Can Empty Space Curve? FRW Cosmology Equations Explored

In summary, the equations for FRW cosmology can be simplified by setting the density and pressure to zero, resulting in a simplified form that depends on the value of k. For k=0, this leads to an ordinary Minkowski space metric. For k=-1, the solution describes a spatially hyperbolic universe with no energy/matter content, where spatial distances increase linearly in time, known as the Milne universe. This universe has zero spacetime curvature and curved 3-dimensional spatial sections. To determine the spacetime curvature, one can calculate the curvature tensor from the metric, which will be zero in inertial, spherical, and cosmological coordinates for the Milne universe. The introduction of mass (or its equivalent
  • #1
copernicus1
99
0
If I start with the standard FRW cosmology equations,
$${\eqalign{
3\dot a^2/a^2&=8\pi\rho-3k/a^2\cr
3\ddot a/a&=-4\pi\left(\rho+3P\right),}}$$
and set [/tex]\rho=P=0[/itex] (or $T^{\mu\nu}=0$), I have
$${\eqalign{
3\dot a^2/a^2&=-3k/a^2\cr
3\ddot a/a&=0.}}$$
The second equation gives $$\ddot a=0,$$ but $$\dot a$$ seems to depend on the value of k.

Namely, if I set k=0, then $$\dot a=0$$ and this leads to an ordinary Minkowski space metric. If I choose k=+1, then a is complex and that doesn't seem physical, but if I set k=-1, then I can get $$3\dot a^2/a^2=3/a^2~~~~\Longrightarrow~~~~\dot a=1~~~~\Longrightarrow~~~~a=t,$$ which I suppose describes a spatially hyperbolic universe (k=-1) with no energy/matter content, where spatial distances increase linearly in time.

Do we just ignore this solution based on the assumption that nonzero k implies the presence of mass/energy by definition, or have I gone wrong in my reasoning somewhere?
 
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  • #2
copernicus1 said:
if I set k=-1, then I can get $$3\dot a^2/a^2=3/a^2~~~~\Longrightarrow~~~~\dot a=1~~~~\Longrightarrow~~~~a=t,$$ which I suppose describes a spatially hyperbolic universe (k=-1) with no energy/matter content, where spatial distances increase linearly in time.

Do we just ignore this solution based on the assumption that nonzero k implies the presence of mass/energy by definition, or have I gone wrong in my reasoning somewhere?

This is the Milne universe, which has curved 3-dimensional spatial sections, and which also has zero spacetime curvature.

George Jones said:
Just a short, redundant comment for clarity.
nutgeb said:
For example, an empty FRW model is characterized by hyperbolic spatial curvature.

Here spacetime is flat and space is curved.

nutgeb said:
For a more plausible kinematic analysis, one might want to compare the supernova data to an FRW Omega=1 spatially flat dust model using light travel distance from time of emission.

Here spacetime is curved and space is flat.

Mass/energy (the stress-energy tensor) is the source of spacetime curvature. Spatial curvature is somewhat arbitrary, i.e., it depends on how the particular 3-dimensional hypersurfaces are chosen.

George Jones said:
You're talking about the Milne universe, which is a a patch of Minkowski spacetime in somewhat unusual coordinates.

Start with Minkowski spacetime in spherical coordinates,

[tex]
ds^2 = dt'^2 - dr^2 - r^2 d \Omega^2 ,
[/tex]

and make the coordinate transformation

[tex]
t' = t \cosh \chi
[/tex]

[tex]
r = t \sinh \chi.
[/tex]
 
Last edited:
  • #3
Awesome thanks. How can I tell what the spacetime curvature is, as opposed to just the spatial curvature?
 
  • #4
copernicus1 said:
Awesome thanks. How can I tell what the spacetime curvature is, as opposed to just the spatial curvature?

Calculate the spacetime curvature tensor (from the metric). Note that if all the components of a tensor are zero in one coordinate system, then they are all zero in all coordinate systems.

In this case, because the components of the Minkowski metric are constant in an inertial coordinate system, the components of the curvature tensor are all zero in inertial coordinates, are zero in spherical coordinates, and are all zero in cosmological coordinates for the Milne universe.
 
  • #5
Thanks this is very helpful.
 
  • #6
Without going through the mathematics as posted above, it would appear that a theoretical totally empty universe with nothing in it would have no mass to "bend" it. Once light is introduced to observe this, a theoretical "mass equivalent" m=E/c^2 is brought in, which would change that and curvature would be effected.
 

1. Can empty space curve?

According to the theory of general relativity, empty space can indeed curve. This is because space and time are not separate entities, but rather part of a four-dimensional space-time fabric that can be affected by the presence of mass and energy. The curvature of this fabric can be observed through the effects of gravity.

2. How is FRW cosmology related to the curvature of space?

FRW cosmology, or the Friedmann–Lemaître–Robertson–Walker model, is a mathematical framework used to describe the large-scale structure and evolution of the universe. It is based on the assumption that the universe is homogeneous and isotropic, meaning that it looks the same in all directions and at all points in time. The equations used in FRW cosmology take into account the curvature of space and the effects of gravity on the expansion of the universe.

3. What are the FRW cosmology equations?

The FRW cosmology equations are a set of equations derived from Einstein's field equations of general relativity. They describe the expansion of the universe by taking into account the curvature of space and the amount of matter and energy present in the universe. These equations are used to study the history and fate of the universe.

4. How does FRW cosmology explain the expansion of the universe?

FRW cosmology explains the expansion of the universe through the concept of the Hubble parameter, which describes the rate at which the universe is expanding. This parameter is affected by the amount of matter and energy present in the universe, as well as the curvature of space. The equations used in FRW cosmology show that the expansion of the universe is accelerating, and this is attributed to the presence of dark energy.

5. What are the implications of FRW cosmology on our understanding of the universe?

The FRW cosmology equations have helped us gain a better understanding of the universe and its evolution. They have allowed scientists to make predictions about the age, size, and fate of the universe based on observations and measurements of its properties. Additionally, FRW cosmology has led to the discovery of dark energy, which has revolutionized our understanding of the forces that govern the behavior of the universe.

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