Show that the energy is conserved in this field/metric

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In summary, the conversation revolved around finding a solution to a problem involving the derivative of energy with respect to time and the application of the formula p * p = -m². One attempt was to find a Killing vector and see its relation to energy, but it is uncertain if this will be helpful. The conversation also mentioned a relevant Killing vector or isometry, and suggested reviewing section 3.8 on symmetries and Killing vectors in Sean Carroll's textbook. The question was posed on how equations (3.161), (3.162), (3.167) and (3.168) in this section would be modified if the geodesic equation is replaced by equation (6.122) in the problem statement.
  • #1
LCSphysicist
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Homework Statement
...
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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  • #2
I admire your modesty in putting this under "introductory" physics homework!
 
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  • #3
PeroK said:
I admire your modesty in putting this under "introductory" physics homework!
Moved.
 
  • #4
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?
 

Related to Show that the energy is conserved in this field/metric

1. What is the concept of energy conservation in a field/metric?

The concept of energy conservation in a field/metric refers to the principle that energy cannot be created or destroyed, but can only be transformed from one form to another. This means that the total amount of energy in a system remains constant over time.

2. How can energy conservation be shown in a field/metric?

Energy conservation can be shown in a field/metric through mathematical equations and calculations. By analyzing the changes in energy within a system and accounting for all forms of energy present, it can be demonstrated that the total energy remains constant.

3. Why is it important to prove that energy is conserved in a field/metric?

Proving that energy is conserved in a field/metric is important because it helps us understand and predict the behavior of physical systems. It also serves as a fundamental principle in many scientific fields, such as mechanics, thermodynamics, and electromagnetism.

4. What are some examples of fields/metrics where energy conservation is observed?

There are many examples of fields/metrics where energy conservation is observed, such as gravitational fields, electric fields, magnetic fields, and fluid flow fields. In all of these examples, the total energy within the system remains constant, even as energy is transformed between different forms.

5. Can energy conservation be violated in a field/metric?

No, energy conservation cannot be violated in a field/metric. This is a fundamental law of physics and has been extensively tested and confirmed through experiments. However, energy can appear to be lost or gained if it is not properly accounted for or if there are external factors at play, such as friction or external forces.

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