<p>This is a confusing question - your title mentions GR, but of course the age of the universe is entirely derived as a result of using GR to solve for the dynamics of the expanding universe.</p>

<p>The text of your question is talking about time dilation and the effects of <em>special</em> relativity (a subset of GR). Here there is a point to be addressed. It is not a good way to think about the explansion of the universe in terms of things receding from each other at ever increasing velocities. It is better to think about space expanding. It is also not possible to apply special relativity except in local inertial frames.</p>

<p>It can and has been shown that the redshift-distance relationship <em>cannot</em> be simply interpreted as the doppler shift due to things moving away from us at ever increasing speeds - since this of course predicts a maximum recession velocity that approaches the speed of light. In fact, beyond redshifts of $z=1.5$, the "recession velocity" in all viable models exceeds the speed of light. What actually happens is that light emitted at some earlier epoch travels across expanding space to get to us and this expansion increases the wavelength of the light.</p>

<p>Essential reading: <a href="https://arxiv.org/pdf/astro-ph/0310808.pdf">Davis &amp; Lineweaver (2003)</a>.</p>

CC BY-SA 3.0
2016-10-06T11:47:37.877Z
I updated question to say GR/SR instead of just GR.
iMerchant
CC BY-SA 3.0
2016-10-06T11:58:42.030Z
@iMerchant that doesn't change the fact that SR is not of any importance here. I reckon you're not going to get a better answer than Rob's.
user1991
CC BY-SA 3.0
2016-10-06T15:17:29.693Z
It's all a bit above my head, but here's another article, @iMerchant, which I have linked to before on similar questions:  https://arxiv.org/abs/0707.0380v1 .  For example: http://physics.stackexchange.com/a/223697/10319
Fattie
CC BY-SA 3.0
2016-10-06T11:12:56.733Z
2016-10-06T11:12:56.733Z
ProfRob
<p>Rob Jeffries gives a good response to this question, but I wanted to go through the basic outline of how the age of the universe is calculated, just so you can see how it works more or less. Be warned though, I'm only giving you the highlights and you'll either have to accept what I'm saying or fill in the blanks yourself.</p>

<p><strong>The Friedmann Equation</strong></p>

<p>As with anything in physics, when you want to analyze something, you need an equation to represent it. Think of the <a href="https://en.wikipedia.org/wiki/Friedmann_equations">Friedmann Equation</a> as the "equation of the universe". It's kind of crazy (to me) to believe, but yes there is an equation that represents the entire universe. The Friedmann equation is directly derived from the GR field equations, however it is quite possible to derive them from Newtonian mechanics (and a little hand waving). A good source for the Newtonian derivation can be found in Ryden's <a href="http://atlas.physics.arizona.edu/~kjohns/downloads/lsst/Ryden_IntroCosmo.pdf">Indroduction to Cosmology</a>, which is a good undergraduate level cosmology textbook.</p>

<p>There are many forms of the Friedmann equation<sup>1</sup> and it can be represented numerous ways (especially depending on the universe model), but I think a convenient form is the following:</p>

<p>$$\bigg(\frac{\dot{a}}{a}\bigg)^2 = H_0^2 \Big(\Omega_{m,0}a^{-3} + \Omega_{r,0}a^{-4} + \Omega_{\Lambda,0} + \Omega_{k,0}a^{-2}\Big) \equiv H_0^2E^2(a)$$</p>

<p>The terms in this equation are defined as follows.</p>

<ul>
<li>$a(t)$ The size of the universe at time $t$, compared to its size now.</li>
<li>$H_0$ The value of Hubble's constant for the current time. Nominally taken to be $\sim68\:\mathrm{km/s/Mpc}$.</li>
<li>$\Omega$ The "density" of the various types of matter/energy in our universe. The subscript $0$ indicates that it is the density for the universe at the present time. For these, $m$ represents matter (both normal and dark), $r$ represents radiation (i.e., light), $\Lambda$ represents the cosmological constant (i.e., vacuum energy), and $k$ is the curvature of our universe (i.e. energy in the space-time fabric itself).</li>
</ul>

<p>This equation may seem scary, but effectively it is telling you is how the size of the universe, has evolved over time based on the current content of the universe. If you know precisely how much matter, radiation, and dark energy you have in the universe right now (or more precisely the density), you can calculate the history of the universe!</p>

<p><sup>1</sup><sub>Technically there is more than one equation, but we'll look at an important form of one of them.</sub></p>

<p><strong>The Age of the Universe</strong></p>

<p>Remember, $a$ is a function of time in the equation above. This means, with a little bit of rearranging and integrating, we can calculate the age of the universe.</p>

<p>$$t=\frac{1}{H_0} \int_{0}^{1} \Big(\Omega_{m,0}a^{-1} + \Omega_{r,0}a^{-2} + \Omega_{\Lambda,0}a^2 + \Omega_{k,0}\Big)^{-1/2} da$$</p>

<p>To calculate the age of the universe, $t$, you have the non-trivial task of measuring good values for the constants and then doing the integral above.</p>

<p>As a trivial case, you can assume we live in a flat, matter-only universe - no light, no dark energy, no curvature. In that case $\Omega_{m,0}=1$ and $\Omega_{r,0}=\Omega_{\Lambda,0}=\Omega_{k,0}=0$. You'll wind up with $t=2/(3H_0)$ which amounts to ~10 billion years old. This type of universe is known as an Einstein-de Sitter Universe and is obviously <em>not</em> the universe we live in.</p>

<p><strong>A Note on Dark Energy</strong></p>

<p>Technically the equations above are a bit more complicated because I haven't accounted for dark energy. I won't go into the specifics, but one can choose to add components to the equation above which include the dark energy equation of state parameter. Generally this parameter is denoted by $w$. The concepts are more or less the same, the equations just become uglier.</p>

<p><strong>Conclusion</strong></p>

<p>With all of the above, I think I can really address your question. </p>

<blockquote>
  <p>Does the age of the universe take into account GR/SR?</p>
</blockquote>

<p>In short, yes. The Friedmann equation, as I said above, is derived directly from the GR equations. Therefore, we're using an equation to represent the universe which directly results from assuming GR is an accurate theory of gravity in our universe.</p>

<blockquote>
  <p>Since everything is moving very fast away from us, doesn't relativistic effects from high speeds come into play here? If something is moving quite fast, time slows down, affecting how "old" it is. Is all this taken into account when calculating the age of the universe?</p>
</blockquote>

<p>Yes and no. As I said, the age of the universe is ultimately derived from GR so anything GR/SR related is accounted for. However, it seems to me that your understanding of how the universe's age is calculated is somewhat simplistic (and there's nothing wrong with that - its a complicated process!). In calculating the age of the universe, we actually don't care about how fast some far-flung galaxy may be moving away from us. We don't even look at those galaxies. What we really look at is the Cosmic Microwave Background (CMB) Radiation. By looking at this radiation, we're able to work out the $\Omega$ parameters more or less (and the dark energy parameter $w$) without every having to concern ourselves with recessional galaxy speeds. Hopefully you can see that the concepts of receding galaxies and time dilation are moot points when it comes to determining the age of the universe.</p>

CC BY-SA 3.0
2016-10-06T15:22:33.953Z
@JoeBlow What you're asking is a fundamentally different question, and most of my answer was trying to show that these types of considerations aren't relevant to calculating the age of the universe. My answer is essentially that "it doesn't affect the age of the universe". What's more the twin example is markedly different from two galaxies receding due to space expansion. In the twin case, they're moving *through* space. In the galaxy case, *space is moving*, but as far a SR is concerned, the galaxies don't move through space wrt each other (aside from peculiar motion).
zephyr
CC BY-SA 3.0
2016-10-06T15:52:04.457Z
"What you're asking is a fundamentally different question" Fair enough, you're quite right. I should ask it separately.  Later:  I did ask a new related question!  :)
Fattie
CC BY-SA 3.0
2016-10-07T02:02:00.433Z
+1 for including those cool looking equations that I have no idea how u type them out on here. It looks really impressive and almost artistic. As for what they actually mean, my eyes glaze over like a dog trying to comprehend blowing bubbles or something.
iMerchant
CC BY-SA 3.0
2016-10-07T02:12:36.347Z
@iMerchant Thanks! I hope that even if the equations seem daunting, the point behind them was not lost on you. To make the equations, surround your text with the $ character and use Latex, a typesetting language designed for creating nicely formatted text/equations as you see above. There are plenty of tutorials online to learn Latex from. Try playing around with Latex [here](http://arachnoid.com/latex/).
zephyr
CC BY-SA 3.0
2016-10-07T02:24:22.840Z
@zephyr - I'm allergic to Latex. :-p But I now know how you do it. Very cool.
iMerchant
CC BY-SA 3.0
2016-10-06T14:15:02.467Z
2016-10-06T14:15:02.467Z
zephyr
<p>It is generally accepted that the age of the universe is approximately 12-15 billion years old based on the speed of the expansion of the universe. Since everything is moving <em>very</em> fast away from us, doesn't relativistic effects from high speeds come into play here? If something is moving quite fast, time slows down, affecting how "old" it is. Is all this taken into account when calculating the age of the universe? I don't know if I'm making any sense to anyone else or not. </p>

CC BY-SA 3.0
2016-10-06T15:35:48.167Z
This was probably answered below, but the expansion of space doesn't cause the time dilation of relativity.  It causes red shift, but not time dilation.
userLTK
CC BY-SA 3.0
2016-10-07T01:18:20.277Z
@userLTK That's not completely accurate. When we observe phenomena that have been redshifted, like the duration of supernovae and reverberation mapping of AGN, the stretching of the total signal also spreads out the arrival time of its parts, giving an apparent time dilation. Gravitational redshift, a related effect, and how it affects time's flow is also necessary for getting GPS right.
Sean Lake
CC BY-SA 3.0
2016-10-07T04:35:04.427Z
@SeanLake  You're right on apparent time dilation and I'm not sure if they can be told apart in any practical way, so you make a good point.
userLTK
<universe><observable-universe><units><age>
Does the age of the universe take into account General Relativity / Special Relativity?
iMerchant
# Does the age of the universe take into account General Relativity / Special Relativity?

(Asked by: username_0 on 2016-10-06)

It is generally accepted that the age of the universe is approximately 12-15 billion years old based on the speed of the expansion of the universe. Since everything is moving *very* fast away from us, doesn't relativistic effects from high speeds come into play here? If something is moving quite fast, time slows down, affecting how "old" it is. Is all this taken into account when calculating the age of the universe? I don't know if I'm making any sense to anyone else or not.

**Tags:** universe, observable-universe, units, age
> **Comments:**
> **username_1**: This was probably answered below, but the expansion of space doesn't cause the time dilation of relativity.  It causes red shift, but not time dilation.  
> **username_2**: @userLTK That's not completely accurate. When we observe phenomena that have been redshifted, like the duration of supernovae and reverberation mapping of AGN, the stretching of the total signal also spreads out the arrival time of its parts, giving an apparent time dilation. Gravitational redshift, a related effect, and how it affects time's flow is also necessary for getting GPS right.  
> **username_1**: @SeanLake  You're right on apparent time dilation and I'm not sure if they can be told apart in any practical way, so you make a good point.  

---

## 2 answers

### Answer by username_6 on 2016-10-06. Score: 11 | Accepted answer

Rob Jeffries gives a good response to this question, but I wanted to go through the basic outline of how the age of the universe is calculated, just so you can see how it works more or less. Be warned though, I'm only giving you the highlights and you'll either have to accept what I'm saying or fill in the blanks yourself.

**The Friedmann Equation**

As with anything in physics, when you want to analyze something, you need an equation to represent it. Think of the [Friedmann Equation](https://en.wikipedia.org/wiki/Friedmann_equations) as the "equation of the universe". It's kind of crazy (to me) to believe, but yes there is an equation that represents the entire universe. The Friedmann equation is directly derived from the GR field equations, however it is quite possible to derive them from Newtonian mechanics (and a little hand waving). A good source for the Newtonian derivation can be found in Ryden's [Indroduction to Cosmology](http://atlas.physics.arizona.edu/~kjohns/downloads/lsst/Ryden_IntroCosmo.pdf), which is a good undergraduate level cosmology textbook.

There are many forms of the Friedmann equation1 and it can be represented numerous ways (especially depending on the universe model), but I think a convenient form is the following:

$$\bigg(\frac{\dot{a}}{a}\bigg)^2 = H\_0^2 \Big(\Omega\_{m,0}a^{-3} + \Omega\_{r,0}a^{-4} + \Omega\_{\Lambda,0} + \Omega\_{k,0}a^{-2}\Big) \equiv H\_0^2E^2(a)$$

The terms in this equation are defined as follows.

* $a(t)$ The size of the universe at time $t$, compared to its size now.
* $H\_0$ The value of Hubble's constant for the current time. Nominally taken to be $\sim68\:\mathrm{km/s/Mpc}$.
* $\Omega$ The "density" of the various types of matter/energy in our universe. The subscript $0$ indicates that it is the density for the universe at the present time. For these, $m$ represents matter (both normal and dark), $r$ represents radiation (i.e., light), $\Lambda$ represents the cosmological constant (i.e., vacuum energy), and $k$ is the curvature of our universe (i.e. energy in the space-time fabric itself).

This equation may seem scary, but effectively it is telling you is how the size of the universe, has evolved over time based on the current content of the universe. If you know precisely how much matter, radiation, and dark energy you have in the universe right now (or more precisely the density), you can calculate the history of the universe!

1Technically there is more than one equation, but we'll look at an important form of one of them.

**The Age of the Universe**

Remember, $a$ is a function of time in the equation above. This means, with a little bit of rearranging and integrating, we can calculate the age of the universe.

$$t=\frac{1}{H\_0} \int\_{0}^{1} \Big(\Omega\_{m,0}a^{-1} + \Omega\_{r,0}a^{-2} + \Omega\_{\Lambda,0}a^2 + \Omega\_{k,0}\Big)^{-1/2} da$$

To calculate the age of the universe, $t$, you have the non-trivial task of measuring good values for the constants and then doing the integral above.

As a trivial case, you can assume we live in a flat, matter-only universe - no light, no dark energy, no curvature. In that case $\Omega\_{m,0}=1$ and $\Omega\_{r,0}=\Omega\_{\Lambda,0}=\Omega\_{k,0}=0$. You'll wind up with $t=2/(3H\_0)$ which amounts to ~10 billion years old. This type of universe is known as an Einstein-de Sitter Universe and is obviously *not* the universe we live in.

**A Note on Dark Energy**

Technically the equations above are a bit more complicated because I haven't accounted for dark energy. I won't go into the specifics, but one can choose to add components to the equation above which include the dark energy equation of state parameter. Generally this parameter is denoted by $w$. The concepts are more or less the same, the equations just become uglier.

**Conclusion**

With all of the above, I think I can really address your question.

> Does the age of the universe take into account GR/SR?

In short, yes. The Friedmann equation, as I said above, is derived directly from the GR equations. Therefore, we're using an equation to represent the universe which directly results from assuming GR is an accurate theory of gravity in our universe.

> Since everything is moving very fast away from us, doesn't relativistic effects from high speeds come into play here? If something is moving quite fast, time slows down, affecting how "old" it is. Is all this taken into account when calculating the age of the universe?

Yes and no. As I said, the age of the universe is ultimately derived from GR so anything GR/SR related is accounted for. However, it seems to me that your understanding of how the universe's age is calculated is somewhat simplistic (and there's nothing wrong with that - its a complicated process!). In calculating the age of the universe, we actually don't care about how fast some far-flung galaxy may be moving away from us. We don't even look at those galaxies. What we really look at is the Cosmic Microwave Background (CMB) Radiation. By looking at this radiation, we're able to work out the $\Omega$ parameters more or less (and the dark energy parameter $w$) without every having to concern ourselves with recessional galaxy speeds. Hopefully you can see that the concepts of receding galaxies and time dilation are moot points when it comes to determining the age of the universe.

> **Comments:**
> **username_6**: @JoeBlow What you're asking is a fundamentally different question, and most of my answer was trying to show that these types of considerations aren't relevant to calculating the age of the universe. My answer is essentially that "it doesn't affect the age of the universe". What's more the twin example is markedly different from two galaxies receding due to space expansion. In the twin case, they're moving *through* space. In the galaxy case, *space is moving*, but as far a SR is concerned, the galaxies don't move through space wrt each other (aside from peculiar motion).  
> **username_5**: "What you're asking is a fundamentally different question" Fair enough, you're quite right. I should ask it separately.  Later:  I did ask a new related question!  :)  
> **username_0**: +1 for including those cool looking equations that I have no idea how u type them out on here. It looks really impressive and almost artistic. As for what they actually mean, my eyes glaze over like a dog trying to comprehend blowing bubbles or something.  
> **username_6**: @iMerchant Thanks! I hope that even if the equations seem daunting, the point behind them was not lost on you. To make the equations, surround your text with the $ character and use Latex, a typesetting language designed for creating nicely formatted text/equations as you see above. There are plenty of tutorials online to learn Latex from. Try playing around with Latex [here](http://arachnoid.com/latex/).  
> **username_0**: @zephyr - I'm allergic to Latex. :-p But I now know how you do it. Very cool.  

---

### Answer by username_3 on 2016-10-06. Score: 11

This is a confusing question - your title mentions GR, but of course the age of the universe is entirely derived as a result of using GR to solve for the dynamics of the expanding universe.

The text of your question is talking about time dilation and the effects of *special* relativity (a subset of GR). Here there is a point to be addressed. It is not a good way to think about the explansion of the universe in terms of things receding from each other at ever increasing velocities. It is better to think about space expanding. It is also not possible to apply special relativity except in local inertial frames.

It can and has been shown that the redshift-distance relationship *cannot* be simply interpreted as the doppler shift due to things moving away from us at ever increasing speeds - since this of course predicts a maximum recession velocity that approaches the speed of light. In fact, beyond redshifts of $z=1.5$, the "recession velocity" in all viable models exceeds the speed of light. What actually happens is that light emitted at some earlier epoch travels across expanding space to get to us and this expansion increases the wavelength of the light.

Essential reading: [Davis & Lineweaver (2003)](https://arxiv.org/pdf/astro-ph/0310808.pdf).

> **Comments:**
> **username_0**: I updated question to say GR/SR instead of just GR.  
> **username_4**: @iMerchant that doesn't change the fact that SR is not of any importance here. I reckon you're not going to get a better answer than Rob's.  
> **username_5**: It's all a bit above my head, but here's another article, @iMerchant, which I have linked to before on similar questions:  https://arxiv.org/abs/0707.0380v1 .  For example: http://physics.stackexchange.com/a/223697/10319  

---

