Post by LaurenceClarkCrossenPost by Python"Laurence", what is your level of education in maths? Just asking.
It doesn't matter because math can't bend space. Anyone who thinks it
can is incompetent in physics. For example, Einstein.
What an idiotic statement! :-D
Math does obviously not curve space.
In GR it is matter and energy that curve spacetime.
GR is a mathematical model which is able to predict what
will be measured in space and time in the real world.
That's the only point with theories of physics.
The validity of a theory depend on its ability to
correctly predict what will be measured in experiments.
GR is thoroughly tested and never falsified.
Remember this statement of yours?
"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.
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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 may meet.
Conclusion:
If mass is present, spacetime is curved.
But remember, spacetime is an entity in the mathematical model GR.
It is meaningless to ask if "spacetime" really exist.
The point is that the mathematical GR correctly predicts
what will be measured in the real space and time.
--
Paul
https://paulba.no/