# B Negative curvature

1. Feb 27, 2017

### mieral

I know negative curvature of spacetime is close to impossible.. but reading about dark energy and how it is repulsive... I'm trying to find illustrations of what would happen when spacetime curvature is negative (locally as whole cosmos having negative curvature is different concept than local negative curvature, right?)..

or beside negative curvature.. what form must the stress energy tensor be so instead of objects attracting via gravity.. they become repulsive...

I'm not trying to spread disinformation but just want to imagine how the geodesics and curvature would look like if there were antimass and negative curvature locally.

I know antimass doesn't exist. But I just want to visualize how the spacetime diagram would look like when it is negative. Or does the question make any sense at all? Just give me a youtube link for some illustration.. thank you.

2. Feb 27, 2017

### A.T.

Tidal gravity pulls free falling objects apart radially. No need for dark energy to have some negative spacetime curvature.

3. Feb 27, 2017

### mieral

I'm not argung about tigal graviy pulls free falling objects radiclly or other stuf. What I'd like to know are what variables and changes in equation for that to happen.. producing negative curvature and objects are repelled. Does it make to have really negative curvature and the stress energy tensor adjusted certain way. In the end, the question is whether it is feasible or not

4. Feb 27, 2017

### PAllen

It is trivially feasible. In 'empty space' connect two large masses by a sufficiently rigid very low mass rod. Then, in between them will be an area of negative curvature where test bodies will separate from each other.

5. Feb 27, 2017

### pervect

Staff Emeritus
I suspect that the OP may be thinking of negative spatial curvature as is talked about in cosmology, rather than negative space-time curvature.

So I suspect that AT and PAllen may be talking about a different notion of curvature than the OP is. Especially since the OP doesn't recognize the fact that particles being pulled apart by tidal forces is associated with a negative component of the Riemann curvature tensor, which is what is motivating PAllen's and AT's response.

Spatial curvature is often confused with space-time curvature, and it's frequently talked about in the context of cosmology as a single number (though I'm hazy on why it's only a single number, I'm not recalling the details at the moment.) But I think there may be some confusion in that AT & PAllen may be answering what the OP asked using the technical definitions of the terms he used, but the intent of the OP might well be different.

6. Feb 27, 2017

### mieral

No. I'm talking about local negative space-time curvature. My thinking (wrong or not) is that local positive space-time curvature can make moon be closer to the earth in geodesics with positive curve spacetime causing "gravity"... so could local negative space-time curvature make the moon be farther away from the earth in geodesics with negative cuved spacetime causing repel of gravity. Or is this not enough and the stress energy tensor must be in certain way? How.

Was Pallen describing this repelling of gravity. If, as he described, you have two larges objects connected with a sufficiently rigid very low mass rod, would the area between them have negative curvature.. so if you send a ball to the rod.. would it repel the rod.

Or if you still can't understand the above. What must you do to the spacetime curvature, stress energy tensor to make the moon be repel from the earth where although the geodesics is straight but the curve is negative and objects got separated (creating the effect of gravity repulsion).

Again, I know gravity repulsion is not possible. But just want to understand what conditions or adjustments in General Relativity to make this occur.

7. Feb 27, 2017

### Staff: Mentor

Sure - the point on the moon that is closest to us (right in the middle of the disk during a full moon) and the point on the opposite side are experiencing a repulsive force between them. The moon is made of solid rock which doesn't stretch much, but we've been able to measure the distortion. If we could saw the moon in half along the line between the side that faces us and the hidden side, the more distant hemisphere would move away from the earth.

You may object that this is not a "repulsive" force; both sides of the moon are attracted towards the earth, and all that's going on is that the near side is attracted more strongly than the far side. That is indeed the standard Newtonian explanation.... but note that you have (probably without noticing) selected coordinates in which the the earth is at rest to come up with that explanation. Work this problem in coordinates in which the center of the moon is at rest and you'll come up with a different explanation, one that recognizes that the negative curvature between the two halves of the moon.

8. Feb 27, 2017

### mieral

Is there a website with java where you can adjust the curvature of space from positive to negative and see parallel geodesics converge (gravity) or diverge (repel of gravity). I know the latter doesn't occur.. but I just want to know if spacetime diagram can theoreticaly cause diverging of geodesics (how does this look like.. there must be one website amongst hundreds out there with this).

Remember we always have opposites.. like positive vs negative, day vs night.. man vs woman.. so i just wonder how positive vs negative curvature looks like locally. Thank you.

9. Feb 27, 2017

### Staff: Mentor

Imagine yourself in a small windowless room that is in freefall towards the surface of a planet. It's freefall (at least until the room hits the surface) so if you hold an object out in front of you and let go, it will float in front in you, just as you've seen in so many videos from astronauts in space. Now suppose you position one small object near the ceiling, another near the floor, and four more near each of the four walls. Look closely though, and you will see the objects near the walls moving towards one another and the objects at the floor and the ceiling moving away from one another.... The spacetime around the planet is has negative curvature in the "vertical" direction and positive curvature in the "horizontal" direction, and this is about as local of a situation as we can imagine.

(And if you're wondering why I put the scare-quotes around the words "horizontal" and "vertical"? Someone watching from outside would call those directions "radial" and "tangential", a better way of thinking about the global spacetime around the planet).

10. Feb 28, 2017

### A.T.

See case C in post below. That is negatively curved space-time along a radial line outside of a massive body.

11. Feb 28, 2017

### mieral

How small can negative curvature be induced in middle of positive curvature? Supposed, for example, just for sake of illustration.. you have a tiny stone made of antimass. If it floats up or repel gravity of earth.. can we say it can induce tiny negative curvature in middle of the positive curvature created by the earth. Again this is just an example.. I know you will say there is no antimass and i'm not arguing anything. .

If localized negative curvature has minimum size. Does it mean a tiny stone of antimass that can be held inside the hands can induce mile size negative curvature so everything a mile around it would float up or repel earth? Let me emphasize this is just for sake of illustration to understand how negative curvature can occur in middle of positive curvature and not to spread disinformation which this web site so hate. Thanks.

12. Feb 28, 2017

### A.T.

The curvature created by the Earth (outside of the Earth) is already negative. See above.

13. Feb 28, 2017

### martinbn

There are already several answers and in my opinion they are all good and illuminating, but it seems that none satisfies you. I think it is because you questions a very vague and probably you don't know what you are asking. Can you clarify what you mean by curvature and by it being negative? Do you mean the Riemann tensor and some components being negative, do you mean scalar curvature, do you mean something else?

14. Feb 28, 2017

### mieral

Well.. I'm avoiding a word from the start because it is a taboo word and physicists hate it. But for now let's use the word and just try to hold off any automatic hate for it.

Gravity = mass = positive curvature = normal stress energy tensor
Antigravity = antimass = negative curvature = altered stress enegy tensor

I know physicists hate the word "antigravity" because it is not the mainstream and it's a word for crackpots. But I need to understand how General Relativity can accommodate matter to repel each other (and why not if it can't). You may say there is no such thing as antigravity. But I want to understand why General Relativity can forever only accommodate gravity that attracts. I've searched for this all over the net but couldn't find the explanations. A Scientific American article or others that explain why it can't would be illuminating.. this would enable more understanding of General Relativity. So can anyone give a good summary why it can't? You may say there is no antimass. But just supposed you directly make the curvature be certain shape (like negative curvature).. can it cause opposite of gravity attraction or repulsion?

15. Feb 28, 2017

### A.T.

16. Feb 28, 2017

### martinbn

You still haven't explained what you mean by curvature.

You have the spacetime, which is a four dimensional manifold. You talk about its curvature. What do you mean by curvature (mathematically)?

17. Feb 28, 2017

### mieral

I'm still trying to visualize or imagine how a four dimensional manifold has positive or negative curvature. I'll search in the webs more before asking so my questions would make more sense.

But remember Einstein Field Equations are kinda like solutions. You can enter certain input and you have black hole solutions. So I'm kinda asking what if you input matter that is opposite to matter.. would it have repulsive gravity. If there is a arxiv paper about this.. do share it.

I just read now so repulsive gravity is not a fringe thing at all.

https://phys.org/news/2011-04-antimatter-gravity-universe-expansion.html

If antimatter or stuff really antigravitate.. what actually happens to the geodesics.. illustrations appreciated.

18. Feb 28, 2017

### A.T.

I can't even visualize a four dimensional manifold without curvature. But for simple cases like radial fall you don't need all four dimensions.

19. Feb 28, 2017

### mieral

http://sns.ias.edu/~malda/sciam-maldacena-3a.pdf

In the above.. Maldacena mentioned anti-di Sitter space has negative curvature. So negative curvature automatically has repulsive gravity right? But how do you imbed small localized negative curvature within a big positive curvature like the earth.. say the negative curvature is caused by a basketball size antimatter (protective by say some kind of bubble to avoid detonation when it is in contact with matter) that just shoots up from the ground on earth? I just need an illustration how the imbedding occurs. I'll draw it tomorrow to correct where I could have imagined it wrong.

20. Feb 28, 2017

### timmdeeg

It seems you are making your life harder than this issue requires.
Imagine two particles which are falling radially towards a central mass, being separated by some radial distance. The lower particle being closer to the mass falls faster than the upper one at a given time. By this their distance increases accelerated or in other words their geodesics are accelerating away from each other, just as shown in #15 "Negative Curvature".
Now imagine two particles with the same radial distance, which are falling towards the center of a mass, being separated in the tangential direction. By this their geodesics are accelerating towards each other, see #15 "Positive Curvature".

Note that most presumably antimass behaves like mass regarding the curvature of spacetime. Two anti-hydrogen atoms would attract each other (a few have been created so far). According to present knowledge repelling gravity is caused by vacuum energy, the cosmological constant or dark energy resp.

21. Feb 28, 2017

### Staff: Mentor

It has both. You are thinking of "curvature" as a single number, but it's not. In a general four-dimensional spacetime, twenty independent numbers (the independent components of the Riemann curvature tensor) are needed to describe curvature. These numbers can be split into two sets of ten numbers, the Weyl curvature and the Ricci curvature; only the latter is tied to stress-energy by the Einstein Field Equation (see below).

Even in a vacuum, spherically symmetric spacetime, such as the spacetime around a massive body like the Earth (i.e., not in the interior but the vacuum region around it), there are at least two curvature numbers (the radial and tangential tidal gravity), and they have different signs (the radial number is negative and the tangential number is positive). This curvature is entirely Weyl curvature, because Ricci curvature is zero in any vacuum region (in the absence of a cosmological constant--see below).

These statements apply specifically to Ricci curvature; they do not apply to Weyl curvature. Also, "stress energy tensor" here has to include the cosmological constant: a positive cosmological constant (in the usual sign convention) corresponds to negative curvature, i.e., "antigravity".

22. Feb 28, 2017

### mieral

Does dark energy only occur in between galaxies because it's the only place where spacetime can be entirely negative curved (so gravity entirely repulsive)?

For sake of discussion. What would happen if concentrated dark energy were acquired and send to earth and stored in a lab. Would it anti-gravitate upward to the sky or fall to the ground? and how to you draw the geodesics of these. This is asked just to understand more about the versatility of spacetime and the different form it may take.. not because of science fiction or crackpotty.

23. Feb 28, 2017

### Staff: Mentor

No. It's everywhere (at least in the simplest model, which right now we have no reason to think is not the correct one). But its effects are only measurable over large distances.

You can't concentrate dark energy (again, in the simplest model). It's the same density everywhere.

Dark energy itself doesn't move at all. Its density is the same everywhere. It causes objects that are in free fall (i.e., acted on by no forces) to accelerate away from each other. More precisely, it causes a form of spacetime curvature (tidal gravity) that does that (negative curvature).

24. Feb 28, 2017

### Comeback City

Would this only be observable on VERY large scales, or is it noticeable if, say, 2 objects are free-falling to Earth side-by-side?

25. Feb 28, 2017

### Staff: Mentor

No. By "large scales" I mean "hundreds of millions of light-years or more".