Anyway, I want to get into the nature of space and time and try to explain why the “nothingness” that spacetime really is makes relativity inevitable.
Space is the vast emptiness between the stars, the black void that fills infinity. There really isn’t anything there. The same is true of time–there is nothing there to grasp, nothing streaming passed us as clocks tick away.
This may seem obvious but it can be hard to wrap one’s mind around–we often stumble into the trap of thinking of this “nothingness” as something there. It can go so far as to be given a name: the aether. Or a coordinate system. We often project coordinate systems onto space from our minds, supposing that space is composed of an origin with axes stretching out in each of the three dimensions.
Most of the time we realize this is just a mental construct and that there is really nothing there. But sometimes we don’t. Sometimes we think there really are these invisible lines and origins and dimensions, and that this is what space is made of. Some of us become committed to such a view and become known as absolutists or objectivists. But I think the majority understand that these coordinate systems are an artefact of human abstraction.
With this understand in mind, we have no trouble conceding that there is no single absolute coordinate system that describes space. Which of the three coordinate systems below is a more “accurate” depiction of space?
You could even have coordinate systems that differ by scale:
None of these depictions are any more accurate or “true” to space than any other. It is completely arbitrary which one you choose. And the sole reason behind this is that these coordinate systems are mental constructs–we impose them on the world–but there is nothing there demanding that we impose them, or that “best fits” one of our projected coordinate systems over another. There is nothing there. So we can impose any damn coordinate system we want!
What about coordinate systems like these?
Yes, these are indeed candidates for valid coordinate systems depicting space–funky and strange as they are, and perhaps unnecessary, they too could work just as legitimately as any other spatial coordinate system. The only criteria a coordinate system has to meet is that for any given point in space and time, there is a definitive mapping of that point onto the coordinate system in question, no more and no less than a single and exact location in the coordinate system that defines the point’s value (x=1, y=2, z=3, t=4). This point may inherit different values depending on the coordinate system, but such values will be no less definitive and unambiguous in one coordinate system than in any other. For example, have a look at the two coordinate systems below:
%20(4,3)%20d.jpg)
The point highlighted in red in both diagrams is the same point in space, but in the coordinate system on the left, it’s value is (3, 2) and in the coordinate system on the right, it’s value is (4, 3). But either way, it works.
(Interestingly, we are coming full circle, talking about the patterns of space time once again).
In fact, there really isn’t any reason that space should be structured according to our customary “right-angled” style of coordinate system. The fact that “horizontal,” “vertical,” and “depth” feel so natural to us is more a result of our biology and history of adaptation to our environment. There is a much greater abundance of objects in our world that stand up straight (people, trees, sign posts) and objects that lie flat (the surface of water when it’s still, table tops, the horizon). This gives rise to neurons in the visual areas of the brain becoming more exercised and developed for processing “horizontal” and “vertical” information, and thus it seems more natural to us to suppose that “horizontal” and “vertical” (and based on those, “depth” too) are fundamental elements of the geometry of space, more fundamental than diagonal, or curved, or non-linear. But this is us. It is our biology and how we’ve adapted to our environment.
Now what if we take time into consideration. Time is a dimension interwoven into any coordinate system worth its salt that bears the same relation to the three spatial dimensions as those three spatial dimensions bear to each other–they are all spacetime dimensions intersecting each other at an origin at orthogonal angles. This type of coordinate system is known as a Minkowski coordinate system, Minkowski being the famous Russian mathematician who invented the idea of the 4D spacetime continuum, the notion that time really ought to be treated, at least mathematically, as a 4th spatial dimension, or rather that the 3 spatial dimensions ought to count as temporal dimensions, or that all 4 as “spacetime” dimensions. And mathematically, this makes sense (subjectively, it’s a bit of a different story, but that can be chocked up to the peculiarities of our biology and neural wiring).
Einstein banked relativity on this idea and this established the rule that whatever can be said of space can be said of time. Whatever applies to one applies to the other.
So take these two coordinate systems:
Note that the two points in the left coordinate system share the same horizontal value. The same points in the right coordinate system don’t. So are they on the same horizontal or not? The answer is: there’s no fact of the matter. The one coordinate system is just as arbitrary as the other. Now if time can be treated just like a spatial dimension, then a similar principle should hold for two events occurring at the same time. If we allow the vertical axis to represent time and the horizontal axis to represent space, and the two points two events occurring at the same time, then we see that there is no reason why we can’t switch to the coordinate system on the right and say that the left event actually occurred before the right event.
Most of you will recognize this as the relativity of simultaneity. The arbitrariness of spacetime coordinate systems explains it.
Recall the discussion James and I got into about differing time dilations depending on whether one was at the rear of the train versus the front. Also recall how I said this desynchronization of time between the rear and the front also nicely accounts for length contraction (the rear is further ahead on the track because we are seeing where it will be in a few seconds, and the front is further behind because we are seeing where it used to be a few seconds ago).
You might also recall I posted a video from which this is a frame:
Notice that from m-pov (the angled train), when a passenger looks to the rear, he is also looking back into the past, and when he looks to the front, he is also looking ahead into the future. That’s the first part in the paragraph above. In other words, from m-pov, the faster the train travels, the more it “twists” into the dimension of time. This is time dilation. It is just like the right coordinate system above. The left coordinate system is s-pov. In s-pov, the train is not seen as “twisted”–it is seen as straight, but at the expense of its resting length. That is to say, as the twisted train passes through what to s-pov looks like the horizontal “present”, the front end will pass through a point closer to that where the rear will pass through–in other words, the front and the rear are closer together along this horizontal present in s-pov.
Now where gravity is concerned, these principles are no different. No coordinate system is any more valid for describing a gravitational field than any other. James questions the reality of UUGFs. Is there really a gravitational field pulling the world back from the point of view of m-pov, or does all that GR amounts to say that we can only treat it as if it were a UUGF? Well, if he understood the basic principle that all coordinate systems are equally valid in describing spacetime, then he’d know what I mean by “there is no fact of the matter.” A coordinate system in which it is the train accelerating while the world stays still is no more accurate a depiction than that in which the train stays still and the world is pulled back by a UUGF. Some might recall I did say that all gravity is is a warping of spacetime–well, it is a warping that looks like this:
The vertical axis is time and the horizontal axis is space. Notice that as an object moves up in time (and all objects move in time), it slowly accelerates to the right due to its following the warped geodesics.
There is no reason this coordinate system couldn’t fit the scenario of the accelerating train any more than a linear one (i.e. one in which the vertical lines are straight and at right angles to the horizontal). Any point in the one coordinate system for which a value can be definitively and unambiguously determined can be so definitively and unambiguously determined in the other.
In fact, you could say that we’ve been falling in some arbitrary direction into a gravitational field since the beginning of time–everything has, just like the world falling into a gravitational field in the train scenario–and you wouldn’t notice a difference. A person in a free falling elevator, Einstein says, will feel no inertia from the fall. It will be as if he were suspended in space, floating in the air, with no forces acting upon him. He could suppose he were falling in any which direction he wanted–up, down, left, right, etc.–and he wouldn’t know which was right. It’s not much different with respect to some arbitrary UUGF while we’re anchored to the Earth. Being anchored onto the ground, we certainly do feel the Earth’s gravity, but who’s to say we haven’t been falling in some arbitrary direction into a UUGF all this time–we and the Earth together. We wouldn’t notice the difference. Feeling like we’re not falling comes natural to us, of course, and is the basis on which we suppose we aren’t in a UUGF, but there was a time when believing the Earth was flat came natural to us, or that the Sun revolved around the Earth came natural to us, and we would have thought the alternatives ridiculously counter-intuitive. But today we’re used to thinking in terms of these alternatives and it isn’t so counter-intuitive any more. Knowing that the Earth is round and that it revolves around the Sun is second nature to us now, because we have been taught to think like this since we were young. If we were taught to think that the entire universe has been pulled into a UUGF since the dawning of time, believing this would also come natural to us.
This is a critical point for understanding relativity. It’s really nothing more than Newton’s first law: What’s the difference between a universe containing only a single object at rest and a universe containing that same single object but moving at a constant velocity. Nothing! There is no substance to space, nothing for objects to “pass through,” and so the two situations are indistinguishable (this, incidentally, is what I think is Newton’s greatest insight–that an object in motion remains in motion because it is equivalent to being at rest and no one demands explanations for why things at rest remain at rest). The relativity of simultaneity is a bit trickier and took Einstein to figure it out, but I stand by what I said in the OP to this thread: it was his deepest insight.
The relativity of simultaneity might be explained by a simple analogy. Take your favorite scene from any movie. Take Gandolf from Lord of the Rings when he tells the Balrog “You shall not pass!” Now ask yourself, “What was I doing when Gandolf told the Balrog ‘You shall not pass!’?”–not “What was I doing when I saw that scene in the movie,” but “What was I doing when it actually happened?” Most would say this question makes no sense. It didn’t actually happen. Gandolf and his friends live in a completely different universe from us, a universe which doesn’t actually exist. Fair enough, but notice what this means–it means there is no simultaneity between events in the real world and events in Middle-Earth. They don’t happen “at the same time” because there is nothing to associate events in the one world with events in the other, nothing to link them up in time. Instead, there is a great big undeniable “nothing” that separates our worlds. Well, is the “nothing” of space and time that separates events in the real world any different? Well, it’s probably different in a few ways, but not on this score I say. The “nothingness” that is space and time are such that there is no spatial or temporal connection between the events that happen within it. One object is a complete world unto itself, like Middle-Earth, relative to another. Whatever happens with one has no bearing on what happens with another. Now, I’d be able to say this and get away with it if it weren’t for light and other force carrying particles. It seems that light indeed bridges the gap between events suspended in space and time. Light is emitted from one event, takes some time to travel through space, and arrives at another event. Then we can form a connection between the times when each event happened. Light, and other force carrying particles, are the element that connect our worlds together and makes them all part of one reality. But without light, there is no fact of the matter about where everything is with respect to each other or when everything happened. Light makes the connection, it establishes simultaneity between one event and another relative to a specific reference frame. But there is no simultaneity before that happens. There are no actual “lines” taken from some absolute natural coordinate system connecting events together in time and space. There is only nothingness–the void that makes everything that follows from Einstein’s relativity possible and necessary.
