# Why is the universe described in terms of Euclidean space and not Minkowski spacetime?

The universe is described as an infinite Euclidean space in cosmology. Why isn't it treated as Minkowski spacetime?

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## 1 Answer

In standard cosmology, the spatial part of the universe is described by a flat Euclidean space. The enveloping spacetime is described by a robertson-walker spacetime (plus small perturbations). What on Earth is a Robertson-Walker spacetime?

To do this, I'm going to draw an analogy:

Well, imagine that you're constructing a 3-D shape, and you want to build it out of unit circles. If you're building a cylinder, all you need to do is define the radius of your cylinder as $R$, expand out your cylinder to the proper radius by multiplying lengths on your unit circle by $R$, and then stack the circles on top of each other, and blam! There is a cylinder.

What if you instead wanted to make a cone? Well, then you know that the radius of the circle at any height $h$ is given by $R(h)=R_{b}\left(1-\frac{h}{H}\right)$, where $H$ is the height of the cylinder, and $R_{b}$ is the radius of the base. Then, to construct your cone, you merely need to stack the circles of the appropriate radius on top of each other, and there's your cone!

Now, to make a robertson-walker spacetime, you do the same basic thing. At every constant $t$, you have a 3-D Euclidean space (there are other options, but the observationally correct one is flat space), and then you stack them on top of each other, expanding your distances by an amount $a(t)$ at a given time. All you have to do is figure out what the function $a(t)$ is, and then you're done. It turns out that, quite generally, it's a requirement that, sometime in the past, $a(t)$ must take the value of zero, so there's the Big Bang. You can get a few other quick results with minimal thinking, too, such as how quickly matter should densify in the past.

But the important thing to note is that the function $a(t)$ changes the geometry pretty radically--you can get a cone instead of a cylinder, or a lot of other shapes. We need to do real general relativity to figure out what form $a(t)$ takes, and that is a bit beyond the scope of this question.

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Do you have any links to resources where I can learn more about Robertson-Walker spacetime? Using Google I mostly find stuff about Robertson-Walker metric. It sounds like those pictures you see on posters depicting the evolution of the universe from a small point, expanding like a wide cone (due to inflation) then a narrower cone (normal expansion) and ending open (because time potentially continues forever) after the part labeled 'current time' or 'modern day' or whatever. This is a quantized treatment of spacetime? This is why gravity becomes a problem when we go earlier than inflation? –  Ocsis2 Apr 9 '12 at 18:23
The Robertson-Walker metric is the distance function in the Robertson-Walker spacetime. The two are inextricably tied to each other. Almost any reference on general relativity will discuss the Robertson-Walker spacetime. If you are curious about some of it's particular properties, I can answer here. It is exactly the model that inspires those pictures that you see on posters depicting the evolution of the universe. The reason why gravity becomes a problem is exactly because $a(t)$ necessarily has a zero, which implies infinite density, which necessitates that quantum gravity has to turn on –  Jerry Schirmer Apr 9 '12 at 20:47
What exactly is the RW spacetime used for in GR? Was it created to solve some particular problem or to answer some specific need? Also, if it's not too much trouble could you check my latest comment on the reply in this submission: link (my comment is the last one on the page under the reply by John Rennie) –  Ocsis2 Apr 10 '12 at 5:18
If you make a few simplifying assumptions about the distribution of matter in the universe, e.g. it's isotropic and uniformly distributed, you can solve the GR equations and the result is the RW metric. The RW spacetime is just the geometry described by the RW metric. –  John Rennie Apr 10 '12 at 8:00
Ah, thanks that clears it up. I was always under the impression that time was only quantized in certain theories like loop quantum gravity. But the RW metric seems to give time the quantized treatment as well. Would you say that there is some degree of quantized treatment of time in all GR physics? Is time not quantized in quantum physics? I don't know why I was under the assumption that this wasn't a normal thing, but perhaps it's because that's the way things are in quantum mechanics. Is loop quantum gravity popular among physicists who have to deal with GR often? –  Ocsis2 Apr 10 '12 at 9:02