Exploring Smoothness of Spacetime

[Pythagorean]

Is Spacetime Smooth?

Smooth: infinitely differentiable

If there were a limit to the differentiability of matter's motion through time, I'd assume it would be at the quantum level (where particles are not actually point particles).

Example:

When I accelerate in my car, the value of my acceleration does not go from 0 to a. There is a non-zero jerk, the rate of change in acceleration. I'm fairly sure that I can also, with my human senses, detect a non-zero change in jerk (i.e. a higher nonzero derivative). My senses are not fine enough to detect much higher derivatives of motion, but I intuitively suspect that it would take infinite energy to move something in a spacetime that were not smooth.

Is there a limit to the differentiation of motion through space with respect to time?

 

[malawi_glenn]

as far as we/I know, yes, it is smooth

 

[fleem]

Everything we know about space-time was learned through classical experiments. Only certain attributes of space-time appear to apply to QM, and classical space-time is incompatible with the results of certain QM experiments. For example, spooky action at a distance is spooky (i.e. paradoxical) only if we insist that classical space-time applies fully to everything in QM.

The basis of classical space-time is the presumption that intervals, dimensions, etc. all exist regardless of whether there are any events occurring. That is, we consider space-time a field within which events occur. We presume two clocks run at the same speed because they are immersed in the same space-time "field", and somehow that field has an attribute called "time" to which both clocks are subject.

That view is fine for classical mechanics. However, consider that an interval that has no events marking its ends, is immeasurable. And saying something exists but is immeasurable, is unscientific. From this we can easily presume that space-time is simply the average effect of many interrelated intervals (which DO have events at their ends), and that there is no space-time where there are no events.

Those two macroscopic clocks are not synchronous because they are immersed in the same space-time. They are synchronous because they are trading myriad particles (virtual photons). Thus they have many consecutive states in common, and it is causality that keeps them synchronous, not some field called "time". Two tiny freezing cold clocks will be less synchronized because they trade less information. Thus a machine in a closed system is welcome to run infinitely fast from one interaction (decoherence/trade of information) with the universe to the next. That's why entangled particles (which are, by definition, in a closed system) are welcome to act like local particles and not worry about causality--because there is no space-time interval between them until they are again measured (interact with) the rest of the universe.

 

[Naty1]

For everyday observations and calculations, the classical view of smooth spacetime is fine and proven to very precise tolerances...BUT!

If you are interested in a quantum based view in response to your question, since you posted it in a quantum forum, see this thread:

https://www.physicsforums.com/showthread.php?p=2259041#post2259041, Is there a limit to frequency. There I have numerous main stream sources referenced which say space and time ARE discrete...

There IS a contradiction with relativity, referenced at the end of my post # 13 in that thread and an explanation of the conflict is referenced in Wikipedia. .

 

[malawi_glenn]

Your sources are wikipedia and popular science books?

 

[WaveJumper]

As far as modern physics goes, spacetime seen through our senses is not the "spacetime" that exists independent of our experience.

If we want to probe deeper and have a more complete answer, we have to invoke relativity and qm and we will have to invoke consciousness and try to explain why our perceptions differ so much with the experimental results(which is a battlefield of interpretations in physics).

Were you talking about the spacetime as seen in our subjective experience or the "spacetime" that we presume must be fundamental? If it's the latter, i don't think current physics has much to say, unless a string theorist wants to present the latest trends towards understanding the true structure of spacetime.

 

[Naty1]

Quantum foam as described by Wikipedia gives a good perspective, I think, on spacetime at small distances: http://en.wikipedia.org/wiki/Quantum_foam


It's not much more than this:

...Since energy curves spacetime according to Einstein's theory of general relativity, this suggests that at sufficiently small scales the energy of the fluctuations would be large enough to cause significant departures from the smooth spacetime seen at larger scales, giving spacetime a "foamy" character. However, without a theory of quantum gravity it is impossible to be certain what spacetime would look like at these scales, since it is thought that existing theories would no longer give accurate predictions in this domain.

 

[malawi_glenn]

i) Quantum foam is still quite a theoretical mumbo jumbo, it is a difference in speaking about what we know and what theories are out there.

ii) you stated elsewhere that you are not a trained physicists, then who do you know that the wiki article gives a good perspective? The article uses terms like "fabric of spacetime", not a very scientific term..

 

[Naty1]

Your sources are wikipedia and popular science books?

absolutely!
Hawking, Brian Greene, Lee Smolin, Paul Steinhardt, Neil Turok, Leonard Susskind...are good enough for me...I'll take their interpretations of the advanced mathematics over my own limited understanding anytime...sometimes arXiv...

What relaible sources do you suggest??

I like to refer to Wikipedia since others can generally have quick and easy accses, perhaps in their native language...

 

[malawi_glenn]

Your sources are wikipedia and popular science books?

absolutely!
Hawking, Brian Greene, Lee Smolin, Paul Steinhardt, Neil Turok, Leonard Susskind...are good enough for me...I'll take their interpretations of the advanced mathematics over my own limited understanding anytime...sometimes arXiv...

What relaible sources do you suggest??

I like to refer to Wikipedia since others can generally have quick and easy accses, perhaps in their native language...

Well, they have interpreted to non-physics people, that is the "problem" of pop-science argument.

 

[Naty1]

Malawi,
as a trusted source of knowledge on this forum, attacking valid theories of mainstream, world renowned physicists, or me personally, will not help me or other forum participants learn new points of view.

If I have misquoted or misinterpretated the authors I've referenced, by all means let me know so I can do better next time...

A lot of others here help me learn, and that's all I'm trying to do for others. People are free to read the sources I've referenced and make their own interpretations...or not.

Remember, this is supposed to be FUN!

 

[malawi_glenn]

The point I am making is that you called those sources "mainstream", and that we must differentiate from what is known and not and if "quantum foam" etc. are Nessiscary vs. Possible.

Now, where did I attack a valid theory? How is a theory valid? well.. if it makes sense with experimental data. So...

 

[Pythagorean]

Were you talking about the spacetime as seen in our subjective experience or the "spacetime" that we presume must be fundamental?

I didn't necessarily mean the spacetime of general relativity (though it may apply for all I know). It's just that when we discuss velocity, acceleration, and higher derivatives, we're discussing motion through space with respect to time. So either space or time alone would be insufficient to describe smoothness. For instance, dx/dx and higher derivatives of space with respect to itself aren't very useful, and dy/dx can be shown to not be smooth (a cliff edge, for instance). Orthogonal space coordinates are generally taken to be independent of each other, anyway. So we know we want to differentiate space with respect to time (or vice versa?).

Those two macroscopic clocks are not synchronous because they are immersed in the same space-time. They are synchronous because they are trading myriad particles (virtual photons). Thus they have many consecutive states in common, and it is causality that keeps them synchronous, not some field called "time". Two tiny freezing cold clocks will be less synchronized because they trade less information. Thus a machine in a closed system is welcome to run infinitely fast from one interaction (decoherence/trade of information) with the universe to the next. That's why entangled particles (which are, by definition, in a closed system) are welcome to act like local particles and not worry about causality--because there is no space-time interval between them until they are again measured (interact with) the rest of the universe.

This is where my "paranoia" about making the statement "spacetime is smooth" comes from. If i remember correctly, in QM, particles tend to "move" from point A to point B without crossing the distance between (but that may just be a failed laymen interpretation from my pre-college years). This might cause lots of problems for any expectations of smoothness.

 

[jambaugh]

Operationally speaking space-time is not physically real. It is the manifold of parameters we use to describe relationships between physical events (which are real). Since we choose to use "smooth" parameters space-time is smooth.

Note that we model gravitation by describing the curvature of space-time but the only real = observable part of this is the paths (causal chain of events) taken by physical objects. Remember that Einstein's equivalence principle goes both ways, (gravitational) dynamic forces are equivalent to geometry but also geometry is equivalent to (gravitational) dynamic forces. Putting it all on geometry is just one of a continuum of possible gauge choices we can make.

I think this is one of the problems with much of current quantum gravity research. Trying to quantize (come up with a quantum mechanical description of) space-time is like trying to quantize . . . oh say the complex plane. These are both abstract mathematical constructs and not physical objects. Rather you quantize something like the hydrogen atom or (we can hope) a particle orbiting around a black hole.

 

[Naty1]

jambaugh:

Operationally speaking space-time is not physically real.

Why do you think that?

What does "physically real" mean to you?

I'm wondering if it is any less real than light or mass or gravity, for example. Seems like we really don't know what any of them really are...

 

[jambaugh]

jambaugh:

Why do you think that?

What does "physically real" mean to you?

I'm wondering if it is any less real than light or mass or gravity, for example. Seems like we really don't know what any of them really are...

Light hits you in the eyes, mass can hit you in the head, these are real. Gravity is the reason the mass hit you in the head if it is an apple falling from a tree. Gravity is a component of the dynamics of light and apples and electrons. We experience these things directly or indirectly.

Let me put it this way... the number 3 is not real it is a mental (mathematical) construct but an essential one in describing real things such as 3 apples. Space and time are not real but likewise essential mathematical constructs in describing real things such as the dynamic relationship between those three apples when we for example juggle them.

 

[Pythagorean]

Note that we model gravitation by describing the curvature of space-time but the only real = observable part of this is the paths (causal chain of events) taken by physical objects.

And even then, we only observe points along the path. Whether it's our eyes or electronic sensors, there's some resolution limit.

Remember that Einstein's equivalence principle goes both ways, (gravitational) dynamic forces are equivalent to geometry but also geometry is equivalent to (gravitational) dynamic forces. Putting it all on geometry is just one of a continuum of possible gauge choices we can make.

I feel like the question is independent of the gauge. Whether we're accelerating upwards in an elevator or being pulled down by gravity, the nature of the motion of objects through space should apply universally.

(edit) by the way, this reminds me of an experiment in which a gauge symmetry is broken:
http://arxiv.org/ftp/cond-mat/papers/0602/0602591.pdf

I think this is one of the problems with much of current quantum gravity research. Trying to quantize (come up with a quantum mechanical description of) space-time is like trying to quantize . . . oh say the complex plane. These are both abstract mathematical constructs and not physical objects. Rather you quantize something like the hydrogen atom or (we can hope) a particle orbiting around a black hole.

As you've implied, there is a lot of symbolism going on here. As physicists, gaining intuition through the mathematics is one of our more interesting career obligations. The complex plane itself is not a physical object, but if we have a physical system that utilizes the mathematics of the complex plane, we can begin to develop intuition about how complex vectors arranged in this plane represent physical processes.

Thank you for an interesting reply!

 

[malawi_glenn]

in QM, particles tend to "move" from point A to point B without crossing the distance between (but that may just be a failed laymen interpretation from my pre-college years). This might cause lots of problems for any expectations of smoothness.

The wave-function is continuous and smooth though. And the particles don't "move", it is meaningless to ask where the particles were before any measurement.

 

[malawi_glenn]

jambaugh, I agree with you in many points, hats off.

But it is possible that a discrete description of space-time MIGHT be more conceivable for describing physics at "smaller" levels that we are aware of today -> just as we might have more than 3 spatial dimensions, there might be more of them, but they are so small that the effect of them are undetectable so far.

 

[WaveJumper]

The wave-function is continuous and smooth though. And the particles don't "move", it is meaningless to ask where the particles were before any measurement.

Yes, if qm is complete the continuous "movement" of large ensemble of particles(classical macro system) is an illusion or a very weak approximation. The true classical "motion" of a tennis ball as described by quantum theory is like this:

If this emoticon :shy: is a tennis ball

:shy: (tennis ball is now here; a moment later it ceases to exist at this point) -- :shy: (tennis ball appears now over here, perhaps at plank length intervals, a moment later ceases to exist at this point) -- :shy: (tennis ball appears now here and a moment later ceases...)...


Since the original question was posted in this subforum, i don't think we can infer knowledge about the structure of "physical" space, as what we already know from QM about it is anti-realist in nature.

 

[Pythagorean]

The wave-function is continuous and smooth though. And the particles don't "move", it is meaningless to ask where the particles were before any measurement.

I agree, the abstract theory of QM wouldn't satisfy this question, it would have to be an observed phenomena that contradicts a smooth spacetime.

In my first post, though, I guessed that it would take infinite energy if spacetime was not smooth (i.e. it would take an infinite impulse for the cases of acceleration if it did not approach it's values gradually) and nobody has commented or confronted on it. Would that be sufficient to say that spacetime is smooth?

 

[Fredrik]

Your questions seem a bit odd to me. Since you're talking about differentiability, I have to assume that you are talking about mathematical models of spacetime, rather than space and time in the real world. The mathematical model is a smooth manifold in all the current theories. They are by definition, so so it's not something you would try to prove.

Are you talking specifically about the differentiability of the curve that a particle takes through spacetime? In that case, you have already restricted the question to be about classical theories, because in QM, motion isn't described by curves in spacetime. (Bohm's version of QM may be an exception, but I don't know enough about it to talk about it). And in the classical theories, you can certainly assume that all curves that represent motion are infinitely differentiable. You can also choose not to do that. It doesn't really matter.

Maybe you're asking because you know that all of the current theories are "wrong", and want to know if something (what?) is smooth in the correct ultimate theory of everything. How can we answer that when we don't even know if such a theory exists?

 

[Nick Bruno]

... i kno little, but this thread is interesting.

i like the part in the beginning when u are all talkin about interaction of stuff, it reminds me of star wars and using the force.

 

[DrFaustus]

Pythagorean -> I think Frederik is spot on with his reply. I'll give you my perspective on this.

As a physicist, you observe events (an apple falling, the moon orbiting, electrons scattering...) in the "real world" and try to understand that. And the way you do that is by making some assumptions and then working out the consequences. If the consequences agree with what you see in the "real world", you claim your assumptions were correct. And so far, the assumption of spacetime smoothness has proved useful in the description of the world. (And I say description, not explanation as you have not really explained your assumptions. In fact, your assumptions are only correct until proven wrong, i.e. experiments not agreeing with your predictions. This is what's behind Frederik's distinction between the "real life" spacetime and the "model" we're using.)

So our understanding of the spacetime structure as of now is that it is smooth.

Will this be so forever? Most probably not, if some sort of quantum gravity is to eventually emerge. But it is only then that you will be able to "scientifically" claim that spacetime is not smooth.

 

[Pythagorean]

Fredrik, DrFaustus:

I'm asking more from an experiential point of view. Before we get too into metaphysics and the discussion of where math and reality fail:

when I accelerate in my car, there's a higher derivative called jerk that I know is there (i.e., I don't go from acceleration, a = 0 to a = c (some constant). I approach c gradually from zero.

I also find it difficult to believe that when I jerk, the jerk goes from some j = 0 to j = b instantly. So there's a higher derivative. I've also posed a question (twice now) about whether it would even be possible for the experiential space-time that I'm talking about to not be smooth. It would take infinite energy to snap from a = 0 to a = c, and intuitively, I feel like this applies to the higher derivatives of motion as well.

I probably should have posted this in classical physics, but I felt like the the finite size of subatomic particles and the discretization of energy would play into the question.

 

[Fredrik]

Nothing you added now made the question any clearer in my mind, or made me want to change anything I said before. "Experimental space-time" isn't a well-defined concept, and you still haven't explained if the word "smooth" in your question refers to a property of your spacetime or a property of curves in it.

The "jerk" is well-defined in the mathematical models. In the real world, it corresponds to what? A funny feeling? The derivative of a funny feeling is undefined.

I'm not trying to be a jerk. I'm just trying to make you see that some things just don't make sense outside of a mathematical model.

 

[Pythagorean]

Nothing you added now made the question any clearer in my mind, or made me want to change anything I said before. "Experimental space-time" isn't a well-defined concept, and you still haven't explained if the word "smooth" in your question refers to a property of your spacetime or a property of curves in it.

The "jerk" is well-defined in the mathematical models. In the real world, it corresponds to what? A funny feeling? The derivative of a funny feeling is undefined.

I'm not trying to be a jerk. I'm just trying to make you see that some things just don't make sense outside of a mathematical model.

In that case, you're failing at "making me see". I've actually had this metaphysical discussion in the philosophy forums and I generally represent your side of it. But I don't take such an extreme side as to say that applying mathematics to things you observe is a pointless waste of time. It's possible I just have a better intuition for motion than you do.

Distance, velocity, acceleration, and jerk all make sense to me as successive derivatives, and I can sense these motions when I'm driving in my car or walking around. The higher derivatives are where my nerves aren't so sensitive.

jerk is the change in acceleration with respect to time. Yes, you can feel a jerk, you can feel acceleration going from 0 to x, it's not just an arbitrary funny feeling. It's a lot different than the constant acceleration of gravity, for example. It would also be pretty odd if that jerk was impulsive or heavy-side.

note, also I said experiential, not experimental.

 

[Fredrik]

I don't know what to say then. I can't make sense of your question. "Experiential spacetime" isn't well-defined either. And you still haven't explained if you're asking about your version of spacetime or curves in it.

 

[Pythagorean]

I don't know what to say then. I can't make sense of your question. "Experiential spacetime" isn't well-defined either. And you still haven't explained if you're asking about your version of spacetime or curves in it.

Well, let's stick to the question of infinite energy being required then (as I've brought up twice). I think It might answer the question. I've been avoiding doing the math here, because I don't know how quantum and relativistic effects will play in, but let's do corrections for that later (where and if we can, I realize that qm mathematics won't play directly in, but I'm hoping any macroscopic consequences in reality can be considered.)

x (distance)
v = dx/dt (velocity)
a = dv/dt (acceleration)
j = da/dt (jerk)

to go from 0 to v velocity without a smooth acceleration, we would have to have infinite energy obviously, because it would mean that

a = dv/dt = inf (the slope would be vertical) and that would require infinite force:
F = ma = dp/dt = inf
so velocity must be differentiable.

now a change in acceleration:
j = da/dt = d^2v/dt^2 = inf
means a change in force:
dF/dt = m*da/dt = m*d^2p/dt^2 = inf

and at this point, my weak mathematical foundation fails me. How do I test the plausibility of an infinite change in Force?

 

[Civilized]

as far as we/I know, yes, it is smooth

Just wanting to add another witness to the fact that every mainstream book on quantum mechanics and quantum field theory treats space / spacetime as a smooth manifold.

i) Quantum foam is still quite a theoretical mumbo jumbo, it is a difference in speaking about what we know and what theories are out there.

Well said, all the currently established mainstream theories (the standard model) and the mainstream extensions to these (GUTs, string theory) always treat spacetime as a smooth manifold.

Naty1, Malawi and I have both gone through a mainstream graduate education in physics, and we are telling you that spacetime is smooth in the standard model and in string theory. You are disagreeing with us on the basis of vague statements in wikipedia and popularized books, when we each have shelves full of textbooks that leave no doubt that spacetime is smooth in all of our current physical theories. In my opinion, we need to get some knowledgeable moderators into this dicussion so that we can resolve this disagreement for good.

 

[malawi_glenn]

I think we should leave the "side track" we got into with Naty1, it is clear from the other posts by the OP (Pythagorean) that he had something different in mind :-)

 

[Fredrik]

Well, let's stick to the question of infinite energy being required then (as I've brought up twice). I think It might answer the question. I've been avoiding doing the math here, because I don't know how quantum and relativistic effects will play in, but let's do corrections for that later (where and if we can, I realize that qm mathematics won't play directly in, but I'm hoping any macroscopic consequences in reality can be considered.)

x (distance)
v = dx/dt (velocity)
a = dv/dt (acceleration)
j = da/dt (jerk)

to go from 0 to v velocity without a smooth acceleration, we would have to have infinite energy obviously, because it would mean that

a = dv/dt = inf (the slope would be vertical) and that would require infinite force:
F = ma = dp/dt = inf
so velocity must be differentiable.

now a change in acceleration:
j = da/dt = d^2v/dt^2 = inf
means a change in force:
dF/dt = m*da/dt = m*d^2p/dt^2 = inf

and at this point, my weak mathematical foundation fails me. How do I test the plausibility of an infinite change in Force?

Now you're clearly talking about curves in spacetime (or space). As I explained before, curves in spacetime are not a part of a quantum mechanical description of motion. So your question can only be interpreted as a question about the smoothness of curves that represent motion in a classical theory. In classical theories, you can choose to let your curves be smooth or you can choose to e.g. let them consist of straight line segments (constant velocity on each segment). It doesn't really matter.

It's not a problem to have curves that fail to be smooth at a finite number of points. It would however be a problem to have a curve that isn't smooth anywhere, but there's no reason to consider such curves in classical mechanics.

Also, as I've said before, and as was repeated by Civilized, the underlying manifold is smooth in all the current theories of physics, both classical and quantum. Here the word "smooth" refers to the fact that the coordinate systems that are part of the mathematical structure are such that any function that represents a change from one coordinate system to another is differentiable infinitely many times.

In my opinion, we need to get some knowledgeable moderators into this dicussion so that we can resolve this disagreement for good.

There's more than enough knowledgeable people here already. How would a moderator resolve the disagreement? By repeating what's already been said? By locking the thread?

 

[fleem]

How do I test the plausibility of an infinite change in Force?

No one has done that yet. I'm not sure if they ever will. Bring infinity into anything and you end up with infinities, and infinity is not like other numbers. Its just not fully defined. I'm not sure we can even call it a "number". In fact, physicists agree that infinity is excluded from this sort of thing, hands down. We've discovered various apparent rules of the universe, and one of those rules is that treating infinity like any other number is a no no, and we don't fully know how to treat it--maybe because there ain't no such animal in the first place. The rules of the universe are discovered empirically, and that's why they need tweaking as we learn more, but so far, infinity isn't a part of them. We've never measured infinity, let alone proved how it affects such equations as the one's you talk about.

Perhaps one of Zeno's paradoxes is more in line with what you are talking about? if so, I tend to agree that Zeno's paradoxes are more than just quaint stories. But I think the point of the other posters is that the current (classical) model ignores things like that, right or wrong. If its Zeno's paradoxes, or the indivisibility of QM intervals, then we can talk about that.

 

[Pythagorean]

Now you're clearly talking about curves in spacetime (or space). As I explained before, curves in spacetime are not a part of a quantum mechanical description of motion. So your question can only be interpreted as a question about the smoothness of curves that represent motion in a classical theory.

(edit/foreshadowing: I have an epiphany and concede to one of your former points at the end of this post)
I guess my question is more along the lines of "is it really, though?" regardless of the theory. I start answering my question by building up from my observations, and I can't frame my observations in terms of quantum mechanics, but I want to be aware of any contradictions or insights to the question provided by QM as I suspect there may be.

It's not a problem to have curves that fail to be smooth at a finite number of points. It would however be a problem to have a curve that isn't smooth anywhere, but there's no reason to consider such curves in classical mechanics.

This is kind of my issue with quantum being on the back burner. Subatomic particles have a finite size (or space does, depending on who you talk to) and angular momentum is quantized. The particles are so small that we're not concerned about whether spacetime is actually smooth because there's so many of them and they're so small compared to other values that we just assume infinite and infinitesimal, spacetime might as well be smooth from that point of view. But my question is, is it really, when you get down to it?

This relates to what fleem brought up:

We've discovered various apparent rules of the universe, and one of those rules is that treating infinity like any other number is a no no, and we don't fully know how to treat it--maybe because there ain't no such animal in the first place.

I don't think we've found a genuine infinity in nature yet. I've always held the assumption that they don't exist, but I'm open to any criticism on that assumption.

Perhaps one of Zeno's paradoxes is more in line with what you are talking about? if so, I tend to agree that Zeno's paradoxes are more than just quaint stories. But I think the point of the other posters is that the current (classical) model ignores things like that, right or wrong. If its Zeno's paradoxes, or the indivisibility of QM intervals, then we can talk about that.

That's kind of where I was going with this, I guess, yeah. Didn't really make the Zeno connection though.

I guess the point of my discussion was to not ignore "things like that".

No one has done that yet

now that I start to think of simultaneity and time dilation and whatcrap in the context of actually trying to perform an experiment to answer that question, the original question is beginning to feel meaningless outside the context of classical mechanics, as Fredrik mentioned at one point.

 

[Nick666]

I don't think we've found a genuine infinity in nature yet.

Isnt the infinite energy of the vacuum a genuine infinity in nature ?