## PHYS4445 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.

## PHYS4445 general relativity HELP（EXAM HELP， ONLINE TUTOR）

A person at the origin of an inertial reference frame $S$ observes a rod of proper length $l_0$ moving towards him at a speed $v$. He notes that the rod takes a time $T$ to pass him.
(a) As measured in $S^{\prime}$, the rest frame of the rod, how long will it take for the observer situated at the origin of $S$ take to pass along the length of the rod? [Hint: answering this question does not require a knowledge of special relativity.]
(b) Assuming that when the front end of the rod passes the observer, the clocks in $S$ and in the rest frame $S^{\prime}$ of the rod are both set to read zero, fill in the missing information in the following space-time coordinates $(x, t)$ and $\left.x^{\prime}, t^{\prime}\right)$ of the two specified events:
(i) Event $E_1$, the front end of the rod passing the observer at the origin of $S$, occurs at $(0, ?)$ in $S$ and $(0,0)$ in $S^{\prime}$.
(ii) Event $E_2$, the tail end of the rod passing the observer at the origin of $S$, occurs at $(0, ?)$ in $S$ and $\left(l_0, ?\right)$ in $S^{\prime}$.
(c) Using the Lorentz transformation, establish a relation between the coordinates of event $E_2$.
(d) Hence prove that the velocity of the rod is given by
$$v=\frac{l_0 / T}{\sqrt{1+\left[\left(l_0 / T\right) / c\right]^2}}$$

(a) The observer situated at the origin of the reference frame $S$ will be measured in $S^{\prime}$ to be approaching at a speed of $v$. Since the rod has length $l_0$ as measured in $S^{\prime}$, this observer at the origin of $S$ will take a time $l_0 / v$ to pass along its length.
(b) (i) Since the clocks on both $S$ and $S^{\prime}$ are synchronized to read zero when the front end of the rod passes the origin of $S$, then event $E_1$ will be observed to occur at $(0,0)$ in $S$.
(ii) The tail end of the rod will pass the observer at the origin of $S$ at time $T$, hence event $E_2$ will occur at the spacetime point $(0, T)$ as measured in $S$. The rod must be assumed to be moving in the negative $x$ direction with respect to $S$, i.e. the frame of reference $S^{\prime}$ (the rest frame of this rod) will be moving with a velocity $-v$ with respect to $S$. With this direction of relative motion, the tail of the rod will be positioned at $l_0$, and, from part (a), will be observed to pass the position of the observer in $S$ at time $l_0 / v$. Hence, in $S^{\prime}$, event $E_2$ will occur at the spacetime point $\left(l_0, l_0 / v\right)$.
(c) Event $E_2$ has coordinates $(0, T)$ as measured in $S$, and $\left(l_0, l_0 / v\right)$ as measured in $S^{\prime}$. Substituting the coordinates of event $E_2$ into either of the the Lorentz transformation equations
$$t^{\prime}=\gamma\left(t+v / c^2 x\right) \quad \text { or } \quad x^{\prime}=\gamma(x+v t)$$
gives $l_0 / v=\gamma T$. Using
$$\gamma=\frac{1}{\sqrt{1-v^2 / c^2}}$$
and solving for $v$ gives the required result, that is
$$v=\frac{l_0 / T}{\sqrt{1+\left[\left(l_0 / T\right) / c\right]^2}}$$

A muon is created at an altitude of $20 \mathrm{~km}$ due to the collision of a cosmic ray particle with a particle in the earth’s atmosphere. Given that the mean lifetime of a muon in its own rest frame is $2.0 \times 10^{-6} \mathrm{~s}$, at what speed would the muon have to be travelling if it were to reach the ground before decaying? Calculate the result using both Newtonian physics and relativistic physics.

According to Newtonian physics, the muon would have to be moving at a speed of $20000 / 2.0 \times 10^{-6}=10^{10} \mathrm{~ms}^{-1}$ in order to reach the earth before decaying.

According to relativistic physics, what we have here is two events: $E_1$, the creation of the particle and $E_2$, the arrival of the particle at the Earth’s surface. We can choose as our reference frames the rest frame of the muon $S^{\prime}$, and the rest frame of the earth, $S$. We further choose the origins of $S$ and $S^{\prime}$ to coincide at the instant that the muon is created, at which time the clocks in the two reference frames are set to zero. Downwards is taken to be the positive $x$ direction. Thus, the coordinates of the two events will be:
$$\begin{array}{ll} E_1: & x_1=0 \mathrm{~m}, t_1=0 \mathrm{~s} \text { in } S, \quad x_1^{\prime}=0 \mathrm{~m}, t_1^{\prime}=0 \mathrm{~s} \text { in } S^{\prime} \ E_2: & x_2=20000 \mathrm{~m}, t_2=? \text { in } S, \quad x_2^{\prime}=0 \mathrm{~m}, t_2^{\prime}=2.0 \times 10^{-6} \mathrm{~s} \text { in } S^{\prime} . \end{array}$$
Applying the Lorentz transformation for the event $E_2$ then gives
$$x_2=\gamma\left(x_2^{\prime}+v t_2^{\prime}\right)$$
where $v$ is the speed of $S^{\prime}$ relative to $S$, i.e. the speed of the muon relative to the Earth. Substituting the data given then leads to
$$20000=\gamma\left(0+v 2.0 \times 10^{-6}\right)$$
which gives
$$\frac{v / c}{\sqrt{1-v^2 / c^2}}=10^{10} / c=333.33$$
which can be readily solved to give $v / c=0.99855$.

## 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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## PHYS4445 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.

## PHYS4445 general relativity HELP（EXAM HELP， ONLINE TUTOR）

1. A reference frame $S^{\prime}$ passes a second reference frame $S$ with a velocity of $0.6 c$ in the $X$ direction. Clocks are adjusted in the two frames so that when $t=t^{\prime}=0$ the origins of the two reference frames coincide.
(a) An event occurs in $S$ with space-time coordinates $x_1=50 \mathrm{~m}, t_1=2.0 \times 10^{-7} \mathrm{~s}$. What are the coordinates of this event in $S^{\prime}$ ?
(b) If a second event occurs at $x_2=10 \mathrm{~m}, t_2=3.0 \times 10^{-7} \mathrm{~s}$ in $S$ what is the difference in time between the events as measured in $S^{\prime}$ ?

(a) The Lorentz transformation is defined by the set of equations
\begin{aligned} x^{\prime} & =\gamma(x-v t) \ t^{\prime} & =\gamma\left(t-v x / c^2\right) \ \text { where } \quad \gamma & =\frac{1}{\sqrt{1-(v / c)^2}} . \end{aligned}
Consequently, if the coordinates of an event in $S$ is $x_1=50 \mathrm{~m}, t_1=2.0 \times 10^{-7} \mathrm{~s}$, and the relative velocities of the two frames is $v=0.6 c$, then
\begin{aligned} x_1^{\prime} & =\frac{1}{\sqrt{1-(0.6)^2}}\left(50-0.6 \times 3 \times 10^8 \times 2.0 \times 10^{-7}\right)=17.5 \mathrm{~m} \ t_1^{\prime} & =\frac{1}{\sqrt{1-(0.6)^2}}\left(2.0 \times 10^{-7}-0.6 \times 50 / 3 \times 10^8\right)=1.25 \times 10^{-7} \mathrm{~s} . \end{aligned}
(b) Using the Lorentz transformation once again for the event $x_2=10 \mathrm{~m}, t_2=$ $3.0 \times 10^{-7} \mathrm{~s}$ in $S$ gives
$$t_2^{\prime}=\frac{1}{\sqrt{1-(0.6)^2}}\left(3.0 \times 10^{-7}-0.6 \times 10 / 3 \times 10^8\right)=3.5 \times 10^{-7} \mathrm{~s} .$$
Thus the time interval between the two events, as measured in $S^{\prime}$, is $t_2^{\prime}-t_1^{\prime}=$ $2.25 \times 10^{-7} \mathrm{~s}$.

1. A spaceship of length $100 \mathrm{~m}$ in its own rest frame $S_A$ passes a second spaceship $S_B$ at a relative speed of $\sqrt{3} c / 2$ and on a parallel course. When an observer at the centre of $S_A$ passes an observer located at the centre of $S_B$, a crew member of $S_1$ simultaneously fires very short bursts from two lasers mounted perpendicularly at the ends of $S_A$ so as to leave burn marks on the hull of $S_2$. The spaceships pass so close to each other that these laser beams travel negligibly short distances. Assume that the two observers are each at the origins of their respective rest frames, i.e. at $x_A=x_B=0$, and that these spatial origins are the midpoint of each of the space ships, and that the clocks are set to read zero i.e. $t_A=t_B=0$ in both reference frames when these two origins coincide (i.e. when the midpoints of each space ship are adjacent to each other.) Further assume that $S_B$ is of sufficient length that the laser beams will strike its hull.
(a) What are the coordinates of the two laser bursts (considered as events in spacetime) in $S_A$ ?
(b) What are the coordinates of these two events as measured in $S_B$ ?
(c) What is the distance between the marks appearing on the hull of $S_B$ ? Is this result an example of length contraction?

(a) The two laser bursts from spaceship $A$ occur at the points $x_{A 1}=50 \mathrm{~m}$ and $x_{A 2}=-50 \mathrm{~m}$ at time $t_A=0$.
(b) The coordinates of these events in the reference frame of the other spaceship are then
\begin{aligned} & x_{B 1}=\gamma\left(x_{A 1}-v t_A\right)=2(50-0)=100 \mathrm{~m} \ & x_{B 2}=\gamma\left(x_{A 2}-v t_A\right)=2(-50-0)=-100 \mathrm{~m} \end{aligned}
and
\begin{aligned} & t_{B 1}=\gamma\left(t_A-v x_{A 1} / c^2\right)=2\left(0-\sqrt{3} \times 50 / 2 \times 3 \times 10^8\right)=-2.9 \times 10^{-7} \mathrm{~s} \ & t_{B 2}=\gamma\left(t_B-v x_{B 1} / c^2\right)=2\left(0+\sqrt{3} \times 50 / 2 \times 3 \times 10^8\right)=2.9 \times 10^{-7} \mathrm{~s} . \end{aligned}
(c) The separation between the marks left on spaceship $B$ is $200 \mathrm{~m}$.
This appears to be an example of ‘length expansion’, but it in fact length contraction as can be seen by taking a closer examination of what is taking place here. Since determining the length of an object in some reference frame requires the simultaneous measurement of the positions of the endpoints of the object, we can consider what is taking place here as being the measurement, in reference frame $S_A$ of the separation of two points on the sides of spaceship $B$, a distance $200 \mathrm{~m}$ apart as measured in frame of reference $S_B$, i.e. this separation has a proper length $l_0=200 \mathrm{~m}$. Simultaneously in $S_A$, the separation between these two points is measured to be $l=100 \mathrm{~m}$. This distance is related to $l_0$ by $l=l_0 / \gamma$

## 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.