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:
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:
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.
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?
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