# Is Spacetime smooth?

1. Jul 6, 2009

### 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?

2. Jul 6, 2009

### malawi_glenn

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

3. Jul 6, 2009

### 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.

4. Jul 6, 2009

### 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. .

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5. Jul 6, 2009

### malawi_glenn

Your sources are wikipedia and popular science books?

6. Jul 6, 2009

### 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.

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7. Jul 6, 2009

### 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:

8. Jul 6, 2009

### 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..

9. Jul 6, 2009

### 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....

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10. Jul 6, 2009

### malawi_glenn

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

11. Jul 6, 2009

### 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!!

12. Jul 6, 2009

### 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....

13. Jul 6, 2009

### Pythagorean

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?).

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.

14. Jul 6, 2009

### 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.

15. Jul 6, 2009

### Naty1

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....

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16. Jul 6, 2009

### jambaugh

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.

17. Jul 6, 2009

### Pythagorean

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

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

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!

18. Jul 7, 2009

### malawi_glenn

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.

19. Jul 7, 2009

### 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.

20. Jul 7, 2009

### WaveJumper

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.

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21. Jul 7, 2009

### Pythagorean

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?

22. Jul 7, 2009

### Fredrik

Staff Emeritus
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?

23. Jul 7, 2009

### 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.

24. Jul 7, 2009

### 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.

25. Jul 7, 2009

### 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.