Can a causal or time-like structure emerge without assuming a metric?

[DavidMartin]

In most relativistic frameworks, time and causality are defined through an underlying spacetime metric with Lorentzian signature.

I’m wondering whether there are approaches where a notion of time ordering or causal structure can emerge prior to assuming a metric structure.

For example, can asymmetry in relational or informational structures be sufficient to define a time-like direction, with a metric description appearing only as an effective or secondary construct?

I’m not looking for a new theory claim, but rather for existing frameworks, models, or references where causality or time is treated as emergent rather than postulated.

Any pointers to known approaches or critical arguments against this idea would be appreciated.

 

[Ibix]

I'd have said SR itself fits the bill, at least if you follow Einstein's original postulates rather than starting by postulating Minkowski spacetime or something like that. You postulate the invariance of the speed of light and the principle of relativity and the causal structure emerges from the properties of the Lorentz transforms that you discover.

 

[DavidMartin]

I agree that in Einstein’s original formulation, causal structure is not postulated geometrically but follows from the Lorentz transformations derived from the two physical postulates.

My interest is precisely in understanding how minimal those postulates can be: whether some form of asymmetry or relational structure could play a role analogous to the invariance of ccc, with metric notions appearing only at a later stage.

Do you know of approaches where the Lorentz structure itself is derived from more primitive relational or informational assumptions?

 

[robphy]

Do you know of approaches where the Lorentz structure itself is derived from more primitive relational or informational assumptions?

AA Robb studied the “conical order” in terms of “before” and “after”.

 

[DavidMartin]

Thanks, that’s helpful.


If I understand correctly, Robb’s “conical order” is essentially an order-theoretic notion of “before” and “after”, defined purely in terms of causal accessibility rather than a metric or clock time.

In that sense, the light-cone structure encodes a primitive asymmetry, and notions like time orientation and Lorentzian geometry can be reconstructed from this causal order rather than postulated upfront.

Would it be fair to say that Robb’s work already points toward causality being more fundamental than the metric, with geometry emerging as a secondary description?

 

[robphy]

Would it be fair to say that Robb’s work already points toward causality being more fundamental than the metric, with geometry emerging as a secondary description?

I think that’s a fair statement.

 

[robphy]

You might be interested in David Malament’s dissertation:
“Does the Causal Structure of Space-Time Determine its Geometry”
https://digitalcommons.rockefeller.edu/student_theses_and_dissertations/496/
and
"The class of continuous timelike curves determines the topology of spacetime"
J. Math. Phys. 18, 1399–1404 (1977)
https://doi.org/10.1063/1.523436
https://pubs.aip.org/aip/jmp/articl...lass-of-continuous-timelike-curves-determines

 

[renormalize]

Would it be fair to say that Robb’s work already points toward causality being more fundamental than the metric, with geometry emerging as a secondary description?

Take a look at https://en.wikipedia.org/wiki/Causal_sets:
"The causal sets program is an approach to quantum gravity. Its founding principles are that spacetime is fundamentally discrete (a collection of discrete spacetime points, called the elements of the causal set) and that spacetime events are related by a partial order. This partial order has the physical meaning of the causality relations between spacetime events."

 

[DavidMartin]

You might be interested in David Malament’s dissertation:
“Does the Causal Structure of Space-Time Determine its Geometry”
https://digitalcommons.rockefeller.edu/student_theses_and_dissertations/496/
and
"The class of continuous timelike curves determines the topology of spacetime"
J. Math. Phys. 18, 1399–1404 (1977)
https://doi.org/10.1063/1.523436
https://pubs.aip.org/aip/jmp/articl...lass-of-continuous-timelike-curves-determines

Thank you robphy, I have so much questions ...

Malament’s result is particularly interesting — if I understand correctly, it shows that under suitable conditions the causal structure determines the metric up to a conformal factor.

So in that sense, causal order already contains most of the geometric information, with only scale left undetermined.

Would it be fair again to say that causal set theory can be viewed as a discrete analogue of this idea, where partial order is taken as fundamental and metric properties emerge statistically?

 

[DavidMartin]

Take a look at https://en.wikipedia.org/wiki/Causal_sets:
"The causal sets program is an approach to quantum gravity. Its founding principles are that spacetime is fundamentally discrete (a collection of discrete spacetime points, called the elements of the causal set) and that spacetime events are related by a partial order. This partial order has the physical meaning of the causality relations between spacetime events."

Take a look at https://en.wikipedia.org/wiki/Causal_sets:
"The causal sets program is an approach to quantum gravity. Its founding principles are that spacetime is fundamentally discrete (a collection of discrete spacetime points, called the elements of the causal set) and that spacetime events are related by a partial order. This partial order has the physical meaning of the causality relations between spacetime events."

I’m trying to understand whether causal order itself might arise from more primitive relational asymmetry.

 

[robphy]

Thank you robphy, I have so much questions ...

Malament’s result is particularly interesting — if I understand correctly, it shows that under suitable conditions the causal structure determines the metric up to a conformal factor.

So in that sense, causal order already contains most of the geometric information, with only scale left undetermined.

Would it be fair again to say that causal set theory can be viewed as a discrete analogue of this idea, where partial order is taken as fundamental and metric properties emerge statistically?

Along these lines (assuming (3+1)-spacetime):
"The causal order C determines the conformal structure of space-time, or nine of the ten components of the metric. The measure on spacetime fixes the tenth component."
- David Finkelstein - "Space Time Code" (1969), Phys. Rev. 184, 1261
https://doi.org/10.1103/PhysRev.184.1261
https://www.davidritzfinkelstein.com/papers/Space-TimeCode.pdf

For causal sets,
Rafael Sorkin describes it as "Order + Number = Geometry"
https://www.einstein-online.info/en/spotlight/causal_sets/ (2006)
See also:
Bombelli, Lee, Meyer, Sorkin "Space-Time as a Causal Set" (Aug 1987) Phys. Rev. Lett. 59, 521
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.59.521
and
Sorkin "Causal Sets: Discrete Gravity" https://arxiv.org/pdf/gr-qc/0309009 (2003)

I haven't kept up with the "state of the art" in causal sets.

Look for papers by Fay Dowker and Sumati Surya.
Sumati Surya (2019) "The causal set approach to quantum gravity"
https://arxiv.org/abs/1903.11544
https://link.springer.com/article/10.1007/s41114-019-0023-1
and
Dowker & Surya (2024) "The Causal Set Approach to the Problem of Quantum Gravity" https://link.springer.com/rwe/10.1007/978-981-19-3079-9_70-1

 

[robphy]

I’m trying to understand whether causal order itself might arise from more primitive relational asymmetry.

This might be of interest:

E. H. Kronheimer & R. Penrose (1967) "On the structure of causal spaces"
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 2 , April 1967 , pp. 481 - 501
DOI: https://doi.org/10.1017/S030500410004144X

Peter Szekeres (1991) "Signal spaces—an axiomatic approach to space-time"
https://www.semanticscholar.org/pap...eres/2e5912578d6f4d56aeff34d567e7257f4d02f73f
Bulletin of the Australian Mathematical Society , Volume 43 , Issue 3 , June 1991 , pp. 355 - 363
DOI: https://doi.org/10.1017/S0004972700029191

 

[DavidMartin]

My question is slightly more “upstream”:

are there approaches where causal order is derived from some underlying relational structure — for example through instability, asymmetry, or symmetry breaking — rather than being postulated as a primitive partial order?

 

[Sagittarius A-Star]

For example, can asymmetry in relational or informational structures be sufficient to define a time-like direction, with a metric description appearing only as an effective or secondary construct?

In classical physics, the only "asymmetric" law, which contains an arrow of time, is the the second law of thermodynamics.

 

[DavidMartin]

That’s a good point — in classical dynamics the second law is indeed the only fundamental time-asymmetric principle.

My question is slightly different, though. I’m not necessarily referring to time-asymmetric laws, but to asymmetric relations. For example, a relation R(a,b) that is not symmetric (i.e. R(a,b)≠R(b,a)).

In such a case, one could ask whether a partial order — and eventually a notion of causal precedence — might emerge from the structure of those relations themselves, independently of thermodynamic irreversibility.

I’m wondering whether any frameworks attempt to derive causal order from such structural asymmetry, rather than from a dynamical arrow like the second law.

 

[Sagittarius A-Star]

My question is slightly different, though. I’m not necessarily referring to time-asymmetric laws, but to asymmetric relations. For example, a relation R(a,b) that is not symmetric (i.e. R(a,b)≠R(b,a)).

Maybe that helps:

This defines a CPT transformation if we adopt the Feynman–Stückelberg interpretation of antiparticles as the corresponding particles traveling backwards in time.

Source:
https://en.wikipedia.org/wiki/CPT_symmetry#Derivation_of_the_CPT_theorem

 

[renormalize]

My question is slightly different, though. I’m not necessarily referring to time-asymmetric laws, but to asymmetric relations.

Can you illustrate your distinction between a law and a relation using examples from physics?

 

[robphy]

My question is slightly different, though. I’m not necessarily referring to time-asymmetric laws, but to asymmetric relations. For example, a relation R(a,b) that is not symmetric (i.e. R(a,b)≠R(b,a)).

Did you look at the Szekeres "Signal Space" reference?
His S relation is reflexive, but not symmetric.

 

[Dale]

I’m wondering whether there are approaches where a notion of time ordering or causal structure can emerge prior to assuming a metric structure.

That is a good question. If you have a pseudo-Riemannian manifold and remove the metric then you are left with a topological manifold.

To get a causal structure you need a partial order. And a topological manifold doesn’t have a natural partial order, so clearly you need something more than the topological manifold.

I guess the question is if you can add something besides a Lorentzian metric. Maybe a connection? I don’t know if you can do that or if the result would have a natural partial order.

 

[DavidMartin]

Did you look at the Szekeres "Signal Space" reference?
His S relation is reflexive, but not symmetric.

That seems close in spirit to what I’m asking about — although in that framework the S relation itself is still taken as primitive.

My question is whether there are approaches where even that non-symmetric signal/causal relation arises from something more primitive, rather than being postulated axiomatically.

 

[DavidMartin]

That is a good question. If you have a pseudo-Riemannian manifold and remove the metric then you are left with a topological manifold.

To get a causal structure you need a partial order. And a topological manifold doesn’t have a natural partial order, so clearly you need something more than the topological manifold.

I guess the question is if you can add something besides a Lorentzian metric. Maybe a connection? I don’t know if you can do that or if the result would have a natural partial order.

So the question becomes: is there any intermediate structure — weaker than a full Lorentzian metric — that is sufficient to induce a natural partial order?

For example, could some non-symmetric compatibility or signaling relation serve that role without already assuming a metric?

 

[DavidMartin]

Suppose one starts not from a partial order, but from a non-symmetric compatibility relation R(a,b)≠R(b,a), with no metric assumed.

Under what conditions would such a relation induce a consistent partial order (i.e. acyclic and transitive)?

Is there any general result characterizing when a directed relational structure can be “promoted” to a causal order?

I am exploring a toy model where directed relational weights dynamically break symmetry and induce orientation. I’m trying to understand what the minimal mathematical conditions are for such a structure to define a genuine causal order. And need a bit more of science en expertise for that.

 

[Dale]

So the question becomes: is there any intermediate structure — weaker than a full Lorentzian metric — that is sufficient to induce a natural partial order?

Yes. My guess would be a connection, but I don’t have the math to figure that out. I have usually started with a metric and derived a metric compatible connection. But I know that in principle connections can be defined that are not metric compatible.

 

[Sagittarius A-Star]

I’m wondering whether there are approaches where a notion of time ordering or causal structure can emerge prior to assuming a metric structure.

Maybe this helps:
https://www.wolframphysics.org/tech...ntial-relation-to-physics/time-and-spacetime/
via:
https://www.wolframphysics.org/technical-introduction/

 

[DavidMartin]

Let me sharpen the question slightly.

Suppose one starts with a directed relational structure R(a,b) that is not assumed to be transitive or acyclic a priori.

Are there known results characterizing when such a structure can dynamically or structurally induce a genuine partial order (i.e. acyclic and transitive), without postulating that order from the outset?

In other words, is there a known mechanism by which a non-symmetric relation becomes an order relation?

 

[DavidMartin]

Can you illustrate your distinction between a law and a relation using examples from physics?

That’s a fair request and didn's saw it sorry.

By a law, I mean a dynamical rule governing evolution — for example Newton’s second law, Maxwell’s equations, or the Schrödinger equation. These specify how physical states change in time.

By a relation, I mean a structural property between elements of a set, independent of any evolution equation.

For example:

• a causal precedence relation (“event A can influence event B”),
• a compatibility relation between configurations,
• or more abstractly, a non-symmetric binary relation R(a,b).

A relation need not describe dynamics; it may simply constrain which elements are connected or comparable.

My question concerns whether such non-symmetric structural relations — prior to specifying any dynamical law — could induce a partial order with causal interpretation.