Geometric Approaches to the Space-Time Continuum

Hello Blog Readers,

For those of you who don’t know, I do a lot of reading in my spare time, and lately I’ve been doing a lot of reading on theories of gravity.  Of course, Albert Einstein’s general theory of relativity remains the most robust and predictive theory on gravity to date.  It posits that what we perceive as gravity is in reality not a “force” as we generally think of the term, but rather the curvature of “the space-time continuum.”  Space-time here represents 4-dimensions (3-dimensional space together with time), making it difficult to visualize or observe the curvature directly (we would need to be able to see in 5 or more dimensions).  So, to picture the idea, we generally have something like the featured image, where space-time is represented as a 2-dimensional grid being curved in the presence of an object.

We live in an amazing time for physics in that we are starting to observe evidence that supports Einstein’s theory.  From Einstein rings to gravity waves, the predictions of relativity seem to be holding true.  Yet, there are still several lingering issues with the theory.  One lingering issue is whether or not the very mechanism behind the theory actually exists: is there such thing as a space-time continuum!?  Or is space something else?

This is not a trivial question to ask.  Prior to Einstein’s theory, the main theory on the makeup of space was that it was some sort of “ether,” akin to a cosmic liquid through which matter was suspended.  And modern mathematicians and physicists are starting to make use of an area of mathematics called “network theory” to predict that there may be no such thing as space at all.  There are only forms of matter, with connections between those forms.  While this idea is difficult to stomach at first (there sure seems to be a lot of empty, black void out there…), that is the very point.  The void simply represents the pathways between potential interactions with matter.  More on this point below.

Anyways, it is this question of the existence of space-time that recently led to me trying to think up a way to test whether space-time actually exists.  This led to a pretty cool thought experiment.  Sometimes it’s fun being a mathematician.  Here’s the experiment:

You have a box.  Inside the box is a complete vacuum: pure, empty space.  Outside the box is also a complete vacuum.  The box has the ability to compress or expand as much as you’d like.  Nothing can escape or enter the box (as far as matter is concerned).  The question is, what happens to the space inside the box as you compress or expand the box?  Is the space trapped within the box, such that it compresses or expands as you compress or expand the box (this would conform with an “ether” theory of space)?  Or, as relativity predicts, does the space “curve” its way around the box as it compresses, such that it makes its way outside the box at some point?  Or is something else going on entirely?

Let’s see if we can visualize this with some math.  As with usual depictions of relativity, we’ll simplify our representation by bringing it down to 2 dimensions.  We have our box sitting in empty space. The orange dot inside the box does not represent matter, but represents a point in empty space.

Default Box

The question becomes, what happens to our orange dot as we compress the box?  First, let’s deal with relativity.  Now with relativity, the space-time continuum is curved by matter, but otherwise represents a uniform coordinate system in which matter can exist and be placed.  Therefore, as the box gets compressed beyond our orange dot in space, the orange dot will ‘curve’ its way around our inescapable box until poof!  The orange dot ends up outside the box.

Relativity Model

If this experiment could be set up and demonstrated, it would be fascinating to behold.  We would watch a marked point in space, on the inside of a seemingly impenetrable box, end up working its way beyond and outside of the box.  The experiment would lend strong support to the existence of a space-time continuum.  It would also have interesting implications on the shape and structure of space (not on the universe, but on space).  It would mean that space can be set up as a grid, such that an object’s position can be placed on the grid.  Currently, the only method we have of judging position is by relating an object to other objects (in other words, we place earth relative to the sun, the other planets, the galaxy, etc., but we don’t really have a way of saying “earth is here, at this particular point in space and time” without referring to these other objects).  Yet, if the experiment yielded the above result, we could in theory someday create a giant grid of the universe.  All of the heavens could be plotted in this grid.

Of course, the above is not the only possibility for what would happen to space as we compress our box.  When I asked the above question in my Facebook feed, I had several friends predict that space would not “escape the box,” but would simply be compressed inside the box (quite easily I might add seeing as it’s empty space).  This is akin to the ether theory of space.

Ether Model

This theory too would have some pretty fascinating implications.  Under this theory, it would be difficult if not impossible to create a coordinate grid for space, because space would behave much as matter itself – something akin to a liquid, but with the interesting caveat that space, unlike matter, could be created and destroyed.  We would therefore still need to place objects by way of reference to other objects.  Yet, at the same time, space could be compressed or expanded at will, and as such would work to fill and meet the needs of any situation.  If this was the case, in theory we could someday harness space-time itself to create universes within universes, or to make pathways that exploit, work around or avoid the curvature of space.

An ether theory of space would be the stuff of science fiction.  Yet, alas, all implications point to space being akin to the relativity model above.  Matter can curve space and time, but it cannot “trap” it for the purpose of expansion or compression.  Space too seems to behave according to Newton’s laws.  It cannot be created or destroyed.  It just is and here we are on earth flying through it all.

Of course, as discussed above, there is one more possibility.  A network theory approach to space-time means that there is no space-time.  There are only objects and the connections between those objects.  Hence, under a network theory our picture looks like this:

Network Model

Here, there is no space, so there is no point in space that we can mark with an orange dot.  There is only our box, existing in…nothing.  We can expand the box and contract it in the…nothingness.  As there is no space, there’s no curvature of space-time as our box moves around in the nothing.  We cannot create or destroy space.  There’s no “edge” of space or the universe.  We cannot order space on a grid (something that is very difficult for our grid-oriented brains to comprehend).  Rather, there is just our box.  If we wish to place the box, we need to find other objects as reference points.

Though existentially this network model is probably the scariest model of all, there’s a lot about it that works with our notions of “space.”  We’ve yet to witness some sort of ether or continuum representing space and time.  Instead, all we see is a void between objects. Our position in the universe cannot be placed, but is in reference to other objects.  We’ve yet to see a shape to space that we can grid.  We don’t know how or why matter curves some unseen thing called “space-time.”  Instead, with theories on the Higgs-Boson, it seems that what we perceive as gravity is indeed an intrinsic force (in the classical sense) relating to certain particles.  Scientists have had to bend over backwards to try to explain why this magical particle interacts with an unseen continuum, rather than simply concluding ‘here’s where the force of gravity comes from.’

In fact, if someone can rectify the observed phenomenon that seem to confirm relativity (stuff like time distortion and Einstein rings) with a network based approach, a more realistic model of our universe will likely result.  Until then though, Albert Einstein and little orange dots of space making their escape from impenetrable boxes remain king, and we can continue to nerd out over the space-time continuum.





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