# How can a quasar be 29 billion light-years away from Earth if Big Bang happened only 13.8 billion years ago? [duplicate]

I was reading through the Wikipedia article on Quasars and came across the fact that the most distant Quasar is 29 Billion Light years. This is what the article exactly says

The highest redshift quasar known (as of June 2011[update]) is ULAS-J1120+0641, with a redshift of 7.085, which corresponds to a proper distance of approximately 29 billion light-years from Earth.

Now I come to understand that the Big Bang singularity is believed to be around 13.8 Billion years ago.

So how is this possible? Does the presence of such a quasar negate the Big Bang Theory?

I'm not a student of Physics and was reading this out of (whimsical) curiosity. Is there something I'm missing here or the "proper distance" mentioned in the fact is a concept that explains this?

A simple google search led me to this article which says the farthest quasar found is 12.9 billion LYs and not 29 billion.
So in the end we have just proven that wikipedia needs more moderation.

In the expanding universe, you have to be a bit careful to define exactly what you mean by distance. The "proper distance" referred to here in that article means the distance measured at the present time. We have to be careful even to define what we mean by that last phrase -- time is relative, you know. But if the universe is approximately homogeneous, then there is a "natural" choice of time coordinate called "cosmic time." If you imagine many, many rulers stretched out between you and the quasar, the proper distance the total length of all of them, added up at the present value of cosmic time.

That's not the same as the distance that the light has traveled, though. There are various fancy general-relativistic reasons why not, but the main idea is a very simple one. That quasar is moving away from us, so it used to be closer to us. The light we're seeing now was emitted when the distance was much shorter, so it didn't have to travel anywhere near 29 billion light-years.

The truth is that that 29-billion light-year figure is calculated based on a certain model of the universe; it's not measured directly. The model it's calculated on is based on the theory of general relativity, and includes the best currently-measured age of the universe. So pretty much by definition, there can't be any contradiction between that distance and the age of the universe.

Yes, there is something you're missing. If you're familiar with special relativity you know that velocities don't add the same simple way they do in Newtonian mechanics. If one spaceship is moving at $c/2$ to the right and another is moving $c/2$ to the left, the relative velocity between them is not $c$, as one might expect, but $4c/5$.

In general relativity the same kind of thing applies to distances as well as velocities. In flat space we define the distance between two objects as the length of the (unique) straight line from one to the other, but in general relativity space can be curved and there is no such thing as a "straight line". The closest analog is a geodesic, which is a smoothly curved path between two points whose length is a (local) minimum. The distance to the quasar you quoted is defined as the length of a certain geodesic connecting it to us.

But there's no reason to expect geodesic distances to behave just like straight-line distances in flat space. In the FLRW metric (which defines the basic shape of the whole Universe in big bang cosmology), there exist geodesics that are longer than the geodesic from the big bang to us used to define the "age of the universe". So having a quasar whose "proper distance" or "comoving distance" from us is 29 billion light years is not a contradiction at all.

BTW, another seemingly contradictory thing you might run into is relative velocities faster than the speed of light. This can happen because relative velocity is also a tricky thing in GR, which is only defined once you decide on a specific path to "parallel transport" the velocity vector along. The relative velocity between two nearby objects is well-defined and can never be greater than $c$, but the relative velocity between two very distant objects (defined a certain way) can be greater than $c$.