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ASTR160 general relativity课程简介
One-semester introduction to general relativity that develops the essential structure and phenomenology of the theory without requiring prior exposure to tensor analysis.
General relativity is a fundamental cornerstone of physics that underlies several of the most exciting areas of current research, including relativistic astrophysics, cosmology, and the search for a quantum theory of gravity. The course briefly reviews special relativity, introduces basic aspects of differential geometry, including metrics, geodesics, and the Riemann tensor, describes black hole spacetimes and cosmological solutions, and concludes with the Einstein equation and its linearized gravitational wave solutions. At the level of Gravity: An Introduction to Einstein’s General Relativity by Hartle.
PREREQUISITES
One-semester introduction to general relativity that develops the essential structure and phenomenology of the theory without requiring prior exposure to tensor analysis.
General relativity is a fundamental cornerstone of physics that underlies several of the most exciting areas of current research, including relativistic astrophysics, cosmology, and the search for a quantum theory of gravity. The course briefly reviews special relativity, introduces basic aspects of differential geometry, including metrics, geodesics, and the Riemann tensor, describes black hole spacetimes and cosmological solutions, and concludes with the Einstein equation and its linearized gravitational wave solutions. At the level of Gravity: An Introduction to Einstein’s General Relativity by Hartle.
ASTR160 general relativity HELP（EXAM HELP， ONLINE TUTOR）
Show that $J^{+}(p)$ is not an open set.
Hint: Suppose $r$ is a point in $J^{+}(p)-I^{+}(p)$. Let us consider a neighborhood of $r$. Then, some points of this neighborhood lie outside of $J^{+}(p)$ and some points lie within $J^{+}(p)$ (see Fig. 29). Therefore, $r$ is not an interior point. Hence, $J^{+}(p)$ is not an open set.
For all subsets $S \subset M$, show that $J^{+}(p) \subset \overline{I^{+}(p)}$.
Hint: Let us consider a Minkowski space with a point $r$ removed as shown in Fig. 30. The points after the deleted point $r$, i.e., on the dotted line are not in $J^{+}(p)$. However, points on the dotted line are in the closer $\overline{I^{+}(p)}$. Hence, $J^{+}(p) \subset \overline{I^{+}(p)}$.
Let $J^{+}(x)$ be closed in $M$. Show that $M$ is causal if $\dot{I}^{+}(x) \cap I^{-}(x)={x}$ for all $x \in M$.
Hint: Given that $M$ is causal. Now let, if possible, $\exists \mathrm{y}(\neq x)$ such that $y \in I^{+}(x) \cap I^{-}(x)$. We know $\dot{I}^{+}(x)=j^{+}(x)$, therefore, $y \in j^{+}(x)$. Since $J^{+}(x)$ is closed in $M$, therefore, $y \in J^{+}(x)$. This means, $x<y$. Similarly, $y \in J^{-}(x)$ indicates $y<x$. This suggests that there is a closed causal curve through $x$. Definitely, this violates the causality. Hence, $y \neq x$ is not true, i.e., $y=x$.
If $\left{\lambda_n\right}$ be a sequence of future inextendible causal curves with a limit point $p$, then $\exists$ a future inextendible causal curve $\lambda$ passing through $p$, may be a limit curve of the $\left{\lambda_n\right}$. Also if $\left{\lambda_n\right}$ be a sequence of time-like curves, then the limit curve $\lambda$ may be only a causal curve.
Textbooks
• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.
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