# A Relativistic Entropic Hamiltonian–Lagrangian Approach to the Entropy Production of Spiral Galaxies in Hyperbolic Spacetime

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**PJ19**”) have demonstrated that it has a Maximum Entropy (

**MaxEnt**) geometry, in the framework of their theory of Quantitative Geometrical Thermodynamics (

**QGT**). The idea of a geometry having an entropy is counter-intuitive but is a consequence of the systematic consideration of the complementary nature of information and entropy that has become accepted since Shannon’s seminal introduction of his “information entropy” [2]. We should note that the concept of a “geometric entropy” is already well accepted in quantum gravity contexts (see, for example, Vacaru et al. [3]) but is not usually treated as an intrinsic property of geometrical structures, although perhaps Quevedo’s “Geometrothermodynamics” [4] is an exception. We should also note not only that Wang et al. [5] have recently specifically investigated the entropy production of certain natural growth processes but also that Pearson et al. [6] have recently shown that a chronometer’s accuracy is proportional to its entropy production. On “accuracy”, Parker and Jeynes [7] (“

**PJ21**”) have also recently established the entropic Uncertainty Principle.

**PLA**): the Principle of Least Exertion (

**PLE**). The PLA is based upon Planck’s constant h (with units of [Js]), and is understood to underpin much of ‘known’ physics (see Coopersmith’s review [8]: in QGT the reduced Planck constant $\hslash $ is isomorphic to 2ik

_{B}, where i

^{2}= −1 and k

_{B}is the Boltzmann constant with units of [J/K]; see Eq.15 of PJ21, [7]). The PLA considers the trajectory of a system’s energy over time, whereas the PLE is entropic in origin (being fundamentally based upon k

_{B}) and describes the spatial ‘trajectory’ (that is, the geometry) of a system and its entropy over space. Thus, the PLE can be understood to underpin the concept of geometric entropy.

_{B}. This describes the mapping of a MaxEnt geometric structure into an incompressible (and therefore conservative) entropic phase space. Together, these fundamental results in the analytical properties of geometric entropy indicate the existence of profound conservation laws operating within the entropic domain; entirely isomorphic to the more familiar energetic conservation laws based upon the PLA.

**MEPP**) have been widely discussed recently. Dewar analyses how these are consistent both with Jaynes’ MaxEnt formalism and Onsager’s linear transport theory, and elegantly shows how Maximum Entropy Production (

**MaxEP**) can quantitatively emerge from within a non-equilibrium statistical mechanical framework (2003 [23], 2005 [24]). Bruers subsequently discusses and amplifies Dewar’s treatment (2007 [25]) with a ‘partial steady state’ MaxEP analysis. The useful mini-review of Martyushev and Seleznev (2014 [26]) also explicitly describes the limitations on the MEPP, thereby demonstrating the invalidity of various “counterexamples” proposed in the literature. Zivieri and Pacini (2018 [27]) use the MEPP in a real biochemical application to living systems.

**Exertion**as a path integral of the entropic Lagrangian. Action is in the energy-and-time or momentum-and-length domains (with units of Planck’s constant), where exertion is in the entropic-momentum and hyperbolic space domain (with units of Boltzmann’s constant). Moreover, just as there is a Principle of Least Action so there is also (an exactly isomorphic) Principle of Least Exertion.

## 2. Hyperbolic Space in QGT

_{3}direction) we use a hyperbolic spacetime, with a space-like dimension q = R ln(x/R), where x is a normal (Euclidean) space measure and R is a (Euclidean) normalising metric which can often be associated with a radius of the structure under consideration; and with the entropic Lagrangian L

_{S}(q, q’, x

_{3}) equations based upon the hyperbolic spacetime dimensions q, and their spatial derivatives q’ with respect to the x

_{3}axis. The associated entropic Hamiltonian H

_{S}(q, p, x

_{3}) is defined (PJ19 [1] Appendix B), as is conventional, using the entropic momentum p. Parker and Jeynes (PJ19 [1], Appendix B: Introduction) note that “in our analysis, the entropic Lagrangian is defined in hyperbolic 3-space q, and its variation is performed with respect to the Euclidean x

_{3}spatial parameter (thus, q’ ≡ ∂q/∂x

_{3}). This is in contrast to the conventional energetic approach, where the kinematic Lagrangian is defined in Euclidean 3-space x, and varied according to the time parameter t”. Thus, the kinematic (“action”) representation has time-like qualities whereas the entropic (“exertion”) representation is more space-like.

_{0}(≡ct) and x

_{3}are conjugate quantities in the Pauli algebra (see PJ19 [1] Appendix A), so that x

_{3}could also be termed ‘geometric time’, in contrast to the conventional (kinematic) time t.

**p**,

**x**} are the conjugate variables (for example, “momentum” and “position”), and t is time (where $\dot{\mathit{x}}\equiv \partial \mathit{x}/\partial t$). But the entropic Hamiltonian is defined for a specific geometry with a preferred axis x

_{3}, and with the entropic conjugate variables {

**p**,

**q**} (“entropic momentum” and “hyperbolic position”) and the special (preferred) axis x

_{3}replacing time. Entropy is space-like where energy is more time-like.

_{3}: thus, the hyperbolic velocity q’ is dimensionless). In QGT the time dimension does not appear explicitly (but it is implicit in the 2nd Law). In addition, in a separate treatment PJ21 [7] show how the canonical relations also define an entropic phase space with properties that obey Liouville’s theorem and associated expressions based on the Poisson bracket.

_{S}/q’ (PJ19 Eq.9b), but when q’ is apparently greater than unity (that is, equivalent to the phase hyperbolic velocity ${q}_{\phi}^{\prime}$, see PJ21 Eq.23) the inverse case applies, and we must use ${q}_{\phi}^{\prime}\equiv p/{m}_{S}$.

_{S}is given by m

_{S}≡ iκ

_{0}k

_{B}(where i

^{2}= −1 and κ

_{0}is a parameter of the system that looks like a “wavenumber”; see PJ19 Eq.9c). That is, the entropic mass m

_{S}is an imaginary quantity, and it scales with Boltzmann’s constant k

_{B}. The parameter κ

_{0}is essentially the system’s wavenumber (or the inverse of the “holographic wavelength”: see PJ21 for a discussion of the holographic principle in this context). It is surprising that something we call “entropic mass” is mathematically an imaginary quantity, but the whole idea of “info-entropy” rests on properties of analytical continuation: that is, complex 4-space is central to this whole formalism. In fact, PJ19 (see their Eq.1b) is explicit about entropy and information being mutually Hodge duals.

_{0}) which, for black holes, is related to the Planck length (see the discussion of the Bekenstein–Hawking Equation in both PJ19 and PJ21) and for alpha particles is related to the diameter of the proton (see [34]).

## 3. The Relativistic Entropic Hamiltonian in QGT

_{S}to be the sum of the kinetic (T

_{S}) and potential (V

_{S}) entropy terms (PJ19 [1] Appendix B Eq.B.13b, p.30: note that the non-relativistic Lagrangian is given in PJ19 Eq.B.40a, and Eq.B.13b of PJ19 is explicitly evaluated in their Equation (Eq.B.40b)):

_{3}direction) is not symmetrical with the other two directions (x

_{1}and x

_{2}).

_{S}is an imaginary quantity, and the conjugate variables of the entropic Hamiltonian are the vectors in hyperbolic 3-space {p, q}, where the (dimensionless) hyperbolic velocity q’ satisfies the canonical relations. That is, T

_{S}is a function of p (and hence q’) alone, and V

_{S}is a function of q alone, as required. However, PJ19 made no distinction between the group q’ and phase ${q}_{\phi}^{\prime}$ hyperbolic velocities (see further in PJ21), where q’ ≤ 1 and ${q}_{\phi}^{\prime}$ ≥ 1, with q’ = 1/${q}_{\phi}^{\prime}$. Both hyperbolic velocities yield the same magnitude for the kinetic entropy T

_{S}, but of different sign due to its logarithmic character. Noting that p tends to be greater than m

_{S}(as calculated for the double-helix DNA forms in PJ19 and also as implied by the analysis below) we assume that the phase hyperbolic velocity is the quantity to be employed in Equation (1), so that the inverse identity for the group hyperbolic velocity is applicable in this case, ${q}_{\phi}^{\prime}=p/{m}_{S}$. Setting V

_{S}= 0 we can rewrite Equation (1) as:

_{S}to zero for Equation (2) just follows the usual simplifications used in Special Relativity. In the case considered by PJ19 (the Milky Way) this simplification is merely formal since they have proved that (in this case) certain approximations are valid, which means that “the hyperbolic accelerations [for the logarithmic double spiral] are therefore all zero indicating the effective absence of any entropic forces or any entropic potentials, V

_{S}= 0” (see PJ19 [1] Appendix B after Eq.B.34c). They go on to say that “it is clear that in hyperbolic space the entropic Hamiltonian of a logarithmic double spiral … is mathematically equivalent to that of a double helix” (for which see also their Appendix B after Eq.B.43: “In [the double-helix] case, the entropic field reduces to [a constant] term … that is … we can equivalently assume V

_{S}= 0”). Note that in the general case, the full (unapproximated) entropic Hamiltonian H

_{S}for the logarithmic double-spiral is indeed a constant of the system (see PJ19 [1] Appendix B and the last line of their Eq.B.40b).

_{S}and p, as described by the Euler–Lagrange Equations (and using Noether’s Theorem) are clearly not interchangeable. Hence, before we can make any progress, we need a credible entropic Hamiltonian that obeys the conventional rules of relativity.

^{2}/2m (where in this case p is the kinematic momentum and m is the inertial mass as usual). In Special Relativity the total energy (ignoring the potential energy terms) is given by ${E}_{0}^{2}={c}^{2}{p}^{2}+{m}^{2}{c}^{4}$ (c is the speed of light as usual), and we have:

^{2}(the rest mass energy) is a background term which plays no part in the classical Lagrangian calculations since it simply differentiates away and can be ignored. Moreover, the kinematic momentum p is incommensurate with the inertial mass m: they have different units. In kinematics, c is needed to make m commensurate with p.

_{S}= 0) let us name the unapproximated relativistic entropic Hamiltonian we seek as “p

_{0}”; the subscript “0” indicating that this is a time-like entropic momentum. Then, taking the logarithm and form in Equation (2) as suggestive, we suppose:

_{0}<< m

_{S}(isomorphic to p << mc in the kinematic case of Equation (3)) then H

_{S}≈ p

_{0}. Thus:

_{S}is imaginary):

_{0}and the entropic momentum p) are now clearly interchangeable. We have a relativistic entropic Hamiltonian! Writing out the 3-vector

**p**in its components we have ${p}^{2}={p}_{1}^{2}+{p}_{2}^{2}+{p}_{3}^{2}$, and

_{μ}where μ = {0,1,2,3}; that is, γ

_{0}is the time-like axis and γ

_{1,2,3}are the space-like axes, using the geometric (Clifford) algebra notation. Note the absence of c in Equation (8). This is because the speed of light c in the kinematic domain is isomorphic to a maximum hyperbolic velocity q’ = 1 in the entropic domain (recall that q’ is dimensionless); moreover, in the entropic domain the entropic momentum is commensurate with the entropic mass.

^{2}/2m is the non-relativistic approximation to the relativistic energy–momentum expression ${E}_{0}^{2}={c}^{2}{p}^{2}+{m}^{2}{c}^{4}$, so the entropic Hamiltonian ${H}_{S}={m}_{S}\mathrm{ln}p/{m}_{S}$ is a non-relativistic approximation to the entropic dispersion relation ${p}_{0}^{2}={p}^{2}+{m}_{S}^{2}$. It is also interesting to note that, although they are non-relativistic approximations, both H and H

_{S}still represent conserved quantities (as according to Noether’s theorem) in their respective Hamiltonian–Lagrangian equations of state; although the full relativistic conservation laws require that it is E

_{0}and p

_{0}that are conserved in their respective kinematic and entropic domains.

_{S}is designated as equivalent to p

_{0}since p is an entropic momentum (denoted as time-like by the subscript “0”). The quantity p

_{0}could therefore be considered as an ‘entropic energy’, although it is not yet entirely clear what such an idea implies, physically (except that it must be positive-definite). At present we restrict ourselves to developing the formalism and leave the interpretation of the isomorphism between the kinematic and the entropic to future work.

## 4. Relationship to Onsager’s Differo-Integral

_{n}is a heat flux or heat flow term (in the three space directions), T is the temperature, and V is the volume. The function $\varphi \left(J,J\right)$ (and its volume integration Φ) is known as the “dissipation function”, and according to [16] is interpreted as a “potential” function for the “mutual interaction of frictional forces”, where such forces are dissipative (entropy producing). Thus, $\varphi $ is positive definite (in accordance with the 2nd Law). Onsager employs the variational principle to demonstrate that the quantity $\dot{S}+{\dot{S}}^{*}-\mathsf{\Phi}\left(J,J\right)$ is a maximum for any system, such that one can write:

_{S}(PJ19 [1], Table 1):

_{3}direction. Thus, we first need to invoke a suitable cross-sectional area A, to make Equations (10) and (11) mutually commensurate. A is constant with respect to γ

_{3}since in hyperbolic space the logarithmic double-spiral behaves as a double-helix (see the context of Equation (2)).

_{3}is across a cross-sectional area A in the γ

_{3}direction, and for our entropic geometries of interest (in this case the ‘cylindrical’ geometry of the double helix) we assume that J

_{1}= J

_{2}= 0. The double helix geometry also implies ∂/∂x

_{1}= ∂/∂x

_{2}= 0. In addition, Onsager’s equation (Equation (10)) is intrinsically based upon temporal derivatives of entropy (and energy), which are not present in Equation (11).

_{0}in place of the original (non-relativistic) entropic Hamiltonian H

_{S}:

_{0}is time-independent; that is, we assume that the entropic momenta (both time-like as well as space-like) of the stable spatial geometries of interest do not change over time (that is, $\mathsf{\delta}\dot{S}=0$). This is true for our maximum entropy systems, and the same is true in conventional kinematics (without dissipation and without the presence of potential fields) where the total energy E

_{0}is a constant of the system. We have also assumed that the velocity quantity ∂x

_{3}/∂t = c is simply the speed of light, which is also a relativistically invariant universal constant.

_{0}is equivalent to the entropy production $\dot{S}$ when made commensurate by the normalising constant c. This is important for the physical interpretation of p

_{0}, thus:

^{n}is the set of basis vectors describing the three spatial coordinates of Minkowski 4-space—see PJ19 Eq.1), which is the key expression linking Onsager’s equation (Equation (9)) to our geometric entropy analysis. We substitute in Equations (12) and (14):

_{3}, and where we also use the relation for the entropic momentum terms (PJ19 Eq.9b): ${p}_{n}={m}_{s}{(\partial {q}_{n}/\partial {x}_{3})}^{-1}={m}_{S}/{q}_{n}^{\prime}$.

_{1}–x

_{2}directions) temperature term T into Equation (15). We also explicitly expand the entropic mass term m

_{S}(m

_{S}≡ iκk

_{B}), also forcing it to be positive-definite as is required for a dissipation function term. We also take advantage of the (Fourier) identity relationship $\mathrm{d}/\mathrm{d}{x}_{3}\equiv \mathrm{i}\kappa $ (PJ19 Eq.15) so that we can write:

_{B}T quantity is clearly an energy term, with the resulting quantity $\partial /\partial t\left({k}_{\mathrm{B}}T/A\right)$ therefore representing an energy flux term. This allows us to identify the following equivalent relations between our entropic geometry and Onsager’s entropy equation:

_{1}= J

_{2}= 0, we find that our relativistic entropic Hamiltonian formalism is therefore exactly equivalent to Onsager’s variational approach. It is worth noting that the dissipation functions in Equations (9) and (14) are closely identified with the ‘entropic mass’ m

_{S}term that we have previously defined; that is, the term $\varphi $ can be understood to be the entropic mass-density flow, whereas Φ is the volume-integrated entropic mass-density flow. Being mass-like in character, it is also clear that all of m

_{S}, $\varphi $ and Φ must therefore also always be positive-definite.

_{0}(the time-like entropic momentum which could be considered analogous to an ‘entropic energy’), which is equivalent to the entropic Hamiltonian, and which we now find is also directly proportional to the entropy production $\dot{S}$ according to Equation (13) and is therefore also positive-definite in accord with the 2nd Law. We noted above that energy is itself a positive-definite quantity: its entropic isomorph p

_{0}is also similarly constrained through the application of the 2nd Law—this is an intriguing relationship between energy’s positive-definite nature and the 2nd Law which we expect future work to illuminate.

_{q}is assumed to be in the appropriate hyperbolic space.

## 5. Conservation of Entropy Production

_{S}is related to the entropic Hamiltonian via the Legendre transformation (PJ19, Eq.11) L

_{S}= 3m

_{S}− p

_{0}.

_{0}is also conserved along with the entropic mass m

_{S}—using the same mathematical and physical reasoning that makes energy, kinematic momentum and inertial mass conserved quantities in Special Relativity. The fact that the entropy production is simply the product of the entropic Hamiltonian with the speed of light c (a universal constant) means that the entropy production $\dot{S}$ ≡ cp

_{0}must have equivalent properties to the entropic Hamiltonian, and consequently should also be a conserved (constant) feature of any entropic system under consideration (provided the usual conditions hold, such as no external potential fields).

## 6. The Entropy Production of an Idealised Spiral Galaxy

**CMB**) temperature is about 3 K: photons at least must flow into the black hole.

_{S}of the galactic structure) that is the dominant part of the entropy production.

_{BH}) where the radius R of the local structure is given by R = r

_{BH}:

_{BH}is given by:

_{BH}, which is given by:

_{BH}= 4.3 × 10

^{6}M

_{⨀}) the Hawking radiation of the central supermassive black hole of the Milky Way is equivalent to a (negative) entropy production of 3.3 × 10

^{−28}J/K·s (Equation (28)), and the entropic momentum component of the entropy production of the idealised Milky Way galaxy is similar: 3.2 × 10

^{−25}J/K·s (Equation (27)).

_{S}of the galactic structure. The entropic mass at the black hole’s Schwarzschild radius is given by (ref. [1]):

_{P}is the Planck length given by:

_{P}= 1.616 × 10

^{−35}m, and c = 3 × 10

^{8}m/s, we obtain the numerical value for the entropy production (dissipation function) of the Milky Way: 1.6 × 10

^{21}J/K·s. That is, the entropic mass component of the entropy production of the idealised Milky Way galaxy is 46 orders of magnitude larger than the entropic momentum component (3.2 × 10

^{−25}J/K·s, Equation (27); comparable to the Hawking radiation, Equation (28)).

## 7. Discussion

_{S}(Equations (16) and (19)). Indeed, it is also noteworthy that in the context of minimum entropy production Ilya Prigogine suggests that “… [entropy] production expresses a kind of ‘inertial’ property of nonequilibrium systems” [39], thus alluding to the fact that there is a ‘mass-like’ aspect to entropy production, which we see here being expressed by the entropic mass m

_{S}. Indeed, Equation (6) indicates that the two commensurate components (the entropic momentum p, and the entropic mass m

_{S}) might intrinsically exhibit different signs and therefore physical behaviours. On the one hand, this is because Equation (6) admits both positive and negative (conjugate) solutions for the entropic mass component m

_{S}; while on the other hand m

_{S}is explicitly imaginary with respect to p. Thus, this indicates an intrinsically different origin and nature to these entropic phenomena: respectively, more time-like (m

_{S}) and more space-like (p) in their origin and behaviours. From this perspective, it is perhaps reasonable to assign the time-like (cm

_{S}) entropy production term to the “dissipation function” Φ, and the space-like (cp) entropy production term to $\dot{S}$ of Equations (9) and (20)—where of course the signs of these terms must be consistent with the frame of reference: either that of the surrounding galaxy or that of the black hole (beyond the event horizon). That is, either side of the hyperbolic boundary, entropy production terms must change sign.

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Parker, M.C.; Jeynes, C.
A Relativistic Entropic Hamiltonian–Lagrangian Approach to the Entropy Production of Spiral Galaxies in Hyperbolic Spacetime. *Universe* **2021**, *7*, 325.
https://doi.org/10.3390/universe7090325

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Parker MC, Jeynes C.
A Relativistic Entropic Hamiltonian–Lagrangian Approach to the Entropy Production of Spiral Galaxies in Hyperbolic Spacetime. *Universe*. 2021; 7(9):325.
https://doi.org/10.3390/universe7090325

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2021. "A Relativistic Entropic Hamiltonian–Lagrangian Approach to the Entropy Production of Spiral Galaxies in Hyperbolic Spacetime" *Universe* 7, no. 9: 325.
https://doi.org/10.3390/universe7090325