How to Change Coordinate Systems in General Relativity

Question

Let me preface by stating that I have no experience with General Relativity. I am working on a project for school that requires a little knowledge of it, so I am hoping to find some help. I do have experience with Special Relativity.

On to the question. I know that one can calculate the age of the Universe using the Lambda-CDM model. After making a few simplifying assumptions, one can find the relation

$$Hleft ( a right )=frac{dot{a}}{a}=H_{0}sqrt{frac{Omega_{m}}{a^{3}}+frac{Omega_{rad}}{a^{4}}+Omega_{Lambda }}.$$

One can numerically integrate to find $t_{0}$, the age of the Universe.

$$t_{0}=int_{0}^{1}frac{da}{aH_{0}sqrt{frac{Omega_{m}}{a^{3}}+frac{Omega_{rad}}{a^{4}}+Omega_{Lambda }}}$$

Now, if I am correct, when performing this calculation for the age, I was working in a co-variant coordinate system (the system that expands with the universe or the system of CMB). For my project, I want to calculate the age of the Universe in a different coordinate system. More specifically, I would like to calculate the age in a coordinate system that is not expanding with the Universe. I know from other articles on here that I cannot use Special Relativity, but I am unsure how to go about this. If someone could show me how to go about this, keeping in mind my knowledge on this subject is very limited, I would be appreciative.

Cinaed Simson · Answer

In short, the same way as before but assume $Omega_{rad}$ is currently very small
$$frac {dot a}{a}=H_{0}sqrt{Omega_{m}a^{-3} + Omega_{Lambda}}$$
which has the solution
$$a(t)=(Omega_{m}/Omega_{Lambda})^{1/3}sinh^{2/3}(t/t_{Lambda})$$
where $t_{Lambda}=2/(3H_{0}sqrt{Omega_{Lambda} })$
Set $a=1$ which gives you $t=t_{0}$ the current age of the Universe.
See the Lambda-CDM model in Wikipedia.

J.G. · Answer

You can write $ds^2$ as $a^2$ times a static metric, introducing a new time coordinate viz. $deta=dt/a$. We say the full metric is conformal to the simpler one (this adjective refers to angle preservation upon the rescaling), so $eta$ is called conformal time. The $eta rthetaphi$ coordinate system fits your criterion.

Note that $deta=da/(a^2 H)$; the conformal age of the universe, in the convention where $a=1$ today, is about $46.8$ Gigayears, which is why the Hubble zone is $93.8$ light Gigayears wide. Thus the conformal age is not the quantity you want! You should still integrate $da/(aH)$ instead, but write $a,,H$ as functions of $eta$ instead of $t$.

How to Change Coordinate Systems in General Relativity

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