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From: clzb93ynxj@att.net (LaurenceClarkCrossen)
Newsgroups: sci.physics.relativity
Subject: Re: Relativity Derives Zero Deflection of Light By Gravity.
Date: Sun, 23 Feb 2025 04:50:18 +0000
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On Sun, 23 Feb 2025 0:05:29 +0000, Ross Finlayson wrote:

> On 02/22/2025 01:30 PM, Paul B. Andersen wrote:
>> Den 21.02.2025 20:37, skrev LaurenceClarkCrossen:
>>>
>>> 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.
>>
>> I have told you before, but you are unable to learn.
>>
>> I will repeat it anyway.
>>
>> 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
>>
>>
>>
>>
>
> 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.

Paul doesn't understand the difference between not understanding and
disagreeing. I disagree with him and you.

Paul is unable to learn.

Very simply, manifolds are not literal spaces. They are only diagrams
representing non-spatial facts as if they were spatial. What you are
speaking about are not surfaces.
Riemannian geometry is about representing non-spatial elements as
spatial diagrammatically. Taken literally, this is a reification
fallacy.

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.

When will you ever understand simple reality? It all boils down to the
reification fallacy, pure and simple.