If one attempts to tackle general relativity, what are the mathematical
prerequisites? Is a course in non-Euclidean geometry necessary?
And tensor algebra?

Mark

Mark-T <mark-t2@lycos.com> wrote:
> If one attempts to tackle general relativity, what are the mathematical
> prerequisites? Is a course in non-Euclidean geometry necessary?
> And tensor algebra?
>
>
> Mark
>

Differential geometry helps a ton. Linear algebra before that.

--
Odd Bodkin — Maker of fine toys, tools, tables

El sábado, 11 de diciembre de 2021 a las 22:13:38 UTC-3, Mark-T escribió:
> If one attempts to tackle general relativity, what are the mathematical
> prerequisites? Is a course in non-Euclidean geometry necessary?
> And tensor algebra?
>
>
> Mark

Follow the General Relativity course from Professor Susskind (https://www.youtube.com/watch?v=JRZgW1YjCKk). Very easy to follow. Most of the math is from Landau-Lifshitz book The Classical Theory of Fields.

On Saturday, December 11, 2021 at 5:13:38 PM UTC-8, Mark-T wrote:
> If one attempts to tackle general relativity, what are the mathematical
> prerequisites? Is a course in non-Euclidean geometry necessary?
> And tensor algebra?
>
> Mark

Differential geometry + its prerequisites. No specific non-Euclidean
geometry is necessary, whatever is needed follows from the relevant
parts of diff. geo.

--
Jan

On Saturday, December 11, 2021 at 10:13:38 PM UTC-3, Mark-T wrote:
> If one attempts to tackle general relativity, what are the mathematical
> prerequisites? Is a course in non-Euclidean geometry necessary?
> And tensor algebra?
>
>
> Mark

Mandatory prerequisites:

1) Cheating, fudging and cooking 101 (Mandatory)
2) Applied Social Engineering and Networking (Mandatory)
3) How to play with polynomials (Optional)
4) The art of plagiarizing (Mandatory)
5) Teamwork Management (Optional)
6) How to make others work for you (Mandatory)
7) How to make deals with establishment (Mandatory)
8) Italian language (Mandatory)
9) How to deal with journalists (Mandatory)
10) Research of Published Scientific Ideas in the last century (Mandatory)

Do this, and you'll be well ahead other retarded who waste their lives playing with a basic set
of 350 non linear differential equations with no physical meaning at all (unless you want to
make a career in CGI 3D animation. In this case, try with quaternions).

On Sunday, 12 December 2021 at 02:13:38 UTC+1, Mark-T wrote:
> If one attempts to tackle general relativity, what are the mathematical
> prerequisites? Is a course in non-Euclidean geometry necessary?
> And tensor algebra?

No. What it takes is studying some other cases of stupidity,
fanatism and brainwashing.

Mark-T wrote:

> If one attempts to tackle general relativity, what are the mathematical
> prerequisites? Is a course in non-Euclidean geometry necessary?
> And tensor algebra?

vectors, tensors, pseudotensors and curvilinear coordinates.

On Saturday, December 11, 2021 at 10:58:30 PM UTC-8, Richard Hertz ate some more shit:
> snip cretinisms<

On 12/11/21 7:13 PM, Mark-T wrote:
> If one attempts to tackle general relativity, what are the mathematical
> prerequisites? Is a course in non-Euclidean geometry necessary?
> And tensor algebra?

A warning: don't use textbooks that are based on coordinates throughout.
For instance, Weinberg, _Gravitation_and_Cosmology_ -- he "defines" a
tensor by the way its coordinate components behave under a coordinate
transform -- that is so wrong-headed it's unconscionable. Also
Eisenhart, _Riemannian_Geometry_.

It was MTW that taught me the important aspects of GR, specifically that
tensors are functions, and the tensor fields of GR are functions on the
manifold and its tangent spaces. Since coordinates are maps of the
manifold to R^N, that makes it explicitly clear that tensors have
nothing to do with coordinates (and are therefore completely independent
of any coordinates). The way tensor components transform under a change
of coordinates is a consequence of that invariance. Moreover, if you
have an understanding of basic physics, MTW is a fun, wide-ranging romp
through much of modern theoretical physics.

Misner, Thorne, and Wheeler, _Gravitation_.

Tom Roberts

On December 11, JanPB wrote:
> On Saturday, December 11, 2021 Mark-T wrote:
> > If one attempts to tackle general relativity, what are the mathematical
> > prerequisites? Is a course in non-Euclidean geometry necessary?
> > And tensor algebra?
>
> Differential geometry + its prerequisites. No specific non-Euclidean
> geometry is necessary, whatever is needed follows from the relevant
> parts of diff. geo.

Differential geometry is required to describe the curvature of space due to gravity,
correct? Which follows from solutions to the field equations. Are those equations
solvable without computer code? Evidently Schwarzchild did so, with his black hole model.

What I've found is pop science treatments of the subject, and advanced
level texts, but nothing that might classify as intermediate general relativity.

Mark

On Saturday, December 11, 2021, Richard Hertz wrote:
> On Saturday, December 11, 2021, Mark-T wrote:
>> If one attempts to tackle general relativity, what are the mathematical
>> prerequisites? Is a course in non-Euclidean geometry necessary?
>
> Mandatory prerequisites:
> 8) Italian language (Mandatory)

What for?

I'm literate in Spanish, another romance language.
Is that close enough?

Mark

On December 12, 2021, Mark-T wrote:
> On December 11, JanPB wrote:
> > > If one attempts to tackle general relativity, what are the mathematical
> > > prerequisites? Is a course in non-Euclidean geometry necessary?
>
> > Differential geometry + its prerequisites. No specific non-Euclidean
> > geometry is necessary, whatever is needed follows from the relevant
> > parts of diff. geo.
>
> Differential geometry is required to describe the curvature of space due to gravity,
> correct?

On this note, another thought...

The usual analogy in the popular literature is the expanding balloon.
The surface is covered with ink blots. We imagine ants on the surface,
unable to access the third dimension.

Then Einstein Ant thinks up a new theory, which predicts expanding, curved
space. Their astronomers confirm this, by watching the ink blots separate.

Then they measure large triangles, using those blots as markers.
(there's something mystical about triangles) They discover space isn't flat,
and calculate the curvature of the surface.

Finally, they deduce the existence of a third dimension of space - the interior
of the balloon. And calculate its volume.

Now, complete the analogy: why can't we infer a fourth spatial dimension,
and determine its volume? Our universe is embedded in that.

We're the ants, are we not?

Mark

On Sunday, December 12, 2021 at 9:16:06 PM UTC-3, Mark-T wrote:

<snip>

> Finally, they deduce the existence of a third dimension of space - the interior
> of the balloon. And calculate its volume.

NO, they (the relativity elite) deform the theories at the pace of their rotten minds!

You can't calculate a REAL VOLUME, because the 4D spacetime is IMAGINARY, either mathematically
or in the sane perception of TRUE PHYSICS!
> Now, complete the analogy: why can't we infer a fourth spatial dimension,
> and determine its volume? Our universe is embedded in that.
>
> We're the ants, are we not?

Some are stupid and arrogant ants, which is the worst and more dangerous kind.

In 104 years, since 1917, only one valid analytical solution for an static point-like mass that
resides in an universe COMPLETELY VOID OF MATTER or ENERGY (ZERO), was developed
for such VERY SPECIFIC scenario. No other manifestation of matter-energy is allowed in this
SIMPLE SOLUTION, so applications for orbits of planets are valid ONLY if such planet is assumed
to be A ZERO MASS POINT-LIKE PARTICLE.

And this solution, actually called Schwarzschild-Hilbert metric, hasn't been superseded by any
other three solutions that were developed since 1962:

1) A non charged but rotating FUCKING DOT (Kerr)
2) Two other solutions for a fucking charged DOT, either static or rotating.

And the FOUR SOLUTIONS apply for a fictitious void universe, where not even 2 Kg asteroid exists.

For the rest of applications in cosmology, and trying to justify BBT, color holes, expansion, etc., patches
with dark matter and energy are the less stupid things used in the last 50 years. Now are talking that
dark energy can be, actually, ANTIGRAVITY.

See? Stupidity everywhere, specially in the feverish mind of relativists, who CRY FOUL when they are called RETARDED!

Re-read Tom's post, and tell me WHERE does he wrote about PHYSICAL PHYSICS!

Nowhere, because he and SR/GR followers of the CULT of relativity believe that MATHEMATICS IS PHYSICS, which is not.

Remember Newton Universal Law of Gravitation?

He stated a very simple equation: F = GMm/r², and it works almost perfectly, for instance, to measure your weight.
Now, watch what Einstein did, to offer as a theoretical proof of GR. He STATED (and asked for Newton's forgiveness) that

F' = GMm/r² [1 + B²/(m²c²r²)] , where B is the angular moment of whatever with a mass m is spinning around M.

So, thanks to Einstein next time you control your weight. Now you have to include your mass m, which is orbiting around
the center of Earth, at height of 6.3 Km and with a period of almost 24 hours.

Your NEW WEIGHT, not measured by any scale at a drugstore YET, will be given in EINSTEINS, not NEWTONS.

Did you EVER questioned yourself WHY there is NOT A SINGLE PHYSICAL UNIT named after Einstein?

You have newton, gauss, pascal, ohm, faraday, volt, ampere, joule, kelvin, etc., but not a single unit einstein!

Why?

Maybe because scientific establishment plays the relativity game to keep retarded in control, but Science&Technology is managed
by more serious people than what anyone would concede. They are not stupid, so let the game "The greatest scientific mind ever"
to be played, while it doesn't disturb current world affairs.

It's more like a convention of Star Trek fans, which gather together to get consolation and some gratification. But, in the end, they
KNOW that are some kind of RETARDED PEOPLE. With relativists, but with much more money involved, happens exactly the same.

Try some real physics, specially on Earth Sciences. There is a lot to do before......

On Saturday, December 11, 2021 at 10:58:30 PM UTC-8, Richard Hertz wrote:
> On Saturday, December 11, 2021 at 10:13:38 PM UTC-3, Mark-T wrote:
> > If one attempts to tackle general relativity, what are the mathematical
> > prerequisites? Is a course in non-Euclidean geometry necessary?
> > And tensor algebra?
> >
> >
> > Mark
> Mandatory prerequisites:
>
> 1) Cheating, fudging and cooking 101 (Mandatory)
> 2) Applied Social Engineering and Networking (Mandatory)
> 3) How to play with polynomials (Optional)
> 4) The art of plagiarizing (Mandatory)
> 5) Teamwork Management (Optional)
> 6) How to make others work for you (Mandatory)
> 7) How to make deals with establishment (Mandatory)
> 8) Italian language (Mandatory)
> 9) How to deal with journalists (Mandatory)
> 10) Research of Published Scientific Ideas in the last century (Mandatory)

Gobbledygook.

--
Jan

On Sunday, December 12, 2021 at 3:55:53 PM UTC-8, Mark-T wrote:
> On December 11, JanPB wrote:
> > On Saturday, December 11, 2021 Mark-T wrote:
> > > If one attempts to tackle general relativity, what are the mathematical
> > > prerequisites? Is a course in non-Euclidean geometry necessary?
> > > And tensor algebra?
> >
> > Differential geometry + its prerequisites. No specific non-Euclidean
> > geometry is necessary, whatever is needed follows from the relevant
> > parts of diff. geo.
> Differential geometry is required to describe the curvature of space

Spacetime, not space.

> due to gravity, correct?

Yes.

> Which follows from solutions to the field equations. Are those equations
> solvable without computer code?

Just like with most realistic PDEs, the closed form solutions are rather
exceptional (see e.g. the Navier-Stokes equations in aviation).

> Evidently Schwarzchild did so, with his black hole model.

Yes, the symmetry of the configuration in this case happens to
reduce the Einstein's PDE to a system of ODEs which is thus solvable
explicitly.

> What I've found is pop science treatments of the subject, and advanced
> level texts, but nothing that might classify as intermediate general relativity.

There is a cool "intermediate" book which is quite old but both "poetic" and
"exact": https://www.amazon.com/Einstein-Theory-Relativity-Fourth-Dimension/dp/1589880447

Otherwise books seem to fall strictly into poetic or hardcore, exclusively.
Anyone knows of an in-between exception like Lieber's book?

--
Jan

On Sunday, December 12, 2021 at 5:13:22 PM UTC-8, Richard Hertz wrote:
> On Sunday, December 12, 2021 at 9:16:06 PM UTC-3, Mark-T wrote:
>
> <snip>
> > Finally, they deduce the existence of a third dimension of space - the interior
> > of the balloon. And calculate its volume.
> NO, they (the relativity elite) deform the theories at the pace of their rotten minds!

The only rotten mind is yours. What a total psychiatric sad case you are.

--
Jan

On Monday, 13 December 2021 at 01:16:06 UTC+1, Mark-T wrote:
> On December 12, 2021, Mark-T wrote:
> > On December 11, JanPB wrote:
> > > > If one attempts to tackle general relativity, what are the mathematical
> > > > prerequisites? Is a course in non-Euclidean geometry necessary?
> >
> > > Differential geometry + its prerequisites. No specific non-Euclidean
> > > geometry is necessary, whatever is needed follows from the relevant
> > > parts of diff. geo.
> >
> > Differential geometry is required to describe the curvature of space due to gravity,
> > correct?
> On this note, another thought...
>
> The usual analogy in the popular literature is the expanding balloon.
> The surface is covered with ink blots. We imagine ants on the surface,
> unable to access the third dimension.
>
> Then Einstein Ant thinks up a new theory, which predicts expanding, curved
> space. Their astronomers confirm this, by watching the ink blots separate.
>
> Then they measure large triangles, using those blots as markers.
> (there's something mystical about triangles) They discover space isn't flat,
> and calculate the curvature of the surface.
>
> Finally, they deduce the existence of a third dimension of space - the interior
> of the balloon. And calculate its volume.
>
> Now, complete the analogy: why can't we infer a fourth spatial dimension,
> and determine its volume? Our universe is embedded in that.
>
> We're the ants, are we not?

And we're waiting for you to enlighten us and make us humans.
Get conscious, man.

Mark-T <mark-t2@lycos.com> wrote:
> On December 12, 2021, Mark-T wrote:
>> On December 11, JanPB wrote:
>>>> If one attempts to tackle general relativity, what are the mathematical
>>>> prerequisites? Is a course in non-Euclidean geometry necessary?
>>
>>> Differential geometry + its prerequisites. No specific non-Euclidean
>>> geometry is necessary, whatever is needed follows from the relevant
>>> parts of diff. geo.
>>
>> Differential geometry is required to describe the curvature of space due to gravity,
>> correct?
>
> On this note, another thought...
>
> The usual analogy in the popular literature is the expanding balloon.
> The surface is covered with ink blots. We imagine ants on the surface,
> unable to access the third dimension.
>
> Then Einstein Ant thinks up a new theory, which predicts expanding, curved
> space. Their astronomers confirm this, by watching the ink blots separate.
>
> Then they measure large triangles, using those blots as markers.
> (there's something mystical about triangles) They discover space isn't flat,
> and calculate the curvature of the surface.

Actually, there’s nothing special about triangles. Quadrilaterals will also
tell you something. But for a PLANE triangle, the sum of the interior
angles is always pi. For triangles on curved spaces, that sum is not pi,
and the ratio of that difference to the size of the triangle tells you
something about the intrinsic curvature.

>
> Finally, they deduce the existence of a third dimension of space - the interior
> of the balloon. And calculate its volume.

Well, that’s not really so. What’s true is that you can associate a local
intrinsic curvature with the reciprocal of a local radius of curvature.
You’d have to make a lot of assumptions to turn that radius of curvature
into a “volume” and it only works for one sign of the curvature.

>
> Now, complete the analogy: why can't we infer a fourth spatial dimension,
> and determine its volume? Our universe is embedded in that.

First of all, don’t assume the embedding dimension is spatial. After all
the curvature is in spacetime, not just space. Secondly, you don’t actually
NEED to assume that the space we’re familiar with is embedded in a higher
dimensional space; the curvature of the 4-dimensional spacetime can be
taken on its own.

>
> We're the ants, are we not?
>
> Mark
>

--
Odd Bodkin -- maker of fine toys, tools, tables

YMark-T <mark-t2@lycos.com> wrote:
> On December 12, 2021, Mark-T wrote:
>> On December 11, JanPB wrote:
>>>> If one attempts to tackle general relativity, what are the mathematical
>>>> prerequisites? Is a course in non-Euclidean geometry necessary?
>>
>>> Differential geometry + its prerequisites. No specific non-Euclidean
>>> geometry is necessary, whatever is needed follows from the relevant
>>> parts of diff. geo.
>>
>> Differential geometry is required to describe the curvature of space due to gravity,
>> correct?
>
> On this note, another thought...
>
> The usual analogy in the popular literature is the expanding balloon.
> The surface is covered with ink blots. We imagine ants on the surface,
> unable to access the third dimension.
>
> Then Einstein Ant thinks up a new theory, which predicts expanding, curved
> space. Their astronomers confirm this, by watching the ink blots separate.
>
> Then they measure large triangles, using those blots as markers.
> (there's something mystical about triangles) They discover space isn't flat,
> and calculate the curvature of the surface.
>
> Finally, they deduce the existence of a third dimension of space - the interior
> of the balloon. And calculate its volume.
>
> Now, complete the analogy: why can't we infer a fourth spatial dimension,
> and determine its volume? Our universe is embedded in that.
>
> We're the ants, are we not?
>
> Mark
>

One more thing on this. While this is a good *starter* to help people
understand some of the key markers of curved spaces, it does a disservice
to the subject in general if it implies this is what general relativity
says our universe is like. Just as an example, the expanding balloon
applies to spaces with positive intrinsic curvature, but the space could
also have negative or zero curvature and still be expanding. A negative
curvature space does not imply any calculable volume of an embedding
higher-dimensional space.

So watch out for things like this that are just intended to break open the
mental shell a little, and don’t take them as more than they are.

--
Odd Bodkin -- maker of fine toys, tools, tables

On 12/12/21 6:16 PM, Mark-T wrote:
> why can't we infer a fourth spatial dimension, and determine its
> volume? Our universe is embedded in that.

No. Not in any sensible manner.

There are theorems about the embedding of a general (3+1)-dimension
semi-Riemannian manifold in higher-dimension manifolds, such that the
embedding is isometric and the outer manifold is flat [#]. The
lowest-dimension outer manifold known has 88+2 dimensions (yes, an
enormous number of spacelike dimensions and two timelike dimensions, the
meaning of which is not obvious).

[#] These are properties you are surely thinking of.

Tom Roberts

On 12/12/2021 8:13 PM, Richard Hertz wrote:
> On Sunday, December 12, 2021 at 9:16:06 PM UTC-3, Mark-T wrote:
>
> <snip>
>
>> Finally, they deduce the existence of a third dimension of space - the interior
>> of the balloon. And calculate its volume.
>
> NO, they (the relativity elite) deform the theories at the pace of their rotten minds!
>
> You can't calculate a REAL VOLUME, because the 4D spacetime is IMAGINARY, either mathematically
> or in the sane perception of TRUE PHYSICS!

At this point, the post was about the third dimension which does exist,
and the 2D ants calculating the volume of the balloon. Do you believe
the ants can or cannot calculate this? Why or why not?

> [snip blithering]

Tell your shrink that your meds aren't working.

On December 13, 2021 at 6:52:25 AM UTC-8, tjrob137 wrote:
> > why can't we infer a fourth spatial dimension, and determine its
> > volume? Our universe is embedded in that.
>
> No. Not in any sensible manner.
> There are theorems about the embedding of a general (3+1)-dimension
> semi-Riemannian manifold in higher-dimension manifolds, such that the
> embedding is isometric and the outer manifold is flat [#]. The
> lowest-dimension outer manifold known has 88+2 dimensions (yes, an
> enormous number of spacelike dimensions and two timelike dimensions, the
> meaning of which is not obvious).

That's moderately mind boggling.

What does isometric mean, in this context?

Mark

On Monday, December 13, 2021 at 4:50:59 PM UTC-3, Michael Moroney wrote:

<snip>

> Tell your shrink that your meds aren't working.

You and Roberts shouldn't drink alcohol at all. Your daily dose of 300 mg antipsychotic drug potentiate
the effect of alcohol 4 times. Re-read what you both posted here after a glass of malbec or merlot.

Like you both drunk a full bottle of wine!

88+2 dimensions for an outer semi-riemannian manifold as the lowest manifold of higher dimension than in GR.

Even the semi-riemannian name is delusional, typical from a relativist, to hide the fact that ct is a fucking dimension!

Drunk, you both. In such state came here to post, after having an embarrassing phone call to a friend whom you contacted
after being apart for 10 years (or a high-school girlfriend).

This thing of the glass of wine while dinning explain a lot, psychos.

Mark-T <mark-t2@lycos.com> wrote:
> On December 13, 2021 at 6:52:25 AM UTC-8, tjrob137 wrote:
>>> why can't we infer a fourth spatial dimension, and determine its
>>> volume? Our universe is embedded in that.
>>
>> No. Not in any sensible manner.
>> There are theorems about the embedding of a general (3+1)-dimension
>> semi-Riemannian manifold in higher-dimension manifolds, such that the
>> embedding is isometric and the outer manifold is flat [#]. The
>> lowest-dimension outer manifold known has 88+2 dimensions (yes, an
>> enormous number of spacelike dimensions and two timelike dimensions, the
>> meaning of which is not obvious).
>
> That's moderately mind boggling.
>
> What does isometric mean, in this context?

It means that the map preserves the lengths of curves, in the formal sense
of a pullback. Pullback is a differential geometry term.

>
>
>
> Mark
>
>

--
Odd Bodkin -- maker of fine toys, tools, tables

On 12/14/21 8:44 PM, Mark-T wrote:
> On December 13, 2021 at 6:52:25 AM UTC-8, tjrob137 wrote:
>>> why can't we infer a fourth spatial dimension, and determine its
>>> volume? Our universe is embedded in that.
>>
>> No. Not in any sensible manner.
>> There are theorems about the embedding of a general (3+1)-dimension
>> semi-Riemannian manifold in higher-dimension manifolds, such that the
>> embedding is isometric and the outer manifold is flat [#]. The
>> lowest-dimension outer manifold known has 88+2 dimensions (yes, an
>> enormous number of spacelike dimensions and two timelike dimensions, the
>> meaning of which is not obvious).
>
> That's moderately mind boggling.
>
> What does isometric mean, in this context?

It means the mapping that embeds the inner manifold in the outer
manifold preserves the metric.

An example: consider holding a basketball in your hand, and idealize it
as a 2-D spherical surface embedded in the 3-D space we inhabit
(ignoring time). This is an isometric embedding, as the same rulers that
measure distance in the 3-D (outer) manifold can be used to measure
distances in the 2-d (inner) spherical surface (of course they must
conform to the surface, so use a flexible tape measure, not a rigid ruler).

Tom Roberts