Interpretation of covariant derivative of a vector field

In summary, the covariant derivative on Riemannian manifolds can be used for parallel transport using the Levi-Civita connection. This can be applied to a smooth curve and a tangent vector by using the equation $\nabla_{\gamma'(s)}l(s)=0$. This can also be extended to a 2-dimensional submanifold by shooting null-geodesics with a given initial direction, resulting in a vector field tangent to those geodesics. The equation $\nabla_l l=0$ can be interpreted as the parallel transport of this vector field along itself, but this may differ from the description given in the script. Additionally, it should be noted that the manifold in question is actually a
  • #1
Pentaquark5
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On Riemannian manifolds ##\mathcal{M}## the covariant derivative can be used for parallel transport by using the Levi-Civita connection. That is

Let ##\gamma(s)## be a smooth curve, and ##l_0 \in T_p\mathcal{M}## the tangent vector at ##\gamma(s_0)=p##. Then we can parallel transport ##l_0## along ##\gamma## by using
$$ \nabla_{\gamma'(s)}l(s)=0 .$$
However, I'm having trouble applying this knowledge to the following case (quote from my script):

Let ##e_0:=\dot{\gamma}## and ##e_1 \perp e_0## be unit vectors. Let further ##S## be the ##2##-dimensional submanifold of ##\mathcal{M}## obtained by shooting null-geodesics with initial direction ##l(0):=e_0+e_1## from all points along the curve ##\gamma##.
Imposing affine parameterisation, this defines on ##S## a vector field ##l## tangent to those geodesics, solution of the equation
$$\nabla_l l=0.$$
I interpret this equation as the parallel transport of ##l## along ##l##, but that does not really coincide with the description given in my script.

Could anybody help me make sense of this equation? Thanks
 
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  • #2
Pentaquark5 said:
I interpret this equation as the parallel transport of lll along lll, but that does not really coincide with the description given in my script.
Why not? A geodesic is a curve whose tangent vector is parallel transported along the curve itself.

Also, you seem to be dealing with a pseudo-Riemannian manifold, not a Riemannian manifold. A Riemannian manifold does not have any non-zero null vectors.
 
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Likes vanhees71
  • #3
Ah, thank you I don't know what went wrong in my head there. And you are of course correct in noting that ##\mathcal{M}## is pseudo Riemannian.

Thanks!
 

Related to Interpretation of covariant derivative of a vector field

1. What is the covariant derivative of a vector field?

The covariant derivative of a vector field is a mathematical operation that describes how the vector changes as it moves along a curved surface. It takes into account the curvature of the surface and adjusts for it, making it a more accurate representation of the vector's behavior.

2. How is the covariant derivative different from the ordinary derivative?

The ordinary derivative only takes into account changes in the vector's magnitude and direction, while the covariant derivative also considers changes in the vector's direction due to the curvature of the surface. This makes the covariant derivative a more precise measure of the vector's behavior on a curved surface.

3. What is the significance of the covariant derivative in physics?

The covariant derivative is essential in general relativity, as it allows for the description of how vectors behave in a curved space-time. It is also used in other areas of physics, such as fluid mechanics and electromagnetism, to describe the behavior of vectors in curved systems.

4. How is the covariant derivative calculated?

The exact calculation of the covariant derivative depends on the specific situation, but it typically involves taking the ordinary derivative of the vector and adding correction terms that account for the curvature of the surface. In some cases, this can be done using tensor calculus and Christoffel symbols.

5. What are some practical applications of the covariant derivative?

The covariant derivative has many practical applications, such as in the study of fluid dynamics, where it is used to describe the flow of fluids on a curved surface. It is also used in computer graphics to model the behavior of light and other physical phenomena on curved surfaces. Additionally, the covariant derivative is used in geodesy to measure changes in the Earth's surface.

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