Could the perceived increasing rate of expansion of the universe be explained by time accelerating, rather than space

If

speed = distance / time

then as time accelerates, speed must also increase.

So, when we look at distant objects in the sky, and they appear to be accelerating away from us, could we actually be witnessing time accelerating, and not space?

UPDATE

When I say time could accelerate, I mean that it could dilate. And that when we observe the effect of that dilation, we mistakenly conclude that distant galaxies are accelerating away from us in space, rather than in time.

• How exactly does time "accelerate" ? And when you say distant objects in the sky appear to be accelerating away from us, what's the reasoning to then jump to "time accelerating and not space" ? – HsMjstyMstdn Mar 6 '18 at 14:14
• Well if your logic holds, then time should decelerate. But I cannot think of a reason why time should slow (Although I even cannot think why space should accelerate). – Yuzuriha Inori Mar 6 '18 at 14:25
• We do live on a massive ball in a solar system in the middle of a galaxy. Everything around us is accumulating more and more mass. Doesn’t mass affect time? Wouldn’t an accumulating mass have changing time dilations? If time were slowing here would it look faster Out beyond the stars? – Bill Alsept Mar 6 '18 at 16:05

First off, I agree with one of the comments that, by your reasoning, one would think that time would need to decelerate to make the speed increase. However, that is neither here nor there (we're talking about time, so neither now nor then?).

The reason we don't consider time accelerating or decelerating is partially because we mostly refer to the proper time in our calculations. Proper time is the time you would measure on a watch. When you see one second tick by on your wristwatch, you know it has been one second for you; nothing can change that. Since we can always rely on proper time behaving the same in any given frame, it makes a useful temporal metric.

Now, let's consider expansion of the universe: Expansion means that in a given unit of time, every fixed distance in space increases by a given amount (in numbers, every second we think any given distance increases by about $2.27\times10^{-16}\%$). Notice that this becomes a bit like compound interest (ironically, the thing Einstein called the most powerful force in the universe) in that larger distances grow by a larger absolute amount. Thus, calling this a "speed" is a bit of a misrepresentation.

But let's humour you anyway. Instead of saying "as distances grow, more total distance is added every second", we could try saying "the total distance between two objects increases by the same amount every second regardless how far apart they are, but the length of one second decreases for objects further away". What would be the difference? Nothing. I can't think of a single difference in testable predictions between these two scenarios. However, the latter scenario is much more complicated and harder to defend (honestly, why would one parsec get the same amount of distance as one million parsecs? All I can think of is "magic"). Additionally, given that we like physical reference frames to be translation invariant, and given that proper time makes a nice and handy metric (not to mention Occam's razor), it's better to stick with what we have; that all distances in space increase by about $2.27\times10^{-16}\%$ every second (except between objects bound by gravity or other forces). It's easier to work with and I don't have to explain it with "magic".

• Occam's razor ! +1. And although time declaration is not "magic", it's just way too counter intuitive to work with. But overall, a good description. :) – Yuzuriha Inori Mar 6 '18 at 16:58

To speak meaningfully of changing the rate at which time flows, we need to define that operationally, i.e., define how we would measure it. For example, in the Hafele-Keating experiment, atomic clocks were initially synchronized, then separated, and when they were reunited, they disagreed. But you're proposing a universal change in the rate at which time flows, and there is no way to measure such a change by comparing clocks.

Mathematically, the way this plays out in general relativity is that you can change the time coordinate in any way you like, and the result is not a physically different spacetime but the same spacetime. For example, if I let $t$ be the clock time on a clock that is moving along with the Hubble flow (i.e., at rest relative to the average of local matter), I can define a new time coordinate $\tau=e^{Ht}$, where $H$ is the Hubble constant. But GR says that the spacetime described in the new coordinates is the same as the spacetime described in the old coordinates.

If you like, you can also define a new metric where the time-time component is varying, e.g., if originally $ds^2=dt^2-\ldots$, then you can define a new metric $ds^2=e^{2Ht}dt^2-\ldots$ But this is again just equivalent to a certain coordinate transformation, and therefore doesn't actually describe a different spacetime. Some changes in the metric are real physical changes, but others are not, and a time acceleration or deceleration by itself is of the type that is not.

So the impossibility of operationally defining a universal acceleration or deceleration of time is baked in to the mathematical formalism of general relativity.