Entropy Scale Factor. This paper proposes a new theory, the…
By: Chris Watson
Tags: Physics, Dark Matter, Black Holes, Astronomy

Possible Connection Between Dark Matter, Dark Energy and Solar Corona Heating

Dark matter, dark energy and solar corona heating are all long-standing problems in astrophysics. They involve physical quantities, including galaxy rotation speeds, the metric expansion of space, and the heat of the solar corona, respectively, that are more than we would expect based of light-based measurements of distance. The common thread of distance raises the possibility that there is an error in how we calculate distance with light. This paper proposes a new theory, the entropy scale factor, that may connect all of the above phenomena. The entropy scale factor describes how the scale of space and time could change depending on how much entropy is enclosed within a boundary. This balanced change in the scale of space and time creates a mismatch between light-based measurements of distance, and other measurements, like field strength. The scale change results in length contraction, which would cause fields to extend further than predicted by current models, possibly explaining the phenomena listed above. This article explains how black holes point towards scale changing with distance. It also argues that the entropy scale factor has not been ruled out by current tests of general relativity, and it suggests directions for future research.

Photo by Alexander Andrews on Unsplash

Introduction

Dark matter was first proposed over 85 years ago, and has been used to explain galaxy rotation dynamics, galaxy cluster dynamics and gravitational lensing around galaxies, among other phenomena. Despite numerous attempts, dark matter has never been directly detected.

Dark energy refers to an unknown kind of energy that causes the metric expansion of space to increase, when general relativity predicts that it should shrink.

The solar corona heating problem centers on explaining how the surface of the sun can be ~5800 K, while the outer atmosphere, the corona, is often over 2 million K. As the sun is the source of the corona’s heat, it is unusual that heat flows from the cooler sun to the much hotter corona, and current theories for corona heating have not been able to explain the magnitude of the difference.

What all these problems have in common is that they use distances from measurements taken using light, and therefore they are susceptible to errors if there is a gap in our understanding of light, and the things that affect it, like spacetime.

In 1974 Stephen Hawking used a semiclassical theory to propose that black holes emit particles. In this theory he combined some properties of quantum mechanics, like quantum fluctuations, with some properties of general relativity, like the black hole horizon. In his theory, Hawking treated the black hole horizon like an unmarked line in space. If this were true, one virtual particle of a pair could cross, leaving the other to be emitted.

It would also have been reasonable to treat black holes as quanta of spacetime. From outside a black hole, there is no way to measure position or time within a black hole, so in this sense they are already indivisible units of spacetime.

If Hawking had treated each black hole as a quantum of spacetime, his theory would have had different results. It would no longer be possible for one virtual particle of a pair to be absorbed, and black holes would not emit radiation. If black holes are quanta of spacetime, they are quanta that change dimensions depending on how much entropy is enclosed within their horizons. If this change in scale is also a property of space outside black holes, then it may explain the phenomena that have been attributed to dark matter, as well as the other astronomical mysteries listed above.

This paper proposes a new theory, the entropy scale factor (ESF), in which there are balanced changes to the scale of space and time depending on how much entropy is enclosed within a volume. It discusses the entropy scale factor in examples involving black holes and the dark matter phenomena, and proposes areas for future study.

Entropy Scale Factor

The entropy scale factor proposes that the scale of space and time change with the amount of entropy enclosed within a boundary. These changes are in addition to the changes in scale, including length contraction and time dilation, predicted by general relativity. In the ESF, as the entropy enclosed within a boundary increases, the scale of space and time within that boundary also increases. This paper will use the Planck length and the Planck time as the standard unit of comparison, due to the importance of the Planck length in black hole thermodynamics.

The ESF is defined by the following equation, which is the Planck length over the Planck length in empty space, or the Planck time over the Planck time in empty space (Eq. 1).

The Bekenstein bound gives the maximum amount of entropy that can be enclosed within a volume. A Schwarzschild black hole with the same radius as the Bekenstein bound exactly saturates the bound, so no more entropy can be added. Because of this relationship, the equations that describe a Schwarzschild black hole can also be used to describe the change in scale that would occur due to the ESF for such a black hole. In this case, the Planck length in the circumference around the black hole is unchanged in scale, compared to the Planck length in empty space, but in the direction perpendicular to the black hole the Planck length has increased to the Schwarzschild radius, and Planck time has increased to the time it takes light to travel that distance. This assumption is made because outside a black hole particles cannot be located more precisely than the Planck length, and inside a Schwarzschild black hole a particle cannot be localized more precisely than the Schwarzschild radius.

Outside a black hole in empty space, the Planck length in the direction of the black hole will decrease with increasing distance from the black hole, as each successively larger sphere has more volume but encloses the same amount of entropy. This will cause length contraction, compared to what is predicted by general relativity, because there will be fewer units of length than expected. This length contraction will be strongest near the black hole, and decrease with distance from the black hole.

If such a change were to occur without a corresponding change in time, it would create an obvious redshift in the spectroscopy from entropic objects, like stars, similar to the way that the metric expansion of space causes a redshifting of light from the early universe. Since such a change has not been reported, a change in the scale of time based on entropy is likely associated with a proportional increase in the scale of time. For time, the entropy scale factor would cause time to run more quickly near entropic objects than predicted by general relativity. This “time contraction” would occur because the Planck time would increase near the black hole, causing there to be fewer units of time than expected. This would result in a blueshifting of light, which would cancel out the reshifting from time contraction.

With these balanced changes to space and time, light coming from near the surface of the black hole will appear similar to what is predicted by general relativity. This would cause investigators who do not correct for the entropy scale factor to make errors in light-based measurements, because they would calculate objects to be further away than they really are (i.e. they are ignoring the length contraction that comes from the scale factor).

Entropy Scale Factor and Translational Symmetry

Another line of reasoning pointing to the ESF is translational symmetry. If black holes are treated as a quantum of space with the Schwarzschild radius equal to a Planck length outside the black hole, then objects could not be translated through the black hole without upsetting their properties [Fig. 1]. The entropy scale factor would fix this asymmetry because the Planck length of all regions of space would depend on the relationship between information and an observer, so the black hole would no longer be treated differently.

Fig. 1 Without the entropy scale factor, if the points on line A are translated through the black hole then the line is changed, breaking translational symmetry.

Entropy Scale Factor and Tests of Gravity

To be a useful theory of nature, the entropy scale factor will have to agree with current tests of general relativity. The ESF does recreate general relativity exactly at the low entropy limit, and many tests of general relativity have been done in low entropy systems. To the author’s knowledge, tests in high entropy systems have not reported on if their data rules out a balanced change to the scale of space and time, as would be expected with this theory.

Another major hurdle is explaining how the ESF could be strong enough to affect the rotation of galaxies, but weak enough that it has not been detected in the movement of our solar system’s planets. There are two reasons why the ESF would be more apparent on an interstellar scale. The first reason is the distance. If this effect has a quick fall off, followed by a long “tail,” then the long distance of interstellar or intergalactic space would make its signal easier to detect. Why might the tail be so long? The space between our galaxy and a neighboring one is filled with light, neutrinos and other particles from each galaxy, so this space will always have its own entropy, in addition to the entropy of the galaxy it contains.

The second reason that the ESF may be more apparent in galactic dynamics compared to solar system dynamics is that stars have much higher entropy than planets. This means that on a galactic scale the ESF would be amplified by all the stars in a given direction, in a way that doesn’t happen with the planets in our solar system [Fig. 2].

Fig. 2 All of the dark circles would affect the space to the right of the white circle, amplifying the effects on space in that region.

Dark Matter and the Entropy Scale Factor

If the entropy scale factor is correct, it may be helpful in explaining the collection of phenomena that have been attributed to dark matter. These phenomena including galaxy rotation rates and gravitational lensing around galaxies.

Galaxy rotation curves show that the outer reaches of galaxies rotate faster than would be expected with general relativity. The entropy scale factor could explain this phenomenon because the length contraction predicted by this theory would mean the stars are not as far apart as currently thought (Fig. 3). This would allow the gravitational fields to extend further than predicted, and may explain galaxy rotation rates.

Fig. 3 Due to the relatively high entropy in the center of the galaxy, the ESF would cause a standard unit of length to be longer in the direction towards the galaxy center (red line), and have less effect around the galaxy circumference (green line). This will result in length contraction in the direction towards the center of the galaxy, because there will be fewer standard units than expected.

Another phenomenon that has been attributed to dark matter is gravitational lensing around galaxies, which is stronger than expected. The strength of gravitational lensing may instead be explained by the ESF, because both the length contraction and time contraction predicted by this theory would increase gravitational lensing around galaxies. These changes in scale would happen because galaxies have high entropy compared to the intragalactic space that surrounds them. The Bullet Cluster is a famous example of gravitational lensing around galaxies. In this system a collision of two galaxy clusters left most of the matter as intragalactic gas in the space between the clusters, and yet gravitational lensing is stronger in the galaxies than for the more massive intragalactic gas. The ESF could explain this phenomenon because the galaxies have higher entropy than the diffuse intergalactic gas, and therefore the effects of the scale factor would be more pronounced in the galaxies.

Solar Corona Heating and Relativistic Jets

One puzzle that may be answered by the entropy scale factor is the solar corona heating problem. The ESF could explain this phenomenon because of time contraction by the surface of sun. The time contraction gradient from the surface of the sun to empty space has the effect of blueshifting light. In this theory, the matching gradient in length contraction cancels out this effect for light, but massive particles respond differently to length contraction. Massive particles, as opposed to light, do not have their energy decrease with the decreasing frequency caused by length contraction, so for these particles the effects of time contraction will predominate, and the time contraction gradient will add thermal energy to the solar corona.

Additionally, length contraction from the entropy scale factor would mean that magnetic fields would extend further into the corona than currently predicted, which could amplify the effects of magnetic reconnection for heating the corona.

By the same mechanisms the entropy scale factor would predict even stronger effects around black holes, due to their higher entropy. These factors may combine to increase the energy of relativistic jets, a phenomenon currently unexplained by general relativity.

Inflation and Dark Energy

Another puzzle that may be solved by the entropy scale factor is inflation. The slow-roll version of the inflation theory postulates that the early universe expanded rapidly due to an inflaton field. This process has been used to explain the evenness of the cosmic microwave background radiation and the density of the universe on a large scale, but it has also been criticized for several reasons, including that the inflaton field has never been detected, the initial conditions for inflation would have to be finely tuned, and that many models predict that inflation would go on forever once it starts. Inflation may not be needed at all in the entropy scale factor model. The early universe was extremely hot and small, which corresponds to very high average entropy. This high entropy would have caused the Planck time to be very long, leading to time running very quickly in the early universe. Time running so quickly due to the ESF may explain phenomena like the limited variations of the cosmic microwave background, without the need for inflation.

The final problem that the entropy scale factor may explain has to do with the metric expansion of the universe. General relativity predicts that gravity will slow the expansion of the universe, but observations indicate that the universe is expanding at an ever increasing rate. Dark energy has been hypothesized as the energy that is driving this expansion, but it may be that these effects are due to the entropy scale factor instead.

The early universe had very high average entropy, as stated above, but the total entropy was very low compared to what it is today. In the examples listed in previous sections, with objects like black holes surrounded by empty space, the changes in space and time are balanced so that a photon emitted near an entropic object will have little change in energy from the ESF. When imaging an isotropic system, like the universe on a large scale, the ESF’s effects on time would predominate, because the one spatial dimension affected in local systems would be averaged out by the two unaffected dimensions. This would make the effects on space smaller than the effects on time, on a global level, due to the random distribution of entropic sources in an isotropic universe.

Applying this principle to our current universe, which is ever increasing in entropy, the entropy scale factor predicts that the scale of the Planck space would increase, but that there would be a faster increase in the scale of the Planck time (time contraction), leading to time to pass slightly more quickly now compared to the past. This effect may contribute to the redshift measurements leading scientists to conclude that the universe is expanding at an increasing rate.

Conclusion

Modified Newtonian dynamics (MOND) is a theory that attempted to explain some of the dark matter phenomena by proposing an increased strength of gravity at very low accelerations. The entropy scale factor also proposes an increase in the strength of gravity, but does so through a change in the scale of both distance and time. The ESF is similar to MOND in that both theories attempt to predict alterations of gravity that have not been ruled out by tests of general relativity. The ESF is different than MOND in that it depends on entropy, instead of mass, which may better explain phenomena like the Bullet Cluster, in which gravitational lensing does not appear to correlate with mass. Additionally, MOND does not predict changes from general relativity at high accelerations, like the ESF does, and could not explain the power of relativistic jets or the solar corona heating problem, or phenomena like the metric expansion of space.

Both loop quantum gravity and string theory have been used to derive the Bekenstein-Hawking equation for black hole entropy. Either of these theories could be adapted to include the ESF, as could any other theory that treats black holes as indivisible units of spacetime.

One criticism of the ESF is that it would need to be finely tuned in order to account for so many astrophysical mysteries. That is, the amount of scale change to account for solar corona heating will have to match up with the amount needed for galactic dynamics, and so on. While this is true, with so few details of this phenomenon worked out there is considerable freedom in building a model to fit different data sources. Additionally, if the ESF is not correct, then it requires belief in a different, and more complicated, network of forces. To reject the ESF one must also believe that the solar corona is heated by a mysterious process that we can’t explain, that there was an undetected inflaton field in the early universe, and unexplained dark energy field in the current universe, that there is dark matter arranged in halos we can’t explain, and that this dark matter is unexpectedly invisible to dark matter detectors. Between these two options, Occam’s razor seems to point towards the entropy scale factor.

The entropy scale factor is unique among physical theories in that it has the potential to connect so many unsolved problems. Numerous studies will need to be done to see if the entropy scale factor agrees with existing data, and can explain unsolved mysteries like the dark matter phenomena in a consistent system. This theory has not had the time for elaboration that has been afforded to theories like MOND or dark matter. It is worth further study because it has not been ruled out by general relativity, and because it has the potential to offer a breakthrough in the understanding of many astrophysical mysteries.

For more information, including a version with a complete list of references, please see my webpage at www.chriswatsonmd.com. If you’d like to help with this project, please email me at chriswatsonmd@gmail.com.