How Can the Universe Be 13.7 Billion Years Old but 46 Billion Light Years Wide?

[Algaib]

If the universe is 13.7 billion years old, then how can the observable universe reach out 46 billion light years, given that nothing can travel faster than light? Is it to do with the expansion of space, which doesn't abide by relativity laws?

 

[edgepflow]

Nothing within space, when starting out below light speed, can reach light speed. However, space itself can expand faster than light.

In fact, basic cosmology shows the distance light could have traveled during the life of the universe is (flat geometery, no cosmological constant): r = 3c X age universe. This gives 41 light years (close to your number)

 

[Chalnoth]

If the universe is 13.7 billion years old, then how can the observable universe reach out 46 billion light years, given that nothing can travel faster than light? Is it to do with the expansion of space, which doesn't abide by relativity laws?

The expansion of space definitely obeys General Relativity. Basically, within General Relativity, relative velocities are only defined locally. That is, I can, within General Relativity, say precisely how fast a car or a beam of light is moving if it is passing right next to me. I cannot, however, say how fast that car is moving if it is very far away. This isn't because the car isn't moving, but instead because the velocity of that car depends entirely upon the numbers I use to describe the space-time between me and the car.

On Earth, for example, if I am standing still with respect to the Earth, and a car passes by me at 60mph, I can say definitively that my relative velocity with respect to that car is 60mph.

But imagine, now, that we are talking not about a car that is right next to me, but instead a car that is on the opposite side of the globe. Now things are no longer so easy: the car really isn't getting any closer or further away from me as it moves, so is it moving with respect to me at all? Or do I only count its motion along the surface of the Earth? It depends upon what you mean. General Relativity is sort of like this when we're talking about far-away galaxies: the curvature between us and the far-away galaxy makes the choice of velocity somewhat arbitrary.

And the fact that this choice is arbitrary has profound consequences. In particular, it means that the speed of light limitation that we see in Special Relativity only applies locally in General Relativity. That is, General Relativity says that no object can ever outrun a ray of light. It cannot, however, make any statement about what the velocity of a far away object will or will not be, because General Relativity says that the velocity of a far-away object is up to our own whims.

Distances are similarly arbitrary. They have to be, in fact, because if there was a definitive way to define distances, then we could use the change in distance over time to define a definitive velocity. But there isn't.

A popular distance measure, however, is the "comoving radial distance". This distance represents the amount of time it would take to bounce light rays between objects if we froze the expansion. Now, at the time the light was emitted from the surface of last scattering (the edge of the observable universe), that surface was, by this measure of distance, only 42 million light years away.

The problem is that at that time, the rate of expansion was so fast that for every meter of space that light ray traveled, the remaining distance was even larger due to the expansion. This ray of light was always traveling at the speed of light as measured by the matter it was passing, but it was still losing ground because our universe was expanding so rapidly.

As time progressed, this rate of expansion slowed. Eventually it slowed so much that the light ray stopped losing ground. However, it had already passed a great deal of distance, so that the stuff that emitted it was left behind long ago. That stuff is, today, sitting at some 46 billion light years away as measured by the comoving distance. But the light ray didn't traverse all that space, because it only started out a mere 42 million light years. Instead, it traversed precisely 13.7 billion light years of space (because that's how long it was traveling, and light rays travel at the speed of light).