Re: Relativity Derives Zero Deflection of Light By Gravity.

On 02/22/2025 01:30 PM, Paul B. Andersen wrote:

Den 21.02.2025 20:37, skrev LaurenceClarkCrossen:

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The only problem is that space is not a
surface, so it can't curve. Non-Euclidean geometry cannot curve space,
so it can't cause gravity.

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I have told you before, but you are unable to learn.
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I will repeat it anyway.
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This rather funny statement of yours reveals that the only
non-Euclidean geometry you know is Gaussian geometry.
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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.
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Note that we must use three coordinates to describe a 2-dimentional
surface.
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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.
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The metric describes the length of a line element.
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The metric describing a flat 2D surface is:
   ds² = dx² + dy²   (if Pythagoras is valid, the surface is flat)
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The metric describing a 2D spherical surface is:
   ds² = dθ² + sin²θ⋅dφ²
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Note that only two coordinates are needed to describe the surface.
The coordinates are _in_ the surface, not in a 3D-space.
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The metric for a "flat 3D-space" (Euclidean space) is:
  ds² = dx² + dy² + dy²   (Pythagoras again!)
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The metric for a 3D-sphere is:
  ds² = dr² + r²dθ² + r²sin²θ⋅dφ²
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Note that only three coordinates are needed to describe
the shape of a 3D space.
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In spacetime geometry there is a four dimensional manifold called
spacetime. The spacetime metric has four coordinates, one temporal
and four spatial.
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The metric for a static flat spacetime is:
  ds² = − (c⋅dt)²  + dx² + dy² + dz²
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If  ds² is positive, the line element ds is space-like,
If ds² is negative, the line element ds is time-like.
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In the latter case it is better to write the metric:
  (c⋅dτ)² =  (c⋅dt)² − dx² − dy² − dz²
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If there is a mass present (Sun, Earth) spacetime will be curved.
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The metric for spacetime in the vicinity of a spherical mass is:
  See equation (2) in
https://paulba.no/pdf/Clock_rate.pdf
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Note that there are four coordinates,  t, r, θ and ϕ
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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.