Gravity as a force and as a curvature

[Nugatory]

i don't see anything in that FAQ about accelerating observers …

Hmmm... yes, and that's an argument for fixing the FAQ. The essential difference between SR and GR is that GR works in non-flat spacetimes (curvature tensor doesn't vanish, regardless of course of choice of coordinates) as well as flat spacetimes (curvature tensor does vanish, again regardless of choice of coordinates). Historically, however, this fact has been obscured for several reasons:
- The equivalence principle established an intuitive connection between acceleration and gravitation before the formulation of GR.
- Before the formulation of GR, there was very little motivation to distinguish between coordinate systems that were funny because of how they were defined (for example, the non-intertial Rindler coordinates that lead to a horizon in a perfectly ordinary flat spacetime that is just as easily spanned by by bone-stock Minkowski x/y/z/t coordnates) as opposed to funny because the underlying spacetime curvature effects.
- SR is nearly always explained using inertial frames because they're generally simpler.

 

[Naty1]

PeterDonis: post #43 You posted...

...was enough to tell {Einstein} that Ricci curvature only, not Weyl curvature, was what coupled to the SET, which already gives you a lot of the right features of the dynamics.

ok, the Ricci tensor measures the kind of curvature that is produced by local masses... the relative volume change of a geodesic ball etc,etc.

How does one obtain/derive that physical insight? I know its true, but so far the insight eludes me.

 

[PeterDonis]

the Ricci tensor measures the kind of curvature that is produced by local masses... the relative volume change of a geodesic ball etc,etc.

Yes. This is just a matter of geometry; see below.

How does one obtain/derive that physical insight? I know its true, but so far the insight eludes me.

I'm not sure exactly what thought process led Einstein to the (incorrect but close) 1913 field equation, $R_{ab} = T_{ab}$. But the definition of the Ricci tensor is purely geometric; if you've accepted that spacetime is a geometric object, it's just a simple geometric fact that the Ricci tensor is what describes things like the volume change of a geodesic ball. And the fact that that volume change should be driven by the presence of matter is just Newtonian gravity: gravity "pulls" geodesics inwards towards the center of the gravitating mass. So I think there's a fairly straightforward line of reasoning that gets you to some close relationship between the Ricci tensor and the stress-energy tensor.

 

[WannabeNewton]

How does one obtain/derive that physical insight? I know its true, but so far the insight eludes me.

The Ricci tensor completely determines the evolution of the expansion scalar of a shear-free, twist-free time-like congruence. The expansion scalar of course is nothing more than the volume change of a sphere Lie transported along an integral curve of the congruence. See Raychaudhuri's equation.

 

[Naty1]

See Raychaudhuri's equation.

ok, that helps...no wonder it did not seem 'obvious'...a bit beyond my ability to 'intuit' without such mathematical hipper dipper.

For those interested:

http://en.wikipedia.org/wiki/Raychaudhuri's_equation#Intuitive_significance

 

[Naty1]

One of the OP's questions:

(b) Concerning the red shift and blue shift: It shows that photons coming out of gravitation zone, looses energy and hence the wavelength becomes longer and causes red shift and vice-verse causing the blue shift. Here the energy of the photon is affected by gravity while IN THE ABOVE CASE IT IS NOT. AM I GETTING SOMETHING WRONG?

pervect replies in post #10:

In the Newtonian analysis, photons do gain or loose energy as it falls. In the GR analysis, the "energy-at-infinity" of the photon is a constant as it falls.

With the GR defintion of "energy-at-infinity" in the GR analysis it's the clocks that measure the photon's frequency that are affected and which cause the red and blue shift.

There is another meaning of "energy" in GR, the locally measured energy. This does red and blue shift as the photon falls, and is probably more similar to the Newtonian notion of energy with which you are familiar. But the local energy is not a conserved quantity, while the "energy-at-infinity" is conserved (well, there's some fine print - it's conserved for those space-times in which it can be defined, like static space-times).

Am I correct in understanding this description applies to both the OP's scenarios?

Seems like a given static observer at a given altitude who observes a different energy in the radially infalling scenario of the OP also observes photons to have slightly different energies, different frequencies, during the transit near a massive object.

Am I also correct in understanding that the 'energy at infinity' concept doesn't apply to cosmological scenarios... where there is expanding space?