An experiment (based on the Relativistics Astrophysics & Cosmology course offered by Cambridge’s Part III Physics program) on using Quantumplations to generate explainer video solutions for problem sheets.
Problem: Calculate the gravitational redshift between a point at infinity and the surface of \((a)\) the Earth, \((b)\) the Sun, \((c)\) a white dwarf and \((d)\) a neutron star, making reasonable assumptions. Bonus: Could any of these redshifts be measured? If so, how? (You
will need to consider whether any features exist which could be used to measure the
redshift, and if so, whether or not the redshift would be visible in the presence of other
broadening effects).
Solution:
Problem: In the lectures an “Einstein Tower” experiment was discussed and it was shown that a freely-falling observer who drops from rest from above the top of the tower (height \(h\)) at the moment when the photon begins climbing up from the bottom, finds that to first order in \(gh/c^2\) there is no change in the photon energy between top and bottom of the tower. Generalize this analysis to the case where instead of falling from rest, the (unfortunate) observer is projected downwards with an initial speed \(u\). Explain why these results support the Strong Equivalence Principle.
Solution:
Problem: Find the geodesic equations on the surface of a sphere, radius \(a\), using the standard \((\theta,\phi)\) coordinate system. Show that these equations are satisfied by great circles. What are the geodesics on the surface of a cylinder?
Solution:
Problem: It is sometimes said that an astronaut falling into a black hole would feel nothing peculiar as they crossed over the event horizon. Discuss whether this is true, and how the statement depends upon the black hole mass, using the following approach. First, use the Newtonian force law to estimate the tidal force affecting two compact objects each of mass \(m\), joined by a taut wire of length \(\ell\), falling radially into a distant black hole of mass \(M\). The tidal force derived using this Newtonian approach is in fact equal to the proper result derived using full general relativity (Optional exercise: demonstrate this using a suitable notion of force in GR), and can therefore be applied all the way up to the hole. Thus show that at the horizon the wire must support a tension of:
\[\frac{mc^6\ell}{8G^2M^2}\]
Using this, estimate numerically the tidal force on a human being at the event horizon
for (a) a \(1M_{\odot}\) mass black hole, and (b) a black hole of mass \(108M_{\odot}\).
Solution: