What is Rieman for a conformal metric?

1. Mar 13, 2015

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.

2. Mar 13, 2015

Matterwave

3. Mar 13, 2015

Kurvature

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......

4. Mar 14, 2015

Matterwave

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

5. Mar 14, 2015

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.

6. Mar 14, 2015

Kurvature

P.S. :
Lest someone think Im 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.

7. Mar 14, 2015

Staff: Mentor

Maxima is free:

http://maxima.sourceforge.net/

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

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.

8. Mar 14, 2015

Staff: Mentor

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

9. Mar 14, 2015

Kurvature

[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
Rtt is proportional to k/a2

10. Mar 14, 2015

Kurvature

[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.

11. Mar 14, 2015

Matterwave

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

12. Mar 14, 2015

Kurvature

[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.

13. Mar 14, 2015

Staff: Mentor

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.

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

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.

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.

14. Mar 14, 2015

Kurvature

Moderator note: post edited to clean up quote tags.

[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.

[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

[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.

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.

Last edited by a moderator: Mar 14, 2015
15. Mar 14, 2015

Staff: Mentor

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.

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.

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.

16. Mar 14, 2015

Kurvature

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

[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.

[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.

[Kurvature]
Why it is of interest to me is a matter over your head

Last edited by a moderator: Mar 14, 2015
17. Mar 14, 2015

Staff: Mentor

Why not?

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

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.

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

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

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

Last edited: Mar 15, 2015
18. Mar 15, 2015

Kurvature

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

[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?

[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.

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

[Kurvature]
"Odd", you gotta 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.

[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

[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.

Last edited by a moderator: Mar 15, 2015
19. Mar 15, 2015

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.

20. Mar 15, 2015

bahamagreen

Just a guess...
For α=0
Pma,k coincides with Pk
For k=1
Pma,k(φ) = Δgφ - α<∇ƒ,∇φ> + ((nα-n+2)/(2(n-2)) x (αΔgƒ + ((nα-n+2)/(2(n-2))R(g))φ

21. Mar 15, 2015

Kurvature

[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

22. Mar 15, 2015

Staff: Mentor

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.

Yes, I am fine with that.

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.

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.

23. Mar 16, 2015

Kurvature

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

[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?

[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]

Last edited by a moderator: Mar 16, 2015
24. Mar 16, 2015

Staff: Mentor

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).

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 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 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.

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.

25. Mar 17, 2015

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: