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Yes. All the standard cosmological solutions are homogeneous and isotropic.Smattering said:Is there a solution of the Einstein field equations for a completely homogeneous universe where all the mass/energy is uniformly distributed?
Yes. All the standard cosmological solutions are homogeneous and isotropic.Smattering said:Is there a solution of the Einstein field equations for a completely homogeneous universe where all the mass/energy is uniformly distributed?
Smattering said:as ##r=0## is not part of the domain, the metric is simply undefined at ##r=0##.
Smattering said:Is there a solution of the Einstein field equations for a completely homogeneous universe
bcrowell said:Yes. All the standard cosmological solutions are homogeneous and isotropic.
PeterDonis said:As bcrowell says, the standard cosmological solutions are homogeneous and isotropic. However, "homogeneous" here means spatially homogeneous--more precisely, there is a family of observers in these spacetimes (these observers are called "comoving" observers) to whom, at every instant of their time, the spatial slice of the universe at that time is homogeneous (and isotropic).
Smattering said:the cosmological solutions are just ignoring any local deviations and treat the universe as entirely homogeneous not only on cosmic scales but also locally?
PeterDonis said:If by "homogeneous" you mean "homogeneous in space and time", i.e., the density of mass/energy is the same everywhere and also at all times, then there is a solution with that property called the Einstein static universe. However, this solution is unstable against small perturbations, like a pencil balanced on its point; any tiny fluctuation in the density of mass/energy anywhere in the universe will cause it to either expand or collapse.
Smattering said:does it imply that in an entirely homogeneous and isotropic universe, we could fine-tune the cosmological constant such that the curvature of spacetime completely disappears and it behaves like Minkowski spacetime?
PeterDonis said:No. The only way to get flat Minkowski spacetime is to have zero density of mass/energy everywhere and zero cosmological constant everywhere.
The Einstein static universe is still a curved spacetime, even though it doesn't expand or contract with time. One manifestation of this is that spatially, the Einstein static universe is a 3-sphere, not a Euclidean 3-space.
Smattering said:O.k., then when they say that *our* universe is approx. flat on large scales, this only means that our universe is too large to measure the curvature with the available methods?
The Einstein field equations relate the curvature to the stress-energy tensor. The cosmological constant can be treated as one term in the stress-energy tensor. There are other terms as well, such as a term for dark matter and one for baryonic matter. Given the curvature of a manifold, there are theorems that in some cases uniquely determine the topology that is consistent with that curvature. This is one such case.Smattering said:So which property leads to this topology? Is it the homogenity and isotropy, or is it the cosmological constant, or is it a combination of both?
bcrowell said:When people say that our universe is approximately flat on large scales, they're referring to *spatial* flatness only. The Riemann tensor measures the curvature of spacetime, not just space. Our universe's spacetime is not even approximately flat. Spacetime curvature is how GR describes gravity. A universe with flat spacetime would be one in which there are no gravitational effects whatsoever.
The Einstein field equations relate the curvature to the stress-energy tensor. The cosmological constant can be treated as one term in the stress-energy tensor. There are other terms as well, such as a term for dark matter and one for baryonic matter. Given the curvature of a manifold, there are theorems that in some cases uniquely determine the topology that is consistent with that curvature. This is one such case.
Smattering said:I was aware that our universe can certainly not be completely flat due to the fact that we can observe lots of gravitational effects all around us. The question was only related to the curvature on very large scales.
Smattering said:otherwise the geodesic line of an object would be independent of its relative speed
Smattering said:I have to confess that until now I would have thought that a curvature of spacetime would also imply a spatial curvature in most cases.
PeterDonis said:I'm not sure I understand what you mean by this. If you are trying to describe geodesic deviation--the fact that geodesics that are parallel at some particular point don't stay parallel--then yes, that is the definitive sign of spacetime curvature.
Whether space is curved depends on how we split up spacetime into space and time (see above). There is no absolute sense in which space is curved or flat.
Smattering said:Let's assume I am throwing stones in a vacuum (jsut to get rid of aerodynamic effects). Let's further assume that I throw all the stone in the eaxctly same direction and angle:
Smattering said:If the curvature was purely spatial, then I would expect all the stones to follow the exactly same trajectory indepently of their initial velocity, wouldn't I?
Smattering said:in reality, the trajectories differ with the initial velocity.