ASTR 3740 Relativity & Cosmology Spring 2025 Problem Set 3: Questions you posed.

1. 5 votes. If a more gravitationally strong object approached a black hole, what would happen?
2. 4 votes. Why can't white holes exist? Or can they?
3. 4 votes. If you are falling feet first into a supermassive black hole, and you point a flashlight toward your feet, will you ever see the light reach them? Explain.
4. 19 votes. Imagine you hold a mirror at arms' length and jump into a black hole. What do you see in your reflection as you approach the Schwarzschild radius? At what point do you stop seeing your reflection?
5. 8 votes. You are plunging feet-first into a black hole. What is the maximum number of images of the back of your own head that you might see?

Outside black holes
6. According to the no-hair theorem, black holes are only characterized by charge, mass, and spin. Give two reasons why highly charged black holes (probably) don't exist in nature?
7. If you were in front of a black hole and there was an odd star that was forming a ring around the black hole, is that because the gravity from the black hole is bending the matter of the star into that shape? Or is it something else? Explain.
8. What happens if a black hole is not accreting mass from outside?
9. What is the minimum distance from the center singularity of a black hole where circular orbit is stable?
10. If a rotating black hole is spinning, is it possible that at a high enough angular velocity to hover at a fixed position closer to the event horizon than for a Schwarzschild black hole? True or false?
11. Do black holes not emit light because the matter and light within the event horizon is so red shifted that it becomes colorless/black? Similar to the ‘A Slower Speed of Light’ simulation where moving backwards near the speed of light would cause objects surrounding you to disappear, becoming completely black. Why don't black holes emit any light at all past the event horizon?
12. If a blue star is behind a black hole and the blue star is spinning, would its direction of spin appear unchanged when the Einstein ring comes into observation?
13. Why does an outside observer never see an object actually cross the event horizon?
14. Near the horizon of an extremal Reissner-Nordstrom black hole, space-time resembles AdS₂ × S². Given the connection between AdS spheres and holography, could extremal black holes hold the key to understanding quantum gravity? What might it mean for the information paradox?
15. Suppose an advanced spacecraft with ultra-fast processors was to attempt circumnavigation through the photon sphere. Would they be likely to succeed?
16. If you are initially stationary barely within the horizons of two equally sized and equally distant supermassive black holes, what will happen to you, and will you be able to see anything outside the horizons?
17. Can a black hole's mass actually decrease over time?
18. Which property of black holes is described in the Kerr metric but omitted from the Gullstrand-Painleve metric? How does this unique characteristic of the Kerr metric distort spacetime surrounding the black hole as compared to the Gullstrand-Painleve metric?
19. Consider the formation of a black hole from a collapsing (giant) star. If material that enters a black hole – to an outside observer – appears to freeze at the event horizon and become redshifted, then wouldn't we be able to see the image after its formation process? Thus knowing what a black hole was formed from, breaking the no hair theorem (black holes cannot be distinguished beyond their mass, spin, and electric charge)?
20. All circular orbits within the innermost stable circular orbit of a black hole are unstable and will quickly spiral into the black hole unless a great force is applied on an object to stay out. True or false?
Tidal forces
21. The larger a black hole is, the stronger its tidal forces are near the event horizon. True or false?
22. Given that the tidal force for a black hole goes as M/(r²) (where M=mass and r=radius), how is it possible that smaller mass black holes exert a larger tidal force on infalling objects than larger mass BHs?
23. Are the tidal forces at the horizon stronger for a stellar mass black hole or a supermassive black hole like Sag A* (which will rip you apart faster as you approach the horizon)?
Falling into a black hole
24. If light attempting to escape from the horizon is trapped stationary (if at a perfectly outward trajectory), would you see the final image of the collapsing star that formed the black hole as you fell?
25. What happens if you fall into a black hole feet first and what would you see in the process?
26. If you launch a probe into a black hole, what would you see, observing the images from said probe?
27. If an observer were to fall into a black hole, if they did not have gridlines to aid them, would they ever realize they passed the event horizon?
28. When looking at a Schwarzschild surface we know nothing, not even light, can escape. Is it true that this surface is then completely black? If not, explain.
29. If you enter a black hole, you cannot escape it, but can you accelerate enough to remain at rest relative to the stars (in other words, can your rockets counteract the gravity of the black hole enough such that you stop falling towards the singularity)?
30. If you and a friend were to fall into a black hole from opposite sides at precisely the same moment, would you be able to give each other a high five right before hitting the singularity?
31. If you were to fall into a black hole at a point where everything is blue shifted significantly as you look outward, is it possible for a person to also fall in afterwards and when they arrive to you they red shift to a normal image/do not appear blue shifted?
32. If you are free falling into a Schwarzschild black hole could you use powerful rockets/propulsion to stop yourself crossing the event horizon once you're close to it? Why or why not?
33. If you fall into a black hole, do you see yourself about to land at a singularity? Why?
34. True or False - When approaching a black hole's singularity, and observer would view said singularity as a single point in space.
35. What causes the mass inflation instability at the inner horizon of a black hole?
36. Imagine you are at a stable orbit around a black hole, very far away from it (and not on earth), travelling at a very low speed. Unfortunately, an evil scientist somehow accelerated the entire earth towards a black hole along an unstable orbit path. When the earth is at around 2 Schwarzschild radii from the singularity, a hero turns on super thrusters that are enough to make the earth stay in this orbit! (a) You observe the earth to be orbiting the black hole at his radius, how is it distorted in your view at different locations along its trajectory? (b) The thrusters get turned off for a split second oh no! The distance from the singularity has now decreased to 1.5 Schwarzschild radii from this fault. Is there any chance the earth can still orbit?
37. As you fall radially inward towards a black hole, you gaze out towards the universe one last time. You notice that as you fall in, your view of the universe becomes increasingly concentrated into a bright circle. Why is that?
Wormhole
38. If a stable Schwarzschild wormhole suddenly formed and connected our universe to another, what evidence might tell us about its presence?
39. Why is it not likely that a black hole creates a corresponding white hole that can act as a wormhole to another space and/or time?
40. You would be able to survive a journey through a wormhole formed between a black hole and white hole if the black hole were massive enough. True or false?
41. The Schwarzschild metric admits negative square root as well as positive square root solutions for the geometry! What do the negative and positive square root solutions represent?
42. Is causality violated in the mirror interpretation of the Schwarzschild wormhole?
Hawking radiation
43. Consider the super massive black hole Sagittarius A* with a mass of ∼4.3×10⁶ solar masses and the ∼15 solar mass black hole Cygnus X-1. Which black hole has the higher Hawking luminosity? How much larger is the luminosity?
44. Hawking radiation, the quantum radiation given off by black holes, allows black holes to be slightly luminous. Find the Hawking luminosity of a black hole with a temperature of 15 K and a Schwarzschild radius of 3 km. Assume the Hawking temperature does not exceed the rest mass energy of any particle type.
45. How does the evaporation time of a black hole depend on its mass? What is the approximate maximum mass a black hole can have in order to evaporate in less time than the age of the universe? How much longer would it take a 30 solar mass black hole to evaporate compared to a 1 solar mass black hole? Show work.

Updated 2025 Mar 3