by Vrunda Macwan
Vrunda Macwan did their undergrad from St. Xavier’s College, Ahmedabad. This research was done as a partial requirement for their degree in the final year, under the supervision of Mr. Ashok Joshi and Dr. Pankaj Joshi from Ahmedabad University. Vrunda’s research interests lie in general relativity, black hole physics, gravitational waves, and mathematical physics. Outside of academia, they regularly keep switching and learning new hobbies besides reading novels and watching shows. Right now they’re learning pottery.
Theory
In 2019, astronomers revealed the image of a black hole which is at the centre of the galaxy M87 using the Event Horizon Telescope (EHT). But long before that, almost a century ago, Einstein, with his theory of general relativity, predicted the existence of black holes as well as something called “photon rings” around them. After the first image of black hole was revealed, it got the scientists in the field much more curious about these so-called “photon rings or photon spheres”.
As the name suggests, it is a ring created by photons. The properties of this ring depend on the object it is orbiting. One way of categorising black holes is by their geometry. In this case, we have selected a black hole that is a perfect sphere, non-rotating, and uncharged. These types of black holes are called “Schwarzschild black holes”.
Now coming back to photon rings and their formation. Suppose light rays – streams of photons are travelling from infinity to the black hole in question. There are three possibilities of what can happen to a singular photon.
- The gravity of the black hole sucks the photon towards itself
- The gravity at that particular distance is not strong enough and hence the photon loses the trajectory and goes into the eye of the observer.
- The distance of the photon from the black hole is just right enough to create a balance between the above two possibilities. Hence the photon starts orbiting around the black hole.
The third possibility ends up creating a ring or a sphere of photons around the black hole which is known as a photon sphere. The distance at which this occurs is R = 3MG, where R is the radius, M is the mass of the black hole being studied, and G is the gravitational constant. When R > 3MG, the photon escapes its trajectory and goes into the eye of the observer. When R< 3MG, it gets sucked inside the black hole.
Methodology
To derive this R=3MG equation, we first begin with deriving the geodesic equation of a Schwarzschild black hole. A geodesic is usually the shortest path that can be taken in any geometry. To solve the equation, we go back to classical mechanics, specifically, lagrangian mechanics. The Euler-Lagrangian equation of motion helps to find the shortest path between two points. From this, we get the geodesic equation for a Schwarzschild black hole. Further, we calculate the orbit equation that the photons are going to follow.
Now we derive something called effective potential. As the name suggests, it is the effective, or the net sum of all the potentials that are acting on the system. In this system, there is gravitational force as well as centrifugal force acting. These are combined into a single potential formula which reduces the problem to a one-dimensional problem. When a photon is travelling from infinity, it is considered as a central force problem. We can use the effective potential to understand the motion of photons. Using the laws of calculus, the first derivative of the effective potential formula gives us the R=3MG.
There comes a point where the total energy of a photon is equal to effective potential. When the total energy of a photon is more than the effective potential, the photon will escape from the black hole, probably into the observer’s eye. Meanwhile, if the total energy is less than the effective potential, then the photon will get sucked inside.
Scientists around the world have been studying this phenomenon and figuring out many things. One of them is a black hole shadow. Due to the presence of this photon sphere, a shadow is cast on the interior of the photon sphere which has come to be known as a black hole shadow. This can be used to determine the properties of the black hole or any compact object that we’re studying. One can also do vice versa, from the known properties of the black hole to find the orbit of the photon sphere.
The results from the study of photon spheres is huge. This is an important field of study in astrophysics, especially in this era of telescopes that reveal the real images of black holes. Not to mention, it provides another experimental test of general relativity.
Edited by Sumeet Kulkarni
Featured Image credit: Event Horizon Telescope, via Wikimedia Commons
