The Fourth Dimension

We have this idea in our heads that the universe consists of three-dimensional space with three-dimensional objects flying through it.  Makes sense, that’s certainly what it looks like and feels like.

But there is something weird going on, and that weird something has to do with the speed of light.  And what’s weird about it is, that light has a speed limit.

You know about velocity, right?   First of all, velocity has something to do with time.  The velocity of an object is how much distance it can cover in a certain amount of time.  60 miles on one hour is a statement of velocity.   Distance and time.

You know that all velocity is relative. When I was taking driver’s ed, I remember they taught me that if you’re facing a head on collision with another car, you should hit the brakes. That didn’t make any sense to me. I figured I should speed up. That way I would win. The other car would go flying off into space and I’d be fine. Not so. A head on collision, where both cars are going 50, is just like running into a wall at 100mph. The only velocity that matters is the relative velocity between the two vehicles.

Let’s have some fun with relative velocity. Let’s say you want learn how to hit a golfball really far. You want to work on your speed, you want that ball travelling as fast as you can right off the tee. Let’s say the absolute most you can get out of a golfball, if you hit it perfectly with your longest driver, is 200mph. Any ideas on how you could up your speed?

Here’s one. Lay a mat down on the deck of an aircraft carrier, and launch golf balls toward the bow. If the carrier is going 35mph, your golfball is now going 235, right?

Being a semi-practicing redneck, I have one more idea. Let’s say we park my pickup truck facing backwards on the stern of the aircraft carrier. I’m in the bed with my club and a golf ball. My buddy floors it, in reverse. I hit the ball. Now I have a boat going 35mph, a pickup going 40, and a golfball that is now traveling 275mph. Awesome!

Now, let’s try the same trick with light.

I park the pickup in the backyard and measure the velocity of light coming out of the headlights at 670,616,629 mph.

Park the truck on the deck of the carrier. Facing forward this time. Now how fast is the light traveling? It’s the same: 670,616,629 mph.

OK, punch it. Get that truck up to 80mph. Carrier’s going 35, what velocity is the light going? Uh oh. Same answer. 670,616,629 mph

The speed of light does not obey the rules of relative velocity. It has a maximum speed. And it’s even weirder than that; as it turns out, nothing can travel faster than the speed of light.

If you’re a sci-fi fan like I am, that’s hard to choke down. Can’t be that hard to build a hyper-lightspeed spaceship, can it? Here’s how I would do it. I would build a spaceship that would go like 99% of the speed of light. Then I’d put an afterburner on it. Next time I went out, I’d run it up to 99%, and then just click it onto afterburner. Presto, warp factor 2, right?

Wrong. Nothing can go faster than the speed of light.

What gives?

The answer is deceptively simple. I say “deceptively” because it took us 30,000 years to figure it out. But we did eventually. Here’s the answer: space is not really three-dimensional. It has to have four dimensions.  That’s the only way the whole thing makes sense.

There are two ways of looking at it.  One is to look at time as the fourth dimension of space.  As a former amateur astronomer, I’m used to looking at it that way.  Stars are really far away.  Light can only travel so far in a given amount of time, but it’s pretty durn fast.  So one way of measuring really big distances is to think about how far light can travel in a year.  Looking at it that way, the nearest star is four light-years away.  With the naked eye you can see things that are over 2 million light-years away.

Convenient, but it is kind of weird.  Because you soon realize that, when you peer into your telescope, you aren’t really looking out into space.  You’re looking back into time.  The light coming to us from the Andromeda galaxy started its journey 2.5 million years ago.  We have no idea what it looks like today, if it’s even still there.  We are looking back into the past.  This is why astronomers think of time as being the fourth dimension of space.

The other way to look at it is to look at the speed of light as a dimension of space.  I think this makes  a bit more intuitive sense, especially if the topic of discussion is the nature of time.

There was a famous mathematician at the turn of the century named Hermann Minkowski, who one-upped Einstein by making a model of four-dimensional space that mortal humans could actually understand.   To give you an idea of how smart Minkowski is, he was Einstein’s calculus teacher when young Albert was in college.  Minkowski thought the guy was a dope.  So there you go.

Minkowski’s plan was to pretend that space is an object.  That you could stand out there wherever God lives and look down on the universe.  Now, if space were three-dimensional, I reckon it would look something like a Rubik cube.  What would four-dimensional space look like?

Question: can we even imagine things in more than three  dimensions?  

Not too difficult to model such things mathematically.   Consider this image, which should be familiar from your high-school geometry class:

 

[plane]

 

This is a plane, with two number lines or axes, the “x” axis and the “y” axis.  Any point on this plane can be described by two numbers, (x,y) corresponding to those two axes.  Thus, this could be called a two-dimensional space.  

Now, what if “x” and “y” are planes, instead of lines?  To locate a point in this space, you220px-Intersecting_planes.svg  would need not two, but four coordinates: (a,b,c,d).  So this is a four-dimensional space.  Easy, right?

Turn this image around so you are seeing the planes edge-on, it should look sort of something like you recall from geometry class.

Check this out.  What does this thing look like, in our four-dimensional space?

If you said, “a cube,” you’re wrong.  No, the space does not define four corners.  Among2d_4d square other things, a cube has eight corners.  Moreover, you can define any point on the surface of a cube with three coordinates.  Because a cube is three-dimensional.

Now, we could project an image of our four-dimensional object onto three-dimensional space, in much the way you might project a three-dimensional object onto a two-dimensional screen by shining a light that would cast a shadow.   The image would be distorted, and we would have to rotate it to average out the distortions.  To be sure, in some projections it might look like a cube.  But not always.

Give up?

So do I.  I can’t imagine things in four dimensions, and neither can you.  And neither can anybody else.  Three is all we get.

Is that enough?  We’ll see.

Here’s the plan.  First, we are going to pretend that the earth is flat.  We are going to mash the three dimensions of space we are all used to — sideways, back-and-forth, up-and-down — into two dimensions.  Squash em flat.

Next, we are going to add the speed of light as a third dimension.  And then we are going to draw a picture of this three-dimensional model on a piece of paper.

Now, the image is going to be distorted.  Any time you draw a picture of something it will be distorted; you’re going to have to represent depth some kind of way, using shading or foreshortening or something.  This image is going to be extra distorted because we left out a dimension of space.  We are going to fix that with rotation.

You know what rotation means.  If I show you a these two images of a Ducati, and ask you to imagine what it would look like from the side, with you going 120mph through the alps with a smokin hot Italian woman on the back, you would have to close your  eyes, form a model of the bike in your head, and rotate it.  And then add the woman.     

 

Let’s start with our smashed-down, two-dimensional, flat world.  Now we are going to introduce the speed of light.  So pretend that you’re standing there on the flat earth, and that you shine a flashlight up into the sky.

7-28-13-HiltonHead-Beach-MilkyWay-original

If you’re standing indoors, the flashlight will make a circle of light on the ceiling.  If you go out to the foyer, where the ceiling is two stories up, it’ll make a bigger circle. If you stand inside the Vehicle Assembly Building at the Kennedy Space Center, it’ll make a really big circle.  If you’re like Kelly DeLay and you can shine your flashlight all the way up to the Milky Way, it’ll make a really big circle.  You can see that the circle of light expands as a cone.  How fast the cone propagates is limited by the speed of light.

If you shine the light up, we’ll call that the future light-cone.  If you shine it down, we’ll call that the past light cone.  Here you go:

Basically what you have here is a model of space-time, which is bounded by the speed of  light.  Remember, nothing can go faster than the speed of light.  So nothing outside your past light-cone could have affected you in the present moment.  Anything outside that light cone doesn’t exist as far as you’re concerned.  Likewise, you can’t travel faster than the speed of light either.  So nothing you do can possibly affect anything outside of your future light cone.

Now, remember.  It’s isn’t quite this simple. We lost a dimension of space.  So to make up for that, we are going to “rotate” the image.  We aren’t rotating it in a circle though.  We are moving the flat earth from the bottom of the picture to the top.  You sort of have to imagine three-dimensional space filling in, and understand that there could be a past and future light cone arising out of any point in that three-dimensional space.  Busy, but not impossible to imagine.

Minkowski called this rotation “time,” but it’s not clear to me that he meant time was a real thing.  The time it takes to rotate the plane of the earth from the bottom of the figure to the top is an artifact of the projection.  If he could have figured out how to draw the picture in four dimensions, he wouldn’t have to account for time at all.  Now, this was a hundred years ago, and we know more about how space is actually structured.  And that image may be righter than we used to think.  But the way he treats time in this image is a very helpful concept, and it’s been proven valid.  If you have a GPS in your car, and it gets you within a thousand yards of your destination, then Minkowski was right.  

We’ll think about what things might look like in Minkowski’s world, and what that means, in the next post.

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