Show that the energy is conserved in this field/metric

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To show that energy is conserved in the given metric, it is suggested to explore the concept of Killing vectors and their relationship to energy conservation. The initial attempts involving the time derivative of energy and the formula p * p = -m² were not fruitful. Reference to Sean Carroll's textbook indicates that relevant equations related to symmetries and Killing vectors can provide insight. Specifically, equations from section 3.8 may need to be modified to accommodate the geodesic equation presented in the problem. Understanding these relationships is crucial for solving the energy conservation question effectively.
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Homework Statement
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Relevant Equations
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1614783348225.png

I would like it very much if someone could give a hint on how to start this question.

In particular, I tried to find the derivative of energy with respect to time, but that was not enough.
Then I tried to apply the formula p * p = -m², but that also didn't get me anywhere.
These were my two attempts, I imagine there is another way but I haven't been able to find it yet

This is the metric:
1614783555247.png


The "magnetic charge" P is zero, at least i think so.

I thought in another way, try to find any Killing vector and see its relation with the energy, but i am not sure if this is will be helpful
 
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I admire your modesty in putting this under "introductory" physics homework!
 
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PeroK said:
I admire your modesty in putting this under "introductory" physics homework!
Moved.
 
Herculi said:
Homework Statement:: ...
View attachment 279056
This problem is from Sean Carroll's textbook.
I thought in another way, try to find any Killing vector and see its relation with the energy, but i am not sure if this is will be helpful.
Yes, there is a Killing vector or isometry that is relevant to this problem.

Review section 3.8 on symmetries and Killing vectors. Equations (3.161), (3.162), (3.167) and (3.168) in this section are for particles following geodesics. How would these equations be modified if the geodesic equation [(3.44) or (3.161)] is replaced by equation (6.122) in the problem statement?
 
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