[math]\text{General Relativity: Flamm's Paraboloid}[/math]
[math]\text{The exterior Schwarzschild metric space-time curvature solutions for a radius greater than the Schwarzschild radius } (r > r_{s})[/math]
[math]\text{can be visualized as mathematical parametrics. There is an additional Euclidean z-axis metric polar dimension w,}[/math]
[math]\text{which has no physical reality and is not part of space-time.}[/math]

[math]\color{blue}{ \text{Any discussions and/or peer reviews about this specific topic thread?}}[/math]

[math]\color{blue}{\text{"I think of the horizon at midnight, the sky and sea blurring together." - Sophie Hardcastle}}[/math]
 

 

 

Edited March 14, 20223 yr by Orion1

Any discussions and/or peer reviews about this specific topic thread?

"I think of the horizon at midnight, the sky and sea blurring together." - Sophie Hardcastle

 

 

 

Edited March 14, 20223 yr by Orion1

Why have you got 2 threads on this one subject ?

Any discussions and/or peer reviews about this specific topic thread?

"I think of the horizon at midnight, the sky and sea blurring together." - Sophie Hardcastle

Reference:
Wikipedia - Schwarzschild metric - Flamm's paraboloid: (ref. 1)
https://en.wikipedia.org/wiki/Schwarzschild_metric#Flamm's_paraboloid
Wikipedia - Schwarzschild radius: (ref. 2)
https://en.wikipedia.org/wiki/Schwarzschild_radius
Wikipedia - Schwarzschild metric: (ref. 3)
https://en.wikipedia.org/wiki/Schwarzschild_metric
Wikipedia - Boyer-Lindquist coordinates - Line element: (ref. 4)
https://en.wikipedia.org/wiki/Boyer-Lindquist_coordinates#Line_element
Science Forums - General Relativity: Four Exterior Metric Solutions - Orion1: (ref. 5)
https://www.scienceforums.net/topic/126549-general-relativity-four-exterior-metric-solutions/
Wikipedia - Photon sphere: (ref. 6)
https://en.wikipedia.org/wiki/Photon_sphere
Wikipedia - Astrophysical jet: (ref. 7)
https://en.wikipedia.org/wiki/Astrophysical_jet
Wikipedia - Accretion disk: (ref. 8)
https://en.wikipedia.org/wiki/Accretion_disk
Wikipedia - Black hole - Innermost stable circular orbit: (ref. 9)
https://en.wikipedia.org/wiki/Black_hole#Innermost_stable_circular_orbit_(ISCO)
 

Mathematica 7 - General Relativity - Flamm's Paraboloid source code notebook attachment:

General Relativity - Flamm's Paraboloid.nb

Mathematica 7 - General Relativity - Schwarzschild Black Hole Toy Model source code notebook attachment:

General Relativity - Schwarzschild Black Hole Toy Model.nb

33 minutes ago, studiot said:

Why have you got 2 threads on this one subject ?

Please merge the threads if there is a protocol violation.

 

 

Edited March 14, 20223 yr by Orion1

Would any students or professors be interested in preserving these published exterior solutions in their academic library archive to survive the author's and server's longevity and for the future study of these exterior solutions by students or professors?

Any discussions and/or peer reviews about this specific topic thread?

"I think of the horizon at midnight, the sky and sea blurring together." - Sophie Hardcastle

Edited June 17Jun 17 by Orion1

Can i get to know that why this topic doesnt get any discussion related to it?

17 hours ago, Dhillon1724X said:

Can i get to know that why this topic doesnt get any discussion related to it?

I understand you are young.

It reaches a point when you have more knowledge of a particular area that you lack peers to challenge you.

There is where Orion1 is... again many people fear math and don't like overthinking.

23 hours ago, Dhillon1724X said:

Can i get to know that why this topic doesnt get any discussion related to it?

Because it’s pretty much standard textbook material, and thus already well known. There’s simply not a lot to discuss here.

16 minutes ago, Markus Hanke said:

Because it’s pretty much standard textbook material, and thus already well known. There’s simply not a lot to discuss here.

Well,Yes.
You are right.

19 hours ago, Dhillon1724X said:

Can i get to know that why this topic doesnt get any discussion related to it?

The thing I don't like about the trampoline 'visualisations' is that they can be misleading.

It is not the grid lines that are distorted.

These are set when the coordinate system is set and cannot be distorted.

In fact distortion is not the right word at all.

The diagrams are as shown simple because at least one of the axes is straight, which is not the case with natural axes.

6 minutes ago, studiot said:

The thing I don't like about the trampoline 'visualisations' is that they can be misleading.

It is not the grid lines that are distorted.

These are set when the coordinate system is set and cannot be distorted.

In fact distortion is not the right word at all.

The diagrams are as shown simple because at least one of the axes is straight, which is not the case with natural axes.

I get that it's unrepresentative at a certain level of understanding, but can you think of a better metaphor for those of us that don't speak maths fluently?

Just now, dimreepr said:

I get that it's unrepresentative at a certain level of understanding, but can you think of a better metaphor for those of us that don't speak maths fluently?

See if you can make anything from this video. (6 minutes)

I think he does the key points rather well.

Note that he introduces two important vector fields, both of which are direction fields

A note to @Dhillon1724X

This also shows the key universal driver that has relevance to your chortons and to the Higgs field., if you look hard enough.

1 minute ago, studiot said:

This also shows the key universal driver that has relevance to your chortons and to the Higgs field., if you look hard enough.

Thanks for sharing,i will watch it and give feedback soon.
For now i have to study a bit for next PD Test.

As an Update-
I failed in Maths PD Test.
I did silly mistakes.

Its hard to explain to Parents about what i am learning now,they will judge on based on marks.
So now i am in a big trouble,Its first time i ever failed.

17 minutes ago, studiot said:

See if you can make anything from this video. (6 minutes)

I think he does the key points rather well.

Note that he introduces two important vector fields, both of which are direction fields

I'll be honest, it struck me as a verbose version of the same metaphor.

My understanding is, the second tick rate is invariant (my apologies if I've used that word incorrectly, but it seems appropriate), so whatever frame one is in, only seems distorted from the outside.

Just now, dimreepr said:

I'll be honest, it struck me as a verbose version of the same metaphor.

My understanding is, the second tick rate is invariant (my apologies if I've used that word incorrectly, but it seems appropriate), so whatever frame one is in, only seems distorted from the outside.

There are lots of 2D and 3D coordinate systems each denoting a different space referenced by those coordinate systems.

In real life we only possess the one such system - the physical space we are in.

So we have to make do with that and imagine the spaces that the others refer to.

The equipotentials are not part of real space so we substitute a real world coordinate to draw the graph.

Does this help ?

Edited July 28Jul 28 by studiot

6 minutes ago, studiot said:

There are lots of 2D and 3D coordinate systems each denoting a different space referenced by those coordinate systems.

In real life we only possess the one such system - the physical space we are in.

So we have to make do with that and imagine the spaces that the others refer to.

The equipotentials are not part of real space so we substitute a real world goordinate to draw the graph.

Does this help ?

I'll have to give it some thought, bc I seem to be missing something important...

39 minutes ago, studiot said:

See if you can make anything from this video. (6 minutes)

I think he does the key points rather well.

Note that he introduces two important vector fields, both of which are direction fields

A note to @Dhillon1724X

This also shows the key universal driver that has relevance to your chortons and to the Higgs field., if you look hard enough.

As far as i can see,It relates to energy distribution across the field.
it can also helps in revealing field structures.
It has many other uses too.

It can be a blueprint for force and mass too.

Just now, dimreepr said:

I'll have to give it some thought, bc I seem to be missing something important...

Just now, Dhillon1724X said:

As far as i can see,It relates to energy distribution across the field.
it can also helps in revealing field structures.
It has many other uses too.

It can be a blueprint for force and mass too.

Yes a blueprint for other physical quantities.

But ones when you attemp to discuss relativity drawing those trumpets amounts to saying that the equipotential lines distort the actual coordiantes you used to draw them.

This is different from the previous video from the same author showing contour lines on a map ( this appeared briefly on the second one as well)

So remember the plot you are creating is say length, length, mass or length length, force or whatever - space that is purely imaginary

1 hour ago, studiot said:

Yes a blueprint for other physical quantities.

But ones when you attemp to discuss relativity drawing those trumpets amounts to saying that the equipotential lines distort the actual coordiantes you used to draw them.

This is different from the previous video from the same author showing contour lines on a map ( this appeared briefly on the second one as well)

So remember the plot you are creating is say length, length, mass or length length, force or whatever - space that is purely imaginary

I will try to learn more and implement it.

Here is another point of view.

Here is a graph I posted recently in another thread

Here is another version of the same graph

And here is yet another version

don't worry about the numbers, just look at the shapes.

All three have their own special uses ie they tell you different things about a first order chemical reaction.

In other words the map is not the terrority.

19 minutes ago, studiot said:

Here is another point of view.

Here is a graph I posted recently in another thread

Here is another version of the same graph

And here is yet another version

don't worry about the numbers, just look at the shapes.

All three have their own special uses ie they tell you different things about a first order chemical reaction.

In other words the map is not the terrority.

If i am right then they are showing values increasing or decreasing with respect to Time.
The first one is related to Radioactive decay,second one is telling about completion of a reaction and the third one is new for me.

Just now, Dhillon1724X said:

If i am right then they are showing values increasing or decreasing with respect to Time.
The first one is related to Radioactive decay,second one is telling about completion of a reaction and the third one is new for me.

When I say the graphs are all the same, I mean that they are all of the same equation that governs the process that produces the graph.

Chemically, radioactive decay is a first order reaction.

I am using first order because there are other orders for different chemical reactions.

The first graph shows how the quantity or concentration of the reactant diminishes or falls as the reaction proceeds.

The second graph shows how the product builds up (the reaction never quite finishes do you know the term asymptote ?)

Because the the equation is very simple it can be solved - which means we can determine a specific formula defining the quantity ot reactant or product at any given time.

The important point here is that using that formula we can adjust the axes to give a highly preferable straight line.

The equation concerned is a differential equation and is the least useful, though most general point of view.

[math]Rate = \frac{{dQ}}{{dt}} = - kQ[/math]

Now relativity says that you can either adjust the axes or adjust the curve depending upon the point of view you wish to take.

This is the connection to relativity.

Now can you tell me the difference between those three graphs and this one, which also has two rectangular axes and a plot.

And then I will explain why this difference is crucial to understanding all the fancy maths that is being bandied about, like General Relativity, QM, tensors and many other things.

Edited July 28Jul 28 by studiot

4 hours ago, Dhillon1724X said:

Thanks for sharing,i will watch it and give feedback soon.
For now i have to study a bit for next PD Test.

As an Update-
I failed in Maths PD Test.
I did silly mistakes.

Its hard to explain to Parents about what i am learning now,they will judge on based on marks.
So now i am in a big trouble,Its first time i ever failed.

Ha, silly mistakes is what my son struggled with in maths, too. He seemed to panic a bit when faced with an unfamiliar problem and rush at it. I used to tell him to pretend to be German, stay calm, take his time and not panic. It helped a bit. It was lack of confidence I think. Anyway, you are not alone and it can be fixed.

42 minutes ago, exchemist said:

Ha, silly mistakes is what my son struggled with in maths, too. He seemed to panic a bit when faced with an unfamiliar problem and rush at it. I used to tell him to pretend to be German, stay calm, take his time and not panic. It helped a bit. It was lack of confidence I think. Anyway, you are not alone and it can be fixed.

Reminds me of the meme: Tell someone you love them today, because life is short, but SHOUT it at them in German, because life is also terrifying.

Um, relatively speaking, of course.

30 minutes ago, Phi for All said:

Reminds me of the meme: Tell someone you love them today, because life is short, but SHOUT it at them in German, because life is also terrifying.

Um, relatively speaking, of course.

Well that’s the thing. In my experience, mostly in Hamburg it’s true, Germans speak softly and with deliberation, are not afraid of silences, and when they say something they have generally thought it through. I put it down to constructing sentences with the verb at the end.