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The closest we can get to seeing that “edge” is the cosmic microwave background, whose light comes from almost as far as the particle horizon. So we can define the observable universe to be a sphere of about 45 billion light-years in radius, centered on us. In actual fact, since the universe has been expanding all that time, something just close enough to send its light to us 13.8 billion years ago is now much farther away-approximately 45 billion light-years. Knowing that the universe is about 13.8 billion years old, logic would tell you that the particle horizon must be a sphere of radius 13.8 billion light-years. This defines the particle horizon, and it’s the farthest out we can observe anything at all, even in principle. A distance at which, if a light beam started there at the first moment, it would take the entire age of the universe to reach us. Since light takes time to travel, and more distant objects are, from our perspective, farther in the past, there has to be a distance corresponding to the beginning of time itself. We define this as being the farthest we could possibly see, given the limitations of the speed of light and the age of the universe. The “observable” part refers to the region within our particle horizon. The present-day observable universe is probably bigger than you think. Reprinted by permission of Scribner, an imprint of Simon and Schuster, Inc. For one fairly natural definition of velocity, there are galaxies we observe that are now and always have been receding from us at a velocity greater than c.Excerpted from The End of Everything by Katie Mack. The second statement is at best an oversimplification because relative velocities of distant objects are not well defined in general relativity. The first statement is incorrect because the expansion of the universe can’t be measured with a single velocity. You may sometimes see statements that cosmological inflation caused the universe to expand faster than c, or that the edge of the observable universe occurs at the place where the Hubble law gives a velocity equal to c. Locally, general relativity is the same as special relativity. Because relative velocities of distant objects aren’t well defined in general relativity, there is no way to extend special relativity’s prohibition on v>c to distant objects in general relativity. Therefore it doesn’t make sense to worry about whether such a velocity is greater than c. In fact, general relativity allows us to assign absolutely any value we like to A’s velocity relative to B it simply isn’t a well-defined thing. If we like, we can use certain measures of distance and time (see: How are time and distance measured in cosmology?) and verbally describe A and B as moving relative to one another at a rate found by taking the change in distance divided by the change in time. If we like, we can verbally describe the situation by saying that both galaxies are at rest, but the space between them is expanding. General relativity does not have a uniquely defined way of talking about the velocity of galaxy A relative to galaxy B if they are at cosmological distances from one another. Can they be receding from one another at a speed greater than c? This question requires relativity. Now suppose we fix our attention on two specific galaxies. For similar reasons, it doesn’t make sense to ask for “the” velocity of expansion of the universe. The velocity will be different if we pick a different pair of atoms. A velocity can only be defined if we first specify which two atoms in the metal we’re talking about. When a piece of metal expands, we can’t describe its overall expansion using a velocity in units of meters per second. This is exactly what happens, for example, when a piece of metal expands because it has been heated. All intergalactic distances are increasing by the same scaling factor in any given interval of time. Hubble’s observation therefore implies v=Hd, where v is the relative velocity of two galaxies, H is a number that is the same for all galaxies, and d is the distance between the two galaxies. A redshift of, say, 0.037% indicates that a galaxy is moving away from us at almost exactly 0.037% of the speed of light.

To keep things simple, let’s start by thinking about how this would be interpreted if we didn’t know about relativity, so that velocity and distance can be defined as we expect in Newtonian mechanics. The expansion of the universe was originally discovered by Hubble, who found that the redshifts of galaxies were proportional to their distances from us. To see why, let’s start by thinking about how we know the universe is expanding. Neither of these questions actually makes sense in the form in which it was posed.