# Contracted Christoffel symbols in terms determinant(?) of metric

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• George Keeling
In summary, M. Blennow's book presents problem 2.18 which shows that the contracted Christoffel symbols can be expressed as the partial derivative of the logarithm of the square root of the metric tensor. This can be shown for both diagonal and non-diagonal metrics.
George Keeling
Gold Member
TL;DR Summary
Should I try to prove this with non-diagonal metric?
M. Blennow's book has problem 2.18:
Show that the contracted Christoffel symbols ##\Gamma_{ab}^b## can be written in terms of a partial derivative of the logarithm of the square root of the metric tensor $$\Gamma_{ab}^b=\partial_a\ln{\sqrt g}$$I think that means square root of the determinant of the metric tensor (and it does in the next question). I don't know how to take a square root of a tensor.

I start with $$\partial_a\ln{\sqrt g}=\frac{1}{2}\partial_a\ln{g}=\frac{1}{2}\frac{1}{g}\partial_ag=\frac{1}{2}g^{-1}\partial_ag$$and$$\Gamma_{ab}^b=\frac{1}{2}g^{bc}\left(\partial_ag_{cb}+\partial_bg_{ac}-\partial_cg_{ab}\right)$$If the metric is diagonal it is pretty easy to show in ##n##-dimensions that those are both the same as$$\frac{1}{2}\sum_{i=1}^{i=n}{g^{ii}\partial_ag_{ii}}$$Should I be trying to prove ##\Gamma_{ab}^b=\partial_a\ln{\sqrt g}\ ## for a non-diagonal metric too?

(Carroll has a similar exercise which is restricted to a diagonal metric. Perhaps he is just not so cruel!)

Oops! After a refreshing nights sleep I remembered that ##\partial_ag=gg^{bc}\partial_ag_{bc}## and saw the light. So ##\Gamma_{ab}^b=\partial_a\ln{\sqrt g}## is true for any old metric.

## 1. What are contracted Christoffel symbols?

Contracted Christoffel symbols are a set of mathematical objects used in differential geometry to describe the curvature of a space. They are defined as the components of the connection in a coordinate basis, and they represent the change in a vector field as it is parallel transported along a curve.

## 2. How are contracted Christoffel symbols related to the determinant of the metric?

The contracted Christoffel symbols can be expressed in terms of the determinant of the metric. In fact, the contracted Christoffel symbols are the components of the inverse of the metric multiplied by the partial derivatives of the determinant of the metric.

## 3. What is the significance of contracted Christoffel symbols in differential geometry?

Contracted Christoffel symbols play a crucial role in differential geometry as they provide a way to measure the curvature of a space. They are used in various mathematical models, such as Einstein's theory of general relativity, to describe the behavior of gravity.

## 4. How are contracted Christoffel symbols calculated?

Contracted Christoffel symbols can be calculated using the metric tensor and its derivatives. The formula for calculating the contracted Christoffel symbols involves taking the inverse of the metric, multiplying it by the partial derivatives of the metric, and then summing over the appropriate indices.

## 5. Can contracted Christoffel symbols be used in other fields of science?

Yes, contracted Christoffel symbols have applications in various fields of science, such as physics, engineering, and computer science. They are used in the study of curved spaces, as well as in the development of mathematical models for physical phenomena. They are also used in computer graphics and image processing to calculate the curvature of surfaces.

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