Entropic gravity promises big but fails to deliver
One of the most bizarre theories of gravity to come out in recent years is that it is an entropic force.
One of the most bizarre theories of gravity to come out in recent years is that it is an entropic force.
Entropic forces don’t exist at the microscopic level. Instead, they are caused by entropy increasing.
The simplest example of an entropic force is a rubber band. The rubber polymers are long strings of atoms that curl up into a state of maximal entropy. When you stretch the rubber, those curled up molecules get elongated, which takes them to a state that has lower entropy. When you let go, entropy reasserts itself and the atoms curl up again.
One physicist, Erik Verlinde, believes that Newton’s theory of gravity works the same way.
This is important since, if gravity doesn’t exist as a force, then it may not need a real quantum description at the microscopic level. Since gravity is notoriously difficult to make into a quantum theory, this might be good news for string theorists like Verlinde.
Mathematician John Baez describes the physics behind it.
Entropic Forces
In 2009, Erik Verlinde argued that gravity is an entropic force. This created a big stir---and it helped him win about…johncarlosbaez.wordpress.com
Since we know that Newton’s gravity is a consequence of Einstein’s description of space and time and how it bends and curves in the presence of matter, entropic gravity says that spacetime itself is “emergent” from entangled states of quantum wavefunctions. In other words, what we experience as space, time, and gravity is all just a bunch of bits being added to and subtracted from different entanglement states.
John Wheeler wrote about “it from bit”, meaning everything including space and time comes from quantum information. As far back as 1963, Wheeler suggested some kind a “pre-geometry” that underlies space and time.
Verlinde won about $6.5M in research funding for his idea, which is a ridiculous amount for a theoretical physicist working on speculative ideas. He also attracted a lot of media attention, far more than those of his peers with more modest proposals.
Verlinde later turned his theory to explain dark matter. He derived the famous MOND, or Modified Newtonian Dynamics, from a modification of the gravitational force thanks to the competition between short range and long range entanglements.
More bits. Dark bits.
Given all the enthusiasm for this idea, I decided I needed to pour cold water and explain why Verlinde’s entropic gravity isn’t as exciting as it first appears.
Verlinde’s Entropic Gravity
Firstly, what does Verlinde’s Entropic Gravity say?
Verlinde makes a very strong assertion that the Newtonian gravitational force can be cast as purely an entropic force, meaning that the force is proportional to the temperature times the change in entropy over distance.
Hence, Verlinde’s gravity requires a well-defined concept of both entropy and temperature. The entropy and temperature imply the existence of a thermal heat bath.
You can think of a thermal heat bath as being some kind of source of potentially infinite energy at a fixed temperature or, conversely, of some isolated system at fixed energy. For example, a warm solution into which a grain of pollen is plunged so that its Brownian motion can be observed.
Other theories of entropic gravity propose that there are entropy and temperature-like quantities that behave as their analogues do in classical thermodynamics. These are uncontroversial. Verlinde says that is not so. Instead, a real temperature and entropy based on microscopic quantum entanglements in the vacuum create spacetime.
Verlinde’s theory also introduces “holographic screens” which are analogous to event horizons of black holes but exist everywhere in spacetime. These screens are thermal heat baths and separate emerged spacetime from non-emerged spacetime, macroscopic from microscopic.
Verlinde asserts that these screens lie on equipotential surfaces, surfaces where the Newtonian gravitational potential is all the same. This is similar to a topographic map where lines are drawn at fixed altitudes but in three dimensions.
All these features together: a real definition of entropy and temperature, thermal heat bath, and holographic screens on equipotential surfaces are unique to Verlinde’s formalism.
Verlinde’s formalism can best be compared to another entropic force: osmosis. When reading his papers, you get the sense that this is the analogy that best fits the concept of gravity as an entropic force.
Osmotic pressure is derived from the tendency for particles to move across a semi-permeable membrane from a region of low to one of high concentration. This is one way that water is filtered (via “reverse” osmosis which is just osmosis).
In Verlinde’s theory the semi-permeable membranes are the holographic screens and particles are drawn to cross them following an entropy gradient. This causes them to “want” to enter regions of high concentration of particles; hence, we get an attractive force.
Verlinde connects his theory to quantum information using a well-defined temperature called the Unruh temperature, a concept from the Unruh effect. This is not the first time the Unruh temperature has been used in a theory of entropic gravity. It was used by Jacobson back in 1995. But this is the first theory to try to use it to explain Newtonian forces across holographic screens.
The Unruh effect says when an observer accelerates in a vacuum that observer will observe a thermal heat bath of particles, while a stationary one will observe none. This is a consequence of quantum field theory and means that the vacuum is actually observer dependent.
To put it simply: acceleration generates a temperature from the vacuum.
Thus, the effect connects gravity, which relates to acceleration via Einstein’s equivalence principle, to quantum fields to thermodynamics. The Unruh effect has a well-defined temperature for that heat bath that fits into Verlinde’s derivation of his temperature as being a “real”, observable temperature of quantum fields.
Conservative and Entropic Forces
Gravity is a conservative force because it is the gradient of a potential. This means that the work gravity does on any object only depends on the initial and final positions of the object, not on the path it takes.
On Earth, the work gravity does on an object falling from a height H to ground 0 only depends on H. The object can fall down, sideways, zigzag. As far as gravity is concerned, it doesn’t matter.
In order for gravity to be an entropic force and conservative at the same time, the temperature has to be a function of the gravitational potential.
Unfortunately, Verlinde chooses to make the temperature depend on the acceleration or gradient of the potential. This is to connect it to the Unruh temperature and its associated acceleration. He needs this in order to explain why acceleration can arise from temperature and not just vice versa.
Requiring gravity to be both conservative and defining the temperature as the Unruh temperature gets Verlinde’s theory into trouble because now he requires a very specific form of the Newtonian potential where the gradient (force) has level sets that are the same as the potential’s. This is a very weird formulation that blows up when you add more particles.
In the two body scenario, in order to generate an attractive force with his holographic screens, Verlinde also needs a negative entropy. Negative entropies are not impossible in modern statistical mechanics although they don’t exist in classical thermodynamics, so we can give Verlinde that.
What is worse is that the entropies and temperatures for the two particles are different from one another, which means that they act as if they are different, isolated systems.
In the N-body case, it gets even worse because you need separate ones for each ordered pair of particles. That means that the number of isolated thermal systems blows up according to the square of the number of particles.
For me, this kills the appeal of Verlinde’s proposal since anyone can define gravity as entropic in any way they want if every pair of particles has its own temperature and entropy.
In Verlinde’s initial articles, he only looked at the “test particle” case, so these issues didn’t come up since there was only one particle. The N-body case continues to be a problem even in his latest work on dark matter alternatives.
This does not pour cold water on all proposals for gravity as an entropic force, not even all those that use the Unruh temperature, only those that define the Newtonian force in terms of it. Verlinde, however, is dependent on the effect.
Verlinde’s theory says the Unruh effect is not an acceleration causing a temperature but a temperature causing an acceleration, and that is a central feature.
Can Verlinde nevertheless dispense with the Unruh temperature dependency? Yes, absolutely. It is trivial to represent N-body gravity as an entropic force with a single temperature and entropy. Retaining the holographic screen proposal still requires negative entropy, but that is not as problematic.
Other researchers such as Jacobson and Padmanabhan have developed less ambitious theories of gravity. These attempt to reconstruct Einstein’s general relativity in a field theoretic framework, but they rely on virtual horizons and observers, not the real temperature and holographic screens that Verlinde’s theory needs.
Jacobson’s thermodynamic gravity for example represents the Einstein equations as thermodynamic equation of state and, like Verlinde, uses the Uhruh temperature. Unlike Verlinde, he does not address Newtonian force laws or particles at all and so N-body scenarios are irrelevant. Thus, the Uhruh can be connected to horizons in a much more coherent relativistic context in this theory.
Twelve years on, Verlinde’s theories still provoke but have failed to provide a coherent theory of gravity.
Friction and Fluctuation in Entropic Gravity
A number of papers going back to the 1960s object to entropic gravity proposals on principle.
Richard Feynman in 1964 criticized an entropic gravity theory in a lecture. He suggested that the same microscopic fluctuations that create spacetime in such a theory would also create drag on objects moving through space. Albert Einstein pointed this out in his 1905 paper on Brownian motion. Essentially, anything subject to thermal energy exchanges with a heat bath also experiences drag when passing through that heat bath because of friction. This would mean that the planets would experience friction with spacetime and slow down.
Classical gravity does not behave this way because it is a conservative force. Instead, any path under gravity is reversible, meaning that you can construct paths where entropy is constant. This seems to suggest that you can’t have friction since that wouldn’t allow for reversible paths. The momentum lost to spacetime friction would be unrecoverable.
Fundamentally, friction is the tendency for energy to dissipate into all degrees of freedom. If you have a planet moving through space, a huge amount of the energy is in the motion of the planet in a single direction, a single degree of freedom. Friction would imply a tendency for that energy to dissipate into the other degrees of freedom, robbing the planet of its momentum.
If spacetime is emergent and gravity an entropic force, a body made of many, many particles traveling through spacetime would be colliding with microscopic fluctuations all the time and losing energy to them. Hence, the energy of all those particles would be expected to gradually dissipate into the microscopic realm in the same way that an engine runs down because its energy dissipates into friction and becomes heat.
It turns out that when you connect gravity to quantum field theory this actually happens from Hawking radiation, the production of quantum particles from spacetime itself. Most people are familiar with it as a phenomenon related to black holes, but all curvature in spacetime produces Hawking radiation. Hence, Hawking radiation can be thought of as dissipation from spacetime and likewise an object traveling through spacetime will produce Hawking radiation which will cause it to lose mass. From a particular observer’s vantage point, that means losing momentum too.
Thus, Feynman’s objection is true, but Hawking radiation is so weak that it has no appreciable effect on planets, stars, or anything that isn’t a smallish black hole over even cosmological time scales. The planets wouldn’t last long enough for us to observe them losing momentum.
Visser, Matt. “Conservative entropic forces.” Journal of High Energy Physics 2011.10 (2011): 1–22.
Padmanabhan, T. “Equipartition of energy in the horizon degrees of freedom and the emergence of gravity.” Modern Physics Letters A 25.14 (2010): 1129–1136.
Jacobson, Ted. “Thermodynamics of spacetime: the Einstein equation of state.” Physical Review Letters 75.7 (1995): 1260.