What is the Geometry of Spacetime Near the Big Bang?

[Imax]

What did spacetime look like some time after the big bang?

1) Euclidian space. Imagine yourself in empty space, no mass, with (x,y,z) coordinates ranging from –infinity to +infinity. Somewhere within this infinite space was a volume with high matter-energy density from the Big Bang.

2) Compact Lorentzian. Spacetime was so badly bent from this high matter-energy density, that spacetime itself was finite, forming a Compact Lorentzian Manifold. This was associated with closed curves, including closed spacelike, lightlike, and timelike curves.

3) Compact Space. Similar to #2, but only space was compact. Time followed a global hyperbolic, ranging from –infinity to +infinity. Not all coordinates needed to be compact:

http://arxiv.org/PS_cache/arxiv/pdf/1104/1104.0015v2.pdf

 

[Drakkith]

I don't believe #1 is accurate. Had you existed right after the big bang, you would be surrounded by a very dense amount of matter and energy that extended to infinite distance in all directons.

 

[Imax]

Hi Drakkith:

I can’t see #1 because I can’t see how it can explain expansion. At first, I thought maybe #2, but closed timelike curves are a problem.

Maybe #3?

 

[Drakkith]

I don't think that spacetime was really any different right after the big bang compared to now. Other than size at least. Maybe someone more knowledgeable than myself can chime in on this.

 

[Fredrik]

1. Definitely not. A Euclidean geometry of space is possible, but matter would have the same properties (including density) everywhere in space.
2. I can't make sense of this. Spacetime is either compact or not compact. Spacetime doesn't have properties associated with a specific value of the time coordinate.
3. A compact space is possible, but the time coordinate would satisfy t>0 at all events.

The simplest answer given by GR is obtained by assuming a) that spacetime can be sliced up into a 1-parameter family of spacelike hypersurfaces that we can think of as space at different times, and b) that each time t, space is homogeneous and isotropic. The solutions that are consistent with these assumptions are called FLRW spacetimes.

 

[George Jones]

3) Compact Space. Similar to #2, but only space was compact. Time followed a global hyperbolic, ranging from –infinity to +infinity. Not all coordinates needed to be compact:

http://arxiv.org/PS_cache/arxiv/pdf/1104/1104.0015v2.pdf

3. A compact space is possible, but the time coordinate would satisfy t>0 at all events.

The simplest answer given by GR is obtained by assuming a) that spacetime can be sliced up into a 1-parameter family of spacelike hypersurfaces that we can think of as space at different times, and b) that each time t, space is homogeneous and isotropic.

An interesting text reference that relates to this is Chapter 15, Spatially Homogeneous Universe Models, from the book Einstein's General Theory of Relativity: With Modern Applications in Cosmology by Øyvind Grøn and Sigbjorn Hervik.

Imax, when linking to something from something from the arXiv, please link like

http://arxiv.org/abs/1104.0015.

This link has useful information, like whether a paper has been submission to a journal, a journal reference if published, and the arXiv version of the paper. The text for the full paper is easily accessible from this type of link. In this case, it is arXiv version 2 of the paper, and there is no indication that the paper has been submitted for publication.

 

[Imax]

2. I can't make sense of this. Spacetime is either compact or not compact.

In #2, spacetime is compact with no boudary conditions. It's like those early crappy video games. If you go too fare rigth, you show up on the left. See Poincaré Dodecahedral.

 

[Fredrik]

In #2, spacetime is compact with no boudary conditions. It's like those early crappy video games. If you go too fare rigth, you show up on the left. See Poincaré Dodecahedral.

What bothered me about what you said in #2 was the word "was". Spacetime doesn't have properties that change with time.

 

[Imax]

What bothered me about what you said in #2 was the word "was".

Oh. I used "was" because I find it easier, conceptually, to think of what spacetime could have look like at some time after the BB. The high matter-energy density must have had some influence on how spacetime looked like.

Spacetime doesn't have properties that change with time.

Agree. That’s were I was going with this. Excluding any possible bizarre effects, the topology of spacetime today should look like spacetime near the BB (but there may be more of it).

What's giving me problems is #3. It seems to be artificially decoupling space and time (i.e. in SR, spacetime is like a fabric, ...).

 

[Fredrik]

(but there may be more of it).

There may not. That was my point. There may be more of space at an earlier time, e.g. in the sense that space at time t is a 3-sphere with a t-dependent radius. What you're saying (at least the way you're saying it) is like saying that the set of real numbers have different properties "at" 1 than at 13 (how does any statement that ends with "at 13" apply to the set of all real numbers?). It's like saying that "this calendar year" means something different in August than it did in February. If you're talking about properties of a hypersurface defined by a value of t, then you're talking about properties of "space at time t", not spacetime.

What's giving me problems is #3. It seems to be artificially decoupling space and time (i.e. in SR, spacetime is like a fabric, ...).

It's not entirely artificial, and it's not really a decoupling. The assumption that spacetime can be sliced up into spacelike hypersurfaces that we can label by a real parameter t, is satisfied by the spacetime of SR too, and probably (I'm guessing) all solutions of Einstein's equation that have been found. It would be artificial to simply postulate that one of the ways to do the slicing is better than the others, but to find the FLRW spacetimes, you only assume that it's possible to do the slicing in a way that makes each slice homogeneous and isotropic. This is motivated by what we see in telescopes. It turns out, once we have found the solutions, that other slicings wouldn't make space homogeneous and isotropic. But it doesn't really decouple space and time, because there's nothing that says that the coordinate system that was postulated to exist is the only one we're allowed to use.

 

[Imax]

It's not entirely artificial, and it's not really a decoupling. The assumption that spacetime can be sliced up into spacelike hypersurfaces that we can label by a real parameter t, is satisfied by the spacetime of SR too, and probably (I'm guessing) all solutions of Einstein's equation that have been found. It would be artificial to simply postulate that one of the ways to do the slicing is better than the others, but to find the FLRW spacetimes, you only assume that it's possible to do the slicing in a way that makes each slice homogeneous and isotropic. This is motivated by what we see in telescopes. It turns out, once we have found the solutions, that other slicings wouldn't make space homogeneous and isotropic. But it doesn't really decouple space and time, because there's nothing that says that the coordinate system that was postulated to exist is the only one we're allowed to use.

Can I consider spacetime as being embedded in a 5 manifold? That could make it easy to slice spacetime into spacelike hypersurfaces.

 

[WannabeNewton]

Can I consider spacetime as being embedded in a 5 manifold? It could make it easier to make slices in a spacelike hypersurface.

Not within the context of GR, no. It only deals with intrinsic properties of 4 - manifolds. Why would embedding the 4 - manifold in higher dimensional space make the process of defining space - like hypersurfaces easier?

 

[Imax]

Metrics are good on distances and angles but they are lousy on time.

 

[Fredrik]

Can I consider spacetime as being embedded in a 5 manifold? That could make it easy to slice spacetime into spacelike hypersurfaces.

No, but you can consider it to be embedded in a spacetime with constant metric with about 90 dimensions.

https://groups.google.com/group/sci...b1e76f76fef77?lnk=st&q&hl=en#3b4b1e76f76fef77

Not sure if that's useful though.

 

[WannabeNewton]

No, but you can consider it to be embedded in a spacetime with constant metric with about 90 dimensions.

https://groups.google.com/group/sci...b1e76f76fef77?lnk=st&q&hl=en#3b4b1e76f76fef77

Not sure if that's useful though.

That's insanely cool but I wonder why is it not possible to embed the 4 - manifolds in $\mathbb{R}^{8}$ as per Whitney's theorem (all differentiable manifolds are Hausdorff and second - countable no?).

 

[Imax]

Maybe 90 degrees of freedom is too many :)

What I’m looking for is something like:

$$\frac{\partial t}{\partial \alpha}$$
where $\alpha$ is some parameter. If time follows a global hyperbolic, then it’s likely that:

$$\frac{\partial t}{\partial \alpha}=0$$
regardless of $\alpha$.

Boring ;)

 

[Imax]

#1 is unlikely, so its #2 or #3. In both #2 and #3, time follows a timelike geodesic. The difference is that in #2 time can have a curvature, but in #3 time is globally Minkowski.

 

[LaurieAG]

1) Euclidian space. Imagine yourself in empty space, no mass, with (x,y,z) coordinates ranging from –infinity to +infinity. Somewhere within this infinite space was a volume with high matter-energy density from the Big Bang

The attached image is a 2D representation of the light paths flowing directly towards a stationary observer who is over 1 galactic year away from 2 sources rotating around a common galactic centre. These twisted paths exist now from source to observer in Euclidian space with the only mass/gravity being involved is that of the rotating sources. This direct link will continue in real time while the observer is stationary until the path from the source expires, is blocked or is distorted.