# Curvature of Spacetime related to mass and expansion

1. Jun 10, 2012

### Phys00

Hi,
I was wondering if anyone could clarify something for me. I have been reading about the curvature of Spacetime and have come across a few things in articles in conjunction with de Sitter and Anti de Sitter spaces "Negative curvature corresponds to an attractive force" and "Positive background curvature means Universe is expanding at an increasing rate". Could someone clarify if positive matter produces positive or negative curvature and if more rapid expansion does cause negative curvature?

Thanks

2. Jun 10, 2012

### Staff: Mentor

The curvature in de Sitter (dS) and Anti-de Sitter (AdS) spacetimes is due to a cosmological constant, not to matter; there is no matter in either spacetime. A positive cosmological constant (dS) causes an accelerating expansion (this is the currently accepted cause for the accelerating expansion of our universe, though it's often called "dark energy" instead, but it's basically the same thing). A negative cosmological constant (AdS) causes an accelerating contraction.

3. Jun 10, 2012

### Phys00

Thanks. When matter is present, however, how then does the curvature change i.e. positive or negative?

4. Jun 10, 2012

### kmarinas86

Obviously your question as it relates to mass, has nothing to do with de Sitter spacetimes.

I would like to first say that there is no reason to believe that only a de Sitter cosmology can describe what we have seen.

With that said:

As far as local curvature is concerned (as might be implied by an object, such as that having mass), our cosmologists only care about positive curvature or none at all.

"Positive" curvature may refer to geometry such as that surrounding gravitational lenses ().

That is the kind of stuff cosmologists think of.

Not the below:

The possibility of local saddle geometries )( for spacetime appears to be completely ignored in our world's science.

Maybe there is a form of matter that can cause curvature of space to bend the other way )( locally. Did we rule that out?

No. We think such possibilities out of our models by making simplifying assumptions which exclude them, or simply ignore them. This is how cosmologists arrived at the cosmological principle (http://en.wikipedia.org/wiki/Cosmological_principle) which is assumed by most as a prori and still controls the practice of cosmology despite well-known facts that our solar system, and even our galaxy, have "habitable zones".

Don't expect a good answer to this question from PF as to whether curvature due to positive mass can only be positive or negative.

It's not available here.

Ask people sometime in the future, when our scientists might be compelled to explanations such as the existence of local negative curvature of spacetime )( despite presence (or perhaps even as the result) of "positive" mass.

Maybe some of them will think a few moments about similarity of lensing materials and properties of "curved space" that perhaps suggest that the latter notion is simply an artifact of not understanding the complexities of the medium that we call "vacuum".

Last edited: Jun 10, 2012
5. Jun 10, 2012

### Staff: Mentor

Anything we would call "matter" will add an attractive component to the "gravity" that is observed. That is, if we add "matter" to the Anti-de Sitter spacetime, it will add to the accelerating contraction caused by the negative cosmological constant. If we add "matter" to de Sitter spacetime, it will work against the accelerating expansion caused by the cosmological constant.

The exact results in the latter case will depend on how much matter is present; if there is enough, it can cause the "universe" to stop expanding and re-collapse, but if not, the expansion may slow down for a while, but eventually, as the matter thins out due to the expansion, its effect will decrease compared to the effect of the cosmological constant, which stays the same as the universe expands. This is pretty much what is thought to have happened in our universe a few billion years ago; the matter had thinned out enough by then that the effects of the cosmological constant began to dominate the dynamics, so the expansion started to accelerate (it had been decelerating up to that point).

6. Jun 11, 2012

### Ich

The presence of a positive energy density fluid contributes a positive spacetime curvature. As Peter Donis said, for normal matter, this also means attraction.
In the case of a cosmological constant or, generally, Dark Energy, things are different: Spacetime curvature behaves the same way, but the negative pressure of such a "substance" "inverts" ist gravitational effect.
That's why in de Sitter space positive curvature is tied to repulsive gravitation (accelerated expansion) while in AdS you have negative curvature and attraction.

7. Jun 11, 2012

### kmarinas86

Well, PeterDonis, what do you say about what you said earlier?

http://en.wikipedia.org/wiki/Anti_de_Sitter_space

Who explains that we can get curvature from nothing?

It's also interesting how the meaning of positive and negative curvatures for these cosmological spacetimes managed to divorce themselves from their meanings in geometry (e.g. for dS, positive curvature is associated with repulsion, and for AdS, negative curvature is associated with attraction).

Last edited: Jun 11, 2012
8. Jun 11, 2012

### Phys00

So from what's been said, would it be correct to say that positive curvature in spacetime implies attraction, whereas in terms of positive de Sitter space it implies repulsion?

9. Jun 11, 2012

### Staff: Mentor

I was asked what would happen if we *added* matter; that makes things different from the case with no matter, which is what I was describing earlier. Technically, if you add matter, the spacetime is no longer "de Sitter" or "anti-de Sitter", strictly speaking, because those spacetimes have no matter in them. But there are certainly valid solutions of the Einstein Field Equation that have both matter *and* a cosmological constant (positive or negative). Those are what I was discussing in my later post.

There is a cosmological constant present, which isn't "nothing".

Read carefully: de Sitter spacetime has a positive *cosmological constant*; anti-de Sitter spacetime has a negative *cosmological constant*. That does *not* mean they have positive or negative *curvature*, respectively. Cosmological constant and curvature are two different things.

Try this Wikipedia page, which specifically discuss dS as a spacetime, i.e., a model of a "universe":

http://en.wikipedia.org/wiki/De_Sitter_universe

Last edited: Jun 11, 2012
10. Jun 11, 2012

### Staff: Mentor

No. See my reply to kmarinas86. Don't confuse the cosmological constant with curvature.

11. Jun 11, 2012

### Staff: Mentor

I see on re-reading that the Wikipedia articles we've been linking to have exactly this confusion. I would not take what they say about "curvature" at face value.

12. Jun 11, 2012

### Staff: Mentor

Just to follow up and clarify how the stuff talked about on those Wiki pages fits together, take a look at the Wiki page on the Friedmann equations:

http://en.wikipedia.org/wiki/Friedmann_equations

In particular, the second Friedmann equation, which I will write in units where G = c = 1 and with slightly different notation:

$$\frac{1}{a} \frac{d^{2} a}{dt^{2}} = - \frac{4 \pi}{3} \left( \rho + 3 p \right) + \frac{\Lambda}{3}$$

This equation is valid for any spacetime whose metric can be written in the Friedmann-Robertson-Walker (FRW) form; de Sitter and anti-de Sitter spacetime are such spacetimes, as we will see. In this equation, $a(t)$ is the "scale factor", and the LHS of the equation describes the acceleration of the scale factor--positive LHS means accelerating expansion or decelerating contraction, negative LHS means decelerating expansion or accelerating contraction. (The scale factor itself is always positive, so the sign of the LHS is entirely due to the sign of d^2a/dt^2.)

On the RHS, $\rho$ is the matter density and $p$ is the pressure due to matter (here "matter" includes radiation, for example, the photons in the CMBR), and $\Lambda$ is the cosmological constant. dS spacetime is just the special case $\rho = p = 0$ and $\Lambda > 0$, while AdS is the special case $\rho = p = 0$ and $\Lambda < 0$.

Since the first term is negative and the second is positive, you can see that positive matter density and pressure, or a negative cosmological constant, will contribute to decelerating expansion or accelerating contraction. Conversely, a positive cosmological constant will contribute to accelerating expansion or decelerating contraction. Combinations of these will work as described in my previous post.

Sometimes, the term $\rho + 3 p$, if positive (which it is for all known forms of matter and radiation), is referred to as "positive curvature" or "positive curvature due to matter". I believe this is the sense in which that term was being used on the Wiki pages. But as you can see, what it really means is positive density and pressure due to matter.

13. Jun 11, 2012

### A.T.

Actually the local space time-curvature near (but outside) a big mass is negative, like the saddle. Geodesics are diverging there (tidal forces). It's only when you include the mass in the considered region that you get an overall positive curvature, which causes geodesics on opposite sides of the mass to converge.

14. Jun 11, 2012

### kmarinas86

What is the range of such curvature? You said "(tidal forces)". I take that to mean that you think that the "effective range" of the negative curvature depends on the "effective range" of tidal forces. Is this the correct interpretation of your statement?

If I understand correctly, if hyberbolic curvature were local, yet ranging much further than tidal forces, even a small negative curvature would have a profound distortion on the appearance on the universe. Furthermore, such an effect might go long undetected because, unlike positive curvature, it wouldn't cause light paths to cross in the way that we see "gravitational lenses", so it would cause far fewer, if any, artifacts. It would also make things appear much further than they really are. It would be like having a bunch of wide angle lenses distributed across the vacuum of space, making everything look farther way. Do you think that cosmologists will have any chance of ruling this possibility out?

Last edited: Jun 11, 2012
15. Jun 11, 2012

### kmarinas86

I was invoking some of the things said by Ich.

16. Jun 11, 2012

### Staff: Mentor

For some reason I didn't see that post until now. I'll respond to it below.

As A.T.'s posts indicate, it isn't that simple. Curvature can be different in different directions. As A.T. noted, the key is that the region you are looking at has to include the matter; if you just look at the vacuum region outside the matter, the spacetime curvature is negative, not positive.

A better single quantity to look at to see the effect of positive energy density for normal matter (and radiation) is the LHS of the Friedmann equation; as I posted before, the LHS is *negative* for positive energy density of normal matter and radiation. So what you are calling "positive curvature", for normal matter and energy, leads to a *negative* LHS of the Friedmann equation.

I clarified what I meant by that in my post on the Friedmann equation. See above.

In the version of the Friedmann equation that I wrote down, the effect of a cosmological constant is in a separate term; there is no "energy density" or "pressure" ascribed to it. However, some physicists prefer to define an energy density and pressure due to the cosmological constant: a positive constant has positive energy density and an equal negative pressure ("equal" in units where G = c = 1), and a negative constant has negative energy density and an equal positive pressure.

With these definitions, you can just look at the first term on the RHS of the Friedmann equation to describe everything: a positive cosmological constant will have rho + 3p negative (the opposite of normal matter), while a negative cosmological constant will have rho + 3p positive (the same as normal matter). The Wiki page on the FRW metric discusses this:

http://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric

As far as curvature goes, since geodesics are diverging in de Sitter space (positive cosmological constant), that would imply *negative* curvature, not positive, at least in some directions, just as it does in the vacuum region outside a massive object. So I"m not sure that I would attribute the accelerating expansion with a positive cosmological constant to "positive curvature".

17. Jun 13, 2012

### Ich

That comment was meant to be read in the context of cosmological models, where the fluid is assumed to be homogeneously distributed.

Right.
Right. As you can see, that's what I have done in my post. I also explained what you say in the next paragraph.

Read the wikipedia article on de Sitter space. It has positive scalar curvature. That's exactly the interesting point here, that curvature doesn't go along with gravitational attraction.
You can see this clearly in the simplified versions of the Friedmann equations: while the source for attraction is $\rho +3p$, the source for spatial curvature is only $\rho$.
(Side Note: FRW spatial curvature is coordinate dependent, you get the relevant spatial sectional curvature by setting H=0 in the respective equation.)

Last edited: Jun 13, 2012
18. Jun 13, 2012

### Staff: Mentor

So you meant spatial curvature, not spacetime curvature? And you also note (correctly) that spatial curvature is coordinate dependent, because (putting it slightly differently than you did) you can "cut" spacelike slices through the spacetime in different ways?

I was talking about spacetime curvature, which is what correlates with the "source" of gravity in GR, via the EFE. Spatial curvature does not; you have yourself pointed out a key reason why, that spatial curvature is coordinate dependent. The "source" of gravity in GR is not coordinate dependent; it's described by an invariant geometric object, the SET. If we include the cosmological constant, then that term also can be viewed as a "source" of gravity, by defining an "effective" energy density and pressure for it, and hence an "effective" SET.

But none of that has anything to do with spatial curvature. I would definitely *not* recommend trying to draw any connection between spatial curvature (as opposed to spacetime curvature) and the "type of gravity" present ("attractive" or "repulsive", accelerating vs. decelerating expansion); that connection doesn't even hold in FRW spacetimes with *zero* cosmological constant. All of those spacetimes have decelerating expansion ("attractive" gravity), with positive energy density, but they have all three different types of spatial curvature (even if we restrict to the spacelike slices of constant comoving time): positive, zero, and negative.

19. Jun 13, 2012

### TrickyDicky

You all seem to be mixing space curvature with 4-spacetime curvature. dS is usually considered as a positive curvature lorentzian 4-space.

20. Jun 13, 2012

### Ich

No, I've written "scalar curvature", which is 4D. De Sitter has positive spacetime curvature.

I added spatial curvature to get the link with the Friedmann equations, where you see the difference in the source terms. It's important that you see the difference between sectional curvature and "space curvature".
The former is quite independent of coordinate choices and essentially gives you spatial curvature as you'd measure it actually building a wheel and comparing its diameter with its circumference. That's what I'm talking about.
The latter depends on the 3D slice you call "space". This slice depends on the relative motion of the fundamental observers, which is why H² is present as a negative curvature contribution. That's not what I'm talking about, but it is what you find in the Friedmann equations.

21. Jun 13, 2012

### Staff: Mentor

Ok. But the 3-metric of a spacelike slice also has a "scalar curvature", which is why I was confused.

Ok, that makes your terminology clearer.

By H^2 I assume you mean the LHS of what the Wikipedia page calls the "first Friedmann equation":

$$\frac{1}{a^{2}}\left[ \left( \frac{da}{dt} \right)^{2} + k \right] = \frac{1}{3} \left( 8 \pi \rho + \Lambda \right)$$

Yes, that depends on how you cut the spacelike slices; the form written above, like the standard form of the FRW metric, assumes that you use slices of constant comoving time, so that comoving observers are at rest in the spatial coordinates. But the "curvature contribution" here is the k term, right? Which can be positive, 0, or negative, depending on the specific parameters of the model.

22. Jun 13, 2012

### Ich

Without the k, that's right. But actually I mean
$$H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3}\rho - \frac{kc^2}{a^2},$$
a little bit later on the same page.
Yes, but I prefer to read this equation differently: k is the net curvature, consisting of two contributions. The first is the effect of mass density, a positive contribution. The second is the effect of using comoving observers to span space, a negative contribution if the universe is expanding. Only the second term depends on the exact choice of coordinates and can be "defined away" by looking at "static" slices. The mass contribution remains, and that's how I interpreted the statements cited in the OP: positive density = positive curvature. Attractive/repulsive gravity is a different matter, as it is influenced also by pressure.

23. Dec 13, 2012

### darkthoughts

This idea interests me in relation to the large-scale structure of the universe, i.e. the coalescence of galaxies into filaments and sheets, bounding large voids. Could there be a very slight, broad-scale spacetime curvature across the large voids, that is opposite to, and compensatory to, the relatively sharp and localised spacetime curvature caused by the concentrations of matter (i.e. galaxies, galactic clusters) in the filaments and sheets? If so it could be posited that expansion of the universe is a consequence of the non-uniform aggregation of matter in it and that "dark energy" is actually just extremely tenuous, locally inverse gravity fields in the vast swaths of universe containing no matter.

We don't know for sure if the Universe's spacetime field as a whole is flat or slightly curved. But whatever it is, it must surely be fixed and not changeable over time, the Universe considered as a whole. It then follows that local warping of parts of the Universe as a result of the coalescence of matter could not alter the sum curvature (or flatness) of the whole Universe, so it would have to be offset by compensatory inverse warping, diffused across the rest of the Universe.

Last edited: Dec 13, 2012