# How does spacetime curve?

1. Jul 2, 2012

### mijfin

Hi all, I don't know much about physics other than what I've read about it and learned from an introductory astronomy course. I've been told to imagine spacetime like a rubber sheet that bends in when you drop something with mass on it, and I've seen diagrams of how this works. What I have trouble grasping, though, is that I've always thought of this space as being empty, without any mass in it- basically as nothingness. So my question is, what is it that allows spacetime to curve like this? Hopefully I don't have to be a physics major to understand the answer :P

2. Jul 3, 2012

### tensor33

Thats a good question. From what I can tell, you're thinking of spacetime as a backdrop for events that occur in the universe. That is how physicists thought also. That is
, before Einstein came along. His theory of General Relativity (which has been proven numerous times) proposed that the matter in the universe actually affects the spacetime around it. You can think of it like this. In empty space, the shortest path between two points is a line. But in the vicinity of a massive object like a star, the shortest distance between two points isn't a line, it's actually a curved line.This is what physicists mean when they talk about the curvature of spacetime. It's hard to explain this concept without math so this is the best I can do.
I hope it helps
P.S.: Welcome to the forum!

3. Jul 3, 2012

### Staff: Mentor

Hi mijfin, welcome to PF!
Well, you have to have some mass/energy somewhere, at least as a boundary condition, otherwise spacetime is flat. What allows spacetime to curve is the presence of energy, momentum, and pressure. These quantities are components of what is called the stress-energy tensor, and it is the "source" of gravity in GR. Here are a couple of links to start with:

http://en.wikipedia.org/wiki/Stress–energy_tensor
http://en.wikipedia.org/wiki/Einstein_field_equations

4. Jul 4, 2012

### TobyC

Aren't there non-trivial solutions to Einstein's equation with no mass/energy? I'm pretty sure you can have curvature without any mass/energy present in principle. A zero einstein tensor doesn't imply a flat metric. Gravitational wave solutions for example should satisfy this property?

5. Jul 4, 2012

### Staff: Mentor

That is what I meant by the "boundary condition" comment. You can set up a manifold without any stress energy simply by excluding any regions with stress energy from the manifold. then the effect of the matter is entirely contained in the boundary conditions.

6. Jul 4, 2012

### bobc2

mijfin, you've posed a quite provocative question. And it's not clear whether you are intending to probe a philosophical aspect of the curvature and matter/stress-energy problem. You seem to probing details about how something "causes" or influences space-time to curve.

Apparently physicists have abandoned any attempt to assign a cause (in a fundamental philosophical sense) to the space-time curvature. First, physicists do not appear to regard space-time as something that exists as a four-dimensional structure. That is, from this point of view you should not picture a structure with actual physical curvature. It's just all mathematical relationships. Einstein's equations do give us mathematical relationships between geometry-like expressions and physical-like objects, but these are just mathematical relationships. You can make flippant comments like, "Mass tells space how to curve and space tells mass how to move..." But these kinds of statements don't really seem to provide enligtenment.

Now, there is a concept known as the "block universe" model which affirms a 4-dimensional structure with real physical curvature. But this model is not generally accepted by physicists on this forum. In any case, this kind of model eliminates causality completely--except as a mathematical relationship useful as a tool in organizing rules of physics. These rules, in the final analysis, are to be taken only as useful mathematical tools--concepts applied in the practice of physics.

7. Jul 4, 2012

### Staff: Mentor

bobc2, can you support this statement with references?

bobc2, you are confusing non-acceptance of your particular "block universe" model with non-acceptance of spacetime as a physical entity. The two are not the same. It is perfectly possible to regard spacetime as a real, 4-dimensional structure, without adopting your version of the "block universe" model.

8. Jul 4, 2012

### Fredrik

Staff Emeritus
What you're really asking here is "Why is GR a good theory?" *. Unfortunately, the only thing that can answer that is a better theory of gravity. Since it seems to be hard as **** to find such a theory, I can assure you that the first time someone writes down an answer, it won't be understood by any physics majors. I would be surprised if there's an answer that can be understood by undergraduate students.

*) Compare this to how someone who studies Newton's theory might ask "How does a massive object know when another massive object has been moved?" The only good answer that we know, is provided by GR and involves spacetime curvature.

9. Jul 4, 2012

### bobc2

It would take quite a while to round up the collection of references to satisfy you. Here is one quick quote that I think would find agreement among most of the physicists of this forum.

"General Relativity", Robert Geroch, page 12: If you like, "four dimensions" is just a convenient way of describing the world and thinking about the world, nothing more. Is the "fourth dimension" real? It should now be clear, from these remarks and from the discussion of "reality" in chapter 1, that physics will not answer such a question, and that the attitude of physicists will be that such a question is not germane...

I think mijfin should follow the suggestions provided by DaleSpam and Fredrik, because that is the way physicists have pursued General Relativity--that's what we have as a theory. That's the real physics--it's what physicists do.

If we are both referring to real existing 4-dimensional physical structures, I think they are the same. Could you explain the distinction you have in mind?

Last edited: Jul 4, 2012
10. Jul 4, 2012

### pervect

Staff Emeritus
The only thing you need to distinguish the geometry of a plane from the geometry of the surface of a sphere is a ruler.

Given that space supports rulers, you have all the tools you need to measure and define curvature. To consider the curvature of space-time, you additionally need clocks as well as rulers, as you need to be able to measure time intervals as well as spatial intervals.

The detailed technical definitions of exactly how curvature is a tensor do get rather mathematical and abstract. But with a tiny bit of knowledge of basic plane geometry, you should see some important differences between the 2d geometry of the surface of a sphere and the 2d geometry of a plane, even though both of them are "empty". What is important is that you are able, conceptually, to mark points on the "empty" space - or the "empty" space-time - if you choose, so that you can measure distances.

The quickest way , perhaps, to tell the difference between a plane and the surface of a sphere is to consider the sum of the angles made by a triangle. Howeer there are other ways that do not involve measuring angles, ways that only require measuring distances, such as measuring the diagonals of a square.

A little bit of thought should convince you that if you have a four-sided figure with four equal sides , and two equal diagonals, that the diagonals on a plane are sqrt(2) times the sides on a plane, and that they are NOT sqrt(2) times the sides if you draw the same square (four equal sides, two equal diagonals) on the sphere.

So, you don't need anything particularly "mystical" to determine whether your geometry is flat or not. It is both necessary, and sufficient, to have a ruler and a way of marking points to determine whether a geometry is flat or not.

11. Jul 4, 2012

### Staff: Mentor

"The question is not germane" is not the same as "the answer is no".

I agree. How does what you're saying about the "block universe" relate to that?

That would involve rehashing all the previous threads we've had on this topic. But briefly, when I say spacetime is a real 4-dimensional structure, I mean we have a 4-dimensional *model*, some of which we have confirmed against reality and some of which we haven't. When you say spacetime is a real 4-dimensional structure, you mean that events are "real" that are spacelike separated from us, meaning that the events you say are "real" may never actually happen (because some cause from outside our past light cone may change them). In other words, you think that believing in a 4-dimensional spacetime commits you to asserting the reality of the *entire* spacetime at once; I don't.

12. Jul 5, 2012

### bobc2

If physicists did affirm the existence of a "real" 4-dimensional universe they would consider the question germane. They do not, and the question is not germane.

You are talking about 4-dimensional space-time in the context of philosophical "presentism." I don't think the forum members are interested in this kind of discussion.

13. Jul 5, 2012

### Staff: Mentor

The question is not germane because the word "real" in this case doesn't correspond to any physical property or any experimental result. Suppose you say that the entire 4-dimensional spacetime continuum is "real", and I say it isn't. What experiment could we run to decide who is right?

If, OTOH, you ask "is spacetime curvature real", the answer is yes, because we can run experiments that show its existence. More precisely, we can show that tidal gravity exists, and tidal gravity is the same thing as spacetime curvature. You could still object that my saying tidal gravity = spacetime curvature is an "interpretation", but that's an argument about words, not physics.

As another example, if you ask "can spacetime curvature propagate dynamically as waves", the answer is yes, at least provisionally--the waves are gravitational waves, and we are running experiments now to try and detect them.

So whether or not "spacetime is real" depends on what you mean by "real". That's why physicists don't view the question as germane as it stands; you have to modify the question to something that can be linked to an experimental result. If you do that for spacetime, the answers to the corresponding questions, as above, are "yes". So in that sense, physicists do think spacetime is "real".

No, I'm not. Nothing I've said requires "presentism". I'm merely pointing out that, as above, there are multiple ways to interpret the statement "spacetime is real".

14. Jul 5, 2012

### bobc2

You seem to be reincarnating logical positivism and operationalism. When physicists talk of these matters they know quite well what they are envisioning when speaking of an external physical reality. The prototype is what we observe in our everyday world, physical objects occupying a 3-dimensional space.

It is easy to extend the concept to 4-dimensional objects occupying a 4-dimensional space-time. We refer to that concept as a block universe. The physics philosophers often represent four basic concepts of reality as represented by the diagrams below.

These are not abstract concepts. If you are not presenting an external reality concept of this kind, you probably have in mind something abstract and more in the line of a mathematical construct rather than an external physical reality of the kind motivated by our observation of a simple external physical 3-dimensional world.

And that’s fine. We can go off into lots of ways to discount a physical reality of the kind that the pedestrian physicist might contemplate. I’ve mentioned several times before about my philosophy of physics professor who maintained that the class room down the hall did not exist until he walked down the hall and opened the door (we physicists in the class just looked at each other with puzzlement).

So, again, there is probably nothing fruitful about pursuing this kind of discussion, particularly when the forum members are probably tiring of it and the monitor is about to pull the plug.

Appologies to mijfin for getting so far off point. I certainly did not intend to hijack your topic.

Last edited: Jul 5, 2012
15. Jul 5, 2012

### Staff: Mentor

Apparently we *are* going to rehash all those other threads. But I'll try to keep it short since I have nothing much to say on all this beyond what I've said before.

Let me ask a simple question about the four different "basic concepts" you describe: how do we tell which one is right? If your answer is that all of them are consistent with our current knowledge, then you've basically punted; the question of which one is right is simply not one we can answer. Which is fine (in fact it's basically the position taken in the quote from Geroch you gave earlier), but then you shouldn't be talking as if one particular answer is generally accepted, when the whole point is that *no* particular answer is generally accepted, because we simply don't know enough.

16. Jul 5, 2012

### zonde

Then you might want to try this Einstein's marble table analogy that explains it in somewhat different way http://www.marxists.org/reference/archive/einstein/works/1910s/relative/ch24.htm

Interesting thing to notice is that rubber sheet analogy and marble table analogy are not equivalent. Say you can't model surface of sphere with marble table analogy and from other side if you would have conditions so that little rods shrink in some area and do not connect any more you can't use rubber sheet analogy to describe this situation.

17. Jul 5, 2012

### zonde

What is the point of 4th concept? It is obvious that you would instantly notice the change if you would swap time dimension with spatial dimension. We would see extended spaghetti like structures along that direction that we do not observe along ordinary space directions. So time dimension is qualitatively different.

18. Jul 6, 2012

### Staff: Mentor

The fourth one is simply wrong, as we have discussed at length previously. If the fourth one were correct then light would not propagate.

19. Jul 6, 2012

### bobc2

I didn't think I was talking as though any particular one was accepted. I was just noting my impression that you seemed to be opting for presentism while using a description more consistent with a block universe. But, you have clarified that.

My personal hunch about what is implied by aspects of time dilation and length contraction in special relativity and curved space in general relativity is not relelvant here. I was trying to convey to mijfin a general attitude among physicists about reality and the implications for cause and effect.

20. Jul 6, 2012

### bobc2

DaleSpam, in that model a photon of light would simply be a very long 4-dimensional filament oriented so as to bisect X1 and X4 coordinates for any observer. I don't think we know enough about the fundamental nature of time nor consciousness to know whether this concept is plausible. But discussion of that is way off topic here and has been determined to be of no value to physics by our forum monitor.