# Where the scale factor a(t) appears in the metric

• I
In summary: If so, what is it "the other speed", then?To me "the other speed" is something more consistent with the speech Einstein gave in Leyden, in 1922.In 1922, Einstein said that the speed of light is not the only speed in the universe. He also said that "the other speed" is the speed of light in a different reference frame of space and time. So in summary, the flat Minkowsky spacetime has a disturbing variation where a metric with "a scale factor" exists.
Hello,
I was enjoying Zee's book on GR when I noticed the location of this "a(t)" thing in the metric sound quite disturbing to me.

BTW: I experience the same annoyance and went down to the same conclusions, when I watched a related Theoretical Minimum lesson...Here's the setup, the flat Minkowsky spacetime (with the c=1 convention):

Here's the disturbing variation, a metric with "a scale factor":

now: can you see "where" that "a(t)" is, in the equation, and what also there should be nearby, when dropping the c=1 convention?

so: is it just me, or a(t) is just the inverse of the speed of light, changing in time?This view would be consistent with what those people are saying, right?

Let alone it's really what the equations say: I'm not reading the equations wrong, am I?PS: That's just using Susskind's "math autopilot", isn't it?
Cheers

is it just me, or a(t) is just the inverse of the speed of light, changing in time?

When you say "the speed of light", you have to be careful in curved spacetime, because you now have to distinguish two speeds that were the same in flat spacetime:

(1) The coordinate speed of light: this is just ##d\vec{x} / dt## for a light beam. You are correct that this varies with ##a(t)## in the spacetime under discussion. But note that this happens whether you put the ##c## back into the metric or not. All the ##c## does is change the units of the ##t## coordinate; it doesn't change any physics.

(2) The locally measured speed of light: this is the speed an observer at a given event would measure for a light ray passing him at that event. This is always ##c## (or ##1## in the units Zee was using); it does not change with ##a(t)##.

hi Peter,

thanks for your reply, but I'm not satisfied with this explanation.First of all, "measuring the speed of light" is quite nonsense, or tautology,
given our units of measure for space and time are defined upon light speed c,
by using multiples of the T and the Lambda of a reference radiation.
(see meter@wiki and second@wiki - "Caesium fountains" are currently used).Given that, the "locally measured speed" of light is tautologically c:
the "local observer" would have "defined his spacetime" with that same c!BTW: this is the same mechanism by which SR works, when changing from
an inertial frame of reference to another: both the observers "define
their spacetime" by an ideal "Caesium fountain" which is "at rest" with them

(the definition of the second requires this "at rest", pls. double check that).

It's just the relativistic doppler effect between those 2 caesium fountains,
which "produces" the Lorentz boost, in the observers' coordinate systems
(it's not hard to clarify this once and for all - pls. double check that).I hope you'd agree with the above,
regarding the "locally measured c"
being tautological, do you ?

PS: "if so", part 1

If so, that's why "that speed" is always c!

If so, what is it "the other speed", then?

To me "the other speed" is something more consistent
with the speech Einstein gave in Leyden, in 1922.
PPS: "if so", part 2

If so, then maybe also h(t) or G(t), for Planks units' sake.

My educated guess will be then for h(t), of course.

That's because c(t) and h(t) are consequences of some energy density
(defined against a volume of space and in 3 volumes of spacetime) of that
"continuum without which both GR and SR are unthinkable"
(he used other words, but that's the core meaning)I hope this would make you mumble a little, at least, does it?
Cheers,

First of all, "measuring the speed of light" is quite nonsense, or tautology,
given our units of measure for space and time are defined upon light speed c

If this bothers you (it shouldn't, because our choice of units is a human convention, not physics), you can just substitute "measurement of dimensionless physical constants that determine our definition of c". Which means measuring the fine structure constant ##\alpha##, since that's the key dimensionless physical constant for electromagnetism, and the one that governs the physical measurements using atomic clocks that are the basis for the SI definition of c. The physics is still the same: local measurements anywhere in a curved spacetime will still yield the same values for physical constants, regardless of the fact that the coordinate speed of light varies. So the variation in the coordinate speed of light does not affect any physics.

Given that, the "locally measured speed" of light is tautologically c:
the "local observer" would have "defined his spacetime" with that same c!

If you mean that what we are calling "the speed of light" c (the fixed number 299,792,458 in SI units) is really just a conversion factor between time units and space units, i.e., between what clocks measure and what rulers measure, and is therefore really a property of spacetime itself, yes, I agree. And then the rebuttal to your claim is simply that this property of spacetime is the same at every event, and is not the same as the coordinate speed of light that varies in a curved spacetime.

what is it "the other speed", then?

It is ##d\vec{x} / dt##, the derivative of the spatial coordinates of a light beam with respect to the time coordinate. It is a property of the coordinates you choose, not something physical.

That's because c(t) and h(t) are consequences of some energy density

Now you are not talking about what our physical theories actually say, but about a speech (not a peer-reviewed paper, i.e., not a valid source for PF discussion) in which Einstein was stating personal opinions, not confirmed scientific theories. That is irrelevant to this discussion.

I hope this would make you mumble a little, at least, does it?

Not at all. See above.

sorry Peter,

I didn't explained my point clearly.Let's work in Planks units, so c=1, ok?

That way no SI, no human related units, plain "ones", ok?Pls, take a better look at the definition of meter and second, so that you realize that:

- 1 unit of time proportional to the period:
"second is the duration of 9 192 631 770 periods of the radiation"
- 1 unit of length proportional to the wavelength
"metre is defined as the distance traveled by light in a vacuum in 1/299 792 458 seconds"

As you can see, 1/c is that 1/299 792 458, by definition,
and there's another "magic number": 9 192 631 770.

Both those magic numbers are "arbitrary",
while proportionality of second to period
and of meter to wavelength is "natural".
Now, can you imagine a "Caesium fountain" throwing out its "special radiation" in all directions in space?

Can you visualize the EM waves propagating out of it, say isotropically?

That's what defines an observer's spacetime, by defining the units using the waves.

It's actually easy to visualize the wavefronts: imagine you're dropping a drop of water
in a pond every unit of time: the drops are time-ticking, so the distance (in time)
between two peaks (a period) is the unit of time.
For the unit of space just take the wavelength.

Do you agree this is the way spacetime metric gets "physically implemented"?

Then SR will follow: just let another "Caesium fountain" in the game.
First draw a picture with both "at rest", then one with "one moving".

If you use the pond/droplets picture, remember "it's always the moving source".
If you still can't get my point think about micro-bats and their echolocation.

Or even better, pretend being a micro-bat wanting to do some physics.

How is it "physically implemented"?

Why do they have 2 kinds of Doppler effects?PS: microbats' "universe" ends up where the atmosphere ends.
That "sounds" like a De Sitter horizon for them.Hope anyone can see my point...

...this point first, then the a(t) thing.Cheers,

Let's work in Planks units, so c=1, ok?

Planck units are ##\hbar = 1##, not ##c = 1##, but I understand you want to use units where ##c = 1##, that's fine.

Hope anyone can see my point...

All I see is that you are restating (in somewhat idiosyncratic language) the standard physics of how atomic clocks work and how EM radiation propagates. Nothing you say in this post has the slightest relevance to the topic of this thread.

PeterDonis said:
Planck units are ##\hbar = 1##, not ##c = 1##, but I understand you want to use units where ##c = 1##, that's fine.

of course, then G=1 too
PeterDonis said:
All I see is that you are restating the standard physics of how atomic clocks work.
Nothing you say in this post has the slightest relevance to the topic of this thread.

Well, I opened this thread to discuss a(t) and c(t),
you mentioned the distinction between "coordinate speed"
and "measured speed", about which we have quite different perspectives.To me,
you're taking for granted
you can build a reference frame
out of thin air.

I was just trying to explain to you
not how atomic clocks work,

but how they are used to build reference frames,
and the implications.But that seems to hard for me to explain this clearly.

Such a shame!
Cheers

you mentioned the distinction between "coordinate speed"
and "measured speed", about which we have quite different perspectives.

Mine is not a "perspective", it's how GR works. Your choice of coordinates does not affect any physics. If you disagree with that, that is not a "perspective", it's contradicting GR.

To me,
you're taking for granted
you can build a reference frame
out of thin air.

We haven't talked about "reference frames" at all. You asked about the scale factor a(t) and the coordinate speed of light c(t) in a particular coordinate chart. The coordinates in that chart don't describe physical measurements; they are just 4-tuples of numbers used to label events (with certain requirements for one-to-one mapping and continuity).

I was just trying to explain to you
not how atomic clocks work,

but how they are used to build reference frames,
and the implications.

Atomic clocks don't measure coordinate time ##t##, in cosmological or any other coordinates. They measure proper time ##\tau## along their worldlines. So does any clock. You have not asked about proper time at all, only about coordinate time. So nothing about atomic clocks has anything at all to do with the question you asked in your OP.

I think you expect that the metric will be the flat Minkowski metric. This is a different example.

sorry, I can't believe it's so hard to understand...

take ANY mono-chromatic, spherical light source,
on large scale (h=0) and no gravity (G=0):

when you turn the light on, it "implements" a reference frame,
by defining the unit(s) of space (wavelength) in all directions,
while relating (with c) those units with the unit for time (period),
with perfect correspondence to today's (=post 1905) metre and second

can you figure out that?shall I mention that spacetime metric is an abstract concept,
which has multiple, different physical implementations?

humans have a light-based implementation,
microbats a sound-based one, quite different

can you understand that?
do any funny consequences pop up to your mind?
cheers

Last edited:
I can't believe it's so hard to understand...

I understand what you're trying to say. I just also understand that it has nothing to do with the original question you asked in this thread.

humans have a light-based implementation,
microbats a sound-based one

This is nonsense. The speed of sound is not a relativistic invariant the way the speed of light is. Your posts are verging on personal theory at this point.

Further discussion does not seem fruitful. Thread closed.

## 1. Where does the scale factor a(t) appear in the metric?

The scale factor a(t) appears in the metric as a coefficient for the spatial components of the metric tensor. It is typically denoted as a(t) or sometimes as the letter R.

## 2. How does the scale factor a(t) affect the metric?

The scale factor a(t) affects the metric by determining the overall size and shape of the universe. As the scale factor changes, the distance between objects and the curvature of space-time also change. It is a key component in understanding the expansion of the universe.

## 3. What is the significance of the scale factor a(t) in cosmology?

In cosmology, the scale factor a(t) is used to describe the expansion of the universe. It is a key parameter in many cosmological models and theories, such as the Big Bang theory. The value of a(t) also provides insights into the age and fate of the universe.

## 4. Can the scale factor a(t) be measured?

Yes, the scale factor a(t) can be measured using various methods in cosmology, such as observations of the cosmic microwave background radiation or the redshift of distant galaxies. These measurements can help determine the current value of a(t) and how it has changed over time.

## 5. How does the scale factor a(t) relate to the expansion rate of the universe?

The scale factor a(t) is directly related to the expansion rate of the universe. As a(t) increases, the expansion rate also increases, meaning the universe is expanding at a faster rate. Conversely, as a(t) decreases, the expansion rate slows down. This relationship is described by the Hubble's law.

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