Finding znew for Covariant Conservation

In summary, by using the unit vector z=(0,0,0,1) to form the tensor z^\mu z^\nu, it is easy to check that ∂_\μ( z^\mu z^\nu=0 in Minkowski spacetime. However, the equation needs to be generalized to curved spacetime, resulting in the equation ∇μ ( znew^\mu znew^\nu)=0. The challenge is finding znew, which should be related to the original z. Can anyone offer assistance?
  • #1
gentleboy
2
0
suppose we have unit vector z=(0,0,0,1), we can use it to form a tensor [itex]z^\mu z^\nu [/itex],
it is easy to check that [itex]∂_\μ( z^\mu z^\nu[/itex]=0 in Minkowski spacetime,
now I want to generalize this equation to general curved spacetime, so that
∇μ ([itex] znew^\mu znew^\nu[/itex])=0.
But I am not sure how to find znew, which should be related to the original z.
Any one can help me? Thanks
 
Last edited:
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  • #2
Welcome to PF! Could you please mark up your equations using LaTeX so they're more readable? Here's an example: [itex]z^\mu[/itex]. To see how I did this, click on the QUOTE button underneath mypost.
 
  • #3
each time when I tried to submit the revision, the web kept saying it is too short, need to lengthen it to at least 4 characters, i do not know what does that mean.
 

FAQ: Finding znew for Covariant Conservation

What is "Finding znew for Covariant Conservation"?

"Finding znew for Covariant Conservation" is a scientific method used to determine the value of the parameter znew in order to achieve covariant conservation in a physical system. This is important in many areas of physics, such as particle physics and fluid dynamics, where maintaining conservation laws is crucial for accurate predictions and simulations.

Why is covariant conservation important?

Covariant conservation is important because it ensures that fundamental physical laws, such as conservation of energy and momentum, are upheld in a system. This allows for more accurate and reliable predictions and understanding of natural phenomena.

How is "Finding znew for Covariant Conservation" performed?

The process of finding znew involves using mathematical equations and calculations to determine the correct value for znew that will result in covariant conservation. This may involve solving differential equations or using numerical methods to find the optimal value.

What are the applications of "Finding znew for Covariant Conservation"?

One of the main applications of finding znew for covariant conservation is in particle physics, where it is used to accurately simulate and predict the behavior of subatomic particles. It is also used in fluid dynamics, astrophysics, and other areas where conservation laws are important.

Are there any limitations to "Finding znew for Covariant Conservation"?

One limitation of this method is that it may not always be possible to find a single value for znew that satisfies all conservation laws. In some cases, a range of values may need to be used or additional assumptions may need to be made. Additionally, this method may become more complex and time-consuming for systems with many interacting components.

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