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Thread: Heat Flow in Special and General Theory of Relativity

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    Heat Flow in Special and General Theory of Relativity

    The relativistic heat energy equation was from Fourierís law of hear induction was:



    In which the heat energy density is defined in the following way first in simple Cartesian coordinates,



    However, the d'Alembert operator just involves an extra term and is nothing too complicated,



    I noticed that an entropy can be formed in the following way: The heat energy from Fourier's law is just



    As we already established, but we have changed it slightly for the squared díAlembertian - this allowed us to have the definition of the element volume. And it was noticed there may be a definition of the entropy from this



    and an irreversible entropy production as



    and production density



    Without the time derivative and in the simple Cartesian coordinate system the construction simply looks like



    And the flow in and out of a classical system is



    The flow in and out of a classical system is



    To finish up, letís just take a walk through some of the fundamental equations you would come to expect when investigating heat flow. If the temperature at any point is changed, the local gradient heat flow is



    We will use this equation in a moment - the heat energy per unit area is,



    we can create a gradient such that



    Then we can retrieve the definition of the heat flow equation in terms of the Ricci curvature again , keep in mind, is the thermal conductivity. In the case above, the curvature has replaced the definition of the gradient . If the temperature at any point changed, the local gradient heat flow is

    Multiply through by and we get,



    Which is the heat flow. After reading work by Arun and Sivaram suggesting that curvature flows, just like heat, they presented a three dimensional case and it made me think about a Relativistic Ricci flow equation satisfying four dimensions. I proposed that takes the form,



    (where we have used the squared notation to make sure we know we are talking about the operator in terms of squared derivatives).

    cont. next page

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    The Ricci flow is the heat equation for a Riemannian manifold. Arun and Sivaram suggest an equation of the form:



    In which is the scalar curvature. Later we will see another form in which I had an idea in which there was a diffusion equation for (the scalar gravitational potential). I came to find that the equation I had proposed already existed in literature, for instance, for the heat flow per unit area we have:



    must also be modified in such a way that:



    where is the spacelike time. One application of understanding curvature in terms of the heat flow came in the form of replacing the divergence with the respective christoffel symbol (or connection)



    This is not such an unusual thing to do, since it seems that such approaches where already in literature, one such example was the renormalized Ricci flow



    Suggested by another author - here you can see the additional term has the Ricci scalar curvature playing the role of the squared space derivatives.
    Now… It was possible to construct all those idea's in the context of curved spacetime. The kind of equation I wanted to study was



    The original form of this equation even considered a mass parameter which took the form of the dispersion, which as a heat equation, would tell you also how that system would have interacted with a certain type of medium... a good example is reflection. Or even absorption. I’ll show that at the end. The idea was simple - I searched for a solution for the flow which has the appearance of



    The idea would invoke a relativistic Newton-Poisson equation of the form



    And this equation in curved spacetime looks like:





    Which is a diffusion equation with respect to in the language of general relativity. It turned out after some more investigation that such an idea was already suggested, the saving grace was I haven’t found anyone write it in this form.

    The last expression in the equation is simply related to variation of action with regards to the metric. Without the potential, that relationship looks like



    The rules for the operator in curved space is





    I also promised to show what it would look like with a mass parameter. That takes the form of:



    In which



    Is the dispersion relation, in which is the wavenumber and the angular frequency. This is a relativistic relationship.
    In the context of curved space in general relativity, the Ricci flow turns into





    And this wraps up the investigation into the Ricci flow and other idea’s which followed.

    **REFERENCES**
    Relativistic heat conduction - Wikipedia (https://en.wikipedia.org/wiki/Relati...eat_conduction)
    https://arxiv.org/ftp/arxiv/papers/1205/1205.4624.pdf (https://arxiv.org/ftp/arxiv/papers/1205/1205.4624.pdf)
    *copyright infringing links taken out*

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    Can you please give reason why we need to read this?
    Just dumping two posts full of calculus here without an explanation is not appreciated.
    Also I took out two (non-functioning) links to a paper server that infringes copyright, which we cannot condone on CQ.
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    Quote Originally Posted by tusenfem View Post

    Can you please give reason why we need to read this?
    You don't need to read it, no one is forcing you to do anything. As for why it has been posted, I wondered if anyone would be interested in the discussion of ricci flow and heat flow.

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    Quote Originally Posted by Dubbelosix View Post
    You don't need to read it, no one is forcing you to do anything. As for why it has been posted
    When a moderator addresses you in a color identified for use in moderation, the last thing you should do is respond with an attitude.

    I wondered if anyone would be interested in the discussion of ricci flow and heat flow.
    Then you should do people the courtesy of starting your threads with an introductory paragraph to outline the topic, otherwise it looks like you're simply holding forth for no particular reason.
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    Yes, what is unclear from the above are:

    1) why do you find this interesting? Introductory comments like that are generally used when introducing topics. I'm not saying it isn't interesting, I'm saying it's hard to tell.

    2) why do you regard it as new, or even do you regard it as new? The physics being used appears to be standard, so is it merely a repackaging of what can be found in more carefully written sources, or are their insights here which are claimed to be new? New insights are of great value, but that only stresses the importance of clarifying what is in fact new here, and if nothing is new, why it is important to draw attention to what is getting missed.

    3) why do you use language where you talk about "energy" when your expressions are those of "energy flux"? For example, your second equation is not, as claimed, an expression for "heat energy density" as made obvious from the fact that a huge constant T leads to zero in that expression. Indeed, many of your expressions use language that suggests they are changes in energy, but actually they are rates of changes of energy, making it hard to follow when the language that describes the equations does not correctly connect to what the equations say. So are you making errors here, or are you just being imprecise with your language? Then you appear to mix up a change in entropy with expressions that involve rates of change of energy, which doesn't make sense. Finally, at one point you appear to take a 1/T from outside an integral to inside the integral, but T is not constant and you have not stated that the integral is intended to be over an infinitesmal volume (indeed, if over an infinitesmal volume, what would be the point of using integral notation in the first place?)

    Having answers to these questions would make casual readers, like myself, more likely to spend the necessary time to understand the comments, and even to believe they are correct.
    Last edited by Ken G; 2018-Jun-03 at 03:29 PM.

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    Quote Originally Posted by Ken G View Post
    Yes, what is unclear from the above are:

    1) why do you find this interesting? Introductory comments like that are generally used when introducing topics. I'm not saying it isn't interesting, I'm saying it's hard to tell.

    2) why do you regard it as new, or even do you regard it as new? The physics being used appears to be standard, so is it merely a repackaging of what can be found in more carefully written sources, or are their insights here which are claimed to be new? New insights are of great value, but that only stresses the importance of clarifying what is in fact new here, and if nothing is new, why it is important to draw attention to what is getting missed.

    3) why do you use language where you talk about "energy" when your expressions are those of "energy flux"? For example, your second equation is not, as claimed, an expression for "heat energy density" as made obvious from the fact that a huge constant T leads to zero in that expression. Indeed, many of your expressions use language that suggests they are changes in energy, but actually they are rates of changes of energy, making it hard to follow when the language that describes the equations does not correctly connect to what the equations say. So are you making errors here, or are you just being imprecise with your language? Then you appear to mix up a change in entropy with expressions that involve rates of change of energy, which doesn't make sense. Finally, at one point you appear to take a 1/T from outside an integral to inside the integral, but T is not constant and you have not stated that the integral is intended to be over an infinitesmal volume (indeed, if over an infinitesmal volume, what would be the point of using integral notation in the first place?)

    Having answers to these questions would make casual readers, like myself, more likely to spend the necessary time to understand the comments, and even to believe they are correct.
    I find it interesting because no matter how hard I have looked, no one I have found has demonstrated a heat equation in curved four dimensional space. That is not meant to mean of course no one has done it before.

    What do you mean about energy, are we talking about , which is defined as a heat energy?

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    Quote Originally Posted by Dubbelosix View Post
    What do you mean about energy, are we talking about , which is defined as a heat energy?
    Among the many points I raised, one was that you claimed your second equation was for "heat energy density." That obviously cannot be true, for the reason I gave.

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    Quote Originally Posted by Ken G View Post
    Among the many points I raised, one was that you claimed your second equation was for "heat energy density." That obviously cannot be true, for the reason I gave.
    No you have the units wrong. The Fourier heat flow is actually as a heat flow per area. That means when you square the operator you pick up an extra derivative and is indeed, a density.

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    Ask yourself this: what is your "heat energy density" if T is huge and constant in space and time.

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    Quote Originally Posted by Ken G View Post
    Ask yourself this: what is your "heat energy density" if T is huge and constant in space and time.
    Is it that you are struggling with the notion of a heat energy density? There are many examples in physics where heat may as well be seen as an energy or content of energy. These relationships are found through the equipartition (assuming I have factors right). So yeah, heat densities almost will certainly exist.

    If you want you can call it a volumetric heat flow. But heat is not so different to energy, especially when understood through the equipartition theorem which relates temperatures to average energies. As for huge and constant temperatures, let's just say, the equation cannot describe energy through the usual equipartition, but I drew on this to show they are in some ways related.

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    Quote Originally Posted by Dubbelosix View Post
    Is it that you are struggling with the notion of a heat energy density? There are many examples in physics where heat may as well be seen as an energy or content of energy. These relationships are found through the equipartition (assuming I have factors right). So yeah, heat densities almost will certainly exist.
    The problem is not with the concept of heat energy density, the problem is that your equation is not a correct equation for an energy density. You just had to answer my question-- the answer is, your equation gives zero if T is large and constant. Of course that's not correct for a heat energy density.
    If you want you can call it a volumetric heat flow.
    Ah, so there's the rub. A "heat flow" is certainly not the same thing as an "energy density." So that's what I was talking about-- the language you were using to describe the equations did not make sense, which made it hard to understand if you were simply using imprecise language, or if it meant the equations were wrong.

    But heat is not so different to energy, especially when understood through the equipartition theorem which relates temperatures to average energies.
    The issue is not with "heat" vs. "energy," it is with "density" versus "flow." Those are not the same at all. The problem recurred when you started putting dots over the Q. Normally, that implies a time derivative, so now you are talking about a time derivative of a heat flow, but you then equate it to a rate of change of entropy, which also does not make sense.
    As for huge and constant temperatures, let's just say, the equation cannot describe energy through the usual equipartition, but I drew on this to show they are in some ways related.
    Why wouldn't the equation describe energy if the temperature was huge and constant? That doesn't make sense either. The things you are saying simply don't make sense, so it's hard to judge what you are trying to do with those equations.

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    Quote Originally Posted by Ken G View Post
    The problem is not with the concept of heat energy density, the problem is that your equation is not a correct equation for an energy density. You just had to answer my question-- the answer is, your equation gives zero if T is large and constant. Of course that's not correct for a heat energy density.
    Ah, so there's the rub. A "heat flow" is certainly not the same thing as an "energy density." So that's what I was talking about-- the language you were using to describe the equations did not make sense, which made it hard to understand if you were simply using imprecise language, or if it meant the equations were wrong.

    The issue is not with "heat" vs. "energy," it is with "density" versus "flow." Those are not the same at all. The problem recurred when you started putting dots over the Q. Normally, that implies a time derivative, so now you are talking about a time derivative of a heat flow, but you then equate it to a rate of change of entropy, which also does not make sense.
    Why wouldn't the equation describe energy if the temperature was huge and constant? That doesn't make sense either. The things you are saying simply don't make sense, so it's hard to judge what you are trying to do with those equations.

    Sorry I could not reply earlier.

    I'm sorry it doesn't make sense, but I am right. In fact, you were more right to say the terminology was imprecise, rather than wrong.

    It certainly is a heat density, or energy density (we've agreed we can go between terminology here) but while it has been known as a volumetric heat flux, a more common name is a heat flux density. I don't know why you were hung up on this density issue, because it certainly is a density. A little advice, whenever you encounter a meaningful numerator weighted by a volumetric denominator, you almost always have a density term. For instance is a spin density, surface density or even a charge density, just to name a few.

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    Quote Originally Posted by PetersCreek View Post

    Then you should do people the courtesy of starting your threads with an introductory paragraph to outline the topic, otherwise it looks like you're simply holding forth for no particular reason.
    I'm sorry, normally I would write an introductory and I should have in this instance now I look back.

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    Quote Originally Posted by Dubbelosix View Post
    It certainly is a heat density, or energy density (we've agreed we can go between terminology here) but while it has been known as a volumetric heat flux, a more common name is a heat flux density.
    But even that statement is imprecise, because a heat flux density is not an energy density, it is an energy flux density. If you insist on using the words incorrectly, you cannot expect people to try to understand what you are saying. I don't even know what you think the units of your expression would be, because the language isn't correct.
    I don't know why you were hung up on this density issue, because it certainly is a density.
    It's a density, yes, but it's not an energy density, it is an energy flux density. If you cannot tell the difference between those things, no one will ever be able to understand what you are trying to say. The difference is not insignificant-- later you start taking time derivatives, so you need to be able to tell if you are tracking a rate of change of entropy, or a rate of change of entropy flux. Those are just completely different things, with totally different ramifications. In fact, it really doesn't make much sense to take a time derivative of an energy flux, when you are trying to look for entropy generation (that would be more like a divergence of an entropy flux, which is totally different). So the exposition is hopelessly confusing due to the absence of correct terminology, and no one can judge if your equations are correct because no one can tell what you think the equations are saying. And then there's the matter of pulling the 1/T inside the integral, which is probably just plain wrong, but again it's hard to say how wrong it is until it is clear what you are saying.
    Last edited by Ken G; 2018-Jun-07 at 09:52 PM.

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    Quote Originally Posted by Ken G View Post
    But even that statement is imprecise, because a heat flux density is not an energy density, it is an energy flux density. If you insist on using the words incorrectly, you cannot expect people to try to understand what you are saying. I don't even know what you think the units of your expression would be, because the language isn't correct.
    It's a density, yes, but it's not an energy density, it is an energy flux density. If you cannot tell the difference between those things, no one will ever be able to understand what you are trying to say. The difference is not insignificant-- later you start taking time derivatives, so you need to be able to tell if you are tracking a rate of change of entropy, or a rate of change of entropy flux. Those are just completely different things, with totally different ramifications. In fact, it really doesn't make much sense to take a time derivative of an energy flux, when you are trying to look for entropy generation (that would be more like a divergence of an entropy flux, which is totally different). So the exposition is hopelessly confusing due to the absence of correct terminology, and no one can judge if your equations are correct because no one can tell what you think the equations are saying. And then there's the matter of pulling the 1/T inside the integral, which is probably just plain wrong, but again it's hard to say how wrong it is until it is clear what you are saying.
    I am not playing this game either.

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    You are talking... rubbish. I don't think I will be coming back here.

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    Quote Originally Posted by Dubbelosix View Post
    You are talking... rubbish. I don't think I will be coming back here.

    Okay, you keep on insulting people here, let me help you help you.
    Infraction
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    Yeah, there's not much I can do for someone who thinks the difference between energy density and energy flux density is "rubbish." For anyone who wants to know that difference and doesn't already, an energy density is an energy per volume, and an energy flux density is an energy per second per area. So to get a flux density, you take a volume density and multiply by some characteristic speed at which it is moving. The implied speed for a heat flux is some kind of diffusion speed, not explicit in the above equations but important all the same-- not rubbish. The concept is like one thousand times simpler than expressions in relativity.
    Last edited by Ken G; 2018-Jun-08 at 01:51 PM.

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    Quote Originally Posted by Ken G View Post
    Yeah, there's not much I can do for someone who thinks the difference between energy density and energy flux density is "rubbish." For anyone who wants to know that difference and doesn't already, an energy density is an energy per volume, and an energy flux density is an energy per second per area. So to get a flux density, you take a volume density and multiply by some characteristic speed at which it is moving. The implied speed for a heat flux is some kind of diffusion speed, not explicit in the above equations but important all the same-- not rubbish. The concept is like one thousand times simpler than expressions in relativity.
    But thanks for your attempts at education, Ken. One would hope that, upon return, the OP would proceed with a more humble acceptance of the constructive criticisms offered.

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    Quote Originally Posted by Ken G View Post
    Yeah, there's not much I can do for someone who thinks the difference between energy density and energy flux density is "rubbish."


    No not at all. You challenged it and I ended up agreeing, the terminology was imprecise, with that much you were right. However, I was not wrong. Then your continuation of the argument even though I attempted to reach level ground, failed. The difference between the energy density and the energy flux density, is a very small argument but you wanted to make it bigger than what it was.

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    Quote Originally Posted by Dubbelosix View Post
    No not at all. You challenged it and I ended up agreeing, the terminology was imprecise, with that much you were right. However, I was not wrong.
    Well here's the problem: I still have no idea if you think you are talking about energy density or energy flux density. So instead of just saying you are right, say what you want to say, and say it right.
    The difference between the energy density and the energy flux density, is a very small argument but you wanted to make it bigger than what it was.
    Nonsense, a difference between two very different concepts is never "a very small argument." Say what you want to say, and say it right, so people will know what you are trying to say. No one can possibly judge the truth in your remarks until they know what you are saying, and so first, you must know yourself what you are saying.

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