Gravitation/general relativity (minus math)

[Eolill]

So, hi. I am taking a course (astrophysics) for which we are required to write a paper (very freely, about 5 pages, no real limitations) about a subject related to, well, astrophysics. I chose gravitation because I guess I have always been fascinated by it.

I realized rather quickly that this would take me into general relativity, but it took me a while to realize just how big of a problem this would be to someone with my limited mathematical background. I know calc, trig, geometry etc and some very limited linear algebra, but it's not viable for me to learn tensor calculus or more advanced lin. alg. right now. Instead, I would like a more conceptual understanding of the subject. I have had troubles, however, finding anything that isn't either so dumbed down so as to be simply trivia, or so full of mathematical jargon that I can't understand what is going on.

My current understanding of the subject is this: I understand the gravitation/acceleration equivalence, and additionally I understand that in GR gravitation is not a force but instead accredited to the curvature of spacetime, meaning that particles are actually moving in straight lines even though they appear curved in 3D space. "Curvature of spacetime tells matter how to move, matter tells spacetime how to curve". I understand that in dealing with GR one uses very small scales, and everything is local, since only locally is flat timespace a good approximation for curved timespace, and that the math for vectors and coordinates and everything becomes very complicated.

As you can see, my current understanding is sketchy and shaky (I'm hoping I didn't get anything wrong; if I did, please correct me) and easily summarized. I have a few specific questions and a request for a general explanation of the subject suited to my level; I'm not an idiot interested in trivia or in repeating big words like a parrot, but my mathematical background is not enough to, at this point, tackle the tensor calculus and stuff.

My questions are these:

1) Timespace is supposedly "curved". A 2D surface can only be curved in 3 dimensions; spacetime is 4 dimensions and should, then, only be "curved" in 5D. Yet, in the paper by Baez/Bunn (http://arxiv.org/PS_cache/gr-qc/pdf/0103/0103044v5.pdf" ), they said that we need not involve higher dimensions; indeed, that the tensor calculus took care of it without going to higher dimensions. So how is it curved then? What am I not getting?

2) Is gravitation a force or not? How could it be, if the particles are just going in the straightest possible line and are not "attracted" to each other? But if it is not, then why are we trying to unify gravitation with the other forces and trying to find force carriers for it and stuff?

3) How does this "straight line" constitute acceleration? The particle should move with constant velocity anyway, should it not, and only the direction change?

4) If one walks in a perceived straight line on the 2D surface of a ball, the line is straight in 2D but actually curved in 3D. Yet, in GR, the line which is straight in the higher dimension (4D) seems curved in the lower (3D). While I intuitively feel that if straight in lower dimension can mean curved in higher, then the opposite should also be possible, I can't think of a good example offhand. Am I misunderstanding this and should be looking at it in a "locally straight" kind of way instead?

5) Does anyone know the string theory take on gravitation? (this doesn't belong here, I know, but even so).


Thank you very much for your time and patience. ^^

/Anna

 

[Mentz114]

Hi,
I'll start the ball rolling by trying to answer

1) Timespace is supposedly "curved". A 2D surface can only be curved in 3 dimensions; spacetime is 4 dimensions and should, then, only be "curved" in 5D. Yet, in the paper by Baez/Bunn (http://arxiv.org/PS_cache/gr-qc/pdf/0103/0103044v5.pdf), they said that we need not involve higher dimensions; indeed, that the tensor calculus took care of it without going to higher dimensions. So how is it curved then? What am I not getting?

You need to find out about special relativity so you get the concept of 4 dimensional spacetime with the Lorentzian metric. A metric tells us how to calculate distances, and in the Lorentzian case, space and time have opposite signs so the distance ( or proper length/time ) is given in differential terms as

[tex]ds^2 = c^2dt^2-dx^2-dy^2-dz^2[/tex]

You can reverse the signs on the RHS and still have the same structure.

The curvature in GR is intrinsic and can occur when the differentials on the RHS have coefficients that are functions of position.

[tex]ds^2 = \sum g(t,x,y,z)_{ab}dx^adx^b[/tex]

The indexes here refer to the 4 dimensions, not powers. So x⁰ is t, x¹ is x and so on. The distribution of matter and energy tell us how to calculate the g's, which is where the gravity is.

 

[Mentz114]

Part 2:

2) Is gravitation a force or not? How could it be, if the particles are just going in the straightest possible line and are not "attracted" to each other?

As you say, how could it be ? In GR when the g's are determined, we can use them to calculate all the possible paths that exist for matter and light. Solving equations of motion for a certain case then requires determining which of those ( infinity) of paths is the one that belongs to our case. Very different from the Lagrangian/Hamiltonian mechanics where the motion is played out in phase space and forces push things around.

 

[diazona]

1) Timespace is supposedly "curved". A 2D surface can only be curved in 3 dimensions; spacetime is 4 dimensions and should, then, only be "curved" in 5D. Yet, in the paper by Baez/Bunn (http://arxiv.org/PS_cache/gr-qc/pdf/0103/0103044v5.pdf" ), they said that we need not involve higher dimensions; indeed, that the tensor calculus took care of it without going to higher dimensions. So how is it curved then? What am I not getting?

Actually, a 2D surface can be curved in ways that cannot be represented in 3D space. (At least, I know a 4D surface can be curved in ways that cannot be represented in 5D space, and I think the same idea is valid down to 2D) You just never hear about them because it's impossible to draw them (and they're usually not that interesting). This is where the tensor algebra becomes useful - the metric tensor gives you a useful way to describe curvature without having to draw a surface in some higher-dimensional space. That's what Mentz114 was getting at ([tex]g_{\alpha\beta}[/tex] is the metric tensor).

2) Is gravitation a force or not? How could it be, if the particles are just going in the straightest possible line and are not "attracted" to each other? But if it is not, then why are we trying to unify gravitation with the other forces and trying to find force carriers for it and stuff?

In physics we have this thing called duality, which just means that one physical phenomenon can be described by two different theories. An example would be electric and magnetic fields - when a magnet moving by a wire loop induces a current in it, you can describe it either using the change of magnetic flux through the loop or by transforming to a different reference frame where the magnetic field becomes an electric field which directly pushes the charges around. (Well something like that anyway) Or a famous example: an electron (for example) can be represented as either a particle or a quantum-mechanical wave(function). Gravity is just another example of this - we have two theories of gravity, one which describes it as a force and the other which describes it as curvature of spacetime. (Currently the former - Newton's theory - is somewhat imperfect, but physicists still hold out hope that that can be fixed)

3) How does this "straight line" constitute acceleration? The particle should move with constant velocity anyway, should it not, and only the direction change?

Well, for one thing, since spacetime is curved, "straight lines" are not straight anymore ;-) We call them geodesics, i.e. the shortest possible path(s) between two points in spacetime. Another thing to consider, though, is that particles actually do move through spacetime with a constant "4-speed," which is basically related to c, the velocity of light. The thing is, for most particles, most of this velocity is devoted to traveling through time rather than through space, and since we the observers are traveling through time along with them, we don't notice the time part of the velocity. It's only when it changes spacetime-direction, so that the 4-velocity has a component in a spatial dimension, that we notice the particle moving.

4) If one walks in a perceived straight line on the 2D surface of a ball, the line is straight in 2D but actually curved in 3D. Yet, in GR, the line which is straight in the higher dimension (4D) seems curved in the lower (3D). While I intuitively feel that if straight in lower dimension can mean curved in higher, then the opposite should also be possible, I can't think of a good example offhand. Am I misunderstanding this and should be looking at it in a "locally straight" kind of way instead?

"Locally straight" may be a better way to think about it - as I mentioned before, in curved spacetime there are generally no truly straight lines. In a flat (non-curved) spacetime, you could have straight 4D lines, and those correspond to free particles moving at constant velocity in straight 3D lines through space.

5) Does anyone know the string theory take on gravitation? (this doesn't belong here, I know, but even so).

String theory reportedly predicts (or "postdicts" perhaps) a particle that has the properties one would expect to see in the force carrier for gravity. (Just as the photon is the carrier for the electromagnetic force, the graviton is the carrier for gravity) That's what got people excited about it as a theory of quantum gravity. But the calculations are so complicated it's taken people 20 (30? 40?) years of effort and they still haven't figured it all out...

 

[Eolill]

Thank you both very much! I believe I understand better now (or, for a couple of things, at least I understand what it is I don't know). ^^ You were very helpful.

 

[Vanadium 50]

Some advice? Don't write a paper on something you don't understand.

It's possible to understand the details of a theory and then condense it into a simplified sketch. It's not possible to understand only a simplified sketch and expand it into the full theory.

 

[VKint]

Actually, a 2D surface can be curved in ways that cannot be represented in 3D space. (At least, I know a 4D surface can be curved in ways that cannot be represented in 5D space, and I think the same idea is valid down to 2D) You just never hear about them because it's impossible to draw them (and they're usually not that interesting).

True--although there are some examples of this that are quite interesting, in fact. The most well-known is the Klein bottle, a surface which cannot be embedded in three-dimensional Euclidean space. It can be shown, however, that any [tex] n [/tex]-dimensional "surface" (manifold) can be embedded in [tex] 2n [/tex]-dimensional Euclidean space (e.g., the Klein bottle can be embedded in 4D space).

The relevant point is that the kind of curvature we care about in relativity is, as has been pointed out, "intrinsic," that is, having to do only with the metric properties of the space. Intrinsic properties of a space do not depend on any embeddings you choose; conversely, there are examples of surfaces that look very different when embedded in 3D, but have the same intrinsic curvature. As an example of the difference between "extrinsic" and "intrinsic," consider a regular curve in 2D. Perhaps you've encountered the "curvature" of such a curve before (usually denoted [tex] \kappa [/tex]); it measures how fast the unit tangent vector changes. But the curve has no intrinsic curvature, in the sense that a one-dimensional person living inside the curve would not be able to tell the difference between his world and a straight line. On the other hand, a 2D person living inside a sphere could tell that the sphere is curved, by constructing a large triangle and measuring how much the sum of the angles deviates from [tex] 180^{\circ} [/tex].

As I mentioned above, there are examples of surfaces with the same intrinsic curvature (usually called Gaussian curvature in the context of surfaces) that look very different. For example, the helicoid and the catenoid have the same Gaussian curvature at corresponding points (we say that they are isometric), but look quite different. (See <http://www.ma.utexas.edu/users/cpatters/m210e07.html> for a helicoid, and <http://www.indiana.edu/~minimal/maze/catenoid.html> for a catenoid.)

 

[Eolill]

Could anyone explain the way gravitational waves (which arise when, for instance, two neutron stars are orbiting each other) and gravitomagnetism (when a heavy body spins and drags spacetime around with it) work?