User talk:Cretu

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Let $$(D,g), (\bar{D},\bar{g})$$ be two (Pseudo-)Riemannian charts in $$\mathbb{R}^n$$. Let $$\mathrm{f}: (D,g) \rightarrow (\bar{D},\bar{g})$$ be an isommetry. Let $$J^i_j (x)=\frac{\partial{} \mathrm{f}^i}{\partial x^j}(x)$$ be the Jacobian.

We have the formula (by definition of an isommetry):

$$ g_{ij}(x)=(\mathrm{f}^* \bar{g})_{ij}(x)=J^k_i(x) J^l_j(x) \bar{g}_{kl}(f(x)) $$

A tensor of rank $$(0,2)$$ (for simplicity) (on the charts $$D,\bar{D}$$ is a collection of smooth functions $$\bar{T}_{ij} (y),T_{ij}(x)$$ that satisfy the transformation property (using a pullback $$\mathrm{f}^*$$:

$$ (\mathrm{f}^*\bar{T}_{ij})(x)=\bar{T}_{kl}(f(x)) J^k_i(x) J^l_j(x) =T_{ij}(x) $$

We now define tensor densities. A tensor density of weight $$W$$ is a collection of smooth function $$\mathbf{\bar{T}}_{ij} (\bar{g}(y),y),\mathbf{T}_{ij} (g(x),x) $$ that satisfy the transformation property:

$$ (\mathbf{f}^* \bar{\mathbf{T}}_{ij})(x)=\bar{\mathbf{T}}_{kl}(\bar{g}(f(x)),f(x)) J^k_i(x) J^l_j(x) =\det{J}^{-W}(x)\mathbf{T}_{ij}(g(x),x)=\mathbf{T}_{ij}(\bar{g}(f(x)),x) $$

Consequently, every tensor density of weight W can be written as:

$$ \mathbf{T}_{ij}=\sqrt{\det{g}(x)}^W T_{ij} $$

with an ordinary tensor $$T_{ij}$$.

The collection of Tensor densities $$\bar{\mathbf{T}}_{ij}(\bar{g}(y),y)=\bar{M}_{ij}(y), \mathbf{T}_{ij}(\bar{g}(f(x)),x)=M_{ij}(x)$$ are thus tensors (But not over the same manifold M as g, instead it is a tensor over the section of T2M defined by g. There is no (a priori) connection defined on that manifold. TR 09:24, 28 October 2011 (UTC)).
 * A tensor is a collection of multilinear maps depending only on the position x. As the dependence on $$g$$ is by position only, this does define a tensor as it is multilinear. Anything that looks like a tensor is a tensor. Cretu (talk) 13:17, 28 October 2011 (UTC)

Using the covariant derivative, we obtain:

$$ \nabla_k \bar{\mathbf{T}}_{ij}=\nabla_k \bar{M}_{ij}=\partial_k \bar{M}_{ij}+ \Gamma^a _{ik} \bar{M}_{aj} + \Gamma^a_{jk} \bar{M}_{ia}=\sqrt{\det{g}}^W (W \Gamma^a _{ak} \bar{T}_{ij} + \partial_k \bar{T}_{ij}+\Gamma^a _{ik} \bar{T}_{aj} + \Gamma^a_{jk} \bar{T}_{ia}) $$

In the next step, we will show that the definition in the article proves non well defined in connection with the Levi-Civita connection.

If we had instead:

$$ \nabla_k \bar{\mathbf{T}}_{ij}=\sqrt{\det{g}}^W \nabla_k \bar{T}_{ij} $$

we can derive a paradox. Thus this definition can not be true. Recall the property of the Levi-Civita connection:

$$ \nabla_k g(e_i,e_j)=g(\nabla_k e_i,e_j)+g(e_i,\nabla_k e_j) $$

We consider the special case $$d=1, i=j=k=1$$ (i.e. one dimension):

$$ \nabla_1 g(e_1,e_1) = 2 \Gamma^1_{11} g(e_1,e_1)=g_{11} g^{11} \partial_1 g_{11} $$
 * Question: How do you define ei?
 * If these are coordinate frame fields, then they are not proper vector fields and the covariant derivative is not defined for them, and the above statement is false.
 * If they are proper vector fields then (det g)1/2 != g(e1,e1) (The RHS is a scalar, while the LHS is not.) The are only numerically equal if e1 coincides with the coordinate frame vector.TR 09:11, 28 October 2011 (UTC)


 * The ei are the basis vectors of tangential space. $$e_i=\partial_i$$ (We are in euclidean charts). $$g$$ ist a matrix with entries $$g=g_{11}=g(e_1,e_1)$$. The determinant is $$g_{11}=g(e_1,e_1)$$. The determinant is a smooth function. Such is the right hand side. Their values are equal on the whole chart. Your mistake is that the determinant is in fact a scalar as it satisfies $$\det{g}(x)=\det{g}(f^{-1} (y))$$. Any arbitrary function $$f$$ satisfies $$f=h*f'$$ for suitable choices, yet it doesn't make $$f$$ anything else than a smooth function. Cretu (talk) 13:17, 28 October 2011 (UTC)

according to the definition of the Christoffel symbols. In 1 dimension, however:

$$ g_{11}=\frac{1}{g^{11}} $$

Thus:

$$ \nabla_1 g(e_1,e_1)=\partial_1 g(e_1,e_1) $$

However, as we are in one dimension, we have:

$$ \det{g}=g_{11}=g(e_1,e_1) $$

Thus:

$$ \nabla_1 det(g)=\nabla_1 g(e_1,e_1)=\partial_1 g(e_1,e_1)\neq 0 $$

as $$g_{11}$$ is an arbitrary non zero smooth function. Which is an immediate contradiction to the definition in the article.