Not sure I quite understand the question Yuri, so please bear with me. If there is invariance with respect to coordinate transformation, then by definition it is independent of the coordinate system? 

There are a number of like ways to illustrate relativity, for instance through 'reductio ad absurdum' scenarios, but they're a bit unwieldy, or cumbersome, for the purposes of explaining relativity in the framework of, say, a physics classroom, so that the most straightforward way is probably to demonstrate independence of actual physical values from the way those values are measured?

Very good question, Yuri!

For what are coordinates at all, Chris?

The meaning of "coordinateless form" is to vague, also was is "strange" about them. It would be great if you would share some details on your thoughts.

The fact that they were used by Newton or Einstein, is of historical consideration. That one can imagine today something else, doesn't make coordinates strange or obsolete, it only makes them a special case of a more general formulation if someone manage to envision one.

It seems to me that nobody understands the statement of the problem. Let me explain this in a simple example. In the middle school I studied the Euclidean geometry, using Russian textbook by Kiselev. This textbook does not contain terms "coordinate" and  "coordinate system" at all. But the Eucuclidean geometryhas been presented properly.  In contemporary geometry the coordinateless presentation does not used. The relativity principle is a geometrical principle. At any rate, it is connected with the space-time geometry. Apparently, the connection of the relativity with transformation of coordinates goes from the space-time geometry, presented in the coordinate representation. If the geometry is presented in the coordinateless form, then,  maybe, the relativity principle can be formulated without a reference to coordinate transformation.

You can use, of course, coordinates, Valentin, but for what?

Without coordinates means without vector space, Yuri. Group theory can be formulated without vector space. For relativity it is SO(4,1). The problem is: Can we introduce distance without vector space or not?

Regards,

Eugene.

The distance between two points of the pointset, where the geometry is given, is the only  comprehensive quantity, which describes the geometry in coordinateless way. The vector AB is an ordered set {A,B} of two points A,B. See detailsfor instance in "Geometry without topology as a new conception of geometry." Int. Jour. Mat. & Mat. Sci. 30, iss. 12, 733-760, (2002), (Available at http://arXiv.org/abs/math.MG/0103002 ).

Dear Eugene, let me add, that presentation of the relativity principle in the coordinateless form needs a presentation of the space-time geometry in the coordinateless form. Unfortunately, nobody can present the space-time geometry in the coordinateless form. If one make this, one obtains a monistic conception of the space-time, geometry, which is described completely in terms of metric and only in terms of metric. As a result the space-time is described by one structure ST. In the nonrelativistic case the event space is described by two structures T and S. The structure T describes a time interval between any two events of the event space. The structure S describes a spatial interval between any two events of the event space. Three structures T,S and ST are not independent. The structure ST describes the time-space interval between any two events of the event space. The Mikelson experimet showed that there is only one structure in the event space (there is no absolute simultaneity and the structure T is absent).  Thus, the relativity principle is formulated as follows. There is only one structure ST in the event space. This statement does not refer to properties of the event space (space-time). It is true also in the general relativity, where the space-time is curwed. Such a formulation of the relativity is very simple, but it needs a good knowledge of geometry.

I am trying to study your papers, Yuri. Some time is needed for this.

About metric. Ostrovski (student of Hilbert) has proved in 20-th, that in number's fields only two possibilities to introduce metric (norm, distance) exist. Real numbers and p-adic numbers. Vladimirov (former director of Steklov institute) tried to incorporate p-adic numbers into physics. May be it will be interesting for You.

Regards,

Eugene.

Yuri,

Does not Einstein's equations of general relativity impart covariance? and does not this property garantee that the solutions be independent on the metric used, i.e. what we usually refer to as "background independence"?

Now the best one can do seems to be to replace space and time by operators – I know most of you don't like a time operator, but it can nevertheless be defined on the interval (-∞, +∞) which the conjugate partners "permit" in general (Klein-Gordon and Dirac's eq.).

So we have then two simultaneous equations to solve, i.e. the one for energy-momentum and the conjugate one for time-space.

Now come the surprise: Following this formulation – keeping QM in the picture – yields unequivocably the Schwartzschild metric, which suggests that QM and GR together do not subscribe to a coordinate less form of relativity.

Erkki,

Yuri is working on this subject since sixties. Please, see his works.

Regards,

Eugene.

Yes Eugene,

I will first hear his answer ...

Last absatz in your comment is unclear, Erkki.

About operators. Where are they acting?

Einstein considered only one metric, but exist several.

There are a lot to think about, Erkki.

Regards,

Eugene.

Dear Eugene and Erkki,

I should reccomend you to see the paper  "Nature of some conceptual problems in geometry and in the particle dynamics". It is my new paper. It has beensubmitted to EJTP. It can be found on my web site http://gasdyn-ipm.ipmnet.ru/~rylov/nscpgpd.htm 

Yuri,

I will look up your paper, while you consider my posting.

Yuri,

I have followed some of your important questions on RG, particularly the exchanges with Marcel. A lot what you say there goes above my understanding. However, my experiences may contain some ideas that you may find, although trivial, fitting some of your fundamental questions.

Regarding metrics, our continuation of quantum mechanics beyond the self-adjoint domain prompts the understanding that such a non-Hermitian extension is commensurate with a linear space representation with a non-definite metric – and here we see no coordinate system or explicit metric.

Another example is the analogy between gravitational interactions and the Gödel paradox (commensurate with the Schwarzschild metric). Also here we see no explicit representation of any metric.