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

[DaveC426913]

$$ds^2=-dt^2+dx^2+dy^2+dz^2$$

Am I over-interpreting, or is this (almost) the pythagorean formula for a 4-dimensional cube?
And as long as t is very small, it degenerates to a 3D cube....

 

[Dale]

Am I over-interpreting, or is this (almost) the pythagorean formula for a 4-dimensional cube?
And as long as t is very small, it degenerates to a 3D cube....

It is the spacetime version of the arc length formula. So it could be considered the “length” of the line segment across the diagonal of an infinitesimal 4D cube. Just like in the Pythagorean theorem $A^2+B^2=C^2$ you can consider $C$ to be the length of the diagonal across a rectangle of sides $A$ and $B$.

 

[PeroK]

Then how are gravitational waves formed if spacetime cannot literally bend and ripple?

I used to have a book of physics essays. In the early days of GR, some physicist made this point and criticised the theory of GR on the basis that it contradicted SR and in his view implied an ether. It was the same idea that in order to have geometric properties spacetime must have a physical substance.

But, an alternative approach, is to be guided by nature. We don't assume a priori that geometric properties demand physical susbstance. Instead, we keep an open mind and, as far as GR is concerned, we find spacetime has geometric properties without the need for spacetime to have physical substance.

If you don't accept that, then essentially you are deaf to what nature is telling you and preferring your own a priori reasoning.

The history of modern physics, from Newton to Maxwell to Einstein to Quantum Mechanics is a lesson in looking at nature first before drawing conclusions about how nature must be. If we didn't do that we would still be laboring under the misapprehensions of Aristotle and there would be no modern physics.

 

[Herman Trivilino]

You shouldn’t need advanced math to explain how spacetime works.

If it takes you two hours to drive to grandma's house, a distance of 150 km, then you drive at an average speed of 75 km/h. You've combined space and time in a calculation. That's an example of how spacetime works.

For a full explanation of how spacetime works you do indeed need advanced graduate-level math.

 

[Demystifier]

But if space is literally the absence of matter or physical properties

This is not what space is. First, space is not the same thing as empty space, so it does not need to be without matter. Second, which seems to be the main source of your confusion, even if there is no matter, why do you think that space does not have physical properties? In fact, what exactly do you mean by physical properties?

If you try to find precise answers to such questions, you might realize that the notion of "physical property" can me much more general than you thought. Any property that has some role in the science called "physics" deserves a title of "physical property". Physical does not need to mean tangible, or even measurable. Physics to a large extent is a theoretical science working with mental concepts. Space, time, and spacetime are such theoretical mental concepts. You cannot measure space directly, but you can think of space directly. The fusion of space and time into spacetime should be understood as a mental fusion, not as a material fusion.

 

[Herman Trivilino]

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. :)