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# 物理代写|广义相对论代写General Relativity代考|Inverse frame field and inverse metric

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## 物理代写|广义相对论代写General Relativity代考|Inverse frame field and inverse metric

Using the inverse frame field, the definition of the metric becomes
$$e_i^a(x) e_j^b(x) g_{a b}(x)=\delta_{i j}$$
which shows explicitly that the inverse frame field represents two vector fields $e_i^a(x)=\left(\vec{e}1(x), \vec{e}_2(x)\right)$ that at each point define an orthonormal frame in the tangent space as in Figure 3.5. That is, $$\vec{e}_i(x) \cdot \vec{e}_j(x)=\delta{i j}$$
If we invert the $2 \times 2$ matrix $g_{a b}$, we obtain a matrix that is called $g^{a b}$ (sometimes ‘contravariant metric’), and has the same information as $g_{a b}$. Using this we can write, from the last equation,
$$g^{a b}(\mathrm{x}) e_a^i(\mathrm{x}) \delta_{i j}=e_j^b(x)$$
This shows that the ‘space’ indices $a, b \ldots$ can be consistently raised and lowered with the metric and its inverse, while the ‘internal’ indices $i, j \ldots$ can be consistently raised and lowered with the Kronecker delta $\delta_{i j}$.

Notice that the metric information about the surface (lengths, angles…) is in the metric field (or the frame field), not in the coordinates. The Cartesian coordinates $X^i$ are defined in such a way that they immediately give distances, but not so for general coordinates $x^a$ : these have no metric information.

## 物理代写|广义相对论代写General Relativity代考|Coordinate change and invariant geometry

If we change coordinates on the surface, it is an easy exercise [do it!] to show that the frame field changes as follows:
$$e_a^i(x) \rightarrow \tilde{e}a^i(\tilde{x})=\frac{\partial x^b}{\partial \tilde{X}^a} e_b^j(\mathrm{x}(\tilde{x}))$$ and the metric field changes as follows: $$g{a b}(x) \rightarrow \tilde{g}{c d}(\tilde{x})=\frac{\partial x^a}{\partial \tilde{X}^c} \frac{\partial x^b}{\partial \tilde{X}^d} g{a b}(x(\tilde{X})) .$$
Two metrics related by this transformation describe the same surface, in different coordinates. Therefore what matters for the geometry of the surface is not the field $g_{a b}(x)$, but rather the equivalence class of these fields under the equivalence defined by (3.30). These equivalent classes are called two-dimensional geometries. For instance, a (metric) sphere is a two-dimensional geometry and can be described by different $g_{a b}(x)$.

## 物理代写|广义相对论代写General Relativity代考|Inverse frame field and inverse metric

$$e_i^a(x) e_j^b(x) g_{a b}(x)=\delta_{i j}$$

$$g^{a b}(\mathrm{x}) e_a^i(\mathrm{x}) \delta_{i j}=e_j^b(x)$$

## 物理代写|广义相对论代写General Relativity代考|Coordinate change and invariant geometry

$$e_a^i(x) \rightarrow \tilde{e} a^i(\tilde{x})=\frac{\partial x^b}{\partial \tilde{X}^a} e_b^j(\mathrm{x}(\tilde{x}))$$

$$g a b(x) \rightarrow \tilde{g} c d(\tilde{x})=\frac{\partial x^a}{\partial \tilde{X}^c} \frac{\partial x^b}{\partial \tilde{X}^d} g a b(x(\tilde{X})) .$$

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