What is Rieman for a conformal metric?

[Kurvature]

Hi...
The ordinary plain vanilla conformal metric in spherical coordinates is:

ds² = a(t)²[dt² - dr²/(1 - kr²) - r² (dΘ² + sin²(Θ) d(φ)²)]

where a(t) is a function of time only.

I am trying to find out what Rieman, Ricci and the Scalar Curvature are
for this common metric when k=1 and a-dot and a-double-dot are zero.

Would it be published anywhere? Is it available on the Internet?

Could anyone check their Mathematica Notebook and tell me?

Alsmost all the tensor components are ZERO if a-dot and a-double-dot = 0 But I
suspect that there are a few crucial non-zero components proportional to k/a²
(particularly in the Ricci diagonal).
As you know this was famously so in the Einstein tensor, which allowed Einstein
to determine the radius of the Universe.

 

[Matterwave]

Isn't that just the FRW metric? You can probably just google the FRW metric and the tensors should be recorded somewhere. These notes have the non-zero Ricci tensor components: http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll8.html

 

[Kurvature]

Isn't that just the FRW metric? You can probably just google the FRW metric and the tensors should be recorded somewhere. These notes have the non-zero Ricci tensor components: http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll8.html

No, its NOT the FRW metric. It is formally identical to the FRW metric if
you write FRW in terms of "conformal time"... but you'd never on Earth
locate a publication of Rieman, Ricci and R in terms of conformal time.

Someone with a Mathematica Notebook could plug in the metric and tell
you the answer in a couple of minutes. Sob...Sigh...

 

[Matterwave]

Ah, I only glanced quickly and didn't notice the a(t) modified the time coordinate as well. Sorry.

 

[Kurvature]

Ah, I only glanced quickly and didn't notice the a(t) modified the time coordinate as well. Sorry.

The FRW metric describes Gravity. The Conformal Metric describes God.
I'll just have to wait for some rich kid with a Mathematica Notebook to
run the conformal metric thrugh Ricci and find out if the diagonal components
are proportional to k/a² similar to the Einstein tensor.

 

[Kurvature]

The FRW metric describes Gravity. The Conformal Metric describes God.
I'll just have to wait for some rich kid with a Mathematica Notebook to
run the conformal metric thrugh Ricci and find out if the diagonal components
are proportional to k/a² similar to the Einstein tensor.

P.S. :
Lest someone think I am daft, I forgot to mention that while FRW describes
the curvature of objective spacetime wch. is Gravity, the Conformal metric
describes the curvature of subjective spacetime (aka "reality") said
phenomena being popularly referred to as "God".
God is the Ricci Curvature of reality.

 

[PeterDonis]

some rich kid with a Mathematica Notebook

Maxima is free:

http://maxima.sourceforge.net/

The ctensor package in Maxima is designed precisely for this kind of computation.

while FRW describes
the curvature of objective spacetime wch. is Gravity, the Conformal metric
describes the curvature of subjective spacetime (aka "reality") said
phenomena being popularly referred to as "God".
God is the Ricci Curvature of reality.

Do you have a reference for this? It doesn't look like physics to me. If it isn't, it's off topic for this forum.

 

[PeterDonis]

As you know this was famously so in the Einstein tensor, which allowed Einstein
to determine the radius of the Universe.

What is this a reference to? I don't understand.

 

[Kurvature]

Maxima is free:

http://maxima.sourceforge.net/

The ctensor package in Maxima is designed precisely for this kind of computation.

[Kurvature]
Thanks profoundly for that tip! I downloaded it, unzipped the .tar file, but I can't
figure out how to start the program... can't even find an .exe file... how do you
get the thing to run? Are there any Usenet newsgroups discussing Maxima ?



Do you have a reference for this? It doesn't look like physics to me. If it isn't, it's off topic for this forum.

[Kurvature]
That was just an aside. I'm here to find out what the diagonal components
of Ricci are for the well known Conformal Metric with k=1. If I can get
Maxima to start I should be able to find out... my guess is that at least
R_tt is proportional to k/a²

 

[Kurvature]

What is this a reference to? I don't understand.

[Kurvature]
The FRW metric contains the same scale factor a(t) as the Conformal Metric.
Einstein used Fiedmans metric to solve the "cosmological problem" in 1915-20
and showed that for maximum expansion in a closed universe a(t)_max = R the
radius of the universe... this is a famous calculation and is in every intorductory
textbook.

 

[Matterwave]

[Kurvature]
The FRW metric contains the same scale factor a(t) as the Conformal Metric.
Einstein used Fiedmans metric to solve the "cosmological problem" in 1915-20
and showed that for maximum expansion in a closed universe a(t)_max = R the
radius of the universe... this is a famous calculation and is in every intorductory
textbook.

Didn't Einstein believe in a static universe, and introduced the cosmological constant to make it as such?

 

[Kurvature]

Didn't Einstein believe in a static universe, and introduced the cosmological constant to make it as such?

[Kurvature]
Yes, but that won't work for a static universe because it's unstable.
On page 112 of Einstein's book The Meaning of Relativity Einstein
says: "The mathematician Friedman found a way out of this dilemma."
And thus the FRW metric was born.

 

[PeterDonis]

The FRW metric contains the same scale factor a(t) as the Conformal Metric.

Not really. In the usual FRW metric, $a$ is a function of FRW coordinate time, which is the same as proper time for comoving observers.

In the conformal metric, $a$ is a function of conformal time, which is a different time coordinate. So the two $a$'s are different, even though they happen to be designated by the same letter.

Einstein used Fiedmans metric to solve the "cosmological problem" in 1915-20

Reference, please? Friedmann did not discover his metric until 1922.

for maximum expansion in a closed universe a(t)max = R the
radius of the universe... this is a famous calculation and is in every intorductory
textbook.

Yes, that's true. Are you just trying to duplicate that calculation?

Also, Einstein made this calculation as part of trying to justify his static universe model, which, as you note, is not really reasonable because it's unstable.

On page 112 of Einstein's book The Meaning of Relativity Einstein
says: "The mathematician Friedman found a way out of this dilemma."

Yes, and the way out was to allow the metric to be dynamic, i.e., to allow the scale factor $a$ to be a function of time. That means $\dot{a} \neq 0$ and $\ddot{a} \neq 0$. So what you appear to be trying to do, based on your OP, doesn't really have anything to do with Friedmann's solution.

 

[Kurvature]

Moderator note: post edited to clean up quote tags.

Not really. In the usual FRW metric, $a$ is a function of FRW coordinate time, which is the same as proper time for comoving observers.

In the conformal metric, $a$ is a function of conformal time, which is a different time coordinate. So the two $a$'s are different, even though they happen to be designated by the same letter.

[Kurvature]
WRONG ! Both the FRW and Conformal metric are written in proper time.
The FRW metric can be rewritten by substituting in the conformal time and
that will make the FRW metric formally identical to the conformal metric,
but that is NOT what we are talking about here. You are mistaken.

Reference, please? Friedmann did not discover his metric until 1922.

[Kurvature]
Reference to what? I'm not here to tete a tete about standard textbook material.
If you are looking for someone to argue with, please look elsewhere.I
I am way too busy

Yes, that's true. Are you just trying to duplicate that calculation?

[Kurvature]
I told you what I am here to do. I am trying to find out
what the components of the Ricci tensor are for the given
Conformal metric.

Also, Einstein made this calculation as part of trying to justify his static universe model, which, as you note, is not really reasonable because it's unstable.

[Kurvature] I already said that.

Yes, and the way out was to allow the metric to be dynamic, i.e., to allow the scale factor $a$ to be a function of time. That means $\dot{a} \neq 0$ and $\ddot{a} \neq 0$. So what you appear to be trying to do, based on your OP, doesn't really have anything to do with Friedmann's solution.

[Kurvature] Someone else asked me about the Friedmann solution
and that is the answer I gave him.
I've already told you several times what I'm trying to do
which is find out what the Ricci tensor is, and the Rrieman tensor
for the given Conformal metric.
You seem to be looking for an argument and I've already told you
I'm way too busy with serious business.
I'm not about to argue with you, as they say.

 

[PeterDonis]

Both the FRW and Conformal metric are written in proper time.

Not with the line element you wrote down in your OP. In that line element, coordinate time is not the same as proper time (at least, it isn't for comoving observers), and $a$ is a function of coordinate time. If you don't want to call that coordinate time "conformal time", that's fine, but it does happen to be pretty standard terminology for a conformal metric.

Reference to what?

To your claim that Einstein used Friedmann's metric in the period 1915-1920. He couldn't have, since Friedmann did not discover his metric until 1922.

I am trying to find out
what the components of the Ricci tensor are for the given
Conformal metric.

And I was trying to understand a bit more about why that is of interest to you. But it's not a big deal either way.

 

[Kurvature]

Moderator's note: post edited to clean up quote tags.

Not with the line element you wrote down in your OP. In that line element, coordinate time is not the same as proper time (at least, it isn't for comoving observers), and $a$ is a function of coordinate time. If you don't want to call that coordinate time "conformal time", that's fine, but it does happen to be pretty standard terminology for a conformal metric.

[Kurvature]
You don't know what you're talking about. The metric that I posted is not and cannot be a real
physical metric and it CERTAINLY is NOT the FRW metric written using "conformal time" because
that IS a real physical metric. And coordinate time in FRW is proper time.
The Rieman metric is a strictly mathematical construct with NO physical content. and one can
write ANY ad hoc metric and plug it into Rieman and compute the Rieman components.
FRW written using conformal time IS NOT the "conformal metric", the metric in my original
post IS by definition the conformal metric.

To your claim that Einstein used Friedmann's metric in the period 1915-1920. He couldn't have, since Friedmann did not discover his metric until 1922.

[Kurvature]
Ok, my claim is that: Einstein used Friedmann's metric in the period 1915-1922
Don't try and nit-pic harrass me on this thread or I'll file a complaint with the list owner.

And I was trying to understand a bit more about why that is of interest to you. But it's not a big deal either way.

[Kurvature]
Why it is of interest to me is a matter over your head
and frankly none of your business.

 

[PeterDonis]

The metric that I posted is not and cannot be a real
physical metric

Why not?

it CERTAINLY is NOT the FRW metric written using "conformal time"

Yes, you've said that. I wasn't claiming that it is.

coordinate time in FRW is proper time.

For comoving observers, yes, this is true, but it has nothing to do with the claim of yours that I was responding to, which was that the conformal metric--the one you wrote in your OP--was "written in proper time". In the metric you wrote in your OP, coordinate time is not the same as proper time for comoving observers (observers with constant spatial coordinates). That is trivial to show from the line element you wrote.

The Rieman metric is a strictly mathematical construct with NO physical content

Really? That seems like an odd claim considering how often Riemannian and pseudo-Riemannian metrics are used in physics.

one can
write ANY ad hoc metric and plug it into Rieman and compute the Rieman components.

Yes, that's true. What does it have to do with the topic of this thread?

my claim is that: Einstein used Friedmann's metric in the period 1915-1922

Which still doesn't make sense since, as I said, Friedmann did not discover his metric until 1922.

 

[Kurvature]

Moderator's note: edited to clean up quote tags.

Why not?

[Kurvature]
Because there is obviously no such thing as a mass configuration that could
produce a universal time dilation such as a²(t) dt²
which appears in my OP metric. Are you serious?

Yes, you've said that. I wasn't claiming that it is.

[Kurvature]
The term "conformal metric" is highly misused. You have
to define what you mean by a conformal metric. I define it
as the metric given in my OP.

For comoving observers, yes, this is true, but it has nothing to do with the claim of yours that I was responding to, which was that the conformal metric--the one you wrote in your OP--was "written in proper time". In the metric you wrote in your OP, coordinate time is not the same as proper time for comoving observers (observers with constant spatial coordinates). That is trivial to show from the line element you wrote.

[Kurvature]
Nor is your statement correct that my metric is the FRW metric
written with "conformal time". They formally look the same
but they are vastly different.

Really? That seems like an odd claim considering how often Riemannian and pseudo-Riemannian metrics are used in physics.

[Kurvature]
"Odd", you got to be kiddin! So is the Pythagorean theorem, that doesn''t
mean the Pythagorean theorem is a physics principle, it's not,
it's a purely mathematical principle.

Yes, that's true. What does it have to do with the topic of this thread?

[Kurvature]
I've told you 4 times now, my OP metric is not the FRW metric written
with conformal time or any other physical metric, which does NOT mean
that I can't plug it into a purely mathematical formulla such as the Rieman
tensor and determine the components... which is what this thread is all
about. Why its about that, doesn't concern you.

Which still doesn't make sense since, as I said, Friedmann did not discover his metric until 1922.

[Kurvature]
Look, you've adequately demonstrated that you're an argumentative, counter productive
harrasser who will go to any lengths, usually even to criticizing punctuation, typos, or
in this case irrelevant dates, ANYTHING to harrass people. You make constant reference to
other people's "claims" which drives you into a writ of fellous jage as inspector Clousoue used
to say.. The Internet is swarming with people like you. I'm not going to respond
to your posts any further, you have nothing of any interest to say.

 

[Matterwave]

In GR, a conformal transformation is one which can be expressed as $g=\Omega^2 g$ where $\Omega$ is a scalar function and is called the conformal factor. Is the metric in the OP not a conformal transformation of some well known metric? If not, why then is it called the "conformal metric" according to you? If it is a conformal transformation from some well known metric (e.g. the FRW metric), then can you not simply find the Ricci and Riemann for that well known metric and find what they are after a conformal transformation? Wald has a section on conformal transformations in the appendix and in there you can find all the formulae for finding the curvature tensors after a conformal transformation has been performed.

 

[bahamagreen]

Just a guess...
For α=0
P^m_a,k coincides with P_k
For k=1
P^m_a,k(φ) = Δ_gφ - α<∇ƒ,∇φ> + ((nα-n+2)/(2(n-2)) x (αΔ_gƒ + ((nα-n+2)/(2(n-2))R(g))φ

 

[Kurvature]

In GR, a conformal transformation is one which can be expressed as $g=\Omega^2 g$ where $\Omega$ is a scalar function and is called the conformal factor. Is the metric in the OP not a conformal transformation of some well known metric? If not, why then is it called the "conformal metric" according to you? If it is a conformal transformation from some well known metric (e.g. the FRW metric), then can you not simply find the Ricci and Riemann for that well known metric and find what they are after a conformal transformation? Wald has a section on conformal transformations in the appendix and in there you can find all the formulae for finding the curvature tensors after a conformal transformation has been performed.

[Kurvature]
You're probably on to something, I may be banging my head against the wall for nothing.
The metric posted in my OP is easily recognizable as a simple conformal transformation
of the standard mundane spherically symmetric closed universe metric where k=1. Obviously
a(t) is the so-called "radius of the universe" and that terminology is used for this metric as well
as the FRW metric by everybody. We note in passing that the FRW metric is "similar but different"
from this conformal metric.
A closed spherically symmetric universe cannot have zero curvature whether it be Rieman,
Ricci,or scaler R. In fact, if a-dot and a-double-dot.are zero the curvature has to be proportional
to 1/a² since this is the fundamental (Gaussian) curvature of a sphere.
Therefore, I expect the Ricci scalar curvature to be 1/a² even in this 4 dimensional
case. But what I need to do is plug this metric into an algebraic curvature calculator and prove it
and check for any other miscellaneous non-zero components of.Rieman.
Finally, he would certainly think that the curvature tensors for the ordinary mundane plain-vanilla
spherical closed universe would be well known but I haven't seen them

 

[PeterDonis]

there is obviously no such thing as a mass configuration that could produce a universal time dilation such as a2(t) dt2 which appears in my OP metric.

You might want to calculate the Einstein tensor for this metric before making this claim, to see whether that Einstein tensor, when divided by $8 \pi G / c^4$, gives a reasonable stress-energy tensor. Maxima can do this for you.

Also, interpreting $a^2 dt^2$ as a "universal time dilation" may not be correct. $t$ is just a coordinate; it doesn't have to have any direct physical meaning. To extract the physical meaning of the metric you wrote down, you need to look at invariants.

You have to define what you mean by a conformal metric. I define it as the metric given in my OP.

Yes, I am fine with that.

Nor is your statement correct that my metric is the FRW metric written with "conformal time".

I didn't make such a statement. You have stated that the metric in your OP is only formally the same as the FRW metric written in a conformal coordinate chart. I am not disputing that.

that doesn''t mean the Pythagorean theorem is a physics principle, it's not

If you mean that no real triangle exactly satisfies the Pythagorean theorem, because real objects are not the same as idealized mathematical objects, then what you say is, strictly speaking, correct. But by this definition, no mathematical equation can be a "physics principle". Yet we use mathematical equations all the time in physics. How is that possible if they're not expressions of physics principles? Physical objects don't have to satisfy the mathematical equations exactly in order for the equations to express physics principles.

 

[Kurvature]

Moderator's note: edited to clean up quote tags.

You might want to calculate the Einstein tensor for this metric before making this claim, to see whether that Einstein tensor, when divided by $8 \pi G / c^4$, gives a reasonable stress-energy tensor. Maxima can do this for you.

[Kurvature]
Now that's interesting. The scalar Einstein tensor is equal to the scalar Ricci curvature which I've just found out is 6/a(t)².
for this metric. Setting that equal to the mass density ρ (low speed approximation) gives 6/a(t)² = kρ . So it does appear
that there is some kind of mass distribution (dust?) that will cause a true conformal expansion? Any further thoughts about that?

Also, interpreting $a^2 dt^2$ as a "universal time dilation" may not be correct. $t$ is just a coordinate; it doesn't have to have any direct physical meaning. To extract the physical meaning of the metric you wrote down, you need to look at invariants.

[Kurvature]
Umm... in the FRW metric the scale factor a(t) serves only to "stretch" or "dilate" tor "expand" he spatial dimesion,
but in the conformal (my OP) metric it is "stretching" or "dilating" or "expanding" the time dimension equally so. I don't
see why you wouldn't call that a "universal time dilation"?[/QUOTE]

 

[PeterDonis]

Setting that equal to the mass density ρ (low speed approximation) gives 6/a(t)2 = kρ .

I don't think you can assume that the low speed approximation is valid. That's equivalent to assuming that there is zero pressure, but if there is zero pressure, then every component of the Einstein tensor except the 0-0 component should be zero. Maxima should be able to tell you whether that's true or not (I don't think it is).

So it does appear that there is some kind of mass distribution (dust?)

Yes, it appears that there is a mass distribution (at least, it appears that there is for $k$ nonzero), but as above, I don't think it has zero pressure, so it would not be termed "dust".

in the FRW metric the scale factor a(t) serves only to "stretch" or "dilate" tor "expand" he spatial dimesion

In a particular coordinate chart, yes. But there is also a particular set of observers that that coordinate chart is adapted to, who have a particular physical property: they see the universe as homogeneous and isotropic. The time coordinate of this coordinate chart is the same as proper time for those observers; that is what makes the physical interpretation of this chart so straightforward.

in the conformal (my OP) metric it is "stretching" or "dilating" or "expanding" the time dimension

In the particular coordinate chart in which you wrote the metric, yes. And observers who are at rest in this chart will see the universe as homogeneous and isotropic. However, the time coordinate in this chart is not the same as proper time for those observers. So there isn't a direct physical interpretation of your conformal metric the way there is for the standard FRW metric. See below.

I don't see why you wouldn't call that a "universal time dilation"?

Before drawing any conclusions about "time dilation", I think you should compute the proper time for observers at rest in the chart in which you wrote your metric in the OP. That should be key to any physical interpretation of what the metric is telling you.

 

[Kurvature]

[Before drawing any conclusions about "time dilation", I think you should compute the proper time for
observers at rest in the chart in which you wrote your metric in the OP. That should be key to any
physical interpretation of what the metric is telling you.[/QUOTE]

[Kurvature]
You keep insisting there is no "universal time dilation" in the OP metric, but I actually have mathematical
evidence that there is.
Back on 8/13/2013 Mentz114 generously gave me an pro bono and erudite calculation that there is NO HUBBLE SHIFT
in the simple conformal metric: ds² = a(t)[dx²+dy²+dz²-dt²]
And the reason is that the universal time dilation of the metric slows the clocks down as the universal (FRW-type)
spatial expansion stretches the wavelengths coming from a distant star. Therefore Mentz114 certainly thought the metric
exhibits a "universal time dilation".
This thread and my exchange with Mentz114 can be seen at:
https://www.physicsforums.com/threads/is-there-a-redshift-in-a-conformally-flat-metric-space.705421/
UNFORTUNATELY, I made the mistake of blurting out to Mentz114 where the metric came from
(said blurt also to be seen at the above URL) and said I was going to publish his result and give him full credit at
which point he got so alarmed that I might be crazy that he told me not ot mention his name, "ever" and
the thread was closed for further comment shortly after!
So, I am treading on the same thin ice again as we speak.
But, you can see from that thread I do understand the the "fortuitous" fact that coordinate time
in FRW happens to be proper time, whereas I'm not sure it is in the OP. Whether or not this makes
any difference in my "applied physics application" of the result, I don't know... but I certainly will find
out sooner or later.
Meanwhile thanks a million for steering me to Maxima... its the greatest gadget I own, second only
to my '94 Merc !

 

[PeterDonis]

Mentz114 certainly thought the metric exhibits a "universal time dilation".

I don't think so. He did show that there is no redshift of light signals between comoving observers (observers at rest in the coordinates in which the metric is written), but that is not the same thing as "universal time dilation", and I don't think Mentz114 was claiming that it is.

thanks a million for steering me to Maxima

You're welcome! It does save a lot of drudgery in computation, and it's a lot more reliable at computing curvature tensors and such than I am.

 

[Kurvature]

I don't think so. He did show that there is no redshift of light signals between comoving observers (observers at rest in the coordinates in which the metric is written), but that is not the same thing as "universal time dilation", and I don't think Mentz114 was claiming that it is.

[Kurvature]
We don't think so. It certainly is evidence of a universal time dilation
in terms of coordinate time. If the distance between the star and the
observer is increasing with coordinate time due to the FRW-type space
expansion, then what else could possibly nullify the Hubble redshift?

 

[PeterDonis]

It certainly is evidence of a universal time dilation in terms of coordinate time.

But, once more, coordinate time is not a physical thing; it's just a label. The physics is contained in invariants--things that are independent of your choice of coordinates. If time dilation is present, there should be some invariant that shows it. Just looking at coordinate time is not enough.

 

[Kurvature]

But, once more, coordinate time is not a physical thing; it's just a label. The physics is contained in invariants--things that are independent of your choice of coordinates. If time dilation is present, there should be some invariant that shows it. Just looking at coordinate time is not enough.

[Kurvature]
The OP metric contains an obvious FRW-type spatial expansion which is known to be real.
Stars at enormous distance are moving away from us at enormous velocities because of it
which causes enormous red shifts. And yet Mentz114's calculation shows there is no
red-shift in the OP metric.
Everyone I've shown the result to immediately concurs that the
a(t)² dt²
term in the metric represents a universal time dilation (of ad hoc and mysterious origin)
which slows all the coordinate clocks thus nullifying the red-shift. I don't see what you
find so illogical in that observation?
On the other hand, your pointing out that OP coordinate time is not proper time
does both concern and baffle me. Are you saying that there really is a red-shift if
the observer uses proper time?

 

[PeterDonis]

Mentz114's calculation shows there is no red-shift in the OP metric.

I should clarify that I have not checked that calculation; I have only skimmed through the thread you linked to.

I don't see what you find so illogical in that observation?

Well, if we just look at coordinate time and the $a(t)$ factor, there is one obvious issue. $a(t)$, from the metric, gives the ratio of proper time to coordinate time at a given instant $t$ of coordinate time, i.e., an interval $dt$ of coordinate time corresponds to an interval $a(t) dt$ of proper time (for an observer at rest in these coordinates). But we do not know the functional form of $a(t)$. For all we know, $a(t)$ could be constant (indeed, that case is what you were asking about in the OP in this thread). Without knowing the functional form of $a(t)$, I don't see how we can know whether it describes "time dilation" or not.

(Note that similar remarks apply to the spatial part of the metric; you are saying the metric describes "expansion", but without knowing the functional form of $a(t)$, I don't see how we can know whether it does or not.)

Also, you appear to be assuming that $a(t)$ increasing with $t$ corresponds to "time dilation". But if $a(t)$ is increasing with $t$, then the amount of proper time elapsed in a given interval of coordinate time is increasing--i.e., that proper time clocks are "running faster" relative to coordinate time. "Time dilation" is usually used to describe a situation where proper time clocks are "running slower" relative to coordinate time.

Are you saying that there really is a red-shift if the observer uses proper time?

I have not done the calculation, so I can't say. But I think proper time is the appropriate standard to use, not coordinate time.