Re: Newton: Photon falling from h meters increase its energy.

Den 17.01.2025 22:20, skrev LaurenceClarkCrossen:

On Thu, 16 Jan 2025 19:35:26 +0000, Paul.B.Andersen wrote:
 

Den 15.01.2025 06:31, skrev LaurenceClarkCrossen:

Euclidean geometry is for planes, and non-Euclidean geometry is for
other surfaces, such as spheres. Space is not a curved surface.

>
One can but admire your sharp observation that space is not a surface.

It is sad that you can't recognize that non-Euclidean geometry applied
to space is a reification fallacy because space is not a surface.

This rather funny statement of yours reveals that the only
non-Euclidean geometry you know is 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 describing a flat 2D surface is:
   ds² = dx² + dy²   (if Pythagoras is valid, the surface is flat)
The metric describing a 2D spherical surface is:
   ds² = dθ² + sin²θ⋅dφ²
Note that only two coordinates are needed to describe the surface.
The coordinates are _in_ the surface, not in a 3D-space.
----------
The metric for a "flat 3D-space" (Euclidean space) is:
  ds² = dx² + dy² + dy²   (Pythagoras again!)
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.
----------
In spacetime geometry there is a four dimensional manifold called
spacetime. The spacetime metric has four coordinates, one temporal
and four spatial.
The metric for a static flat spacetime is:
  ds² = − (c⋅dt)²  + dx² + dy² + dz²
If  ds² is positive, the line element ds is space-like,
If ds² is negative, the line element ds is time-like.
In the latter case it is better to write the metric:
  (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:
  See equation (2) in
https://paulba.no/pdf/Clock_rate.pdf
Note that there are four coordinates,  t, r, θ and ϕ
--------------------
So to your parallel lines which you claim have to meet in curved space.
Two points:
1. In spacetime geometry, it is spacetime that is curved.
2. What is a "line"? In Euclidean geometry we would say
    "a straight line". A more precise expression is a "geodesic line".
In spacetime geometry the definition of "geodesic line" is rather complicated.
But all free falling objects, including photons, are moving along
geodesic lines. So let us consider light beams (the trajectory of
a photon).
Far out in space, where spacetime is quite flat,
we have two parallel light beams.
These light beam pass on either side of the Sun,
where spacetime is curved.
The light beams are gracing the Sun, and will be
gravitationally deflected by 1.75".
The light beams will then meet 274 AU after they passed the Sun.
Parallel geodesic lines will meet.
--
Paul
https://paulba.no/