Den 23.02.2025 05:50, skrev LaurenceClarkCrossen:
On Sun, 23 Feb 2025 0:05:29 +0000, Ross Finlayson wrote:
On 02/22/2025 01:30 PM, Paul B. Andersen wrote:
>
This rather funny statement of yours reveals that the only
non-Euclidean geometry you know is Gaussian geometry.
>
>
Mostly those are all piece-wise and broken one way or the
other with regards to invariance and symmetry, these
"non-Euclidean" geometries, with regards to something
like "Poincare completion", which in the theory of continuous
manifolds, has that it's a continuous manifold.
Thank you for conveying Paul's comment to me. But wait. That doesn't seem to be anything he said to me. I don't find it in the search
function. I don't recall noticing it, and I do not have a lot of time
for this forum.
Yes it was written for you. Twice.
Paul doesn't understand the difference between not understanding and
disagreeing. I disagree with him and you.
You were obviously ignorant of the fact that there are
other non-Euclidean geometries than Gaussian geometry.
Loosely explained, Gaussian geometry is about surfaces in 3-dimentinal
Euclidean space. The shape of the surface is defined by a function
f(x,y,z) where x,y,z are Cartesian coordinates.
Note that we must use three coordinates to describe a 2-dimentional
surface.
Riemannian geometry is more general.
Loosely explained, Riemannian geometry is about manifolds (spaces)
of any dimensions. The "shape" of the manifold is described by
the metric.
The metric describes the length of a line element.
The metric for a "flat 3D-space" (Euclidean space) is:
ds² = dx² + dy² + dy² (Pythagoras!)
The metric for a 3D-sphere is:
ds² = dr² + r²dθ² + r²sin²θ⋅dφ²
Note that only three coordinates are needed to describe
the shape of a 3D space.
It is not possible to disagree about this:
Riemannian geometry can describe curved 3D space.
Fact!
Paul is unable to learn.
Very simply, manifolds are not literal spaces.
A 3D manifold is literally a 3D space.
They are only diagrams
representing non-spatial facts as if they were spatial.
Read this again and try not to laugh! :-D
What is a "non-spatial fact" ?
Which "non-spatial facts" are represented as if they were spatial?
What you are
speaking about are not surfaces.
Indeed! I am speaking about 3d spaces
Riemannian geometry is about representing non-spatial elements as
spatial diagrammatically. Taken literally, this is a reification
fallacy.
A reification fallacy? :-D
So Riemannian geometry is as treating abstract "non spatial entities"
as if they were real "spatial diagrammatically".
Make sense, doesn't it? :-D
Riemannian geometry is mathematics.
It is correct by definition. Like Euclidean geometry.
And there is no such thing as "disagreeing" with
a mathematical definition.
Curves in manifolds are not curves in spaces. Non-Euclidean geometry
cannot bend space or even describe bent space. There is no such thing
because space is not a surface. Space-time fabric is not space.
So let's talk about "spacetime".
First:
Theories of physics such as Newtonian Mechanics [NM], The Special
Theory of Relativity [SR] and The General Theory of Relativity [GR]
are mathematical models of Nature (or the reality or whatever)
they are not Nature. It is meaningless to ask if the entities
in the theory "really exist".
The only test of the validity of a mathematically consistent theory
is if its predictions are in accordance with measurements.
"Spacetime" is an entity in GR, and spacetime geometry is mathematics,
so the following is correct by definition:
In spacetime geometry there is a four dimensional manifold called
spacetime. The spacetime metric has four coordinates, one temporal
and three spatial.
The metric for a static flat spacetime is:
ds² = − (c⋅dt)² + dx² + dy² + dz²
or since ds² = - (c⋅dτ)²
(c⋅dτ)² = (c⋅dt)² − dx² − dy² − dz²
If there is a mass present (Sun, Earth) spacetime will be curved.
The metric for spacetime in the vicinity of a spherical mass is:
ds² = -(1-2GM/c²r)c²dt² + (1/(1-2GM/c²r))dr² + r²(dθ² + sin²θ⋅dϕ²)
or:
(c⋅dτ)² = (1-2GM/c²r)c²dt² - (1/(1-2GM/c²r))dr² - r²(dθ² + sin²θ⋅dϕ²)
Note that there are four coordinates, t, r, θ and ϕ
Bottom line:
It is a fact that the entity spacetime in the mathematical model GR
will be curved if there is a mass present (Sun, Earth).
You are however free to believe that the predictions of GR not
are in accordance with real measurements.
But then you must also believe that all the physicists involved
in all the experiments that have confirmed GR, don't know what
they are doing or are frauds.
Is that what you believe? :-D
-- Paulhttps://paulba.no/
Relativity Derives Zero Deflection of Light By Gravity.
Re: Mass of photon? What it can't exist.