Defining a black hole is challenging, as it doesn’t have a single, straightforward definition, applicable to every object that we may call a black hole. However, black holes are among the most fascinating and heavily researched objects in theoretical high-energy physics. To gain some insight, let’s explore the Schwarzschild metric, which we introduced in one of our previous posts. This metric allows us to measure the distance between two infinitesimally close points in four-dimensional spacetime. But something unusual happens at two specific values of the radial coordinate r=r_H and r=0. At these points, certain components of the metric become infinite.
Is this a problem? It depends. At r=0 , spacetime indeed “breaks down,” leading to what’s known as a singularity. However, at r=r_H, this apparent issue isn’t a true singularity. With a different choice of coordinates, the infinity disappears, meaning r=r_H is not physically problematic in the same way as r=0. We’ll delve deeper into the concept of singularities in upcoming posts.
Interestingly, the radius r_H also represents a critical distance known as the “event horizon.” This is the boundary beyond which not even light can escape the black hole’s gravitational pull. Historically, the concept of a critical radius for massive objects dates back to the 18th century. Both John Michell and Pierre-Simon Laplace suggested that a sufficiently dense star would have a gravitational pull so strong that it could trap light. Although their work predated the theory of general relativity, which gives us the modern framework for black holes, their intuition from classical mechanics and second cosmic velocity indeed resulted in the correct value for the critical radius. Nowadays, we understand that this is mostly a consequence of dimensional analysis. If you’re curious, you can explore Michell’s original letter here:
https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1784.0008,
where he describes objects with properties resembling those of black holes. Note that their argument anticipated the objects whose existence was only directly confirmed through observations over 200 years later.