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I am the first in the world

Reality Check said:
This seems to be the entire point of the story.
You are assuming that just because you use x and y for 2D Cartesian coordinates in flat space that a metric for non-flat space that has x and y in it is using 2D Cartesian coordinates.
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I will try and explain how coordinates and metrics work one more time using what I hope will be an intuitive enough explanation for you to grasp.

Imagine you are an ant on a flat 2D plane. You like using Cartesian Coordinates so you start making a grid ala Decartes. Being an ant you can not necessarily know you are on a flat 2D plane for sure so you do some measurements (in the sense of my previous post) and find out, yep, for at least the part of the plane you have mapped out, the space is flat.

Now imagine you are an ant on the surface of an elliptic paraboloid. Here is a link covering such a surface: http://en.wikipedia.org/wiki/Paraboloid. So you go about making grids in the 2D Cartesian way you did before. You do some measurements and find out that for the part of the surface you have mapped that the space is not flat.

In the two cases the same type of coordinate system was used, 2D Cartesian, but the expression you will get for what the metric will be is different. The ant could have used 2D polar coordinates, a whole lot of other kinds of coordinate systems. The type of the coordinate system does not change just because you embed it in a new space.

I am really saying what is in line with conventional thought here RC, so here is a reality check for you, you do not understand metrics. Think about them more carefully and maybe do some reading, because, between the two of us, you are the one that is taking the quack position on this one.

Reality Check said:
You do not seem to understand what I stated so here is another try:
Your question was not clear enough for any answer . Thus I did not answer it. I did not say yes. I did not say no.
There was no disagreement with logic because there was no understandable question to be answered.

What I have already sid was:
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What you sid is you like to repeat yourself. I said I will hold you to what you have said, not that you said yes or no. Do you mind if I hold you to your word that in the last post you said "I did not say yes. I did not say no.", or would that be pushing things? We do have a quote mechanism on these posts for a reason, no? Unless you think the generally agreed upon rules of debate should not be applicable to you.

Vorpal, let me get back to you on your post in a bit. I just woke up... Maybe when it is my night-time I will try and explicate things.
 
alfa1 said:
In my view the flag of Transylvania: Red, Green, Blue - because these are the primary colors.
Those who designed the flags did not care about science!
For example the Romanian flag BYR (semn rau) is not good Blue, Yellow, Read; the frequency is descending from high frequency(blue) to low frequency(red).
The German flag is not good too because the frequency is descending…
http://partidultransilvania.tk
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r920835fbd98.jpg
 
tensordyne said:
Imagine you are an ant on a flat 2D plane. You like using Cartesian Coordinates so you start making a grid ala Decartes. Being an ant you can not necessarily know you are on a flat 2D plane for sure so you do some measurements (in the sense of my previous post) and find out, yep, for at least the part of the plane you have mapped out, the space is flat.

Now imagine you are an ant on the surface of an elliptic paraboloid. Here is a link covering such a surface: http://en.wikipedia.org/wiki/Paraboloid. So you go about making grids in the 2D Cartesian way you did before. You do some measurements and find out that for the part of the surface you have mapped that the space is not flat.

In the two cases the same type of coordinate system was used, 2D Cartesian, but the expression you will get for what the metric will be is different. The ant could have used 2D polar coordinates, a whole lot of other kinds of coordinate systems. The type of the coordinate system does not change just because you embed it in a new space.
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I don't know what you think "Cartesian" means, but whatever it is, it's not what it means to me. Cartesian coordinates make sense only in flat space.

Wiki defines "Cartesian" as a set of coordinates that measure the distance to two perpendicular lines. Think about latitude and longitude. Those measure the (geodesic) distance to two "straight lines" (the geodesics of zero latitude - the equator - and zero longitude) that cross at right angles. And yet, those coordinates more closely resemble polar coordinates than they do Cartesian. For example, and the north and south pole they reduce to polar coordinates.

I am really saying what is in line with conventional thought here RC, so here is a reality check for you, you do not understand metrics. Think about them more carefully and maybe do some reading, because, between the two of us, you are the one that is taking the quack position on this one.
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No, I don't think so. Before accusing people of being "quacks", you might learn a little bit more about the topic. It's clear you're a beginner in this area. A little less arrogance might serve you well.
 
tensordyne said:
...
Now imagine you are an ant on the surface of an elliptic paraboloid. Here is a link covering such a surface: http://en.wikipedia.org/wiki/Paraboloid. So you go about making grids in the 2D Cartesian way you did before. You do some measurements and find out that for the part of the surface you have mapped that the space is not flat.
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Really basic stuff but as already pointed out: Cartesian coordinates are only defined for flat space.
What your ant is doing is finding out that it is invalid to use Cartesian coordinates in a curved space.
 
coordinate system types versus metrics.

:boxedin:

Well, let's see what Vorpal and Albert Einstein have to say on the matter of coordinate systems and geometry (metrics).

In post #45 by Vorpal: "Coordinates are independent of geometry."

Einstein as Quoted in Gravity by MTW: "Why were another seven years required for the construction of the general theory of relativity? The main reason lies in the fact that it is not so easy to free oneself from the idea that coordinates must have an immediate metrical meaning."

If coordinates are not supposed to have an immediate metrical meaning, then why should the thing that gives the coordinates, the coordinate system, have an immediate metrical meaning? It does not have to, and that is the point.

I could go on. I could use the definition in wikipedia of cartesian coordinates with the qualifier that no assumption as to geometry is used and show that for instance using said type of coordinate system on a sphere does not give a Euclidean metric.

I could even point out that abstractly one can map a sphere to a plane (stereographic projection I think it is called) and use Cartesian Coordinates on the plane so that the points on the sphere for half the sphere (have to use coordinate charts because of coordinate based singularities, a theorem on this exists as well from what I understand) correspond in a one-to-one and onto way. If you equip the plane with the proper metric it will give the same answers (in that patch) as the sphere gives. Geodesics will match, areas will match, etc. etc.

If you have a problem with mapping, then I can just point out my previous post to show how one could locally talk about using, for instance, 2D Cartesian Coordinates on a surface and how that can result in non-Euclidean metrics.

Point is sol invictus and Reality Check, there is a more abstract way to look at coordinate systems and metrics :jaw-dropp. In essence, you are complaining because I am using this more abstract notion and telling me I can not use it (and telling me I am a newbie for using higher-level abstraction, which is pretty funny).

So do as Einstein did is my suggestion, free yourself from the idea that coordinate systems imply a specific geometry. If not, well, I would rather debate with others who do get this idea.
 
tensordyne said:
I could even point out that abstractly one can map a sphere to a plane (stereographic projection I think it is called) and use Cartesian Coordinates on the plane so that the points on the sphere for half the sphere (have to use coordinate charts because of coordinate based singularities, a theorem on this exists as well from what I understand) correspond in a one-to-one and onto way. If you equip the plane with the proper metric it will give the same answers (in that patch) as the sphere gives. Geodesics will match, areas will match, etc. etc.
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Of course. I use that mapping (and related ones) all the time - literally on a day-to-day basis. FYI, those coordinates cover everything except a single point (the "north pole" of the sphere if you visualize it sitting on the plane), not half the sphere. The "point at infinity" of the plane is the north pole.

I would not call the coordinates in that plane Cartesian, because the metric isn't flat. If you want an unambiguous term that indicates the metric can be written in the form (some function of the coordinates)*(flat metric) - where obviously the (flat metric) can be written in Cartesian coordinates - it's "conformally flat". The sphere is one example.

Point is sol invictus and Reality Check, there is a more abstract way to look at coordinate systems and metrics :jaw-dropp. In essence, you are complaining because I am using this more abstract notion and telling me I can not use it (and telling me I am a newbie for using higher-level abstraction, which is pretty funny).
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No, I am complaining because you are using technical terms incorrectly. I am fairly certain your misunderstandings go well beyond a misuse of language, but it's hard to be certain.

There is nothing very mysterious about all of these coordinate systems. As Vorpal has tried to explain to you, coordinates are nothing more or less than a way of labeling points. Changing from one coordinate system to another requires a little bit of math, and given the coordinates and the metric, the geometry is fully determined (modulo a few advanced subtleties that go far beyond the level of this conversation, like singularities, event horizons, etc.).

But there's really nothing more mysterious here than there is in changing from polar to Cartesian coordinates in the plane (even event horizons can be understood in almost exactly those terms). The laws of physics are invariant under such transformations for obvious reasons - physics cannot depend on arbitrary labels.
 
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tensordyne said:
If you have a problem with mapping, then I can just point out my previous post to show how one could locally talk about using, for instance, 2D Cartesian Coordinates on a surface and how that can result in non-Euclidean metrics
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Your previous post was how using 2D Cartesian Coordinates on a surface can result in the realization that the surface cannot be described by 2D Cartesian Coordinates..
Your ant on a curved surface tried to use 2D Cartesian Coordinates (e.g. drew a triangle) and found that their 2D Cartesian Coordinates were not 2D Cartesian Coordinates (i.e. the angles in their triangle did not sum up to 180 degrees).

tensordyne said:
Point is sol invictus and Reality Check, there is a more abstract way to look at coordinate systems and metrics :jaw-dropp. In essence, you are complaining because I am using this more abstract notion and telling me I can not use it (and telling me I am a newbie for using higher-level abstraction, which is pretty funny).
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The point is tensordyne, that there is a more abstract way of looking at coordinate systems and metrics - that is to forget about coordinate systems and use coordinate-free notation:jaw-dropp!
The essence is that you are using a more concrete (not more abstract) and what looks incorrect notion for coordinate systems, e.g. that Cartesian coordinates that are only defined for flat space can be used for any space.
 
tensordyne--
On what's a 'Cartesian coordinate', the term is generally reserved for flat manifolds. Though in a few places the term can be used where there's technically not a metric at all; the only usage for specifically non-flat manifolds involved transforming from any coordinates adapted to rotational symmetry in the same way as from polar to Cartesian in Euclidean space, e.g., this occurs in Petersen's Riemmanian geometry. Though it's intuitive, it should be considered intentional abuse.

There's already a term for the specific case of coordinates in which the metric takes the Cartesian form at a point, as you have in post #112: Riemann normal coordinates. For a unit tangent vector v at that point, one can take the geodesic with that initial velocity, and therefore identify every point on it with the pair (λ,v), where λ is the length along the geodesic, or in general an affine parameter. For an orthonormal basis in the origin's tangent plane, v has components (vα), so the (λ,v) can be assigned coordinates (λvα). This is one way to think about Riemann normal coordinates.

In a small enough neighborhood, this makes sense and behaves analogously to Cartesian coordinates in every way relevant to what you're trying to do in post #112--if I understand you at all. Riemann normal coordinates define a locally inertial frame, and so correspond operationally to the measurements you're talking about, and the metric at the origin does take the form you require.

Regarding your earlier desire for spherical coordinates to have the radial parameter correspond to distance, there's also a standard term for something close: polar coordinates (regardless of dimensionality; one way to define them would be analogously to RNCs except taking ordinary spherical coordinates in the origin's tangent plane while treating it Euclidean). Though once again this means the metric at the origin takes the block-diagonal form dr²+{bleh}, and so does not actually represent radial distance in general.

Hopefully that clarifies some things.

tensordyne said:
Well, let's see what Vorpal and Albert Einstein have to say on the matter of coordinate systems and geometry (metrics).

In post #45 by Vorpal: "Coordinates are independent of geometry."

Einstein as Quoted in Gravity by MTW: "Why were another seven years required for the construction of the general theory of relativity? The main reason lies in the fact that it is not so easy to free oneself from the idea that coordinates must have an immediate metrical meaning."
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If by coordinates, one simply means either a coordinate chart or a coordinate chart with extra differentiable compatibility criteria with other charts (as many define 'coordinate system', see post 55), then it is absolutely the case that coordinates are independent of geometry.

If the situation one has in mind is encountering an expression for ds² in terms of some variables, then since one is defining coordinates, metric, and a relationship between them, then under every reasonable interpretation, they are interdependent--you have an equation that relates them.

If by 'coordinates', you mean what you call 'coordinate system', then I'm honestly not sure, because I don't have a very clear idea of what that is. However, if I take the guidance of your post 112, then your 'coordinate system'='Riemann normal coordinates', in which case they are tightly connected, by definition, as the geodesics carry geometrical information, in our case both affine and metrical.

Reality Check already complained that one of your questions to him was vague. But I think it wasn't just the one.
 
If I am reading you right, a consequence of your theory is that energy can also get diverted from the real to the virtual universe.

But doesn't that violate the conservation of matter/energy theory?
 
:DE=mc^2
E=1/2mv^2

Therefore mc^2=1/2mv^2
Hence c^2=1/2v^2
c= Sqrt(1/2)v
c=0.25v

So you are driving always at a quarter of speed of light.

Derived from your equations..... Except that you introduced another factor of two....

Prepare for lots of speedeing tickets.:D
 
In Quantum mechanics my theory explains: " The amplitude of the wave is proportional with the speed of the particle: A =k*v " Adrian Ferent
 
alfa1 said:
In Quantum mechanics my theory explains: " The amplitude of the wave is proportional with the speed of the particle: A =k*v " Adrian Ferent
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That's a very peculiar thing to try to 'explain'. What wave has an amplitude that is proportional to the speed of a particle?
 
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