How do space and time fuse together to form “spacetime?”

[Mister T]

But if space is literally the absence of matter or physical properties, and time has no physical properties, how do the two merge to form the 4-dimensional structure called spacetime?

You are over-thinking the concepts here. Take a tape measure and measure the distance between two locations. That's the amount of space between them. Take a stop watch and measure the amount of time that passes between two events. That's the amount of time between them.

Just as when you use something like the Pythagorean theorem you can calculate a distance knowing two or three different dimensions. $r=\sqrt{(x^2+y^2+z^2)}$. You're using Euclidean geometry.

There are other types of geometry. It's useful to use time as a dimension when doing calculations in relativity. You calculate something called the interval, for example a spacelike interval would be $s=\sqrt{(x^2+y^2+z^2-t^2)}$.

In your mind you have somehow misconstrued the concept, like by reading popsci books or articles rather than studying the physics. There are no shortcuts to understanding, popsci accounts attempt to create a shortcut to learning but they often have the opposite effect.

 

[pervect]

TL;DR Summary: If neither space nor time have any physical properties of matter underlying them, how can space and time merge to form spacetime?

Here is the definition of spacetime?

“In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.”

But if space is literally the absence of matter or physical properties, and time has no physical properties, how do the two merge to form the 4-dimensional structure called spacetime?

Spacetime can bend and ripple like a physical object so it seems like spacetime should have physical properties of matter underlying it that allow for the bending and rippling, but it doesn’t. So how do space and time “fuse” to form spacetime if neither have physical properties?

I'd suggest that the warmup problem, described in "The Parable of the Surveyor", could help. There's an online version at http://spiff.rit.edu/classes/phys150/lectures/intro/parable.html, you can also find a slightly different treatment in Taylor & Wheeler's book "Space-time physics", which has a free download of the second edition (not the most current), https://www.eftaylor.com/spacetimephysics/. As always, a textbook reference is more reliable than a web reference, but the web reference is often more convenient.

The warm-up problem is to ask "Why do we consider north-south displacements (henceforce NS) and east-west displacements (henceforth EW) to be of the same nature when we are surveying flat land as opposed to conceptually separate entities?

We could say it's "obvious" - but it's not, really. It's just something we are used to doing.. A related problem would be to ask "what happens if we treated NS and EW differently"? The basic answer, as explained in the parable, is that the theoretical framework becomes more cumbersome, though with enough effort we could perform the calculations this way. It's just unwieldy compared to the treatment when they are unified.

The rather deep answer that is discussed by the parable. is that there is an invariant quantity, "distance", given by the pythagorean theorem, that combines NS and EW displacements into a unified geometric entity, one that is independent of the observer, i.e. an invariant quantity. It's the existence of this invarfiant quantity that underlies the idea that NS and EW don't need to be treated as different entities.

Now, lets move from the "parable of the surveyor" into the problem we were interested in originally, the unification of space and time.

When we move to special relativity, the notions of "distance" and "time" acquire an observer dependent nature, because of effects such as Lorentz contraction and time dilation. So "spatial displacement" and "time displacement" are no longer observer independent quantites, they're not invariants.

But - rather similar to the pythagorean theorem that unifies NS and EW displacements into the invaraint geometrical concept of "distance", there is an analogous formulation of special relativity that combines spatial displacements and time displacements to generate a quantity known as the "Lorentz interval". While spatial displacements and time displacements are observer dependent, the Lorentz interval is independent of the observer. It is invariant.

My goal in this post is not to fully explain the Lorentz interval, but to introduce the concept and give a few references, and to suggest how this answers your question.

The math of the Lorentz interval is no more complicated than the math in the Pythagorean theorem.

To fully understand the concepts will most likely take some study, beyond reading this post, which is basically simply motivational. The issue in understanding it are not complicated mathematics - the math is simple. It's the conceptual issues that take some thought and study.

 

[sbrothy]

Advanced math? Like algebra?

Yeah, I more or less gave up when letters showed up in the equations. :)