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PHYSICS262课程简介
Einstein’s General Theory of Relativity is a basis for modern ideas of fundamental physics, including string theory, as well as for studies of cosmology and astrophysics. The course begins with an overview of special relativity, and the description of gravity as arising from curved space. From Riemannian geometry and the geodesic equations, to curvature, the energy-momentum tensor, and the Einstein field equations. Applications of General Relativity: topics may include experimental tests of General Relativity and the weak-field limit, black holes Schwarzschild, charged Reissner-Nordstrom, and rotating Kerr black holes, gravitational waves (including detection methods), and an introduction to cosmology (including cosmic microwave background radiation, dark energy, and experimental probes). Prerequisite: PHYSICS 121 or equivalent including special relativity.
Terms: Aut | Units: 3
Instructors: Graham, P. (PI) ; Abdel-Aziz, A. (TA)
Schedule for PHYSICS 262
Prerequisites
General relativity is a theory of gravitation developed by Albert Einstein between 1907 and 1915. The theory of general relativity says that the observed gravitational effect between masses results from their warping of spacetime.
By the beginning of the 20th century, Newton’s law of universal gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses. In Newton’s model, gravity is the result of an attractive force between massive objects. Although even Newton was troubled by the unknown nature of that force, the basic framework was extremely successful at describing motion.
PHYSICS262 General Relativity HELP（EXAM HELP， ONLINE TUTOR）
A particle in Minkowski space travels along a trajectory:
$$
\begin{aligned}
& x(\tau)=\alpha \tau^2 \
& y(\tau)=\tau \
& z(\tau)=0
\end{aligned}
$$
(a) What are the spacelike components of the 4 -velocity, $U^i$ ?
Solution:
The spacelike components of four velocity is
$$
U^i=\frac{\partial x^i}{\partial \tau}=(2 \alpha \tau, 1,0)
$$
That’s correct! The 4-velocity of a particle in Minkowski space is given by:
$$
U^i=\frac{d x^i}{d \tau}
$$
where $\$ x^{\wedge} i \$$ are the spacetime coordinates of the particle and $\$ \backslash$ tau $\$$ is its proper time.
In this case, we have:
$$
x^i=(x, y, z)=\left(\alpha \tau^2, \tau, 0\right)
$$
So the spacelike components of the 4-velocity are:
$$
U^i=\frac{d x^i}{d \tau}=(2 \alpha \tau, 1,0)
$$
Well done!
Using the relation $U \cdot U=-1$, compute $U^0$.
Solution:
The inner product of the four velocity vector $U^\mu=\left(U^0 U^1 U^2 U^3\right)$ is
$$
\begin{aligned}
U \cdot U=-\left(U^0\right)^2+\left(U^1\right)^2+\left(U^2\right)^2+\left(U^3\right)^2 & =-1 \
\Longrightarrow-\left(U^0\right)^2+4 \alpha^2 \tau^2+1+0 & =-1 \
\Longrightarrow U^0 & = \pm \sqrt{2+(2 \alpha \tau)^2}
\end{aligned}
$$
This is the timelike component of velocity four vector.
To find $\$ U^{\wedge} 0 \$$, we can use the relationship between the 4-velocity and the spacetime interval $\$ d s^{\wedge} 2 \$$ :
$$
d s^2=-c^2 d \tau^2=-U \cdot U d \tau^2
$$
where $\$ c \$$ is the speed of light. For a massive particle, we have $\$ \mathrm{ds}^{\wedge} 2<$ $0 \$$, so $\$ U \backslash c \operatorname{dot} U=-1 \$$.
Using the spacelike components of the 4-velocity that you calculated earlier, we have:
$$
\begin{aligned}
U \cdot U & =-\left(U^0\right)^2+\left(U^1\right)^2+\left(U^2\right)^2+\left(U^3\right)^2 \
& =-\left(U^0\right)^2+4 \alpha^2 \tau^2+1+0 \
& =-1
\end{aligned}
$$
Solving for $\$ U^{\wedge} 0 \$$, we get:
$$
U^0= \pm \sqrt{2+(2 \alpha \tau)^2}
$$
This is the timelike component of the 4-velocity, which can be interpreted as the rate at which proper time \$\tau \$ elapses for the particle as measured by an observer at rest in the reference frame in which the particle is moving.
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