Friday, January 18, 2008

Defending Spacetime

Mark Chu-Carroll has posted on Good Math, Bad Math another post responding to a relativity denier. I respect Mark a lot and appreciate all the writing he does, but in this case I think his argument is weak. If you'll allow, I'd like to give my own thoughts on both the original article—entitled Nasty Little Truth About Spacetime Physics— and Mark's response.

In short, the author claims that relativity predicts motion is impossible, thus relativity is wrong, or at least not a valid representation of reality. Why is motion impossible in relativity? I'll get there in a bit.

I will say that I totally agree with Mark's characterization of the author—who might be named Louis Savain, but since I'm not sure I'll just say "the author." Primarily, he throws out a lot of insults and ad hominem attacks on physicists and mathematicians without discussing a single claim these people have made. He seems to be deeply offended by their ideas but never explains why.

The heart of the matter is the claim that movement through spacetime is impossible, thus spacetime must not exist. I couldn't possibly type this argument myself without my fingers breaking themselves in protest, so I'll just copy and paste for you.

Why is motion in spacetime impossible? It has to do with the definitions of space and time and the equation of velocity v = dx/dt. What the equation is saying is that, if an object moves over any distance x, there is an elapsed time t. Since time is defined in physics as a parameter for denoting change (evolution), the equation for velocity along the time axis must be given as v = dt/dt which is self-referential. The self-reference comes from having to divide dt by itself. dt/dt always equals 1 because the units cancel out. This is of course meaningless as far as velocity is concerned.
That's basically the whole thing. Because velocity through the time dimension must be defined in such a way, the notion of velocity through a time dimension is silly!

Wait, wait, wait. Do I spy a circular argument? It's so rare to see those in the wild, let us pause and admire its terrible yet majestic beauty!


Okay, we're done. Of course, velocity is not defined that way. The conclusion that velocity through a time dimension is meaningless is embedded in the author's definition of "velocity." He's really close to being correct, but missing some important points. To explain, we need a few basic facts about relativity, which I will present in a whirlwind tour.

  1. Distances in spacetime are given by the spacetime interval,
    ds2 = dx2 + dy2 + dz2 - (ct)2
    Where x, y, and z are the ordinary 3 spatial dimensions, t is time, and c is the speed of light in vacuum. Note that ct has units of distance. See Introduction to special relativity

  2. An important consequence of special relativity is time dialation. For a stationary observer, the time of a moving object runs slower. The stationary time, or "proper time," t0 is related to the moving time, t, by a factor γ, equal to 1/√(1-v2/c2). Speaking in math rather than words, time dialation says
    t0 = t/γ
    See Introduction to special relativity and Fox's cogent derivation.

  3. Velocities in spacetime are derivatives of the spactime interval ds taken with respect to proper time dt0.

Now that we know those things, what can we say? Well, let's ignore the space velocities like dx/dt0 and focus on the time. The velocity in that dimension is given by d(ct)/dt0. Differentiating the time dialation equation, given in 2., we see
dt/dt0 = γ
And so the velocity in the time dimension,
d(ct)/dt0 = cγ
This is a perfectly sensible result. The author says velocity in the time dimension is a unitless constant. It is neither. It is in units of meters per second, as a velocity should be, and since γ is dependent on (relative) velocity it is not constant.

You can say some really interesting things at this point. Since your relative velocity with yourself is zero—if not, you've got problems—you are always moving through time at speed c. You are literally hurtling into the future at the speed of light. For outside observers who are moving relative to you, your time slows down as you move faster. I won't get much deeper, but if you're interested, there are loads of interesting things to read about relativity. Google "special relativity" or go to any library; under Library of Congress it should be QC 1-75 or QC 170-197, under Dewey I think it's 560.

Now that we've broken this argument on the wheel of sense-making, I think we should look at Mark CC's response to it. Honestly, I can't tell what he's trying to say. I'll reproduce his argument for you.
Think of simple line drawn on graph paper: y=3x+2. Can we ask how fast y changes relative to x? According to the argument of our crackpot, the answer is no. Because how can we define motion in x? The line is already drawn on the page. x doesn't move. Any motion in x must be defined in one of three ways. It could be motion in x relative to something outside of the system that we're measuring, but that doesn't make sense: we're talking about measurement inside the system - we can't define that in terms of something that has nothing to do with the system we're looking at. We could define motion in x in terms of motion in y - but since motion in y is defined in terms of motion in x, that's circular - and that's clearly no good. And finally, we could define motion in x in terms of motion in x. But that's circular. So you can't look at the rate of change along that line, because "change" is meaningless on the line - any definition of it is nonsensical.
He seems to be approaching this argument from a purely formal math perspective, which I don't think is a fruitful avenue. It's not the math this guy is getting wrong; indeed, given his incorrect definitions, his math was completely correct. No, he got the physics wrong.

What can we say from the article about the author's motiviations? We know he's vigorously attacking the notion of spacetime, but why? From my perspective, what he's really fighting against is wormholes and time travel, those pop physics buzzwords par excellence. He really seems to hate those concepts, and the people who popularize them. Let me give you an example or two.
It is important that people see relativity for what it is, a mathematical trick for the prediction of macroscopic phenomena involving the motion of bodies in a spatial coordinate system. Spacetime is an abstract mathematical construct, that is all. The other stuff (motion in spacetime, time travel, advanced and retarded waves, wormholes, etc...), is pure hogwash. It is so trivially proven wrong in fact, that it is insulting to the lay public, the same public that funds most scientific projects.

The nasty and shocking little truth is that time does not change, a million wormhole and time travel fanatics wearing their little Klingon and Ferengi outfits notwithstanding.

I think the author misunderstands "velocity in the time dimension" as meaning "one is free to move about in time just like one is in space" and thus spacetime arises the ire the author directs at time travel. However, that's just not true. As discussed above, you do have a little control over your time-velocity: as you go faster it slows down. Some more enthusiastic physicists have come up with novel, possibly unfalsifiable ways of adjusting one's time-velocity to be faster forward or even negative, allowing travel to the distant future or the past. Is it true? Eh, maybe. We need the evidence to see, and there isn't going to be any for a long time for many time travel ideas.

However, just because our author hates time travel so much doesn't make it wrong. And just because he confuses spacetime with time travel doesn't make it wrong either.

To close, I'd like to blast you with several quotes from the article. Most show the blinding arrogance of the author, but the first is funny for a special reason. The author has obviously read enough about relativity to know that a four-dimensional spacetime makes sense and solves real-world problems. However, he has what he believes is a silver bullet against a time dimension, so he has this to say.
There are very good reasons to suppose that there is a fourth dimension in which matter is moving at c in one direction. This fourth dimension, if it exists, is certainly not time since a temporal dimension is illogical as I have shown above. It would have to be a spatial dimension. I will have more to say about this in a future addition to the site.
Sadly that addition never came. I really wanted to know what he would say, but alas.

Also, in the author's bizzaro-physics where spacetime doesn't exist, gravity—which is the warping of spacetime by matter—needs a cause...
By now the reader should realize that there is no such thing as spacetime and that gravity does not have anything to do with the curvature of a physical spacetime. There is something else out there that causes bodies to fall, without a doubt, something physical, something material. Over the last century or so, relativists have steadfastly and sometimes deviously rejected any suggestion that space is not empty and that there is a need to invoke some sort of material substance or aether to explain phenomena like gravity. Given that spacetime is a fictitious math construct, it is obvious that it cannot possibly account for gravity. [Emphasis mine]
I'll quote the late Perry DeAngelis when I say, "The ether?" We'll not even go there today.

The last is a section called "Isn't it Amazing?" in which the author really needs to apply Occam's Razor. Either every professional physicist is wrong, or he is.
Isn't it amazing that Dr. Kip "Wormhole" Thorne and his time travel colleagues at Caltech and elsewhere can claim that the mathematics of general relativity does not forbid time travel even though it does exactly that?

Is it not also amazing that physicist Julian Barbour felt it necessary to write an entire book to demonstrate "The End of Time" when it can be shown in a single sentence? Why should it take an entire book to convince the old school that there is no time dimension? It is not as if one is trying to deprogram a cult. Or is it?

Isn't it strange that Dr. Thorne, Dr. Wheeler, Dr. Deutsch, Sir Stephen and company were not aware that nothing can move in spacetime? Being the celebrated mathematicians that they are, one would suppose it would be their business to know and understand something so trivial that it can be explained to high school kids. After all, it is not as if there is not a single physicist in the world who knows about this. I know of many who do. Could not just one of them write a line to Dr. Thorne or Sir Stephen and alert them of their error? How did they get their time travel papers past peer review? How did they get so darn famous? Did I hear someone say fraud? Or is it just plain incompetence and crackpottery?

And isn't it interesting that Sir Stephen was present at a recent symposium at the famous Caltech Institute in Pasadena, California, honoring the scientific achievements of Dr. Thorne? Birds of a feather? You bet!

UPDATE: Via a comment on Mark CC's article, we can see that the author—who is named Louis Savain—has been pushing exactly the same argument for at least twelve years. For sooth, what I just linked was posted to our fair intertubes in 1996. Something tells me I won't get through to him with this little blog post.