Getting from Space and Time to Space-time

03 April 2017

Are space and time two separate entities? Or are they just different dimensions of one thing—the space-time continuum? And what difference does it make if they are? Those are just some of the questions we discuss this week, as we dig into the nature of the space-time continuum.

At one level, the basic structures of space and time seem straight-forward. Space has three dimensions—east/west, north/south, and lower/higher. Time has just one dimension, and a fixed direction.  In space, you can move in many different directions.  But time has a single direction.  It flows from the past, through the present, into the future, along a single line.

Now common sense sees space and time as independent of each other, rather than merged together into a single continuum. It also sees each as absolute. No matter where in the universe you measure from, my birth in Sandusky Ohio occurred 11 years after and 800 miles east of John’s birth in Lincoln Nebraska, to borrow John’s example. Common sense is not alone in this. Newton saw things in much the same way. He saw space as a really big -- in fact, infinite -- free-standing container—separate and distinct from the material universe it contains. Space, on his view, is related to things in space in something like the way a coffee cup is related to the coffee it contains. If you were to pour the coffee out of the cup, the cup would remain, with its structure entirely intact.  Just as an empty coffee cup still has a structure, so too does empty space.

Time is also a container, on the Newtonian view.  Time contains the entire spatial manifold—that is, space with all of its material contents. The spatial manifold moves through time as a single unit. If we go back to the cup,  first we have the coffee in the cup and then we have the cup sitting on a timeline. The evolution of the universe is the state of the coffee changing as the cup, with the coffee in it, moves along the timeline.

This is a pretty and compelling picture. It accords well with our commonsense experience of space and time.  But it turns out to be almost completely wrong.  Newton’s contemporary and rival, Leibniz, was perhaps the first to reject it—not so much on empirical, scientific grounds, but on philosophical grounds. He thought that the picture was worse than false. He thought that it was incoherent. According to Leibniz, Newton’s picture violated the principle of sufficient reason.  That’s the idea that for everything that exists, there must be a sufficient reason that explains why it is as it is and not some other way.

One of his many arguments for this conclusion turned on our imagining the material universe as a whole shifted one light year to the right in absolute space, rather like moving the furniture in your living room to the right, while keeping the arrangement intact. The crucial problem is that since you can’t directly observe absolute space, you’ve got no fixed frame of reference—like the walls and the floors of your room—to measure the movement against.

But from this it follows that you couldn’t tell if you were in the shifted universe or the unshifted one.  That is to say, there would be no discernible distinction between one position in absolute space and another (if such a thing happened to exist).  The point is not just that we can’t figure out where in absolute space the universe is.  The point is that the very idea of absolute space is an empty idea, at least according to Leibniz.  But I won’t recapitulate the full details of that argument here, since it would take us too far afield.

So let's set absolute space and time aside as philosophically problematic, at the very least.  I hasten to add, though, that absolute space is, in the end, less empirically problematic than Leibniz might have imagined.  Of course, relativity is often and rightly said to reject both absolute space and absolute time. But rejecting absolute space and time still doesn’t quite get us to the space-time continuum.  To get to that destination, we need to talk about light. Light is, of course, the fastest thing in the universe.  But the main thing about light is that that it travels at the same velocity relative to all observers.  It's this last property that gives light a very special role with respect to the space time continuum. You can think of the constancy of light on analogy with the rigidity of a ruler. You wouldn’t want a ruler that varied in size as you moved it from object to object. A ruler, if it is to function as a ruler, has to be rigid!  The constancy of light can be thought of as a form of rigidity that makes it suitable as the intrinsic measuring rod of the universe.

Not everything that moves about in the universe has this same property. In fact, most moving things do not!  Suppose I’m standing still at the line of scrimmage on a football field.  You are lined up at wide receiver.  At the snap of the ball, you take off running downfield at the blazing speed of 20 miles per hour. I throw the football at a healthy 60 mph. I see the ball moving away from me at 60 mph, you see it approaching and then perhaps overtaking you at 40 mph. But light is different. If I cast a beam of light downfield, then even if I am standing still and you are running your tail off to catch up with it, we’ll each see the light moving away at the same speed—the speed of light.

Once we view light as the measuring rod of the universe, it's this property that gets us to the idea of space-time.  To see how, imagine that you’re standing at the center of a speeding train car. I’m standing on a platform watching you go by. When we pass each other, a flash of light is emitted from the center of the car. To you, the front and back of the car are located at fixed distances, equidistant from the source of light. You judge that the light reaches the two ends simultaneously. But I see the rear of the train moving toward the flash point and the front of the train moving away from it. Now this is where those weird properties of light—the ones that make it suitable as a measuring stick of the geometry of the continuum—come in. Since the speed of light is the same in all directions for all observers, to me the light headed for the back of the train covers less distance than the light headed for the front. So your measuring stick tells you that the flashes strike at the same time. Mine tells me that they strike at different times.

Notice that not only do we disagree about simultaneity, we also disagree about distances travelled. This means that we disagree about the length of the car. And here’s the kicker, we’re both right. This makes simultaneity relative. It makes length relative too. But there is one thing that NOT relative. The space time continuum itself. Indeed, it is as absolute, in its own way, as Newton imagined space and time each to be. It’s just that the space-time continuum turns out to have a radically different geometry than Newton (and common sense) took space to have.

This is no doubt mind-boggling stuff. But we hope you tune in and join the conversation and help us unravel the mysteries of space, time and space-time.

#### ClassicalMatter

Thursday, April 6, 2017 -- 10:40 PM

There is a simple explanation

There is a simple explanation for special relativity: it is a consequence of the fact that matter consists of waves (stationary particles are standing waves). Suppose that there is an absolute (i.e. Galilean) space and time , and ask "How would measurements by different observers be related if all of their measurements use waves with a fixed speed?" The answer is that all of their measurements would be consistent with special relativity. For example, if submarines made all of their measurements (including time) via sonar, then measurements made by different submarines would be related by Lorentz transformations with respect to the underwater sound speed (neglecting displacement of the water by moving subs). You can find this derivation at http://www.classicalmatter.org/UnderwaterRelativity.htm. Of course, we know that matter does satisfy the Lorentz-invariant wavelike equations of quantum mechanics, and special relativity is a consequence of those equations.
(In your show it was mentioned that Maxwell's equations led Einstein to his theory of relativity. It is worth mentioning that Maxwell derived his equations from simple elastic solid models in ordinary Galilean space-time.)