To the best of our current knowledge, there does not appear to be a “south pole” of time: That would be a Big Crunch, which it now appears is not going to happen.
And Leo Bloom, no offense intended, but have you been partaking a bit much of the holiday cheer? Your posts usually look a lot more coherent than that.
That much I’ve got. Time was not, then time was. But the shape of time was apparently dramatically different in the first “moments”, was curved … how much and in what way does that dimensional metaphor play out?
And of course we are not in the realm of evidence I don’t think, but if the time dimension is curved and anything sphere-like, then how would we know that there is another side beyond the equator until we are on that side of it?
Note: I am not proposing this; I am just trying to wrap my head around what the metaphor if taken in a literal sense, would look like from our perspective at different points.
The North Pole analogy sort of works this way:
In coordinates of latitude and longitude there is a singularity at the North Pole because longitude becomes indeterminate there. However the North Pole in terms of the geometry of the Earth it is nothing special, so the pathological behaviour is in the coordinates rather than the actual geometry of the Earth. The North Pole is therefore an example of a coordinate singularity.
At the big bang singularity certain coordinate invariant quantities blow up to infinity, so the pathological behaviour is independent of coordinate system and is a fundamental property of the geometry of cosmological spacetime. So it is not a coordinate singularity (the usual term I believe to describe any intrinsic singularity in spacetime would be “gravitational singularity”)
One approach to quantum gravity that at one point of time was very much en vogue in theoretical physics (Hawking was one of it it’s leading proponents) involved viewing the real pseudo-Riemannian manifold used to describe spacetime as a “section” of a complex Riemannian manifold. In this approach one result was that singularities like the big bang singularity can become properties of the coordinate systems describing the real pseudo-Riemannian section and not an intrinsic property of the complex Riemannian manifold of which that section was a part. Hence, skirting technical and interpretational problems, in this view the big bang is relegated to being a mere coordinate singularity, like the North pole.
I’m not sure the analogy makes a whole lot of sense outside of this specific context as in the normal GR pov the North Pole isn’t a good analogy for the big bang singularity.
No. I re-read it today in the cold light of dawn.
Reads fine, and is coherent and does, in short short, express clearly something about which I’ve thought in many differing contexts. Is the discussion so polarized that it isn’t?
Also, I dislike SD posts that start with such smugness. But this OP sounds nipple, though.
Yes, he sounds nipple. According to Apple auto-correct.
Obviously this stuff can and does rapidly get above my math level, so my questions are no doubt a bit less than insightful. My ignorance however never stops me from asking! My first reaction is that the north pole(s) is/are not arbitrary, be it based on magnetic north or rotational north, but I get the idea that you are trying to draw/explain a distinction between origin coordinates that are the basis of the math vs ones that emerge as a consequence of the math … thing is your distinction seems to basically dismiss the analogy of time a geometric coordinate rather than explaining in comprehensible terms what time being curved would look like to observers within the spacetime, (inclusive of the POVs of ones looking backwards to the very first micromoments after the Big Bang, and as one experienced a curve that inflects moving forward, if one had a lifetime measured in those many billions of years).
This is a very different issue than time as a function of relativity (if my crude understanding holds) as those aspects depend on having different frames of reference and there is, to the best of our knowledge, no other frame of reference that applies to the complete universe. There is no external time yardstick (other arguably than the speed of light). OTOH are we, to some degree, a different frame of reference observer when we look back at events approaching the Big Bang? Is our perspective/measurement of the passage of that time different than would be there had been someway to measure the time from the perspective of being within those “moments”?
And would where we sit in a hypothetically curved time dimension impact what we conclude about whether expansion is speeding up or slowing down as we look at distant areas?
Again, I am merely trying to wrap my puny brain around the image of spacetime as physical complete “thing” - a thing that is in spatial terms believed to be expanding at an ever increasing rate but was felt might someday might re-contract. And I get that as well as I think it is gettable without a full handle on all the math. The question is about the shape and size (so to speak) of that time dimension when thought of a geometric entity and its being, to my crude analogy making, a dependent rather than an independent variable. In my head flipping time from the X with spatial dimensions as the Y (spatial dimensions expand and maybe someday contract as a function of time) to time as the Y to spatials’ X (time varies as function of the size of the universe). Neither of course makes any intuitive sense but from what is said the latter does seem to be true for at least the very farthest back we can extrapolate to.
The earth actually has four (eight) poles: the axis of diurnal rotation, located at a fixed point called 90°N/S; the axis of precession, which I think is located 90° to the plane of the heliocentric ecliptic (not a fixed point on the surface); the axis of wobble, which probably passes through the equator (not a fixed point); and the axis of the local barycenter, which is at 90° from the orbit of the moon’s ecliptic (not a fixed point). There might be a fifth one related to the slippiness of the crust, but I am not sure how that would be defined.
The North Pole is based on the rotation of the Earth and as the magnetism of the Earth is related to its rotation the magnetic North Pole approximately coincides. It’s clear this is why we choose the North Pole to be a special point in coordinates of longitude and latitude, however neither the rotation or magnetic field are geometric properties of the Earth, which is why I said it was not geometrically special (taking geometric as its modern meaning, rather than the meaning suggested by its ancient Greek root). If we were to be over-fastidious, we could say that the flattening of the Earth at the poles due to the rotation of the Earth might make a case for the North Pole being geometrically special. However point was just to illustrate the difference between a coordinate singularity and a singularity which results from the geometry of a surface.
The reason I was dismissive of the analogy outside of the context described in my last post is that for the following reason:
The big bang singularity represents a kind of unavoidable incompleteness in the manifold. I.e. the wordlines representing the history of observers all abruptly and unavoidably end (unavoidably at least within the limitations placed on standard GR big bang models) at a finite point in their past at the big bang singularity. Contrast this to the North Pole where curves on the surface of the Earth extend through the North Pole without any problem. Therefore the reason we can’t define “before the big bang” is due to a fundamental and unavoidable geometric incompleteness, whereas we can’t define “North of the North Pole” simply because of the definition of “North”. The only real analogy to be drawn is that both the questions “What was before the big bang?” and “What is North of the North Pole” don’t have answers in the contexts in which they are being asked, which to me means the analogy offers no real insight.
I would also myself avoid the concept of "time being curved"as it does not well describe what is actually going on. However I can sort of make sense of it on a heuristic level and it simply manifests itself as expansion, i.e. the tendency of the proper distance of objects to increase with time.
These are two questions I don’t see as being relevant as you can approach them in different ways, but ultimately the incompleteness or finiteness in the history of any given observer is fundamental. Or in other words I could write a lot here, but it would just be a sidetrack.
One geometric analogy may be to see spacetime as a circular cone (though of course a cone is a 2D surface and spacetime is a 4D manifold with a slightly odd metric signature - so the analogy has severe limitations - for example the surface of a cone can have no intrinsic curvature). Cosmological time is analogous to to the lines that go from the apex and are orthogonal with its base. Each circular section along one of these lines could be seen as a spatial slice (not that a spatial slice need to be topologically compact like a circle). As you go from the apex to the base the circular sections become bigger like and expanding Universe. Also there is a singular/degenerate point at the apex representing the big bang (though note the big bang singularity is actually a curvature singularity as opposed to a conical singularity which is more analogous to the apex of a cone, but as I said the analogy has severe limitations!).
This is cool. Thanks. Nice info in thread drift.
So to your understanding and explanation time is completely linear, flat, so to speak, beginning at the apex of a 4-D “cone” and orthoganol from apex to base.
Still another poster, whose understanding of this stuff is also far greater than mine, presented a different perspective:
And he is not the only one I have heard characterize time as other than the straight line dimension from the top of the cone view that you describe.
FWIW I have found this discussion in physorg forums that discusses part of the same question:
But not sure if there was a consensus response …
Clearly though there is the concept that time curves even within the universe as it exists currently.
So what happens to a theoretical clock at a point extremely close to the Big Bang, when all of the universe is in a very small space? It must be slowed, i.e. be curved.
FWIW, back to the op, I also found this article by Gabriele Veneziano a theoretical physicist at CERN who developed string theory in the late
1960s, about a string theory construct that supports time existing before the Big Bang.
Like I said, the cone is a limited analogy.
I can sort of see why someone might say “time is curved” in a heuristic sense as, taking a spatially flat Universe, the curvature of spacetime only manifests itself in standard cosmological coordinates when you bring cosmological time into the equation i.e. if you suppress the time coordinate to take a 3D spatial slice you get a nice flat space. However the statement “the time dimension is curved” doesn’t have a whole lot of meaning on a more technical level.
It may help to understand a bit more about time as a physical concept (rather than purely as a coordinate) in general relativity:
In spacetime each hypothetical observer can be represented as a world-line in spacetime. A word-line is much like the line you see on a time-displacement graph (e.g. classic time-displacement graph for projectile motion), except of course rather than being drawn on a 2D Euclidean plane it exists as a curve in 4D spacetime which is a bit more complicated setting as it has a rather funky metric signature and may also have curvature (and may have different topology too). Word-lines aren’t the only kind of curves that you can exist in spacetime, but they are easily identifiable as they form a particular class of curves in spacetime called timelike curves (that curves can be sorted into different classes is a result of the metric signature).
Now the amount of time experienced by an observed between two events is analogous to the arc length of their worldine between the two events. This is called “proper time” and it is basically the time that a hypothetical perfect wristwatch worn by the observer would record.
If our observer is free-falling (i.e. not subject to any non-gravitational force) then his world line is a geodesic, which if you like is a generalized straight line (that is to say geodesics in Euclidean spaces are what we call straight lines). If the observer is subjected to some force then their world-line will not be a geodesic.
The amount of time an observer experiences between two events is going to depend, among other things, on both whether their world-line is a geodesic or not and the curvature of the spacetime that word-line intersects between the two events.
So my first problem with trying to make sense of “the time dimension is curved” is are we talking about the curvature of spacetime or the curvature of the word-line? Both will be different for different word-lines. Assuming the former (i.e. curvature of spacetime), you can’t really just say “well this is the curvature of time, this is the curvature of the x dimension, etc etc.”, the curvature of spacetime doesn’t apply to “a dimension” it applies to spacetime.
Let me assure you “Universe time” and “Current time” are not concepts you will likely encounter outside of that post. An observer in the early universe can only experience time in the early Universe and an observer at the present time can only experience time at the present time so the idea is nonsensical. When we talk about the age of the Universe it is almost always in terms of “cosmological time” which is the proper time (see above) experienced by an observer who is at rest to the frame where the CMBR is isotropic. This is also the maximum time that any observer at a given event would’ve experienced since the big bang.
As a side note sometimes we do use conformal time which did pass slower than cosmological time in the early Universe. However conformal time is useful for a specific and quite esoteric reason (i.e. the conformally mapping the spacetime to another spacetime) and it’d be difficult to argue that it was the actual passage of time from the pov of an actual observer.
Gravitational time dilation only exists as a solid concept in asymptotically flat static spacetime. Big bang models are neither asymptotically flat or static so at the very least we’d have to come up with a completely new way of defining gravitational time dilation. It must be said that we can observe the Universe in the past by looking at faraway corners where the light has taken a long time to travel to us and we do observe it to be red-shifted which is the same as being “slowed-down”, however rather than trying to create a new concept of time dilation of dubious usefulness we ascribe this to the expansion of the Universe.
I’ve always thought that is a bad analogy. There are at least three spots on Earth called “the North Pole.” Stand at any of them, there are two directions you can point, and call it North.
What’s north of the North Pole? Two North Poles.
That doesn’t work. You need to start by defining what you mean by “north”. If you mean geographic north, then the north pole is the geographic north pole, and there’s no point further geographic-north than that. If you mean magnetic north, then the north pole is the magnetic north pole, and there is no point further magnetic-north than that. Changing definitions halfway through an argument is just sloppiness, not cleverness.
Well that’s the point, isn’t it. That’s exactly why the analogy fails. Hawking doesn’t specify which North Pole, or which definition of North he means.
I do want to thank you for trying to explain it although I must admit that I cannot quite parse all of what you wrote.
Not sure though about dubious usefulness though on the last bit, even though my brain cannot wrap itself completely around the concepts. It seems to me that what I am hearing from the experts is that time changes as consequence of other factors and is of different nature in different coordinates of the universe’s spacetime. It further seems to me that understanding the cosmological nature of time at points infinitesimally near various sorts of singularities, including black holes and the Big Bang (different from each other as they may be, and I understand your point that what applies at one sort does not necessarily apply to another) should inform the meaning of what we observe of those limit-near to singularity events and not be of dubious value at all.
Is the concept of time being a function of those aspects of the universe that give rise to spatial dimensions and mass as much as the other way around, rather than a linear measuring stick from apex to base, not one that is utilized?
BTW, Peter Morris you already made that comment in post 9.
Why does it matter? Only those deliberately looking to obfuscate the issue would think he’s not referring to the everyday usage of 90 degrees north latitude. And even they have to pick only one of the other definitions; as soon as they do there is a fixed north point from which everything else is south. There is no possible physical ambiguity because the reference is to a coordinate system.
Yes, English words often have more than one meaning. That is clearly not the same as saying that any or all those meanings are applicable in a given context. If Hawking talks about dogs and frogs, the odds are incredibly high that he is talking about animals and not andirons and violin bow nuts.
Hawking doesn’t have to specify which north pole he means, because the analogy works exactly as well regardless of which one is meant. Pick a pole, any pole, but no matter which you pick, you can’t go north of it.
On the contrary. The most common meaning of North Poleis “the point in the Northern Hemisphere where the Earth’s axis of rotation meets its surface.” That point is only *approximately *equal to 90 degree latitude, and moves over time.
The point is that “north Pole” is ambiguous, which makes it a bad analogy.
From your link:
What does it mean though to say time ran slower in the past than today? At its most basic time dilation is a scheme for comparing different clocks, but if we say time ran slower in the past that suggests we are comparing (or at least can compare) the rate of the same clock at different times. That’s why I say it is nonsensical.
[quick note: “co-moving” as used below means that an object/observer is in the frame where the CMBR appears isotropic, we will assume (slightly incorrectly, but not massively so) that we are co-moving.]
Red shift/blue shift (I hesitate to use the phrase “Doppler shift” as that usually has more specific connotations) is how we compare clocks by and as I said in fact in the big bang model, in an expanding Universe, faraway “co-moving” objects (and due to the finite speed of light it means we are also observing them in the past) appear to be red-shifted. Hence from a purely visual point of view if we were to observe a clock on such an object it would indeed appear to run at a slower rate than our own. It may seem then it is perfectly sensible to talk about time dilation in the past vs the present by looking at faraway clocks, but unfortunately it is not that easy.
In relativity we don’t tend to take the total red shift/blue shift factor as the time dilation factor. For example in special relativity the gamma factor which describes time dilation is taken from the relativistic correction to the Doppler factor rather than the Doppler factor itself. Hence in special relativity when a clock is travelling towards an observer it will actually be blue-shifted (i.e. visually it will appear to run faster than their own clock), however we still say that it is slowed down from the point of view of the observer compared to their own clock as it doesn’t (visually) run quite as fast as it would without the relativistic correction.
How then in our cosmological scenario do we decide which portion of the red shift is due to time dilation and which portion is due to “other factors”? Well the answer is that we apportion all of the red-shift of co-moving objects to “other factors”, the “other factors” specifically being the expansion of the Universe. The reason for this choice is lies in the coordinates we choose and that in these coordinates the cosmological time corresponds to the proper time of the class of co-moving observers. It could be rightly pointed out that spacetime coordinates are arbitrary, however some coordinates are less arbitrary than others and we have actually specifically chosen the cosmological spacetime such that it possess these spatially homogeneous and isotropic coordinates.
Now as you’ve pointed out, gravitational time dilation is an “thing” in relativity, so why doesn’t it apply in this situation? The answer is that its zone of applicability is actually quite narrow, applying to certain situations which are neat enough (or more properly, possess the right symmetries) for it to make sense as a coherent concept. It’s not that in other situations clocks all run at the same rate regardless of what gravity/spacetime is doing, it’s just they are too messy for something as nice and as neat as gravitational time dilation to emerge.
The two ingredients required for gravitational time dilation to be defined are asymptotic flatness and staticity (i.e. the property of being static).
An asymptotically flat spacetime is one in which is typically characterized by an isolated collection of mass or masses, e.g. a star in otherwise empty space. This first ingredient arguably is not absolutely essential, but what it provides is a a notion at infinity of there being no gravity so that we can compare and say “a clock runs slower here than it would at infinity where there is no gravity by this amount”.
A static spacetime is one that, if you like is strongly-independent of time or more specifically there exists coordinates where the space coordinates are independent of the time coordinate and vice versa. The reason this is needed is that the independent time coordinate can be compared to the proper time of a clock at the independent space coordinates in a way that is consistent for all clocks in the spacetime. Without this ingredient the comparison becomes arbitrary and inconsistent.
In standard cosmological models, spacetime is neither asymptotically flat or static, so gravitational time dilation is not a concept encountered.
