sobriquet <***@yahoo.com> wrote or quoted:
[Spacetime from Wikipedia]
Post by sobriquet"In physics, spacetime is a mathematical model that fuses the three
Yes, it's a model! And that model usually uses real numbers
for the four dimensions. So we could study the real numbers for
their topological properties. But we would learn something about
physics only to a limited extend, because it's a model, not the
real thing. So, we would learn something about our model. To study
the real world, one needs experimental devices like the LHC.
The model of space as R^3 or spacetime as R^4 is fine for
intermediate orders of magnitude the size of a man. But it
probably is not accurate anymore on a scale very much larger
or very much smaller than that.
Post by sobriquetQuantities like matter and energy initially seemed to be continuous,
Yes. To the bare eye and hands they seem to be continuous.
Post by sobriquetsince they typically consist of extremely large collections of discrete
elements (like molecules, atoms or particles).
This is also only true on a certain scale. You could look at
a particle. What is it? Is it like a ball with a hard border,
with something continuous inside, or is it a kind of a cloud
with a soft border? Measurements seems to indicate that it
is neither of these two (see books at the end).
The crucial point is: mathematics knows no limits when
decending down the scale: [0,1] is an intervall, but the tiny
[0, 10^(-10)] is also in interval that has topologically the
same properties. In mathematics, you can the go on to study
[0, 10^-(10^10)], and you get to study [0, 10^-(10^10^10)] and you
can dive down as deep as possible and it all makes perfect sense.
But it physics, we need high energy particle accelerators to study
small distances. Maybe we can study distances of 10^-15 m there.
But not very much smaller. Even with future technologies, there will
always be some limit beyond which you cannot study the structure of
the real space-time. So it makes no sense to talk about it as if it
would be a mathematical space like the R^4 (the model) where one
can always go down arbitrarily to any scale (size) one chooses.
Post by sobriquetSo gradually we came to realize that the concept of a continuous
quantity that can be subdivided indefinitely is unrealistic.
The indefinite subdivision is possible mentally and maybe
in mathematical models, but we do not have instruments to
inspect the real world to arbitrary small scales. There is
and will always be a scale below which we cannot see anything.
So, from the POV of physics, it makes no sense to wonder
about something that is not observable and will never be.
Post by sobriquetWe don't know if this holds as well for time and space, but it might be
that everything in reality that can be quantified ultimately consists of
discrete elements. If that turns out to be the case, that renders
discussions about the concept of continuity irrelevant, because at that
point it seems to be about a concept that only exists in our imagination.
So it's a bit like discussing how many angels can dance on the head of a
pin (assuming that angels don't really exist).
To think of space like a kind of a cellular automaton (as
did Zuse and later Wolfram) is satisfying because it's an
explanation by something that seems to be easy to understand,
but currently there is no indication that reality works this
way. Lattice gauge theories are a tool for computations and
deliver some useful results, but they are a model.
To learn more about reality one has to look at reality; it does
not help to look at mathematical models. And when we study
reality, there are certain limits beyond which we cannot pass.
Here are two recommendable popular science books that give
two nice perspectives on contemporary physics:
"Quantum Field Theory, As Simple As Possible" (2023), A. Zee and
"Why String Theory" (2016), Joseph Conlon.