A Note on Lavos' Shell, Dimensional Travel, and 4D Space

[Jack Kieser]

After reading up on the "paradox" of Lavos' shell and his 1999 AD emergence / 12,000 BC demise, I have come to the conclusion that there is no paradox.  I think we're simply applying theories of time travel (time bastard and TTI) to dimensional travel mistakenly.

Allow me to elaborate.

Here is an illustration of 2D space.




Note that in order to move between points B and C, you only move along one axis, while moving from either B or C to point A results in movement along 2 axes.  This is in line with everything we know about spacial movement.

Now, let's look at 3D space.




Movement between points A, B, and C are preserved as-is from our 2D plane.  Movement between points B - B' and A - A', however, represent 3D transformations.  As I'm sure you all understand so far, this is like jumping in place and is not a protected act; all transformations here are basic 3D planar movements.

These three axes represent three dimensions of planar movement.  Let's add a 4th axis.  This axis, Omega, will represent a 4D space (length, width, depth, and time). 



Each point along axis Omega represents another 3D space.  This is what we call a timeline.  All of the transformations from our 3D space are preserved; they all operate exactly the same.  Any movement parallel to the Omega axis, however, represents a movement along the 4th dimension (time).  These acts, as we know, are protected.  Chrono Compendium has explored all of the mechanics of travel up until (and including) the 4th dimension.  Here's where things get hairy.

We assume that Lavos creates a "pocket dimension" when he arrives on Earth in 65M BC.  The very concept of a "pocket dimension", however, is flawed!  You see, we can represent movement along a single axis as a line; I can move from a point at offset 1 to a point at offset 12 on a line easily.  If I want to move any other way, however, I can't simply explain that this second point exists on a "pocket axis" that somehow inhabits the same axis but not; I have to create another axis to account for this difference in movement, and thus I create 2D space.  This concept applies to dimensional travel.  If we want more space to exist, we have to have room for it, and so to account for travel in 5 directions (length, width, depth, time, AND dimension), we would need to create a 5D space by adding another axis. 

This other axis will represent all of the possible combinations of time and space parallel to each other.  One more illustration.



Note that I have purposefully offset the two lines from each other.  The reason for this is that two dimensions need not be equal.  In 2D space, two points are referenced by their relation to each other and their surrounding space.  This offset is the only relationship between spacial points.  Much in this same way, two dimensions are only related by their offset from each other; nothing else needs to be equal.  The two dimensions need not have the same mass, volume, energy... or time.  Time in one dimension is free to run at a rate inequal to the rate of time in any surrounding  dimensions.  You see, all of time, in 5D space, is condensed into a single point (each point then becoming a dimension with its own personal timeline).  If we think of dimensions this way, then Lavos' non-emergence in 1999 AD when killed in 12,000 BC is no longer a paradox because dimensional travel is not equivalent to time travel; the two kinds of travel are along separate and different axes!

When Lavos burrows into his "pocket dimension" (just a separate dimension from Crono's), he is breaking from 4D space into 5D space.  He isn't traveling along the Omega axis, and as far as we know, only travel along an Omega axis is protected by TTI, since the Omega axis is the only axis that doesn't act as a spacial axis.  Separate dimensions, however, are spacial in nature (they are offsets of space, not time) and so are not subject to TTI.  When Lavos exits his dimension in 1999 AD, he is traveling along the 5th axis again.  You see, all points in time are condensed when moving from 4D to 5D space, so when traveling 5ht dimensionally, you have access to ALL points along the Omega axis.  You never move ALONG it.  Only this movement along the Omega axis is protected (as far as we know), so Lavos' 1999 AD emergence is never protected by TTI! 

Now, how about Lavos' shell?  Again, when Crono and friends travel to Lavos' dimension to fight his shell, they have access to ALL points in time at once; the point of entrance for Crono's party is completely arbitrary.  They have engaged Lavos in his own dimension, however, so when they defeat his shell, it is along his own timeline (an event logged on Lavos' Omega axis).  This act isn't protected either, though.  If Lavos had the ability to travel in time, he could theoretically go back ALONG HIS OWN OMEGA AXIS and prevent Crono's party from defeating his shell, thus allowing him to return to Crono's dimension with his shell intact; why he doesn't, we don't know.  All we know is that when Crono and friends return to Lavos' dimension, they arrive AFTER the arbitrary time they did the first time around.

This allows for Lavos to lose his shell AND his 1999 AD emergence when Crono's party defeats him in 12,000 BC.

[FaustWolf]

Welcome tot he Compendium! I'm moving this to the "Time, Space, and Dimensional Travel" analysis forum for rigorous discussion.

[Jack Kieser]

Thank you very much; long time lurker/reader, first time contributor.  This has been eating away at me for at least 2 weeks now, so I'm glad I could get it out.

[Zergplex]

This seems to be a very interesting theory, my first question is if you have played Chrono Cross and how this theory would hold up against the enhanced amount of dimension hopping you do in that game?

I'm pretty sure that someone had made a case for the inclusion of Dimensional Travelers Immunity/Dimensional Bastard as well. You may want to take a look at those if only for reference, I think I'm about to before I make a larger post about the good/bad parts of this theory.

[Jack Kieser]

Although I never played all the way through Cross personally, I have read all of the time/dimensional travel theorems and essays on CC multiple times; I didn't notice anything that really dealt with the concept of axial dimensional travel like this.  Most of the things I read just treated dimensional gates as having the same properties as time gates and ascribed the same effects onto both kinds of travel.

I'm really hoping some of the more versed people here can shed some light on the subject, though.

[chrono eric]

Zergplex is referring to me and Eske's epic discussion in:

Time Devourer's Defeat Undoes the Fall of Guardia and Lucca's Death
http://www.chronocompendium.com/Forums/index.php/topic,6571.0.html

In which we proved Dimensional Traveller's Immunity and Dimensional Bastard have to exist to avoid paradoxes in the timeline. But I'm not necessarily sure they would apply here to your theory.

I'm having trouble seeing how this 5D view of time and space is different from the 5D view of time and space that me and Eske refer to as "Time Error" in many of our analysis posts, which deals with a 5D move through time and space that is associated with a time travel event to preserve continuity to the timeline. At each point on the "Time Error" or 5D time axis, there exists an entire self-contained universe and self-contained timeline.

For example:

Time Error 0 - or - 5D Time point zero: A time traveller time travels at Time X to Time X-100.

Time Error 1 - or - 5D Time point one: The time traveller emerges at Time X-100. He disappears at Time X later on due to TTI.


In this way, you can see that when individuals time travel, they make a 5D move through time and they never actually emerge in the past of their home timeline. This is what we have been using to analyze a great many things here recently, including Dimensional Bastard and Dimensional Travellers Immunity, and it seems to be very similar to what you are proposing. Could you perhaps elaborate upon the differences between your theory and ours as it relates to time travel (not to Lavos' pocket dimension, as that theory is pretty much abandoned now anyways)?

[Jack Kieser]

Hmm... well, I'm not very much on the up-and-up when it comes to CC's events.  If I understand properly (and feel free to correct where necessary):

*Dalton travels from Zeal to 1000 AD, strengthening the Porre army
*the Porre army defeats Guardia

These actions are standard time travel, and so it's pretty obvious that TTI protects them.

*there are two dimensions: Home and Another
*in both dimensions, Luccia's orphanage burns (an event preceding the split)
*Vann from Home World dimension travels to Another World
*Home Vann travels back in time with Serge to save Kid from a fire
*Another Vann does no traveling

The question is what events, under my theory, are preserved.  According to my theory, the act of Home Vann traveling to Another World is NOT protected, as Home Vann does not travel along an Omega axis to do so.  His act of traveling to save Kid, however, IS protected.  That means that, because of TB and TTI, at the time of the Kid saving, regardless of the state of Home Vann (let's assume for whatever reason that Home Vann is sent to the DBT), Home Vann will appear to save Kid.  This seems pretty basic so far.  You may ask, though, how Home Vann would show up if dimension hopping isn't a protected act.  I can explain that pretty simply, actually.

Refer to my 2D illustration in the OP.  Let's assume that the points on the plane are balls of energy.  Remember that energy in a closed system must remain constant, and that no system can be perfectly closed; though matter cannot be transferred in a closed system, energy can, and energy is just matter in a different form.  Back to the 2D plane.  Would it be a stretch to say that, theoretically, a ball of energy moving from point B to point C is a series of additions and subtractions of energy from space?  I don't think so; energy is taken from point B and added (placed) in point B+1, then the same in the transition from B+1 to B+2, and so on and so forth until the energy reaches point C. 

Let's cut a few steps out for time's sake and move straight to 4D space.  Some people say that time travel breaks conservation of mass/energy because of a lack of causation, but instead of viewing each particular moment in time as a closed system, let's view the entire dimension (including all points along the Omega axis) as a single closed system!  Thus, time travel is simply removing energy (or matter) from one point on the Omega axis and adding it to another point on the Omega axis.  All we're doing is a -1/+1 function, so no matter is created or lost. 

There is a reason that Time Bastard must exist; because systems are closed and matter/energy can only change forms, not be created or destroyed.  When an item is TB'ed, instead of the dimension sending the energy to the DBT (which would equate to destruction, or energy leaving the system), it is just being subtracted from whatever point it is currently at and added to the destination point on the Omega axis.  This is why acts of time travel are protected; TB acts as the mechanism for preserving the system's total energy.  Remember, though, that no system is perfectly closed.  This is why dimensional travel is possible at all, otherwise energy/mass could never leave its home system (dimension).

When we move to 5D space, we can view, much like we did with the move to 4D space, the ENTIRE 5D plane as a single closed system (with all of the systems comprising it reduced to singularities).  When something moves from one dimension to another, however, it's still only moving spatially.  This is why dimensional hopping cannot be protected acts; energy isn't moving through time, it's only moving through space.  What you described with the dimensional traveler hopping dimensions and then traveling in time in the hopped-to dimension shows 5D movement, and if we think of the DBT as another singularity in the 5D plane, connected to all other points (instead of having a separate DBT for every dimension) it's very plausible to assume that, in order for the traveler's actions to be preserved al-la TTI and TB, the energy used to recreate him in the second dimension's past could come from the DBT, which is already a repository for extra energy not directly in use in any of the other closed systems comprising 5D space.  Because none of the 5D points can be perfectly closed systems, we can see a natural mechanism for one of the dimensions being allocated some extra energy that will later be sent back to the DBT, much in the same way a computer allocates global system memory as needed to many separate, closed program strings.

We can also see other mechanisms like this.  If we assume that, instead of the DBT being a repository of data it is a repository for pure energy, we can assign a very important function to individual systems (dimensions):  the allocation of energy in SPECIFIC FORMS.  In the example of the dimension-hopping time traveler, dimension 2 (with the preserved act of time traveling) would need a copy of the traveler regardless of where (or when) he is in 5D space.  He can't just appear, but he can't just come from the energy in dimension 2's system, because it is "closed" by default and, in theory, all energy is allocated somehow.  It's not a simple -1/+1 function this time because the traveler's energy is not held in the home system.  If the DBT is an energy repository, however, the DBT can send raw energy not in use by any other dimension to dimension 2, to be assembled on-the-fly in dimension 2 into the traveler.  When his deeds are complete and he "leaves" for home, dimension 2 disassembles the traveler back into unused energy and returns it to the DBT.  This can be done if all of 5D space is a system unto itself.

[Eske]

@Jack Kieser:   Hey welcome!  Great post - I'm glad to see more people in this section. 


@chrono eric:  His version of 5D is spacial unlike our version of "time error" which is a temporal dimension.  For this theory, Lavos has access to all 4D points from 5D space, but also experiences a personal time.  This personal time would be like our version of "time error".

Back@Jack Kieser:

The question is what events, under my theory, are preserved.  According to my theory, the act of Home Vann traveling to Another World is NOT protected, as Home Vann does not travel along an Omega axis to do so.  His act of traveling to save Kid, however, IS protected.   That means that, because of TB and TTI, at the time of the Kid saving, regardless of the state of Home Vann (let's assume for whatever reason that Home Vann is sent to the DBT), Home Vann will appear to save Kid.  This seems pretty basic so far.  You may ask, though, how Home Vann would show up if dimension hopping isn't a protected act.  I can explain that pretty simply, actually

While I understand what you mean, his appearance in the other dimension disturbs everything around him - certainly not limited to saving Kid.  Therefore he would "appear" as soon as he changes anything in the Omega axis.  I believe you are saying that while travelling between dimensions is not preserved, changes to the Omega axis are preserved, so essentially he will appear as if TTI (or DTI) protected him.

As a rebuttal, I produced a somewhat extreme example in another thread to support DTI.  Here is an edited version of it (corrections) ::

Professor A invents a Time Travel "gate key" and can open time gates anywhere.
Then, 10 minutes later, he invents a machine that creates a Dimensional Vortex that exists in all time periods (like CTDS) and is fixed in place, and can only exist if the machine is operational.
Professor A can lay time gates perfectly on top of the Dimensional Vortex to use both simultaneously.
Professor A indeed uses his devices simultaneously, immediately after both are invented.

From Dimension 1, Time X,   Professor A travels to  Dimension 2, Time X-10.
There, he tells a resident, Person B, to traverse dimensions  to Dimension 1 and kill Professor A before all means of extra/other spatial travel were invented.   

Person B at D2: Time X-10 travels  to     D1: Time X-10 and kills Professor A just seconds after Time Travel is invented.  Dimensional Travel is never invented.    Note that Person B has not time travelled at all.

Now lets add  "1" to Time Error and play this through again.

A (time) gate will open in Time X-10 in D2 no matter what. Professor A (who, by TTI, will always remember inventing dimensional travel as well) will appear 10 minutes into the past, meet the counterpart of Person B and tell him to cross dimensions to kill Professor A.  Professor A will be shocked to see that the dimensional vortex no longer exists.   (If the machine is rendered inoperable, the vortex ceases to exist in all time periods.  like how the CTDS  portals only exist in all time periods for Time Error Y+1 after they are created at Time Error Y.)

That means that no version of Person B crosses dimensions (NOT time) to kill Professor A.  If DTI did not exist,  this would create a loop because if Person B can't kill Professor A, then dimensional travel will exist --  but then Person B would now be able to kill Professor A, negating the existence of dimensional travel - and so on.

[chrono eric]

Dammit Eske I was just about to post that  . Now I have nothing to say of substance right now.

[Jack Kieser]

Alrighty then.  Back to work.

First, I'd like to go into the topic of light cones, just for clarity's sake (just in case anyone in the future reading this hasn't read up).  The concept of a "light cone" is basically the range of effects (in the shape of a cone, geometrically) that a photon of light could possibly have, with the tip of the cone at T0 and anything after that T1, T2, etc.  The accepted theory is that (like you said) ANY changes at all, be it the effects of a person or of a single photon of light, would have a measurable effect on anything within its light cone, instantaneously causing TTI. 

Secondly, we need to discuss the concept of predetermination in CT.  Is there predetermination?  The simple answer is no.  You see, none of the changes made to Crono's timeline actually take place until one of the party members actively effects the timeline.  If this wasn't the case, then CT would not be able to take place.  For instance, it would be impossible to see Lavos destroy the world in 1999 AD from 2300 AD because 2300 AD would never become an apocalyptic future; Crono and Co. would have saved the future in 12,000 BC, so when they traveled to 2300 AD, they'd arrive in the saved future (predestined to be saved back in 12,000 BC).  Thus, no effect is ever observed in a CT-style timeline until an action has taken place.

It is for these reasons that I don't think that interdimensional travel cannot be protected and we cannot apply the light-cone theory of photonic activation of TTI during interdimensional travel because doing so would naturally create an even bigger paradox than Lavos' shell or the 1999 AD discrepancy:  TTI-affected travel TO THE FUTURE.

You see, we know that travel to the past activates TTI, but why?  Past and present are only offsets; their meanings are solely determined by perspective and relativity.  We can't say "only time travel to the past activates TTI" because whose past are we talking about?  The travelers?  It's too arbitrary.  We'd have to include ANY instance of light-cone activation, which would include moving to the future (both directions along the Omega axis).  But if that was the case, then Crono and Co.'s arrival into the ruined 2300 AD future would be a protected act!  If the ruined future is sent to the DBT, how is this the case?  It would create a forward paradox:  Crono is to arrive and take actions (protected actions) in a future that doesn't exist.

Thus, we can't just say that any light-cone activation automatically is a protected act, because the events of Crono Trigger violate this (the Moonlight Parade ending is a clear illustration).  What if (for the sake of argument) we define TTI differently?  What is the purpose of TTI?  Expressly to suppress paradox.  Is travel to the future an act that creates paradox naturally?  Simply put, that would be impossible.  Remember, events are not predetermined in the CT universe.  If I travel to the future, there is no guarantee that anything there will happen for sure.  Let's say that I go from 1000 AD to 1050 AD and record all of the results of sports matches for the previous 50 years, then return to the exact point that I left 1000 AD.  I start betting on sports matches that I (for the sake of argument) would have never bet on before.  The future I went to in order to get those sports results is automatically changed because I'm taking actions that may (or may not) have far-reaching effects on the future.  I could win big on a particular team, which (after finding out about me from reading the "Kid Strikes It Rich On Sports Bet" headline in the paper) effects future outcomes (maybe spurring the team to do better, making them win a match that I had originally recorded as a loss).  If TTI effects future travel, we have a problem, since I'd have to go and record the outcomes of games that may have actually turned out quite differently due to my interference.

Now, since events cannot be predetermined, we could possibly define TTI as immunity or protection given to acts whose absence could cause break in causation (or a paradox).  This way, travel forward cannot be included, since nothing is predetermined (you can't contradict an action that hasn't taken place yet), but any travel in reverse can, since the light-cone activation in the past could have drastic effects (or even incredibly minimal effects) that could possibly prevent the traveler from going to the past in the new timeline. 

If we think of TTI as a natural response of the universe to protect from paradox and breaks in causation, then we see why inter dimensional travel cannot be protected by light-cone activated TTI... since you aren't traveling along the Omega axis backwards, you're effect on the dimension isn't breaking already established causation in the same way that traveling in reverse would do (nothing you could possibly do at that point could effect your future actions after your entrance into the dimension, since you're acting them out for the first time).  If you then (after entering the dimension) travel in reverse, your past light-cone could effect you when you enter the dimension for the first time, causing you to act differently and possibly never travel into the past, which could cause a paradox, what TTI is supposed to suppress.

I'll try to elaborate more in the morning when I'm not exhausted.

[ZeaLitY]

Just for consideration, what about the End of Time? The End of Time is definitely on a scale of Time Error (Gaspar and Spekkio both observe history from there and the party's actions), and so their entry to the End of Time shouldn't be protected by TTI. They enter the End of Time in 2300 A.D. only after witnessing the Day of Lavos recording, and at the end of the game, they also recall events and memories that were only created through the adventure as assisted by their visits to the End of Time.