# How is space-time curved?

1. Feb 5, 2009

### hermy

i recently read A Brief History of Time. it says that the orbits of planets may seem elliptical, but are actually straight in space-time. so how does that happen?? how does space-time curve?? here does time refer to imaginary time??? and what do we really mean by imaginary time? is it taken that way just for mathematical convenience?

2. Feb 5, 2009

### Nabeshin

An analogy that might help you understand how the orbits can "seem" curved is an airplane flying on Earth. Over long distances, the "straight" path of the plane actually looks like a great arc (you know this if you've ever flown from say America to Asia and you fly over the arctic circle).

What this all amounts to is that in flat space the shortest distance between two points is what we see as a straight line. But when space curves, this is no longer the case, and so the objects are simply following the natural trajectory for the curve.

3. Feb 6, 2009

### hermy

thanks nabeshin. i understood that bit, but i still can't understand why space-time should be curved in the first place. isn't spacetime just a combination of 3-d space and time? i think i am going wrong in the concept somewhere... if yes, do correct me.

4. Feb 6, 2009

### marcus

One could also ask why space should be nearly flat.
Why should it be that the angles of a triangle in most cases add up to only a little bit different from 180 degrees?

It's an astonishing coincidence and absolutely requires some scientific explanation.

5. Feb 6, 2009

### DaveC426913

Space-time is curved because that is what the presence of matter does to it via gravity.

"Space tells mass how to move" while "mass tells space how to curve"

Thus is the nature of the universe.

6. Feb 6, 2009

### Nabeshin

Unfortunately, this is the truth of the matter. We don't really know why matter curves space, we just know that it does.

7. Feb 7, 2009

### v2kkim

Why matter curves space is an interesting question, and in the future we might find a little better understand through physics advance like string theory etc, but we will still keep 'why' in our mind. The ultimate pursuit is interesting but might be impossible to get answer for human who has only human-like thinking, for example, we are confined in 3 space dimensions and 1 time dim., but the new string theory says there are more dimensions and gravitation seems to be treated with specialty. That theory says in sub-atomic scale there are many extra dimensions but in larger scale we see only regular dimensions. I am just trying to make connection that come to my mind on 'why'.

8. Feb 7, 2009

### jambaugh

My \$0.02 worth...

To your last question, imaginary time, yes this is a mathematical convenience.

The use of the word "straight" here is not quite right. The analogy is this picture you and another person walking south from the north pole starting in slightly different directions. You both begin moving away from each other but by the time you reach the equator you're distance apart stops growing and starts shrinking until as you reach the south pole you meet. You are walking "straight" in the sense that you neither turn left nor right but your paths are curved in the sense that the distance between you doesn't change uniformly. This is because the surface of the earth is not flat but curved... i.e. curved 2-dimensional space.

In SR space and time are components of 4-dimensional space-time so you can imagine objects traveling at different velocities as traveling in different space-time directions (at a rate of one space-time unit per proper-time unit.) In GR this space-time is curved so as you "walk" in space-time your direction=velocity can change. You are accelerated relative to other object's paths.

9. Feb 7, 2009

### Nabeshin

String theory (if it were correct) might shed some understanding into a unification of forces, but it's not going to get much closer to why things happen. For example, why are things strings instead of points? At a certain point you just have to lay down an initial set of postulates and accept them. For the case of this discussion it's that mass (through gravity) warps spacetime.

10. Feb 7, 2009

### DaveC426913

I can't say why it might be this way of course, but I get the gist of what strings solve.

If particles are ponts (0 dimensions) then the transfer of energy between two points must occur at a rate that is infinite (according to HUP the smaller distance and time, the larger the transfer rate will be).

But strings solve this problem by providing a mechanism whereby the energy transfer is NOT at an infinite rate becaue it occurs over a non-zero distance and a non-zero time.

It is greatly simplified I grant but, if I am indeed understanding it correctly, not only does it resolve the problem of particle interaction but it also reconciles SR/GR with QM quite elegantly in one fell swoop that's within the grasp of a layperson.

11. Feb 8, 2009

### Nabeshin

Yeah, this is what I mean. We find theories which work better, are simpler, more concise, or what have you. But yet your description doesn't give a why beyond "it resolves problems". I think this is actually a very important thing to understand because a lot of people think science attempts to delve into the why of matters, but it really cannot do more than describe (albeit in much more elegance).

12. Feb 9, 2009

### tiny-tim

Hi hermy!

I think you're confusing the spacetime of special relativity with general relativity …

SR spacetime is flat (as you say, it's just a combination of 3-d space and time) …

GR spacetime is curved (locally, it is flat, and looks exactly like SR spacetime, in the same way that the Earth's surface locally looks flat ) …

an orbit is locally a straight line in 4 dimensions, even in GR spacetime, but when you stick the whole orbit together and try to map it onto something you can visualise in 3 dimensions, then the orbit looks curved for the same reason that the 2-dimensional Equator (of the Earth) has to look curved if you try to visualise it in 3 dimensions.

13. Feb 10, 2009

### TalonD

I am a layperson, tell me in simple language, how string theory reconciles sr/gr with qm ?

A question of why....
we can answer some 'why' questions as our knowledge advances. Just because we can't say 'why' mass causes spacetime to curve right now doesn't mean we won't be able to answer that someday. Although, if you think about it, it seems kind of obvious given what we know so far, but the obvious isn't always right especially without a mathematical framework to back it up.

14. Feb 10, 2009

### jambaugh

Disclaimer: I personally think string theory is a dead end but I am not well versed enough to make an informed judgment. That having been said I'll give it a try...

First let me try to give you an idea of gauge field theories. Imagine that at every point in space there is an extra degree of freedom corresponding to an abstract rotation. We can model this rotation as a complex phase $e^{i\theta}$. Phase rotations form a group SO(2)=U(1). If you then want to define a particle's configuration you must give its position and in addition its phase. The thing is that we can redefine the zero phase arbitrarily, and do so at each point in space. We then must consider how phases at different points match up so that we can define the change in phase as a particle moves. This defines a vector which becomes the electromagnetic vector potential. Similarly we can define more involved gauge groups at each point in space and these successfully model the weak and strong forces via SU(2) and SU(3).

Now another way to treat these abstract degrees of freedom is to assume extra dimensions and "do" general relativity in this extended space-time. These are called Kaluza-Klein theories as Kaluza developed such a theory of electromagnetism. You can think of the extra dimension(s) as curled up so small we don't ever see them. Elementary particles are cycling around these continuously and their "momentum" about this cyclic dimension corresponds to their gauge charge. The thing is this is all in the classical domain and a satisfactory quantum theory incorporating GR doesn't yet exist. String theory in part posits that this is because we are still working with point particles moving through these extra dimensions.

String theory and its higher dimensional generalization brane-theory describes a quantum version of vibrating n-dimensional membranes in a larger dimensional (flat) space-time. Since we can view curved space-time as a 4-dimensional hyper-surface in a larger flat space-time one may attempt to quantize curved space-time as such a quantum "brane". We can also add extra dimensions to invoke those gauge charges mentioned above. Similarly matter can be viewed as smaller "sub-branes" manifesting either as topological defects or as particular modes of quantum excitations of this vibrating space-time-gauge "brane".

This is a preliminary and quite naive view of the actual "theories" and as I said I'm not totally up to speed with all the current research. I may also have misrepresented some aspects. I, believing this to be a wrong approach, have not paid enough attention to give a better description. Others may point out mistakes in my description but I hope I've given you a sense of how the string/brane theorists hope to link up GR and QM in this way.
(As to why I see it as a wrong approach that's a topic for another thread.)