应用数学的进一步方法 Further Methods of Applied Mathematics MATH323
But this is the never null condition which we have assumed. Also, $$ \left\langle\frac{d x^{i}}{d s}, \frac{d x^{i}}{d s}\right\rangle=g_{i j} \frac{d x^{i} d x^{j}}{d s d s}=g_{i j} \frac{d x^{i} d x^{j}}{d t d t}\left(\frac{d t}{d s}\right)^{2}=\pm\left(\frac{d s}{d t}\right)^{2}\left(\frac{d t}{d s}\right)^{2}=\pm 1 $$ For the converse, we are given a parameter $t$ such that $$ \left\langle\frac{d x^{i}}{d t}, \frac{d x^{i}}{d t}\right\rangle=\pm 1 . $$ in other words, $$ g_{i j} \frac{d x^{i} d x^{j}}{d t d t}=\pm 1 . $$ But now, with $s$ defined to be arc-length from $t=a$, we have $$ \left(\frac{d s}{d t}\right)^{2}=\pm g_{i j} \frac{d x^{i} d x^{j}}{d t d t}=+1 $$
The following are equivalent for a locally Minkowskian manifold $M$ (a) A coordinate system $\bar{x}^{i}$ is Lorent $z$ at the point $p$ (b) If $x$ is any frame such that, at $p, G=\operatorname{diag}\left[1,1,1,-c^{2}\right]$, then the columns of the change-of-coordinate matrix $$ D_{j}^{i}=\frac{\partial \bar{x}^{i}}{\partial x^{j}} $$ satisfy $\langle$ column $i$, column $j\rangle=\left\langle\boldsymbol{e}{i}, \boldsymbol{e}{i}\right\rangle$, where the inner product is defined by the matrix $G$. (c) $\bar{G}=\operatorname{diag}\left[1,1,1,-c^{2}\right]$
Relativity, one of the greatest theories of twentieth century physics, has dramatically changed mankind’s “common sense” conception of the universe and nature, and many of its conclusions are still difficult to understand today – “simultaneous relativity,” “four-dimensional spacetime,” “spacetime folding,” the “equivalence principle,” and so on. Many of these conclusions are still difficult to understand today – “simultaneous relativity”, “four-dimensional spacetime”, “spacetime folding”, “equivalence principle”, etc. – and have led to many misunderstandings. – and has led to many misunderstandings, even misinterpreting the theory and doing bad things. Most people think that relativity is just an impractical physical theory, words written on paper.
相对论后作业代写
In case the Lipschitz condition is satisfied [38], the system of (1.75) admits a unique solution of the initial value problem $\mathcal{X}(0)=x_{0}$. The solution is called the integral curve of the system (1.75) passing through $x_{0}$. We now consider the family of integral curves passing through various initial points by putting $$ \begin{gathered} x=\xi\left(t, x_{0}\right) \ x_{0} \equiv \xi\left(0, x_{0}\right) \end{gathered} $$ Changing the notation in (1.76i) and (1.76ii), we write $$ \begin{aligned} &\hat{x}=\xi(t, x) \ &x \equiv \xi(0, x) \end{aligned} $$ The original differential equations yield $$ \frac{\partial \xi(t, x)}{\partial t}=\overrightarrow{\mathbf{V}}(\xi(x, t)) $$
