# Different Kinds of Metric

by waterfall
Tags: kinds, metric
 P: 381 Are there different kinds of metric in GR? For instance. I read in http://www.astronomy.ohio-state.edu/...682/notes3.pdf 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?
 Sci Advisor P: 5,162 There are two ways to classify different metrics: (1) different solutions (2) different coordinates Think about the equation ax˛+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)
P: 381
 Quote by tom.stoer There are two ways to classify different metrics: (1) different solutions (2) different coordinates Think about the equation ax˛+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)
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?

P: 5,162

## Different Kinds of Metric

no;

and what is "the normal GR metric"?
P: 381
 Quote by tom.stoer no; and what is "the normal GR metric"?
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?
 P: 381 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.
 P: 1,937 The FRW metric is a 4-D metric. The source you cited is only talking about "the spatial part" of the metric.
 P: 381 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.
P: 381
 Quote by 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.
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?
 P: 1,937 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).
P: 381
 Quote by 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).
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?
 P: 1,937 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.
P: 381
 Quote by 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.
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?
PF Patron
Emeritus
P: 8,837
 Quote by waterfall So both FRW and Schwarzschild metrics are 4D?
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.
P: 381
 Quote by 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).
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?
P: 2,193
 Quote by waterfall 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?
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.