The Gaussian Co-ordinate Systems of General Relativity





The focus here is on foundational issues of fundamental importance, not on cosmetic and pedantic and superficial matters like 'such a statement is unsuitable for a scientist', 'nature comes first', 'be the judge, ...etc.'.

Einstein has justified the connection between space-time variability and gravity many times over and in so many places.

Einstein, also, has defined, in various contexts, the co-ordinate systems of general relativity and illustrated them step by step and with the utmost clarity and precision.

This is only one example:



According to Gauss, this combined analytical and geometrical mode of handling the problem can be arrived at in the following way. We imagine a system of arbitrary curves (see Fig. 4) drawn on the surface of the table. These we designate as u-curves, and we indicate each of them by means of a number. The Curves u= 1, u= 2 and u= 3 are drawn in the diagram. Between the curves u= 1 and u= 2 we must imagine an infinitely large number to be drawn, all of which correspond to real numbers lying between 1 and 2. We have then a system of u-curves, and this "infinitely dense" system covers the whole surface of the table. These u-curves must not intersect each other, and through each point of the surface one and only one curve must pass. Thus a perfectly definite value of u belongs to every point on the surface of the marble slab. In like manner we imagine a system of v-curves drawn on the surface. These satisfy the same conditions as the u-curves, they are provided with numbers in a corresponding manner, and they may likewise be of arbitrary shape. It follows that a value of u and a value of v belong to every point on the surface of the table. We call these two numbers the co-ordinates of the surface of the table (Gaussian co-ordinates). For example, the point P in the diagram has the Gaussian co-ordinates u= 3, v= 1. Two neighbouring points P and P¹ on the surface then correspond to the co-ordinates

P: u, v
P¹: u + du, v + dv,

where du and dv signify very small numbers. In a similar manner we may indicate the distance (line-interval) between P and P¹, as measured with a little rod, by means of the very small number ds. Then according to Gauss we have

ds² = g₁₁du² + 2g₁₂dudv = g₂₂dv²

where g₁₁, g₁₂, g₂₂ are magnitudes which depend in a perfectly definite way on u and v. The magnitudes g₁₁, g₁₂, and g₂₂ determine the behaviour of the rods relative to the u-curves and v-curves, and thus also relative to the surface of the table. For the case in which the points of the surface considered form a Euclidean continuum with reference to the measuring-rods, but only in this case, it is possible to draw the u-curves and v-curves and to attach numbers to them, in such a manner, that we simply have:

ds² = du² + dv²

Under these conditions, the u-curves and v-curves are straight lines in the sense of Euclidean geometry, and they are perpendicular to each other. Here the Gaussian coordinates are simply Cartesian ones. It is clear that Gauss co-ordinates are nothing more than an association of two sets of numbers with the points of the surface considered, of such a nature that numerical values differing very slightly from each other are associated with neighbouring points "in space".
So far, these considerations hold for a continuum of two dimensions. But the Gaussian method can be applied also to a continuum of three, four or more dimensions. If, for instance, a continuum of four dimensions be supposed available, we may represent it in the following way. With every point of the continuum, we associate arbitrarily four numbers, x₁, x₂, x₃, x₄, which are known as "co-ordinates". Adjacent points correspond to adjacent values of the coordinates. If a distance ds is associated with the adjacent points P and P¹, this distance being measurable and well defined from a physical point of view, then the following formula holds:

ds² = g₁₁dx₁² + 2g₁₂dx₁dx₂ ....g₄₄dx₄²

where the magnitudes g₁₁, etc., have values which vary with the position in the continuum. Only when the continuum is a Euclidean one is it possible to associate the co-ordinates x₁ .. x₄ with the points of the continuum so that we have simply

ds² = dx₁² + dx₂² + dx₃² + dx₄²

In this case relations hold in the four-dimensional continuum which are analogous to those holding in our three-dimensional measurements. However, the Gauss treatment for ds² which we have given above is not always possible. It is only possible when sufficiently small regions of the continuum under consideration may be regarded as Euclidean continua. For example, this obviously holds in the case of the marble slab of the table and local variation of temperature. The temperature is practically constant for a small part of the slab, and thus the geometrical behaviour of the rods is almost as it ought to be according to the rules of Euclidean geometry. Hence the imperfections of the construction of squares in the previous section do not show themselves clearly until this construction is extended over a considerable portion of the surface of the table.
We can sum this up as follows: Gauss invented a method for the mathematical treatment of continua in general, in which "size-relations" ("distances" between neighbouring points) are defined. To every point of a continuum are assigned as many numbers (Gaussian coordinates) as the continuum has dimensions. This is done in such a way, that only one meaning can be attached to the assignment, and that numbers (Gaussian coordinates) which differ by an indefinitely small amount are assigned to adjacent points. The Gaussian coordinate system is a logical generalisation of the Cartesian co-ordinate system. It is also applicable to non-Euclidean continua, but only when, with respect to the defined "size" or "distance", small parts of the continuum under consideration behave more nearly like a Euclidean system, the smaller the part of the continuum under our notice.

http://www.wbabin.net/einstein.zip

Re: Stanley 16, The Gaussian Systems of General Relativity September 6 2008 at 7:09 PM Stanley16

Stanley: The focus here is on foundational issues of fundamental importance, not on cosmetic and pedantic and superficial matters like 'such a statement is unsuitable for a scientist', 'nature comes first', 'be the judge, ...etc.'.

Einstein has justified the connection between space-time variability and gravity many times over and in so many places.

Einstein, also, has defined, in various contexts, the co-ordinate systems of general relativity and illustrated them step by step and with the utmost clarity and precision.

This is only one example:

Stanley, Blah...blah...blah! OK! So what is your point Stan? What are you comparing Einstein to? Who are your trying to prove wrong? Why do you keep posting this tripe and never answer any questions? Why does Cinci always follow me and mis-direct the subject? Are you using Einstein to prove Einstein or just copy/pasting for the fun of it? Are you comparing Einstein to Newton? If so, why do you not say just so? If it is 3:00am in China, what time does the 7:00pm train arrive at Grand Central Station N.Y., N.Y.? <That was a softball question Stan! Good luck.

bob s

None of the above explains or justifies or rationalizes in any way or manner the basic assumption at the heart of General Relativity that equates gravitation to the curvature of the hypothetical four-dimensional continuum of this unrealistic theory.


All what Einstein has done, in the fore-mentioned passage, is to state very verbosely that Gauss' co-ordinate systems are geodesic and made up of curves instead of straight lines.

Surely, the Gauss Scheme works as long as the co-ordinates are three and the space remains Euclidean.

But, as soon as the assumption of more than three dimensions is made, matrix algebra comes from the front door and geometry jumps out of the window.


Thus, to continue labeling these symbolic and purely algebraic systems as geometries and geometrical spaces is certainly misleading and useless.

Scientists say the collider is finally ready for an attempt to circulate a beam of protons the whole way around the 17-mile tunnel.
The test, which takes place Wednesday, is a major step toward seeing if the the immense experiment will provide new information about the way the universe works.

http://www.cnn.com/2008/TECH/09/08/lhc.collider/index.html

Well, today, the scientists (the cream of the cream of the human intelligence and genius) at LHC of Geneva have fired the protons all the way across a whopping distance of 17 miles:
http://www.admin.cam.ac.uk/news/dp/2008090801

But so far, the only fact they know for sure is that protons can fly.

Nevertheless, the mystery of all mysteries remains as daunting and mysterious as ever.

How can the protons fly without any wings or feathers or propellers or jet engines made by the chief engineer, Cincirob, in the sixties and the seventies and the eighties of the last century?

That is, my friends, the mystery of all mysteries that shall remain as daunting and mysterious as ever!

http://www.msnbc.msn.com/id/26439957 Controllers checked the alignment of the beam as barriers were removed at each stage of the route. Applause and shouts greeted every report of progress along the 330-foot-deep (100-meter-deep) tunnel — climaxing when the beam made its first full clockwise circuit, less than an hour after it was turned on.

17 miles in one hour is uh...17 mph (well almost), what are they using, a golf cart to push the beam?

Cincirob: Relativity (I assume you mean Einstein) didn't "presuppose" anything but the constancy of the speed of light for inertial observers. Read his paper. Pure babble. And relativity isn't the "cutting edge" of science. It was 100 years ago. Try to keep up.




AAF: No; I mean Relativity; but what is the difference between Einstein and his Relativity anyway? Can you show me what it is without any more humbug on your part, Cincirob; please? And you did it again! Cinci, stop making this ludicrous suggestion over and over again! That paper as written over 102 years ago, and everybody with the slightest interest in physics knows it. So, keep the best of your 'pure babble' for yourself, please; and everybody will love you for it!


Cincirob: Einstein is a person. Relativity is a theory. People theorize or "presuppose". Theories are the result. Except you apparently who thinks relativity is the "cutting edge" of science. Once it's in all the text books and is used in the design of physical systems, it ain't cutting edge anymore. I just call 'em like I see 'em. Babble is babble.



AAF: But he is dead now; he is no longer a person anymore, except perhaps as seen from the reference frame in which his 'Old One' is at rest! And so, his theory of Relativity is what he left us with; and this theory of his presupposes the multiplicity of time in moving frames of reference, because even though Newton's Absolute Time can be made to work within the framework of the theory of Relativity, the theory of Relativity would lose most of its predictions and become largely irrelevant. So, Cinci, please keep your 'pure babble' inside your knapsack or fold it and leave it under your bed before you show up to work!



Aaron: The more he does this, the less relevant he becomes. Eventually the admin will remove the behavior. Then we can all get back to new ideas.



Cincirob: Forgive my brevity, but I'm away from home and paying of this by the minute. The paragraph above discusses the issue of the distance of the stationary observer from the moving observer and the travel time of the signal to the stationary observer. Including this variable greatly complicates the situation. The need to consider it is eliminated by Einstein's 1905 approach where he first shows how clocks within a given frame can be synchronized. Then the moving observers clock is compared to these synchronized clocks locally instead of dealing with the signal travel time delay. This doesn't mean you couldn't work out the principles of relativity by including the travel times but if you have arrived at different conclusions than relativity, you have probably not got it right.



AAF: That is the kind of counter argument that might actually help your Albert's 'very dumb' Relativity; and I like it; so I've looked at it; and I've confiscated it, forgive me, Cinci, for copying it verbatim from there to right here, please! I came, I saw, I conquered: http://www.roman-empire.net/republic/caesar.html

AAF repeats: Aaron: The more he does this, the less relevant he becomes. Eventually the admin will remove the behavior. Then we can all get back to new ideas.

cinci: Maybe he was talking about you.
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Cincirob: Forgive my brevity, but I'm away from home and paying of this by the minute. The paragraph above discusses the issue of the distance of the stationary observer from the moving observer and the travel time of the signal to the stationary observer. Including this variable greatly complicates the situation. The need to consider it is eliminated by Einstein's 1905 approach where he first shows how clocks within a given frame can be synchronized. Then the moving observers clock is compared to these synchronized clocks locally instead of dealing with the signal travel time delay. This doesn't mean you couldn't work out the principles of relativity by including the travel times but if you have arrived at different conclusions than relativity, you have probably not got it right.

AAF: That is the kind of counter argument that might actually help your Albert's 'very dumb' Relativity; and I like it; so I've looked at it; and I've confiscated it, forgive me, Cinci, for copying it verbatim from there to right here, please! I came, I saw, I conquered: http://www.roman-empire.net/republic/caesar.html

cinci: It's not an argument. It's an explanation. You should try explaining something some time instead of just spouting drivel.
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