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A Thought Experiment:

We are in flat spaceime provided with a reference frame—a rectangular Cartesian frame. The coordinate labels[the spatial labels] are visible to us. Each spatial point is provided with a clock—and the different clocks are synchronized wrt to each other. Gravity is now turned on and made to vary upto some final state. During this process of experimentation the physical separations change while the coordinate labels remain fixed to their own positions. [Coordinate separations remain unchanged]. The length and the orientation of a vector changes in this process both in the 3D and in the 4D sense. We are passing through different/distinct manifolds in our thought experiment and if the 4D arc length does not change we are simply having a transition between manifolds for which ds^2 is not changing but the metric coefficients are changing.We consider the option of $ds^2$ changing in this posting.

Query: Our experiment indicates at projective transformations operating in the physical sense[considering changes in the metric and in the value $ds^2$].A time dependent field is being observed where the metric coefficients are not being preserved. Is it important to include projective transformations[concerned with the non-preservation of the metric] in the mathematical framework of GR?

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You aren't talking about what are normally called "projective transformations", you are talking about dynamical coordinate transformations. This is called the "hole argument" in the early GR literature, and in some obnoxiously wrongheaded current philosophy literature. – Ron Maimon Dec 23 '11 at 5:41

What you are talking about are not what are normally called projective transformations, but nonphysical coordinate changes, these are gauge changes in GR. Einstein was famously confused about this for years, beginning in 1913 when he wrote about the "Hole Argument" in General Relativity. He concluded that it is impossible to have generally covariant equations of motion, only changing his mind in 1915 when he realized the hole argument was completely bogus.

The hole argument is similar to your "projective transformations". He imagines a manifold where the physical situation stays the same, but the points get relabelled dynamically. This leads the metric coefficients to change in a certain way, which cannot be determined by the equation of motion, because it is totally arbitrary.

You didn't say it exactly the same--- you considered letting the metric coefficients change with time, so that the coordinate distance changes, but at the end of the day, the length of any two curves stays the same with the appropriate map, so that the transformation is just a coordinate change. So it's the same thing.

To get unconfused about this, it is best to think of the exactly analogous (but much simpler) gauge transformations in EM. Two vector potentials which have the same holonomy (the same E and B fields in spacetime of normal trivial topology) are identified. This identification means that oscillations of the electromagnetic potential of the form $A=\nabla \phi$ are unphysical, and they have no specific heat, they don't move charges, etc.

The coordinate transformations in GR are exactly analogous. They are more interesting, because in GR, unlike in EM, a gauge transformation, a coordinate change, can introduce a horizon where there was none before. This happens when you change coordinates to the frame of a constantly accelerated observer.

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In the suggested experiment[referring to the question] the coordinate labels remain fixed in their own locations.I have clearly stated that the coordinate separations remain unchanged but the physical separations change due to changes in the values of $g_{\mu\nu}$,coordinate labels remaining fixed in their own positions. The points have not been relabelled in any way--but physical separations have changed – Anamitra Palit Dec 23 '11 at 6:41
@AnamitraPalit: But you said a few sentences later that "ds does not change". What ds doesn't change? You either mean that you are doing a shift of metric which is not physical, or you are saying self-contradictory things. – Ron Maimon Dec 23 '11 at 6:42
I have simply said that if ds^2 does not change while the transition[change in the values of $g_{\mu\nu}$] is going on,there would be a serious problem----there would be a physical transition between distinct manifolds where the metric is not being preserved[due to changes in $g_{\mu\nu}$] while the value of the line element remains invariant.In such an event we would have a corresponding mathematical transformation from one manifold to another where the metric is not preserved while $ds^2$ is preserved.So ds^2 should change in these transformations.That is exactly what I wanted to say. – Anamitra Palit Dec 23 '11 at 6:50
@AnamitraPalit: Is it possible to describe your scenario as "a diffeomorphism from the active viewpoint (points move around, but labels move with them) that is not an isometry ($ds^2$ changes)" ? This would probably be more conventional terminology than "projective transformations", which is more often used in connection with projective geometry. – twistor59 Dec 23 '11 at 7:41
@Anamitra: If you aren't talking about hole argument, I don't see what you are talking about at all. So you can change the metric on a manifold to get a different metric manifold. Who cares? How is this a problem? You can continuously deform a metric sphere to a metric egg or a metric box. Why is this a contradiction? – Ron Maimon Dec 23 '11 at 8:51

Interesting discussion. This is my 2 cents contribution:

1- as @RonMaimon stated above, the term "projective transformations" is probably inappropriate and misleading in this context. "Projective" has specific meanings in mathematics, none of which seems to me to be applicable in this case. The OP should perhaps justify the use of the term

2- In GR literature there are examples of the kind of transformations described by the OP, where you have a parametrized family of manifolds M($\epsilon$) with metrics g($\epsilon$) that are related by a $\epsilon$-family of isomorphisms. These transformations are used in (higher order) perturbative GR to covariantly describe perturbations of exact solutions.

For example see: or other papers by David Brizuela and Jose M. Martın-Garcıa

The conclusion is that the language of differential geometry is powerful enough to deal with these kinds of transformations which describe perturbations of the underlying manifold.

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