Lie Derivative in Relativity: Examples & Uses

In summary, the Lie Derivative can be useful in the General Theory of Relativity by generating diffeomorphisms. It is used to find isometries by setting the Lie Derivative of the metric to zero and solving for the vector field, which gives Killing vectors. This is explained in detail in Carroll's GR notes.
  • #1
kent davidge
933
56
Can someone point me some examples of how the Lie Derivative can be useful in the General theory of Relativity, and if it has some use in Special Relativity.

I'm asking this because I'm studying how it's derived and I don't have any Relativity book in hand so that I can look up its application (and I'm not even sure whether it's easy to find out).
 
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  • #2
Diffeomorphisms are generates by lie derivatives. So e.g. to find isometries one puts the lie derivative of the metric to zero and solves for the vector field, giving so called Killing vectors.
 
  • #3
See e.g. Carroll's GR notes .
 

Related to Lie Derivative in Relativity: Examples & Uses

What is a Lie Derivative in Relativity?

A Lie derivative is a mathematical operation used in general relativity to describe how a tensor field changes along a specific direction defined by a vector field. It is used to study how a physical quantity changes as it moves along a specific path in spacetime.

How is the Lie Derivative calculated?

The Lie derivative of a tensor field is calculated by taking the directional derivative of the tensor along the vector field and then subtracting the contribution of the tensor's transformation under the flow of the vector field. This can be expressed as a commutator of the vector field and the tensor field.

What are some examples of Lie Derivative in Relativity?

One example of the use of Lie derivative in relativity is in the study of gravitational waves. The Lie derivative of the metric tensor describes how the curvature of spacetime changes as a gravitational wave passes through it. Another example is in the study of black holes, where the Lie derivative is used to calculate the rate of change of physical quantities such as mass and angular momentum as matter falls into the black hole.

How is Lie Derivative used in practical applications?

Lie derivative is used in many practical applications in general relativity, such as in the study of cosmology, black hole thermodynamics, and the dynamics of gravitational collapse. It is also used in the numerical simulation of gravitational waves and in the analysis of data from gravitational wave detectors.

What are the limitations of Lie Derivative in Relativity?

One limitation of Lie derivative in relativity is that it only applies to smooth, differentiable spacetimes. It also assumes that the vector field is parallel to the geodesic flow of the tensor field. Additionally, the calculation of Lie derivative can become complex and computationally intensive for higher-order tensors.

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