Is the period a physical observable in General Relativity? I am currently seeing the classical tests of GR. To justify the introduction of a test based on the Doppler effect, the professor says that the previous test ( Shapiro and echo-radar test ) is based on non-physical parameters as the radius of Earth's orbit since it changes when coordinates change. 
Furthermore, he introduce the period and eccentricity of the Earth's orbit ( orbital parameters ) because they are physical parameters ( I suppose this means they don't change with change of the metrics ). 
However, I don't see the difference with the radius since making the change $t \rightarrow 2t $ seems to change the period by a factor 2. The only thing I could imagine is that it does not satisfy Einstein equation anymore but I don't think so.
What have I misunderstood? Or am I right?
 A: 
the professor says that the previous test (Shapiro and echo-radar
  test ) is based on non-physical parameters as the radius of Earth's
  orbit since it changes when coordinates change.

I can't be sure your professor really said that. In general terms it's
perfectly true, but it must be taken "cum grano salis" (this is Latin,
the language spoken in ancient Rome and in the Roman empire - it
means "with a grain if salt").
There are coordinates endowed of physical meaning, sometimes exact.
sometimes approximate. This is the case with Earth's orbital radius
(more exactly, major semi-axis). If Schwarzschild's coordinates are
taken within solar system coordinate $r$ is an excellent approximation
to proper distance to Sun's centre. Still better, a circle having $r$
as Schwarzschild radius has proper length $2\pi r$, with no
approximation.
Going a little deeper, whatever coordinates you choose, once metric is
known you're always able to compute invariant quantities. So I
really can't understand that statement.
Your example ($t\to2t$) is rather trivial. There are much more complex
coordinate changes, e.g. mixing time and space, where physical
interpretation may become more demanding. But never impossible. BTW
Einstein's equations have no rôle here, as you can write them in any
coordinates you prefer.
Now consider orbital period. It suffers much the same problem, as it
can measured in several ways and using clocks located in several
places. Two usual cases are


*

*period measured with clocks located on Earth (the most common way,
of course)

*period computed in astronomical time scales (see later).


In any case there will be differences to be accounted for when the
highest precision is required. No one time is more right or more
physical than another. What is necessary is that it have an accurate
operational or theoretical definition, in order to allow transformations
and reduction of data.

As to astronomical time scales, I'll copy from wikipedia
https://en.wikipedia.org/wiki/Geocentric_Coordinate_Time
Geocentric Coordinate Time [...] It is equivalent to the proper time
experienced by a clock at rest in a coordinate frame co-moving with
the center of the Earth: that is, a clock that performs exactly the
same movements as the Earth but is outside the Earth's gravity well.
It is therefore not influenced by the gravitational time dilation
caused by the Earth.
[...]
Because the reference frame for TCG is not rotating with the surface
of the Earth and not in the gravitational potential of the Earth, TCG
ticks faster than clocks on the surface of the Earth by a factor of
about $7.0\cdot10^{-10}$ (about 22 milliseconds per year).
https://en.wikipedia.org/wiki/Barycentric_Coordinate_Time
Barycentric Coordinate Time [...] is equivalent to the proper time
experienced by a clock at rest in a coordinate frame co-moving with
the barycenter of the Solar system: that is, a clock that performs
exactly the same movements as the Solar system but is outside the
system's gravity well. It is therefore not influenced by the
gravitational time dilation caused by the Sun and the rest of the
system.
[...]
Because the reference frame for TCB is not influenced by the
gravitational potential caused by the Solar system, TCB ticks faster
than clocks on the surface of the Earth by $1.550505\cdot10^{-8}$
(about 490 milliseconds per year).

I don't expect you may understand all that I've copied. I did it just
to give you some feeling of how astronomers really work. None of those
times is realy "physical" (re-read their definitions) but nevertheless
they can be and actually are used as basis for extremely accurate
astronomical research.
A: As Elio and others have written, the comment by your professor is not really very meaningful. I would rather like to write about the title of your question. Of course the period of Revolution of Earth is a physical observable, after all you can measure it! Obviously different observers might measure different periods, but this does not mean anything special. Even energy has different values for different observers, but this does not mean that energy is not a physical observable. Adding in GR does not really help much. GR is not a crazy theory, but a very successful physical theory describing our universe. Even the Mayas were able to measure the period of Revolution of the Earth, so it would be amazing if GR could not do the same,only much much better.
