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I am trying to equate the famous Stephen Hawking to some of our other famous scientists and noted that the vast majority have an associated equation with their name. As for example Einstein was the famous E=mc2 and his stress tensor field equation linking matter and energy to space and time. Schrödinger like his one famous Schrödinger's equation. Boltzmann's entropy formula. Heisenberg's uncertainty equation. The Dirac equation. and so on and so forth.

I am thinking he will be remembered for some formula that links quantum mechanics to general relativity and information.

Can anyone venture what this might be? I assume there are other formulas that are perhaps more unclear in the sense that I saw a book by Leonard Susskind to the effect how he made the world safe for quantum mechanics and this was related to Hawking's formulas I assume but frankly the book was over my level of physics training. Maybe someone can comment on how this plays into Hawking's most notable "formula". Assuming such a thing exists.

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    $\begingroup$ Perhaps something related to the fact that the entropy of a black hole is proportional to its surface area? $\endgroup$
    – Dave L Renfro
    Commented Mar 18, 2018 at 21:04
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    $\begingroup$ I will be surprised if he's remembered more for anything else than Hawking Radiation, at least among non-physicists. $\endgroup$
    – user541686
    Commented Mar 19, 2018 at 2:20
  • $\begingroup$ @Mehrdad ALS (no offence) $\endgroup$
    – gen-ℤ ready to perish
    Commented Nov 22, 2019 at 18:17

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One can only speculate what Hawking will be remembered for, but according to NYT in 2002 he apparently expressed a wish to have what he saw as his biggest accomplishment engraved on his tombstone (following the tradition that goes back at least to Archimedes):"Perhaps in imitation of Boltzmann, Dr. Hawking declared at the end of the meeting that he wanted the formula for black hole entropy engraved on his own tombstone". The formula in question is $$S=\frac{kc^3}{4G\hbar}A,$$ where $A$ is the area of the event horizon, $c$ is the speed of light and $k$, $G$, and $\hbar$ are the Boltzmann, gravitational and reduced Planck constants, respectively. It is also often written in the form $S=\frac{k}{4l_P^2}A$, where $l_P:=\sqrt{\frac{G\hbar}{c^3}}$ is the Planck length, and called the Bekenstein–Hawking formula (Bekenstein conjectured the proportionality to $A$). The formula is related to Hawking's work on black hole evaporation ("Hawking radiation") back in 1973-4. It led to the so-called black hole information loss paradox, see (Information) Paradox Regained? by Manchak and Weatherall for a recent status review, and was later linked to 't Hooft's 1993 holographic principle in string theory.

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    $\begingroup$ Hawking seems to have preferred writing it using $h$, viz. $S=\frac{\pi kc^3 A}{2Gh}$. $\endgroup$
    – J.G.
    Commented Mar 18, 2018 at 22:21
  • $\begingroup$ Thank you .....I believe this is the one that would have earned him the Nobel Prize. I had deleted the post...but then you were kind to respond and I promptly accepted ... by the way ...your response also answers my 2nd question about the evolution on the "Information Paradox" your answer has inspired me to do a little more research $\endgroup$
    – Sedumjoy
    Commented Mar 18, 2018 at 22:53
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    $\begingroup$ "Perhaps"?! There is absolutely no way he wasn't deliberately and consciously referencing Boltzmann. $\endgroup$
    – Gareth McCaughan
    Commented Mar 19, 2018 at 10:57
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    $\begingroup$ His tombstone actually reads: HERE LIES WHAT WAS MORTAL OF STEPHEN HAWKING 1942–2018 $$T=\frac{\hbar c^3}{8\pi GMk}$$ $\endgroup$
    – gen-ℤ ready to perish
    Commented Sep 2, 2018 at 7:43
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    $\begingroup$ @gen-z: This is now what the tombstone inscription says: cai.cam.ac.uk/news/memorial-professor-stephen-hawking $\endgroup$
    – Alexandre Eremenko
    Commented Nov 22, 2019 at 1:53

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