# Toward “timeless” continuous-time causal models

I’m a bit at a loss as to where to put this. I know the in­fer­en­tial gap is too great for it to go any­where but here, and I know that the num­ber of peo­ple on LW in­ter­ested in this sub­ject could be counted on one hand. The pre­req­ui­sites would al­most cer­tainly be Time­less Causal­ity and more math­e­mat­ics than any­one is re­ally in­ter­ested in learn­ing.

So, I apol­o­gize in ad­vance if you read this and dis­cover at the end it was a waste of your time. But at the same time, I need peo­ple who know about these things to talk about them with me, to en­sure that I haven’t gone crazy… yet. And most im­por­tantly, I need to know the peo­ple who have done this be­fore, so that I don’t have to do it. Google can’t find them.

## Introduction

There are cur­rently some efforts to gen­er­al­ize the causal mod­els of Pearl to con­tin­u­ous-time situ­a­tions. Most of these at­tempts in­volve re­plac­ing some causal dis­crete vari­ables Xi with time-de­pen­dent ran­dom vari­ables Xi(t). Pos­si­bly due to memetic in­fec­tion from Yud­kowsky, I don’t think this is nec­es­sar­ily the cor­rect ap­proach. The philo­soph­i­cal power of Pearl’s the­ory comes from the fact that it is time­less, that ba’o vimcu ty bu.

In or­der to mo­ti­vate my work­ing defi­ni­tions for where such a time­less con­tin­u­ous-time the­ory will go, I need to go back to clas­si­cal causal­ity and de­cide what a time­less for­mu­la­tion ac­tu­ally means, for­mally. Spoiler: it means re­plac­ing time-de­pen­dent evolu­tion with a global flow on the phase space of the sys­tem. This is more or less in line with what is said in Time­less Physics with re­gard to the glimpse of “quan­tum mist” illus­trated there.

## The role of phase space

What is “time­less­ness”? The first thing I thought of af­ter read­ing the time­less sub­se­quence was, “What does a time­less for­mu­la­tion of the wave equa­tion look like?” First of all, this was the right thought, be­cause the wave equa­tion is what I’ll call (af­ter the fact) “clas­si­cally causal” in a sense to be de­scribed soon. I wouldn’t have seen the time­less­ness in a differ­ent math­e­mat­i­cal model, be­cause not all math­e­mat­i­cal mod­els of re­al­ity pre­serve the un­der­ly­ing phe­nom­ena’s causal struc­ture. On the other hand, this was the wrong thought, be­cause the wave equa­tion is not the sim­plest con­tin­u­ous-time sys­tem that would have led me to this for­mal­iza­tion of time­less­ness. Un­for­tu­nately the one that is eas­ier for me to see (La­grangian me­chan­ics) is harder for me to ex­plain, so you’re stuck with a sub­op­ti­mal ex­pla­na­tion.

The wave equa­tion mod­els all sorts of wave-like phe­nom­ena: light, acous­tic waves, earth­quakes, and so on. If we take the speed of sound to be one (as physi­cists are wont to do), the dis­per­sion re­la­tion is ω2 = k2. Such a dis­per­sion re­la­tion satis­fies the Kramers-Kronig re­la­tion. As it turns out, equa­tions whose dis­per­sion re­la­tion satis­fies this con­di­tion satisfy what I’m call­ing “clas­si­cal causal­ity”, but what is more com­monly known as finite speed of prop­a­ga­tion — or, more phys­i­cally speak­ing, the fact that sig­nals stay within their light cone.

The most com­mon prob­lem as­so­ci­ated with the wave equa­tion is the Cauchy prob­lem. At time zero, we spec­ify the state of the sys­tem: its ini­tial po­si­tion and ve­loc­ity at ev­ery point. Then the solu­tion of the wave equa­tion de­scribes how that ini­tial state evolves with time. From a more ab­stract point of view, this evolu­tion is a curve in the space of all pos­si­ble ini­tial states. This space is com­monly referred to in the spe­cific case of the wave equa­tion as “en­ergy space”, which fur­ther illus­trates why this ex­am­ple is a bit bad for ped­a­gog­i­cal pur­poses. From now on, we’re only go­ing to talk about phase space.

Here is where we can re­move time from the equa­tion. In­stead of think­ing of the wave equa­tion as as­so­ci­at­ing to ev­ery state in phase space a time-de­pen­dent curve is­su­ing forth from it, we’re go­ing to think of the wave equa­tion as spec­i­fy­ing a global flow on the whole of phase space, all at once. In sum­mary, I am led to be­lieve that time­less for­mu­la­tions amount to ab­stract­ing away the time-de­pen­dence of the sys­tem’s evolu­tion as a flow on the phase space of the sys­tem. And to think, this in­sight only took three years to in­ter­nal­ize, pro­vided I’ve got­ten it cor­rect.

## Causal flow

The situ­a­tion for a causal model is harder. In part, be­cause stochas­tic things have shoddy ex­cuses for deriva­tives. For the mo­ment, we’re go­ing to take the eas­iest pos­si­ble con­tin­u­ous-time sys­tem: our causal N vari­ables of in­ter­est, Xi, take only real val­ues. The space of all the pos­si­ble states of the sys­tem is N-di­men­sional Eu­clidean space, which is easy enough to work with. I’m go­ing to im­plic­itly as­sume that causal vari­ables evolve con­tin­u­ously; that is, the side­walk doesn’t go from be­ing com­pletely dry to com­pletely wet in­stan­ta­neously. Things like light switches and push but­tons can still be mod­eled prac­ti­cally by bump func­tions and the like, so I don’t see this as a real limi­ta­tion.

The some­what harder bul­let to swal­low is the as­sump­tion that the ran­dom vari­ables are Marko­vian; that is, they are “mem­o­ryless” in the sense that only the pre­sent state de­ter­mines the fu­ture. Pearl spends some time in Causal­ity defend­ing this as­sump­tion from crit­i­cism that it doesn’t ap­ply to quan­tum sys­tems — I be­lieve this defense is rea­son­able. I be­lieve that causal mod­els are nec­es­sar­ily re­fine­ments of our be­liefs about what is still for the most part a clas­si­cal world, and so the Markov as­sump­tion is not nec­es­sar­ily un­nat­u­ral.

The phase space of N-di­men­sional Eu­clidean space is known as the tan­gent bun­dle, which amounts to hav­ing an ad­di­tional copy of N-space at ev­ery point. Mo­rally speak­ing, the tan­gent bun­dle rep­re­sents all the di­rec­tions and speeds in which the sys­tem can evolve from any given state.

We need some data about how the sys­tem is sup­posed to evolve: what I will call the causal flow. As best as I can cur­rently con­jec­ture, this data should take the form of a “bun­dle” of prob­a­bil­ity mea­sures P, one for each point in N-space, such that each prob­a­bil­ity mea­sure P(x) is defined over the tan­gent copy of N-space at­tached to that point.

By anal­ogy with the pre­vi­ous sec­tion, the time-evolu­tion of the sys­tem is given by Lip­s­chitz-con­tin­u­ous curves in N-space. (Lip­s­chitz-con­tin­u­ous, be­cause if we as­sume they are differ­en­tiable curves, the Markov as­sump­tion goes out the win­dow.) In con­trast with the dis­crete the­ory of causal­ity, and as men­tioned above, we don’t al­low causal vari­ables to “jump” spon­ta­neously, and there is a limit to how sharply they can “turn”.

A use­ful thing to have around would be the prob­a­bil­ity that the sys­tem will evolve from one state to an­other via a spe­cific choice of one of these curves. Lip­s­chitz-con­tin­u­ous curves are rec­tifi­able, and so one can re­ca­pitu­late a sort of Rie­mann sum — if you’re in­ter­ested, I have it for­mally writ­ten down in a .pdf, but the cur­rent for­mat is un­friendly to maths. So for now, you’ll just have to take my word for it when I say I can define the prob­a­bil­ity of the flow fol­low­ing a spe­cific path. From there, it’s just a path in­te­gral to defin­ing the prob­a­bil­ity of get­ting from one state to an­other.

## Where to go from here?

Given this causal flow, d-sep­a­ra­tion should arise as a ge­o­met­ri­cal con­di­tion — but per­haps only a lo­cal one, for the causal struc­ture of the sys­tem can also evolve with time. To in­ter­vene in this sys­tem is to pro­ject it onto a cer­tain hy­per­plane, pre­sum­ably, in some yet-to-be-de­ter­mined way. And fi­nally, there ought to be some way to define coun­ter­fac­tu­als, but my limited math­e­mat­i­cal fore­sight has already run too thin.

BONUS: If you’ve made it this far and can’t think of any­thing else to say, I’m will­ing to Crocker-en­ter­tain prob­a­bil­ities that I’m in­sane and/​or a crack­pot.

• Maybe the in­di­vi­d­ual tra­jec­to­ries of such a sys­tem could be de­scribed as solu­tions to some stochas­tic differ­en­tial equa­tion?

Also by anal­ogy with con­tin­u­ous-time Markov pro­cesses, it might be eas­ier to for­get about in­di­vi­d­ual tra­jec­to­ries and in­stead think about the “flow” of prob­a­bil­ity den­sity in phase space, which can prob­a­bly be de­scribed by a par­tial differ­en­tial equa­tion with­out need­ing to define your own Rie­mann sums, path in­te­grals and such. Or maybe I’m miss­ing some­thing here and you re­ally need cus­tomized ma­chin­ery?

Also Cosma Shal­izi is an ex­pert on both causal mod­els and con­tin­u­ous-time stochas­tic pro­cesses, so maybe you could ask him or look at his work if you haven’t seen it already.

• Also by anal­ogy with con­tin­u­ous-time Markov pro­cesses, it might be eas­ier to for­get about in­di­vi­d­ual tra­jec­to­ries and in­stead think about the “flow” of prob­a­bil­ity den­sity in phase space which can prob­a­bly be de­scribed by a par­tial differ­en­tial equa­tion with­out need­ing to define your own Rie­mann sums, path in­te­grals and such. Or maybe I’m miss­ing some­thing here and you re­ally need cus­tomized ma­chin­ery?

It might work. I haven’t thought about it yet.

Also Cosma Shal­izi is an ex­pert on both causal mod­els and con­tin­u­ous-time stochas­tic pro­cesses, so maybe you could ask him or look at his work if you haven’t seen it already.

I know of him and have read some of his stuff, but the work isn’t in a suffi­ciently sta­ble state to bother an aca­demic with it yet. I need more ev­i­dence that this is the fruit­ful path. I ex­pect it would be difficult to con­vince him of the value of such an effort, since there’s no ev­i­dence yet that it’s even differ­ent from what is be­ing done already.

• This looks re­ally in­ter­est­ing. It is still rather sketchy, though, and my physics is not good enough to be con­fi­dent about how to fill in the de­tails. Would you mind send­ing along your pdf?

One minor quib­ble and then a more gen­eral re­mark:

The some­what harder bul­let to swal­low is the as­sump­tion that the ran­dom vari­ables are Marko­vian; that is, they are “mem­o­ryless” in the sense that only the pre­sent state de­ter­mines the fu­ture.

This is not quite cor­rect as a de­scrip­tion of the SGS-Pearl sys­tem. For SGS-Pearl, a causal sys­tem is Marko­vian rel­a­tive to a graph, not nec­es­sar­ily rel­a­tive to time. That is, the graph­i­cal par­ents of a vari­able screen that vari­able off from all of its non-de­scen­dants. As­sume a dis­crete-time model. There is no re­quire­ment that the graph­i­cal par­ents of a given vari­able live in the im­me­di­ately pre­vi­ous time-slice. We could have de­layed, di­rect cau­sa­tion. For ex­am­ple, if we had X(t=T) --> X(t=T+2) and also Y(t=T+1) --> X(t=T+2), where no other vari­able is a di­rect cause of X(t=T+2). The vari­able X(t=T+2) is in­de­pen­dent of its non-de­scen­dants given X(t=T) and Y(t=T+1).

If you as­sume that all gen­uine, fully un­rol­led causal con­nec­tions have the same time-in­ter­val, then your ver­sion of the Markov con­di­tion lines up with SGS-Pearl.

On its face, stat­ing the usual global causal Markov ax­iom—but as­sum­ing densely-or­dered times, rather than non-densely-or­dered times, is non-ob­vi­ous. Since in such a sys­tem no vari­able (graph-ver­tex) will have any speci­fi­able di­rect causes (par­ents), you can’t sim­ply say that a vari­able is in­de­pen­dent of its non-de­scen­dants con­di­tional on its par­ents.

You also can’t just con­di­tion on a time-slice of an­ces­tors (even as­sum­ing that such a slice screens off the past from the fu­ture), since there might be branch­ing from a vari­able that lives be­tween what­ever slice you pick and the tar­get vari­able. That is, sup­pose the times are densely or­dered, and your tar­get vari­able Z lives at time T2. Now, sup­pose that you con­di­tion on all the vari­ables X at time T1 < T2. Since the or­der­ing is dense, there are times be­tween T1 and T2. For all we know, there might be a vari­able X(un­lucky) that lives at one of the times be­tween T1 and T2 such that X(un­lucky) is a com­mon cause of Z and some non-an­ces­tral, non-de­scen­dant of Z, call it Y. In that case, we ex­pect Y and Z to be as­so­ci­ated in virtue of the com­mon cause X(un­lucky).

Thoughts?

• Would you mind send­ing along your pdf?

I’ll find some­where to stick it and link it to the post.

The some­what harder bul­let to swal­low is the as­sump­tion that the ran­dom vari­ables are Marko­vian; that is, they are “mem­o­ryless” in the sense that only the pre­sent state de­ter­mines the fu­ture.

This is not quite cor­rect as a de­scrip­tion of the SGS-Pearl sys­tem.

Yeah, sloppy writ­ing on my part; the “time” that ap­pears here is only an ob­server’s se­quence of ob­ser­va­tions of the sys­tem’s state. I agree with what you say about dis­crete-time mod­els.

On its face, stat­ing the usual global causal Markov ax­iom—but as­sum­ing densely-or­dered times, rather than non-densely-or­dered times, is non-ob­vi­ous.

What was as­sumed there is not that, be­cause in this de­vel­op­ment I have not got­ten to the point of find­ing a di­rected graph of vari­ables any­where. Pre­sum­ably I’ll need a lo­cal corol­lary of Pearl’s the­o­rem 1.4.1, i.e., ev­ery causal flow model (prob­a­bly sub­ject to some tech­ni­cal re­stric­tions) lo­cally in­duces a graph model with a com­pat­i­ble joint prob­a­bil­ity dis­tri­bu­tion. This has some hope of suc­ceed­ing; if a dis­tri­bu­tion is con­sis­tent with a graph model, then small per­tur­ba­tions of it are also con­sis­tent with it.

This is what I meant to as­sume: that X is a con­tin­u­ous-time markov pro­cess.

• This is what I meant to as­sume: that X is a con­tin­u­ous-time markov pro­cess.

Ah! Yes, that makes sense. I’m look­ing for­ward to read­ing the pa­per.

• I’m look­ing for­ward to read­ing the pa­per.

Sorry, my real life job in­ter­vened and kil­led most of this week. There’s a slight kink in the cur­rent draft that I need to rewrite (it’s not a game-breaker), but that’ll have to wait un­til I get some free time. I also need to find some free web­space some­where; it seems that the Me­gau­pload de­ba­cle kil­led all the free file­send­ing plat­forms.

• My at­tempts at find­ing semi-sta­ble web­space failed. In the mean­while, for the sake of Nisan’s ra­zor, here is a tem­po­rary link to the .pdf. It hasn’t been fixed yet; the end­ing is not very rigor­ous. I prob­a­bly got a bit too ex­cited near the end.

• May I sug­gest work­ing out the graph­i­cal model ver­sion of con­tin­u­ous time Markov chains as an in­ter­me­di­ate step

(e.g. some­thing like this: http://​​boa.unimib.it/​​bit­stream/​​10281/​​19575/​​1/​​phd_unimib_040750.pdf)

• pa­per-ma­chine_2013 no longer be­lieves any­thing he wrote about this last year, and no longer has the re­sources to start again from scratch, due to im­pend­ing ABD sta­tus.

• LaTeX in Less Wrong

Also, the wiki page on us­ing LaTeX in Less Wrong.

It would be ideal if there were a script some­where that eats an aus­tere LaTeX file and spits out an html file.

• Wouldn’t the right solu­tion be to use MathJax?

• Hm, yes. I don’t think I can do that, though, be­cause I can’t put javascript into posts.

• If LW would up­date the page tem­plate to have the script in the html header, I think we’d be set. Isn’t there a site ad­min for this?

I think this is crit­i­cal, be­cause ra­tio­nal­ity in the end needs math­e­mat­i­cal sup­port, and MathJax is re­ally the de facto way of putting math in web posts at this point.

• Some­one once re­quested that Less Wrong im­ple­ment jsMath, and it seems like it was de­clined. I just sub­mit­ted a re­quest for MathJax. I guess we’ll see what hap­pens.

• That seems to ac­com­plish the same thing as John Maxwell’s util­ity.

I have a file that con­tains text in­ter­spersed with many for­mu­las in LaTeX math mode, de­limited by dol­lar signs or what­ever. I’d like some­thing that will re­place those \$-de­limited for­mu­las with html image tags. I’ll prob­a­bly write one my­self when I need it.

• You would be my per­sonal hero for a pe­riod of time not ex­ceed­ing a week, with at most ten pos­si­ble thirty-sec­ond ex­cep­tions dur­ing that pe­riod.

• I tweaked John Maxwell’s util­ity and came up with this thing. It only works for LessWrong posts, not com­ments.

EDIT: Now it works for com­ments too.

• What’s the goal here? To say “yes, it ex­ists, causal­ity does ex­ist for con­tin­u­ous time?” To use it for stuff? Be­cause if it’s the lat­ter I think a lot of loss of gen­er­al­ity is gonna have to hap­pen, par­tic­u­larly about what all these func­tions at each point in phase space are.

• Time is real, so I’m not a fan of time­less­ness. How­ever, while you have a con­ser­va­tive flow in your model, you can still work your way back to his­to­ries and thus to the re­al­ity of time within a his­tory.

Con­sider how Bohmian me­chan­ics does it. You have the Schrod­inger evolu­tion of a wave­func­tion, with a con­served flow of prob­a­bil­ity den­sity. If you chart a tra­jec­tory through con­figu­ra­tion space ac­cord­ing to the gra­di­ent of the phase of the wave­func­tion, you end up with a timelike fo­li­a­tion of (con­figu­ra­tion space x time) with non­in­ter­sect­ing tra­jec­to­ries. Add the usual prob­a­bil­ity mea­sure, and voila, you have a mul­ti­verse the­ory of self-con­tained wor­lds which nei­ther split nor merge, and in which the Born rule ap­plies.

In prin­ci­ple you can do the same thing with one of those time­less-look­ing “wave­func­tions of the uni­verse” which show up in quan­tum cos­mol­ogy. Here, in­stead of H psi = i.hbar dpsi/​dt, you just have H psi = 0 (where the Hamil­to­nian is gen­eral rel­a­tivity cou­pled to other fields). So in­stead of an evolv­ing wave­func­tion on a “con­figu­ra­tion space of the uni­verse”, you just have a static wave­func­tion. But you can still take the gra­di­ent of psi’s phase, ev­ery­where in that con­figu­ra­tion space, and so you can figure out Bohmian tra­jec­to­ries that di­vide up the uni­ver­sal con­figu­ra­tion space into dis­joint self-con­tained his­to­ries.

In prac­tice, things are more com­pli­cated. In gen­eral rel­a­tivity, you dis­t­in­guish be­tween co­or­di­nate time and phys­i­cal time (proper time). The proper time which elapses along a spe­cific timelike curve is an in­var­i­ant, an ob­jec­tive quan­tity. But it is calcu­lated from a met­ric, the ex­act form of which de­pends on the co­or­di­nate sys­tem. You can rescale co­or­di­nate time, ac­cord­ing to some diffeo­mor­phism, but then you ad­just the met­ric ac­cord­ingly, so that dis­tances, an­gles, and du­ra­tions re­main the same. If you ac­tu­ally try to fol­low the pro­gram of Bohmian quan­tum grav­ity that I out­lined, it’s hard to define the wave­func­tion of the uni­verse with­out reify­ing a par­tic­u­lar co­or­di­nate time, a step which is just like hav­ing a preferred frame in spe­cial rel­a­tivity. I sus­pect that the an­swer lies in string the­ory’s holo­graphic prin­ci­ple, which says that a quan­tum the­ory con­tain­ing grav­ity is equiv­a­lent to an­other quan­tum the­ory that doesn’t con­tain grav­ity, and which is defined on the bound­ary of the space in­hab­ited by the first the­ory. In terms of this sec­ond the­ory, the space away from the bound­ary is emer­gent, it’s made of com­pos­ite de­grees of free­dom from the bound­ary the­ory. In the real world, it’s go­ing to be time which is “emer­gent”, from the “renor­mal­iza­tion group flow” of a Eu­clidean field the­ory defined at “past in­finity”. In fact, ex­cuse me while I run away and study the Bohmian tra­jec­to­ries for such a the­ory…

Any­way, bring­ing this back to Judea Pearl: As soon as math­e­mat­ics rep­re­sents a his­tory as a “tra­jec­tory” in a state space, it is already be­com­ing a lit­tle “time­less” in a for­mal sense. Con­sider some­thing as sim­ple as a time se­ries. You can plot it on a graph and now it’s a shape rather than a pro­cess. You can spec­ify its prop­er­ties in a time­less ge­o­met­ri­cal fash­ion, even though one of the di­rec­tions on the graph rep­re­sents time. In talk­ing about flows on state spaces, I don’t think you’re do­ing more than this. So what you’re do­ing is harm­less, from the per­spec­tive of a time-re­al­ist like my­self, but it also doesn’t re­ally em­body the full rev­olu­tion of Ju­lian Bar­bour’s on­tolog­i­cal time­less­ness, which nec­es­sar­ily in­volves both gen­eral rel­a­tivity and quan­tum me­chan­ics. Gen­eral rel­a­tivity makes proper time a phys­i­cal vari­able, and quan­tum me­chan­ics mat­ters by way of many wor­lds: Bar­bour’s mul­ti­verse is one of “many mo­ments” (he calls them time cap­sules). In or­der to in­ter­pret an un­evolv­ing wave­func­tion of the uni­verse, rather than di­vide it up into tra­jec­to­ries, he com­pletely pul­ver­izes it into mo­ments, one mo­ment for each point in con­figu­ra­tion space.

If you want to imi­tate Bar­bour’s time­less­ness, then the cru­cial step is the on­tolog­i­cal one of deny­ing that the mo­ments have a unique past or fu­ture. But if you have a con­ser­va­tive causal flow, you can always string the mo­ments to­gether into spe­cific his­to­ries, like the Bohmian tra­jec­to­ries. Tech­ni­cal difficul­ties for the defi­ni­tion of tra­jec­to­ries only en­ter for rel­a­tivis­tic sys­tems, be­cause you want to avoid reify­ing a par­tic­u­lar co­or­di­nate time. But for non­rel­a­tivis­tic sys­tems, it looks like for­mal time­less­ness in a causal model (in the sense you de­scribe) is just a change of per­spec­tive that’s always available and can always be re­versed.

• Time is real, so I’m not a fan of time­less­ness.

The point of time­less­ness is not to say that time is un­real, merely that it is su­perflu­ous.

It’s difficult for me to fol­low your com­ment. While I’m fa­mil­iar with the the­o­ries you dis­cuss (with the ex­cep­tion of string the­ory and quan­tum cos­mol­ogy), I don’t see how some of them are linked to this. I’m not try­ing to do any­thing so great as unify quan­tum me­chan­ics and gen­eral rel­a­tivity.

As soon as math­e­mat­ics rep­re­sents a his­tory as a “tra­jec­tory” in a state space, it is already be­com­ing a lit­tle “time­less” in a for­mal sense.

Yes.

Con­sider some­thing as sim­ple as a time se­ries. You can plot it on a graph and now it’s a shape rather than a pro­cess. You can spec­ify its prop­er­ties in a time­less ge­o­met­ri­cal fash­ion, even though one of the di­rec­tions on the graph rep­re­sents time. In talk­ing about flows on state spaces, I don’t think you’re do­ing more than this.

Time is no longer “one of the di­rec­tions on the graph”. If you fix a tra­jec­tory, then it comes with it’s own time, but the more in­ter­est­ing ob­ject is the flow, which does not have any sense of time.

We agree that what­ever I’m do­ing is mostly harm­less.

So what you’re do­ing is harm­less, from the per­spec­tive of a time-re­al­ist like my­self, but it also doesn’t re­ally em­body the full rev­olu­tion of Ju­lian Bar­bour’s on­tolog­i­cal time­less­ness, which nec­es­sar­ily in­volves both gen­eral rel­a­tivity and quan­tum me­chan­ics.

That will have to wait for some­one else. I haven’t read Bar­bour, and it sounds hor­rifi­cally difficult.

But for non­rel­a­tivis­tic sys­tems, it looks like for­mal time­less­ness in a causal model (in the sense you de­scribe) is just a change of per­spec­tive that’s always available and can always be re­versed.

Prob­a­bly. But such a thing could still be worth­while.