Light Cones in an Expansive Universe: Exploring GR Problems

[Cancer]

Hi everyone!
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.:
$$ds^2 = -dt \otimes dt + a(t)^2 dx \otimes dt$$
Where $a(t)=t^{1/\epsilon}$ (for $t>0$) and $a(t)=(-t)^{1/\epsilon}$ (for $t<0$).
We take a vector $V^\mu = dx^\mu / d\lambda$ and apply it to the metric as $ds^2 (V,V)=0$.
For $t>0$ we get:
$$dt = \pm a(t) dx \rightarrow x(t) = \pm \frac{t^{1-1/\epsilon}}{1-1/\epsilon}$$
Ok, having all this said (until this point everything is correct, is part of the exercise) here's my problem.
For $\epsilon >1$ we have a decelerating expansion and we get, for instance for $\epsilon = 2$:
$$t = x^2$$
(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 $0 < \epsilon < 1$. In this case, for instance for $\epsilon = 1/2$, we get
$$t=x^{-1}$$
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!

 

[Cancer]

At the beginning I thought that the second case could be because a(0)=0 (even though t=0 is not defined...), so every observer is like conected at t=0... but then I wouldn't understand the first case! Because both are expansions, they should look like similar.

 

[Orodruin]

Did you notice that you also happened to change sign on the $\pm$? If you draw the light cone from any given point (including the constant you neglected), all of the past at t = 0 is going to be included. Similarly, for t smaller than 0, all space at t=0 will be in the future light cone. In particular, for t > 0, this means all of space for t < 0 is within the past light cone.

 

[Cancer]

Yeah, but why that doesn't happen in the first case? :(

In addition, if we take $\epsilon =1 \rightarrow a(t) = t$, then:
$$dt = \pm t dx \rightarrow \ln (t)=\pm x \rightarrow t = \exp (\pm x)$$
That case is also reaaally weird, as you have line cossing in the future and in the past...

Thanks for the answer ;)

 

[sardar]

Hello,

Thank you for sharing your question and thoughts on light cones in an expansive universe. I am a scientist and I would be happy to provide a response.

First, I would like to clarify that the FRW metric you have described is a simplified version of the Friedmann-Robertson-Walker metric, which is used to describe the expanding universe in general relativity. In this simplified version, the metric only considers two dimensions (time and one spatial dimension) and uses a power law for the expansion scale factor.

Now, to address your question about the different light cones for accelerating and decelerating expansions, let us first consider the case of decelerating expansion (ε>1). In this case, the scale factor a(t) increases with time, which means that the distance between two points in the universe increases at a decreasing rate. This leads to a stretching of the light cone, as you have correctly noted.

On the other hand, for accelerating expansion (ε<1), the scale factor decreases with time, which means that the distance between two points in the universe increases at an increasing rate. This leads to a shrinking of the light cone, which is what you have observed in the case of ε=1/2. This can also be seen in the image you have shared, where the red lines on both sides of the axis get closer together as x decreases.

To understand why this happens, we can consider the equation for the scale factor a(t) in the FRW metric, which is given by a(t)∝t1/ε. As t decreases (for ε<1), the scale factor decreases at a faster rate, leading to a shrinking of the light cone. This is in contrast to the case of decelerating expansion, where the scale factor increases at a decreasing rate, leading to a stretching of the light cone.

I hope this explanation helps to clarify the difference in light cones for accelerating and decelerating expansions in an expansive universe. It is important to note that these are simplified models and do not fully capture the complexity of the universe, but they can still provide valuable insights into the behavior of light and space in an expanding universe.

Best regards,