(Summary: In a previous post, I referred to a statement made by Lee Smolin that “time *does* exist,” and that I disagree with him. It has been pointed out to me that this sounds like an extremely odd statement and counterstatement without its original context. I offer in my defense that he started it, and he gave no context either. So here’s a summary of what he meant by that, and in very briefly why I don’t feel confident that it’s correct.

Lee, if you’re reading this, please correct me if I’ve misstated your position!

First of all, there are two things that everyone is willing to agree on. First, at all reasonable large distance scales, say above 10^{-42}cm, the world is governed to an excellent approximation by relativity. In particular, the world appears to be smooth and continuous (a manifold, to be precise) and neither “space” nor “time” is entirely meaningful in its own right. Instead, our macroscopic world has three space directions and one time direction, and the natural symmetries and so on of the world treat these on an equal footing; we can talk about “rotating” between a space and a time direction. (In relativity, that sort of rotation is equivalent to changing one’s velocity) So in daily life, space and time are really very similar.(*)

The second widely agreed thing is that, at some sufficiently short distance, the above description has got to be false. The basic reason is that quantum mechanics (QM) and general relativity (GR) don’t play nicely together. QM says, for example, that the position of a particle is often not known precisely; instead we can say that it has such a probability of being *here*, and such-and-such a probability of being *here*.(*) General relativity, on the other hand (this is Einstein’s theory of gravity) says that the presence of matter stretches space and time sort of like a bowling ball stretches a trampoline.(*) The problem is, if the object is at position number 1, over here, then space at position number 2 has been stretched, possibly even a lot – in fact, if the stretching is big enough, just what we mean by “position number 2” is murky. This means that even discussing what the probability that the particle is at position 2 is not well-posed; the shape of space is a function of the positions of matter, and the positions at which matter can even exist is a function of the shape of space. Something’s got to give. In general, in any situation where both quantum mechanics and gravity are strongly active, the very notion of space as being smooth and continuous falls apart – much more so all the symmetries that connect space and time. This happens (among other times) at high energies, high temperatures (like in the early universe) and very short distances.

So now we come to the question of time. Basically, one of the major open questions in high-energy physics is what it is that replaces the notion of “spacetime” at these very short distances and high energies, and how our ordinary universe emerges from that at low temperatures. (i.e., if we have some sort of situation that looks nothing like space at high temperatures, how is it that when it cools down it ends up looking like spacetime, and not, say, like a jagged ball of fuzz?) There are two major approaches to this question right now.

Approach 1: When spacetime breaks down, all of it breaks down; the state of the universe at these high energies has no notion of any “coordinates” or smooth spaces or symmetries. As the system cools, this system coalesces into a smooth space and all of spacetime, and the symmetries that relate space and time, emerge together. String theory favors this approach.

Approach 2: The symmetry between space and time is, to some extent, an accident. At high temperatures, the notion of space breaks down, but the notion of time stays valid; imagine that space breaks into little shards or plaquettes, but these shards still have a smooth time evolution. As the universe cools down, these plaquettes coalesce to form space, and a nontrivial accidental symmetry emerges at low temperature that makes time – previously something very different from space – behave like just another dimension. Loop quantum gravity is based on this approach. (This is what Lee was referring to when he said “time *does* exist”)

Although the second approach seems a bit strange in this context, it’s important to realize that it has certain advantages: the equations are much easier to deal with (it turns out, for technical reasons, that the breakdown of a spacetime is a bit more subtle than the breakdown of just space) and there are situations elsewhere in physics where a dimension of spacetime emerges out of some quantity that at first looks nothing at all like a dimension.

My personal feeling about this matter, however, is that the former approach is more likely to be correct. The hypothesis that there’s an additional, very deep, symmetry breaking phase, and that the notion of time is preserved to all energies and to all orders, does not seem consistent with what I know of the high-energy properties of physics. I believe for other reasons (entropy bounds, shape-of-the-universe constraints, some instincts based on what we do know about the breakdown of spacetime) that there must be a complete discretization of spacetime at sufficiently high energies, which could not allow a continuous time dimension to survive.

But ultimately, of course, these are just hunches – the only way to verify either of these conjectures is by experiment. Which is why I’ll say for now “I think he’s wrong, but I’m willing to be convinced.”

(*) And there are obviously a lot of subjects scattered throughout this that probably require detailed discussion, but there’s a size limit on both LJ comments and on the reasonable size of a digression on theoretical physics.

(Incidentally, my own research was on noncommutative geometry, and in particular with an eye towards how that may be a property of spacetime just beyond the point of this breakdown – it looks like spacetime starting to break into fuzzy patches that overlap and shift relative to one another, and those patches getting more and more disconnected as the energy goes up. My hunch is that this or something closely related to it is the first stage of the breakdown of spacetime, but that it breaks down even more at higher energies still.)

making sure I understand what you said:

there must be a complete discretization of spacetime at sufficiently high energies, which could not allow a continuous time dimension to survive.As in, if you look at a bitmap image closely enough, it’s a mess, but if you step back far enough that the discrete changes are smaller than your ability to measure them, it looks nice and smooth?

Or more like how the energy released from electrons dropping valences (light emitted from a flame) is in discrete pieces that corresponds to the drop to a lower, more stable orbit? But that at some point, the size of the energy being released compared to the size at which your point of view is becomes so great that everything gets blocky and non-continuous? But at our massive pov, it’s all very smooth, because we can’t tell that it’s a bunch of different discrete bits all mashed together?

(I’m grossly simplifying into HS/first-year engineering college chem/physics). I’ve yet to finish Greene’s “Elegant Universe”…

Like the former. The idea is that the “pixel granularity” is about 10-42cm, although instead of something as nice as pixels there’s probably something quite a bit more complicated going on. The pixel size may also increase dramatically under some circumstances, like near the center of a black hole or in the early universe.

but the disruption in spacetime due to high energies only exists a long as a high concentration of energy exists, but as energy disappates throughout the system (toward equilibrium) then spacetime returns to its “normal” state

which means that the rip in the fabric of spacetime exists only as long as there exists energy to maintain it

btw, you can just tell me that I’m full of shite and I’ll shut it

another tack on whether or not time exists:

my watch works

i exist in the now

and not in the has-been or the will-be

and that’s pretty consistent

if time doesn’t exist

then something keeps me on this continuum

at a fairly consistent pace

observable to me

Another way to look at it is that spacetime is made out of pixels – and as long as you’re working at low enough energies or long enough distances, you don’t notice them. The question is just whether time is pixelated too, or if it stays continuous no matter how closely you look.

The fact that something keeps you on this continuum at a consistent place is essentially the magic fact that somehow or other, when you look at it from far enough away, it looks like a continuum; when you zoom in very closely there’s no obvious reason why it should do that, and even less why the continuum it forms should have nice properties like one time direction and three space directions.

Ok, I think I understand that. Thanks.

Atoms of space and time

Please send your comments to subhendu.das@excite.com

The article “Atoms of Space and Time,” by Lee Smolin, in the January 2004 issue of Scientific American, attempts to answer a very fundamental question: is nature continuous or discrete? However, it appears that the author is already biased toward his hypothesis that nature is discrete.

Is matter discrete?

Smolin says on page 66, “the granularity of matter is old news.” Is that really correct? Has anyone proven that electrons cannot be broken into smaller particles? We have already seen that neutrons can be broken. It is only a matter of time when our technology will break electrons.

On same page Smolin asks, “…does the world evolve in series of tiny steps, acting more like a digital computer?” The digital computer is not a discrete device. Often we casually say that it uses only zeros and ones, but in reality it does not. If you put an electronic probe of an oscilloscope at any pin of a microprocessor you will find a continuous time electrical signal.

Are real numbers discrete?

On page 69 Smolin writes, “…the volume could be any positive real number.” He used such statements in several places. It seems that the author thinks real numbers are discrete. In fact hey are not. In all mathematics we use finite precision representation of numbers. But that is not the true representation. The integer 2 is really the real number 2.0000…000… with an infinite number of zeros. When we say 2.00124 we really mean 2.0012413568…9457… with an infinite number of decimal places, but our number is accurate to only five decimal places, so we do not write the rest of the digits.

Or he should treat all finite precision numbers of his equations as open intervals around the specified point. For example he should treat the integer 2 as an open interval (2-, 2+) where is a very small number.

Is calculus continuous?

Surely Smolin has used lot of calculus to justify his theory. But is calculus continuous or discrete? Consider the Derivative operator D[xn]. It produces nxn-1. Is this a continuous operation? No. If you plot the graphs of xn and nxn-1, you will see a large gap between them. The Derivative operator D peels off a function in discrete steps. This Derivative cannot be used to analyze the discreteness of nature. It is too gross for the author’s subject.

Are limits correct?

The concept of limits was visualized using a figure drawn on paper. In describing this figure we have used two contradictory notions. On one hand we said we are approaching a limit and on other we still used the same gross level view of the figure. When we approach a limit we must also magnify the figure to see exactly what is happening. If we approach 100 times closer to a point we should magnify the figure by 100 times or a thousand times to see the result. To observe microscopic things we need a microscope. It is the same thing as saying that we are moving closer and closer to a distant star and still thinking that the star will maintain the same small size. So we see that the fundamental notion of calculus, the limit, has failed to treat continuous and discrete behavior of nature in a consistent way.

Is Derivative a tangent?

Consider how we proved that the derivative is a tangent. Imagine the figure, a circle, and a line OP, starting at O on the circle, intersecting at Q and extending beyond to P. If we keep the point O fixed and move the point Q closer and closer to O along the circumference, then the line OP will move and become a tangent at O. We see the same fallacy here; we are not using a microscope. As the point Q moves towards O we must magnify the figure. If you do that then you will see that the figure is basically not changing, only becoming bigger and bigger, and the line will never become tangent. Thus to analyze atoms of nature we cannot use this derivative to represent rate of change of variables or tangent to a function. That will lead to inconsistent results.