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If we just look at our local galactic cluster, if all of the galaxies that are a part of it are moving away from each other, and so the overall 'density' of the strength of gravity in the cluster is decreasing over time, does that mean that time itself is also speeding up?

If this is the case, would it be on a noticeable scale, and if so, would there be any way would could measure it happening (seeing as we'd still be experiencing everything at exactly the same rate since the things/people around us would be just as affected by the weakening strength of gravity)?

Let me know if it isn't clear what I'm asking and I'll try to make it more concise, also, be gentle, I'm a layman ;)

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Possibly a duplicate of physics.stackexchange.com/a/43073/2451 , physics.stackexchange.com/q/37629/2451 and links therein. –  Qmechanic Jan 5 '13 at 0:40
The way I read this, it is not a duplicate. Correct me if I'm wrong, but isn't this asking whether time is speeding up because we are sitting in an ever-shallowing potential well? (cf the slowing of time close to a black hole) –  Chris White Jan 5 '13 at 3:50
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2 Answers

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machinemessiah, if you understood hwlin's answer please accept it. It is elegant, correct and really says all that needs saying. However, Wouter raised a reasonable concern that you or others reading this might not be familiar with the concepts taken for granted in hwlin's answer. In that case I offer what may be an intelligible answer, which also addresses the breakdown of the cosmological assumption made by hwlin (and everyone else). Apologies if it is too wordy. :)

A key insight of relativity is that any discussion of physics must be rooted in operational procedures for performing measurements. All invisible concepts in physics - space, time, energy, fields, etc. - are shorthands, or powerful organising principles, which govern relationships between measurements which are made by some procedure, however informal such procedures may be. (Have you ever seen space? I haven't. But I have noticed that I routinely avoid banging my head against the wall, though, when the circumstance warrants it, such contact can be easily arranged. These and many other routine observations support the notion of space as a powerful organising principle. Usually the observations are so routine that we don't even think of them as observations, but observations they are.)

So to your question: is time speeding up? How would you measure such a speed-up? You are correct that if everything else is speeding up in the same way then there is no way to measure a speed-up. Because of this the idea of a global speedup of time is operationally meaningless. It could only make sense to some god-like observer standing outside of our (portion of) spacetime looking in. This is essentially what hwlin told you. You can always come up with a notion of time in cosmology that just "trucks along," without changing pace, and this cosmic clock serves as a good definition for time: it is simple, robust, ticks all the necessary boxes, makes the equations very nice and is hard to disagree with.

Another key insight of relativity is that every reference frame is as good as any other. So for intance, every observer has the right to construct their own clock and call the measurement "time." Though there is a complete democracy of observers, each with their own definition of time, the laws of relativity tell how any two definitions of time, measured by any two well constructed clocks are related. (A bad clock would be a grandfather clock leaning over sideways, for example. No one is required to pay attention to a bad clock.) So now we can ask how clocks in the local cluster of galaxies are related to clocks in the distant universe. This is where it gets interesting.

Cosmologists tend to make an approximation that the whole universe is homogenous. That is, all of the matter is spread evenly throughout the universe. This approximation dramatically simplifies the equations! If the universe is not homogenous then you need to specify a bunch of numbers at every point of space and keep track of how they are all changing. If instead you pretend that every point in space is equivalent then you only need to keep track of one set of numbers! hwlin's reply was in the context of this approximation. It is correct insofar as the homogeneity assumption is correct.

Of course the universe is not homogenous, but on cosmological scales it comes pretty close. hwlin's answer, addressed at these scales, is completely appropriate. Unfortunately you mentioned galaxy clusters which, by definition, violate the homogeneity assumption. So let's see what we can say about the local neighbourhood, where the cosmological assumption breaks down.

Suppose that the cluster is neighboured by a void (perhaps the Local Void? I don't know my extra-galactic geography). In this case clocks inside the cluster run slower than clocks outside the cluster. How to measure this? We can arrange that somebody inside the cluster, "lower down in the gravity well," sends a pulse of light once every second - by their clock - to an observer situated outside the galaxy cluster. In this case there is an observable difference: the observer outside the cluster receives the pulses at intervals greater than one second. He says the the clock of the guy inside is running slower. This is the well known gravitational redshift, and is well established by now. It was observed at two different heights on the Earth in the Pound-Rebka experiment.

An observer inside the cluster could attempt to observe the faint light emitted by extra-galactic hydrogen, which has a characteristic frequency determined by quantum mechanics. What we would see is a subtle blue-shift of radiation coming from the void. This measurement is similar in principle to the measurement of the Lyman-alpha forest. It is also similar to the integrated Sachs-Wolfe effect (thanks Chris White), which measures the photons from the cosmic microwave background rather than nearby voids. I'm not sure if present day observations are able to see it.

If the cluster lost mass for some reason and the mass didn't go into the void, then you could in principle measure a slight decrease of the blueshift. If the mass went into the void it becomes a very complicated thing that depends on the distribution of matter where you are looking. In any case the expansion of the universe wouldn't cause this as the local cluster is held together by the gravity of the galaxies inside it, and the cosmic expansion really only separates unbound systems.

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Regarding seeing the blueshift of photons leaving a void: I can't recall if this has been done directly (I bet exoplanet RV instruments have the sensitivity, but the effect is degenerate with peculiar motion), but we do have the integrated Sachs-Wolfe effect, which basically measures how the depth of the potential well changes over the travel time of the photon. –  Chris White Jan 7 '13 at 7:53
There it is! I was banging my head trying to think of the name. Sachs-Wolfe, Sachs-Wolfe... Thank you. :) –  Michael Brown Jan 7 '13 at 7:56
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No, time is not speeding up. When we talk of "the universe speeding up," we mean that distant galaxy clusters are receding at an ever-increasing rate.

Time is not. The easiest way to see that is to compare the metric of cosmology $ds^2=-dt^2 + a^2(t)dr^2$ with the metric of special relativity $ds^2=-dt^2 + dr^2$. You see that the only difference is that in cosmology, there is this factor $a(t)$. $a(t)$ accounts for the expansion of space; but it does NOT multiply the time component. Thus, cosmological time is basically just trucking along.

I should mention that this is really due to our definition of cosmological time. We could invent some crazy time coordinate that speeds up. In fact, we could define space coordinates that expand even in perfectly static space. Such definitions are not useful and not used in cosmology. Our definitions are a good one, because a (comoving) clock on Earth measures the same time as the one in our equations.

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Note: edited your answer to remove the incorrect factor of $a^2$ from the SR metric. Otherwise your explanation is perfect! –  Michael Brown Jan 5 '13 at 3:21
Ah, thanks for catching that typo! –  hwlin Jan 5 '13 at 3:32
I'm not sure how much of a layman the OP is, but perhaps some expansion is useful/necessary on the subject of metrics and maybe even on the symbols used (depending on the exact knowledge of the OP). For people familiar with these concepts and symbols it's a very clear and concise explanation though. +1 –  Wouter Jan 5 '13 at 11:55
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protected by Qmechanic Mar 6 '13 at 16:53

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