Cosmological model, geodesics question

In summary, the conversation discusses the geodesic equation and its solution for the Christoffel symbols. It is mentioned that the geodesic equation is obeyed and the geodesics are described by a constant value of x^i. The individual also shares their confusion about solving the temporal component equation and discusses using an affine parameter to solve for it. The conversation ends with a question about whether the approach taken was appropriate.
  • #1
binbagsss
1,259
11

Homework Statement



I am unsure of Q3 but have posted my solutions to other parts

cosmomodoel.png


Homework Equations

The Attempt at a Solution



3)
ok so it is clear that because the metric components are independent of ##x^i## each ##x^i## has an associated conserved quantity ##d/ds (\dot{x^i})=0##. (1)

The geodesic equation can be written as ##\frac{d^2x^i}{ds^2}+\Gamma^i_{ab}\dot{x}^a\dot{x}^b=0 ##

I look at the Euler-Lagrange equations and I can quickly show that all Christoffel symbols with an upper index ##^i## are zero and so the above is zero (via comparing the the form above of a geodesic equation and identifying the Christoffel symbols from the e-l equations this way). so the first term is zero i have shown by the KVF and the second term zero. so the geodesic equation is obeyed.

Now here is my probably very stupid question, how is this constant ##x^i ## geodesics that the above geodesic equation describes. . the above geodesic equation would hold for ##\dot{x^i}## constant which implies that ##x^i## is constant ofc, but here I see my thoughts are way of track and i have clearly misunderstood something as just using this line of reasoning the geodesic equation is trivially satisfied...

thanks.
 

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  • #2
The spatial equations yes. You replaced them by the correspondig first integral. You still need to solve the temporal one.
 
  • #3
Orodruin said:
The spatial equations yes. You replaced them by the correspondig first integral. You still need to solve the temporal one.

okay many thanks.
I am unsure what you mean by 'integral ' however.
But I see clearly I have missed out the time -component equation of ##\ddot{x^u}+\Gamma^u_{ab}\dot{x^a}\dot{x^b}=0 ##

I have found that the Christoffel Symbols ##\Gamma^x_{xt} , \Gamma^t_{xt} \neq 0 ## , and ##\Gamma^t_{tt}=0##
And so for there to be a chance of a geodesic, from my above workings, I assume the quesion means to hold all ##x^i## constant and not just one?

This leaves me to look at whether ##\ddot{t}=0##?

To this I check whether there is an affine parameter ##s## such that this is possible. The definition of affine parameter is that ##dL/ds=0##

and so if I write

##L=-\dot{t^2} + K_{x_1} + K_{x_1} + K_{x_1} ##; where ##K_{x_1}## is the KvF associated with the ##x_i## coordinate.
and then differentiate wrt s to get
##0=\ddot{t}##

is this ok?
have I took a long-winded approach?
 
  • #4
binbagsss said:
I am unsure what you mean by 'integral ' however
Not ”integral”, ”first integral”.
 

Related to Cosmological model, geodesics question

1. What is a cosmological model?

A cosmological model is a theoretical framework used to describe the structure and evolution of the universe. It includes various physical laws and assumptions about the universe, and is used to make predictions and explain observations about the universe.

2. How do cosmological models explain the expansion of the universe?

Cosmological models use the theory of general relativity to explain the expansion of the universe. This theory states that the fabric of space-time is constantly expanding, causing galaxies and other objects to move further apart from each other.

3. What are geodesics in the context of cosmological models?

In the context of cosmological models, geodesics are the paths that objects follow in the fabric of space-time. They represent the shortest distance between two points in space-time and are affected by the curvature of space-time.

4. How do geodesics help us understand the structure of the universe?

Geodesics help us understand the structure of the universe by showing how objects move and interact in space-time. By studying the paths of geodesics, we can learn about the curvature of space-time and the distribution of matter and energy in the universe.

5. Can cosmological models be tested and validated?

Yes, cosmological models can be tested and validated through observations and experiments. By comparing the predictions of a model to real-world data, scientists can determine its accuracy and make adjustments if necessary. This process of testing and validating is ongoing as new data and technologies become available.

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