Does time slow down because the universe is expanding at an accelerating rate?
If the universe is expanding at an accelerating rate, such that the galaxies' moving away from each other is accelerated, then time should also slow down.
And when universe will accelerate to the speed of light, then time should stop.
Does time slow down because universe is expanding at an accelerating rate?
Can any physicist explains if this is true or not?
This is going to sound a little pedantic, but "expansion" is different from "speed" or "velocity": Expansion of a volume means that the expanding volume is growing larger, whereas speed refers to the rate of motion of the entire volume toward a point outside it. That's why inflationary cosmologies allow for expansion to occur, in the first instant after their "big bang" or "big bounce", at a rate that's several times the speed of light (-I believe that 6 times the speed of light is a possible rate that's mentioned by the physicist Vilenkin, in his pop. sci. book titled "Many Worlds in One").
Yes, time does run slower for far-away objects, as observed from our point of view; this is a prediction of general relativity. And yes, because expansion accelerates, this time dilation slowly, very slowly, becomes more pronounced (this would happen even if the expansion didn't accelerate, but just continued at the same rate).
This time dilation is a well-known effect, and is always taken into account when doing observations. For instance, when observing distant supernovae, one is often interested in how their luminosities decrease as a function of time. This is called their lightcurve. In order to compare lightcurves at different redshifts, they are usually converted to their restframe, i.e. how they would look if you were "standing next to the supernova" (e.g. Goldhaber et al 2001).
However, time dilation does not work exactly as you seem to think. At galaxy at a redshift of $z$ has its time dilated by a factor of $1+z$, so time runs twice as slow for a galaxy at, say, $z=3$ than for a galaxy at $z=1$. Galaxies with redshifts larger than $z\sim1.5$ recede faster than the speed of light, and time does not at all stop here. Only for $z\rightarrow\infty$, i.e. at the beginning of time at Big Bang, does the time dilation approach infinity.
I disagree with this assessment. Firstly in the context of FRW coordinates the difference the observer slowing of faraway clocks is not attributed to time dilation as all comoving observers' clocks run at the same rate as cosmological time. Secondly even if you did wish to attribute this to time dilation (which is not necessarily wrong) then it would not be because the supernovae are far in space or because they are receding, it is because they are far in time and the light they emitted was from a time when the Universe's scale factor was less than it is today.
@JohnDavis: I agree that the time dilation is not _because_ they're far away. But because being far away means receding fast, their time does run slower. The number of photons received per second from a source at redshift $z$ is smaller by a factor $1+z$ of what is emitted. If with a magically powerful telescope you could observe an alien with a clock that measured cosmological time, then you would see it run slower by the same factor. That's time dilation.
Hubble shift is not a function of the recessional velocity though and it is possible to observe Hubble shift in an object that has zero recessional velocity at both time of emission and observation (albeit the scale factor would have to evolve in a non-realistic way). Nor is red shift seen as exclusively due to time dilation when the distance between objects is changing, for example in SR can appear blue-shifted and hence sped up when inertial motion only is involved, even though there is time dilation.
It is not impossible to attribute Hubble shift to time dilation, though that would be an unconventional way of looking at it and it would be problematic from a conceptional point of view. However if you did you would have to say it was a result of clocks running slower in the past due to a smaller scale factor and not due to recessional velocity.
No no, I don't mean to say that the Hubble shift (by this I assume you mean the shift in wavelength due to the Hubble expansion) is caused by time dilation, but time dilation and redshift is caused by the same reason, namely expansion. You are absolutely right that in principle we can imagine a static universe where light is emitted from a distant galaxy, and on its way down to us the universe suddenly expands by a factor $z$, then stops expanding, in which case we would observe the same effect. It just seems that we're not living in such a universe.
So if I understand you well, you object to my phrasing "time does run faster for far-away objects, as observed from our point of view, because they recede from us at high speeds". I suppose it would be more correct to say "because the Universe has been expanding in the meantime". Am I understanding you correctly?
I object to the use of the term "time dilation". Time dilation is not a term usually applied to this situation as it implies, in its most general sense, the comparison of the proper time of an observer between two events to a coordinate time between those two events. This is what it is in SR and this is what it is when talking about gravitational time dilation. However in this situation we almost always use FLRW coordinates and the proper time for a comoving observer between two events is the same as the FLRW coordinate time, even though co-moving observers observe red-shift in each other.
Ok wait, I just discovered a huge blunder. I wrote "time does run _faster_". I obviously meant "time does run _slower_". I'll edit.
But I think the above error was not what you referred to. As for the term "time dilation", maybe astronomers are more sloppy than physicists. I think we use the term simply to refer to the fact that time runs slower in some other reference frame.
I am more distinguishing between the observed rate of clocks slowing and time dilation. Remember red-shift occurs even in Newtonian physics (and so clocks appeared to run slow), yet time runs at the same rate for all observers.
In general, better refrain from using 'inflation' unless you're talking about the early Universe-inflation. Rather, the Universe is expanding at an accelerated rate.
The crux of this question seems to be in what you call 'the' time. There is no such thing as 'the' time, and I'm not sure what definition of time you're expecting to stop.
Time is perfectly well defined in any restframe and will not be affected by the overall expansion of the Universe, at whatever rate.
Our time does not slow down due to the expansion of the universe. And time will not stop.
However time dilation does occur, and this is something different.
When something is moving at a very high velocity, relative to us, we see time slowing down for it. As more distant galaxies are moving away from us at great speeds, events in those galaxies would (from our point of view) appear to run slower. This is observed in the at which supernovae brighten and fade described in this paper.
The inhabitants of those galaxies would not perceive any change in the rate of time. (rather they would see our clocks slow down, as we are the ones moving fast, from their perspective)
The rate of expansion is increasing (though why this should be is mysterious) but no part of the universe will accelerate to the speed of light, as nothing may travel at the speed of light. So time dilation will remain finite.
I object "no part of the universe will accelerate to the speed of light, as nothing may travel at the speed of light" in conjunction with the expansion of the universe. It's clear that no mass, energy or information can be transferred faster than the s.o.l. But, objects in a distance that's far enough away from any observer at any point in the universe do not "travel" away due to a proper motion but due to space emerging at any point in between. With this space generation no transfer of mass, energy or information is involved, hence, this generation can lead to speeds greater than the s.o.l.