OPhO 2023, Теория — Задача 3. T3: General Relativity

The Alphabet of Spacetime Metrics

In multivariable calculus, we study the generalization of line elements to various coordinate systems. A coordinate system in $\mathbb{R}^3$ can be described by a function $\psi$ that takes a point $(x, y, z) \in \mathbb{R}^3$ and outputs the three coordinates describing the same point in the coordinate system. For example, the $\psi$ describing spherical coordinates would be defined as $\psi(x, y, z) = (r, \theta, \phi)$. As there must be a unique representation of each point in $\mathbb{R}^3$ in the $\psi$ coordinates (ex. $r = 3, \theta = \frac{\pi}{2}, \phi = \pi$ cannot correspond to two points on the 3D space) there must be a one-to-one correspondence between points in $\mathbb{R}^3$ and points described in $\psi$ coordinates. Hence, there’s a well defined inverse function $\psi^{-1}$. In spherical coordinates, for instance, $\psi^{-1}(r, \theta, \phi) = (x, y, z)$.

Suppose we have a coordinate system $\psi(x, y, z) = (q_1, q_2, q_3)$. We ask a very important question: if we make an increment of $dq_i$ in each $q_i$ in the $\psi$ coordinates, what’s the physical distance between the starting and the finishing points in $\mathbb{R}^3$? In other words, what’s the distance between the points $\psi^{-1}(q_1 + dq_1, q_2 + dq_2, q_3 + dq_3)$ and $\psi^{-1}(q_1, q_2, q_3)$? This distance is known as the line element generated by a coordinate system $\psi$. If the coordinate system is orthogonal, meaning the basis vectors $\hat{q_1}$, $\hat{q_2}$, $\hat{q_3}$ are all mutually perpendicular, we know that the physical displacements in taking $q_i \to q_i + dq_i$ are in perpendicular directions in $\mathbb{R}^3$, so we can find the line element by Pythagoras’s theorem.

(a) Show that for an orthogonal coordinate system $\psi$ in an arbitrary $\mathbb{R}^n$, the square of the line element $ds^2$ has the form $\sum_{i=1}^n g_i(q_1, ..., q_n)dq_i^2$ where each $g_i$ is a function of $q_1, ..., q_n$. Find an expression for $g_i$ in terms of the derivatives of $\psi$ or $\psi^{-1}$.

We can think of physical observers as coordinate systems. If an arbitrary observer moving in space-time measures an event $(t, x, y, z)$, the coordinates seen by the observer can be expressed as $\psi(t, x, y, z)$ by some function $\psi$. It turns out that special relativity can be described with the math we’ve developed so far. In particular, we can consider measuring space-time lengths by integrating the line element along paths in space-time. You are given that for any observer with the coordinate system $\psi \equiv (t, x, y, z)$, the line element is given by $ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$

(b) It’s a well-known result of special relativity that space-time lengths are measured to be the same regardless of the frame. Resolve the twin paradox by integrating the line element along the spaceship twin’s world-line.

It turns out that physics in weak gravitational fields can also be described with the math we’ve developed so far. Consider a 1 dimensional system with a time-independent gravitational potential $\Phi(x)$ everywhere.

(c) An important postulate of general relativity states that observers that move solely under the influence of gravity take the path with the longest space-time length. Suppose an observer of this system takes a path parameterized by a variable $\sigma \in [0, \sigma_f]$: $(t(\sigma), x(\sigma), y(\sigma), z(\sigma))$. Show that the line element $ds^2 = -(1 + 2\Phi(x))dt^2 + (1 - 2\Phi(x))dx^2$ generates equations of motion that are consistent with Newton’s laws (neglecting special relativity).

Hint: Refer to the Euler-Lagrange equation, but you do NOT have to perform the derivation to solve this problem.

(d) Derive a statement of time dilation at a point $(x, y, z)$.

The Spacetime Metric of a Black Hole

As we saw in part 1, the Minkowski metric describes geometry of spacetime in special relativity and is given as

In general relativity, the result is more complex, and we describe spacetime via the Schwarzchild metric, a solution of Einstein’s equations under spherical symmetry. The spacetime interval in spherical coordinates can be expressed in terms of proper time $\tau$ and the speed of light $c$ as

where $t$ is the coordinate time, $r$ is the radial coordinate, $\theta$ is the polar angle, $\phi$ is the azithumal coordinate, and $r_s$ is the Schwarzchild radius of a massive body. The Schwarzchild radius is related to the body’s mass $M$ by $r_s = \frac{2GM}{c^2}$ where $G$ is the universal gravitational constant.

General relativity describes gravity as the consequence of curving spacetime. Geodesics generalize straight lines to curved spacetime and are very useful as a result. In the following problem, the radial geodesic equation may be helpful:

where $\Gamma$ represents a Christoffel symbol. The relevant terms of $\Gamma$ are $\Gamma^r_{tt}$ , $\Gamma^r_{rr}$ , $\Gamma^r_{\theta\theta}$ , and $\Gamma^r_{\phi\phi}$. Define $B = 1 - \frac{r_s}{r}$, then we can get:

The Photon Sphere

In 2019, the first image of a black hole was taken in 2019 by the Event Horizon Telescope (EHT) at the center of the galaxy M87. The historic image of the black hole showcased a glowing ring of light surrounding the dark abyss known as the ”photon ring.” In this region, light rays follow closed circular paths due to the strong gravitational pull of the black hole. When a photon follows a closed circular path on the photon ring, it can effectively orbit the black hole multiple times before either escaping to infinity or being captured by the event horizon. This phenomenon is known as the ”photon sphere.”

You may recall the massive outburst of media to the images of M87. Many were already talking about it before the images came out, like Veritasium. Take 10 minutes to watch the video.

In this problem, we will be analyzing the optical effects for an observer next to a black hole. Assume for an idealized case that the black hole has a mass $M = 15M_\odot$ and is uncharged and not rotating.

(e) Prove that the minimum radius of the black hole’s photon ring follows the relationship $r_p = \frac{3}{2}r_s$.

(f) Bill is near the black hole a distance $r_p$ away. Assume Bill’s width is $w = 2\delta$ where $\delta = 0.25$ m. Sketch light ray trajectories showing how Bill could see his own back. Determine the range for the angle of view $\alpha$ enabling this.

(g) Approximately what is the orbital period (in miliseconds) of a singular photon in coordinate time $t$ around the black hole. At what mass of a black hole would this time be greater than the visual reaction time of a human $t_r = 150$ ms (meaning if you closed your eyes, it would take some time for you to visualize your back again)? Is this mass possible?

(h) Propose a method (realistic or not) for Bill to safely get close to the black hole without getting ripped apart by gravity. If you believe there is no possible method, then give your reasoning behind why. You might want to consult various sources for information. To simplify the process and avoid teams searching for data such as ”shear stress of a human”, assume that Bill is a cylindrical object made of a material of your choosing that has a uniform density, a radius $\delta$, and a length of $L = 1.75$ m. Bill possesses exceptional engineering skills, enabling him to construct virtually anything, provided it doesn’t contravene the principles of physics.

(i) Let’s assume that the galaxy has $N_{\mathrm{star}}$ stars, each emitting light isotropically with the same total power $L$. Assume the stars are distributed uniformly throughout a spherical volume of radius $R$. For an ideal case where the path of the light from the stars is not obstructed by any other celestial objects, estimate the expected number of photons from starlight that would contribute to the photon ring or fall in the event horizon per unit time.