Emergent relativity from causal networks

[The Geoff]

I came across this recently: http://arxiv.org/abs/1008.4805v1"

Whilst it's mostly beyond my level, from what I can grasp the suggestion is that SR and Minkowski's spacetime can be derived simply by looking at the universe as a network of discrete causal events (a granular universe on the Planck scale, presumably).

The obvious criticism would be that posing a causal network requires a model of spacetime of some description anyway, so it's a bit chicken and egg? It's an interesting idea though, and one which ties in with Wheeler's "it from bit" and various fringe concepts of the universe being a large scale computation of some form. Anyone with a bit more insight have any thoughts?

(PS - first real post, hello everybody!

 

[bcrowell]

Welcome to PF!

The obvious criticism would be that posing a causal network requires a model of spacetime of some description anyway, so it's a bit chicken and egg?

Why do you say that? A causal network is just a directed graph, isn't it? It doesn't have to be embedded in a background spacetime.

Skimming the paper, it's not clear to me how new or nontrivial it is. It's been known for a long time that SR can be derived from assumptions about the symmetry of spacetime.[Rindler 1979] This paper just seems to be doing the same thing with a discrete grid, but as far as I can tell that's kind of a trivial change.

Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51

 

[bcrowell]

I could be wrong, but I think the crucial issue is whether or not you assume the lines on the graph all have a fixed length. If you don't assume that, then I think you have a purely relational theory, in which the only physical observables are intersections (of world-lines, of field-lines, ...). If you do assume that, then I think the graph is simply an artificial discretization of a smooth spacetime that has a certain metric.

For example, I think the way theories like loop quantum gravity and causal dynamical triangulation work is that you do not assume that each line is one Planck length. In LQG, what ends up being quantized is area, not length.

BTW, I'm going to ask that this thread be moved to the Beyond the Standard Model forum, where I think you'll get better replies from people who understand this kind of thing.

-Ben

 

[The Geoff]

The graph has both spacelike and timelike directions (fig 1, showing a analogue of a traditional light-cone), so the CN is defined as something very similar to Minkowski spacetime, and from this they derive SR. In effect, it seems to replicate Einstein's original work, but from a different set of axioms and suggesting a different interpretation?

As you say, a crucial aspect is whether you consider each causal link to have the same length, in which case you can derive the invariance of c from the theory rather than using it as an postulate.

Does that make any sense whatsoever?

 

[bcrowell]

In effect, it seems to replicate Einstein's original work, but from a different set of axioms and suggesting a different interpretation?

It's definitely not the same axiomatization as Einstein used in 1905. However, I think it may just be a trivial reworking of the axiomatization used in Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51.

As you say, a crucial aspect is whether you consider each causal link to have the same length, in which case you can derive the invariance of c from the theory rather than using it as an postulate.

This is where I would like to get opinions from people on the Beyond the Standard Model forum who know more about this sort of stuff. In some sense the "length" of each line should be zero, since the Minkowski norm of a lightlike vector is zero. There is probably some better way to express this than the way I'm doing it. Maybe a better way of saying it is that if you have a diamond with events labeled A (top), B and C (sides), and D (bottom), they're assuming that a cause at D will propagate via the two paths DBA and DCA and arrive at A simultaneously. This seems to me to show that this is not really a purely relational theory, but just a discretized copy of Minkowski space.

 

[granpa]

if you think of it as a mass spring sysem then the speed at which waves (think light) travel through the net will be less than the speed at which information is passed from node to node. Possibly much much less.
the speed would be a function of the ratio of mass to spring stiffness

 

[bcrowell]

if you think of it as a mass spring sysem then the speed at which waves (think light) travel through the net will be less than the speed at which information is passed from node to node. Possibly much much less.
the speed would be a function of the ratio of mass to spring stiffness

I think what this demonstrates is that you don't want to think of it as a mass-spring system.

 

[granpa]

I don't see your point.
I posted that to show that it was a reasonable way of looking at it.

otherwise you are left with a taxicab metric

 

[Naty1]

Without reading the author's reference (2), I can't tell what's added here either...So Crowell's post #2 seems appropriate:

it's not clear to me how new or nontrivial it is.

I was just listening the Verlinde's talk at the Perimter Institute, courtesy of someone in these forums probably, http://streamer.perimeterinstitute.ca/mediasite/viewer/NoPopupRedirector.aspx?peid=37ff156a-33d5-40d8-9986-d5ec82d96d91&shouldResize=False

and it seems a potentially lot more interesting: starting with less (just entropy) he gets general relativity (space,time and gravity)...so it seems more encompassing...

 

[Naty1]

The author states

We now introduce the main notion of causal network
(CN) as a partially ordered set of events with the par-
tial order representing the causal relation between two
events.

so I'd tend to agree with the OP here:

The obvious criticism would be that posing a causal network requires a model of spacetime of some description anyway, so it's a bit chicken and egg?

Crowell Posts in resposne:

Why do you say that? A causal network is just a directed graph, isn't it? It doesn't have to be embedded in a background spacetime.

I don't understand...Seems to me if you have 'causal events' and an 'order' there has to be space and time...I could not figure out exactly what a directed graph is, but here
http://en.wikipedia.org/wiki/Directed_graph

it notes 'ordered pairs' and looking that up shows 'first and second'...and 'two objects' which seems like time and space must be involved...?

 

[bcrowell]

Why do you say that? A causal network is just a directed graph, isn't it? It doesn't have to be embedded in a background spacetime.

I don't understand...Seems to me if you have 'causal events' and an 'order' there has to be space and time...I could not figure out exactly what a directed graph is, but here
http://en.wikipedia.org/wiki/Directed_graph

it notes 'ordered pairs' and looking that up shows 'first and second'...and 'two objects' which seems like time and space must be involved...?

A directed graph is simply a bunch of dots with line segments connecting some of them, and arrowheads to indicate a direction for each line segment. It's not embedded in a manifold. There are no coordinates. There are no metrical notions built in at all. Maybe when you hear "event," you automatically think of coordinates like (t,x,y,z). There is no such implication built into a bare-bones directed graph.

-Ben

 

[marcus]

Ben is right Naty.

Causets approach has no manifold. Everything is relations among elements of an abstract set (the "events").

It is ingenious how much structure they can recover just from that---dots and arrows. Not embedded.

(there is also a finite chain condition, as I recall---there are axioms so to speak---some minimal assumptions about the structure, but it's pretty basic.)
============

It is just kneejerk prejudice that mathematically representing spacetime and its geometry requires a manifold

Now with 1004.1780 and 1010.1939 it looks like Lqg has pretty much gotten rid of the manifold too.

 

[atyy]

Well, the original post may not have meant spacetime in the usual sense here - manifold or pseudo-Riemannian manifold. He may have wanted causality and Lorentzian signature (or Lorentzian likeness, since there is no metric at the start here) to be emergent. So eg. LQG does not have emergent Lorentzyness, unlike say or http://arxiv.org/abs/0705.4652 or http://arxiv.org/abs/0808.2495 .

 

[bcrowell]

there is also a finite chain condition, as I recall

Can you tell us about that?

 

[czes]

Ben is right Naty.

Causets approach has no manifold. Everything is relations among elements of an abstract set (the "events").

It is ingenious how much structure they can recover just from that---dots and arrows. Not embedded.

(there is also a finite chain condition, as I recall---there are axioms so to speak---some minimal assumptions about the structure, but it's pretty basic.)

It seems to be similar to the Max Tegmarks "Mathemetics Universe" - http://arxiv.org/abs/0704.0646
and
"Special Relativity from Causal Sets" of Kevin Knuth - http://arxiv.org/abs/1005.4172v2

My idea is that the quantum events are created by the interference of the information due to Compton wave which is non-local and ordered according to holographic principle.
The space, time, matter, energy emerges from a simple interference between information "yes" and "no". The charge (+) and (-) is defined after a relation to another quantum event , before it is not defined. The quantum events create quantum vacuum with its virtual particles-antiparticles.
Due Tegmark the causal sets is the simplest idea of the physics - just a mathematics.

 

[Chronos]

An obscure reference that might be of interest: http://arxiv.org/abs/gr-qc/0509120

 

[marcus]

...

(there is also a finite chain condition, as I recall---there are axioms so to speak---some minimal assumptions about the structure, but it's pretty basic.)
...

Can you tell us about that?

The Wippy article is pretty good
http://en.wikipedia.org/wiki/Causal_sets
It lists the four "axioms". What I was calling "finite chain" is the fourth one which they call the "locally finite" condition.

It just says you can't have an infinite chain x<..<..<...<y

I don't follow Causets, but I've seen articles where they did some remarkable math just with those four "axioms". Counting causal chains, taking minima and maxima of lengths forward from x to y and then backward from y to x' and so on. They are able to prove theorems and extract a surprising amount of structure---just with a partial-ordered set plus this finite causal chain condition (my name for it.)

The first three "axioms" are just the usual requirements for a partial order relation
reflexive
antisymmetric
transitive

it is the fourth that gives them a little traction, so they can start recovering something vaguely like geometry. It's philosophically cool, and I admire Rafael Sorkin, but I'm more excited by what's going on here:
1004.1780
1010.1939

BTW you know the PIRSA video archive, well Joe Henson recently gave a talk called Invitation to an Invitation to Causal Sets. He has worked with Fay Dowker and with Sorkin. I recall him saying that Dowker was expected to be giving several lectures at Perimeter about Causets. So he termed his one hour talk an Introduction to Dowker's Introduction. (I hope that she actually does give the series.)
A PhD student of Dowker's just posted a thesis. It used Henson's Cactus simulation tools to simulate causets. Using supercomputer/cluster tools may bring about some progress in Causets, just as Monte Carlo opened up CDT for Ambjorn and Loll in 2004 and led to quite a bit of progress.

 

[marcus]

See this earlier thread:
https://www.physicsforums.com/showthread.php?t=429531

 

[czes]

In Causal Sets there isn't a distance. There is a number of the events which are creating a set or system.
http://en.wikipedia.org/wiki/Partially_ordered_set
The distance is equal a number of events between the points, I think.
Is it true ?

 

[marcus]

In Causal Sets there isn't a distance. There is a number of the events which are creating a set or system.
http://en.wikipedia.org/wiki/Partially_ordered_set
The distance is equal a number of events between the points, I think.
Is it true ?

There is the finite causal chain condition, so the researchers have been able to get concepts like distance (going forwards or backwards in time, timelike distances if you want).

A priori there is no idea of "spatially between". so one cannot naively count the "points between"

But there are "workarounds". By going forwards and then backwards one can get from event A to an event B which is "outside the lightcone". One can take minima. They have devised ingenious methods of coping. I read about this a long time ago and don't want to review it. You need to read something. Try the Wippy article. Even STUDY that article. Understand "sprinkling" and how Lorentz invariance is achieved.

Or someone else will offer a link.
Or wait until 18 October and watch the Fay video on PIRSA. A smart personable articulate and attractive individual as I recall. Causets is not going to run away.
==============

CZES, I will just edit this post to respond to your next. I must go. The orig. post by The Geoff referred a paper by D'Ariano---not a regular Causets author and not much cited in the literature. He considers events in a regular lattice. Normally in Causets you do not make that rather strong assumption---no physical reason to assume that events fall in a regular array like that. So it is a highly atypical paper about something divorced from the main Causets program.

Thread too full of wild speculation, can't stay with it. I offered some suggestions if you want to really find out about Causets. In particular you can follow links in that earlier thread. Or if you stay in this one, you will find other people to discuss other things with. Good luck with your investigation and learning.

 

[czes]

There is the finite causal chain condition, so the researchers have been able to get concepts like distance (going forwards or backwards in time, timelike distances if you want).

A priori there is no idea of "spatially between". so one cannot naively count the "points between"

Is it like in Transactional Interpretation of John Cramer ? An observer may also move forwards or backwards.
As I understand it, an observer doesn't move in a space but from event to event. The space is built of quantum events. Therefore it is quantized and discrete. Is it right ?

 

[granpa]

In Causal Sets there isn't a distance. There is a number of the events which are creating a set or system.
http://en.wikipedia.org/wiki/Partially_ordered_set
The distance is equal a number of events between the points, I think.
Is it true ?

hence information travels through the network according to the taxicab metric (post 8 and 6)
http://en.wikipedia.org/wiki/Taxicab_metric

Taxicab geometry versus Euclidean distance:
The red, blue, and yellow lines have the same length (12) in taxicab geometry for the same route.
In Euclidean geometry, the green line has length 6×√2 ≈ 8.48, and is the unique shortest path.

and hence the need for light to travel slower than information travels through the network

 

[Naty1]

Crowell, Marcus...thanks for the feedback...I could have looked at those Wiki references I used for a thousand years and based on the language used there never come to the viewpoints you posted...and I take yours as corerect...

Extremely difficult, sometimes, to interpret things accurately on one's own...

 

[Max™]

I've been thinking about an approach like this for a while, should be interesting to see how it develops.