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I'm solving some GR problems and I have a question.

The problem is that we have a metric like the FRW metric but in 1+1 dimensions, i.e.:

[tex] ds^2 = -dt \otimes dt + a(t)^2 dx \otimes dt[/tex]

Where [itex]a(t)=t^{1/\epsilon}[/itex] (for [itex] t>0[/itex]) and [itex]a(t)=(-t)^{1/\epsilon}[/itex] (for [itex]t<0 [/itex]).

We take a vector [itex] V^\mu = dx^\mu / d\lambda [/itex] and apply it to the metric as [itex] ds^2 (V,V)=0[/itex].

For [itex]t>0 [/itex] we get:

[tex]dt = \pm a(t) dx \rightarrow x(t) = \pm \frac{t^{1-1/\epsilon}}{1-1/\epsilon}[/tex]

Ok, having all this said (until this point everything is correct, is part of the exercise) here's my problem.

For [itex]\epsilon >1 [/itex] we have a decelerating expansion and we get, for instance for [itex] \epsilon = 2 [/itex]:

[tex] t = x^2 [/tex]

(Up to some constant I don't care)

I can more or less understand this result, it's similar to the light cone in Minkowski's, and it gets more stretched with time as the expansion gets decelerated.

Let's go to the case [itex] 0 < \epsilon < 1[/itex]. In this case, for instance for [itex] \epsilon = 1/2 [/itex], we get

[tex] t=x^{-1} [/tex]

I don't understand this case, the light-cone from the future and the past are not even conected.

http://upload.wikimedia.org/wikipedia/commons/4/43/Hyperbola_one_over_x.svg

Imagine this image but with the red lines in both side of the axis.

Does anyone have an explanation to this fact? I don't see why the accelerating and the decelerating expansions give us so different light-cones!

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# Light cones in an expansive Universe

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