On Wed, 19 Feb 2025 15:28:06 +0000, LaurenceClarkCrossen wrote:What you and the consensus believes is nonsense because the momentum
The mass-velocity relationship prevents photons from having any mass.
Therefore, they cannot be affected by gravity.
Gravity couples to eneregy-momentum (all 10 components of it), not
just the mass (the 00-component).
--
Jan
--
On Wed, 19 Feb 2025 15:28:06 +0000, LaurenceClarkCrossen wrote:I disagree with the basics of the theory. You pretend to mind read.
The mass-velocity relationship prevents photons from having any mass.
Therefore, they cannot be affected by gravity.
Gravity couples to eneregy-momentum (all 10 components of it), not
just the mass (the 00-component).
--
Jan
--
On Wed, 19 Feb 2025 15:28:06 +0000, LaurenceClarkCrossen wrote:p=mv
The mass-velocity relationship prevents photons from having any mass.
Therefore, they cannot be affected by gravity.
Gravity couples to eneregy-momentum (all 10 components of it), not
just the mass (the 00-component).
--
Jan
--
On Wed, 19 Feb 2025 15:28:06 +0000, LaurenceClarkCrossen wrote:Photons have no momentum because they have no mass so they are
The mass-velocity relationship prevents photons from having any mass.
Therefore, they cannot be affected by gravity.
Gravity couples to eneregy-momentum (all 10 components of it), not
just the mass (the 00-component).
--
Jan
--
On Wed, 19 Feb 2025 15:28:06 +0000, LaurenceClarkCrossen wrote:I think its those who think gravity affects anything other than mass who
The mass-velocity relationship prevents photons from having any mass.
Therefore, they cannot be affected by gravity.
Gravity couples to eneregy-momentum (all 10 components of it), not
just the mass (the 00-component).
--
Jan
--
Den 19.02.2025 16:28, skrev LaurenceClarkCrossen:
The mass-velocity relationship prevents photons from having any mass.
Therefore, they cannot be affected by gravity.
There is no way you can be ignorant of the fact that
photons are gravitationally deflected.
So why do you claim it is impossible?
The mass-velocity relationship prevents photons from having any mass. Therefore, they cannot be affected by gravity.
W dniu 20.02.2025 o 19:49, Paul B. Andersen pisze:Thank you for reminding me of Paul's ignorant comment. It's his usual
Den 19.02.2025 16:28, skrev LaurenceClarkCrossen:
The mass-velocity relationship prevents photons from having any mass.
Therefore, they cannot be affected by gravity.
There is no way you can be ignorant of the fact that
photons are gravitationally deflected.
So why do you claim it is impossible?
Maybe he is a hidden foollower of your
idiot guru who thought and postulated the
same.
On Thu, 20 Feb 2025 19:13:51 +0000, Maciej Wozniak wrote:
W dniu 20.02.2025 o 19:49, Paul B. Andersen pisze:Thank you for reminding me of Paul's ignorant comment. It's his usual
Den 19.02.2025 16:28, skrev LaurenceClarkCrossen:
The mass-velocity relationship prevents photons from having any mass.
Therefore, they cannot be affected by gravity.
There is no way you can be ignorant of the fact that
photons are gravitationally deflected.
So why do you claim it is impossible?
Maybe he is a hidden foollower of your
idiot guru who thought and postulated the
same.
appeal to experiment that begs the question of the derivation's
validity. Paul is ignorant of the lack of a derivation for the doubling.
Jan is saying you can have momentum without mass because he's ignorant
that momentum requires mass: p = mv.
Thank you for reminding me of Paul's ignorant comment. It's his usual
appeal to experiment that begs the question of the derivation's
validity. Paul is ignorant of the lack of a derivation for the doubling.
Jan is saying you can have momentum without mass because he's ignorant
that momentum requires mass: p = mv.
So you prefer to believe that your derivation that
photons are not affected by gravity is correct,
and that GR's predictions thus are proven wrong.
So you must be ignorant of the fact that photons are observed
to be deflected by gravity exactly as predicted by GR.
Because you would not claim that "photons cannot be affected
by gravity" if you knew that they are.
Or would you? :-D
Paul
On Wed, 19 Feb 2025 21:51:26 +0000, LaurenceClarkCrossen wrote:Then, you can obviously prove that momentum can exist without mass by
On Wed, 19 Feb 2025 20:39:33 +0000, JanPB wrote:
On Wed, 19 Feb 2025 15:28:06 +0000, LaurenceClarkCrossen wrote:p=mv
The mass-velocity relationship prevents photons from having any mass.
Therefore, they cannot be affected by gravity.
Gravity couples to eneregy-momentum (all 10 components of it), not
just the mass (the 00-component).
--
Jan
--
p= 0v= 0
This is just an elementary school formula which you never progressed
past.
You are like an adult that forever is stuck practising scales while
claiming Bach and Beethoven never existed.
--
Jan
--
On Wed, 19 Feb 2025 21:51:26 +0000, LaurenceClarkCrossen wrote:Everyone can see you cannot explain how anything can have momentum
On Wed, 19 Feb 2025 20:39:33 +0000, JanPB wrote:
On Wed, 19 Feb 2025 15:28:06 +0000, LaurenceClarkCrossen wrote:p=mv
The mass-velocity relationship prevents photons from having any mass.
Therefore, they cannot be affected by gravity.
Gravity couples to eneregy-momentum (all 10 components of it), not
just the mass (the 00-component).
--
Jan
--
p= 0v= 0
This is just an elementary school formula which you never progressed
past.
You are like an adult that forever is stuck practising scales while
claiming Bach and Beethoven never existed.
--
Jan
--
On Wed, 19 Feb 2025 21:42:01 +0000, LaurenceClarkCrossen wrote:I'm not pontificating. I'm only giving my opinion. At least I give
On Wed, 19 Feb 2025 20:39:33 +0000, JanPB wrote:
On Wed, 19 Feb 2025 15:28:06 +0000, LaurenceClarkCrossen wrote:What you and the consensus believes is nonsense because the momentum
The mass-velocity relationship prevents photons from having any mass.
Therefore, they cannot be affected by gravity.
Gravity couples to eneregy-momentum (all 10 components of it), not
just the mass (the 00-component).
--
Jan
--
formula requires mass.
Incorrect.
Mass is a prerequisite.
Incorrect. Why don't you study physics before pontificating idiocies?
Massless particles would
have no momentum.
Nonsense. You simply repeat a bit of elementary school physics that
you never progressed past.
You are very confused. You are arguing from authority
instead of reasoning. That is ad verecundium, a failure to reason.
Gravity does not couple with massless particles.
I'm simply saying that you have no idea what you are talking about.
--
Jan
--
Everyone can see you cannot explain how anything can have momentum
This is just an elementary school formula which you never progressed
past.
You are like an adult that forever is stuck practising scales while
claiming Bach and Beethoven never existed.
--
Jan
--
without mass. Resort to ad hominem is a failure to reason.
Le 20/02/2025 à 23:39, clzb93ynxj@att.net (LaurenceClarkCrossen) a écritOf course, p=mv/sqrt(1-v^2/c^2) still requires mass, and it is mistaken
:
Everyone can see you cannot explain how anything can have momentum
This is just an elementary school formula which you never progressed
past.
You are like an adult that forever is stuck practising scales while
claiming Bach and Beethoven never existed.
--
Jan
--
without mass. Resort to ad hominem is a failure to reason.
Ad hominem attacks are unfortunately far too frequent. We don't explain anything, we attack.
Here, I'm not sure that the attack is even scientifically justified,
because it is wrong to believe that there has been a fantastic evolution
of thought since p=mv.
Certainly the relativists understood that p=mv/sqrt(1-v²/c²) but there
is no need to shout glory to God from the rooftops. They don't even know
why, and most of them believe that it is mass that is relative, and not
the perception of speed.
Don't laugh, friends, I've been reading what scientists and social
networks say for 40 years.
And it's scary.
It's terrifyingly stupid.
R.H.
Le 20/02/2025 à 22:31, "Paul B. Andersen" a écrit :Light remains a mystery. The problem with the refraction solution is
So you prefer to believe that your derivation that
photons are not affected by gravity is correct,
and that GR's predictions thus are proven wrong.
So you must be ignorant of the fact that photons are observed
to be deflected by gravity exactly as predicted by GR.
Because you would not claim that "photons cannot be affected
by gravity" if you knew that they are.
Or would you? :-D
Paul
What is the evidence that photons are deflected by the presence of
matter
in space?
If the space surrounding bodies has the ability to deflect the
trajectory
of bodies, then space is not pure nothingness.
But how can absolute vacuum manage to deflect bodies and photons with
its
little hooked fingers?
What evidence do we have of this?
The deflection of the rays of stars at the solar periphery?
This is not evidence, but a grotesque claim. The sun ejects matter much further than one to two solar radii, and therefore the sun is surrounded
by particles and gases ejected over millions of kilometers (which slowly
fall back to its surface).
Can't these gases and particles, over millions of kilometers, have
refracting capacities?
The same goes for galaxies, probably surrounded by enormous masses of
gas
over thousands of light years. So, couldn't this cause diffraction
rings?
As for black holes, which we can't see, who can prove their existence?
Can't a huge mass of matter, at the center of a galaxy, have all the gravitational characteristics of a black hole without being a black
hole?
Haven't we made physics more metaphysical than physical?
R.H.
Le 20/02/2025 à 22:31, "Paul B. Andersen" a écrit :
So you prefer to believe that your derivation that
photons are not affected by gravity is correct,
and that GR's predictions thus are proven wrong.
So you must be ignorant of the fact that photons are observed
to be deflected by gravity exactly as predicted by GR.
Because you would not claim that "photons cannot be affected
by gravity" if you knew that they are.
Or would you? :-D
Paul
What is the evidence that photons are deflected by the presence of
matter in space?
Den 20.02.2025 20:34, skrev LaurenceClarkCrossen:
On Thu, 20 Feb 2025 19:13:51 +0000, Maciej Wozniak wrote:
W dniu 20.02.2025 o 19:49, Paul B. Andersen pisze:Thank you for reminding me of Paul's ignorant comment. It's his usual
Den 19.02.2025 16:28, skrev LaurenceClarkCrossen:
The mass-velocity relationship prevents photons from having any mass. >>>>> Therefore, they cannot be affected by gravity.
There is no way you can be ignorant of the fact that
photons are gravitationally deflected.
So why do you claim it is impossible?
Maybe he is a hidden foollower of your
idiot guru who thought and postulated the
same.
appeal to experiment that begs the question of the derivation's
validity. Paul is ignorant of the lack of a derivation for the doubling.
Jan is saying you can have momentum without mass because he's ignorant
that momentum requires mass: p = mv.
So you prefer to believe that your derivation that
photons are not affected by gravity is correct,
and that GR's predictions thus are proven wrong.
So you must be ignorant of the fact that photons are observed
to be deflected by gravity exactly as predicted by GR.
The photons are not affected, but space itself is.
The idea of GR was, that gravity is actually an effect of 'curvature',
which is itself caused by gravity.
What gets curved is actually the 'axis of time' local to a certain part
of space.
What we call 'space' has a certain (geometric) relation to the axis of
time, if we regard the axis of time as imaginary and the axes of space
as real.
Then we have i as a factor, by which time gets multiplied and what gives
us three real axes of space.
This space is therefore depending on the direction of time.
Curvature of the axis of time is actually an acceleration in a
space-time diagram, which usually has one axis of time and only one of
type space.
Now this can 'curve' and we get gravity, which is a force, that results
from such curvature.
Now light ('photons') pass through such a distorted space upon force
free straight lines, which are actually curved in spacetime in presence
of a gravitational field.
This pass is NOT curved by gravitation, but by curvature of space.
...
Am Donnerstag000020, 20.02.2025 um 22:44 schrieb Richard Hachel:
Le 20/02/2025 à 22:31, "Paul B. Andersen" a écrit :
So you prefer to believe that your derivation that
photons are not affected by gravity is correct,
and that GR's predictions thus are proven wrong.
So you must be ignorant of the fact that photons are observed
to be deflected by gravity exactly as predicted by GR.
Because you would not claim that "photons cannot be affected
by gravity" if you knew that they are.
Or would you? :-D
Paul
What is the evidence that photons are deflected by the presence of
matter in space?
The photons are not affected, but space itself is.
The idea of GR was, that gravity is actually an effect of 'curvature',
which is itself caused by gravity.
What gets curved is actually the 'axis of time' local to a certain part
of space.
What we call 'space' has a certain (geometric) relation to the axis of
time, if we regard the axis of time as imaginary and the axes of space
as real.
Then we have i as a factor, by which time gets multiplied and what gives
us three real axes of space.
This space is therefore depending on the direction of time.
Curvature of the axis of time is actually an acceleration in a
space-time diagram, which usually has one axis of time and only one of
type space.
Now this can 'curve' and we get gravity, which is a force, that results
from such curvature.
Now light ('photons') pass through such a distorted space upon force
free straight lines, which are actually curved in spacetime in presence
of a gravitational field.
This pass is NOT curved by gravitation, but by curvature of space.
....
THit is good to hear someone here who can plainly explain relativity so
Le 21/02/2025 à 09:25, Thomas Heger a écrit :
The photons are not affected, but space itself is.
The idea of GR was, that gravity is actually an effect of 'curvature',
which is itself caused by gravity.
What gets curved is actually the 'axis of time' local to a certain
part of space.
What we call 'space' has a certain (geometric) relation to the axis of
time, if we regard the axis of time as imaginary and the axes of space
as real.
Then we have i as a factor, by which time gets multiplied and what
gives us three real axes of space.
This space is therefore depending on the direction of time.
Curvature of the axis of time is actually an acceleration in a space-
time diagram, which usually has one axis of time and only one of type
space.
Now this can 'curve' and we get gravity, which is a force, that
results from such curvature.
Now light ('photons') pass through such a distorted space upon force
free straight lines, which are actually curved in spacetime in
presence of a gravitational field.
This pass is NOT curved by gravitation, but by curvature of space.
...
That's what the scientists say.
I can't help but think that this is all a bit unclear.
On Fri, 21 Feb 2025 8:26:26 +0000, Thomas Heger wrote:
Am Donnerstag000020, 20.02.2025 um 22:44 schrieb Richard Hachel:
Le 20/02/2025 22:31, "Paul B. Andersen" a écrit :
So you prefer to believe that your derivation that
photons are not affected by gravity is correct,
and that GR's predictions thus are proven wrong.
So you must be ignorant of the fact that photons are observed
to be deflected by gravity exactly as predicted by GR.
Because you would not claim that "photons cannot be affected
by gravity" if you knew that they are.
Or would you? :-D
Paul
What is the evidence that photons are deflected by the presence of
matter in space?
The photons are not affected, but space itself is.
The idea of GR was, that gravity is actually an effect of 'curvature', which is itself caused by gravity.
What gets curved is actually the 'axis of time' local to a certain part
of space.
What we call 'space' has a certain (geometric) relation to the axis of time, if we regard the axis of time as imaginary and the axes of space
as real.
Then we have i as a factor, by which time gets multiplied and what gives
us three real axes of space.
This space is therefore depending on the direction of time.
Curvature of the axis of time is actually an acceleration in a
space-time diagram, which usually has one axis of time and only one of
type space.
Now this can 'curve' and we get gravity, which is a force, that results from such curvature.
Now light ('photons') pass through such a distorted space upon force
free straight lines, which are actually curved in spacetime in presence
of a gravitational field.
This pass is NOT curved by gravitation, but by curvature of space.
....
THit is good to hear someone here who can plainly explain relativity so everyone can understand. 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. The idea that the concept of curved space is somehow an improvement over the idea of fields of force is mistaken
because it involves the elementary error of reification fallacy.
On Fri, 21 Feb 2025 8:26:26 +0000, Thomas Heger wrote:
Am Donnerstag000020, 20.02.2025 um 22:44 schrieb Richard Hachel:it is good to hear someone here who can plainly explain relativity so everyone can understand. The only problem is that space is not a
Le 20/02/2025 à 22:31, "Paul B. Andersen" a écrit :
So you prefer to believe that your derivation that
photons are not affected by gravity is correct,
and that GR's predictions thus are proven wrong.
So you must be ignorant of the fact that photons are observed
to be deflected by gravity exactly as predicted by GR.
Because you would not claim that "photons cannot be affected
by gravity" if you knew that they are.
Or would you? :-D
Paul
What is the evidence that photons are deflected by the presence of
matter in space?
The photons are not affected, but space itself is.
The idea of GR was, that gravity is actually an effect of 'curvature',
which is itself caused by gravity.
What gets curved is actually the 'axis of time' local to a certain part
of space.
What we call 'space' has a certain (geometric) relation to the axis of
time, if we regard the axis of time as imaginary and the axes of space
as real.
Then we have i as a factor, by which time gets multiplied and what gives
us three real axes of space.
This space is therefore depending on the direction of time.
Curvature of the axis of time is actually an acceleration in a
space-time diagram, which usually has one axis of time and only one of
type space.
Now this can 'curve' and we get gravity, which is a force, that results
from such curvature.
Now light ('photons') pass through such a distorted space upon force
free straight lines, which are actually curved in spacetime in presence
of a gravitational field.
This pass is NOT curved by gravitation, but by curvature of space.
....
TH
surface, so it can't curve. Non-Euclidean geometry cannot curve space,
so it can't cause gravity. The idea that the concept of curved space is somehow an improvement over the idea of fields of force is mistaken
because it involves the elementary error of reification fallacy.
On Wed, 19 Feb 2025 21:45:37 +0000, LaurenceClarkCrossen wrote:If you could reason, you wouldn't insult.
On Wed, 19 Feb 2025 20:39:33 +0000, JanPB wrote:
On Wed, 19 Feb 2025 15:28:06 +0000, LaurenceClarkCrossen wrote:I disagree with the basics of the theory. You pretend to mind read.
The mass-velocity relationship prevents photons from having any mass.
Therefore, they cannot be affected by gravity.
Gravity couples to eneregy-momentum (all 10 components of it), not
just the mass (the 00-component).
--
Jan
--
Unsolicited advice is worthless. Deplatforming is a defense tactic of
ideologues and that relativity is an ideology has been demonstrated by
Peter Hayes in "The Ideology of Relativity: The Case of the Clock
Paradox."
What components couple with gravity that have no mass? Photons, Higgs
bosons, and what else? Particles without mass? I certainly don't accept
that nonsense.
That you think anything but mass couples to gravity disqualifies you.
Nonsense. Pick a different hobby.
--
Jan
--
LaurenceClarkCrossen wrote:
On Fri, 21 Feb 2025 8:26:26 +0000, Thomas Heger wrote:
Am Donnerstag000020, 20.02.2025 um 22:44 schrieb Richard Hachel:it is good to hear someone here who can plainly explain relativity so
Le 20/02/2025 à 22:31, "Paul B. Andersen" a écrit :
So you prefer to believe that your derivation that
photons are not affected by gravity is correct,
and that GR's predictions thus are proven wrong.
So you must be ignorant of the fact that photons are observed
to be deflected by gravity exactly as predicted by GR.
Because you would not claim that "photons cannot be affected
by gravity" if you knew that they are.
Or would you? :-D
Paul
What is the evidence that photons are deflected by the presence of
matter in space?
The photons are not affected, but space itself is.
The idea of GR was, that gravity is actually an effect of 'curvature',
which is itself caused by gravity.
What gets curved is actually the 'axis of time' local to a certain part
of space.
What we call 'space' has a certain (geometric) relation to the axis of
time, if we regard the axis of time as imaginary and the axes of space
as real.
Then we have i as a factor, by which time gets multiplied and what gives >>> us three real axes of space.
This space is therefore depending on the direction of time.
Curvature of the axis of time is actually an acceleration in a
space-time diagram, which usually has one axis of time and only one of
type space.
Now this can 'curve' and we get gravity, which is a force, that results
from such curvature.
Now light ('photons') pass through such a distorted space upon force
free straight lines, which are actually curved in spacetime in presence
of a gravitational field.
This pass is NOT curved by gravitation, but by curvature of space.
....
TH
everyone can understand. 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. The idea that the concept of curved space is
somehow an improvement over the idea of fields of force is mistaken
because it involves the elementary error of reification fallacy.
Gravity does not bend light.
Gravity doesn't bend anything...
it's only...mindbending.
Relativity is Einstin on acid.
purplehaze
spacetime
lucy
in
the
sky
with
diamonds
Am Donnerstag000020, 20.02.2025 um 22:44 schrieb Richard Hachel:
Le 20/02/2025 à 22:31, "Paul B. Andersen" a écrit :
So you prefer to believe that your derivation that
photons are not affected by gravity is correct,
and that GR's predictions thus are proven wrong.
So you must be ignorant of the fact that photons are observed
to be deflected by gravity exactly as predicted by GR.
Because you would not claim that "photons cannot be affected
by gravity" if you knew that they are.
Or would you? :-D
Paul
What is the evidence that photons are deflected by the presence of
matter in space?
The photons are not affected, but space itself is.
The idea of GR was, that gravity is actually an effect of 'curvature',
which is itself caused by gravity.
What gets curved is actually the 'axis of time' local to a certain part
of space.
What we call 'space' has a certain (geometric) relation to the axis of
time, if we regard the axis of time as imaginary and the axes of space
as real.
Then we have i as a factor, by which time gets multiplied and what gives
us three real axes of space.
This space is therefore depending on the direction of time.
Curvature of the axis of time is actually an acceleration in a
space-time diagram, which usually has one axis of time and only one of
type space.
Now this can 'curve' and we get gravity, which is a force, that results
from such curvature.
Now light ('photons') pass through such a distorted space upon force
free straight lines, which are actually curved in spacetime in presence
of a gravitational field.
This pass is NOT curved by gravitation, but by curvature of space.
....
THCurving the (imaginary) axis of time cannot curve real space since space
That's what the scientists say.
I can't help but think that this is all a bit unclear.
Well, possibly.
Le 22/02/2025 à 08:31, Thomas Heger a écrit :Relativity is prestigious because it is not critically examined. If it
That's what the scientists say.
I can't help but think that this is all a bit unclear.
Well, possibly.
Many things scientists say are not very clear.
It would not be a major problem in the history of humanity if they
agreed
to discuss it, and to come back to the less clear points, to doubt them,
or even abandon them.
But they do not want to.
Maybe artificial intelligence will one day become truly intelligent and
truly efficient. Maybe.
It will then impose the logical concepts that today's physicists and mathematicians do not want to hear. Maybe.
This will necessarily involve a lot of astonishment.
I remind you that I managed to rewrite ALL of special relativity with
ALL
the correct equations, just as I denounced "problems" in the basic
concepts of complex numbers taught at school, where things are incorrect
and extraordinarily complicated (we don't even know how to clearly
explain
what i is, and we believe absurdities such as if i²=-1 then i^4=1 since (-a)²(-a)²=a^4).
For 40 years I have been pointing out, here and there, huge blunders of
logic and concept, asking that we review things that seem clear, but are nevertheless false.
Human arrogance invariably answers me that if all this were false, it
would be known.
It would be known?
Are we sure?
R.H.
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.
On Sat, 22 Feb 2025 20:52:31 +0000, Richard Hachel wrote:
Relativity is prestigious because it is not critically examined. If it
were, it would be seen to be nonsense, as thousands of scientists have pointed out.
On 02/22/2025 01:30 PM, Paul B. Andersen wrote:
Den 21.02.2025 20:37, skrev LaurenceClarkCrossen:
The metric for a static flat spacetime is:
ds² = − (c⋅dt)² + dx² + dy² + dz²
On 02/22/2025 01:30 PM, Paul B. Andersen wrote:Thank you for conveying Paul's comment to me. But wait. That doesn't
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.
Le 22/02/2025 à 22:43, clzb93ynxj@att.net (LaurenceClarkCrossen) a écritHercule Poirot would understand that some things are 99% untrustworthy.
:
On Sat, 22 Feb 2025 20:52:31 +0000, Richard Hachel wrote:
Relativity is prestigious because it is not critically examined. If it
were, it would be seen to be nonsense, as thousands of scientists have
pointed out.
The problem is that all those who have tried to contradict or improve
the
theory of relativity have done worse than better.
Absolutely all of them.
Since 1905, and the publication of Poincaré's work, everything that has
been added to it has made the theory drift into horror.
The theory of Minkowskian space, the explanation of the Langevin paradox
by the Doppler effect (but omitting the fantastic theoretical spatial
zoom
by change of reference frame that no one takes into account), black
holes,
spaghetti deformations, white fountains and wormholes, all this has
turned
into the absurd.
Even today, those who seek are looking badly.
And those who are wrong not to seek are just as much lost in useless skepticism.
R.H.
Le 22/02/2025 à 22:43, clzb93ynxj@att.net (LaurenceClarkCrossen) a écrit :
On Sat, 22 Feb 2025 20:52:31 +0000, Richard Hachel wrote:
Relativity is prestigious because it is not critically examined. If it
were, it would be seen to be nonsense, as thousands of scientists have
pointed out.
The problem is that all those who have tried to contradict or improve
the theory of relativity have done worse than better.
Absolutely all of them.
Since 1905, and the publication of Poincaré's work, everything that has
been added to it has made the theory drift into horror.
The theory of Minkowskian space, the explanation of the Langevin paradox
by the Doppler effect (but omitting the fantastic theoretical spatial
zoom by change of reference frame that no one takes into account), black holes, spaghetti deformations, white fountains and wormholes, all this
has turned into the absurd.
On Fri, 21 Feb 2025 8:26:26 +0000, Thomas Heger wrote:
Am Donnerstag000020, 20.02.2025 um 22:44 schrieb Richard Hachel:Curving the (imaginary) axis of time cannot curve real space since space
Le 20/02/2025 à 22:31, "Paul B. Andersen" a écrit :
So you prefer to believe that your derivation that
photons are not affected by gravity is correct,
and that GR's predictions thus are proven wrong.
So you must be ignorant of the fact that photons are observed
to be deflected by gravity exactly as predicted by GR.
Because you would not claim that "photons cannot be affected
by gravity" if you knew that they are.
Or would you? :-D
Paul
What is the evidence that photons are deflected by the presence of
matter in space?
The photons are not affected, but space itself is.
The idea of GR was, that gravity is actually an effect of 'curvature',
which is itself caused by gravity.
What gets curved is actually the 'axis of time' local to a certain part
of space.
What we call 'space' has a certain (geometric) relation to the axis of
time, if we regard the axis of time as imaginary and the axes of space
as real.
Then we have i as a factor, by which time gets multiplied and what gives
us three real axes of space.
This space is therefore depending on the direction of time.
Curvature of the axis of time is actually an acceleration in a
space-time diagram, which usually has one axis of time and only one of
type space.
Now this can 'curve' and we get gravity, which is a force, that results
from such curvature.
Now light ('photons') pass through such a distorted space upon force
free straight lines, which are actually curved in spacetime in presence
of a gravitational field.
This pass is NOT curved by gravitation, but by curvature of space.
....
TH
is not a surface.
Le 22/02/2025 à 08:31, Thomas Heger a écrit :
That's what the scientists say.
I can't help but think that this is all a bit unclear.
Well, possibly.
Many things scientists say are not very clear.
It would not be a major problem in the history of humanity if they
agreed to discuss it, and to come back to the less clear points, to
doubt them, or even abandon them.
But they do not want to.
Maybe artificial intelligence will one day become truly intelligent and
truly efficient. Maybe.
It will then impose the logical concepts that today's physicists and mathematicians do not want to hear. Maybe.
This will necessarily involve a lot of astonishment.
I remind you that I managed to rewrite ALL of special relativity with
ALL the correct equations, just as I denounced "problems" in the basic concepts of complex numbers taught at school, where things are incorrect
and extraordinarily complicated (we don't even know how to clearly
explain what i is, and we believe absurdities such as if i²=-1 then
i^4=1 since (-a)²(-a)²=a^4).
For 40 years I have been pointing out, here and there, huge blunders of
logic and concept, asking that we review things that seem clear, but are nevertheless false.
Human arrogance invariably answers me that if all this were false, it
would be known.
It would be known?
Are we sure?
On Sun, 23 Feb 2025 0:05:29 +0000, Ross Finlayson wrote:
On 02/22/2025 01:30 PM, Paul B. Andersen wrote:
This rather funny statement of yours reveals that the only
non-Euclidean geometry you know is Gaussian geometry.
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.
Loosely explained, Gaussian geometry is about surfaces in 3-dimentinal Euclidean space.
But then you must also believe that all the physicists involved
in all the experiments that have confirmed GR, don't know what
they are doing or are frauds.
R.H.Hercule Poirot would understand that some things are 99% untrustworthy.
I find that most critics of relativity, especially you, don't go far
enough. For example, the photoelectric effect does not prove or call for photons.
The metric for a static flat spacetime is:OOOps! Nasty typo!
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.
W dniu 23.02.2025 o 14:17, Paul.B.Andersen pisze:
Loosely explained, Gaussian geometry is about surfaces in 3-dimentinal
Euclidean space.
Paul, poor halfbrain, it's Euclidean geometry
which is about surfaces in Euclidean space.
Paul has to divert from the derivation to the experiments because heBut then you must also believe that all the physicists involved
in all the experiments that have confirmed GR, don't know what
they are doing or are frauds.
Well, the facts are not to discuss, are they?
If s² is positive, s is space-like,
If s² is negative, s is time-like.
Den 23.02.2025 05:50, skrev LaurenceClarkCrossen:
On Sun, 23 Feb 2025 0:05:29 +0000, Ross Finlayson wrote:
On 02/22/2025 01:30 PM, Paul B. Andersen wrote:
This rather funny statement of yours reveals that the only
non-Euclidean geometry you know is Gaussian geometry.
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.
Yes it was written for you. Twice.
Paul doesn't understand the difference between not understanding and
disagreeing. I disagree with him and you.
You were obviously ignorant of the fact that there are
other non-Euclidean geometries than 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 for a "flat 3D-space" (Euclidean space) is:
ds² = dx² + dy² + dy² (Pythagoras!)
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.
It is not possible to disagree about this:
Riemannian geometry can describe curved 3D space.
Fact!
Paul is unable to learn.
Very simply, manifolds are not literal spaces.
A 3D manifold is literally a 3D space.
They are only diagrams
representing non-spatial facts as if they were spatial.
Read this again and try not to laugh! :-D
What is a "non-spatial fact" ?
Which "non-spatial facts" are represented as if they were spatial?
What you are
speaking about are not surfaces.
Indeed! I am speaking about 3d spaces
Riemannian geometry is about representing non-spatial elements as
spatial diagrammatically. Taken literally, this is a reification
fallacy.
A reification fallacy? :-D
So Riemannian geometry is as treating abstract "non spatial entities"
as if they were real "spatial diagrammatically".
Make sense, doesn't it? :-D
Riemannian geometry is mathematics.
It is correct by definition. Like Euclidean geometry.
And there is no such thing as "disagreeing" with
a mathematical definition.
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.
So let's talk about "spacetime".
First:
Theories of physics such as Newtonian Mechanics [NM], The Special
Theory of Relativity [SR] and The General Theory of Relativity [GR]
are mathematical models of Nature (or the reality or whatever)
they are not Nature. It is meaningless to ask if the entities
in the theory "really exist".
The only test of the validity of a mathematically consistent theory
is if its predictions are in accordance with measurements.
"Spacetime" is an entity in GR, and spacetime geometry is mathematics,
so the following is correct by definition:
In spacetime geometry there is a four dimensional manifold called
spacetime. The spacetime metric has four coordinates, one temporal
and three spatial.
The metric for a static flat spacetime is:
ds² = − (c⋅dt)² + dx² + dy² + dz²
or since ds² = - (c⋅dτ)²
(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:
ds² = -(1-2GM/c²r)c²dt² + (1/(1-2GM/c²r))dr² + r²(dθ² + sin²θ⋅dϕ²)
or:
(c⋅dτ)² = (1-2GM/c²r)c²dt² - (1/(1-2GM/c²r))dr² - r²(dθ² + sin²θ⋅dϕ²)
Note that there are four coordinates, t, r, θ and ϕ
Bottom line:
It is a fact that the entity spacetime in the mathematical model GR
will be curved if there is a mass present (Sun, Earth).
You are however free to believe that the predictions of GR not
are in accordance with real measurements.
But then you must also believe that all the physicists involved
in all the experiments that have confirmed GR, don't know what
they are doing or are frauds.
Is that what you believe? :-D
Am Samstag000022, 22.02.2025 um 20:28 schrieb LaurenceClarkCrossen:
On Fri, 21 Feb 2025 8:26:26 +0000, Thomas Heger wrote:
Am Donnerstag000020, 20.02.2025 um 22:44 schrieb Richard Hachel:Curving the (imaginary) axis of time cannot curve real space since space
Le 20/02/2025 à 22:31, "Paul B. Andersen" a écrit :
So you prefer to believe that your derivation that
photons are not affected by gravity is correct,
and that GR's predictions thus are proven wrong.
So you must be ignorant of the fact that photons are observed
to be deflected by gravity exactly as predicted by GR.
Because you would not claim that "photons cannot be affected
by gravity" if you knew that they are.
Or would you? :-D
Paul
What is the evidence that photons are deflected by the presence of
matter in space?
The photons are not affected, but space itself is.
The idea of GR was, that gravity is actually an effect of 'curvature',
which is itself caused by gravity.
What gets curved is actually the 'axis of time' local to a certain part
of space.
What we call 'space' has a certain (geometric) relation to the axis of
time, if we regard the axis of time as imaginary and the axes of space
as real.
Then we have i as a factor, by which time gets multiplied and what gives >>> us three real axes of space.
This space is therefore depending on the direction of time.
Curvature of the axis of time is actually an acceleration in a
space-time diagram, which usually has one axis of time and only one of
type space.
Now this can 'curve' and we get gravity, which is a force, that results
from such curvature.
Now light ('photons') pass through such a distorted space upon force
free straight lines, which are actually curved in spacetime in presence
of a gravitational field.
This pass is NOT curved by gravitation, but by curvature of space.
....
TH
is not a surface.
What we call 'space' is actually our own past-light cone.Relativity needs 3D space to curve without a surface to bend the path of
It is a picture we receive from the past and therefore not real.
This picture depends on our own position and state of motion.
We regard it as 'real thing' because we can see stars and galaxies and
all sort of other things, which actually do not exist (anymore).
TH
Den 23.02.2025 05:50, skrev LaurenceClarkCrossen:
On Sun, 23 Feb 2025 0:05:29 +0000, Ross Finlayson wrote:
On 02/22/2025 01:30 PM, Paul B. Andersen wrote:
This rather funny statement of yours reveals that the only
non-Euclidean geometry you know is Gaussian geometry.
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.
Yes it was written for you. Twice.
Paul doesn't understand the difference between not understanding and
disagreeing. I disagree with him and you.
You were obviously ignorant of the fact that there are
other non-Euclidean geometries than 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 for a "flat 3D-space" (Euclidean space) is:
ds² = dx² + dy² + dy² (Pythagoras!)
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.
It is not possible to disagree about this:
Riemannian geometry can describe curved 3D space.
Fact!
Paul is unable to learn.
Very simply, manifolds are not literal spaces.
A 3D manifold is literally a 3D space.
They are only diagrams
representing non-spatial facts as if they were spatial.
Read this again and try not to laugh! :-D
What is a "non-spatial fact" ?
Which "non-spatial facts" are represented as if they were spatial?
What you are
speaking about are not surfaces.
Indeed! I am speaking about 3d spaces
Riemannian geometry is about representing non-spatial elements as
spatial diagrammatically. Taken literally, this is a reification
fallacy.
A reification fallacy? :-D
So Riemannian geometry is as treating abstract "non spatial entities"
as if they were real "spatial diagrammatically".
Make sense, doesn't it? :-D
Riemannian geometry is mathematics.
It is correct by definition. Like Euclidean geometry.
And there is no such thing as "disagreeing" with
a mathematical definition.
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.
So let's talk about "spacetime".
First:
Theories of physics such as Newtonian Mechanics [NM], The Special
Theory of Relativity [SR] and The General Theory of Relativity [GR]
are mathematical models of Nature (or the reality or whatever)
they are not Nature. It is meaningless to ask if the entities
in the theory "really exist".
The only test of the validity of a mathematically consistent theory
is if its predictions are in accordance with measurements.
"Spacetime" is an entity in GR, and spacetime geometry is mathematics,
so the following is correct by definition:
In spacetime geometry there is a four dimensional manifold called
spacetime. The spacetime metric has four coordinates, one temporal
and three spatial.
The metric for a static flat spacetime is:
ds² = − (c⋅dt)² + dx² + dy² + dz²
or since ds² = - (c⋅dτ)²
(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:
ds² = -(1-2GM/c²r)c²dt² + (1/(1-2GM/c²r))dr² + r²(dθ² + sin²θ⋅dϕ²)
or:
(c⋅dτ)² = (1-2GM/c²r)c²dt² - (1/(1-2GM/c²r))dr² - r²(dθ² + sin²θ⋅dϕ²)
Note that there are four coordinates, t, r, θ and ϕ
Bottom line:
It is a fact that the entity spacetime in the mathematical model GR
will be curved if there is a mass present (Sun, Earth).
You are however free to believe that the predictions of GR not
are in accordance with real measurements.
But then you must also believe that all the physicists involved
in all the experiments that have confirmed GR, don't know what
they are doing or are frauds.
Is that what you believe? :-D
Den 23.02.2025 05:50, skrev LaurenceClarkCrossen:
On Sun, 23 Feb 2025 0:05:29 +0000, Ross Finlayson wrote:
On 02/22/2025 01:30 PM, Paul B. Andersen wrote:
This rather funny statement of yours reveals that the only
non-Euclidean geometry you know is Gaussian geometry.
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.
Yes it was written for you. Twice.
Paul doesn't understand the difference between not understanding and
disagreeing. I disagree with him and you.
You were obviously ignorant of the fact that there are
other non-Euclidean geometries than 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 for a "flat 3D-space" (Euclidean space) is:
ds² = dx² + dy² + dy² (Pythagoras!)
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.
It is not possible to disagree about this:
Riemannian geometry can describe curved 3D space.
Fact!
Paul is unable to learn.
Very simply, manifolds are not literal spaces.
A 3D manifold is literally a 3D space.
They are only diagrams
representing non-spatial facts as if they were spatial.
Read this again and try not to laugh! :-D
What is a "non-spatial fact" ?
Which "non-spatial facts" are represented as if they were spatial?
What you are
speaking about are not surfaces.
Indeed! I am speaking about 3d spaces
Riemannian geometry is about representing non-spatial elements as
spatial diagrammatically. Taken literally, this is a reification
fallacy.
A reification fallacy? :-D
So Riemannian geometry is as treating abstract "non spatial entities"
as if they were real "spatial diagrammatically".
Make sense, doesn't it? :-D
Riemannian geometry is mathematics.
It is correct by definition. Like Euclidean geometry.
And there is no such thing as "disagreeing" with
a mathematical definition.
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.
So let's talk about "spacetime".
First:
Theories of physics such as Newtonian Mechanics [NM], The Special
Theory of Relativity [SR] and The General Theory of Relativity [GR]
are mathematical models of Nature (or the reality or whatever)
they are not Nature. It is meaningless to ask if the entities
in the theory "really exist".
The only test of the validity of a mathematically consistent theory
is if its predictions are in accordance with measurements.
"Spacetime" is an entity in GR, and spacetime geometry is mathematics,
so the following is correct by definition:
In spacetime geometry there is a four dimensional manifold called
spacetime. The spacetime metric has four coordinates, one temporal
and three spatial.
The metric for a static flat spacetime is:
ds² = − (c⋅dt)² + dx² + dy² + dz²
or since ds² = - (c⋅dτ)²
(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:
ds² = -(1-2GM/c²r)c²dt² + (1/(1-2GM/c²r))dr² + r²(dθ² + sin²θ⋅dϕ²)
or:
(c⋅dτ)² = (1-2GM/c²r)c²dt² - (1/(1-2GM/c²r))dr² - r²(dθ² + sin²θ⋅dϕ²)
Note that there are four coordinates, t, r, θ and ϕ
Bottom line:
It is a fact that the entity spacetime in the mathematical model GR
will be curved if there is a mass present (Sun, Earth).
You are however free to believe that the predictions of GR not
are in accordance with real measurements.
But then you must also believe that all the physicists involved
in all the experiments that have confirmed GR, don't know what
they are doing or are frauds.
Is that what you believe? :-D
Den 22.02.2025 22:30, skrev Paul B. Andersen:
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,OOOps! Nasty typo!
If ds² is negative, the line element ds is time-like.
If s² is positive, s is space-like,
If s² is negative, s is time-like.
S is the interval between two events.
--
Paul
https://paulba.no/
On Sun, 23 Feb 2025 13:17:32 +0000, Paul.B.Andersen wrote:
Den 23.02.2025 05:50, skrev LaurenceClarkCrossen:I disagree with it, as do thousands of excellent scientists. Riemannian geometry can't describe curved 3D space without a surface.
On Sun, 23 Feb 2025 0:05:29 +0000, Ross Finlayson wrote:
On 02/22/2025 01:30 PM, Paul B. Andersen wrote:
This rather funny statement of yours reveals that the only
non-Euclidean geometry you know is Gaussian geometry.
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.
Yes it was written for you. Twice.
Paul doesn't understand the difference between not understanding and
disagreeing. I disagree with him and you.
You were obviously ignorant of the fact that there are
other non-Euclidean geometries than 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 for a "flat 3D-space" (Euclidean space) is:
ds² = dx² + dy² + dy² (Pythagoras!)
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.
It is not possible to disagree about this:
Riemannian geometry can describe curved 3D space.
Fact!
Paul is unable to learn.
Very simply, manifolds are not literal spaces.
A 3D manifold is literally a 3D space.
They are only diagrams
representing non-spatial facts as if they were spatial.
Read this again and try not to laugh! :-D
What is a "non-spatial fact" ?
Which "non-spatial facts" are represented as if they were spatial?
What you are
speaking about are not surfaces.
Indeed! I am speaking about 3d spaces
Riemannian geometry is about representing non-spatial elements as
spatial diagrammatically. Taken literally, this is a reification
fallacy.
A reification fallacy? :-D
So Riemannian geometry is as treating abstract "non spatial entities"
as if they were real "spatial diagrammatically".
Make sense, doesn't it? :-D
Riemannian geometry is mathematics.
It is correct by definition. Like Euclidean geometry.
And there is no such thing as "disagreeing" with
a mathematical definition.
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.
So let's talk about "spacetime".
First:
Theories of physics such as Newtonian Mechanics [NM], The Special
Theory of Relativity [SR] and The General Theory of Relativity [GR]
are mathematical models of Nature (or the reality or whatever)
they are not Nature. It is meaningless to ask if the entities
in the theory "really exist".
The only test of the validity of a mathematically consistent theory
is if its predictions are in accordance with measurements.
"Spacetime" is an entity in GR, and spacetime geometry is mathematics,
so the following is correct by definition:
In spacetime geometry there is a four dimensional manifold called
spacetime. The spacetime metric has four coordinates, one temporal
and three spatial.
The metric for a static flat spacetime is:
ds² = − (c⋅dt)² + dx² + dy² + dz²
or since ds² = - (c⋅dτ)²
(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:
ds² = -(1-2GM/c²r)c²dt² + (1/(1-2GM/c²r))dr² + r²(dθ² + sin²θ⋅dϕ²)
or:
(c⋅dτ)² = (1-2GM/c²r)c²dt² - (1/(1-2GM/c²r))dr² - r²(dθ² + sin²θ⋅dϕ²)
Note that there are four coordinates, t, r, θ and ϕ
Bottom line:
It is a fact that the entity spacetime in the mathematical model GR
will be curved if there is a mass present (Sun, Earth).
You are however free to believe that the predictions of GR not
are in accordance with real measurements.
But then you must also believe that all the physicists involved
in all the experiments that have confirmed GR, don't know what
they are doing or are frauds.
Is that what you believe? :-D
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