# Is three-dimensional time consistent and paradox-free for time/dimension travelers?

This question was edited based on answers already submitted. I am designing a universe with both time travel and alternate realities, but I don't want the whole thing resting on hand-waving and paradox. I understand just enough theoretical physics to get in trouble. To resolve the paradoxes I see with the whole thing, I have put together three dimensions of time, and use discrete time states (universes) so time does not blur together. Time is treated as a three-dimensional analog to three-dimensional space.The entire multiverse is treated like a time object, so the fact it gets wider or thicker doesn't violate conservation. The expanding multiverse displaces existing timespace,so nothing is created, only transformed from unfixed (assumedly high energy) pluripotent reality (primordial chaos) to fixed reality. My understanding is incomplete as to if this should work and not violate generally-understood current theoretical physics, but I need a reality-check to be sure it works. Does this violate physics, and if not, are there paradoxes intrinsic to the design? If this is not the correct place to address this question, please tell me where that is instead of just shutting down the question.

1. Causality: The first dimension of time is causality (linear time). Every universe is a one-dimensional string. Cause follows effect in a classical "timeline." You can't alter your own timeline. Each moment of time represents a discrete time state and a separate strand of time. These strands may differ in spacetime as well as timespace; I'm not sure how it would exactly behave.
2. Synchronicity: The second dimension is synchronicity (horizontal time). All strands are set apart by a moment of time, off-set in a diagonal, so all points of time are happening simultaneously in separate discrete time strings. Somewhen a universe is being born, and theoretically somewhen the last energy of a universe is draining away. The Now is like a wave moving across the strings. You can travel to "your" childhood and change things, but the changes only affect that discrete strand, so when you go home your universe is still the same.
3. Probability: This is my least worked-out dimension and the least critical for plot (vertical time). All observable possible events happen, and the more they are observed, the thicker the observed universe gets (a bit like Schrodinger's cat, a new universe is created where each event happened).
• So, what is the meaning of an event happenning at $x = 17$, $y = 21$, $z = -4$, $t_{\text{causal}} = 102$, $t_{\text{synchro}} = 45$, and $t_{\text{prob}} = - 33\,$? How do you make tridimensional time compatible with plain old boring classical physics? What does the new sexadimensional spacetime look like? The question is full of lofty words, but short on the actual description of how tridimensional time actually works. For example, what's the meaning of the word "simultaneous" in the clause "all points of time are happening simultaneously"? (The usual meaning is "at the same time"...) – AlexP May 17 '20 at 18:16
• One of my chief worries in this idea is that it was incompatible with conventional physics, like gravity, black holes, etc. thus the question. I just dont understand the math well enough to tell if it's nonsense, or if there is underlying principles that allow/account for these variables. – DWKraus May 17 '20 at 21:12
• Why not simply have these other timelines existing at different points along extra spatial dimensions, like a stack of 2D flatlands parallel to each other in 3D space? If you assume some timelines are just like ours but ahead of or behind us chronologically, and there were a way to leave one's own timeline and travel to another, this would allow for a type of apparent time travel, without the causality problems that come from having multiple timelines. – Hypnosifl May 18 '20 at 5:05
• But if the strings are imagined as parallel, the dimension running parallel to each string can still be imagined as time, I'm just suggesting that the "sideways" dimension used in hopping from one string to another can be seen as an extra spatial dimension rather than an extra time dimension. In terms of the math this will avoid the issue of causal loops which seems inevitable when you allow multiple time dimensions in relativity (see my comments here) – Hypnosifl May 18 '20 at 6:57
• Your words indicate vacuousness hiding behind a smorgasbord of scientific terms. I fear your descriptions are inconsistent with themselves, let alone the relationships between the 3 "kinds of time." Time is most definitely not analogous to a spatial dimension, in particular because one cannot control passage of /thru time. This means there's no possible way to map multiple time-dimensions into anything meaningful in a universe that has spatial dimensions. – Carl Witthoft May 18 '20 at 12:54

@DisasterlyDisco raised some very good points regarding the dimensionality of space-time and @AdrianColomitchi also points out fundamental issues with the presentation. With regards to this I'd like to elaborate while providing some insight into how to properly use the terminology of quantum mechanics.

# Intro

In physics, words like "state", "dimension" and "quantum" have very specific mathematical meanings. Thus a system based upon these concepts used incorrectly, to the trained ear, will not sound coherent and hence not realistic.

In quantum mechanics, a "state" refers to a mathematical formation known as a wave-function. The wave function contains information about the probability of that system having particular quantum number. These quantum numbers correspond to things called observable, which we can measure.

Now this is all very abstract (quantum mechanics is notorious for this), and so here's an example which hopefully will clear up how to use the terminology.

# A basic example: demonstrating how to apply quantum principles

Imagine you just have one particle: If this were classical mechanics its easy, you can describe everything about this physics of the particle with a few quantities: where it is at :(x,y,z,t), its momentum ($$p_x$$,$$p_y$$,$$p_z$$) and its total energy: H = T+V, where T is the kinetic energy you can get from its mass and momenta, and V is the potentials that the particle is under the influence of. While this seems complicated, once you know all these quantities, you know everything there is about the particle, furthermore once you know these things, you can predict exactly how the particle will move: its velocity and its acceleration.

This is not true of quantum particles, which means that they must be described with wave-function. Lets look at a really simple example:

Consider just one quantum particle which is not under the influence of any potential. We would describe it with a wave function, to really simplify it lets restrict its motion to one dimension (the x axis):

$$|\Psi \rangle$$

So what is this particle doing? The answer, we don't know, but we can make an educated guess. For starters the particle might be propagating (or moving) in either the +x direction or the -x direction, so we say that its total wave function is the super position of these two "states":

$$|\Psi \rangle = |\phi\rangle_+ + |\phi\rangle_-$$

Now, this does not have the meaning (which is unfortunately the most pop-sci way to talk about this) that the particle is moving both in the +x and -x directions, what this means is that the particle has probabilities of moving in each of the +x and -x directions, and we can't know for sure until we measure the particle.

So we have the wave function, but where is the particle at. I don't have a x coordinate yet. How do I get it? The answer is that its buried in the wave function. So to get it out of the wave function you apply the position operator and perform what is called the "inner product" over an interval [$$x_a$$,$$x_b$$]:

$$\langle \Psi|X\Psi \rangle$$

Now I know the probability that the particle is between $$x_a$$ and $$x_b$$. But I still don't know where it is at. That's because that as good as we can get with quantum mechanics.

$$\langle \Psi|P\Psi \rangle$$

Don't worry about the math of how to do these calculations. That would require taking a university level course in quantum mechanics, and fortunately is not necessary to understand how this work.

So if we can't know the energy here, how can we ever talk about energy in quantum mechanics if we can only get probabilities?

The answer to that is the mystery of quantum numbers: in both quantum and classical systems particles can be anywhere. But in quantum systems, unlike classical systems, they cannot have just any 'ol energy value but must contain specific units of energy which may increment/decrement in discrete levels.

Again however, we can't know for certain the particles exact value, but only until we "measure" it. Measuring doesn't have to involve a scientific instrument, it just refers to an interaction the particle undergoes which collapses the wave function, and the "actual" values can be observed.

Now if you were keen on the math you might have noticed something a little strange here; I stated that taking the inner product of the wave function, when an observable operator gets applied, returns a probability for the observable. But in our example above, the inner product of the wave function would look like this:

$$\langle \Psi | \Psi \rangle$$

which would expand to this:

$$(\langle\phi|_+ + \langle\phi|_- ) (|\phi\rangle_+ + |\phi\rangle_-\rangle)$$

and expanding completely:

$$_+ \langle\phi|\phi\rangle_+ \ + \ _+ \langle\phi|\phi\rangle_- \ + \ _- \langle\phi||\phi\rangle_+ \ + \ _- \langle\phi|\phi\rangle_-$$

Uh-oh...

Our math only works if, under the implication that the states are orthogonal $$\ _+ \langle\phi|\phi\rangle_- \ = \ _- \langle\phi|\phi\rangle_+ = 0$$

But +x and -x are clearly not orthogonal, what gives.

This is because the states of the particle moving in these directions are orthogonal not the actual directions themselves.

# How this applies to the question

I have put together three dimensions of time, and use quantum time states so time does not blur together

Using the example as reference, quantum states refer to wave-functions and not to physical dimensions such as x,y,z. While wave-functions incorporate physical dimensionality into their construction to be sure, they they only apply to the quantum particles themselves. As shown in the example, qunatum mechanics does not imply that individual states blur together until observed, but serve to provide probabilities regarding the actual values of the observables such as position and momenta.

We would not talk about a quantum "time state", just like there is no quantum position states or quantum energy state. There is a quantum state, which may be a super-position of individual mathematically "orthogonal" quantum states.

The entire universe is treated like a time object, so the fact it gets wider or thicker doesn't violate conservation

There about 16 conservation laws of which 6 are always true. If a conservation law is broken, we say that there is conditions for the law to be violated and we refer to this as symmetry breaking. So what is a "time object"? And which conservation laws does being a "time object" prevent the universe from breaking?

My understanding says this should work and not violate generally-understood current theoretical physics [regarding the three dimensions of time]

The usage of the word dimension here, is not consistent with the mathematical definition. The mathematical definition of dimension reference to the most basic element required to define a single point within a mathematical space. Your three "dimensions" are really three restrictions on how time works in your story, lets examine each one of them:

Causality: The first dimension of time is causality. Cause follows effect in a classical "timeline." You can't alter your own timeline. Each moment of time represents a quantum time state and a separate strand of time.

I agree that to form a logical system, you cannot violate causality. Not allowing your characters to alter their own timeline would be a good way to prevent causal paradoxes. However, the last sentence, doesn't really make sense as regards the usage of the physics language. See above regarding the meaning of "quantum state". To illustrate, a moment of time would be a meaningless formation for a quantum state, equivalent to saying "the quantum states of x = 3".

Synchronicity: The second dimension is synchronicity. All points are offset by a moment of time, so all points of time are happening simultaneously in separate quantum time strings. You can "travel" to your childhood and change things, but the changes only affect that time state, so when you go home your universe is still the same.

A "moment of time" as measure by whom? All time is relative to specific frames of reference in our own universe. This does not mean that time does not obey certain rules, it certainly does, but these rules allow for different frames of reference to measure time intervals differently than one another.

What is a "seperate quantum time string"? Is this a fancy way of referring to one universe within the multiverse? If so, how are these multi-verses connected. Look-up Brian Green's options for how multiverses can exist if you are interested in how to construct a multi-verse which doesn't violate the known laws of physics.

Probability: This is my least worked-out dimension and the least critical for plot. All observed possible quantum events happen, and the more they are observed, the thicker the observed universe gets.

What is the observed universe? This is a way of saying multiverse, or talking about the literal observed universe, which IRL consists of all the stars and galaxies we can see from earth. Does this mean that the actual universe somehow gets new universes appended onto it? From the previous descriptions it seems that you want multiverse.

Is the meaning that all possible quantum events occur, just not in the same universe, but spread through out the multiverse? If so, then I'd consider the implications of this. For instance, if a probability of a quantum number, as determined by a quantum state consisting of a basis of two states, is 0.3 for state A, versus 0.7 for state B, does this mean that in one universe state A is the actual state and for another that state B is the actual state (two overall). Or, does this mean there are seven universes with state B, and only three for which state A is the actual.

Also remember, that quantum states are very distinct from human decisions (for which the physics is currently unknown). Common trophes might regard monumental and pivotal moments in history: for instance, a president ordering a nuclear strike. In one universe, the strike is called off at the last moment and the world of tomorrow appears a few years later, in another universe the planet is in the depth of a nuclear winter with human life on the brink of extinction. While it sounds good, even if the multiverse hypothesis is correct, quantum physics does not support these kind of outcomes resulting from the collapse of quantum states.

Another similar concept is found in "random" events being cast as quantum events. So for instance, the villain has the hero on his knees and is ready to vaporize him, but decides to be "merciful" and toss a coin to determine his fate. The coin lands heads and the hero lives to fight another day, but the evil mastermind cackles...knowing the hero's been vaporized in another universe...Except this isn't correct at all. Tossing a coin is a classical event, which only appears random from normal level of baysian inference and is not decided by collapsing quantum states. To fully explain this would require an answer all of it's own.

The expanding universe displaces existing timespace, so nothing is created, only transformed from unfixed pluripotent reality (chaos) to fixed reality.

How is time-space transformed, allowing multiple actualized states to exist simultaneously with the creation of independent realities? And how does this not violate the first point regarding causality? What is the mechanism by which the "unfixed pluripotent reality" becomes "fixed reality"? Does this occur for every quantum event? And if so, how are the other "time strings" kept separate?

# Conclusion

I hope this was just enough quantum mechanics to help you to understand the terminology and demonstrate how the principles are applied without being bogged down with too much math.

• That's pretty thick, but it is much more precise than how I discussed it. Like I said, I understand just enough to kinda understand and get in trouble. Causality in my model is divided into distinct universes (strings) set apart by equal portions (distances?) of time (I'll call them moments). 'Chaos' (primordial, not physics) is a bit like an unobserved particle that can be in any state until fixed, and a particle breaking down or not fills a portion of chaos as a new universe takes shape where both possibilities exist (chaos existed before so no new matter/energy are created) – DWKraus May 18 '20 at 4:29
• This is very helpful in terms of using the appropriate terms to describe what I'm trying to say. There's a lot here, I'll see about revising the question in the morning. – DWKraus May 18 '20 at 4:34

Based on your post it is really hard to answer your question succinctly. This is mainly because what you are describing is big, broad, and vague enough to be somewhat flexible. None the less, let me try to give my answers and then proceed with some clarifying questions of my own.

First and foremost, you ask whether or not your idea violates physics, and to that I'll have to answer No, but also maybe Yes. What you are describing, a 3d time-space (for lack of a better word, not to be confused with space time), is not violating any known physics in so far as it is vague enough that I can't apply my understanding of physics to debunk it. There is enough though that I see some areas that can be clarified and fleshed out better, so that we can arrive at the point where we can apply some physics to it. So let's get to that.

You call this a 3 dimensional time, analogous to 3d space. There are two points I want to hit on here. First, your three dimensions per your descriptions don't seem to span the same "space". Each of your axis in this timespace seem to describe different effects and they are not interchangeable. Compare this to classical 3 dimensional space where we can't truly discern directions as all three dimensions are the same in function. In this way the dimensions as you describe them are to each much more like what the time dimension is to the three space dimensions in 4d spacetime. Both space and time are part of the same spacetime, but the time dimension is discernible from the space dimensions. We can not truly discern left and right from up and down, but forward and backward in time is different in some ways from left and right. Because of this we also call our spacetime a 3+1 dimensional spacetime, and in this way your idea to me describes like a 3+1+1+1 space-causality-synchronicity-probability, more than a 3+3 space-time. (This kinda starts leading down the rabbit hole of parity and time symmetry, which is interesting if unrelated to relativity)

On the note of spacetime, I will put forward my second point. You liken 3d time to 3d space, but do not really touch upon the more modern understanding of our 4d spacetime universe. We know that space and time are just facets of the same spacetime and that transformations in one invariably leads to transformations in the other. So, how does your idea conform to 4d spacetime? How does your time react to gravity and acceleration of massive bodies? As you describe it, your causality dimension seems to be the most analogous to modern time and could easily be shoehorned in as the time dimension in spacetime, but what of your two other dimensions? How would they react? If your setting uses 4d spacetime as a basis (and i don't know if you want to do that, but if you do) then I think you should ponder how all your time dimensions and the classical space dimensions mesh together in a cohesive whole.

It is from the above two points that the "maybe yes" stems as how your setting works within these would determine whether or not they are in violation of physics, as you put. Do note that time is a stupid beast we aren't done wrangling, and that it is one the main reasons why we a still stuck without a joint operation between general relativity modern particle physics. If you truly find something that pleases both then publish that bizz with all haste. And if not, then don't pull out your hair over it.

As for paradoxes I can't really help you yet. It seems like you at least avoid the grandfather paradox with your Synchronicity dimension if I understand it correctly, but honestly it is hard to find paradoxes at the state of your description. I would love to look for them though if you feel like expanding on your idea.

Finally I am curious what you your concepts "time object" and "quantum time state" means in regard to your setting. I would also like to hear more about what you mean about time blurring together and why that is a problem.

This has been a bit of a ramble, I'm sorry about that, but I hope some of it was useful. It is hard to rally talk about the physics of time when that is very rarely the focus of physics. I would love to hear more about your setting though and hope that you will divulge more of your ideas behind your 3d time setting - I love a good time travel story :)

• In my setting, my concern is that if you treat space like a three-dimensional object, altering one part (say, traveling back in time and disrupting a historical period) would distort time around it, like a black hole bending space. With discrete quanta of time, a change to one would have little effect on the others. As to gravity, the problem would come if substantial objects like black holes were in different positions in different quantum states, as the "bend" to time would not match from one state to another and time could overlap as one is distorted in time and another isn't. – DWKraus May 17 '20 at 23:35

reality-check, eh? Nope, the "time dimensions" that you listed can not be independent.

Synchronicity: ... You can "travel" to your childhood and change things, but the changes only affect that time state, so when you go home your universe is still the same

That travel violates the conservation of matter/energy - the atoms that make up you-the-traveller already exist at destination in other combinations.

If you , in your "synchronicity travel", did not take a trip along "causality" dimension, you have no way to make those atoms exists at destination.

If you do need to take the trip in "causality" dimension (e.g. make new atoms at destination) to visit the "syncronicity" one, it means that "casual time" and "synchronous time" values aren't independent, thus can't act as separate dimensions.

If no trips are possible without mixing the "time dimensions", then there are no reasons to assume they exist as such, you don't have any means to test the hypothesis.

• My thought was that the destination dimension is like a separate place in timespace, and functionally it would also be a different place in spacetime, so moving matter from one time place to another doesn't destroy or create anything. The alternate timespace isn't really your childhood, only another place that is identical to your childhood until someone or something changes it. This does, admittedly mean the universe has to be vastly bigger than our current understanding of 3D space. – DWKraus May 17 '20 at 23:53
• @DWKraus See also the QM phase space - not all the points of the phase space are reachable from a given point without violating conservation laws. – Adrian Colomitchi May 18 '20 at 0:16
• @ Adrian Colomitchi I'm afraid I don't understand why bringing your own entropy to another "horizontal" time position is an issue. Your own entropy is constant, and the different timespace is elsewhere/elsewhen, following it's own rules. I'm not challenging, merely not understanding. I did some edits to clarify language. Thank you – DWKraus May 18 '20 at 12:54
• @DWKraus the more details you add for the "synchronicity" the less I understand it. I can't see what a "travel along synchronicity dimension" means, much less how this travel could be realized without affecting "causality". I can't even see what "synchronicity dimension" means, it's like you are proposing "the set of all possible events, regardless of when they happen or what causal relationship they have; but mind you, it cost nothing to slide from one to the other". Sorry, it looks to me more of a math formalism gimmick, in no connection with the reality. – Adrian Colomitchi May 18 '20 at 13:22