Unruh's "Slogan" for GTR
Dravish said:
To quote from ‘Time, Gravity and Quantum Mechanics, by W. G. Unruh’;
“A more accurate way of summarizing the lessons of General Relativity is that gravity does not cause time to run differently in different places (e.g., faster far from the Earth than near it). Gravity is the unequable flow of time from place to place.
That's a very interesting slogan and after a minute's reflection, I like it! Except for the fact that I fear that most newbies will completely misunderstand it, as I think you have done. The key is that the word "unequable" hints that comparing "elapsed times" measured by different observers is not as straightforward as in Newtonian physics, or even as in flat spacetime.
Unruh's first point is one which I find myself making repeatedly to counter misleading language in many bad popular books (and in forums like this one). His second point, well, before you can grasp that you need to have the correct visual intuition in terms of local light cones and distances taken along two curvilinear arcs sharing the same endpoints in a Lorentzian spacetime--- otherwise you'll probably misunderstand what he's trying to get at here. As a self-test, other readers can try to understand a la Unruh the "gravitational red shifting" of signals sent radially from an observer hovering over a Schwarzschild object to a more distant hovering observer. (Hint: divergence of two initially parallel and nearby null geodesics, both cut by two timelike geodesics, the world lines of the two observers.)
Dravish said:
It seems that perhaps gravity and time should be equated
No, no, no! That's certainly not at all what he's trying to suggest!
Dravish said:
I am trying to put together a paper using this perspective
For schoolwork? From "perhaps gravity and time should be equated" I think you need to back up and get some geometrical intuition. I think Geroch,
General Relativity from A to B is just what you need.
intel said:
Can I take this and say that spacetime, space and time are the direct result of particles or any matter which came into existence at t=0, the big bang?
Certainly not! Intel and Dravish, I think you both need to be very very careful about drawing conclusions from verbal descriptions. These can play a valuable role, but to understand gtr you'll need to understand the math. That said, I think the book by Geroch does a fabulous job at getting absolutely the most out of the least mathematical background. Also, posters like pervect and robphy are far more reliable sources than some other regular posters here. As I think they will agree, ultimately, good textbooks (and perhaps local faculty, especially if they specialize in gtr) are still your most reliable sources of information.
"jimbobjames" asked about computing the geodesic equations, say for a line element obtained from a "Monge patch". The most efficient way to do this is via the geodesic Lagrangian, which can be read right off the line element. This is well explained in Misner, Thorne, & Wheeler,
Gravitation and in other books. I've also explained it in great detail in long ago posts to sci.physics.* which can probably be found via Google (search on "Coordinate Tutorial"). And be careful, you did not correctly compute the linearized Schwarzschild line element correctly and your claim that "it only has time curvature" is incorrect; see MTW. I hope you are not going to join the group of posters who insists on pretending to stating misinformation as fact--- if you don't have good reason to be confident, you should sprinkle your computations with "I believe that", or even restrict yourself to asking pervect questions until you know more.
Here's a simple example. Consider H^{1,1} ("two dimensional de Sitter spacetime") in the lower half plane chart with line element
ds^2 = \frac{-dt^2 + dx^2}{t}, \; \; -\infty < t < 0, \; -\infty < x < \infty
Read off the geodesic Lagrangian
L = \frac{ -\dot{t}^2 + \dot{x}^2}{t}
where u \mapsto \dot{u} denotes differentiation with respect to the parameter s of our unknown geodesic. The Euler-Lagrange equations read
<br />
0 = \frac{d}{dt} \, \frac{\partial L}{\partial \dot{t}} <br />
- \frac{\partial L}{\partial t}, \; \;<br />
0 = \frac{d}{dx} \, \frac{\partial L}{\partial \dot{x}} <br />
- \frac{\partial L}{\partial x}<br />
After simplifying, this reduces to
<br />
\ddot{x} = \frac{2 \, \dot{t} \, \dot{x}}{t}, \; \;<br />
\ddot{t} = \frac{ \dot{t}^2 + \dot{x}^2}{t} <br />
from which you can read off the Christoffel symbols, although you don't really need these since at this point you already know the geodesic equations in the given chart. Partially integrating the first equation gives \dot{x} = t^2/B^2 where B is a constant of integration. Plugging this into the second equation gives
t(s) = B \, \sec(s), \; \; x(s) = A + B \, \tan(s)
where A is another constant of integration. (Note circular trig functions, not hyperbolic trig functions!) Eliminating the parameter shows that the timelike geodesics can be characterized as hyperbolic arcs which are orthogonal to t=0, which represents "timelike infinity" in a Penrose-Carter chart. This chart covers only the "top half" of H^{1,1} if you think of an embedding in E^{1,2} as a hyperboloid of one sheet; the "equator" is at "t=-\infty" in this chart.
