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

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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.
 
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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.
 
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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?
 
DavidMartin said:
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”.
 
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?
 
DavidMartin said:
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.
 
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DavidMartin said:
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."
 
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robphy said:
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?
 
  • #10
renormalize said:
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."
renormalize said:
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.
 
  • #11
DavidMartin said:
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
 
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  • #12
DavidMartin said:
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
 
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  • #13
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?
 
  • #14
DavidMartin said:
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.
 
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  • #15
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.
 
  • #16
DavidMartin said:
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:
Wikipedia said:
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
 
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  • #17
DavidMartin said:
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?
 
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  • #18
DavidMartin said:
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.
 
  • #19
DavidMartin said:
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.
 
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  • #20
robphy said:
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.
 
  • #21
Dale said:
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?
 
  • #22
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.
 
  • #23
DavidMartin said:
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.
 
  • #25
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?
 
  • #26
renormalize said:
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.
 
  • #27
Physical interactions and signal propagations in a Lorentzian spacetime (even without initially assuming a metric) yield a causal structure that is empirically found to be transitive and acyclic. This is not a metaphysical assumption but a forensic fact: "There’s something important about how we confirm the structure of spacetime and verify that distant cosmic events are real that often gets overlooked. Every photon, neutrino, and gravitational wave that reaches us is like a time-stamped receipt proving that a specific event happened at a particular moment in cosmic history, complete with a light-speed paper trail to back it up. This isn’t abstract philosophy—it’s more like forensic accounting for spacetime itself.

The Cosmic Microwave Background (CMB) acts as the universe’s master record book. It’s not just leftover glow from the Big Bang—it’s a detailed ledger of the universe’s early transactions. The temperature fluctuations we measure in the CMB are like entries in this ledger, and the polarization patterns act as cryptographic signatures confirming causality. When scientists map the CMB, they’re reconstructing real events from 13.8 billion years ago, events that were causally locked into our past before Earth even existed. This is hard observational data confirming that distant events were real when they happened, that their effects propagated causally to reach us, and that the universe keeps impeccable records.

This evidence matters because it shows us three things about how time works in the universe. First, the past wasn’t erased—it left causal invoices in the form of light, gravity, and neutrinos that we can still detect. Second, the future isn’t pre-rendered like a movie file waiting to play—it’s more like an unsigned contract waiting for physical inputs to determine what happens next. Third, the “now” is the active transaction, a cosmic update tick where the next state gets computed from the previous one.

Consider what happens when we observe a supernova. If distant events weren’t real until we saw them, we’d have some serious problems explaining what we actually observe. Take Supernova 1987A: its neutrinos arrived three hours before its light, even though both traveled for 168,000 years to reach us. If the supernova didn’t explode until we saw it, why did the neutrinos show up first? Did the universe pre-load the neutrino data but forget about the photons? The only sensible conclusion is that the explosion actually happened 168,000 years ago, and the universe broadcast the evidence at light speed, no observation required.

Millisecond pulsars provide another compelling example. These cosmic metronomes tick with near-perfect regularity. If their pulses weren’t real until we observed them, why do their arrival times match general relativity’s predictions down to the nanosecond? Did spacetime fake the pulsar’s rhythm just in case we happened to look? No—the pulses were emitted, traveled through space, and arrived on schedule, proving that distant time is real and events happen whether we observe them or not.

Some people try to use quantum mechanics to claim that reality is fuzzy until measured, but this doesn’t hold up when we look at large-scale cosmic structures. The CMB photons were emitted 380,000 years after the Big Bang, and their temperature fluctuations match predictions from quantum fluctuations during cosmic inflation. If these fluctuations weren’t real until 1965 when scientists first detected the CMB, how did they manage to pre-structure galaxy clusters billions of years earlier? Did the universe pre-compute its own large-scale structure just to trick us? The simpler explanation is that the CMB was always real—its patterns were baked into spacetime long before any observer existed.

Here’s the key insight: we can confirm that distant events and moments in our own history happened at the same time in cosmic terms by matching parts of different histories using measurable signals. When we receive a signal like photons or gravitational waves from a distant source, that signal carries information about the event that emitted it. By analyzing the signal’s properties—wavelength, travel time, redshift—we can reconstruct when that distant event occurred relative to our local history. This establishes that the distant event and a specific moment in our timeline were happening during the same cosmic era, even though they’re separated by vast distances. This isn’t metaphysical speculation; it’s observational physics.

Let’s walk through a concrete example. Imagine a distant planet emits light at its local time. That light reaches Earth much later at our local time. Using the speed of light, the distance between us, and accounting for the universe’s expansion if relevant, we can calculate how long the light took to travel. Therefore, we know the planet existed at a time that corresponds to a specific moment in Earth’s history. This confirms that the planet’s emission event and a particular slice of Earth’s history were happening during the same cosmic timeframe. We have direct evidence of this kind of non-local temporal relationship.

Scientists do this constantly. When measuring supernova light curves, mapping the CMB, tracking galaxy redshifts, or timing pulsar signals, each measurement confirms that distant events were happening during specific epochs of our own cosmic history. This directly contradicts claims that only your local present moment is real, that distant events have no temporal relationship to us until we observe them, or that time is purely local and can’t be compared across space.

The universe’s observable uniformity and smoothness on large scales gives us something special: a natural rest frame defined by the Cosmic Microwave Background. Any observer moving relative to this frame would measure a Doppler effect in the CMB. This uniformity allows scientists to divide spacetime into slices where density, temperature, and expansion rate are uniform everywhere. These slices are surfaces of constant cosmic time—moments when the universe has, on average, the same properties everywhere. They provide a global time coordinate for the universe.

A cosmic “now” is one of these slices: the set of all events across space that share the same cosmic time value since the Big Bang. The universe’s large-scale uniformity provides a standard of rest and a natural clock through its expansion, which together define a sequence of “nows” for the cosmos as a whole. This doesn’t abolish the relativity of time for local physics or for observers moving at high speeds relative to each other—two spaceships passing each other at high speed will still disagree on the timing of distant unrelated events. And we can’t directly measure this cosmic present for distant events due to light’s finite speed, and the concept becomes ambiguous inside horizons like black holes. But this doesn’t change the fundamental point: it provides a physically meaningful way to understand cosmic time.

This framework supports three important ideas. First, cosmological realism: distant events are real and have temporal structure; they’re not conjured into existence by our observation. Second, global coherence: the universe has a cosmic-scale temporal structure that allows meaningful comparisons of different histories. Third, causal continuity: signals carry real information about when things happened, preserving the temporal fabric of reality.

When we receive light from a distant galaxy showing it formed when the universe was two billion years old, that corresponds to the same cosmic timeframe when the Milky Way was forming its first stars. The photon’s journey is a causal thread connecting that past emission to our present detection. The universe’s expansion rate at emission, the travel time, and our reception form an unbroken causal chain. This is how we know Andromeda and the Triangulum Galaxy are real right now in cosmic time, even though we’re seeing them as they were millions of years ago. The universe isn’t just “out there”—it’s causally connected in ways we can measure, reconstruct, and verify. Such reframes the emergence of causal order not as a mathematical abstraction but as a physical necessity forced by the structure of empirical evidence. The universe itself generates a directed relational structure through the emission and reception of what it calls “receipts”—photons, neutrinos, and gravitational waves. These receipts establish a relation R(a,b) meaning “event a causally affects event b via a signal.” No metric or pre‑assumed partial order is needed to define this relation initially; it is simply the observed fact that signals leave one event and arrive at another. The forensic consistency of these receipts—for example, supernova neutrinos arriving before photons, or pulsar timings matching relativistic predictions across billions of years—forces the relation to be acyclic. If it were cyclic, a signal could loop back and contradict the recorded order of arrival. Similarly, transitivity is forced because the receipts chain: if event a sends a photon that triggers event b, and b emits a gravitational wave that reaches us, the combined receipt links a to the observer through b. The CMB functions as the ultimate ledger, encoding causal links from the early universe that have remained transitive and acyclic for 13.8 billion years, structuring galaxies long before any observer existed. Thus the mechanism is not a mathematical postulate but a physical inevitability: once you have interactions that propagate at finite speeds and those propagations leave detectable traces, the resulting network of relations is empirically required to be a partial order. Such explicitly rejects the notion that this order is imposed by consciousness or measurement; rather, it is “baked into spacetime” by the dynamics of propagation itself.".
 
  • #28
@OmniThoughts Please don't post chatbot output here. If any of us wanted to have a conversation with a chatbot then we would just ask them directly.
 
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  • #29
Dale said:
@OmniThoughts Please don't post chatbot output here. If any of us wanted to have a conversation with a chatbot then we would just ask them directly.
I just asked the chatbot to refine my findings into a more professional and much less verbose way to answer the question. Would you like it if I answered via the same info but in my less prfessional way of talking/speech?
 
  • #30
OmniThoughts said:
I just asked the chatbot to refine my findings into a more professional and much less verbose way to answer the question. Would you like it if I answered via the same info but in my less prfessional way of talking/speech?
I would prefer your thoughts.
 
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