Covariant derivative of the metric

In summary, the conversation discusses the covariant derivative of the metric and the exponential of a scalar field, and whether they give zero or not. It is mentioned that the covariant derivative of the exponential of a scalar field will only be zero if the field is constant. It is also suggested to check the explicit definition of how the covariant derivative acts on a (1,1)-type tensor, as it may give zero if the connection is symmetric.
  • #1
vitaniarain
11
0
hello!
just a quick question, does the covariant derivative of the metric give zero even when the indices(one of the indices) of the metric are(is) raised?
also another question not entirely related, does the covariant deriv. of exp(2 phi) where phi is the field, also give zero or not necessarily? thanks
 
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  • #2
Act with the covariant derivative on a kronecker delta. You know that

[tex]
g_{ab}g^{bc} = \delta_a^c
[/tex]

The covariant derivative of your exponential of scalar field is given by a partial derivative per definition; this will only be zero for constant phi.
 
  • #3
if u act on a kronecker delta with the covariant deriv will that give zero? :S
 
  • #4
Well, you know how a covariant derivative acts on a (1,1)-type tensor, right? Just write it down explicitly and check :) I would say that you'll find that it indeed is zero, provided the connection is symmetric.
 
  • #5


The covariant derivative of the metric is a mathematical operation that describes how a vector field changes as it moves along a curved manifold. It takes into account the curvature of the space and ensures that the vector is parallel transported along the path. In general, the covariant derivative of the metric does not give zero, even when the indices of the metric are raised.

Regarding the question about the covariant derivative of exp(2 phi), it depends on the specific metric and the field phi. In general, the covariant derivative of exp(2 phi) will not necessarily give zero, as it also depends on the curvature of the space. However, in some cases where the curvature is zero, the covariant derivative may give zero. It is important to note that the covariant derivative is a mathematical operation and its value depends on the specific metric and field being considered.
 

1. What is a covariant derivative of the metric?

The covariant derivative of the metric is a mathematical tool used in the study of differential geometry and general relativity. It is a way to compute the change in a tensor field along a given direction, taking into account the curvature of the underlying space.

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

The covariant derivative of the metric takes into account the curvature of the space, while the ordinary derivative does not. This means that the covariant derivative is coordinate-independent and can be used in curved spaces, while the ordinary derivative is only applicable in flat spaces.

3. What are the uses of the covariant derivative of the metric?

The covariant derivative of the metric is used in various fields of physics, including general relativity, differential geometry, and quantum field theory. It is particularly important in understanding the behavior of matter and energy in curved space-time, as described by Einstein's field equations.

4. How is the covariant derivative of the metric calculated?

The covariant derivative of the metric can be calculated using the Christoffel symbols, which represent the connection coefficients of the underlying space. These symbols are used to define the geodesic equation, which describes the shortest path between two points in a curved space. The covariant derivative is then computed as the ordinary derivative along this geodesic path.

5. Can the covariant derivative of the metric be visualized?

The covariant derivative of the metric cannot be directly visualized, as it involves abstract mathematical concepts such as tensors and curvature. However, its effects can be seen in the behavior of matter and energy in curved space, which can be visualized through simulations and models.

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