How to measure time in the early universe?

[Cosmology2015]

I would like to study more about the notion of time in modern physics. In particular, I would like to understand better the following question: how does one measure the time in the early universe? To measure time one needs to have clocks. A natural choice to be used as clocks would be particles with mass. The problem is that just after the Big Bang particles with mass did not exist then the idea of use clocks cannot be applied and therefore the notion of time is lost. I read somewhere that there is an area of mathematics called conformal geometry that could be used to understand this problem. Does anyone know references on this topic? The application of conformal geometry in conjunction with the theory of general relativity has been studied at the moment? Any answer would be of great value!

 

[phinds]

You are correct in that there were no (and could not BE any) clocks in the early universe, but it is none-the-less taken that time had the same rate then as now. Certainly Steven Weinberg, who knows more physics than you or I are ever likely to, had no trouble with the concept in his book "The First Three Minutes".

 

[PeterDonis]

A natural choice to be used as clocks would be particles with mass.

A natural choice, yes, but not the only possible one.

I read somewhere that there is an area of mathematics called conformal geometry that could be used to understand this problem.

The universe can be described using "conformal" coordinates, but the "time" in these coordinates is not the same as the proper time of comoving observers, which is what is standardly used to define the "age" of the universe. The relationship between the two times is known, though, so you can compute the latter from the former.

 

[Cosmology2015]

The universe can be described using "conformal" coordinates, but the "time" in these coordinates is not the same as the proper time of comoving observers, which is what is standardly used to define the "age" of the universe. The relationship between the two times is known, though, so you can compute the latter from the former.

This idea is really very interesting. Could you tell me more about how to describe a system using conformal coordinates and how to calculate this relationship between the two times? Is there any reference (books, articles, papers, lectures notes) where I can learn conformal geometry applied to cosmology, in particular in this problem of measuring time in the early universe? Any answer would be of great value!

 

[PeterDonis]

Could you tell me more about how to describe a system using conformal coordinates and how to calculate this relationship between the two times? Is there any reference (books, articles, papers, lectures notes) where I can learn conformal geometry applied to cosmology

Unfortunately I can't find a good simple reference online, but most textbooks on cosmology will discuss this, at least briefly.

The idea is simple: start with the FRW metric in the usual comoving coordinates:

$$
ds^2 = - dt^2 + a^2(t) d\Sigma^2
$$

where I have written $d\Sigma^2$ for a generic line element on a spacelike hypersurface of constant time; for our purposes here we don't care if the geometry of this hypersurface is flat, open, or closed. We then define conformal time $\eta$ as follows:

$$
dt = a d\eta
$$

This transforms the metric to conformal form:

$$
ds^2 = a^2(\eta) \left( - d\eta^2 + d\Sigma^2 \right)
$$

The key difference, as you can see, is that in the conformal form the scale factor $a$ multiplies the entire line element, not just the spatial part. The advantage of this is that light rays always have coordinate speed 1 in these coordinates, just like in ordinary flat Minkowski spacetime.

in particular in this problem of measuring time in the early universe?

Most cosmologists do not believe that conformal time $\eta$ corresponds to the time actually measured by any observers, even in the early universe. It is purely a mathematical convenience. There are some who have tried to construct speculative theories in which observers in the early universe actually measure conformal time $\eta$, not comoving coordinate time $t$, but I'm not aware of any of these gaining any traction.

 

[Cosmology2015]

Unfortunately I can't find a good simple reference online, but most textbooks on cosmology will discuss this, at least briefly.

No problem! In fact, your answer was very enlightening and I am very grateful you have booked your time to answer it. I have another question that involves the second law of thermodynamics. The second law of thermodynamics states that the total entropy of an isolated system always increases over time which means that decreases with time. What concept of time the law is referring to? More precisely, how to measure the entropy in the early universe?

 

[PeterDonis]

The second law of thermodynamics states that the total entropy of an isolated system always increases over time which means that decreases with time.

Which means that what decreases with time? I think you left out a word.

What concept of time the law is referring to?

The modern interpretation is to turn the law around: the direction of increasing entropy defines a thermodynamic "arrow of time", i.e., it defines the direction of increasing time, or at least of thermodynamic time. The reason why this direction of time coincides with the direction of our experience of time is that forming memories is a thermodynamic process that increases entropy, so any events that we can remember must have occurred when entropy was lower than it is now, when we are remembering them.

how to measure the entropy in the early universe?

The entropy of the ordinary matter and energy of the universe is just the entropy of a perfect fluid with a certain temperature.

The entropy associated with the expansion of the universe, and with gravitational clumping of matter, can also be calculated. Most of the entropy increase in the universe is due to these two effects.

 

[Cosmology2015]

Which means that what decreases with time? I think you left out a word.

Indeed! In fact, I wanted to express the fact that if the entropy increases as time passes then it means that the same decreases if one looks back in time. Again, you perfectly understood my question and I would like to thank you for your time. Despite my technical limitations, since I am still learning tensor calculus, general relativity and cosmology, this is a subject that I have a strong interest and I have tried to understand better.

 

[MeJennifer]

In general relativity there are no preferred coordinates so the question would be entirely observer dependent.
Seems to me you are asking for a cosmology question as such this topic should be moved to the cosmology section.

 

[PeterDonis]

Seems to me you are asking for a cosmology question as such this topic should be moved to the cosmology section.

Good point, moved.

 

[Cosmology2015]

Consider the Einstein field equations in the form:

$R_{\mu \nu }-\frac{1}{2}Rg_{\mu \nu }=\frac{8\pi G}{c^{4}}T_{\mu \nu }$

Now applying an appropriate conformal rescaling given by:

$\bar{g_{\mu \nu }}=\Omega ^{2}g_{\mu \nu }$

Replacing in Einstein field equations gives the following expression:

$R_{\mu \nu }-\frac{1}{2}R(\Omega ^{2}g_{\mu \nu })=\frac{8\pi G}{c^{4}}T_{\mu \nu }$

To simplify consider the energy-momentum tensor equal to zero in some region under consideration (vacuum field equations):

$R_{\mu \nu }-\frac{1}{2}R(\Omega ^{2}g_{\mu \nu })=0$

Solving the equation for the conformal factor $\Omega$:

$\Omega ^{2}=\frac{2R_{\mu \nu }}{Rg_{\mu \nu }}$

$\Omega=\sqrt{\frac{2R_{\mu \nu }}{Rg_{\mu \nu }}}$

As my general relativity knowledge is very limited, especially when it comes to the mathematics involved, I wonder if the above expressions are mathematically correct. If so, how can I apply this conformal rescaling to study the physics of early universe? Any answer would be of great value!

 

[Cosmology2015]

Does anyone know of bibliographical references on this topic?

 

[haushofer]

Consider the Einstein field equations in the form:

$R_{\mu \nu }-\frac{1}{2}Rg_{\mu \nu }=\frac{8\pi G}{c^{4}}T_{\mu \nu }$

Now applying an appropriate conformal rescaling given by:

$\bar{g_{\mu \nu }}=\Omega ^{2}g_{\mu \nu }$

Replacing in Einstein field equations gives the following expression:

$R_{\mu \nu }-\frac{1}{2}R(\Omega ^{2}g_{\mu \nu })=\frac{8\pi G}{c^{4}}T_{\mu \nu }$

To simplify consider the energy-momentum tensor equal to zero in some region under consideration (vacuum field equations):

$R_{\mu \nu }-\frac{1}{2}R(\Omega ^{2}g_{\mu \nu })=0$

Solving the equation for the conformal factor $\Omega$:

$\Omega ^{2}=\frac{2R_{\mu \nu }}{Rg_{\mu \nu }}$

$\Omega=\sqrt{\frac{2R_{\mu \nu }}{Rg_{\mu \nu }}}$

As my general relativity knowledge is very limited, especially when it comes to the mathematics involved, I wonder if the above expressions are mathematically correct. If so, how can I apply this conformal rescaling to study the physics of early universe? Any answer would be of great value!

No, they are most certainly not. You cannot take square roots of tensors. You have to trace them first such that you get a scalar equation. Only then you can treat the equation like an ordinary equation, because scalars behave like numbers under the operations of multiplication etc.

I believe one of the appendices of Wald covers conformal transformations in detail.

 

[gianeshwar]

Forming memories I understand decrease entropy as information increased.

 

[Cosmology2015]

No, they are most certainly not. You cannot take square roots of tensors. You have to trace them first such that you get a scalar equation. Only then you can treat the equation like an ordinary equation, because scalars behave like numbers under the operations of multiplication etc.

I believe one of the appendices of Wald covers conformal transformations in detail.

First of all I would like to thank you for your response! I had not paid due attention to the fact that one cannot take square roots of tensors. It really was a mistake. My general relativity knowledge is very limited, especially when it comes to the mathematics involved. How can I trace the tensors first in order to get a scalar equation? I searched how to calculate the square roots of tensors but I did not find much about it. Indeed the appendices of Wald covers conformal transformations in detail. The problem is that the level is too high for me. I will try other bibliographic sources. If you know of other bibliographic references I would be really grateful! I would like to thank you again for your time! Have a nice week!

 

[Cosmology2015]

Forming memories I understand decrease entropy as information increased.

It's a really interesting point of view. But if information decays over time would it mean that somehow the information is lost? It's possible? Do you know of bibliographical references on this topic? It's a very interesting subject! I would like to take the opportunity to wish you a nice week!

 

[enorbet]

Perhaps an ignorant question, but how can we conclude that remembering is always at a greater state of entropy than the thing remembered was at the time? It is my understanding that while the entropy of the overall Universe is increasing we also know that, given locality, entropy can decrease for an interval of time?

 

[Cosmology2015]

Anyone?

 

[PeterDonis]

how can we conclude that remembering is always at a greater state of entropy than the thing remembered was at the time?

You can't. What you can conclude is that the entropy of the thing remembered, plus the entropy of the thing that will store the memory (your brain, a computer disk, whatever), was lower before the memory was formed than after. If you look at the physics of how a memory gets formed, you will see that the basic reason for this is that, in order to store a memory, the previous state of the memory storage has to be thrown away and lost. That makes memory formation an irreversible process and an increase of entropy, even if the entropy of the thing that gets remembered stays the same.