Pressure and stress: since pressure is the flow of momentum through a unit area, the walls 'feel' this pressure as an induced sress. This must have the same units as pressure, hence the stress-energy tensor could be called the pressure-energy tensor. Tension is another name for sress, so we get another 'heuristic' way to think about gravity and pressure: a spring with a weight attached so the system is in equilibrium (i.e. at rest) has an induced tension/stress/pressure. This is negligible compared to the 'pressure' in the time direction (see Baez).
As explanations go, I think this youtube video has quite a lot to say although it's been seen here before): http://www.youtube.com/watch?v=DdC0QN6f3G4. It starts with a Euclidean frame (Newtonian), in which triangles sum to 180[sup]o[/sup]. In this frame, time is 'flat' and so is space: all vertical lines are parallel, and all horizontal lines are parallel. So clocks run at the same rate regardless of height above the surface. But Einstein says that gravity curves spacetime. So we next see a frame in which the horizontal lines remain parallel (as arcs of a circle with the same centre), but the vertical lines are now 'rays' extending from a central point (above frame in the example). So this tells us that there is a way to transform all the points along the path the apple follows, from one frame to the other. For instance, all the horizontal and vertical lines intersect at right angles in both frames, the transform must preserve this angle: time remains orthogonal to space (in one dimension). Furthermore, in Einstein's frame a clock at a smaller radius of curvature has a smaller 'time distance' to follow than a clock at a greater radius. Einstein also says gravity and acceleration are indistinguishable. Hence, the apple can be accelerating under the influence of an external force anywhere in spacetime; when it accelerates, it follows a straight path, and when it has constant velocity it follows a curved path. But the diagrams suggest that the Einstein frame is curved because of the presence of a gravitating mass, this implies that without gravity present, acceleration itself is what curves spacetime. That is, the apple follows a curved path when at rest in a gravitational field, but without the field the path, as such. has unknown curvature (??)
Deleted unfriendly post. Based on what's been going on I'd suggest you pick up a text and find out for yourself.
I have a vague idea what it is. It's part of the curvature of spacetime measured by the Ricci tensor. It's the rate of change in volume of a system of particles which are following curved paths; the paths are curved because the particles are contributing to the local mass density (unless they are free falling in a gravitational field, in which case the field may have a much larger effect than the local density of the particles). Or maybe I need to read some textbooks.
I got a real easy one to read. It starts with the metric solutions to the EFE and the principal of extremal aging [Noether's principal of least action]. All the math required to do this book is basic calculus, algebra, quadratic equation for finding limits, and how to do a weak field approximation for working in the weak field. Most the book is deriving the predictions confirmed by the great tests on GR. Really the most interesting stuff for me. It's the book I always access the free downloads from. Taylor and Wheeler's Exploring Black Holes. Still by far the most interesting book I've read. It was created by Prof Taylor to introduce undergraduates to the science of GR with the intellectual support of Prof Wheeler. May he rest in peace. Most the folks who think they know GR ignore this book. Pretty obvious to me. If you want to learn how gravitational science is done it's good to read this book. If you want to read about the history of gravitational physics Kip Thorne's book is great. Not the treatise on GR but his Black Holes and Time Warps. That book made me want to read a text of GR. I think you could read that book even without all the training associated with calculus. Just learn how to find derivatives and integrations. You can do this while you're working on the problems. The path they take you on is really interesting. Later. Good fortune in your quest. Think about this. Not much will be learned from folks you have some adversarial issue with. No matter how hard they try. It's hard to overcome these adversarial issues in a place like this. Everybody is waiting to ponce like it's a show of weakness to be intellectually honest. That's just my opinion. It's getting real close to my periodic internet forum exit. I should read more.
brucep: it would be nice to be able to afford some good books; but I still have indirect access to a university library. Really I'm interested in developing some good heuristics, in particular this business of curvature. That you can use the parabolic path of say, an apple through space to figure out some things about it, in particular what happens to the time dimension looks like a nice one. That video just shows that an acceleration curve can be 'straightened', all you need to do is give the time axis a radius of curvature. You can then refer everything to the centre of this radius, clocks move through the same angle but different distances (hence time dilation); the surface of the earth has a local curvature (in three dimensions) which still looks flat. But time is curved in the opposite sense, so to speak, and by a lot more (something to do with the speed of light).
I've been waiting for the 2nd edition. This is a link to MIT courseware for Exploring Black Holes. 'Lecture notes' is a group of 10 video lectures associated with projects covered in the course. http://ocw.mit.edu/courses/physics/...-general-relativity-astrophysics-spring-2003/
As time is increasing, the distance between galaxies is increasing. Why don't you read a book or take a class?
So? Does it make the "distance" as 4D-spacetime or it(distance) remains as 3D-space? Is there any "time dilation" with "cosmological expansion"?
More about Newtonian frames of reference. In the free fall example shown in the video, the Newtonian frame includes certain assumptions. One of these is that time, as mentioned, is universal: it('s rate) doesn't change relative to distance from a gravitational centre. Another is that acceleration (g) is constant; this assumption is based on the fact that the vertical distance the apple falls through is much less than the radius of the earth. (The assumption is true for small displacements). So it wouldn't be true if the apple fell through a much larger distance. Therefore, we have the assumptions that locally, the gravitational field is homogenous and directed horizontally 'everywhere', and that clocks at different heights (potentials) record the same intervals of time. In the curved version, clocks at different heights will run at slightly different rates, but over the interval of time in which the apple falls, this difference will be negligible. But, in the diagram the arc lengths appear to have quite different lengths (the difference is not negligible), so what's up with that? The time axis of course, has an arbitrary scale--you could think of either plot as being like a sheet of "something" that gets pulled across the path of the falling apple, so the path is traced out. But if you change the time scale, say by "pulling the sheet through" more slowly, it won't change the ratio between arcs at different radii. Another point: there is no horizontal motion, the apple falls vertically at all times. What if the apple does have a horizontal velocity initially? It will trace out a parabolic path in space as well as in time.
'The range of the Earth's gravity-field within which an apple should free-fall towards the Earth's center' is much smaller than the distance between galaxies. So, within the Earth's gravitational field the effect of 'cosmological expansion' may not be observed. In GR, the 'relativistic effect' is considered, which causes additional time-dilation. In cosmological expansion, there is no 'time-dilation'. It will have a parabolic path in NM(Newtonian Model). In GR its path will depend on whether the 'initial horizontal velocity' is relativistic or not.
What I posted (about the video) doesn't say anything about cosmic expansion. It's about free fall in a local frame where g is constant. The question was about why the arcs have different lengths and how this is related to time dilation. You would need atomic clocks to measure any difference in recorded times over the interval it takes an apple to fall from an apple tree. Yes I know it will trace out a parabolic path in space if it has an inital velocity. Relativistic velocity isn't a consideration--it's about free fall in a Newtonian frame (where all velocities are much less than c).
What the video takes you through is the concept of vectors in a gravitational field. In the Newtonian frame, these are all parallel to the field, which is vertical (post 257 says it's horizontal, ignore that). So adding a horizontal velocity component means orthogonality, and a way to make the physics more general. But there are already some linear 'equations' in the frame. Since acceleration is constant, velocity is v = gt, and horizontal displacement h = vt. Further, dv/dh = 1/t. (Actually, velocity and acceleration are negative, so I've flipped each frame over). Add a horizontal velocity, and something happens in the Einstein frame that doesn't happen in the Newtonian one.
