Erik Verlinde's entropic gravity and optical gravity

Erik Verlinde has suggested that gravity is not a "fundamental force", but guided by entropy.

Erik P. Verlinde
On the Origin of Gravity and the Laws of Newton
https://arxiv.org/abs/1001.0785

Recall the optical coordinates that we introduced in our blog post:

http://meta-phys-thoughts.blogspot.fi/2018/04/the-horizon-is-perfect-mirror-also-in.html

Let us look at the optical coordinates of the Schwarzschild black hole geometry. The circumference around a black hole, as measured in wavelengths λ, grows exponentially when we approach the horizon in λ-steps. This is much faster than how the circumference grows far away, if we take steps to the opposite direction.

In a sense, from the point of view of an electromagnetic wave, the space close to the horizon is much bigger than the R^3 Euclidean space far away.

We could try to explain gravitation as an entropic force. A wave will tend to spread more to the direction where the space looks bigger to it, in some sense, there are more degrees of freedom available for it in that direction.

A counter-argument to the entropic interpretation of gravitation is the elliptic orbit of a planet: if entropy takes the planet closer to sun, then why does the planet then move farther away in a half of its orbit?

In the free particle solution of the nonrelativistic Schrödinger equation, the wavenumber k for a massive particle is proportional to its momentum p. If we have a potential well, the de Broglie wavelength for the particle grows shorter when the particle falls into the well, because the momentum of the particle grows there.

It seems to be a general rule that waves in various settings tend to steer into zones where the wavelength becomes shorter, or the speed of waves is slower. We say that those zones have a lower potential, because they attract waves. We interpret this behavior as due to an attractive force.

When a wave enters a zone where its wavelength grows shorter, the form of the wave tends to become more complex. Can we say that the "entropy" of the wave has increased? Can we measure the entropy of a wave by its Fourier decomposition?

Gravitation as a force seems to be no special with waves, except that gravitation affects all known mass-energy. If gravitation is an "entropic" force, then all forces are "entropic" forces?

In Wikipedia, the formula for the entropic force is derived from the "degrees of freedom" in the border surface (holographic screen) of a zone in space. If mass-energy is contained in the zone, then we can calculate the "temperature" of that mass-energy of the screen.

The gravitational acceleration is set such that the temperature of the screen matches the Unruh temperature of the acceleration. The author of this blog does not know an answer to the following:

Open problem 1. Why the Unruh temperature of the gravitational acceleration should match the temperature of the screen. What is the physical intuition behind that?

Waves tend to steer into the direction of a shorter wavelength

Our optical view of gravity might offer an intuitive explanation why the correct acceleration for gravitation can be obtained from seemingly entropic concepts.

If we have a static observer in a gravitational field, then in the optical coordinates, the space appears stretched on the side of the the lower gravitational potential, that is, the wavelengh of light that we use as the measuring stick is shorter there.

For a static observer, there are more "degrees of freedom" when we go downhill in the potential, because the space appears stretched there and thus bigger. Waves tend to turn to the direction where there are more degrees of freedom available. Here we consider both light-speed waves and also the nonrelativistic wave function of a massive particle.

The static observer sees waves accelerating downhill in the potential. On the other hand, a freely falling observer sees the geometry of the space more flat. The degrees of freedom appear to be symmetric on each side of the freely falling observer. Therefore, he will not observe acceleration of waves.

In the above way, the degrees of freedom in a zone in space is connected to the acceleration of a wave downhill in the potential. But is it a fruitful point of view to claim that the degrees of freedom is the primary thing and the force, or acceleration, is secondary?

Also, rather than referring to degrees of freedom, it is much simpler to say that waves tend to steer to the direction of a shorter wavelength. From the point of the view of waves, the geometry of the space is described with a measuring stick of one wavelength. In that geometry, the waves or rays tend to move in a straight line. But from a static observer's view, that straight line steers downhill in the gravitational potential.

Unruh temperature of reflected waves in a gravitational field?

Open problem 1 above raises the question of how we can associate a temperature to acceleration, or to a gravitational field.

How can we associate a temperature to a specific value of acceleration? The Unruh temperature is one such way. Suppose that a static observer H is in a gravitational field and the gravitational acceleration is g at his location. A static source far away sends light waves toward H.

In the vicinity of H, the form of light waves of a very short wavelength is not much distorted by the gravitational field. Short length waves do experience blueshift when we move downwards, but since that happens under a great number of wavelengths, an individual wave can keep its form quite well.

If we have a very long wavelength wave, its electric field E will appear almost constant in the vicinity of H. We could say that the also a long wave keeps its form well in the vicinity of H, even though the waveform is very much distorted at longer distances. As an extreme case, we may have a static electric field E in the gravitational field. The electric field will have some constant value at each location. There is no reflection, as there is no wave oscillation.

If we have a medium wavelength wave, then its form is considerably distorted in the vicinity of H. We can associate a temperature with the "most distorted" wavelength. Is this the Unruh temperature for acceleration g?

If we have a rocket with acceleration g in flat Minkowski space, then the Unruh temperature tells us the wavelength of the light for which the acceleration, in some sense, distorts the waveform most.

Recall that the amount of "negative frequencies" in the Fourier decomposition of the wave determines how much Unruh radiation we are supposed to observe for a specific acceleration g.

We showed in previous blog posts that Unruh radiation most probably does not exist. What is the interpretation of the Unruh temperature then?

A static gravitational field certainly scatters or reflects back some of an incoming planar wave, because the waveform gets distorted and destructive interference cannot possibly cancel all reflected waves that are dictated by the Huygens principle.

Waves sent by a freely falling source in a gravitational field

Above we assumed that the source of waves is static and far away from the gravitating mass. Waves sent by the source appear as fixed frequency waves for each static observer in the field. For observers further down, the frequency appears higher. The observers observe a blueshift of the waves.

In the Unruh setup, we have an accelerating rocket and we study waves that are sent by an inertial source in the Minkowski space.

To mimic the Unruh setup under a gravitational field, the source of the waves actually should be freely falling in the field, and the rocket is replaced by a static observer. This is equivalent to having a freely falling laboratory where we have a source of waves, and an observer who is static relative to the gravitating mass. Inside the laboratory, the observer appears to accelerate upwards.

The source of waves in this case is like an accelerating ambulance coming downhill the gravitational potential. Its siren sounds as a chirp in the ears of the static observer.

How do such waves behave in the gravitational field?

Waves sent by an accelerating source in a Minkowski space

If we have a static observer in a gravitational field and he receives planar waves from a static source far away, that is analogous to having an accelerating rocket send planar waves to another accelerating rocket behind it. The observer sits in the second rocket.

Static observers in the gravitational field will see some of the waves reflect back. Similarly, observers in the accelerating rockets will see some waves starting to propagate to the direction of the acceleration.

Waves in a time-varying gravitational field

Open problem 3. Assume that we build a wave packet from purely positive frequencies and let it pass through a time-varying gravitational field. The waveform, as calculated by the canonical transformation (see http://meta-phys-thoughts.blogspot.fi/2018/04/does-unruh-radiation-exist.html, Definition 6) may afterwards contain negative frequencies. What is the role of these negative frquencies?

Can a time-varying gravitational field produce photons?

Our critique of Hawking radiation contains a conjecture that a time-varying gravitational field cannot produce photons if there is no ordinary matter present. Ordinary matter is coupled to the electromagnetic field, and can produce photons when the gravitational field pushes its particles around.

If a time-varying gravitational field is a gravitational wave, or it is produced by hypothetical dark matter, then photons cannot be produced, according to our conjecture.

This is at odds with the fact that a time-varying electromagnetic fields, or photons, create a time-varying gravitational field. If there is a coupling to that direction, why not to the other direction?

If we have a small random quantum fluctuation in the electromagnetic field, can a time-varying gravitational field make it to grow to real photons? Can the gravitational field give enough energy?

Let us study an analogous situation. The Dirac electron-positron field is coupled to the electromagnetic field. A time-varying Dirac field will always create a time-varying electromagnetic field.

In the Schwinger process, an electric field will create electron-positron pairs. That is, even a static electric field creates a permanent disturbance of the Dirac field, if the potential difference in the electric field is at least 1.022 MV.

In the coupling of the Dirac field and the electromagnetic field, there is an asymmetry: any small disturbance in the Dirac field always produces photons, but only a relatively large 1.022 MV potential difference in the electric field creates a permanent disturbance to the Dirac field. The quantum of the Dirac field has an energy 511 keV, while a photon can have any energy.

We have a conjecture that electromagnetic waves really are waves in the polarization of virtual electron-positron pairs. There is really just one field, the electron-positron field, whose temporary polarization manifests as photons, and permanent polarization as electrons and positrons. A big difference in the electric potential can turn a virtual electron-positron pair into a real one.

What about the gravitational field?

Conjecture 1. Gravitational waves really are polarization of virtual pairs, one having a positive mass-energy and the other having the opposite negative mass-energy.


Can a strong static gravitational field create pairs of photons?

Suppose that a quantum fluctuation creates a transient pair of wavelength b virtual photons. One photon hass mass-energy E and the other -E. The E photon travels down the gravitational potential. Let its wavelength get shorter by some ratio R. It gains energy from gravitational pull, so that it has energy R * E. If the photon did not have enough energy to be real at the start, it cannot gain enough energy to become real as it travels down.

The -E photon will travel up (?). It gains positive energy from the gravitational push. Its mass-energy will approach zero but can never become zero. Thus, creation of photons by this process seems to be prohibited by quantum field theory.

Can a static gravitational field create electron-positron pairs?

Suppose we have a virtual electron e- with mass-energy E and a positron e+ with mass-energy -E.

The electron will travel downward and gain momentum and mass-energy from the gravitational pull. If it gains the 511 keV, do we have then a real electron?

The positron has the same problem as the photon: its negative mass-energy will diminish, but can never reach zero. It cannot become a real particle.

Looks like a static gravitational field cannot create pairs of particles. The difference to the Schwinger process is that the mass-energy of the negative energy particle can never become positive.

The black hole horizon is a perfect mirror also in classical general relativity?

Our hypothesis of optical gravity suggests that the forming horizon of a black hole will act as a perfect mirror for all incoming waves in any field.

The paper of Jahed Abedi, Hannah Dykaar, and Niayesh Afshordi, in turn, suggests that gravitational waves do indeed reflect back from the horizon:

Echoes from the Abyss: Tentative evidence for Planck-scale structure at black hole horizons

https://arxiv.org/abs/1612.00266

But is it really so that in classical general relativity, arbitrary waves should travel down the geometry of a black hole without reflecting back?

Let us study the Schwarzschild solution of the geometry.

Close to the horizon, the radial speed of light as measured in the global coordinates of the Schwarzschild solution goes as ~ d where d is the Schwarzschild global distance from the horizon and the tangential speed goes as ~ √d.

Definition 1. Let us have a ray of light whose wavelength far away in the Minkowski space is λ. Let the ray of light enter a static gravitational field. We can use the wavelength of the light as a measuring stick for distances. By optical coordinates we mean coordinates where spatial distances are measured in wavelengths λ, that is, we measure spatial distances by the optical path length of optics.


https://en.wikipedia.org/wiki/Optical_path_length

We can choose the wavelength λ of a radially ingoing wave such that, close to the horizon, each one wavelength step even closer will make the radial speed of light to halve, when measured in the global Schwarzschild coordinates. We call such a step a λ-step.

The λ-step will make the tangential speed of light to go to 1/√2 of the previous step. The wavelength λ far away from the black hole is of the order of the Schwarzschild radius.

Let us try to visualize the geometry of the Schwarzschild solution. We measure the radial and tangential distances in terms of the local length of the wavelength λ. By local we mean what is the wavelength of the incoming wave at that location, measured by a static observer. For tangential distances, we imagine that a mirror at a 45 degree angle is used to turn the radial ray of light into tangential.

Let z = 0 in the Schwarzschild solution. We try to visualize the x,y-plane. If we try to keep λ constant in the visualization, we get a kind of surface.

Our surface looks like a funnel with a wide upper end, the flat Minkowski space, and an infinitely deep exponentially widening pipe that descends to the horizon:

____            ____

        \        /

          \    /

           |   |

          /    \
       /          \

     .               .
 .                      .

The pipe is infinitely deep before we reach the horizon. That is, the ingoing wave will have an infinite number of cycles λ before it reaches the horizon.

The width of the pipe grows exponentially when we move downward because the tangential λ decreases exponentially. We cannot really embed the pipe into a 3-dimensional Euclidean space because the circumference grows exponentially at each step λ closer to the horizon. The figure above is only to help imagination, not to be taken literally.

The funnel neck is at its narrowest at 3/2 Schwarzschild radii in the global Schwarzschild coordinates.

https://en.m.wikipedia.org/wiki/Fermat%27s_principle

Fermat's principle states that a ray of light can travel a path which is a stationary point of the optical length of the path, with respect to small variations in the path. A ray of light can have a circular orbit around the black hole at that radius.

Abedi et al. assume in their paper that part of outgoing waves are reflected back roughly at the narrowest neck. Abedi et al. consider the LIGO black holes which are rapidly spinning and have the Kerr solution while we have been looking at non-spinning Schwarzschild black holes.

Open problem 2. Where does the backreflection of outgoing waves actually occur? The geometry is not flat anywhere close to the horizon. The narrow neck in the figure is not special in that respect. In optical gravity, the optical density grows smoothly as we approach the horizon. Why would the narrow neck somehow produce more backreflection than other points around it? What does optics tell us? How does the spinning affect the backreflection?


Open problem 2 can probably be solved with a numerical simulation.

As an aside, note that if we measure the depth of the pipe with a ruler (= proper length) then the pipe is not infinitely deep. If an observer is at distance d from the horizon in global Schwarzschild coordinates, then the proper distance from the horizon is 2 * √(d * r_s), where r_s is the Schwarzschild radius. An ingoing wave will be blueshifted close to the horizon. That is why the pipe is infinitely deep if we measure in wavelengths λ.

If we have an incoming planar wave, how much of its energy can travel down the pipe? The Huygens principle tells us that we can calculate a new place for a wavefront by assuming that each point in the old wavefronts is a new source of oscillation.

Close to the horizon, the space is very much curved. If we try to draw a square where each side is λ and the upper line is horizontal, then the sides differ very much from the radial direction. The lower line would be √2 λ if we put the sides radially.

          λ
   _________
   \             /    A "square" close to the horizon
 λ  \_____/  λ

          λ

According to the Huygens principle, there will always be waves reflected back if there is not a total destructive interference of the reflected waves. If a planar wave proceeds in the flat Minkowski space, the reflected waves are completely canceled out by destructive interference.

Intuitively, it is likely that the destructive interference cannot cancel out reflected waves in the very much curved geometry close to the horizon. A numerical simulation may confirm this.

The reflection would be very strong for waves of length λ, that is, waves whose length far away from the black hole is of the order of the Schwarzschild radius. The reflection might be up to 10 % (?).

For shorter waves the reflection is less, but since the distance to the horizon is infinite in terms of λ, even a small ratio of reflection will, after an infinite number of iterations, reflect everything back.

There is some uncertainty, how exactly we should simulate waves in a curved spacetime geometry. Our optical gravity hypothesis is one possibility.

Could it be that all energy of the downgoing waves will eventually be reflected back?

Open problem 3. If planar waves hit a black hole, will all wave energy be reflected back before the downgoing waves reach the horizon?


If the wavelength is of the order of the Schwarzschild radius, then Problem 3 can be solved with numerical methods. An analytic solution is unlikely.

For short waves, the Bogoliubov transformation might offer a way to estimate the reflection. We can use geometric optics to trace the wave a certain distance closer to the horizon. Then do the Fourier decomposition of the wave in a freely falling reference frame. The wave will appear as a chirp in such a frame. Its decomposition contains negative frequencies. The negative frequency waves can be interpreted as reflected waves that are traveling upwards.

Open problem 4. Is there a depth where a sizeable portion of downgoing waves has been reflected back? Is the Planck length related to this depth in some way?


Open problem 5. If we have a static observer very close to the horizon, then any falling particle will have an almost infinite mass-energy as observed by the observer. How do these almost infinite masses affect the local geometry?

Critique of resonant cavity of Abedi et al.

Abedi et al. calculated that the echo repeat time should be of the order 0.1 seconds in the LIGO data. That is, if we assume reflection points at about 1.5 Schwarzschild radii and at 1 Planck length proper distance from the horizon.

In our model, the backreflection of waves may happen smoothly and not be concentrated at the narrow neck at 1.5 Schwarzschild radii.

Abedi et al calculated that in the LIGO data, the Schwarzschild distance from the horizon is of the order 10^-74 meters when the proper distance from the horizon is the Planck length 1.6 * 10^-35 meters. They expect the reflection to happen around that position.

Gravitational waves that the LIGO can observe have a long wavelength, that is, of the order of our λ above. Each one wavelength λ step towards the horizon halves the Schwarzschild distance from the horizon. If we start at a Schwarzschild distance of, say 10 kilometers from the horizon, it will require roughly 250  λ-steps to get to 10^-74 meters from the horizon in Schwarzschild coordinates.

If we assume a 10 % reflection at each λ-step, then the horizon has reflected most of the wave back already at 10  λ-steps, not 250, as Abedi et al. assume.

The author of this blog will next study the Kerr solution and the backreflection at 1.5 radii.

How do black holes orbit and merge in optical gravity?

https://en.wikipedia.org/wiki/No-hair_theorem

The name of the black hole no-hair theorem is misleading. Actually, it is a conjecture. It claims that after a black hole is formed, it can be described, for all practical purposes, with the mass, the angular momentum, and electric charge: that is, with just three real numbers.

Optical gravity, on the other hand, implies that the inside of the horizon is essentially frozen, but the Newtonian gravitational pull of the matter inside horizon can still be felt by an outside observer. Thus, the entropy of a forming black hole is large in optical gravity. We cannot describe a black hole with just three real numbers.

A problem in the optical gravity hypothesis is how do we model the merger of a binary black hole? LIGO has provided us with some empirical data of mergers.

In an optical black hole, the local speed of light inside the forming event horizon is zero. If we bring two black holes together, how do their forms adjust in the merger?

To explain the orbiting of two black holes, we have to add the following hypothesis to optical gravity:

Hypothesis 1. The local speed of light for signals between two spacetime points inside or close to the forming black hole horizon is essentially zero, but the forming black hole can move collectively and rotate collectively.


Hypothesis 1 is required by Lorentz invariance. If a horizon would stop and stick at "one position" in space, a black hole would have an infinite inertial mass.

Open problem 2. In optical gravity, what type of global transformations are allowed to deform the mass inside or near a forming horizon? Inside or very close to the horizon, the effective Newton or Coulomb force between masses that are close together, is essentially zero. All strictly local change has stopped from the point of view of a global observer. But there could be global deformations to the wave function of the system that still could happen and shape the insides of a forming horizon.


Open problem 2 exists also in traditional general relativity. We cannot map the proper time of an observer inside of the forming horizon to the proper time of an outside observer, because no signal can reach the outside.

If we have a cigar-shaped star and make a black hole out of it by collapsing a dust shell on it, does the cigar-shaped gravitational field persist or is it deformed to be spherical? If it becomes spherical, then an outside observer would see the mass inside the horizon to "move", but how can that happen if no signal can come out from inside the horizon?

Echoes of gravitational waves are evidence for optical gravity?

http://backreaction.blogspot.fi/2018/04/a-black-hole-merger-merger-merger.html

Sabine Hossenfelder's blog post mentions an interesting result from LIGO:

https://arxiv.org/abs/1612.00266

Gravitational waves seem to bounce off from the horizon of a black hole.

Our blog post on April 20, 2018 introduced the optical gravity hypothesis:

http://meta-phys-thoughts.blogspot.fi/2018/04/optical-theory-of-gravity.html

The hypothesis implies that the forming horizon is a perfect mirror for all fields - it has an infinite optical density for all fields. That could explain why gravitational waves bounce back.

But is the reflection of gravitational waves from the horizon actually predicted by standard general relativity?

UPDATE: Yes, it is! See the post:

http://meta-phys-thoughts.blogspot.fi/2018/04/the-horizon-is-perfect-mirror-also-in.html?m=1

Actually, optical gravity is not a new theory of gravitation - it is a new interpretation of classical general relativity. Therefore, all consequences of optical gravity can also be derived in classical general relativity.

Optical gravity does clarify some vague aspects of general relativity: optical gravity claims that the information in the Newton force propagates at the global speed of light while the Einstein equation leaves the speed vague. Also, optical gravity describes the structure of a forming black hole, while there are various views of what Einstein equation says.

Optical gravity may be amenable to quantization while the Einstein equation has presented insurmountable problems.

Pauli exclusion principle comes from the repulsion between electrons?

According to the spin statistics theorem, particles with a half-integer spin are fermions, that is, they obey the Pauli exclusion principle.

But that does not explain why electrons are fermions in the first place.

An electron has a relatively strong magnetic moment. If we have two electrons whose spins are in opposite directions, the electric repulsion between them is partly canceled.

If the distance is just 10^-15 m, then the magnetic attraction between two electrons is still just 10^-20 of the electric repulsion.

The magnetic attraction may be the underlying reason why every quantum state in a stationary atom can accommodate both an electron with spin +1/2 and spin -1/2. The magnetic attraction only works well in a system of two electrons, not three. That is why a state can hold exactly 2 electrons.

The force of magnetic attraction is too weak. Why can two electrons then fit on each quantum state?

But why the Pauli exclusion principle? Consider the particles in a box model of quantum mechanics. If 3 electrons would be in the ground state, then we would have an electron cloud that is denser in the middle in the box, and furthermore, the magnetic attraction does not help in keeping all 3 electrons close to each other.

To achieve a smoother distribution of the electron cloud in the box, one has to populate energy levels above the ground state.

Todo: prove that the minimum energy is obtained with 2 electrons on each level.

If we would have an atomic nucleus that is orbited by negatively charged bosons (hypothetical boson electrons), then all those bosons would fall into the lowest possible orbit? Classically, if we have a positive charge in the middle and compensating negative charges, those negative charges come as close to the positive charge as they can get.

Conclusion: we cannot explain the Pauli exclusion principle in an atom by the electric repulsion alone.

The Bell inequality does not require faster-than-light communication

Suppose that we have a source that produces pairs of correlated photons, such that the polarization of the photons is opposite to each other.


Polarization                            Polarization
filter   <-----photon 1   photon 2----> filter

We measure the polarization of each photon relative to a polarizarion filter. The output is either +1 or -1. The filters can be turned so that they are at an angle relative to each other.

https://en.m.wikipedia.org/wiki/Bell%27s_theorem

The famous theorem by John Bell states that the distribution of measurement results versus the angle cannot be reproduced if we assume that each photon already had its polarization determined before the measurements.

Some researcher claim that this shows there must be some kind of faster-than-light communication between the measuring devices.

Their claim is based on the following:

Assumption 1. The measuring devices make the wave function of the pair of photons to "collapse" at each end of the measurement apparatus. The measurement result can then be compressed to a single binary number +1 or -1 and sent to the scientist to tabulate and analyze.


Assumption 1 already highlights what is the error in this thinking: we make the wave function of the apparatus to collapse before the data is in the head of the observer, the scientist.

If we cut off part of the wave function of a system before the observer interacts with the system, then, of course, that will lead to strange behavior, like the need for faster-than-light communication or loss of unitarity.

The correct way to treat this experiment is to let the wave function propagate to the measurement devices and then to the head of the scientist. The interference pattern, that is, the table of measurement results, is formed in his head. There is no need for faster-than-light communication.

It is like a complex double-slit experiment where the screen is the head of the scientist. It is the screen where we finally let the wave function to collapse, if we use the Copenhagen interpretation.

If we use the Bohm model, then the hidden markers of the photons will sail on the wave function all the way to the head of the scientist where the markers finally "hit the shore" and determine in which of the many alternative worlds the observing subject will live in after the experiment. In the Bohm thinking, the wave function does not collapse at all. All the alternative worlds continue to exist. The markers just choose in which world our observing subject will be afterwards.

Open problem 2. The observing scientist receives the measurement outcomes from the different ends A and B of the apparatus at different times. For the interference pattern to form, the wave function must not collapse before the scientist has the outcome from both A and B in his head. How do we formalize this? Can the Bohm model help?


https://en.m.wikipedia.org/wiki/Entropy_of_entanglement

Entanglement is traditionally seen as a purely quantum phenomenon, but we can also interpret it classically. Entanglement just means that a classical wave was born in a single place in spacetime in a single process. When the classical wave propagates, there is correlation between its values in different points of spacetime. This correlation is called entanglement.

Entanglement entropy in quantum mechanics has the following meaning: if we have a combined system C & D in a pure state, and we measure C as accurately as we can, what is the von Neumann entropy of the mixed state in D after that?

Open problem 3. Can we really measure half of the system first and make its wave function to collapse, and later compare the result to the measured values in the rest of the system? That would require faster-than-light communication in the Bell inequality experiment. Does entanglement entropy have any physical meaning?