Christoffel Symbol Ansatz for 4D Spacetime

In summary, we have discussed the Ansatz metric and Christoffel symbol of a D+1 dimensional AdS spacetime and how to find the Christoffel symbol with two indices. However, this notation has not been seen in any articles or books and may not be a standard notation used in differential geometry.
  • #1
darida
37
1
Ansatz metric of the four dimensional spacetime:

[itex]ds^2=a^2 g_{ab}dx^a dx^b - du^2[/itex]

where:

[itex]a,b=0,1,2[/itex]

[itex]a(u)=[/itex]warped factor

Christoffel symbol of a three dimensional AdS spacetime:

[itex]\Gamma^{c}_{ab}= \frac{1}{2} g^{cd}(∂_b g_{da} + ∂_a g_{bd} - ∂_d g_{ba})[/itex]

Now how to find [itex]\Gamma^{a}_{b}[/itex]?
 
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  • #2
First, what are the [itex]g_{ab}[/itex]? Are they functions of the "x"s only or also of u? Clearly the Christoffel symbols depend on exactly how they depend on the coordinates. Second, what do you mean by "a(u)"? Is it that the "a" in "[itex]a^2[/itex]"? You are already using "a" as an index. Surely "a(u)" is not an index so it would be better to use a different symbol.
 
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  • #3
Ok I fix them:
darida said:
Ansatz metric of [itex]D+1[/itex] dimensional spacetime:

[itex]ds^2=a^2 g_{ij}dx^{i} dx^{j} + du^2[/itex]

where:

Signature: [itex]- + + +[/itex]

Metric [itex]g_{ij} \equiv g_{ij} (x^{i})[/itex] describes [itex]D[/itex] dimensional AdS spacetime

[itex]i,j = 0,1,...,(D-1) = D[/itex] dimensional curved spacetime indices

[itex]a(u) = [/itex] warped factor

[itex]u = x^{D} [/itex]

[itex]D = [/itex] number of spatial dimensional

Christoffel symbol of [itex]D[/itex] dimensional AdS spacetime:

[itex]\Gamma^{k}_{ij}= \frac{1}{2} g^{kl}(∂_{j} g_{li} + ∂_{i} g_{jl} - ∂_{l} g_{ji})[/itex]

Now how to find [itex]\Gamma^{i}_{j}[/itex]?
 
  • #4
What does [itex]\Gamma^{i}_{j}[/itex] mean, i.e., what is a [itex]\Gamma[/itex] with two indices?
 
  • #5
George Jones said:
What does [itex]\Gamma^{i}_{j}[/itex] mean, i.e., what is a [itex]\Gamma[/itex] with two indices?

I don't know that's why I asked :confused:

*edit:

Well, one said that

[itex]\Gamma^{\rho}_{\mu\nu} = ...\Gamma^{i}_{j}[/itex]

[itex]R_{\mu\nu} = ...R_{ij}[/itex]

where:

[itex]\mu, \nu, \rho = (D + 1)[/itex] dimensional curved spacetime indices

[itex]R_{ij} = \Lambda_{D} g_{ij}[/itex]

[itex]\Lambda_{D} = [/itex] cosmological constant
 
Last edited:
  • #6
Can you give specific references? In what articles or books have you seen this notation?
 
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  • #7
It's not in any articles/books. I just met someone who told me that :(
 
  • #8
Just giving my two cents, but I have never seen, in my study of differential geometry, a Christoffel symbol with 2 indices. The closest I can think of would be the the connection one-forms which are a set of 6 one-forms, and so they are sometimes written with two (anti-symmetric) indices...but...that's usually written in a way different notation.
 
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  • #9
Oh okay
 

1. What is the Christoffel Symbol Ansatz for 4D Spacetime?

The Christoffel Symbol Ansatz for 4D Spacetime is a mathematical technique used in the study of General Relativity. It involves using Christoffel symbols, which are mathematical objects that represent the curvature of spacetime, to describe the behavior of particles in a four-dimensional space.

2. How is the Christoffel Symbol Ansatz used in General Relativity?

In General Relativity, the Christoffel Symbol Ansatz is used to calculate the equations of motion for particles and fields in a curved spacetime. This allows us to understand how the curvature of spacetime affects the behavior of matter and energy.

3. What is the difference between the Christoffel Symbol Ansatz and the Christoffel Symbol?

The Christoffel Symbol Ansatz is a mathematical technique that uses the Christoffel symbols to describe the behavior of particles in a four-dimensional spacetime. The Christoffel Symbol is a mathematical object that represents the curvature of spacetime and is used to calculate the equations of motion in General Relativity.

4. Can the Christoffel Symbol Ansatz be applied to higher dimensional spacetimes?

Yes, the Christoffel Symbol Ansatz can be applied to higher dimensional spacetimes. The concept of Christoffel symbols and their use in describing the curvature of spacetime remains the same, regardless of the number of dimensions. However, the calculations become more complex as the number of dimensions increases.

5. Why is the Christoffel Symbol Ansatz important in the study of General Relativity?

The Christoffel Symbol Ansatz is important because it allows us to understand the effects of gravity on particles and fields in a curved spacetime, as described by General Relativity. It is a crucial tool for making predictions and calculations in this field of study.

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