spacetime appears continuous to us but is fundamentally discrete. The number of causal set elements in a region gives rise in the continuum approximation to what we experience as the spacetime volume of the region and the order gives rise in the continuum approximation to the spacetime causal order.

Discrete spacetime: Things happen, they just happen in a partial order – p. 13/2

A nice way to represent a causal set is by thinking of it as a directed graph and drawing its Hasse diagram. For example:

x v y z w PAST FUTURE

Elements are vertices and relations are edges. Element x precedes (“is in the past of”) y and y precedes z. Transitivity implies that x precedes z so that relation does not need to be drawn in. The irreducible relations are called links. Element w is unrelated to any other. In a causal set that could be our visible universe there would be of the order of 10240 elements with a correspondingly large and complex web of relations.

Discrete spacetime: Things happen, they just happen in a partial order – p. 14/2

Causal sets are rich enough to be spacetime

Back in continuum spacetime, worldlines are restricted: from a particular Here-and-Now, there are some regions of spacetime my worldline can lead me to and some that are physically inaccessible to me. The reason is the familiar one: nothing can travel faster than

• light. So, roughly speaking, from my Here-and-Now, I can’t get to There-and-Now because I

have to take time to get There, at least as much time as light would take.

R P γ Q

Discrete spacetime: Things happen, they just happen in a partial order – p. 15/2

The past and future lightcones of the point P, given by the metric. Q is in the causal future of P (and P is in the causal past of Q) and we write Q ∈ J+(P). R is causally unrelated, or spacelike, to P. This lightcone structure exists similarly at every point of spacetime and all together, the data

• f which points are in the causal futures of which other points is known as the causal

structure of the spacetime. This causal order is a partial order (if the spacetime is well-behaved): if we define a binary relation < by P < Q whenever Q ∈ J+(P) then it is transitive and acyclic.

Discrete spacetime: Things happen, they just happen in a partial order – p. 16/2

Order + Number = Geometry

Powerful theorems by Hawking and Malament show that the causal structure of a spacetime together with a volume measure on spacetime fixes the metric (the geometry). Theorem: Let (M, g) and M′, g′) be strongly causal spacetimes, and let f : M → M ′ be a causal bijection (i.e. f(x) < f(y) iff x < y). Then f is a smooth conformal isometry. So it is plausible that a causal set can furnish the geometrical information required to be approximated by a continuum spacetime at large scales: the order provides the spacetime causal order and the number of elements provides the volume measure.

Discrete spacetime: Things happen, they just happen in a partial order – p. 17/2

Classical Sequential Growth

Causal set theory has a well-defined kinematical structure which in principle could furnish us with a discrete underpinning to spacetime. What about dynamics? What “Laws of Motion” do causal sets obey? Will they pick out the manifold-like causal sets from all the (vast majority of) non-manifold-like causal sets? We have, due to the work of Rideout and Sorkin, a family of dynamical “Laws of Motion” or, better, “Laws of Growth” for causal sets. These can’t be the final answer because they are not quantum mechanical but only classically stochastic, but they are an important stepping stone to the real dynamics. Example: Transitive Percolation (blackboard...)

Discrete spacetime: Things happen, they just happen in a partial order – p. 18/2

Discrete General Covariance

The Block Universe was a result of the General Covariance of GR, in other words the independence of physics from any choice of coordinates on spacetime. In particular, no physical import is attached to the time coordinate: the physical time that elapses along a worldline is not the difference in the time coordinates of its endpoints. There is no “global time”. But it seems that in the Sequential Growth models, there is a sort of “global time” which is the linear order of birth of the elements. Discrete General Covariance, then, is the principle that no physical meaning should attach to these birth order labels. In the continuum General Covariance manifests itself in the invariance of the action functional (which gives rise to the equations of motion) under changes of coordinates. What is the analogue here? The probability of obtaining a given finite causet at stage n of the growth process should not depend on the birth order of the elements. Example: transitive percolation.....

Discrete spacetime: Things happen, they just happen in a partial order – p. 19/2

Discussion Points

Whether or not the Block Universe is a problem is a personal view. Whether or not Sequential Growth models do restore the reality of the passage of time to physics is a personal judgment. Maybe this is a non-solution to a non-problem! (And, moreover, we don’t have the proper quantum dynamics yet.) Nevertheless, Sequential Growth models embody the notion of “becoming”: the birth process is there in the model. The birth process does not happen “in time” but the causal structure that results from the process is time. There is no physical sense in which either of two elements which are not ordered in the resulting causal set happened first. Their birth order is not physical. “Becoming” and lack of a global time peacefully co-exist in these models. Things happen, but in a partial order. In a Sequential Growth Model, the causal past of any newly born element is real: reality accumulates, like sediment, with the events of the past fixed and unchanging and the future as yet unrealised potentiality.

Discrete spacetime: Things happen, they just happen in a partial order – p. 20/2