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
A familiar example of a causal set is a family tree. My uncle James Lynn Rideout has compiled
an extensive database of 8000+ of our relations, which is depicted here. I am number 735. My uncle is 693. It
is plotted using the convention of spacetime diagrams in which the oldest
ancestors appear at the bottom.
Note that the only parent-child relationship is considered. The many disconnected people are spouses for whom he did not know the parents or children.
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.
For example, 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 causal
set. It is this `hyper-connectivity' that allows them to
maintain Lorentz invariance in the presence of discreteness. (In the case of causal sets, it is the mean valence of the Hasse diagram which is important, not of the causal relation itself.)
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 (or causets) 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. 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'.
The growth process is depicted by the poset of finite causal sets, sometimes called poscau, in which two causets are related (in the poset's order) if one can reach the other by a sequence of sequential growth. (Equivalently, a causet A comes before, or precedes, B, if B contains a past-closed subcauset which is isomorphic to A.) I have generated diagrams of poscau for all causets up to four elements, and five. The green or blue arrows indicate gregarious children (those which arise when the new element appears unrelated to any others), and red arrows timid children (the child which occurs when the new element appears above the entire parent causet). The numbers attached to the edges give the "number of ways to transition from the parent to the child". All these are explained in detail in my paper with Rafael Sorkin on the subject, or in my Ph.D. thesis, both of which are available on arXiv.org.
[These diagrams are available in a variety of file formats, colors schemes, and licenses. Please write me at the above email address to inquire on details. I grant permission to use the images in presentation slides, provided the attribution and copyright notice remain intact. Please also write if you detect an error!]
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
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.
Some VRML images of causal sets embedded into three dimensions that I made in 1997.
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,
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.
Mine is 4, eg: Rafael Sorkin -> Graham Brightwell -> Peter Fishburn -> Paul Erdös
"The fear of the LORD is the beginning of wisdom..."
[Much is lost in the translation from Herbew to English, for example in the word "fear" and the rendering of the name "LORD", however it may be regarded as the beginning of my journey towards knowledge and understanding. (See also the preceding verses of Proverbs 1.)]
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