Distances don't make sense.

The Flat earth geographics do not make a lot of sense to me. Google the distance between London and Montreal, the distance should be approximately 5277km. Google the distance between Cape Town and Buenos Aires, the approximate distance should be 6877km. However according to a flat earth world map, the distance between Cape Town and Buenos Aires should be over twice the distance between London and Montreal.

Incorrect.

Here.

Quote from: Jane on November 25, 2016, 10:35:03 AM

I've got a map 

Using a metric I've derived before on this forum:

Quote from: Jane on September 01, 2016, 01:50:14 PM

The Earth is defined by the x and y axes, with the North Pole (for the sake of tradition) at 0, and the South at infinity. An infinite plane is used so that if you reach the South, you can come out the far side. Once the metric is used, the distance shouldn't actually be infinite. This is just our set.
Longitude is 0 along the line y=0, latitude is zero on the circle of radius 1 centred at the North pole.

Now then, our spectacularly awful looking metric. For points P₁=(x₁,y₁), P₂ = (x₂,y₂):

[jsTex]d(P_1,P_2) = \cos^{-1} \left( \sin \left(\tan^{-1} \left(\frac{x_1^2 + y_1^2 -1}{2x_1} \right) \right) \sin \left(\tan^{-1} \left(\frac{x_2^2 + y_2^2 -1}{2x_2} \right) \right) +\cos \left(\tan^{-1} \left(\frac{x_1^2 + y_1^2 -1}{2x_1} \right) \right) \cos \left(\tan^{-1} \left(\frac{x_2^2 + y_2^2 -1}{2x_2} \right) \right) \cos \left|\tan^{-1} \left( \frac{y_1}{x_1}  \right) - \tan^{-1} \left( \frac{y_2}{x_2} \right) \right| \right)[/jsTex]

The following map gives over half the Earth. The rest can be extended out, but it grows in size.



But remember, we're in non-Euclidean space so you have to calculate distances with the above metric.

Did you just.. take the metric induced by the projection of the sphere? 

Did you just.. take the metric induced by the projection of the sphere? 

  I suppose I did.  Can you explain to me why that is the wrong approach? 

Quote from: Kami on February 18, 2017, 12:48:52 AM

Did you just.. take the metric induced by the projection of the sphere? 

  I suppose I did.  Can you explain to me why that is the wrong approach?

Sorry, I meant disputeone's post, not yours


One could argue that the distances given by google are not correct (if they were then a round earth would not be debatable, right?). So can you verify that the distances given by google are the correct ones?

Quote from: Melodicmarc on February 18, 2017, 01:00:10 AM

Quote from: Kami on February 18, 2017, 12:48:52 AM

Did you just.. take the metric induced by the projection of the sphere? 

  I suppose I did.  Can you explain to me why that is the wrong approach?

Sorry, I meant disputeone's post, not yours


One could argue that the distances given by google are not correct (if they were then a round earth would not be debatable, right?). So can you verify that the distances given by google are the correct ones?

Naturally 

As you can see Jane had to do backflips to make it work.

It's based on JD's non euclidean flat earth.

Quote from: Kami on February 18, 2017, 03:20:57 AM

Quote from: Melodicmarc on February 18, 2017, 01:00:10 AM

Quote from: Kami on February 18, 2017, 12:48:52 AM

Did you just.. take the metric induced by the projection of the sphere? 

  I suppose I did.  Can you explain to me why that is the wrong approach?

Sorry, I meant disputeone's post, not yours


One could argue that the distances given by google are not correct (if they were then a round earth would not be debatable, right?). So can you verify that the distances given by google are the correct ones?

Naturally 

As you can see Jane had to do backflips to make it work.

It's based on JD's non euclidean flat earth.

But it still does not work.
Why is it so hard to get a map of the flat earth.  It should be very easy.

Quote from: disputeone on February 18, 2017, 04:21:16 AM

Quote from: Kami on February 18, 2017, 03:20:57 AM

Quote from: Melodicmarc on February 18, 2017, 01:00:10 AM

Quote from: Kami on February 18, 2017, 12:48:52 AM

Did you just.. take the metric induced by the projection of the sphere? 

  I suppose I did.  Can you explain to me why that is the wrong approach?

Sorry, I meant disputeone's post, not yours


One could argue that the distances given by google are not correct (if they were then a round earth would not be debatable, right?). So can you verify that the distances given by google are the correct ones?

Naturally 

As you can see Jane had to do backflips to make it work.

It's based on JD's non euclidean flat earth.

But it still does not work.
Why is it so hard to get a map of the flat earth.  It should be very easy.

It does work (I assume, I haven't double checked)
You can manipulate one space to another.
It is merely projecting the surface of a sphere onto a flat plane.

Incorrect.

Quote from: disputeone on February 17, 2017, 06:53:04 PM

Here.

Quote from: Jane on November 25, 2016, 10:35:03 AM

But remember, we're in non-Euclidean space so you have to calculate distances with the above metric.

Can't we forget non-Euclidean space. That's something purely in the mind of John Davis and Jane likes to confuse things by demonstrating her prowess in maths.

I will agree that, according to Einstein's GR, spacetime is curved (non-Euclidean)
but the only "curvature" of space-like in our vicinity is to change the apparent curvature of the earth by a minute amount:

9   Conclusion
This exercise produced several expressions of relativistic curvature for solar system objects.  The true diameters of the Sun and Earth are 4.1 km and 4.4 mm greater, respectively, than one would expect from applying Euclidean geometry (C = πd) to the observed surface of these bodies.

From Calculations on space-time curvature within the Earth and Sun

Note too, that even this minute curvature (about 1 part in 2,900,000,000) is in the radius-of-curvature - it does not "roll a flat surface into a ball", no way!

Dragging in non-Euclidean space unnecessarily just confuses the whole issue.

Can't we forget non-Euclidean space.

NEVAR!!!

Quote from: Canadabear on February 18, 2017, 04:31:31 AM

Quote from: disputeone on February 18, 2017, 04:21:16 AM

Quote from: Kami on February 18, 2017, 03:20:57 AM

Quote from: Melodicmarc on February 18, 2017, 01:00:10 AM

Quote from: Kami on February 18, 2017, 12:48:52 AM

Did you just.. take the metric induced by the projection of the sphere? 

  I suppose I did.  Can you explain to me why that is the wrong approach?

Sorry, I meant disputeone's post, not yours


One could argue that the distances given by google are not correct (if they were then a round earth would not be debatable, right?). So can you verify that the distances given by google are the correct ones?

Naturally 

As you can see Jane had to do backflips to make it work.

It's based on JD's non euclidean flat earth.

But it still does not work.
Why is it so hard to get a map of the flat earth.  It should be very easy.

It does work (I assume, I haven't double checked)
You can manipulate one space to another.
It is merely projecting the surface of a sphere onto a flat plane.

Correct.

Canadabear you made yourself look silly there.

Quote from: JackBlack on February 18, 2017, 04:33:45 AM

Quote from: Canadabear on February 18, 2017, 04:31:31 AM

Quote from: disputeone on February 18, 2017, 04:21:16 AM

Quote from: Kami on February 18, 2017, 03:20:57 AM

Quote from: Melodicmarc on February 18, 2017, 01:00:10 AM

Quote from: Kami on February 18, 2017, 12:48:52 AM

Did you just.. take the metric induced by the projection of the sphere? 

  I suppose I did.  Can you explain to me why that is the wrong approach?

Sorry, I meant disputeone's post, not yours


One could argue that the distances given by google are not correct (if they were then a round earth would not be debatable, right?). So can you verify that the distances given by google are the correct ones?

Naturally 

As you can see Jane had to do backflips to make it work.

It's based on JD's non euclidean flat earth.

But it still does not work.
Why is it so hard to get a map of the flat earth.  It should be very easy.

It does work (I assume, I haven't double checked)
You can manipulate one space to another.
It is merely projecting the surface of a sphere onto a flat plane.

Correct.

Canadabear you made yourself look silly there.

Jane's map does not work,  that's what I said.
And if the earth is flat,  it should be easy to generate a working map but the not one existing.

The map works, how many more people have to tell you.

As Rab has pointed it it proves more than anything that you can twist numbers to do pretty much anything.

This is important to consider.

Jane's map does not work,  that's what I said.
And if the earth is flat,  it should be easy to generate a working map but the not one existing.

Except, it does.
It is a projection of a flat plane in non-Euclidean space into a flat surface in Euclidean space.
You cannot directly measure the distance, you need to consider how the space is mapped.

Quote from: rabinoz on February 18, 2017, 05:58:48 PM

Can't we forget non-Euclidean space.

NEVAR!!!

Please show some evidence or even a reason why our space might be non-Euclidean more than given by solutions to GR.
Of course those desparate to maintain disputes may disagree.

Do you think that this "computer" used non-Euclidean geometry to compute, on a desk calculator, the orbital data for John Glenn's adventure.
From The woman whose math sent John Glenn to orbit says goodbye to a 'good man'.

I like the old KISS philosophy where you don't unnecessarily drag in esoteric theories, especially if they have no basis!

BTW Katherine Johnson was the computer, the desk calculator was just for the number crunching!

I got nothing solid.. haha you got me.

The non-euclidean flat earth idea is fascinating tho imo.

Edit.

Actually, from what I understand Johns theory relies on space being exactly as curved as GR predicts, take for example satellites, they are moving in straight lines in curved space.

http://www.feynmanlectures.caltech.edu/II_42.html

If spacetime around the earth is curved then perhaps our reality is a curved earth when in actuality it is flat in curved space?

What would happen, as a thought experiment, if we were to straighten out the spacetime where earth is located?

What would be the shape of the earth then?

I agree this is more of a philosophical question than anything else.


Damn I'm good.

I got nothing solid.. haha you got me.
The non-euclidean flat earth idea is fascinating tho imo.

Actually, from what I understand Johns theory relies on space being exactly as curved as GR predicts, take for example satellites, they are moving in straight lines in curved space.

http://www.feynmanlectures.caltech.edu/II_42.html

Some do call it straight lines, but no more than geodesics (Great Circle Routes) on the surface of a sphere are straight lines.
That is why I prefer to call those paths geodesics rather than straight lines, simply because people interpret straight lines as straight lines in Euclidean (flat) space.

In GR it is spacetime that is curved, not necessarily space. As I tried to explain in a previous post, space in the vicinity of earth is curved, but so minutely that I doubt it is measureable. And it does not tend to curl a flat surface, but only change the apparent curvature by about 4.4 mm in a diameter of 12,742 km - big deal!
So satellites do orbit in space exactly as we are used to. It is only in spacetime that their paths are geodesics.

That is why orbital calculated using Newtonian Mechanics, with relatively (usually only SR) only used as a minor correction.
Just as well, when you look at the complexity of GR calculations,  where there are usually no closed solutions.

Mass tells space-time how to curve, and space-time tells mass how to move.

And that's a simplification, as it is not just mass but mass and energy that "tells space-time how to curve".

If spacetime around the earth was curved then perhaps our reality is a curved earth when in actuality it is flat in curved space?
What would happen, as a thought experiment, would happen if we were to straighten out the spacetime where earth is located?
What would be the shape of the earth then?
I agree this is more of a philosophical question than anything else.

Damn I'm good.

Yes, good at "thought experiments", which Einstein used a lot, but then he had a lot of evidence and ideas from others behind him.
And when he had his ideas he went to other experts for assistance, such as Herman Minkowski
"Minkowski taught at the universities of Bonn, Göttingen, Königsberg and Zürich. At the Eidgenössische Polytechnikum, today the ETH Zurich, he was one of Einstein's teachers."

  Yes, Einstein wasn't that bad either! 

Quote from: disputeone on February 19, 2017, 12:17:00 AM

I got nothing solid.. haha you got me.
The non-euclidean flat earth idea is fascinating tho imo.

Actually, from what I understand Johns theory relies on space being exactly as curved as GR predicts, take for example satellites, they are moving in straight lines in curved space.

http://www.feynmanlectures.caltech.edu/II_42.html

Some do call it straight lines, but no more than geodesics (Great Circle Routes) on the surface of a sphere are straight lines.
That is why I prefer to call those paths geodesics rather than straight lines, simply because people interpret straight lines as straight lines in Euclidean (flat) space.

In GR it is spacetime that is curved, not necessarily space. As I tried to explain in a previous post, space in the vicinity of earth is curved, but so minutely that I doubt it is measureable. And it does not tend to curl a flat surface, but only change the apparent curvature by about 4.4 mm in a diameter of 12,742 km - big deal!
So satellites do orbit in space exactly as we are used to. It is only in spacetime that their paths are geodesics.

That is why orbital calculated using Newtonian Mechanics, with relatively (usually only SR) only used as a minor correction.
Just as well, when you look at the complexity of GR calculations,  where there are usually no closed solutions.

Quote from: John Wheeler

Mass tells space-time how to curve, and space-time tells mass how to move.

And that's a simplification, as it is not just mass but mass and energy that "tells space-time how to curve".

Quote from: disputeone

If spacetime around the earth was curved then perhaps our reality is a curved earth when in actuality it is flat in curved space?
What would happen, as a thought experiment, would happen if we were to straighten out the spacetime where earth is located?
What would be the shape of the earth then?
I agree this is more of a philosophical question than anything else.

Damn I'm good.

Yes, good at "thought experiments", which Einstein used a lot, but then he had a lot of evidence and ideas from others behind him.
And when he had his ideas he went to other experts for assistance, such as Herman Minkowski
"Minkowski taught at the universities of Bonn, Göttingen, Königsberg and Zürich. At the Eidgenössische Polytechnikum, today the ETH Zurich, he was one of Einstein's teachers."

  Yes, Einstein wasn't that bad either! 

But it (curvature of spacetime in GR) does curve a flat surface, see our solar sytem for example, curvy planets everywhere, curvy orbits, curvy sun.

As much as I like curves, I have to admit that gravity (curvature of spacetime) will curve a flat surface into a sphere, this has been demonstrated extensively here when it was proven that an infinite plane could hold a stable gravitational field. Anything smaller than infinite and it would collapse into a sphere.

Therefore, I would disagree the curvature of spacetime does tend to curve flat objects.

Einstein was brilliant, so brilliant in fact he was called crazy by many of his peers. Just like John Davis, you have to remember Einstein proposing GR was taken to be absurd at first.

The non-euclidean flat earth idea is fascinating tho imo.

I somewhat like it as well, but prefer calling it by its simpler name:
The non-flat flat Earth.

Actually, from what I understand Johns theory relies on space being exactly as curved as GR predicts, take for example satellites, they are moving in straight lines in curved space.

It is somewhat more complicated than that, and it is the time part which curves to point towards Earth. The space part is still basically flat.
What he tries to use is the equivalence principle, but that then requires that Earth orbits the Satellite, as those 2 points of view cannot be distinguished except by invoking the math and force required which indicates that the satellite doesn't have the mass required to remain stationary while Earth orbits it.

As much as I like curves, I have to admit that gravity (curvature of spacetime) will curve a flat surface into a sphere, this has been demonstrated extensively here when it was proven that an infinite plane could hold a stable gravitational field. Anything smaller than infinite and it would collapse into a sphere.

Therefore, I would disagree the curvature of spacetime does tend to curve flat objects.

Einstein was brilliant, so brilliant in fact he was called crazy by many of his peers. Just like John Davis, you have to remember Einstein proposing GR was taken to be absurd at first.

While it is true that an infinite plane has a stable gravitational field, it doesn't match that observed by reality.

The gravitational field of a flat plane does not decay with increasing distance. Instead it remains constant.

That is not what is observed in reality.
If it was, geostationary orbits simply wouldn't work, (or at best would require a completely different altitude), and tidal forces would not exist to lock the moon into resonance.

Quote from: disputeone on February 19, 2017, 12:17:00 AM

The non-euclidean flat earth idea is fascinating tho imo.

I somewhat like it as well, but prefer calling it by its simpler name:
The non-flat flat Earth.

Quote from: disputeone on February 19, 2017, 12:17:00 AM

Actually, from what I understand Johns theory relies on space being exactly as curved as GR predicts, take for example satellites, they are moving in straight lines in curved space.

It is somewhat more complicated than that, and it is the time part which curves to point towards Earth. The space part is still basically flat.
What he tries to use is the equivalence principle, but that then requires that Earth orbits the Satellite, as those 2 points of view cannot be distinguished except by invoking the math and force required which indicates that the satellite doesn't have the mass required to remain stationary while Earth orbits it.

Thanks.

I will have to check kami's math but I think that the gravitational field can decay with distance to the flat plane however only vertically. That is directly up from it.

I will have to check kami's math but I think that the gravitational field can decay with distance to the flat plane however only vertically. That is directly up from it.

The gravitational field over a uniform infinite plane is vertically down and constant in magnitude everywhere.

Plane of Mass
g = 2 π G ơ
Note that this is a constant: the acceleration due to gravity of an infinite plane of mass is independent of the distance from the plane!

In his science fiction novel 2010: Odyssey Two, author Arthur C. Clarke describes a large rectangular slab that has been build by an alien race and placed in orbit around Jupiter. Astronauts are able to calculate the mass of the slab by placing a small spacecraft near the center of the large face and timing it to see how long it takes to fall to the surface of the slab. By approximating the slab as an infinite plane, they use Eq. (22) to find the acceleration; from that and the falling time, they can calculate the mass. (Actually, Dr. Clarke got the wrong answer in the book. You may want to find the book and see if you can calculate the correct answer.)

From Department of Physical Sciences and Engineering, Prince George’s Community College, Gauss’s Law for Gravity

Yes, you check that out.

Thanks.

So x lightyears from the infinite plane would still experience an acceleration of 9.8m/s ² directly "down"?

Thanks.

So x lightyears from the infinite plane would still experience an acceleration of 9.8m/s ² directly "down"?

Yes, but only if your infinite flat plane is much greater than x lightyears! Infinite, means greater than any number you can imagine, then infinitely bigger than that.

Mind you, I have never said that I agree with the concept of an infinite flat plane earth. The idea is completely ridiculous.

Just to top it off, if that plane is not quite infinite, it is unstable and will collapse into a not quite infinite sphere in not quite infinite time!
The flat plane of a flat earth would not need to be really infinite to adequately approximate the infinite flat earth,
This sort of thing

The Potato Radius: a Lower Minimum Size for Dwarf Planets
Summary: Gravitational and electronic forces produce a correlation between the mass and shape of objects in the universe. For example, at an average radius of ~ 200 km – 300 km, the icy moons and rocky asteroids of our Solar System transition from a rounded potato shape to a sphere. We derive this potato-to-sphere transition radius -- or “potato radius” -- from first principles. Using the empirical potato radii of asteroids and icy moons, we derive a constraint on the yield strength of these bodies during their formative years when their shapes were determined. Our proposed ~200 km potato radius for icy moons would substantially increase the number of trans-Neptunian objects classified as “dwarf planets”.

From The Potato Radius: a Lower Minimum Size for Dwarf Planets.

The limit for rocky planets is only about 300 km, so you see the Flat Earth cannot countenance Newtonian Gravitation, no matter how well proven!

Depends what type of infinity you are talking about.

Yeah I know the plane has to be infinite I was in the original thread, I was like "but but, muh gravitational collapse" I was wrong..

I agree we are not equipped to contemplate true infinity.

Credit to Jane for schooling me on infinity.

Thanks.

I will have to check kami's math but I think that the gravitational field can decay with distance to the flat plane however only vertically. That is directly up from it.

Assuming it is an infinite plane in Euclidean space, it can't.

A field (like gravity, magnetism (although that is more complex due to it being a dipole), electrostatics, etc), can be represented by field lines, which indicate the direction a particular type of object will be accelerated towards. For electrostatics it is a positive charge, for gravity it is a massive object (i.e. one with mass). The strength of the field is indicated by the density of the lines (i.e. number of lines per unit area).

The lines only start (or end) at things which generate the field (charges), and spread evenly around it.

For a point charge (or things which can be represented as a point, such as a homogeneous sphere when outside the sphere), the field lines all converge into a single point, or spread out from that point.
Thus, if you have field f0, at radius r0, and n lines, these can be related as n=k f0 4 pi r0^2, where k is a constant relating field strength to number of lines per unit area.
If you now consider another location, at ri, with field strength fi, then, noting that the number of lines is constant, you get the relation:
k fi*4 pi ri^2=k f0*4 pi r0^2
thus:
fi ri^2=f0 r0^2

Thus:
fi=f0 r0^2/ri^2
and thus the field is proportional to 1/r^2.

For an infinite line, it now just spreads in 2 dimensions, around a circle, and thus you get fi=f0 r0/ri, and a field proportional to 1/r.

For an infinite plane, there is no where for it to spread out to, so the lines must keep the same density as they go away from the object and thus the field remains constant.

Quote from: disputeone on February 19, 2017, 03:36:32 PM

Thanks.

I will have to check kami's math but I think that the gravitational field can decay with distance to the flat plane however only vertically. That is directly up from it.

Assuming it is an infinite plane in Euclidean space, it can't.

A field (like gravity, magnetism (although that is more complex due to it being a dipole), electrostatics, etc), can be represented by field lines, which indicate the direction a particular type of object will be accelerated towards. For electrostatics it is a positive charge, for gravity it is a massive object (i.e. one with mass). The strength of the field is indicated by the density of the lines (i.e. number of lines per unit area).

The lines only start (or end) at things which generate the field (charges), and spread evenly around it.

For a point charge (or things which can be represented as a point, such as a homogeneous sphere when outside the sphere), the field lines all converge into a single point, or spread out from that point.
Thus, if you have field f0, at radius r0, and n lines, these can be related as n=k f0 4 pi r0^2, where k is a constant relating field strength to number of lines per unit area.
If you now consider another location, at ri, with field strength fi, then, noting that the number of lines is constant, you get the relation:
k fi*4 pi ri^2=k f0*4 pi r0^2
thus:
fi ri^2=f0 r0^2

Thus:
fi=f0 r0^2/ri^2
and thus the field is proportional to 1/r^2.

For an infinite line, it now just spreads in 2 dimensions, around a circle, and thus you get fi=f0 r0/ri, and a field proportional to 1/r.

For an infinite plane, there is no where for it to spread out to, so the lines must keep the same density as they go away from the object and thus the field remains constant.

Awesome Jack, thanks heaps.