Are we accelerating in the universe?

[instanton]

OK, I'm not an expert in cosmology but I'm quite curious about it. Current observations tells us that the universe is accelerating (in what sense I don't quite understand). On the other hand, the Friedman equation says that the scale factor a(t) has a positive second time derivative provided the equation of state of the whole universe obays some condition (e.g. \rho+3 p < 0 if the whole matter content in the universe is an ideal fluid). What bothers me is that the time t in Friedman equation is only the FRW coordinate time, not the proper time. So a(t) has a possitive second time derivative does not gurantee that the universe is accelerating irrespective of observers. In particular, in FRW coordinates, observers sitting at the origin of FRW coordinates should not be accelerating in t, because the actual radial coordinate should be a(t) times r, rather than just a(t). on the other hand, we can easily infer that other (comoving) observers sitting at other places of the universe should percieve us as being accelerating in their owh FRW coordinate (with the origin located at their own place). If the other observers are not comoving, they might found the evolotion of the universe very different from what we observe. So, what it really means by the statement that the universe is under acceletating expansion? Are we accelerating ourselves in the universe?

 

[marcus]

...tells us that the universe is accelerating (in what sense I don't quite understand)... So, what it really means by the statement that the universe is under acceletating expansion? ...

It doesn't mean ordinary acceleration of ordinary motion. It simply means that the second time deriv. a"(t) of the scalefactor a(t) is positive.

The scalefactor is most simply understood if you think of a network of observers who are all stationary relative to the CMB and have a common idea of time that goes with being at "CMB rest". That means the present moment in time (t_present) has a definite meaning and we can normalize the scalefactor by setting a(t_present) = 1.

The distances that are increasing are distances between stationary observers---observers who are at rest with respect to the CMB. The simple fact that a"(t) > 0 merely means that distances between stationary observers are growing at an increasing rate.

Ordinary acceleration has a definite direction, but the mere fact that a"(t) > 0 is not associated with any direction. The familiar idea of motion is normally associated with some destination that you are getting closer to, but a galaxy participating in the cosmic expansion of distances is not approaching any destination. It is not getting closer to anything, just farther apart from everything. It is a change in geometry (not motion within a fixed geometry.)

Since the "acceleration" has no preferred direction it should not be thought of as acceleration of ordinary motion. But that is OK. We often speak of changes that don't involve directed motion as speeding up or slowing down--- in technology, economics, global climate change, or changes in human and other species populations. Just a change in the rate of change.

 

[bcrowell]

Hi, instanton,

Welcome to PF!

The FRW t coordinate is not the same as the proper time of every possible observer, but it *is* the same as the proper time of a comoving observer.

FRW models are homogeneous, so there is nothing special about the coordinate origin.

The equivalence principle says that any observer can be considered as either accelerating or nonaccelerating, and this is perceived the same as a gravitational field. Comoving observers are special because they perceive zero gravitational field.

Here is a coordinate-independent way of stating the fact about acceleration. Let L be the distance measured between comoving galaxies A and B by laying a chain of comoving rulers between them, and let t be the time on a comoving clock. Then d^2L/dt^2>0.

-Ben

 

[marcus]

Yes, a network of "comoving" observers is for practical purposes another way of talking about observers who are at CMB rest.

Maybe we should say what the practical criterion for telling if you are comoving or not would be. The observer looks around and measures the temperature of the CMB in all directions. If there is an obvious Doppler hotspot then he is moving relative CMB.

A moving observer sees a Doppler dipole in the radiation map, significantly hotter (bluer) ahead of him and colder (redder) behind.

It's an approximate notion, since there is some noise or random fluctuation in the temperature map. And there is another criterion involving galaxy redshifts that doesn't depend on the CMB.