Hello folks,(adsbygoogle = window.adsbygoogle || []).push({});

this is going to be a bit longish, but please bear with me, I'm going nuts over this.

For a term paper I am working through a paper on higher dimensional spacetimes by Andrew, Bolen and Middleton. You can http://arxiv.org/abs/0708.0373" [Broken].

My problem/confusion is in calculating the Riemann tensor from the given metric (basically a Robertson-Walker metric with additional dimensions), which reads:

[itex]

{\mathrm{d}s}^2 = -{\mathrm{d}t}^2 + a^2(t) \left[ \frac{{\mathrm{d}r}^2}{1-K r^2} + r^2 \left( {\mathrm{d}\theta}^2 + \sin^2\theta {\mathrm{d}\phi}^2 \right) \right] + b^2(t) \gamma_{m n}(y) {\mathrm{d}y}^m {\mathrm{d}y}^n

[/itex]

whereais the scale factor for the usual spacial dimensions andbthe same for the extra dimensions, their amount being some integerd. [itex]\gamma_{m n}[/itex] is the part of the metric in the extra dimensions.

The assumptions are that both parts of the dimensions are flat. This means of course thatK=0. And for the (of course maximally symmetric) Riemann tensor of the extra dimensions, [itex]R_{abcd} = k (\gamma_{ac}\gamma_{bd} - \gamma_{ad}\gamma_{bc})[/itex], this meansk=0.

They now go on to compute the components of the Riemann tensor, one of which reads:

[itex]

R_{a0a0} = a \ddot{a}

[/itex]

where the indexa(not that clevery chosen, if you ask me) is any of the indices in the usual spacial dimensions, that isa=1,2,3.

OK, this is what they do. I hope you've been following so far.

Now I'm trying to get these results myself. My professor said that the flatness of the extra dimensions means that [itex]\gamma_{m n}[/itex] is "essentially the unit matrix". Oh well, since I didn't (and still don't) have anything else to go on, I chose to believe and use that notion, as well as settingK=0.

(In order to be able to calculate something at all, I added two extra dimensions to the usual four. That means that I set

[itex]

\gamma = \left(

\begin{array}{cc}

1 & 0 \\

0 & 1

\end{array}

\right)

[/itex]

So with this ansatz I calculated the Riemann components, but I found that while

[itex]

R_{1010} = a \ddot{a}

[/itex]

as in the paper, for the second spatial index I get

[itex]

R_{2020} = r^2 a \ddot{a}

[/itex]

and

[itex]

R_{3030} = r^2 \sin^2\theta \ a \ddot{a}

[/itex]

for the third. I think you can see the pattern here.

So, guided by this pattern and as a form of sanity check I dumped the Robertson-Walker inspired metric in favor of a Minkowski metric for the first 4 dimensions while leaving [itex]\gamma[/itex] as it was (in other words, I useddiag(-1,1,1,1,1,1)as the metric). And guess what? I get theexact same resultas they do in the paper, as was to be expected by this point.

I hope you can see why I am confused by this. It would seem to me that they used a Robertson-Walker metric to get the Riemann tensor for a Minkowski metric. What is it that I'm missing here?

And need I say this everything gets infinitely more complicated when I leave [itex]\gamma[/itex] undefined and actually make it dependent ony, as they say in the original definition of the metric (see above)? I just don't know how I would impose the "flatness" of the additional space as a restriction on [itex]\gamma[/itex]. I'm completely stumped there.

In the same vein: When my teacher said that a flat spacetime essentially means a unit matrix I didn't think much about it, but it doesn't make a whole lot of sense to me, considering that the (spatial part of the) Robertson-Walker metric is certainly not the unit matrix, not even when you have a flat space (K=0). So why did she say that, or rather, what did shereallymean by that? (She's away for some time now, so I can't ask her).

OK, this is my long story.

Any ideas, I could really use them. Thanks in advance.

regards,

/W

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Robertson-Walker metric in higher dimensions (and problematic Riemann tensor)

**Physics Forums | Science Articles, Homework Help, Discussion**