Black holes are 230 years old

On this day in history, 230 years ago, the world learned of the idea of a black hole.
The first known place in which a black hole is described is in a letter written by John Mitchell which was sent to Henry Canvendish in 1783. In the letter Mitchell writes:

If the semi-diameter of a sphere of the same density as the Sun in the proportion of five hundred to one, and by supposing light to be attracted by the same force in proportion to its [mass] with other bodies, all light emitted from such a body would be made to return towards it, by its own proper gravity. -Mitchell

The letter was written on the 26th of May 1783. It was then read at a meeting of the Royal Society in London on the 27th of November of the same year. Finally it was published in the society’s journal, Philosophical Transactions, Royal Society, London, on the 1st of January 1784 [aside: The Royal Society, a group of natural philosophers and scientists, formed on the 28th of November 1660, making it over 350 years old!].
Mitchell reached his conclusion by using Newton’s Laws. Sir Isaac Newton had published his laws of motion and gravitation via Principia Mathematica in the year 1687, about a century before Mitchell’s letter to Cavendish. You can use Newton’s laws to ask the question: how fast do I have to be moving in order to escape the gravitational pull of Earth? The answer is found by equating your Kinetic Energy (Ek, the energy associated with your moving away from the planet) and your gravitational potential energy (Eg, the energy associated with Earth pulling on you):

(1)   \begin{equation*} Ek=Eg \end{equation*}

(2)   \begin{equation*} {1 \over 2} mv^2 = {GMm \over r} \end{equation*}

where M is the mass of the Earth, m is your mass, v is your speed, G is the universal gravitational constant, and r is the radius of the Earth. Playing with the equation a bit (and entering in the M and r of Earth), you get:

(3)   \begin{equation*} v=\sqrt{{2GM \over r}}=11.2 km/s \end{equation*}

Therefore, in order to escape Earth’s gravity, you need to have a starting speed of 11.2 km/s. This is called, happily, ‘escape velocity.’ You can use this equation on any object whose mass and radius you know (escape velocity for: the Moon=2.4 km/s, Mars=5.0 km/s, and so on).
Mitchell’s mental leap was to ask the equation: ‘what if an object’s escape velocity was the speed of light?’ What would the ratio of mass to radius of that object be? The speed of light was measured at the time to be 295 000 km/s (Bradley, 1728). So, crunching the numbers, Mitchell found that an object with the same density of the Sun, but with a radius 500x bigger, would have a surface escape velocity equal to the speed of light. Therefore, ‘all light emitted from such a body would be made to return towards it, by its own proper gravity.’
Astounding! This was the first time anyone had ever supposed there to be an object that would have a gravitational potential well deep enough to capture light. Unfortunately, Mitchell’s work was not uncovered until the 1970s. Until that time, Pierre Laplace was considered the first to propose the idea.

A slight addendum to the story: Mitchell based his proposal on the idea that light would be attracted to matter via Newton’s laws. This was accepted at the time but research into particle vs. wave theory of light in the 1800s showed that light could not act in such a way. This squashed the idea of black holes (or ‘dark stars’ as they were referred) until a gentlemen by the name of Albert Einstein developed General Relativity and showed that light follows the curved space created by a massive object. This reinstated a scientific basis for the idea of a black hole.
John Mitchell was a polymath: geologist, mathematician, physicist, astronomer, and more. He contributed deeply to many fields (including plate tectonic theory, and magnetism).

Suggested Reading:

Philosophical Transations – The original letter written to Cavendish
American Museum of Natural History – John Mitchell and Black Holes
Astronomy Society of Edinburgh – Black Holes History
The American Physical Society – John Mitchell Anticipates Black Holes

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