# How to Calculate the Metric and Christoffel Symbols for GR Flat and Curved Space

• RestlessRiver
In summary: I'm not sure if there are any other online sources that I could recommend. I don't think Sean Carroll's lecture notes are freely available.
RestlessRiver
1.a) Concider the 2-space consisting of a spherical shell at constan radius, r. In polar coordinates the line element on the surface can be written (a, b,∈ 1,2)

## Homework Equations

calculate gab, Γabc, R1212, R2121, R11, R22, R

## The Attempt at a Solution

I don't have an attempt on a solution cause I honestly have no idea how. Our teacher has said that it's possible to calculate the Christoffel sybols and the metric but he never showd how =/

really apprisiate any help

thanks a lot

Well, do you not have any textbook or something where the formula for Christoffel symbol is given in terms of the metric tensor. There is such a formula, and you will need to know it.

We got the defenition of the christoffelsymbol

Γrsa=½gal(glr,s+gls,r-grs,l)

and in flat space gab=(1 0 0 0 || 0 -1 0 0 || 0 0 -1 0 || 0 0 0 -1)

and R is Riemann's tensor

I'm supposed to calcuate the same thing for 2-space of the cylinder and then determin which is flat and which is curved, and I know it's the cylinder that is flat and the sphere that's curved.

And we only have recomended books, we don't need to buy them, and for a poor student like me, well yea, I don't have the money to buy the books.

Yes, so you have the formula.You can now use it to find all the desired quantities. Similarly for the cylinder you can write down the metric tensor for a cylinder and work it out. I suppose you know the condition for a manifold being curved or not?

Well, if you can't buy books, there are some online resources. Sean Carroll's lecture notes on GTR are I think freely available. There may be other stuff at a more elementary level, you can look around.

## 1. What is the difference between flat and curved space in general relativity?

In general relativity, space is described as either flat or curved. Flat space is the absence of any curvature, while curved space is characterized by the presence of curvature caused by the presence of mass or energy. This curvature is what we experience as gravity.

## 2. How does general relativity explain the motion of objects in space?

In general relativity, the motion of objects in space is explained by the curvature of space caused by the presence of mass or energy. This curvature affects the path of objects, causing them to follow curved trajectories rather than straight lines.

## 3. How does the concept of spacetime relate to the curvature of space in general relativity?

In general relativity, spacetime is a four-dimensional mathematical model that combines space and time into a single entity. The curvature of space is described by the bending of spacetime, which is caused by the presence of mass or energy.

## 4. Can you give an example of a phenomenon that can only be explained by the curvature of space in general relativity?

The bending of light by massive objects, known as gravitational lensing, is a phenomenon that can only be explained by the curvature of space in general relativity. This phenomenon occurs when the path of light is affected by the curvature of spacetime caused by a massive object.

## 5. How does the concept of spacetime curvature impact our understanding of the universe?

The concept of spacetime curvature has greatly impacted our understanding of the universe by providing a more accurate description of gravity and the motion of objects in space. It also helps explain various phenomena, such as the expansion of the universe and the behavior of black holes.

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