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.
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
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
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:
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