3-d versus 4-d spacetime curvature

[DiracPool]

A second SR question that has been on my mind lately is that of hyperbolic nature of Minkowski space. The fact that the invariant interval, or lines of constant delta S trace out a hyperbola according to the equation, ##x^2-(ct)^2=S^2##, is fascinating to me and seems to imply that space-time has a negative curvature.

However, from cosmology I hear that it seems as though the debate over whether the universe has positive, negative, or flat curvature is favoring the "flat" solution according to observations. Does this observation strictly apply only to the 3 dimensional "space only" picture of the universe, or is it a general statement about the nature of spacetime in general?

I remember Penrose addressing this question in the Road to Reality at some point but don't have access to the book to reference it.

 

[PeterDonis]

The fact that the invariant interval, or lines of constant delta S trace out a hyperbola according to the equation, ##x^2-(ct)^2=S^2##, is fascinating to me and seems to imply that space-time has a negative curvature.

One can interpret the word "curvature" in a way that makes this statement true (for example, one could look at the curvature of a hyperboloid of constant spacelike interval from the origin), but the term "spacetime curvature" in GR means something different. In GR terms, Minkowski spacetime is flat; it has zero curvature.

from cosmology I hear that it seems as though the debate over whether the universe has positive, negative, or flat curvature is favoring the "flat" solution according to observations. Does this observation strictly apply only to the 3 dimensional "space only" picture of the universe, or is it a general statement about the nature of spacetime in general?

The "positive, negative, or flat" question applies to spacelike slices of constant FRW coordinate time; basically, these are slices of constant time for observers who see the universe as homogeneous and isotropic.

FRW spacetime itself (i.e., the spacetime of the universe, in the simplified model used as a first approximation in cosmology) is curved regardless of whether the spacelike slices of constant time have positive, negative, or zero curvature. This is because the universe is expanding--more precisely, it's because the rate of expansion of the universe (meaning, the rate of change of the scale factor with respect to FRW coordinate time) is changing. Up to a few billion years ago, the rate of expansion was decreasing, which indicates positive spacetime curvature (more precisely, positive curvature "in the time dimension", so to speak); but now the rate of expansion is increasing, which indicates negative spacetime curvature. Again, this is true regardless of the curvature of spacelike slices of constant time.

 

[pervect]

A second SR question that has been on my mind lately is that of hyperbolic nature of Minkowski space. The fact that the invariant interval, or lines of constant delta S trace out a hyperbola according to the equation, ##x^2-(ct)^2=S^2##, is fascinating to me and seems to imply that space-time has a negative curvature.

"Curvature" as used popularly is a vague word. In terms of the Riemann tensor, Minkowskii space is flat. It seems to me that if we replace "line" with "geodesic", the parallel postulate also applies to Minkowskii space. The closest thing I have to a formal proof of this is that parallel transport is path-independent in Minkowskii space (this is also related to flatness, in fact the path dependence of parallel transport can be taken as a technical definition of curvature).

This leads to the interesting question - if it's not the parallel postulate that separates Minkowskii geometry from Euclidean geoemtry, what postulate of Euclid's does Minkowskii geometry fail?

It's not entirely clear to me what the answer to this question is, my best guess is that it's number 4, "That all right angles are equal to one another." In particular, I believe that the "right angle" between a timelike geodesic and a spacelike geodesic isn't "equal" to the angle between two spacelike geodesics. This is mostly based on treating space-time angles as rapidities, I got this idea of angle from some posts here on PF, rather than a textbook, so I'm not sure how common the notion is.

I suspect that the problem really needs a more modern axiom system than Euclid's to really do a good job of answering this question - while I know that different axiomizations exist (for instance from Hilbert and Tarskii), I'm not aware of the formal details.

 

[PeterDonis]

It seems to me that if we replace "line" with "geodesic", the parallel postulate also applies to Minkowskii space.

I'm not sure that's true--more precisely, I think whether it's true depends on which formulation of the parallel postulate you adopt, which amounts to saying that formulations that are equivalent in Euclidean geometry are not equivalent in Minkowski geometry.

The formulation that seems to apply in Minkowski geometry is the one that says parallel lines never meet; with appropriate definitions of parallel timelike and null geodesics, this works for them just as it does in the obvious way for spacelike geodesics.

However, the formulation Euclid actually used in the Elements, IIRC, was a different one: it said that if two lines are cut by a third line in such a way that the included angles on one side of the third line sum to less than two right angles, then the two lines meet at some point on that side. But this requires a definition of "right angles" that can be applied to timelike and null geodesics as well as spacelike ones, and as you point out, that could be a problem.

We could use the Minkowski metric to define a "right angle" between a timelike and spacelike geodesic (this is the definition you appear to be implicitly using), and with this definition, I believe the original formulation would work (it basically says that two inertially moving objects that are moving towards each other will meet in the future, and two such objects moving away from each other must have met in the past). But I'm not sure how you would define "right angles" for null geodesics in a way that made the original formulation work, since using the Minkowski metric null geodesics are orthogonal to themselves.

 

[pervect]

I'm not sure that's true--more precisely, I think whether it's true depends on which formulation of the parallel postulate you adopt, which amounts to saying that formulations that are equivalent in Euclidean geometry are not equivalent in Minkowski geometry.

The formulation that seems to apply in Minkowski geometry is the one that says parallel lines never meet; with appropriate definitions of parallel timelike and null geodesics, this works for them just as it does in the obvious way for spacelike geodesics.

That's the intuitive formulation I was using.

However, the formulation Euclid actually used in the Elements, IIRC, was a different one: it said that if two lines are cut by a third line in such a way that the included angles on one side of the third line sum to less than two right angles, then the two lines meet at some point on that side. But this requires a definition of "right angles" that can be applied to timelike and null geodesics as well as spacelike ones, and as you point out, that could be a problem.

We could use the Minkowski metric to define a "right angle" between a timelike and spacelike geodesic (this is the definition you appear to be implicitly using), and with this definition, I believe the original formulation would work (it basically says that two inertially moving objects that are moving towards each other will meet in the future, and two such objects moving away from each other must have met in the past). But I'm not sure how you would define "right angles" for null geodesics in a way that made the original formulation work, since using the Minkowski metric null geodesics are orthogonal to themselves.

My current thinking on angles is that for a right-angled Euclidean triangle with the length of the hypotenuse equal to one, an angle alpha implies that the sides of the triangle are cos(alpha) and sin(alpha), where alpha can vary between -pi and pi. Then we would have cos^2(alpha) + sin^2(alpha)=1.

For a Minkowskii "right-angled triangle", assuming again that space and time are "at right angles", with the hypotenuse (the Lorentz interval) being 1, I am by analogy assuming an "angle" of alpha would imply that the sides of the triangle are cosh(alpha) and sinh(alpha). Alpha, however, is unbounded in this case, so the concept of "angle", while analogous, doesn't seem to behave exactly as the Euclidean concept, as we now have "infinite" angles. The "length" (Lorentz interval) of the hypotenuse is now cosh^2(alpha) - sinh^2(alpha)=1.

I'm afraid my thinking on the topic isn't terribly rigorous, it would be illuminating to see a full mathematical treatment of the issue in terms of some modern axiom system for Euclidean geometry. I still hope the ideas might be useful even if they are a bit speculative. It may turn out that the assumptions I have made above aren't the "best" ones for fitting the geometry into the axiom system.