Covariant Derivative: Different for Vectors, Spinors & Matrices?

In summary, a covariant derivative is a mathematical operation that allows for the differentiation of vector or tensor fields on a curved space. It takes into account the curvature of the space and adjusts the usual derivative to account for this curvature. The covariant derivative is different for vectors, spinors, and matrices due to their different transformations under a change of coordinates. Its purpose is to extend the concept of differentiation to curved spaces and calculate rates of change and gradients of fields. It is closely related to the concept of parallel transport and is used in general relativity, particle physics, and other areas of physics to describe quantities in a curved space.
  • #1
thehangedman
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2
The covariant derivative is different in form for different tensors, depending on their rank.

What about other mathematical entities? The electromagnetic field A is a vector, but it has complex values. Is the covariant derivative different for complex valued vectors? And what about spinors? Matrices?
 
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  • #2
For spinors you have to use the spin connection to define the covariant derivative, because tensors don't transform under spinoral representations of the Lorentz group. A matrix is not a tensor or spinorial object in general, so the cov. derivative on a matrix is not defined until you specify the spinorial/tensorial character.
 

What is a covariant derivative?

A covariant derivative is a mathematical operation that allows for the differentiation of a vector or tensor field along a specific direction or path on a curved space. It takes into account the curvature of the space and adjusts the usual derivative to account for this curvature.

How is the covariant derivative different for vectors, spinors, and matrices?

The covariant derivative is different for vectors, spinors, and matrices because each of these objects transforms differently under a change of coordinates on a curved space. The covariant derivative takes into account these different transformations and adjusts accordingly to maintain the correct behavior of the object under differentiation.

What is the purpose of the covariant derivative?

The purpose of the covariant derivative is to extend the concept of differentiation to curved spaces. It allows for the calculation of rates of change and gradients of vector and tensor fields on a curved space, taking into account the effects of curvature.

How is the covariant derivative related to the concept of parallel transport?

The covariant derivative is closely related to the concept of parallel transport, which refers to the idea of moving a vector or tensor along a curved space while keeping its direction and magnitude constant. The covariant derivative is used to calculate the change in the vector or tensor due to the curvature of the space during this parallel transport.

What are some applications of the covariant derivative in physics?

The covariant derivative is used extensively in general relativity, which is a theory of gravity that takes into account the curvature of space-time. It is also used in particle physics to describe the behavior of particles and their interactions in a curved space-time. Additionally, the covariant derivative is used in other areas of physics, such as fluid mechanics and electromagnetism, to describe quantities in a curved space.

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