ask a physicist

[freyar]

I'm presuming this is obtained from the measurements of the energy density, which makes for a flat universe.

How do we get from that to the conclusion that there aren't joins? Does the lack of uniformity in directions (diagonals are longer than the perpindiculars) make that physically impossible? Or is it considered too strange?

(From a topologists point of view, a standard construction is to take a unit square and to identify the opposing sides to make a new quotient space of the original unit square. That makes for a flat, unbounded, yet finite, space. The space is not uniform in all respects.)

The energy density of the universe is (within measurement error) the amount that makes the universe spatially flat, yes. Due to the cosmological constant, the expansion is accelerating, again as mentioned earlier. [I am hesitant to use the phrase "open universe" since that had implications up until the discovery of the cosmological constant that no longer apply.]

However, even a flat universe can have a nontrivial topology. Your example of gluing the edges of a square together is a good one --- that's the mathematical definition of a 2D torus. So (the spatial part of) our universe could be a flat 3D torus or one of several other alternatives. (That's assuming it's actually exactly flat; there are other alternatives if it's positively or negatively curved.) How do we know if the universe is infinite or a finite torus/other example? We actually have to go measure. There are several groups that have looked at the cosmic microwave background (the oldest visible light); if the universe is finite and small enough, we'd be able to see repeated patterns due to seeing the same spot in the early universe from several directions (think about different ways you can shoot the same area on the screen in the old Asteroids game). So far, we don't see any repeated patterns like that, which means that the universe is either infinite or else finite but so big we can't see all the way around it.

There could be some local joins or curves (wormholes and black holes and such). But, on a broad scale, there is a point to be made: Mass/energy causes spacetime to curve. As far as we know, mass is the *only* thing that makes spacetime curve. Real physical spacetime cannot have arbitrary topological configurations *without* mass to make it so. This is why we say that the low energy density means spacetime is flat, at least within the observable universe.

This isn't really correct. There are finite, _flat_ spaces with nontrivial topology that solve Einstein's equations with _zero_ mass or energy density. A torus is an example. And our universe could have one of those spatial topologies. Black holes and wormholes do have curvature, though, so they require mass/energy.

 

[freyar]

The curvature is an artifact of the embedding, not of the join.

GR tells us about local topologies, not about global ones. To make conclusions about the global topology, additional evidence is needed.

In the second link posted above, several experiments are described, with the result being, at that time, "we don't know". However, there may be more recent results, as newer experimental data (satellite measurements) should now be available, compared with what was available to the authors of that paper.

Thx!
TomB

Yeah, just to emphasize this, TomB is exactly right here. There are a number of flat 3D spaces with nontrivial topology, ie, which are multiply connected. As he says, the Einstein equations, or really any field equation you want to write down for gravity, is going to be a local equation and won't tell you about the topology. You have to go measure it separately. So even if you want to shrink the error bars to zero and say that the universe is spatially flat (but expanding), you have to go do a separate measurement to tell if there is nontrivial topology. As far as we can tell so far, the universe is indistinguishable from an infinite one, but that could be the same as finite but bigger than our ability to see until the universe is a whole lot older.

 

[Herobizkit]

How might one calculate the gravitational effects, if any, of multiple moons on a theoretical Earth-like planet? Would such moons affect said Earth in any other perceptible way (like, for example, what would tides be like with multiple moons?)

 

[freyar]

How might one calculate the gravitational effects, if any, of multiple moons on a theoretical Earth-like planet? Would such moons affect said Earth in any other perceptible way (like, for example, what would tides be like with multiple moons?)

You can just add up the gravitational effects of the individual moons using Newton's law of gravitation.* Assuming normalish moon sizes and distances, tides would really be the main noticeable effect on the "host" planet. And you're right to think the tides would be more complicated; basically, you have to add the tidal effects of all the moons together at any given time. So, as the moons change positions around the planet relative to each other (sometimes they are lined up or not), the tides will be add up or cancel out to an extent. This is kind of like how the relative positions of the sun, earth, and moon affect our real tides --- tides are stronger then they make a line and weaker when the earth is at the vertex of a right angle. But the pattern in time would be more complicated with multiple moons.

*I'm assuming normal-sized moons. If your "planet" and "moons" are neutron stars, though, you'd probably have to think about general relativity. Then things get hairy mathematically.

 

[Umbran]

This isn't really correct. There are finite, _flat_ spaces with nontrivial topology that solve Einstein's equations with _zero_ mass or energy density. A torus is an example.

Well, be careful there - the torus (and the 9 other such topologies that satisfy the conditions) do it by *combining* curvatures. The torus is locally flat, globally flat, but "regionally" curved - on the inside of the torus, around the hole, you have negative curvature (you see the typical "saddle" of negative curvature around the donut hole), and the outer side shows positive curvature. The sum total effect is net zero.

We tell ourselves that, in theory, the thing can come about by having a density parameter of 1. But, that can be done two ways - by having it be 1 *everywhere*, or having it be a *net* 1 with it being below 1 in some regions, and above that in others. This latter is argued to be a more reasonable expectation for how you get a universe of that shape, as it is hard to imagine how the regional curvature physically rises without the appropriate energy density.

 

[freyar]

Well, be careful there - the torus (and the 9 other such topologies that satisfy the conditions) do it by *combining* curvatures. The torus is locally flat, globally flat, but "regionally" curved - on the inside of the torus, around the hole, you have negative curvature (you see the typical "saddle" of negative curvature around the donut hole), and the outer side shows positive curvature. The sum total effect is net zero.

Absolutely not. You're talking about extrinsic curvature, the curvature of the embedding of a 2D torus in 3D flat space. The torus itself has zero intrinsic curvature, and that's what GR cares about. We can see the curvature of the torus easily: a 2D torus has metric ds^2 = dx^2 + dy^2 for 0<= x <2 Pi R_x and 0<= y <2 Pi R_y with x and y both identified at their edges (meaning that all functions on (x,y) have periodicity 2 Pi R_x in the x direction and 2 Pi R_y in the y direction). As you know, that's the flat metric.

If we're thinking of (the spatial part of) our universe as a 3D torus, we shouldn't be thinking of it as embedded in a larger-dimensional flat space, since that higher-dimensional space might not exist. The torus defined by this type of mathematical quotient is perfectly flat everywhere. This you can trust me on --- several of my research papers deal with tori. Of course, if you want an independent source, there's always the wikipedia article. The first section deals with the embedding of a 2-torus in 3D flat space, then you get the mathematical definitions, then generalizations, etc.

 

[Landifarne]

Interesting question from one of my AP Chemistry kids (12th grader, dreamer-stoner type, but the guy often has intriguing ideas):

My student understands that, when discussing electron orbitals in an introductory manner, we are not terribly concerned that indivdual orbitals (and their corresponding nodes) in a polyelectronic atom ignore the effects of all the atom's other electrons/orbitals [i.e. we assume that the actual orbitals and nodes that do exist more-or-less resemble a simple superimposing of the individually calculated orbitals].

So, he asked me a month ago whether (again, assuming the above) we could gain any insight by ignoring the orbital shapes/structures and, instead, focussed on how the nodes interact/interrelate.

I paused for a while and thought about how x-ray diffraction patterns give insight into crystal lattice structures and molecular structures (interferometry). There must be an analogy there, somewhere...

Anyway, any of you have thoughts on what my kid was getting at? Is there some kind of "inverted" Schrodinger Equation that would focus more on the characterization of the nodes?

 

[freyar]

Interesting question from one of my AP Chemistry kids (12th grader, dreamer-stoner type, but the guy often has intriguing ideas):

My student understands that, when discussing electron orbitals in an introductory manner, we are not terribly concerned that indivdual orbitals (and their corresponding nodes) in a polyelectronic atom ignore the effects of all the atom's other electrons/orbitals [i.e. we assume that the actual orbitals and nodes that do exist more-or-less resemble a simple superimposing of the individually calculated orbitals].

So, he asked me a month ago whether (again, assuming the above) we could gain any insight by ignoring the orbital shapes/structures and, instead, focussed on how the nodes interact/interrelate.

I paused for a while and thought about how x-ray diffraction patterns give insight into crystal lattice structures and molecular structures (interferometry). There must be an analogy there, somewhere...

Anyway, any of you have thoughts on what my kid was getting at? Is there some kind of "inverted" Schrodinger Equation that would focus more on the characterization of the nodes?

That is an interesting thought. I'm not aware of anything that really just deals with the nodes, though there are a lot of different approaches in quantum chemistry, so it's possible I just don't know of one. But I'm not terribly sure it would be useful, either. You really need the whole wavefunction in some kind of approximation to get the energy levels. Measurement-wise, knowing where the nodes are might help, but I don't know of a way to make that measurement in analogy to diffraction, either.

The big issue is the approximation that you can just use hydrogen-like orbitals filled up one at a time without interaction. That generally doesn't work very well at all quantitatively. It gives some rough guidelines for qualitative chemical behavior (hence the periodic table), but the numbers don't come out well.

 

[Umbran]

This you can trust me on

I don't need to. It isn't like I don't have my own texts and notes - just sometimes I should double check them before writing. I was conflating two things from a course years ago, without realizing it. Carry on.

 

[freyar]

I don't need to. It isn't like I don't have my own texts and notes - just sometimes I should double check them before writing. I was conflating two things from a course years ago, without realizing it. Carry on.

No problem, happens to all of us from time to time.