# David P. Rideout

### Perimeter Institute for Theoretical Physics

31 Caroline St N
Waterloo, Ontario N2L 2Y5
```Office: 256      voice: +1 (519) 569 7600 x6562
fax:              7611
Email: drideout at perimeterinstitute.ca
```

## Causal Sets

Causal sets are discrete partially ordered sets, which are postulated to be a discrete substratum to continuum spacetime. The order gives rise to macroscopic causal order, while the discreteness or 'counting' gives rise to macroscopic spacetime volume. Given that causal structure is sufficient to reproduce the conformal metric, and discreteness provides the remaining volume information, it is reasonable to expect that causal sets alone possess sufficient structure to reproduce the entire continuum spacetime geometry.

A good introduction for 'laypersons' can be found at the Albert Einstein Institute's "Einstein online" site.

### Can a discrete lattice really be Lorentz invariant?

The `usual' discrete structures which we encounter, e.g. as discrete approximations to spatial geometry, have a `mean valence' of order 1. e.g. each `node' of a Cartesian lattice in three dimensions has six nearest neighbors. Random spatial lattices, such as a Voronoi complex, will similarly have valences of order 1 (or perhaps more properly of order of the spatial dimension). Such discrete structures cannot hope to capture the noncompact Lorentz symmetry of spacetime. Causal sets, however, have a `mean valence' which grows with some finite power of the number of elements in the causet set. It is this `hyper-connectivity' that allows them to maintain Lorentz invariance in the presence of discreteness.

Below is a demonstration of the Lorentz invariant character of causal sets. The top left image is a square region of 1+1 Minkowski space, into which has been sprinkled 4096 points. To the right is a blow up of a small region of the original region. The bottom left image shows the same points as viewed by an observer moving at v=-4/5. The same region (by the v=0 observer's coordinates) is blown up on the right. The arrangement of points is not literally the same, of course, but it is also a random Poisson sprinkling, of the same density.

### Sequential growth dynamics

The dynamics of causal sets can be expressed in terms of a sequential growth process, in which the causal sets grows, one element at a time, from the empty causet (causal set). This `time' in not some physical external time, but is `purely gauge'. One might regard physical time as being `embodied in the growth', rather than the growth `occuring in time'.

Below are some Hasse diagrams of random causal sets generated by the transitive percolation dynamics. The colors of the links do not play an essential role. (The purple links connect elements on neighboring 'layers', where the layer of an element is the length of the longest past directed chain which ends at that element. The green links span multiple layers.)

The following movies depict sequential growth of a region of 2+1 Minkowski space:
slow and fast and fast, and larger

### Some miscellaneous links related to causal sets

Link to Rafael Sorkin's Sixtieth Birthday Celebration.

David Meyer's thesis on the dimension of causal sets is available from MIT. If you have trouble obtaining it feel free to drop me a note. I also have paper copies of Luca Bombelli's and Alan Daughton's theses.

A topical school on causal sets will be held at Imperial College London from 18-22 September 2006. Registration is free.

Some VRML images of causal sets embedded into three dimensions that I made in 1997.

## Cactus

Cactus is a computational framework which greatly facilitates large scale collaborative computational projects, by separating computational details from 'the physics'. (Cactus can and is used in many fields besides physics, e.g. bioinformatics.) I am part of the development community for the framework itself, and am building modules (called 'thorns') for doing discrete quantum gravity computation within the Cactus framework.

## Erdös Number

Mine is 4, eg: Rafael Sorkin -> Graham Brightwell -> Peter Fishburn -> Paul Erdös

## Personal

Picture of my wife Yvonne and I:

Would you like to be able to control your personal computer, or should somebody else decide what it does? If the former then support the Free Software Foundation: