What Is the Proper Time of the CMB and Its Implications?

[D.S.Beyer]

TL;DR

Relative to today, was time passing faster, slower, (or neither) at the moment of last scattering?

Did some searches through these forums but didn't find this exact question. I'm sure it's already been asked, but I just missed it, my apologies. Please link.

I’ll try and ask this question in 3 different ways, and maybe the idea behind it will become apparent. I know that semantics can really throw off a physics discussion.

1 : Relative to today, was time passing faster, slower, (or neither) at the moment of last scattering?

2 : How much of the red shift of the CMB is attributed to a shift due to time dilation/contraction? If any?

3 : Hypothetically, if a clock was around at the moment of last scattering, and its light is just now reaching us, would the hands on that clock be ticking faster or slower (or neither) than a clock on present day earth?

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A few ways I've thought about answering this :

So... the impetuous for this question is the repeated statements that the early universe was hot and dense (relative to today), which to me sounds like such a state of matter would cause some (relative to us, now) distortions in the spacetime metric.

I've tried to approach this before through the vein of 'gravitational time dilation' and gravitational potential, but got dead ends.
I've also tried to think about this as comparing the spacetime metric then and now, but it sort of spiraled into a GR 2-body problem.
"You can't define a gravitational potential in a non-stationary spacetime, and the FLRW spacetime used in cosmology is non-stationary. So this question can't be phrased in a way that makes sense in GR terms."Along the way I've learned about the Sachs-Wolf Effects, NISW and ISM (both early and late). These are amazing, but don't make a relationship between the matter/energy density at the surface of last scattering and the matter/energy density today. (NISW calcs the gravitational redshift against the average background temperature and the hotter or cooler inhomogeneities at the moment of last scattering. ISM calcs redshifts that are modified by passing thru grav wells which are effected by universal expansion)

Now I'm thinking the approach may be to dig into the energy density of the Friedman equations, and compare then and now based on the scale factor.
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Clearly I'm not a trained physicists and just casually interested in these topics.
What is a better way of asking this questions, and/or going about finding the answer?

 

[Vanadium 50]

Relative to today, was time passing faster, slower, (or neither) at the moment of last scattering?

One second per second, same as it is today.

 

[Ibix]

There's no meaningful way to compare. Gravitational time dilation is measurable because you put two clocks one above the other and let them exchange light signals to zero themselves, then exchange light signals to determine that one is faster than the other.

A clock now cannot exchange signsls with a clock yesterday, let alone with a clock several billion years ago. It can only receive them, which just tells you that the signal is redshifted.

To paraphrase myself, I know what it means to compare how my watch and yours are running. I don't know what it means to compare how my watch is running today with how it was running yesterday.

 

[PeroK]

This whole idea got a bit of an airing recently:

https://www.physicsforums.com/threads/what-happens-to-time-as-space-is-expanding.1001016/

 

[Nugatory]

Relative to today, was time passing faster, slower, (or neither) at the moment of last scattering?

Time passes, tautologically, at the rate of one second per second. Yes, you will often hear time dilation described as "time slowing down", but that model only works in the flat spacetime of special relativity, and even there it is more misleading than informative. In the curved spacetimes of geeral relativity and cosmology, it is completely unworkable; it's one of the first things we unlearn when we take up general relativity.

So as it stands your question isn't well-formed. You might want to back up and look at two treatments of time dilation in special relativity using a different perspective than you'll find in most intro textbooks:
1) Explain time dilation between two moving clocks in terms of relativity of simultaneity instead of the standard "time slows down for moving clocks" explanation. A happy side effect is that you will have a clear resolution of the apparent paradox that when A and B are moving relative to one another, both find the other clock to be running slow.
2) Explain the twin paradox in terms of the "distance" (length of spacetime interval, amount of proper time) through spacetime traveled by each twin. This is the geometric approach championed by Taylor and Wheeler's "Spacetime Physics" (available free online) and is an essential prerequisite to general relativity.Once you've been through that you'll have the language needed to ask your question and understand the answers.

 

[Nugatory]

I know what it means to compare how my watch and yours are running.

Indeed you do, but even that comparison doesn't mean quite what it sounds like; there's an implied simultaneity convention that you're quite aware of of but that a casual reader of this thread might not notice until after they've fallen into the pit.

 

[Ibix]

there's an implied simultaneity convention that you're quite aware of

Indeed. Setting that up is what all the exchanging light signals does. And Ixm assuming a timelike Killing field, or at least a timelike nearly-Killing field (a severely maiming field? Sorry...), in order to be able to define gravitational time dilation at all.

 

[D.S.Beyer]

This whole idea got a bit of an airing recently:

https://www.physicsforums.com/threads/what-happens-to-time-as-space-is-expanding.1001016/

Wow. I'm going to have to take a moment or two to get through this thread, but it does get to the heart of a lot of what I'm interested in.

 

[D.S.Beyer]

There's no meaningful way to compare. Gravitational time dilation is measurable because you put two clocks one above the other and let them exchange light signals to zero themselves, then exchange light signals to determine that one is faster than the other.

A clock now cannot exchange signsls with a clock yesterday, let alone with a clock several billion years ago. It can only receive them, which just tells you that the signal is redshifted.

To paraphrase myself, I know what it means to compare how my watch and yours are running. I don't know what it means to compare how my watch is running today with how it was running yesterday.

This blows my mind that this is such a difficult question, but in a good way.

Why is the exchange of signals important? Can't it just be a one way thing?
For example (and in wild hypothetical land), if I take a telescope in my backyard and zoom in on a clock that is strapped to the body of a high orbit GPS satellite, don't I literally see the satellites clock running slower?

...actually don't answer that. Let me dig around in that other thread, and also try and think of a way to frame this question in a way that @Nugatory suggests.

 

[PeterDonis]

if I take a telescope in my backyard and zoom in on a clock that is strapped to the body of a high orbit GPS satellite, don't I literally see the satellites clock running slower?

First, over an entire orbit, the clocks on GPS satellites actually run faster than clocks on the ground; they are in high enough orbits that the effect of being in a higher gravitational potential outweighs the effect of their orbital speed. If you used the ISS as an example instead, then those clocks do run slower over an entire orbit than ground clocks (because the orbit is much lower).

Second, as far as what you would actually see through a telescope if you watched a clock on a satellite, you would sometimes see the clock running faster and sometimes see it running slower, because what you see is due to the relativistic Doppler effect. (For example, consider the satellite coming over the horizon in the west, flying overhead, and then going below the horizon in the east; as it comes over the horizon and comes towards you, you see its clock running faster; then, as it passes overhead and goes back down over the horizon, you see its clock running slower.) If the Earth were transparent and you could watch the clock on the satellite over an entire orbit, you would see the "faster" and "slower" times balance out to the effect over an entire orbit that I described above (faster for GPS satellites, slower for the ISS).