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The diameter of my observable universe is 90 billion ly measured in proper distance. But isn't lenght a relative quantity in the theory of Relativity? Could an observer moving at a different velocity than me measure a different diameter for my observable Universe?

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If the observer has always been moving at a high speed, then yes, s/he will measure a different value than you. But if s/he accelerated to a high speed just now, you would obtain the same result.

The reason is that the radius of the observable Universe is defined as the distance light has had the time to travel since the Big Bang, and that does not depend on your instantaneous velocity.

However, if an observer started moving at, say, 0.866 times the speed of light when the Universe was born, then in the reference frame of that observer, the Universe would only be $\sqrt{1 - 0.866^2} = $ 1/2 the age measured by you, i.e. 6.9 Gyr. At this age, the radius of the observable Universe was only ~39 Gly, i.e. ~0.8 times the present value.

Note that in order for this result to be exact, the observer would have to slowly accelerate. The reason is that in an expanding Universe, a given "peculiar" velocity wrt. the surrounding matter will slowly decrease, because the comoving coordinate system expands with the Universe. For instance, an observer starting out with $v = 0.866c$ at the time the cosmic microwave background was released 380,000 yr after the Big Bang, would only be moving at 80 km/s today if s/he didn't keep accelerating.

If the observer were clever, s/he would probably realize that s/he did't live in an asymmetric universe where all matter seemed to be whirling by at close to the speed of light, but rather that the Universe were quite isotropic, but s/he had somehow acquired a high speed with repect to everything else. It would then be possible to deduce the "true" age of the Universe.

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  • $\begingroup$ So if a civilization passed us by, which has been traveling at 0.866 c since soon after Big Bang, the visible universe would look 1/5 smaller to them as it does to us next to them at the same time? They would make different measurement of the CMB? (This is the weirdest thing I've heard since that stuff about galaxies looking larger the further away they are when z is big). $\endgroup$ – LocalFluff Jul 30 '15 at 17:25
  • $\begingroup$ (1−0.8662²)= 1/2 Maybe it is true in astronomy, I can't tell anymore. But it is the concept that is the issue anyway. $\endgroup$ – LocalFluff Jul 30 '15 at 17:31
  • $\begingroup$ @LocalFluff: Ha ha I know that astronomers are notorious for rounding up or down to nice numbers, but this was just a mistake: I was missing a square root. Thanks for spotting it! $\endgroup$ – pela Jul 30 '15 at 21:25
  • $\begingroup$ Yes of course I got that to begin with. A square root here or there doesn't change much. But what do you really mean with a speedy passer by here observing another size of the visible universe? Something wrong is hard here, either your formulation, or my brain (or in worst case: reality). $\endgroup$ – LocalFluff Jul 30 '15 at 21:42
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    $\begingroup$ I think it's sound. In the speeding civilization's reference frame, everything moves fast, so everything's time is dilated. $\endgroup$ – pela Jul 30 '15 at 21:59
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What would an observer see if they were moving at nearly the speed of light? They would see distances along the direction of motion appear shorter, but perpendicular to that direction distances would appear normal (ie. the same as observers in the non-moving frame). "Non-moving", you say angrily. "All motion is relative. How can there be a non-moving frame?" Well, there is a special frame relative to the mean of all of the mass in the visible universe and in this frame the Microwave Background Radiation is uniform over all directions in the sky. This is a natural non-moving (and non-rotating) frame for doing cosmology called the comoving frame.

Now back to the moving frame. There, one observes all nearby galaxies in the direction of motion as blueshifted and all nearby galaxies behind as redshifted. Therefore, Hubble's Law would be a function of direction and would be pretty complicated unless one made all the appropriate relativistic corrections. If the observers did not know relativity, and just go ahead and simply correct Hubble's Law according to measured velocities and velocity independent distance measures, they would conclude that the universe is shaped like an oblate ellipsoid (too short in 1 direction and correct in the other 2 directions, from our point of view). But, in fact, to them it actually is that shape and they actually can travel more quickly between stars and galaxies in that 1 direction than we can (at least in their forward direction). Of course, if they land anywhere in a typical galaxy then they will be switching to the comoving frame and see a full sized spherical universe.

They will also observe a Microwave Background that is quite a bit warmer in one direction and colder in the opposite direction and 2.7K in the circle between these. If they know relativity they will understand why this is and why the density of galaxies is higher in one dimension and why Hubble's Law is so complicated. But yes, their universe will be, in a real sense, smaller (in one dimension) than ours.

Galaxies do all have some motion in the comoving frame and this is called peculiar velocity. The Milky Way is being pulled toward Andromeda galaxy at roughly 50 km/s and the whole Local Group is pulled toward the Virgo Cluster (the center of the Local Supercluster) at 100 - 200 km/s. Galaxies in the Virgo Cluster have velocity dispersion of 700 km/s, so some move at about twice that. And a large region around is moving at ~ 500 km/s due to superclusters at large distances. But, since these peculiar velocities tend to add up to less than 2,000 km/s, no alien race is going to be at relativistic speeds unless they are in a spaceship of their own devise.

One more issue. On the time scale of a billion years or several, this moving frame will move to a location in space that is moving at a similar velocity and so they will merge with the non-moving (but expanding) frame. So, slowly their oblate universe is transforming to a spherical one, unless they have a means to continuously accelerate.

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