Basic questions about general relativity concepts

[Fredrik]

Hi Ich. Sorry about that. I have to do something else right now, but I'll try to write something a few hours from now. I think I just forgot about this thread because I have a lot of other things on my mind, and also other threads going on.

 

[Fredrik]

I agree. You can trace this tendency back to a repulsive acceleration due to DE. Generally, if ä>0 in an exactly homogeneous universe, there is a tendency to expand. If ä<0, there's a tendency to contract. The important thing - and my point - is: this tendency is totally independent of H. It doesn't matter if the universe is contracting or expanding. What matters is if it does so in an accelerated way.

This looks wrong to me. I would say that there's a "tendency to expand" even when $\ddot a=0$. The reason is the geometry in the solar system is much better approximated by something like a Schwarzschild solution (where there is no expansion of space) than by a FLRW solution, because matter isn't distributed homogeneously and isotropically at these scales. This implies that measuring devices don't expand and that intergalactic distances do. It's precisely the fact that measuring devices don't expand (or that they at least do it at a much smaller rate) that makes redshift observable.

After taking a quick look at the papers bcrowell linked to and the discussion between you and him, I would say that the answer to the question of what causes the redshift depends on what coordinate system we're using. If we're using a FLRW system, the cause is indeed that the light expands along with the cosmological expansion (while measuring devices don't). If we're using our local inertial frame (in which the simultaneity lines are geodesics that are tangent to a hypersurface of constant FLRW time), the cause is a doppler shift.

 

[Ich]

This looks wrong to me. I would say that there's a "tendency to expand" even when $\ddot a=0$.

Ok, I'll derive my points of view here, it's just a few lines.
Take the radial equation of free motion in a FRW universe (slow speed limit is enough):
$$2\dot a \dot r + a \ddot r = 0$$
Pick an origin and switch to "proper distance coordinates" $x=a\,r$ where
$$\ddot x = \ddot a r + 2\dot a \dot r + a\ddot r$$
Combined you get
$$\ddot x = r\ddot a$$
which becomes zero for $\ddot a = 0$. With this equation it's easy to calculate the effect of expansion on bound systems. This is my first point.

Next step is not to take $\ddot a$ as the cause of that perturbation term in the equation of motion, but to look for a common cause of the perturbation and $\ddot a$. Insert the second Friedmann equation,
$$\frac{\ddot{a}}{a} = - \frac{4\pi G}{3}(\rho + 3p)$$
you get
$$\ddot x = -x\frac{4\pi G}{3}(\rho + 3p)$$
which shows that the perturbation is nothing else than the gravitation of an effective mass density $\rho + 3p$. This is not surprising:
Birkhoff's theorem tells us that, if we regard a ball around an arbitrary origin, the dynamics inside the ball is completely unaffected by the outside universe ($\dot a, \ddot a$ cannot alter the dynamics). Any changes have to come from within the ball.
To calculate the dynamics, remove all the matter/energy inside the ball. You're left with flat space. Then add local energy again as a perturbation, you get the complete dynamic behaviour, with the Newtonian approximation sufficient for almost every purpose. This is my second point.
It's as if the outside universe doesn't exist, which is a necessary consequence of Birkhoff's theorem.