This may be considered a companion piece to my earlier post Pulp Physics, which paid homage to Quentin Tarantino (and Princeton), while this is triggered by a spontaneous epiphany:
Body Heat + Fatal Attraction = Black Hole Thermodynamics.
Body Heat is a 1981 American neo-noir erotic thriller film written and directed by Lawrence Kasdan in his directorial debut. It stars William Hurt, Kathleen Turner, and Richard Crenna, and features Ted Danson, J. A. Preston, and Mickey Rourke. The film was inspired by Double Indemnity(1944).
Body Heat was the first noir film that I recall seeing, and to this day, I am exceptionally alert in a bar when drinking a dirty (‘filthy’) Martini!
It also can serve as a title to Planck’s derivation, that kicked-off the quantum revolution, where he managed to side-step the ultraviolet catastrophe, an embarrassment created by classical analysis of black-body radiation.
Fatal Attraction is a 1987 American psychological thriller film directed by Adrian Lyne from a screenplay written by James Dearden, based on his 1980 short film Diversion. The film became a huge box office success, grossing $320.1 million against a $14 million budget, becoming the highest-grossing film of 1987 worldwide. At the 60th Academy Awards, it received 6 nominations: Best Picture, Best Director, Best Actress (for Close), Best Supporting Actress (for Archer), Best Screenplay Based on Material from Another Medium, and Best Film Editing.
What better movie title for Black Holes? The simplest derivation, using classical physics, is to find the radius (about a point mass of M) at which the escape velocity becomes the speed of light, c. This is also called the Schwarzschild radius, of the event horizon, of a Black Hole (that has mass M, no charge, and is not spinning), derived properly using Einstein’s General Relativity (GR).
GR is not difficult to understand. It is just tedious to wade through the notation, especially when one ploughs through it the first time.
In the spirit of Śrīdhara Brāhmaṇa, let me attempt to demystify it.
Step1a: Gravitational pull (by earth) feels no different from Acceleration in space (in a rocket) to a person that is inside a box with no windows.
This is called “equivalence principle” and Einstein has said that this epiphany “was the happiest moment of his life.”
Step1b: The track of an accelerating body in a space-time diagram is a curve (rather than a straight line, which implies constant velocity).
Putting it together:
Gravity is curvature in space-time.
Now, to more technical stuff.
Step 2a: Curvature analysis in 4-dimensions with (locally Euclidian geometry) was developed by Riemann, who I have discussed previously in the context of Topology of Mutated Driver Pathways. This requires 4×4 matrices (‘Tensors’) in modern notation (but due to it being symmetric matrices, only 10 are independent values).
Step 2b: Einstein replaced “locally Euclidean metric” with “locally Lorentzian metric” to accommodate his Special Theory (that I have previously discussed as “the most consequential application of Pythagoras Theorem”😏).
Step 2c: Derivatives in flat surface need to be replaced by covariant (and contravariant) derivatives in curved space, because we need to account for second-order effects.
Step 2d: At low velocities and low mass, we need to match classical (Newtonian) gravitation.
This leads us to (a coupled system of partial differential equations with 10 independent variables):
The space-time curvature is proportional to the energy density.
In the dynamic case, following Wheeler:
Space tells mass how to move; mass tells space how to curve.
These set of equations imply that the universe cannot be stationary, but because gravity is attractive, would lead to continuous shrinking. Since, at that time, it was inconceivable that the universe can be anything but stationary, Einstein added a ‘cosmological” constant to counter this shrinking. Just a few years later, thanks to the work of Hubble and others, it was shown that the universe was not stationary. So, Einstein wanted to remove this constant, considering it “his biggest blunder.” That is:
He was for it before he was against it.
Putting it indelicately:
Einstein is the John Kerry of Physics.
In the final decade of the 20thCentury (as I have discussed in Science and Speculation), it was observed that our universe is actually accelerating in its expansion. So the “standard cosmological model” now contains the constant, and is said to represent the effect of Dark Energy, and so:
Einstein’s biggest blunder was actually saying that it should be removed!
At this point, you may also find yourself agreeing with Carlo Rovelli, Seven Brief Lessons on Physics:
A university student attending lectures on general relativity in the morning and others on quantum mechanics in the afternoon might be forgiven for concluding that his professors are fools or neglected to communicate with each other for at least a century. In the morning the world is curved space where everything is continuous; in the afternoon it is flat space where quanta of energy leap.
To this duo of contradictory theories, let us add another theory that has had its own history of incoherent development:
We now believe that heat is due to motion. Motion of what? In an electromagnetic field, this is like studying a gas of photons. But we know photons are quantum mechanical, and so have both particle properties (as in photo-electric effect where it is all-or nothing with respect to absorption by an electron) as well as wave properties (double-slit experiment).
Suppose we ask:
What happens when a photon is captured by a black hole?
The energy and entropy outside the Black Hole have decreased, so the energy and entropy of the black hole must increase.
If the energy of the black hole increases, its mass increases, and so curves space more, increasing its event horizon, and so its surface area.
Curiously, the form of the increase of the surface area of the event horizon is similar to that seen in classical thermodynamics when one looks at increase in entropy. So:
Entropy of Black Hole is proportional to the Surface Area of its Event Horizon.
But if a Black Hole has entropy, it must have temperature.
If it has temperature, and it is higher than its surrounding environment, then it should radiate.
If it radiates, then it loses energy. That is:
Black Holes evaporate.
But nothing can escape a Black Hole.
So, how can it radiate?
This brings us to another branch of physics, called quantum field theory (QFT), that I have also previous covered building on the nursery rhyme “One Little Monkey Jumping on the Bed”.😏
Vacuum is not empty. It is nervously active. Particle-antiparticle pairs spontaneously appear and disappear.
Now suppose such a pair appears just outside a Black Hole.
Suppose the anti-particle of this pair is sucked in. The particle flees.
This fleeing particle is the Black Hole radiation.
It did not come from inside a Black Hole, but from the outside, close to the event horizon.
No such radiation has been seen yet. Many reasons have been given, and I am reminded of a quote of Chuck Yeager:
At the moment of truth, there are either reasons or results.
Thanks for the post Sridhar! Little known fact: Carlo Rovelli started his faculty career at Pitt.
Didn’t know that!