You're welcome!
Yianni said:
Thanks for that, it clarified a lot.
Always good to hear!
Yianni said:
The reason I mention the idea that space itself is moving was because in Brian Greene's book The Fabric of the Cosmos he suggests that if the universe is expanding uniformly on the large scale, then no time dilation should occur between people on distant galaxies assuming their velocities relative to one another are zero other than the velocity caused by the expansion of the universe; i.e. they look at each other and their watches remain in sync.
Right, well I think of that scenario as a picture, but in words I'd put it like this: the world lines of the dust particles form a congruence (family) of timelike geodesics which fill up the spacetime model (an FRW dust solution of the Einstein field equation). Furthermore, there is a unique family of "spatial hyperslices" for these observers, Riemannian three-manifolds which are everywhere orthogonal to the world lines of the dust. (In general, if the dust were "swirling" it would not be possible to find a family of spatial hyperslices.) For concreteness, here is the "line element" of the FRW dust with E^3 hyperslices:
<br />
ds^2 = -dT^2 + T^{4/3} \, \left( dx^2 + dy^2 + dz^2 \right), \;<br />
0 < T < \infty, \, -\infty < x, \, y, \, z < \infty<br />
Visualizing this requires a four-dimensional figure, but since all spatial directions are equivalent in this model (we say it is "isotropic"), let's just set z=0. Then you can visualize the "map" (as in representation of a curved space like the surface of the Earth) --- or, to use the correct term, the "coordinate chart"--- defined by this line element as the upper half space 0 < T < \infty, \, -\infty < x, \, y <\infty, with the plane T=0 representing the Big Bang. Then the world lines of the dust have the simple form x=x_0, y=y_0, i.e. "vertical half-lines", and the orthogonal slices have the form T=T_0, i.e. "horizontal planes". To study radio signals exchanged by the dust particles, we recall that the world line of a "photon" is a null geodesic, so we can set ds=0 in the line element, which defines the "distance" in a small piece of our coordinate chart. Here comes an example of why first year college calculus is fun and useful! From the line element we obtain
<br />
dT = T^{2/3} \, \sqrt{dx^2 + dy^2} = T^{2/3} \, dr<br />
or dr/dT = T^{-2/3}, so that the "world sheets" formed by taking all the signals from some event on the world line of some dust particle have the form of a surface somewhat like a paraboloid, whose sides gets steeper as T grows. In flat spacetime the analgous surface would simply be a half-cone with constant slope unity, so the shape of these distorted light cones (as drawn in our coordinate chart) is a consequence of the fact that at the hyperslices for a later time T_1 > T_0, a coordinate difference x_1-x_0, where y = y_0, \, T=T_1, corresponds to a larger distance, \Delta s = T_1^{2/3} \, \Delta x > T_0^{2/3} \, \Delta x. This is the Hubble expansion: the dust particles move away from one another, but at a slower and slower rate, as time increases. Note that even though in our chart the world lines appear to be maintaining the same "horizontal distance", appearances are misleading, just as appearances can be misleading in a map of the Earth which typically misrepresents distances.
There is a very nice picture of the "distorted light cones" described above in the excellent popular book
The First Three Minutes by Steven Weinberg, BTW. It is quite possible to understand the geometry even before studying differential calculus if one properly interprets such a picture!
The point is that this is coordinate chart is said to be "comoving with the dust particles" in that the coordinate planes T=T_0 correspond to the spatial hyperslices orthogonal to the world lines of the dust particles, and also differences \Delta T taken along the world line of a dust particle correspond to intervals of proper time as measured by an ideal clock carried by that dust particle. Indeed, we can imagine that observers riding on a small group of nearby dust particles synchronize their clocks by exchanging light signals, and although this gets a bit tricky, it is clear enough that a successful synchronization would yield values of the coordinate T. (Abstractly, a coordinate is just a monotonic function on some manifold, i.e. which has nonzero gradient. A choice of such functions on some "neighborhood", such that their gradients are nowhere parallel, gives a local coordinate chart defined on that neighborhood.)
If we imagine "time running backwards", then as T \rightarrow 0 "from above" we see that the distance between any pair of dust particles (that is, the "horizontal distance" between their world lines) must decrease to zero, and if we compute the density of dust from the Einstein tensor in this solution, we find \rho = \frac{1}{6 \, \pi \, T^2} showing that indeed the density blows up as T \rightarrow 0. So we can even say that T is a kind of "universal time" for the dust particles in which "time zero" corresponds to the "Big Bang".
The hyperslices T=T_0 in this model have the line element (put dT=0 in the spacetime line element)
<br />
ds^2 = T_0^{4/3} \, \left( dx^2 + dy^2 + dz^2 \right), \;<br />
-\infty < x, \, y, \, z < \infty<br />
which is the line element of euclidean three-space, so we can say that the hyperslices in this FRW model are "locally flat", meaning locally isometric to euclidean three-space, E^3. In fact, they are globally isometric to euclidean three-space in the model I have described, but with a seemingly simple change (identify the "vertical plane" x=1 with x=0) we obtain a model with spatial hyperslices which are locally flat but globally isometric to "cylinders". We could also have hyperslices which are locally flat but isometric to "flat tori". If this intrigues you, see Jeffrey Weeks,
The Shape of Space.
This cosmological model is highly idealized: the dust particles roughly correspond to galaxies, but of course real galaxies are clumped into "clusters", there are large "voids" free of galaxies, and so on, and the galaxies are not really "pointlike" but have complex and interesting substructures (such as our Solar system). Nonetheless this simple model succesfully reproduces the basic features of the observed Hubble expansion (it does even better at very large distances if we add a small "Lambda terms" to the right hand side of the Einstein field equation to the term representing the dust). It is worth mentioning that it is possible to construct more sophisticated cosmological models which are "perturbations" of our FRW dust model but which attempt to model inhomogeneities in the distribution of galaxies or to otherwise improve on the FRW dust models.
Yianni said:
I suppose I mentioned it because I thought the idea behind 'spacetime curvature' was that if everything in a region is accelerating uniformly, then instead of describing those objects as accelerating, you described the space as such. I.e. the space is permeated by theoretical clocks and rulers at every real number along each of the three spatial axis, which are accelerated by gravity, and this is called space. I suppose that idea was just plain wrong though!
Well, there might be something correct lurking in there, but I hope the above helps in appreciating how specialists in gravitation physics tend to think of this kind of scenario.
Yianni said:
I'm only just finishing High School but my maths is pretty decent (I don't mean I actually know that much, I live in Australia and the standard of maths in schools here is quite appalling, but I can pick up on ideas pretty easily).
Can't be worse than (generic) American schools! (But that's a topic for general discussion). I have the impression you can probably get the right "picture" from the right books.
Yianni said:
Could you recommend any textbooks for learning both SR and then GTR? Or one for learning SR and then one for at least some sort of light introduction to the maths of GTR? I'm learning some of the ideas in SR from the Feynman lectures, but I think I really need a textbook to get into the nitty-gritty of it. Also, to learn GTR, what maths would be necessary before even opening up the textbook? Obviously some level of calculus and vector maths, but what else?
Linear algebra would be very very useful. Gtr uses the languages of tensors, which are slight generalizations of the "linear operators" of linear algebra. Also, linear operators can be represented by matrices, but this representation depends upon a choice of "basis", which sets up the idea of tensor fields as something you can write down in terms of some choice of coordinates, and also the notion of "orthonormal basis" sets up the idea of "frame fields"
I think the textbook by D'Inverno,
Understanding Einstein's Relativity, might be ideal for you; coupled with Feynman's Lectures it just might give you just enough str to get started on gtr. (Maybe not, since I just remembered that Feynman's discussion of Minkowski geometry is a bit murky.) Last I knew, this was available in paper back new for perhaps U.S. $40.00 Alternatively, you could get Edwin F. Taylor and John Archibald Wheeler,
Spacetime Phyhsics, Freeman, 1992 (make sure to make a table of the trigonometry appropriate to Minkowski geometry, called "hyperbolic trig" and to compare it with standard high school trig, or "elliptical trig"), and L.P. Hughston and K.P. Tod,
An introduction to general relativity, Cambridge University Press, 1990, which cost about U.S. $25.00 new, last I checked. All these books will no doubt be available more cheaply used from InterNet booksellers.
Yianni said:
And would the textbook itself explain the "mathematical machinery of the theory of differentiable curves in Lorentzian manifolds," or would this need to be learned before beginning?
Both the gtr textbooks I mentioned aim to be "self-contained" relative to a standard undergraduate curriculum. I would expect that even a bright high school student with some calculus and some linear algebra will find some topics too challenging at first, but you could still learn a very great deal, I think!
(Feynman's Lectures are a great way to learn physics, BTW! I also really like Blandford and Thorne,
Applications of Classical Physics, which is available free at
www.pma.caltech.edu/Courses/ph136/yr2004/index.html The authors are leaders in physics; Thorne and Wheeler are also coauthors with Misner of the classic graduate level textbook
Gravitation, which you can save for later. And I am very glad to see a high school student who appreciates how much better standard textbooks are for learning topic T than arbitrary websites, which tend to be full of misinformation--- the Cal Tech website I just mentioned being a notable exception!)
