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

Am Samstag000025, 25.01.2025 um 12:46 schrieb Paul.B.Andersen:

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

It is a bad idea to give time a dimension of the same kind as space.
I suggest to take imaginary numbers for time and real for space.
Than a spacetime-diagramm would be kind of an Argand-diagram, where the axis of time is orthogonal to the axis of space (space axes are multiplied by i).
Since space has three dimension and not only one, we nee to multiply the Argand-diagram by three ('pump it up') and would come to a construct called 'complex four vectors', also known as 'Bi-Quaternions'.
These have eight components in four groups, which each are complex numbers.
This seems to be the structure of spacetime, about which I have written this 'book':
https://docs.google.com/presentation/d/1Ur3_giuk2l439fxUa8QHX4wTDxBEaM6lOlgVUa0cFU4/edit?usp=sharing
TH
...