# Is the Universe spinning?

1. Jul 15, 2013

### Orion1

According to my understanding of the LCDM Friedmann–Lemaître–Robertson–Walker metric, the Universe under such a metric model is homogeneous and isotropic, and does not have any spin rotation.

Friedmann–Lemaître–Robertson–Walker metric:
$$c^{2} d\tau^{2} = -c^{2} dt^2 + a(t)^2 \left(\frac{dr^2}{1 - k r^2} + r^2 d\theta^2 + r^2 \sin^2 \theta \; d\phi^2 \right)$$

Galaxies spin, stars spin, planets spin, and particles spin. So, why not the whole Universe?

Why would the Universe Big Bang not have any strong angular momentum and rotational energy on a vast scale?, or at least powerful vortices?

Reference:
Friedmann–Lemaître–Robertson–Walker metric - Wikipedia
Solving Einstein's field equations - Universe - Wikipedia
Detection of a Dipole in the Handedness of Spiral Galaxies with Redshifts z ~ 0.04 - Longo - Department of Physics, University of Michigan
Is the Universe Spinning? - news.discovery.com

2. Jul 15, 2013

### Mordred

Although there have been some anomalies of motion, such as the one you mentioned. On the larger scale, the observations do not suggest that the universe is spinning. Or rather its not conclusive enough to say that it is or isn't with any degree of certainty. This is also a question that crops up from time to time. If you search the forum for "dark flow" you many also find some decent discussion. It is possible however we still don't have the conclusive evidence one way or the other.

Coincidentally the metric you posted doesn't support a spinning universe. The only link you supplied as support is the pop media article and the arxiv article. I seem to recall a counter paper to this but have been unable to locate it thus far, currently seeing if I still have it in my archive

Last edited: Jul 15, 2013
3. Jul 15, 2013

### Mordred

It may bve interesting to note one of the authors later papers. In the one you posted the findings challenge the cosmological pinciple as a rotation denotes a preferred location ie not all locations would observe the same thing. However this later paper by the same author concludes that the universe is at sufficient scales homogeneous and isotropic.

http://arxiv.org/ftp/arxiv/papers/1305/1305.0464.pdf

4. Jul 15, 2013

### WannabeNewton

You can have a homogenous rotating universe (Godel universe) but it won't be isotropic. Observers at every location will observe the same anisotropy due to the preferred direction in space. Anyways, the comoving observers in the FLRW metric clearly undergo no rotation because their 4-velocity field $\xi^{a}$ is hypersurface orthogonal to a one-parameter family of space-like hypersurfaces (the homogenous and isotropic time slices of the universe, with time referring to proper time according to the comoving observers) and the existence of such a foliation requires that $\xi^{a}$ have no rotation. More explicitly, $\omega^{a} = \epsilon^{abcd}\xi_{b}\nabla_{c}\xi_{d}$ is the twist of $\xi^{a}$; now $\epsilon_{ajkl}\omega^{a} = -6\xi_{[j}\nabla_{k}\xi_{l]} = 0$ because of hypersurface orthogonality so $\epsilon^{ejkl}\epsilon_{ajkl}\omega^{a} = -6\omega^{e} = 0$ i.e. the comoving observers have vanishing twist and, as a result, the rotation $\omega_{ab} = \epsilon_{abcd}\xi^{c}\omega^{d} = 0$ as well.

We had a thread on the exact same topic a while back: https://www.physicsforums.com/showthread.php?t=696477

5. Jul 16, 2013

### Orion1

Godel metric in Minkowski coordinates: (ref. 1)
$$ds^2= \frac{1}{2\omega^2} \, \left( -\left( dt + \exp(x) \, dz \right)^2 + dx^2 + dy^2 + \frac{1}{2} \exp(2x)\, dz^2 \right)$$

The Godel metric listed on Wikipedia is in Minkowski coordinates. For my tensor examination, I require it to be in spherical coordinates, I could not locate a spherical coordinate form in Google search engine, so I attempted the transformation myself:

Godel metric in spherical coordinates: (ref. 2)
$$c^{2} d\tau^{2} = \frac{1}{2 \omega^2} \left( -c^2 dt^2 + dr^2 + r^2 d\theta^2 - \frac{e^{2r} r^2 \sin^2 \theta}{2} \; d \phi^2 - 2c e^{r} r \sin \theta \; d\phi \; dt \right)$$

Is this transformation correct?

Reference:
Godel metric - Wikipedia
Metric tensor - General Relativity - Wikipedia

Last edited: Jul 16, 2013
6. Jul 16, 2013

### Chalnoth

These things (except particles) tend to spin noticeably because they have collapsed from much larger clouds of essentially randomly-moving particles. A tiny bit of random net rotation of a larger cloud turns into a much more noticeable rotation after collapse.

Our universe as a whole hasn't collapsed: it's been expanding.

Is there some net rotation? Probably.* But our universe is so large, and getting larger all the time, that there's basically no way for it to be noticeable.

* One caveat here is that relativity may potentially make this issue moot. The problem is that if you have a spinning object, it tends to drag space around it. So if the whole universe has some net rotation, will it just drag space around it in such a way that that rotation is completely undetectable? I think it might, but I'm not certain.

7. Jul 16, 2013

### Mordred

wow its been awhile since I've seen the Goedel form without Lambda.

I had to dig deep into my archive for this article lol. You will probably find everything you need in this paper. By the way this includes the Goedel form with the cosmological constant.

http://arxiv.org/pdf/gr-qc/0511015v1.pdf

edit: lol just noticed its referenced on the Goedel wiki reference page as well. wuld have saved me some hassle ah well

Last edited: Jul 16, 2013
8. Jul 17, 2013

### Orion1

According to (ref. 1, pg. 3, eq. 2.3) and (ref. 1, pg. 5, eq. 2.15) and (ref. 2, pg. 3, eq. 1), $a$ is a constant defined as:
$$a^2 = \frac{1}{2 \omega^2} = - \frac{1}{2 \Lambda}$$

The criteria between $\omega$ (angular velocity) and $\Lambda$ (cosmological constant) for these Godel metric models:
$$\omega^2 = - \Lambda$$

Godel metric in spherical coordinates and with cosmological constant:
$$c^{2} d\tau^{2} = - \frac{1}{2 \Lambda} \left( -c^2 dt^2 + dr^2 + r^2 d\theta^2 - \frac{e^{2r} r^2 \sin^2 \theta}{2} \; d \phi^2 - 2e^{r} r \sin \theta \; c \; dt \; d\phi\right)$$

Is this transformation correct?

Reference:
Godel's Metric And Its Generalization - Ozgoren
Reflections on Kurt Gödel - Wang

9. Jul 17, 2013

### Orion1

Wikipedia Godel metric...

The Godel metric given by Wikipedia:
$$ds^2 = \frac{1}{2 \omega^2} \, \left( -\left( dt + \exp(x) \, dz \right)^2 + dx^2 + dy^2 + \frac{1}{2} \exp(2x)\, dz^2 \right)$$

The Godel metric given by (ref. 1, pg. 3, eq. 2.2)
$$ds^2 = a^2 \left[ -(dx_0 + e^{x_1}dx_2)^2 + dx_1^2 + \frac{e^{2x_1}}{2} dx_2^2 + dx_3^2 \right]$$

If I overlay coordinates $(x_0,x_1,x_2,x_3)$ with $(t,x,y,z)$ the result is:
$$ds^2 = a^2 \left[ -(dt + e^{x}dy)^2 + dx^2 + \frac{e^{2x}}{2} dy^2 + dz^2 \right]$$

The metrics do not appear to match, please clarify?

Reference:
Godel's Metric And Its Generalization - Ozgoren

10. Jul 17, 2013

### Staff: Mentor

Transpose y and z, and look again.

11. Jul 17, 2013

### Mordred

Currently working out in the field typing from my phone. There are a couple of properties of the Godel universe you should also consider.

1) universe is static
2) there are no timelike closed null geodesics.
3) the frame dragging and its lightcone path effects.

“the compass of inertia”. this is a key aspect of the Godel universe

another key property concerning light paths quoted from the link I provided.

Godel’s rotating cosmological model [12], [13] is one of the most interesting solutions
of Einstein’s field equations with negative Lambda-constant, particularly in view of its
contribution to our understanding of rotation in relativity and its signs of causality
breakdown due to the existence of closed timelike curves

later on

Godel’s stationary solution of Einstein’s field equations with cosmological constant
describes the gravitational field of a uniform distribution of rotating dust matter, where
- loosely speaking - the gravitational attraction of matter and the added attractice force
of a negative lambda-constant is compensated by the centrifugal force of rotation

a lot of the above is shown in the Ricci tensers. Just some key aspects to look at and consider. Looks like Nugatory is helping with the metrics your working on

It will be interesting to see how you plan on incorperating expansion into a static model witrh frame dragging

Wish I could help more than that but work calls.

Last edited: Jul 17, 2013
12. Jul 17, 2013

### WannabeNewton

The godel universe is not static, it is only stationary. Stationary means the space-time has a time-like killing vector field and static implies the time-like killing vector field is hypersurface orthogonal; the latter condition would actually obstruct the global time-like congruence represented by the dust particles from rotating. It can be shown that there exists no static solution to the Einstein field equations generated by dust.

13. Jul 17, 2013

### Mordred

ah your correct I misread the one paper,

edit: I located another reference that may be handy, I particularly like the visualizations presented in this paper. The causality violations of the Godel universe is also intriging. When I get more time I will probably look deeper into the metrics. It will give me something new to study. Need a break from perturbation field self studies lol

http://arxiv.org/pdf/1303.4651v1.pdf

however my question still stands particularly in regards to this quote from the reference I posted.

"In his paper [8] Godel admitted that his rotating universe cannot serve as a model of
the universe we live in since it does not contain any redshift for distant objects accounting
for an expansion as required by Edwin Hubble’s law of 1929"

Last edited: Jul 17, 2013
14. Jul 17, 2013

### Mordred

Found the paper with the referenced quote above.

http://fuchs-braun.com/media/91ac4f6879c27351ffff8191blackf0.pdf

As the the physical meaning of the solution proposed
in this paper, it is clear that it yields no red shift for
distant objects. For, by using the transformation (I)
defined in the proof of the properties (1) and (2), one
proves immediately that light signals sent from one
particle of matter (occurring in the solution) to another
one arrive with the same time intervals in which they
are sent.

15. Jul 17, 2013

### WannabeNewton

If you're interested in looking at the differential geometry of the Godel space-time, see chapter 3 of Malament's notes (which were turned into a text on the foundations of general relativity): http://www.socsci.uci.edu/~dmalamen/bio/GR.pdf

16. Jul 17, 2013

### Mordred

Thanks I'll definetely look through it, at a quick glance it looks like a great article to study.

17. Jul 17, 2013

### WannabeNewton

I hope you enjoy it. I recently bought the aforementioned text and I love it thus far. Rotation in general relativity is quite a deep subject area.

18. Jul 17, 2013

### Orion1

Transpose y and z and the metrics are equivalent.

Are such coordinate transpositions allowed in General Relativity metrics, and therefore both metrics from post #9 are correct? or is my symbolic coordinate substitution method and interpretation incorrect?

Last edited: Jul 17, 2013
19. Jul 17, 2013

### Staff: Mentor

Yes. x, y, z, and t are just variable names with no significance until you plug them into a metric or use them in a coordinate transformation. You just have to be consistent in how you use them.

It's fine. Somewhere back in this thread you said "If I overlay coordinates (x0,x1,x2,x3) with (t,x,y,z)...."; all you needed to do was to overlay (t,x,z,y) instead. You could have overlaid "dog", "cat", "sheep", and "cow" if you had wanted.

20. Jul 17, 2013

### Oldfart

If the universe is found to be spinning, would that be evidence that the universe is not infinite?

21. Jul 17, 2013

### WannabeNewton

Not necessarily. For example the Godel universe is infinite and rotating.

22. Jul 18, 2013

### Mordred

Just a side note curiosity, I wonder how close the characteristics between the Godel universe and Poplowskii's torsion, and spinor characteristics match up in similarity. Its another model that can have rotation related characteristics.

http://www.physics.indiana.edu/~nipoplaw/publications.html [Broken]

its been awhile since I read any of his papers, some of his metric applications I found intriguing at the time.

Last edited by a moderator: May 6, 2017
23. Jul 18, 2013

### Chronos

24. Jul 18, 2013

### Orion1

Godel metric in spherical coordinate form...

Godel metric in Minkowski coordinate form $(dt,dx,dy,dz)$:
$$ds^2 = a^2 \left[ -(dt + e^{x}dy)^2 + dx^2 + \frac{e^{2x}}{2} dy^2 + dz^2 \right]$$

Godel metric in expanded Minkowski coordinate form $(dt,dx,dy,dz)$:
$$ds^2 = a^2 \left[ -dt^2 + dx^2 - \frac{e^{2x}}{2} dy^2 + dz^2 - 2 e^x \; dt \; dy \right]$$

Godel metric in spherical coordinate form $(dt,dr,d\theta,d\phi)$:
$$c^{2} d\tau^{2} = - \frac{1}{2 \Lambda} \left( -c^2 dt^2 + dr^2 - \frac{e^{2r} r^2}{2} \; d \theta^2 + r^2 \sin^2 \theta \; d\phi^2 - 2 r e^{r} \; c \; dt \; d\theta \right)$$
Is this transformation correct?

Reference:
Godel metric - Wikipedia
Metric tensor - General Relativity - Wikipedia
Godel's Metric And Its Generalization - Ozgoren

Last edited: Jul 18, 2013
25. Jul 18, 2013

### Mordred

I don't see a problem with that derivative, however Nugatory is far better at calculus than I. Some things I have to practice more often lol.