# Entropic forces and the 2nd law of thermodynamics

15/01/10 02:21

Let me address some other confusions in the blog discussion. The fact that a force in entropic does not mean it should be irreversible. This is a complete misunderstanding of what it means to have an entropic force. This is why I added section 2 on the entropic force. For a polymer the force obeys Hooke's law, which is conservative. No doubt about that.

Just last week we had a seminar in Amsterdam on DNA. Precisely the situation described in section two was performed in lab experiments, using optical tweezers. The speaker, Gijs Wuite from the Free University in Amsterdam, showed movies of DNA being stretched and again released. These biophysicist know very well that these forces are purely entropic, and also reversible. The movies clearly showed reversibility, to a very high degree. In fact, I asked the speaker specifically about this, and he confirmed it. They test this in the lab, so it is an experimental fact that entropic forces can be conservative.

So please read section 2, study it and read it again, and think about it for a little longer. When the heat bath is infinite, the force is perfectly conservative. For the case of gravity the speed of light determines the size of the heat bath, since its energy content is given by E=Mc^2. So in the non relativistic limit the heat bath is infinite. Indeed, Newton's laws are perfectly conservative. When one includes relativistic effects, the heat bath is no longer infinite. Here one could expect some irreversibility. In fact, I suspect that the production of gravity waves is causing this. Indeed, a binary system will eventually coalesce. This is irreversible, indeed. This all fits very well. Extremely well, actually. Of course, when I first got these ideas, I worried about too much irreversiblity too. I knew about the polymer example, but had to study it again to convince myself that entropic forces can indeed be reversible.

Another useful point to know is that when a system is slightly out of equilibrium, it will indeed generate some entropy. But a theorem by Prigogine states that the dynamics of the system will adapt itself so that entropy production is minimized. Yes, really minimized. This may appear counterintuitive, but I like to look at it as that it seeks the path of least resistance. So this means that there will in general not be a lot of entropy generated. At least, the system will do whatever it can to minimize it.

By the way, it is true that the total energy of a system of two masses is given by the total mass. But if one then takes the entropy gradient to be proportional to the reduced mass, one again recovers the right force. I thought of putting that in the paper, but I think it is kind of trivial. This confusion was not to difficult to solve.

Another point that may not be appreciated is that the system is actually taken out of equilibrium. If everything would be in equilibrium, the universe would be a big black hole, or be described by pure de Sitter space. Only horizons, no visible matter. If a system is out of equilibrium, there is not a very precise definition of temperature. In fact, different parts of the system may have different temperatures. There is no problem also with neutron stars. In fact, physical neutron stars do not have exact zero temperature. But the temperature I use in the paper is one that is associated with the microscopic degrees of freedom, which because there is no equilibrium, is not necessarily equal to the macroscopic temperature.

In fact, the microscopic degrees of freedom on the holographic screens should not be seen as being associated with local degrees of freedom in actual space. They are very non local states. This is what holography tells us. In fact, they can also not be only related to the part of space contained in the screen, because this would mean we can count micro states independently for every part of space, and in this way we would violate the holographic principle. There is non locality in the microstates.

Another point: gravitons do not exist when gravity is emergent. Gravitons are like phonons. In fact, to make that analogy clear consider two pistons that close of a gas container at opposite ends. Not that the force on the pistons due to the pressure is also an example of an entropic force. We keep the pistons in place by an external force. When we gradually move one of the pistons inwards by increasing the force, the pressure will become larger. Therefore the other piston will also experience a larger force. We can also do this in an abrupt way. We then cause a sound wave to go from one piston to the other. The quantization of this sound wave leads to phonons. We know that phonons are quite useful concepts, which even themselves are often used to understand other emergent phenomena.

Similarly, gravitons can be useful, and in that sense exist as effective "quasi" particles. But they do not exist as fundamental particles.

Just last week we had a seminar in Amsterdam on DNA. Precisely the situation described in section two was performed in lab experiments, using optical tweezers. The speaker, Gijs Wuite from the Free University in Amsterdam, showed movies of DNA being stretched and again released. These biophysicist know very well that these forces are purely entropic, and also reversible. The movies clearly showed reversibility, to a very high degree. In fact, I asked the speaker specifically about this, and he confirmed it. They test this in the lab, so it is an experimental fact that entropic forces can be conservative.

So please read section 2, study it and read it again, and think about it for a little longer. When the heat bath is infinite, the force is perfectly conservative. For the case of gravity the speed of light determines the size of the heat bath, since its energy content is given by E=Mc^2. So in the non relativistic limit the heat bath is infinite. Indeed, Newton's laws are perfectly conservative. When one includes relativistic effects, the heat bath is no longer infinite. Here one could expect some irreversibility. In fact, I suspect that the production of gravity waves is causing this. Indeed, a binary system will eventually coalesce. This is irreversible, indeed. This all fits very well. Extremely well, actually. Of course, when I first got these ideas, I worried about too much irreversiblity too. I knew about the polymer example, but had to study it again to convince myself that entropic forces can indeed be reversible.

Another useful point to know is that when a system is slightly out of equilibrium, it will indeed generate some entropy. But a theorem by Prigogine states that the dynamics of the system will adapt itself so that entropy production is minimized. Yes, really minimized. This may appear counterintuitive, but I like to look at it as that it seeks the path of least resistance. So this means that there will in general not be a lot of entropy generated. At least, the system will do whatever it can to minimize it.

By the way, it is true that the total energy of a system of two masses is given by the total mass. But if one then takes the entropy gradient to be proportional to the reduced mass, one again recovers the right force. I thought of putting that in the paper, but I think it is kind of trivial. This confusion was not to difficult to solve.

Another point that may not be appreciated is that the system is actually taken out of equilibrium. If everything would be in equilibrium, the universe would be a big black hole, or be described by pure de Sitter space. Only horizons, no visible matter. If a system is out of equilibrium, there is not a very precise definition of temperature. In fact, different parts of the system may have different temperatures. There is no problem also with neutron stars. In fact, physical neutron stars do not have exact zero temperature. But the temperature I use in the paper is one that is associated with the microscopic degrees of freedom, which because there is no equilibrium, is not necessarily equal to the macroscopic temperature.

In fact, the microscopic degrees of freedom on the holographic screens should not be seen as being associated with local degrees of freedom in actual space. They are very non local states. This is what holography tells us. In fact, they can also not be only related to the part of space contained in the screen, because this would mean we can count micro states independently for every part of space, and in this way we would violate the holographic principle. There is non locality in the microstates.

Another point: gravitons do not exist when gravity is emergent. Gravitons are like phonons. In fact, to make that analogy clear consider two pistons that close of a gas container at opposite ends. Not that the force on the pistons due to the pressure is also an example of an entropic force. We keep the pistons in place by an external force. When we gradually move one of the pistons inwards by increasing the force, the pressure will become larger. Therefore the other piston will also experience a larger force. We can also do this in an abrupt way. We then cause a sound wave to go from one piston to the other. The quantization of this sound wave leads to phonons. We know that phonons are quite useful concepts, which even themselves are often used to understand other emergent phenomena.

Similarly, gravitons can be useful, and in that sense exist as effective "quasi" particles. But they do not exist as fundamental particles.

# Essential points of the paper

13/01/10 07:39

THE ESSENTIAL NEW POINTS OF THE PAPER

I have noticed another point of the paper that is not appreciated in blog discussions. For many years, there have been previous works in the literature that discuss the similarity between gravity and thermodynamics. In particular in Jacobson's work there is a clear statement that if one assumes the first law of thermodynamics, the holographic principle, and identifies the temperature with the Unruh temperature, that one can derive the Einstein equations. This is a remarkable result. Yet it is already 12 years old, and still up to this day, gravity is seen as a fundamental force. Clearly, we have to take these analogies seriously, but somehow no one does.

I studied the previous papers very well, and know about them for years. Many people have. We have seen a recent increase in papers following Jacobson, and extending his work to higher derivative gravity, and so. But from all of these papers, I did not pick up the insights I presented in this paper. What was missing from those papers is the answer to questions like: why does gravity have anything to do with entropy? Why do particles follow geodesics? What has entropy to do with geometry?

The derivation of Jacobson does not take in to account the fact that the mass of an object and therefore its energy can change due to the displacement of matter far away from it. There is action at a distance hidden in gravity, even relativistically. The ADM and Komar definitions of mass make this non-local aspect of gravity very clear. This non local aspect of gravity is precisely what the holographic principle is about.

Jacobson's argument is ultra local, and assumes the presence of stress energy crossing the horizon. But there is no statement about an entropic force that is influencing particles far away from the horizon. My point of view is an attempt to take a much more global view, and map out the information over a bigger part of space, even though initially I can only do that for static space times.

The statement that gravity is an entropic force is more then just saying that "it has something to do with thermodynamics". It says that motion and forces are the consequence of entropy differences. My idea is that in a theory in which space is emergent forces are based on differences in the information content, and that very general random microscopic processes cause inertia and motion. The starting point from which this all can be derived can be very, very general. In fact we don't need to know what the micro-scopic degrees of freedom really are. We only need a few basic properties.

For me this was an "eye opener", it made it from obscure to obvious. It is clear to me know that it has to be this way. There is no way to avoid it: if one does not keep track of the amount of information, one ignores the origin of motion and forces. It clarifies why gravity has something to do with entropy. It has to, it can not do otherwise.

When I got the idea that gravity and inertia emerge in this way, which is close to half a year ago, I was really excited. I felt I had an insight that makes clear what gravity is. But I decided not to publish too quickly, also to allow time to make it more precise. But also to see if the idea that gravity is entropic would still appear to me as new as exciting as my first feeling about it. And it does. Now, almost half a year later, I still feel that way.

For instance, the similarity between the entropic force for a polymer and gravity is a real clue to something important. The fact that it fits in well with an adapted version of the work of Jacobson gives additional support. The derivation of the Einstein equations is not really knew, in my mind, since it technically is very similar to the previous works. And I agree that the other line of the paper that discusses inertia is heuristic, and leaves some important gaps. But nevertheless I decided to publish it anyway, because I think this approach to gravity is the right one, it is different, very different from everything that is done today.

Everyone who does not appreciate that this view is different from previous papers are missing an essential point. If space is emergent, a lot more has to be explained than just the Einstein equations. Geodesic motion, or if you wish, the laws of Newton have to be re-derived. They are not fundamental. This has not been discussed anywhere, not even noted that it is the case.

If the previous papers had made the emergence of gravity so clear, why are people still regarding string theory as the final theory of quantum gravity? Somehow, not everyone was convinced that these similarities mean something, or at least, people had no clear idea of what they mean.

Some people may think that when we develop string theory further that eventually we will learn about this. I am not sure that string theory will necessarily take us in the right direction, if we keep regarding the definition in terms of closed strings as being microscopically defined, or may be equivalent to some other formulation. And not if we keep our eyes closed for emergent phenomena. Gravitons can not be fundamental particles in a theory of emergent space time and gravity.

So what is the role of string theory, if gravity is emergent? I discussed this at some level in the paper. It should also be emergent, and it is nothing but a framework like quantum field theory. In fact, I think of string theory as the way to make QFT in to a UV complete but still effective framework. It is based on universality. Many microscopic systems can lead to the same string theory. The string theory landscape is just the space of all universality classes of this framework. I have more to say about it, but will keep that for a publication, or I will post that some other time.

Of course, I would have liked to make things even more clear or convincing. In this paper, I use heuristic and you might say handwaving arguments. The issue of motion: why is the acceleration a that I introduced equal to the second time derivative of the position? If one assumes the equivalence principle, it is clear. Also coordinate invariance would be enough. But I do not have a very precise way of seeing how that emerges. How to go from just information to a Lorentzian geometry in which general coordinate invariance is manifest. Some assumptions have to be made.

But again, this are questions that others have not been even started to think about. These are questions that have not been even addressed by previous works. But they are essential. When one really understands this well, there should be no doubt that gravity is emergent and forces are driven by entropy.

This is the essential idea, which is really new and important, and which in my view justifies this level of reasoning, certainly in a first paper. It is clear that this is not the final paper on this subject. This is also my own view. I clearly did not answer all of the questions. In fact, my approach probably raises more questions than it answers. But it should be obvious that these questions are important, very fundamental and their answers should lead us in a completely new direction. Our theories will have to based on new paradigms.

I find all this still very exciting and will continue to work in this direction.

And remember, quantum mechanics was also not developed in one paper. Do you think de Broglie knew exactly what he was talking about? Leaps based on intuition are sometimes necessary. They are an important part of progress in science, even if they do not immediately give complete finished theories of Nature.

I have noticed another point of the paper that is not appreciated in blog discussions. For many years, there have been previous works in the literature that discuss the similarity between gravity and thermodynamics. In particular in Jacobson's work there is a clear statement that if one assumes the first law of thermodynamics, the holographic principle, and identifies the temperature with the Unruh temperature, that one can derive the Einstein equations. This is a remarkable result. Yet it is already 12 years old, and still up to this day, gravity is seen as a fundamental force. Clearly, we have to take these analogies seriously, but somehow no one does.

I studied the previous papers very well, and know about them for years. Many people have. We have seen a recent increase in papers following Jacobson, and extending his work to higher derivative gravity, and so. But from all of these papers, I did not pick up the insights I presented in this paper. What was missing from those papers is the answer to questions like: why does gravity have anything to do with entropy? Why do particles follow geodesics? What has entropy to do with geometry?

The derivation of Jacobson does not take in to account the fact that the mass of an object and therefore its energy can change due to the displacement of matter far away from it. There is action at a distance hidden in gravity, even relativistically. The ADM and Komar definitions of mass make this non-local aspect of gravity very clear. This non local aspect of gravity is precisely what the holographic principle is about.

Jacobson's argument is ultra local, and assumes the presence of stress energy crossing the horizon. But there is no statement about an entropic force that is influencing particles far away from the horizon. My point of view is an attempt to take a much more global view, and map out the information over a bigger part of space, even though initially I can only do that for static space times.

The statement that gravity is an entropic force is more then just saying that "it has something to do with thermodynamics". It says that motion and forces are the consequence of entropy differences. My idea is that in a theory in which space is emergent forces are based on differences in the information content, and that very general random microscopic processes cause inertia and motion. The starting point from which this all can be derived can be very, very general. In fact we don't need to know what the micro-scopic degrees of freedom really are. We only need a few basic properties.

For me this was an "eye opener", it made it from obscure to obvious. It is clear to me know that it has to be this way. There is no way to avoid it: if one does not keep track of the amount of information, one ignores the origin of motion and forces. It clarifies why gravity has something to do with entropy. It has to, it can not do otherwise.

When I got the idea that gravity and inertia emerge in this way, which is close to half a year ago, I was really excited. I felt I had an insight that makes clear what gravity is. But I decided not to publish too quickly, also to allow time to make it more precise. But also to see if the idea that gravity is entropic would still appear to me as new as exciting as my first feeling about it. And it does. Now, almost half a year later, I still feel that way.

For instance, the similarity between the entropic force for a polymer and gravity is a real clue to something important. The fact that it fits in well with an adapted version of the work of Jacobson gives additional support. The derivation of the Einstein equations is not really knew, in my mind, since it technically is very similar to the previous works. And I agree that the other line of the paper that discusses inertia is heuristic, and leaves some important gaps. But nevertheless I decided to publish it anyway, because I think this approach to gravity is the right one, it is different, very different from everything that is done today.

Everyone who does not appreciate that this view is different from previous papers are missing an essential point. If space is emergent, a lot more has to be explained than just the Einstein equations. Geodesic motion, or if you wish, the laws of Newton have to be re-derived. They are not fundamental. This has not been discussed anywhere, not even noted that it is the case.

If the previous papers had made the emergence of gravity so clear, why are people still regarding string theory as the final theory of quantum gravity? Somehow, not everyone was convinced that these similarities mean something, or at least, people had no clear idea of what they mean.

Some people may think that when we develop string theory further that eventually we will learn about this. I am not sure that string theory will necessarily take us in the right direction, if we keep regarding the definition in terms of closed strings as being microscopically defined, or may be equivalent to some other formulation. And not if we keep our eyes closed for emergent phenomena. Gravitons can not be fundamental particles in a theory of emergent space time and gravity.

So what is the role of string theory, if gravity is emergent? I discussed this at some level in the paper. It should also be emergent, and it is nothing but a framework like quantum field theory. In fact, I think of string theory as the way to make QFT in to a UV complete but still effective framework. It is based on universality. Many microscopic systems can lead to the same string theory. The string theory landscape is just the space of all universality classes of this framework. I have more to say about it, but will keep that for a publication, or I will post that some other time.

Of course, I would have liked to make things even more clear or convincing. In this paper, I use heuristic and you might say handwaving arguments. The issue of motion: why is the acceleration a that I introduced equal to the second time derivative of the position? If one assumes the equivalence principle, it is clear. Also coordinate invariance would be enough. But I do not have a very precise way of seeing how that emerges. How to go from just information to a Lorentzian geometry in which general coordinate invariance is manifest. Some assumptions have to be made.

But again, this are questions that others have not been even started to think about. These are questions that have not been even addressed by previous works. But they are essential. When one really understands this well, there should be no doubt that gravity is emergent and forces are driven by entropy.

This is the essential idea, which is really new and important, and which in my view justifies this level of reasoning, certainly in a first paper. It is clear that this is not the final paper on this subject. This is also my own view. I clearly did not answer all of the questions. In fact, my approach probably raises more questions than it answers. But it should be obvious that these questions are important, very fundamental and their answers should lead us in a completely new direction. Our theories will have to based on new paradigms.

I find all this still very exciting and will continue to work in this direction.

And remember, quantum mechanics was also not developed in one paper. Do you think de Broglie knew exactly what he was talking about? Leaps based on intuition are sometimes necessary. They are an important part of progress in science, even if they do not immediately give complete finished theories of Nature.

# Logic of the paper

12/01/10 00:50

The paper is not technical, but some background is needed, more than just being able to read the text and the equations. The text explains the logic, but apparently some important points are misunderstood. Clearly, I should do a better job in making them more clear. But it is my impression that the misunderstanding is partly due to a lack of background or a difference in reference frame. Because the logic of the paper is being misrepresented in some reports, I add here some clarifications.

So here is an attempt to address some of the points that I think are not appreciated or generally understood.

The starting point is a microscopic theory that knows about time, energy and number of states. That is all, nothing more. This is sufficient to introduce thermodynamics. From the number of states one can construct a canonical partition function, and the 1st law of thermodynamics can be derived. No other input is needed, certainly not Newtonian mechanics. TIme translation symmetry gives by Noether's theorem a conserved quantity. This defines energy. Hence, the notion of energy is already there when there is just time, no space is needed.

Temperature is defined as the conjugate variable to energy. Geometrically it can be identified with the periodicity of euclidean time that is obtained after analytic continuation. Again there is nothing needed about space. Temperature exists if there is only time.

It is possible to introduce other macroscopic variables that are associated with a finite but still large subset of the microstates. Let us denote such a variable by x, at this point this is just some arbitrary choice. It can be any macroscopic variable that singles out a collection of the microscopic states. So specifying x in addition the to energy gives more detailed description of the microscopic states, but nothing more. So it is not even necessary to think about x as a space coordinate. Nevertheless one can define a number of microstates denoted by Omega(E,x) for given energy E and for a given value x for this macroscopic variable x.

Next one can introduce a formal variable called F and introduce in the partition function as the thermodynamical dual to x. Following just standard statistical physics (I avoid the word mechanics, since Newton's law is not necessary) one can obtain the 1st law of thermodynamics. dE=TdS -Fdx. This makes clear that F is a generalized "force", but it has nothing to do with Newton's law yet. It is defined in terms of entropy differences. The macroscopic force that is obtained in this way has no microscopic origin in terms of microscopic field. The force is entirely a consequence of the amount of configuration space and not mediated by anything. There is no space yet.

The meaning of the statement that space is emergent is that the space coordinates x can be viewed as examples of such macroscopic variables. They are not microscopically defined, but just introduced as a way of singling out part of the available micro states. It is my impression that not all readers have understood or appreciated this essential point. Hence, if the number of states depend on x there can be an entropic force, when there is a finite temperature. This is all, nothing more. Again, for this point I don't need to assume Newtonian mechanics. It does not exist yet in this framework.

The other central point paper is that if one chooses a macroscopic coordinate x that corresponds to a fixed position in a non-inertial frame, that Newton's law of inertia F=ma will be the consequence of such an entropic force. It has to be. There is no other way it can arise, simply because x is not a microscopic variable. It is obvious. Nevertheless, it is a fundamental new insight that has not been noted before. This is not an empty or circular statement. It says something about the way that the function Omega(E,x) should behave as a function of x. All this can be derived and defined without the input of Newtonian mechanics.

The other formulas presented in the paper are just there to illustrate that indeed it is possible to get gravity from this kind of reasoning, and that it is consistent with the ideas of holography. But the main point concerns the law of inertia. The derivation of the Einstein equations (and of Newton's law in the earlier sections) follows very similar reasonings that exist in the literature, in particular Jacobson's. The connection with entropy and thermodynamics is made also there. But in those previous works it is not clear WHY gravity has anything to do with entropy. No explanation for this apparent connection between gravity and entropy has been given anywhere in the literature. I mean not the precise details, even the reason why there should be such a connection in the first place was not understood.

My paper is the first that gives a reason why. Inertia, and hence motion, is due to an entropic force when space is emergent. This is new, and the essential point. This means one HAS TO keep track of the amount of information. Differences in this amount of information is precisely what makes one frame an inertial frame, and another a non-inertial frame. Information causes motion.

This can be derived without assuming Newtonian mechanics.

So the logic of the part of the paper dealing with inertia is:

microscopic theory without space or laws of newton-> thermodynamics -> entropic force -> inertia.

The part that deals with gravity assumes holography as additional input. But this is just like what has been done before. It is also not the main point of the paper. Gravity in a way does not exist in Einstein's theory either. But one would like to recover the gravity equations. The logic here is

thermodynamics + holographic principle -> gravity.

The obvious question is: where does the holographic principle come from? Of course, it was extracted from the physics of black holes. But the holographic principle can be formulated without reference to black holes or gravity. Hence, it can be taken as a starting point, from which one then subsequently derive gravity. Again, this part is in essence not new. Jacobson followed exaclty the same logic.

This way of turning the logic of an existing argument around is done more often in physics, and it is known to lead to much more clear formulations of a theory. One example that comes to mind is the way that Dirac used the result of Heisenberg that p and q do not commute, which was obtained in some roundabout way, and made it in to the starting point for quantum mechanics. This is how it is being taught today. Another is how Einstein changed the logic used by Lorentz and used the constancy of light as a postulate. Lorentz tried to explain why it was constant by using concepts like Lorentz contraction. In fact, Lorentz theory and Einsteins are equivalent. The only difference is in the chosen starting point. The reason we give credit to Einstein is because everything follows in a much more simple fashion from his postulated starting point.

In my paper I claim that gravity follows in a very simply fashion from holography, but that the direction the other way is much more complicated. One has to be able to switch one's perspective, and then the logic becomes more clear. So don't use your usual reference frame with Newtonian mechanics in the back of your mind, but let it go first.

Anyhow, I hope this clarifies some points, and removes some of the misunderstandings .

So here is an attempt to address some of the points that I think are not appreciated or generally understood.

The starting point is a microscopic theory that knows about time, energy and number of states. That is all, nothing more. This is sufficient to introduce thermodynamics. From the number of states one can construct a canonical partition function, and the 1st law of thermodynamics can be derived. No other input is needed, certainly not Newtonian mechanics. TIme translation symmetry gives by Noether's theorem a conserved quantity. This defines energy. Hence, the notion of energy is already there when there is just time, no space is needed.

Temperature is defined as the conjugate variable to energy. Geometrically it can be identified with the periodicity of euclidean time that is obtained after analytic continuation. Again there is nothing needed about space. Temperature exists if there is only time.

It is possible to introduce other macroscopic variables that are associated with a finite but still large subset of the microstates. Let us denote such a variable by x, at this point this is just some arbitrary choice. It can be any macroscopic variable that singles out a collection of the microscopic states. So specifying x in addition the to energy gives more detailed description of the microscopic states, but nothing more. So it is not even necessary to think about x as a space coordinate. Nevertheless one can define a number of microstates denoted by Omega(E,x) for given energy E and for a given value x for this macroscopic variable x.

Next one can introduce a formal variable called F and introduce in the partition function as the thermodynamical dual to x. Following just standard statistical physics (I avoid the word mechanics, since Newton's law is not necessary) one can obtain the 1st law of thermodynamics. dE=TdS -Fdx. This makes clear that F is a generalized "force", but it has nothing to do with Newton's law yet. It is defined in terms of entropy differences. The macroscopic force that is obtained in this way has no microscopic origin in terms of microscopic field. The force is entirely a consequence of the amount of configuration space and not mediated by anything. There is no space yet.

The meaning of the statement that space is emergent is that the space coordinates x can be viewed as examples of such macroscopic variables. They are not microscopically defined, but just introduced as a way of singling out part of the available micro states. It is my impression that not all readers have understood or appreciated this essential point. Hence, if the number of states depend on x there can be an entropic force, when there is a finite temperature. This is all, nothing more. Again, for this point I don't need to assume Newtonian mechanics. It does not exist yet in this framework.

The other central point paper is that if one chooses a macroscopic coordinate x that corresponds to a fixed position in a non-inertial frame, that Newton's law of inertia F=ma will be the consequence of such an entropic force. It has to be. There is no other way it can arise, simply because x is not a microscopic variable. It is obvious. Nevertheless, it is a fundamental new insight that has not been noted before. This is not an empty or circular statement. It says something about the way that the function Omega(E,x) should behave as a function of x. All this can be derived and defined without the input of Newtonian mechanics.

The other formulas presented in the paper are just there to illustrate that indeed it is possible to get gravity from this kind of reasoning, and that it is consistent with the ideas of holography. But the main point concerns the law of inertia. The derivation of the Einstein equations (and of Newton's law in the earlier sections) follows very similar reasonings that exist in the literature, in particular Jacobson's. The connection with entropy and thermodynamics is made also there. But in those previous works it is not clear WHY gravity has anything to do with entropy. No explanation for this apparent connection between gravity and entropy has been given anywhere in the literature. I mean not the precise details, even the reason why there should be such a connection in the first place was not understood.

My paper is the first that gives a reason why. Inertia, and hence motion, is due to an entropic force when space is emergent. This is new, and the essential point. This means one HAS TO keep track of the amount of information. Differences in this amount of information is precisely what makes one frame an inertial frame, and another a non-inertial frame. Information causes motion.

This can be derived without assuming Newtonian mechanics.

So the logic of the part of the paper dealing with inertia is:

microscopic theory without space or laws of newton-> thermodynamics -> entropic force -> inertia.

The part that deals with gravity assumes holography as additional input. But this is just like what has been done before. It is also not the main point of the paper. Gravity in a way does not exist in Einstein's theory either. But one would like to recover the gravity equations. The logic here is

thermodynamics + holographic principle -> gravity.

The obvious question is: where does the holographic principle come from? Of course, it was extracted from the physics of black holes. But the holographic principle can be formulated without reference to black holes or gravity. Hence, it can be taken as a starting point, from which one then subsequently derive gravity. Again, this part is in essence not new. Jacobson followed exaclty the same logic.

This way of turning the logic of an existing argument around is done more often in physics, and it is known to lead to much more clear formulations of a theory. One example that comes to mind is the way that Dirac used the result of Heisenberg that p and q do not commute, which was obtained in some roundabout way, and made it in to the starting point for quantum mechanics. This is how it is being taught today. Another is how Einstein changed the logic used by Lorentz and used the constancy of light as a postulate. Lorentz tried to explain why it was constant by using concepts like Lorentz contraction. In fact, Lorentz theory and Einsteins are equivalent. The only difference is in the chosen starting point. The reason we give credit to Einstein is because everything follows in a much more simple fashion from his postulated starting point.

In my paper I claim that gravity follows in a very simply fashion from holography, but that the direction the other way is much more complicated. One has to be able to switch one's perspective, and then the logic becomes more clear. So don't use your usual reference frame with Newtonian mechanics in the back of your mind, but let it go first.

Anyhow, I hope this clarifies some points, and removes some of the misunderstandings .