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# 物理代写| Transition to Curved Spaces 相对论代考

## 物理代写

3.8 Transition to Curved Spaces
We can readily generalise the foregoing discussion to tensors on curved spaces o Riemannian manifolds. A Riemannian space, roughly speaking, can be thought o as a $n$ dimensional generalisation of a two dimensional surface on which a metric has been defined. Any Riemannian space can be embedded in a sufficiently higl dimensional Euclidean space-the maximum dimension required is $n(n+1) .$ The same result also holds for a pseudo-Riemannian space of given signature which can be embedded in a flat space with appropriate signature (Eisenhart (1926)). For example, a 3 -sphere can be embedded in the 4-dimensional Euclidean space-one does not need 6 dimensions because it happens to be a highly symmetric space. Consider a 4 dimensional Euclidean space with Cartesian coordinates $x^{i}, i=1,2,3,4$, then the equation $\left(x^{1}\right)^{2}+\left(x^{2}\right)^{2}+\left(x^{3}\right)^{2}+\left(x^{4}\right)^{2}=1$ describes a 3 dimensional unit sphere equation $\left(x^{1}\right)^{2}+\left(x^{2}\right)^{2}+\left(x^{3}\right)^{2}+\left(x^{4}\right)^{2}=1$ describes a 3 dimensional unit sphere embedded in 4 dimensions. Another example is the 4 dimensional Schwarzschild If one thinks of a curved space as a subspace of a higher dimensional flat space, then
${ }^{2}$ For the advanced reader: Contravariant vectors (tangent vectors) and covariant vectors (1-forms) belong to different vector spaces, namely, the tangent space and the cotangent space respectively. However, the metric tensor defines an isomorphism between them by $V_{i}=g_{i j} V^{j}$ and its inverse $V^{i}=g^{i j} V_{j}$. This has been demonstrated pictorially in the example.
$3.8$ Transition to Curved Spaces
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it inherits the metric – the induced metric-from that flat space. See Exercise $3.10$ for a 2 dimensional space (surface) embedded in 3 dimensional Euclidean space. Since we did not put any restriction on dimension of the spaces, all the foregoing discussion on vectors and tensors is valid for $n$ dimensional Riemannian manifolds.
However, embedding the curved space in a higher dimensional Euclidean space is not necessary-we can have a Riemannian space in its own right. We only need to prescribe a set of $n$ coordinates $x^{i}$ or a coordinate system in general on a subset of the space with sufficiently smoothness properties. This is called a chart. We generally require several charts to cover the space. In fact a curved space cannot be covered by a single chart. Now suppose we have another chart $x^{\prime j}$, then on the intersection of the two charts, we will have $x^{i i}$ s as functions of $x^{j}$ and vice-versa. In the intersection region of the two charts, the vectors and tensors follow exactly the same transformation laws as discussed earlier. Our analysis therefore is local and restricted to a chart or to an intersection region of charts. In this introductory book, we will restrict ourselves to local analysis. What about the metric? The metric must be prescribed. Since we do not have any Euclidean space in the background, the metric has to be given. It may be positive definite or indefinite as may be required from physical considerations. In relativity, both special and general, we have a 4 dimensional manifold on which the metric is indefinite.

Given these considerations, the foregoing discussion on vectors and tensors applies equally well to general $n$ dimensional Riemannian spaces.

3.8 向弯曲空间的过渡

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