# Flat Universe and Gravity

1. Sep 11, 2007

### PhanthomJay

It seems to me that current thinking leans toward an expanding and accelerating universe that is neither positively nor negatively curved, but rather, essentially, 'Flat'. How does this thinking not contradict with the fact that gravity, which surely exists, is the curvature of spacetime in the presence of mass? In other words, how can a universe be flat if we know, from gravity, that it is curved??

2. Sep 11, 2007

### Wallace

Here is an analogy that may help you understand this. Consider the following situation: you are located at equal distance between two equal mass objects and the positions of you and the three objects all lie along a straight line. What is the gravitational force on you? (note that the answer is the same regardless of whether you consider Newtonian Gravity, General Relativity or any other theory of gravity). This demonstrates that in principle multiple obejcts can curve space-time but the net result for some situations is the same as space-time with nothing in it.

This is not quite the case for a 'flat' universe. The solution to General Relativity for a flat expanding universe (solutions to the equations are known as 'metrics') is in general different from the metric of a space-time with nothing in it (empty, special relativistic space-time), however the 'spatial' part of the metric is flat, time is however curved. Whether the universe is flat, closed or open depends on the average energy density. To be flat the Universe must have a precise 'critical' density of total energy. Any less and the universe is open and more makes it closed.

Note that a closed universe does not neccessarily collapse to a big crunch in the future and nor does a a flat or open universe neccessarily expand forever, it depends on the energy content. If dark energy, such as a cosmological constant or something like it exists (at it appears to at this stage) then the universe will continue to expand at an increasing rate forever, even if it was 'closed' (slightly less dense than the critical density) as is possible given current data which indicates that the universe is quite close to being flat but may be slightly curved one way or the other.

3. Sep 12, 2007

### PhanthomJay

OK, thanks, the analogy leads me to believe that spacetime is curved near mass, and when I find myself in between 2 equal masses and equidistant from each, that I'm simply on the 'top' of the curve above the 2 valleys, that is, on a 'horizontal' tangent to the curve; in this respect, my spacetime is flat. I suppose, then, that with mass scattered throughout the universe, that the net effect of these cancelations result in the overall flatness of space? But this brings me to another question regarding 'local' curvature, which i will post separately.

4. Sep 13, 2007

### Wallace

Just don't take my little analogy too literally, the full detail is more complex than that as for instance you can have material scattered evenly throughout the Universe but that alone does not lead to spatial flatness. It turns out you have to have the correct average density of material, as well as being distributed evenly.

This means that in fact in almost every location in our Universe space is not flat in that for instance the curvature of space-time produced by the Sun keeps the Earth in orbit and that made by the Earth keeps us from floating away. When we say 'the Universe is flat' this means that on large scales, once the mass has been smoothed then space is flat.

5. Sep 13, 2007

### hellfire

This view is not correct because it would imply that every model with homogeneous and isotropic distribution of mass density is flat.

The curvature of spacetime is determined by the energy-momentum tensor (the distribution and flows of energy and momenta) of matter. If there is some matter content, the curvature of spacetime will be different from zero and spacetime will be curved. Flat spacetime has zero curvature and you can get it as a possible solution to the equations of general relativity without any matter content.

However, even with matter content flat spacetime can be used to describe physics with a negligible gravitational force. For example, the scattering experiments in the standard model of particle physics. Gravitation is not considered in this model due to its feebleness and, therefore, spacetime can be assumed to be flat. You have to note also that space is always flat in a flat spacetime.

But a flat space does not mean that spacetime is flat. The best example is cosmology. There gravitation plays a fundamental role and cannot be neglected. Matter content different from zero will imply a curved spacetime. However, you can have models with matter content, curved spacetime, but flat space. In the FRW cosmological models the curvature of space is determined by the energy density of matter and the expansion rate of space. The greater the expansion rate, the greater the density required to make space flat.

Last edited: Sep 13, 2007
6. Sep 13, 2007

### PhanthomJay

Thanks,this is now a second response in this post that distinguishes between 'space', 'time', and 'spacetime'....the former being possibly flat, and the latter two, curved. Hitherto, I was led to believe that space and time were inseparable, hence the word 'spacetime' as opposed to the 'space and time' as separate entities. Comment?

7. Sep 13, 2007

### hellfire

Space and time are part of a single physical entity that couples to matter via Einstein equations. However, the separation between space and time is necessary in order to describe the world we observe, formulating well defined initial value problems and making predictions. In most of the physically relevant space-times one can foliate space-time into spatial hypersurfaces that do not intersect each other and that are labeled uniquely by a time coordinate. If this condition is not given causal paradoxes may appear.

Last edited: Sep 13, 2007
8. Sep 13, 2007

### PhanthomJay

Thanks. You guys sure know your stuff!