# Different Kinds of Metric

1. Mar 2, 2012

### waterfall

Are there different kinds of metric in GR? For instance. I read in http://www.astronomy.ohio-state.edu/~dhw/A682/notes3.pdf [Broken] there the FRW Metric is about:

"In 1917, Einstein introduced the first modern cosmological model, based on GR, in which the spatial metric is that of a 3-sphere:"

Why. Can the GR metric be based on 2D, 3D, 3-sphere, 3-cube? Can you give other examples?

What would happen to the EFE if you change the metric?

Btw.. is the FRW metric about space or about spacetime?

Last edited by a moderator: May 5, 2017
2. Mar 2, 2012

### tom.stoer

There are two ways to classify different metrics:
(1) different solutions
(2) different coordinates

Think about the equation ax²+bx+c=0. There are two solutions x1,2 (this corresponds to 1) and of course you can make a change of coordinates x'=x-a (this corresponds to 2)

3. Mar 2, 2012

### waterfall

Let's take the example of the FRW Metric. How does it differ to the normal GR metric? I think the FRW metric is about space, while the GR metric is about spacetime. Is this distinction correct?

4. Mar 2, 2012

### tom.stoer

no;

and what is "the normal GR metric"?

5. Mar 2, 2012

### waterfall

I think the normal GR metric involves 4-D spacetime (mathematical space) with differential manifold. While the FRW metric is about physical space (the space where we live). If not, what is the distinctions between the two?

6. Mar 2, 2012

### waterfall

And there's the Schwarzschild metric which I think is 4D. I think what's unique about the FRW metric is its 3D or our space compared to others.

7. Mar 2, 2012

### Matterwave

The FRW metric is a 4-D metric. The source you cited is only talking about "the spatial part" of the metric.

8. Mar 2, 2012

### waterfall

Let's take this definition at wiki for a good start:

"In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric. When a topological space has a topology that can be described by a metric, we say that the topological space is metrizable.

In differential geometry, the word "metric" is also used to refer to a structure defined only on a differentiable manifold which is more properly termed a metric tensor (or Riemannian or pseudo-Riemannian metric)."

I'd like to know what is the difference between the FRW Metric in the universe and say the Schwarzschild metric in black hole. Just an intuitive grasp or simple distinctions will do.

9. Mar 2, 2012

### waterfall

I have an intuitive grasp of what are manifolds.. which is map without coordinates or more of topology. So I guess the GRW metric and Schwarzschild metric is more about topographic or the shape of the manifold?

10. Mar 2, 2012

### Matterwave

A particular metric gives you the distance between infinitessimally close points. The FRW and Schwarzschild metrics are two different metrics which are both solutions for the EFE's for two different conditions. Schwarzschild metric is a so called vacuum solution which is valid outside a spherical mass distribution. FRW metric is a solution for a homogeneous and isotropic distribution of dust (perfect fluid).

11. Mar 2, 2012

### waterfall

So both FRW and Schwarzschild metrics are 4D? I thought the FRW metric is like the metric used in say defining a chair.. or 3D metric.. what do you call 3D metric then?

12. Mar 2, 2012

### Matterwave

The FRW metric is definitely 4-D.

$$ds^2=-c^2dt^2+a(t)^2(\frac{dr^2}{1-kr^2}+r^2d\Omega^2)$$

Notice the presence of dt in there.

13. Mar 2, 2012

### waterfall

Ok thanks I guess the opening paper confused me when it said "In 1917, Einstein introduced the first modern cosmological model, based on GR, in which the spatial metric is that of a 3-sphere:"

How about a black hole, is its spatial metric also that of a 3-sphere? What other examples where the spatial metric is not a 3-sphere?

14. Mar 2, 2012

### Fredrik

Staff Emeritus
Every solution of Einstein's equation is a metric for a 3+1-dimensional spacetime. The FLRW metrics are the ones that are consistent with the assumption that spacetime can be "sliced" up into a 1-parameter family of 3-dimensional spacelike hypersurfaces that are homogeneous and isotropic. These hypersurfaces can be thought of as "space, at different times".

The metric of spacetime induces a Riemannian metric on each of these hypersurfaces. It's convenient to use these metrics to distinguish between the main classes of FLRW solutions.

15. Mar 2, 2012

### waterfall

But from linearized gravity concept (where curved spacetime = flat spacetime + spin-2 fields) which you are familiar with in our discussions in the other threads. FRW metric can be modelled as Flat spacetime + spin-2 fields. Since the flat spacetime has a version called Milne universe. Then FRW Metric is really just Milne Universe + Spin-2 fields. Do you agree or object to this and why?

16. Mar 2, 2012

### Nabeshin

The spatial metric is certainly not a 3-sphere for a black hole! For a schwarzschild hole,
$$dl^2=\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2d\Omega^2$$

In the case of negative curvature of the universe, the spatial metric is not a 3-sphere either. In the case of a flat universe, the spatial metric is not a 3-sphere. The spatial part of the Alcubierre metric is not a 3-sphere. I could keep going.

17. Mar 3, 2012

### tom.stoer

flat spacetime + spin-2 field ist not applicable here!

18. Mar 3, 2012

### waterfall

Why? Please tell me the reason so I can sleep at night because I've been thinking of this for many days. Thanks.

19. Mar 4, 2012

### tom.stoer

There are three classes of FRW metrics, namely k=±1,0; only k=0 corresponds to a flat spacetime. You cannot describe a globally (topologically) different spacetime e.g. the closed k=+1 type by considering a flat k=0 type plus small oscillations.

Think about a function (for k=0)

f(t) = ωt

and try to find a small fluctuation that deformes it to (for k=+1)

f(t) = sin(ωt)

This is what you need to deform the k=0 to the k=+1 type. Of course this is impossible with a small fluctuation.

20. Mar 4, 2012

### waterfall

So the solution to this is that Strings don't need to first be Ricci-flat and add spin-2 field. The solution is to directly solve for the FRW from first principle without going to this Ricci-flatness?

The FRW thing also proves that the plain particle physics approach of using flat spacetime plus spin-2 fields = curved spacetime wont work. Therefore at least String theory and LQG need to be the minimum approach of quantum gravity?

21. Mar 4, 2012

### tom.stoer

When consideruing classical FRW I was talking about classical GR; it shows that even classical "flat spacetime + small oscillations" does not work for all classes of solutions of GR. It's the same with strings: they find a huge number of classes of "classical solutions" and consider small oscillations for each class. The big difference is that for different "classical solutions" one has to consider different oscillations, i.e. the particle content is different.

It's hard to compare ST and LQG here.

ST always uses these "classical solutions" and constructs a quantum theory on top of them. So for each class you get a quantum theory, but in a sense you don't have the big picture, i.e. no unique quantum theory (string theory) describing all these classes at once.

In LQG it's exactly the other way round. They claim to have a unique quantum theory of gravity (w/o particle physics) w/o the need to introduce these classes. But at the samepoint this is the weak point of LQG b/c they are not able to derive the classical solutions, i.e. they don't have a way to extract the classical physics with different spacetimes we know from GR (there are only very restricted models or methods from which they get rather specific spacetimes).

22. Mar 4, 2012

### waterfall

Are you saying that at this point in time in our String Theory. We still don't know how to even derive at the FRW metric using Perturbative String Theory and it can only be solved by going into the ongoing non-perturbative AsD/CFT path?

23. Mar 4, 2012

### tom.stoer

The FRW metric can definitly not be derived using flat space + perturbative strings; I don't know which realistic vacuum solutions have been constructed so far, but AdS is not the right way to go b/c AdS is not realistic; with a positive cosmological constant one needs dS instead, for FRW w/o cc again something different.

Last edited: Mar 4, 2012
24. Mar 4, 2012

### waterfall

atyy, I can't send private message to you because you somehow disable the feature. I just wanted to ask you what exact pages in the MTW book Gravitation one can find the statements that the FRW Universe satisfy harmonic coordinates and that all spacetime that is covered by harmonic coordinates can be decomposed into flat spacetime + spin-2 field. Exact pages please. Thanks.

25. Mar 4, 2012

### atyy

The reference I gave was Weinberg, not MTW. But in fact, I misread it, so I don't know if the FRW solution can be expressed in harmonic coordinates. Also, it seems that in 1963, Feynman was unsure if gravity as a field on flat spacetime could handle cosmology http://books.google.com/books?id=tnpOP4JH-J8C&source=gbs_navlinks_s (p187).

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