General Relativity, Schwarzschild's Metric, and Applications

[The Wanderer]

Homework Statement

I have been trying to understand the actual applications and mathematics behind Einstein's Field Equations. I have watched a two hour long lecture on how they were derived and have pretty much understood it, however I still don't know how to actually "use" Einstein's Field Equations. I ran into Schwarzschild's Metric and I believe the first thing you do is you solve for metric tensor so for simplicity sake I am going to chose his metric. Now say if I wanted to calculate the time it takes for the Earth to orbit the sun, how would I go about this? I know you have to somehow use geodesics, but I'm not entirely sure how. If you don't want to go through all the work that is fine as long as you point me to something to read. Also I noticed that the Schwarzschild's metric equation looks a lot like the arc length formula. Is this correct? Can it somehow be rearranged to express it as an integral, then? Also, could you help me by telling me what mathematics I need to understand to further grasp this topic. I haven't been formally taught Calculus, but these are the topics that I understand/know.

• derivatives
• what a definite and indefinite integral is
• how to solve an indefinite integral
• don't know how to solve a definite integral rigorously, but know how to with a calculator as an aide
• understand what a differential equation is, but not how to solve for one. (I think you use a differential field or something, seen them and I could guess how you generate one)
• partial derivatives
• gradients

Homework Equations

Schwarzschild's Metric
ds^{2} = (1 - \frac{r_s}{r})^{-1}dr^{2} + r^{2}(dθ^{2} + sin^{2}θd\varphi^{2})-c^{2}(1 - \frac{r_s}{r})dt^{2}

The Attempt at a Solution

Not exactly sure if I can show an attempt at a solution, but I do not need to be "spoonfed", just point me in the right direction and tell me what I need to learn and what to read. I do, however, learn from examples the best as I am a visual learner, but it isn't required. Thank you so much for the help.

 

[BruceW]

hi, welcome to physicsforums :)
yeah, it is the formula for the small change in arc-length (squared) as you move around in spacetime. And why is this related to the metric? because:
$ds^2 = g_{\mu \nu} dX^{\mu} dX^{\nu}$
so this essentially tells us about the metric $g_{\mu \nu}$
Now, since it is $ds^2$ you need to take square root, then choose a path which you want the integral to go over, then you will get the arc length along that path. And as you've probably heard, the Earth would move on a geodesic, so we want to just consider the geodesic paths http://en.wikipedia.org/wiki/Schwarzschild_geodesics Also, you might have read, the arc-length along the worldline of some object is equal to the time as measured by that object. So, the arc-length along the path of the Earth would tell you the time which has passed according to someone on earth. (neglecting the gravitational effect of the earth).

edit: and about something to read... you probably want to look for a good introductory book on relativity. I have not found much stuff on the internet, wikipedia is not terrible, but not great either. Also, I guess it would be good to go over Calculus topics again. But on the other hand, to some extent if you find you are having trouble with a specific mathematical topic, you can then find out more about it.

 

[The Wanderer]

Thank you for welcoming me. :) Ok, so I have to take the "partial" integral of the metric with respect to one variable, treating the others as constants. Correct? And I don't need the arc length between say the moon and the Earth if I know the radius. And I have to check my notes on the world line (sounds familiar but I can't remember it's exact meaning), but my intuition is guessing that what I have to do is calculate the arc length with respect to phi from 0 to 2pi. I'm probably wrong because that's probably not what the world line is. I'm on my phone right now so unless you reply before I get home I'll fix that and make sure. Perhaps you could answer another question I've been seeing. How do you calculate the christoffel symbols? I know how they are derived I just don't know how to calculate them. Again you don't have to rigorously show how they are calculated, just point me in the right direction. And if you or anyone knows of a good read on what tensors exactly are, I would appreciate it. I know that vectors and scalars are tensors, but I'm having trouble actually conceptualizing what they are as a whole or what I am actually doing with these tensors. I just cannot connect intuitively and understand what they are. Thank you so much for the help.