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Does the topology of the universe mix the space and time dimensions? Or would it be nonsense to have timelike loops on a cosmological scale? (Like with the toru

Does the topology of the universe mix the space and time dimensions? Or would it be nonsense to have timelike loops on a cosmological scale? (Like with the torus example–one loop in a spatial coordinate, the other in a time coordinate). They're talking about *three*-dimensional manifolds so I assume it's an automatic no, but I don't have the background to understand why it's an obvious no.

pavel_lishin 2y ago
I also share your assumption, and also share your ignorance! I have no idea. Closed time-like loops are possible (mathematically, at least), but I don't know if that's something that can be a consequence of the universe's shape.
soulofmischief 2y ago
It's important to understand what "time" is. To construct a clock, we must have some sort of device which exhibits exact periodicity, and use it as a frame of reference. This is the only way we can measure the passing of "time", via referencing some periodic phase change. Modern atomic clocks observe the oscillation of states in a Cesium atom to define the Second. The quantum nature and energy potentials involved at this atomic scale lead to an exact, measurable periodicity. Einstein essentially describes a "clock", wherein a particle (or EM wave) periodically travels between two spatially distinct points A and B. First it travels from A -> B, and then back from B -> A, striking a sensor on both sides. That is one pulse. We would then reference all timekeeping via some coefficient of that pulse. Let's assume this particle is traveling at the constant "speed of light", the highest possible speed for a massless particle that we have observed. In a "stationary" frame of reference, the amount of "time" it takes for the particle to go from A -> B should be the same as B -> A, so we can say 1 pulse is 2AB, twice the distance of A to B. But what happens if A and B are both moving in the direction of A -> B at the same speed? Well, we could imagine that it would take the particle longer to travel from A -> B than before, but quicker to travel from B -> A by the same factor. Yet, when we conduct such an experiment from the same co-moving reference point (when we move alongside A and B at the same speed), we discover that sensors on both ends confirm that the "time" it takes for the particle to travel from A -> B is still the same exact amount of time that it takes to travel back from B -> A. What gives? How can this be possible, if the particle is not able to increase its speed while traveling from A -> B, since it already is traveling at the maximum observed speed of light? Well, when we instead measure the particle's travel from a stationary reference point outside A and B, we do in fact measure the results we expect. It does take longer to travel from A -> B than it does from B -> A. What??? How can we get two different measurements? Isn't there a single, objective reality? Well, it turns out, if we measure the distance from A to B while in the stationary outside frame, the distance shrinks with proportion to the speed of the moving A & B system. So because the distance shrinks, the "time" it takes to complete a round trip actually shortens. The particle doesn't gain more speed; it just has to travel less. But when we measure the distance while comoving with A and B, we find it hasn't shrunk. To make things worse, imagine A and B are inside a box. How would they ever know how fast they are really moving? Maybe they seem still, but are they orbiting a star? A galaxy? A cluster? So we can only discern movement by observing from an outside frame of reference, in the context of two or more distinct frames/objects. We can only measure a relative speed of any given object. From any particular frame, the "speed of light" seems to hold. That was the mental experiment which led Einstein to uncover the principle of special relativity. And we have since experimentally confirmed both spatial and temporal dilation. Because if space is dilating, our perception of time must be dilating as well, seeing as how the entire measurement of "time" in this scenario is rooted in the spatial distance traveled by the particle. Does this make sense? This is why we have "spacetime". Time is a direct consequence of measuring the spatial difference between two states of the universe. It's crazy, it's weird, and it asks the question of "what is the speed of light, and why is it relative?", but it's internally consistent and experimentally verified. To answer your question about timelike loops on a cosmological scale: the energy involved in maintaining the stability of such a system would be astronomically insane. Even the smallest of theoretical wormholes are rifled with issues concerning temporal stability. Under a very particular, theoretical construction of the universe, stable closed timelike curves could be possible, but it's not likely. Further reading: On the Electrodynamics of Moving Bodies https://www.fourmilab.ch/etexts/einstein/specrel/www/ https://www.fourmilab.ch/etexts/einstein/specrel/www/ Don't be scared to take a peek at the paper, it involves some light maths, but is largely conceptual and surprisingly digestible.
- drdeca 2y ago
  I don't see what the first part of your comment brings to the last part of your comment, the part which addresses the question. I think the question asked is maybe something along the lines of, "If we consider the pseudo-Riemannian manifolds with signature (3,1), considered up to topological equivalence, are all the topological differences between these, determined by the 3-dimensional spacelike slices (independent of choice of foliation)?"
  - soulofmischief 2y ago
    I took it as OP seeking to understand how the topology of space and time are intertwined, and asking whether a spatially closed universe would also be somehow temporally closed. With this in mind, I felt like establishing the exact relationship between time and space might explain how a spatially closed universe would not necessarily be temporally closed, as the measurement of time involves the measurement of relative distance over multiple frames, which wouldn't necessarily change when measuring at the "boundaries" of a closed spatial loop (since these boundaries are themselves relatively defined) Apologies to OP if I misunderstood the question.
- jiggawatts 2y ago
  An error in this type of thinking is that it assumes that the observer isn’t also made of light. In our physical reality all scientific instruments are made of matter, the behaviour of which is almost entirely determined by electromagnetic interactions. Any effect that changes the light clock somehow also changes co-moving observers identically! It’s like asking an a 2D character drawn on a rubber sheet if they think that the things they see on the rubber sheet are changing as that sheet is uniformly stretched. Relativity was formulated in a time before we were even sure that atoms exist.
  - perihelions 2y ago
    I don't think you can have computational objects on lightlike (null) trajectories? If light is interacting with light (to have computation), it's effectively moving slower than the vacuum speed of light and the information is not on a null trajectory.
antognini 2y ago
Yes, it does. Gödel found a solution to the Einstein field equations that shows that it is possible to have closed timeline curves on a cosmological scale under particular conditions. (The universe is rotating and has a carefully chosen value for the cosmological constant.) https://en.wikipedia.org/wiki/G%C3%B6del_metric https://en.wikipedia.org/wiki/G%C3%B6del_metric

pavel_lishin 2y ago

I also share your assumption, and also share your ignorance! I have no idea. Closed time-like loops are possible (mathematically, at least), but I don't know if that's something that can be a consequence of the universe's shape.

soulofmischief 2y ago

It's important to understand what "time" is. To construct a clock, we must have some sort of device which exhibits exact periodicity, and use it as a frame of reference. This is the only way we can measure the passing of "time", via referencing some periodic phase change. Modern atomic clocks observe the oscillation of states in a Cesium atom to define the Second. The quantum nature and energy potentials involved at this atomic scale lead to an exact, measurable periodicity. Einstein essentially describes a "clock", wherein a particle (or EM wave) periodically travels between two spatially distinct points A and B. First it travels from A -> B, and then back from B -> A, striking a sensor on both sides. That is one pulse. We would then reference all timekeeping via some coefficient of that pulse. Let's assume this particle is traveling at the constant "speed of light", the highest possible speed for a massless particle that we have observed. In a "stationary" frame of reference, the amount of "time" it takes for the particle to go from A -> B should be the same as B -> A, so we can say 1 pulse is 2AB, twice the distance of A to B. But what happens if A and B are both moving in the direction of A -> B at the same speed? Well, we could imagine that it would take the particle longer to travel from A -> B than before, but quicker to travel from B -> A by the same factor. Yet, when we conduct such an experiment from the same co-moving reference point (when we move alongside A and B at the same speed), we discover that sensors on both ends confirm that the "time" it takes for the particle to travel from A -> B is still the same exact amount of time that it takes to travel back from B -> A. What gives? How can this be possible, if the particle is not able to increase its speed while traveling from A -> B, since it already is traveling at the maximum observed speed of light? Well, when we instead measure the particle's travel from a stationary reference point outside A and B, we do in fact measure the results we expect. It does take longer to travel from A -> B than it does from B -> A. What??? How can we get two different measurements? Isn't there a single, objective reality? Well, it turns out, if we measure the distance from A to B while in the stationary outside frame, the distance shrinks with proportion to the speed of the moving A & B system. So because the distance shrinks, the "time" it takes to complete a round trip actually shortens. The particle doesn't gain more speed; it just has to travel less. But when we measure the distance while comoving with A and B, we find it hasn't shrunk. To make things worse, imagine A and B are inside a box. How would they ever know how fast they are really moving? Maybe they seem still, but are they orbiting a star? A galaxy? A cluster? So we can only discern movement by observing from an outside frame of reference, in the context of two or more distinct frames/objects. We can only measure a relative speed of any given object. From any particular frame, the "speed of light" seems to hold. That was the mental experiment which led Einstein to uncover the principle of special relativity. And we have since experimentally confirmed both spatial and temporal dilation. Because if space is dilating, our perception of time must be dilating as well, seeing as how the entire measurement of "time" in this scenario is rooted in the spatial distance traveled by the particle. Does this make sense? This is why we have "spacetime". Time is a direct consequence of measuring the spatial difference between two states of the universe. It's crazy, it's weird, and it asks the question of "what is the speed of light, and why is it relative?", but it's internally consistent and experimentally verified. To answer your question about timelike loops on a cosmological scale: the energy involved in maintaining the stability of such a system would be astronomically insane. Even the smallest of theoretical wormholes are rifled with issues concerning temporal stability. Under a very particular, theoretical construction of the universe, stable closed timelike curves could be possible, but it's not likely. Further reading: On the Electrodynamics of Moving Bodies https://www.fourmilab.ch/etexts/einstein/specrel/www/ https://www.fourmilab.ch/etexts/einstein/specrel/www/ Don't be scared to take a peek at the paper, it involves some light maths, but is largely conceptual and surprisingly digestible.

drdeca 2y ago
I don't see what the first part of your comment brings to the last part of your comment, the part which addresses the question. I think the question asked is maybe something along the lines of, "If we consider the pseudo-Riemannian manifolds with signature (3,1), considered up to topological equivalence, are all the topological differences between these, determined by the 3-dimensional spacelike slices (independent of choice of foliation)?"
- soulofmischief 2y ago
  I took it as OP seeking to understand how the topology of space and time are intertwined, and asking whether a spatially closed universe would also be somehow temporally closed. With this in mind, I felt like establishing the exact relationship between time and space might explain how a spatially closed universe would not necessarily be temporally closed, as the measurement of time involves the measurement of relative distance over multiple frames, which wouldn't necessarily change when measuring at the "boundaries" of a closed spatial loop (since these boundaries are themselves relatively defined) Apologies to OP if I misunderstood the question.
jiggawatts 2y ago
An error in this type of thinking is that it assumes that the observer isn’t also made of light. In our physical reality all scientific instruments are made of matter, the behaviour of which is almost entirely determined by electromagnetic interactions. Any effect that changes the light clock somehow also changes co-moving observers identically! It’s like asking an a 2D character drawn on a rubber sheet if they think that the things they see on the rubber sheet are changing as that sheet is uniformly stretched. Relativity was formulated in a time before we were even sure that atoms exist.
- perihelions 2y ago
  I don't think you can have computational objects on lightlike (null) trajectories? If light is interacting with light (to have computation), it's effectively moving slower than the vacuum speed of light and the information is not on a null trajectory.

antognini 2y ago

Yes, it does. Gödel found a solution to the Einstein field equations that shows that it is possible to have closed timeline curves on a cosmological scale under particular conditions. (The universe is rotating and has a carefully chosen value for the cosmological constant.) https://en.wikipedia.org/wiki/G%C3%B6del_metric https://en.wikipedia.org/wiki/G%C3%B6del_metric