# Is causality a total order?

I've read that it is physically not possible to violate causality defined as a total order on the spacetime graph. So I was wondering if at least causality can be broked down to a partial order and if phenomenon in the universe exist that induce such an order.

Meaning if we have a set of events $E=\{A,B\}$ and an existing total order $A<B$, is it possible to create $A\leq B$, such that our successor function $f:\mathbb{R}^2 \rightarrow\mathbb{R}^2$ that maps an event in time to another event in the future $\{e_0,t_0\}\mapsto\{e_1,t_1\}$ could map an event to another event without changing the time variable $f(e_0,t_0)\mapsto\{e_1,t_0\}$. So that the event $A$ that triggers $B$ either happend before or simultaneous with $B$. I don't expect that $f(B,t_0)\mapsto\{A,t_1\}$ is possible.

I hope my question is understandable.

• Special relativity does not allow for a total order, but any which way you look like it, I don't think that an order structure for events alone is enough to do reasonable physics. Field theory basically gives you this partial order structure for free, anyway, and you can't get around it, if you want to make physical predictions. – CuriousOne Jan 2 '16 at 0:17
• I kind of get what you're asking, but I'm curious as to the source of the first sentence. Spacetime is partially ordered, with timelike separated events being comparable and spacelike separated events not being comparable. – user10851 Jan 2 '16 at 10:54
• to set a total order, one must be super observer. It's a corollary of is there an universal time ? – user46925 Jan 2 '16 at 19:33

The causal structure of spacetime forms a partial ordering, defined as --- given two points in spacetime $p,q\in\mathcal{M}$ --- we have $q\leq p$ if and only if $p$ lies in the causal future of $q$.

Why partial order and not total? Because the order is defined by (1) take a geodesic $\gamma$ connecting the two points, (2) consider its proper length $s=\int_{\gamma}\mathrm{d}s$, (3) if the sign of $s\leq 0$ (in $-+++$ signature) and temporally $q$ precedes $p$, then $q\leq p$.

If $p$ and $q$ were lightlike separated, then $s=0$ and a partial order could determine causal ordering of events but a total order could not.

Remarkably, the entire metric (up to some arbitrary conformal factor) can be recovered from this! See:

• Hawking, S. W. and King, A. R. and McCarthy, P. J., " A new topology for curved space–time which incorporates the causal, differential, and conformal structures". Eprint
• Malament, D. B. (1977). "The class of continuous timelike curves determines the topology of spacetime". Journal of Mathematical Physics 18 (7): 1399–1391. DOI:10.1063/1.523436

This has been used as the basis for an approach to quantum gravity, called "Causal Set Theory" as Stephen Anastasi has noted.

Partial orders are permitted in Causal Set Theory (Raphael Sorkin) and it does make predictions, which relate to an information universe at least, and these might be regarded as physical. That said, at present such predictions are rather limited. Separately, violation of Bell's inequality, in which the result of an experiment in one part of the universe immediately affects (in a highly confined circumstance) the results in other areas of the universe, predicted by quantum mechanics but not permitted by special relativity, suggests that a partial ordering might relate. There is also some consideration of closed timelike curves (see for example 't Hooft's discussion in "A Locally Finite Model for Gravity." of Gott, J.R.: Phys. Rev. Lett.66, 1126 (1991) and Ori, A.: Phys. Rev. D 44, R2214 (1991) which suggest the possibility of backwards causation (though this might have been sorted since he wrote that paper in 2008). So, if I am understanding your question correctly, partial order is a possibility as far as competing theories now stand.