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We have a very special clock that has existed since the dawn of time. Its purpose is to measure the age of the universe. It is always very far from any massive body or gravitational field and it is always held stationary with respect to the cosmic background radiation.

A clock in a gravitational field will run more slowly because of the time dilation due to gravitational potential. A clock that had moved at some point in its history would suffer time dilation because of its non-zero velocity.

Is it possible that any other clock could ever run faster than our very special isolated stationary clock? So will our very special clock measure the highest possible value for the age of the universe?

Roger Wood
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  • Possible duplicate: https://physics.stackexchange.com/q/495821/123208 – PM 2Ring Mar 17 '22 at 06:41
  • Bear in mind that the comoving frame of the CMB moves with the Hubble flow, so 2 bodies separated by a constant comoving distance are separating at a speed of ~70 km/s per megaparsec, in terms of proper distance. – PM 2Ring Mar 17 '22 at 06:47
  • @PM2Ring Yes, this is a duplicate. Thanks for the link. – Roger Wood Mar 17 '22 at 06:50
  • @PM2Ring Maybe the answer is not completely obvious. There will be time associated with an average density of the universe and a time associated with a large void. The former would be a single unique well-defined time, but the latter will be a longer time although it will depend on the size of the void. – Roger Wood Mar 17 '22 at 23:18
  • Sure, but the amount of time dilation is very small, unless the gravitational potential is huge, so it's dwarfed by other uncertainties in our estimate of the age of the universe (around 40 million years). Eg, using the formula I posted here, a clock dilated only by the Sun's potential at 1 AU loses ~9.87 years per billion years. (The Schwarzschild radius of the Sun is ~2953.25 m). – PM 2Ring Mar 18 '22 at 08:33
  • @PM2Ring I've gone a bit more extreme with a void of 1 billion light-years (9.46E+24 m) and an average density of 6E-27 kg/m^3 for the universe. Then GM/rc^2 gives +0.04% which is 5 million years longer for the age of the universe, but, yes, still small compared with the uncertainty. – Roger Wood Mar 18 '22 at 19:32

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The fastest clock will be one that is at the center of a large void in the universe. This will be slightly faster than a "standard" clock embedded in an extended region of average mass density which will suffer more gravitational time dilation. (this answer just formalizes the exchange of comments with @PM 2Ring)

Assuming the universe is infinite and homogeneous, etc. with mass density, $\rho$, we can take the average gravitational potential as our zero reference. The potential at the surface of a large spherical void is thus $+GM/r$ and the potential at its center is $\Phi = +{3 \over 2}GM/r $. Here $M = \rho.{4 \over 3} \pi r^3$ is the missing mass. This gives $\Phi = +G\rho.2\pi r^2 $ (this seems to involve the surface area of the void - is that a coincidence?)

The ratio of the time-rates is $ {t_{fastest} /t_{standard} } \approx 1+\Phi/c^2 = 1+ (G/c^2) \rho . 2\pi r^2 $

Taking the average density of the universe as $\rho = 6 \times 10^{-27} kg/m^3 $ and using the Giant Void as an example (radius ~0.6 billion light years = $5 \times 10^{24} $ meters), we get $ {t_{fastest} /t_{standard} } \approx 1.0007$.

So the conclusion is that a clock in the middle of the Giant Void would run 0.07% faster than a standard clock. This is 10 million years over the age of the universe (but is still small compared with the uncertainty on that age.)

Roger Wood
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