Three questions related to the principle of general covariance in GR

[Michael_1812]

1. A (presumably) simple question:

We are used to think that the affine connections emerge whenever one wants to
differentiate a vector (tensor, spinor) on a curved manifold in general relativity. Now suppose that we are still on a flat background of special relativity, though in a curvilinear grid. Will the connections show up there?

2. Now comes a tougher one:

According to the (weak) equivalence principle, at each point of the general-relativity manifold there exists a reference frame (the freely falling one) wherein the motions are the same as in an inertial frame of the special relativity.

According to the principle of general covariance, all the fundamental laws of physics stay form-invariant under a transition between any two coordinate charts (provided the derivatives in these laws are covariant and are taken with respect to a true scalar - the interval). Stated alternatively, the fundamental laws can be expressed in a coordinate-independent way.

How do these two fundamental principles square?

I think it was Weinberg's book where I saw a sentence saying that the principle of general
covariance is a mathematical implementation of the equivalence principle. Why is that??

3. ... and the toughest one:

Above, when formulating the equivalence principle, I used the wording "at each point of the general-relativity manifold". Is this wording at all legitimate? I mean: is it right to assume that each point of the manifold is a physical event? I am asking, because we have gauge invariance, so it may happen that a whole orbit corresponds to one and the same event.
Please correct me if I am wrong.

Many thanks for your time and help,
Michael

[George Jones]

1. A (presumably) simple question:

We are used to think that the affine connections emerge whenever one wants to
differentiate a vector (tensor, spinor) on a curved manifold in general relativity. Now suppose that we are still on a flat background of special relativity, though in a curvilinear grid. Will the connections show up there?

Yes. And they even show up in Galilean/Newtonian mechanics in curvilinear coordinates.

Sorry for not getting back to you. I kept putting this off in the hope of writing a more detailed answer than I give above.

[atyy]

2. Now comes a tougher one:

According to the (weak) equivalence principle, at each point of the general-relativity manifold there exists a reference frame (the freely falling one) wherein the motions are the same as in an inertial frame of the special relativity.

According to the principle of general covariance, all the fundamental laws of physics stay form-invariant under a transition between any two coordinate charts (provided the derivatives in these laws are covariant and are taken with respect to a true scalar - the interval). Stated alternatively, the fundamental laws can be expressed in a coordinate-independent way.

How do these two fundamental principles square?

I think it was Weinberg's book where I saw a sentence says that the principle of general
covariance is a mathematical implementation of the equivalence principle. Why is that??

I believe Weinberg does say GC=EP BUT he does not define GC as you do, and says that GC as normally defined is physically empty. (But check his book to be sure.)

I think it's something like:
GC (as normally defined, not Weinberg's def) is physically empty.
EP implies gravity can be geometrically formulated, but does not lead uniquely to GR - see eg. Newton-Cartan theory or Nordstrom's second theory.
GR requires one more principle: "no prior geometry" (MTW).
And GR in full form does not have particles travelling on geodesics - it's a field with "metric" symmetry, whose equations of motion are derived from a Lagrangian, and matter is just other fields, which are also derived from a Lagrangian.

Edit: In addition to "no prior geometry", GR also requires an assumption about the highest order derivative that enters the field equations.

[atyy]

Above, when formulating the equivalence principle, I used the wording "at each point of the general-relativity manifold". Is this wording at all legitimate? I mean: is it right to assume that each point of the manifold is a physical event? I am asking, because we have gauge invariance, so it may happen that a whole orbit corresponds to one and the same event.
Please correct me if I am wrong.

It's legitimate. Strictly speaking, you should say isometry equivalence class of manifolds and metrics or something like that, but it's taken for granted you can just make true statements about one member of the equivalence class. See eg. Ellis & Hawking p56.

[atyy]

Edit: In addition to "no prior geometry", GR also requires an assumption about the highest order derivative that enters the field equations.

Sean Carroll's http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll4.html gives more detail. A quick summary of the essential points. Suppose we already decided from the EP that the "metric" should become a field that models gravity. Then we should presumably seek a Lagrangian whose terms contain the metric. Because the Lagrangian is a scalar, a further restriction on these terms is that they should be scalars. This allows Lagrangians such as Eq 4.76. To obtain the Hilbert action 4.55, a further restriction is made to include only terms containing up to 2 derivatives of the metric.

Also useful:
Luca Bombelli, http://www.phy.olemiss.edu/~luca/Topics/grav/higher_order.html.
Donoghue, http://arxiv.org/abs/gr-qc/9512024, especially Eq 21.

[atyy]

Hmmm, I didn't realise that there are many subtleties about the relationship between metric theories of gravity and the equivalence principle:

http://arxiv.org/abs/0707.2748
Theory of gravitation theories: a no-progress report
Thomas P Sotiriou, Valerio Faraoni, Stefano Liberati