General Relativity, Schwarzschild's Metric, and Applications

In summary, the conversation discusses the use and mathematics behind Einstein's Field Equations, particularly in relation to Schwarzschild's Metric. The individual is seeking guidance on how to calculate the time it takes for the Earth to orbit the sun using geodesics and the arc length formula. They also mention needing help understanding and solving differential equations and partial derivatives. The conversation ends with a request for resources on tensors and a better understanding of their concept.
  • #1

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
[itex]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}[/itex]

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.
 
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  • #2
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.
 
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  • #3
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.
 
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1. What is General Relativity?

General Relativity is a theory of gravity proposed by Albert Einstein in 1915. It describes the relationship between matter and the curvature of space and time, showing that gravity is not a force between masses, but rather the result of the curvature of space and time caused by massive objects.

2. What is Schwarzschild's Metric?

Schwarzschild's Metric is a mathematical equation used in General Relativity to describe the curvature of space and time around a spherically symmetrical mass. It was discovered by Karl Schwarzschild in 1916 and is important for understanding the behavior of black holes.

3. What are some applications of General Relativity?

General Relativity has numerous applications in modern science and technology. It is used in the Global Positioning System (GPS) to accurately measure time and distance, in cosmology to understand the structure and evolution of the universe, and in astronomy to study the behavior of massive objects such as black holes and neutron stars.

4. How does General Relativity differ from Newton's theory of gravity?

Newton's theory of gravity, proposed in the 17th century, states that gravity is a force of attraction between two masses. In contrast, General Relativity describes gravity as the curvature of space and time caused by the presence of massive objects. It is a more accurate and comprehensive theory that can explain phenomena that Newton's theory cannot, such as the bending of light by massive objects.

5. Is General Relativity a proven theory?

Yes, General Relativity has been extensively tested and has been proven to be an accurate description of the behavior of gravity. Its predictions, such as the bending of light and the existence of black holes, have been confirmed through numerous experiments and observations. However, it is still an active area of research and there are ongoing efforts to further test and refine the theory.

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