The Time Interval Only Set, Part 2

1.1. Introduction

The time interval description finds several arguments that do not apply to the one moment time description, which is the usual one including in present physics. The definition and arguments for the time interval description in contrast to the Newtonian tangent approach, are defined in and originate from (Hollestelle, 2020). These arguments have been studied in earlier publications, while intervals themselves have been subject of philosophy earlier. Some of the results in this paper are time interval description only results, this is indicated within the text.

Within the time interval description one can disregard the infinitesimal limits of the tangent method of Newton. However, within Hamiltonian time independent events, results for both the time interval description and the one moment time description are the same.

A main concept applied in part 1 is the time interval description Noether charge, derived for a star-source emission cloud interpreted as one collective, to be a unit for the simultaneous wave emission energy measure (Hollestelle, 2024), (Noether, 1971). The description of the wave emission energy for the emission propagation surface applies the closed sphere-like surface energy concept originating from an earlier paper (Hollestelle, 2017).

The Main Question

A question remaining from part 1, when wave propagation surfaces are to be exemplary to describe overall physical phenomena, what is to be exemplary for such surfaces. The example in comment B is the main result of this paper, for this question. This example is the complete version to the ‘wrinkled’ wave propagation surface proposed in (Hollestelle, 2021, 2022). Two decisive theorems, propagation theorem 1 and 2, are included in par. 1.7, while their proof is included in comment B. This paragraph includes the introduction of a time interval gravitation constant.

The Outline of This Paper Is as Follows

Within three comments at the end of the paper, are included the following subjects, comment A: zero or non zero mass for complementarity wave particles, comment B: an example approximation for the wave emission propagation surface, comment C: The method to approach the main question, surface measure preserving specific Lorentz transformations.

Asymmetry

Not considering other arguments, asymmetry for time interval ∆t relates to asymmetry within time interval equilibrium. One moment time average < t >||∆t = t0 equal to ‘multiplication unit’||t is defined to correspond with time interval ∆t0 = ‘multiplication unit’||∆t, and from ‘multiplication zero’ units correspondences can be found for any set units and set ‘zero’ units and a relation for ∆t and time interval Noether charges and structure constants, (Hollestelle, 2024). In this way asymmetry, non zero symmetry, within the time interval description is related to several of the results in this paper. Comment 1. Equilibrium can be proposed to relate to invariance of some parameters where other and different parameters are variable, following (Arnold, 1989). In this way time interval equilibrium can be included in and become part of Curie’s principle which refers to the occurence of asymmetry in cause and effect, (Curie, 1894).

Curie’s principle relates asymmetry in one domain, the latter, to asymmetry in another domain, the former. This paper, both part 1 and part 2, concerns time intervals and the subject is finite asymmetric time intervals, which implies a time dependent Hamiltonian.

Wave Emission

Internal interaction, par. 5. and comment A, seems to be similarly relevant for acquiring gravitational mass from zero mass for the wave particle. Zero mass photons and wave emission by time development propagating away from the source, do not acquire mass by internal interaction. A related, albeit different, argument is from (Hollestelle, 2018). In this reference’s discription, due to internal interaction, outgoing photons, starting with zero mass, do not acquire mass and the Higgs mechanism does not apply to these photons. The opposite process, ingoing or free falling wave particle, the zero mass photon or non zero mass wave particle, can acquire mass from internal interaction, without the Higgs mechanism.

Within par. 3.8 several theorems are gathered that consider the time interval set and star source emission cloud collectives within the time interval description. These theorems, theorem 3b and 3c in particular, claim the addition property is valid for time interval quantity sets. The addition property being valid for the star source wave emission cloud, is crucial to regarding this addition property universal when applied to any collective cosmology.

Non Zero Equilibrium, and Acquired Energy

The time interval wave emission energy, comment A, was confirmed to be valid and equal to wave emission described with qm fields, (Sakurai, 1978), (Hollestelle, 2021). Defined is time interval non zero equilibrium, par. 4. and 5, different from time interval zero equilibrium such that where wave emission energy remains confirmed by measurements, gravitation energy can be different from it’s equivalence relation in comment A, or otherwise the acquired extra energy related to the non zero equilibrium situation is independent and not to be interpreted to be an addition for gravitation energy or wave emission energy.

1.2. Current Parameters, Object Rulers

The introduction and application of current parameters in this paper is new for the time interval set. Current parameters are introduced to be similar to an object ruler to describe a stationary object occurring in space-time. Measuring implies counting current parameter bias or scale for the object. Current parameters describe an object that is not reduced to a one moment time coordinate, with one value, or space coordinates, the object is at a place, meaning it has a non zero dimensional domain in 3+1-dim. space time. They resemble an invariant ‘displacement’ field in field theory.

Comment 2. Current parameters, different from space time coordinates, are a special type of property indications. One can define properties according to their symmetries. One symmetry is whether property values can be negative or positive or both. For space time the properties are the coordinates within the one moment time description. However within the time interval description there are differences, e. g. time intervals depend on counting time in two ways, even though the time interval set is 1-dim. Time intervals, e. g. [tb, ta], apply one moment time parameters tb < 0 and ta > 0, where 0. t0 = ‘multiplication zero’||t from the one moment time parameter set. Current parameters, interpreted to be object ruler, can not have such domains, rather 0. t0 = ‘multiplication zero’||t^^ is the set lower bound, the domain being chosen with this property. This is due to their definition depending on counting. In this way temperature T is an object ruler, even while not directly related to other object rulers, or current parameter t^^, comment B, and mass similarly is an object ruler.

Definition 1. Any object ruler is defined with its domain being a one sign only set. Counting starts from ‘addition zero’, when calibrated. One counts by successive addition. A recent discussion of numbers related to counting is found in (Murawski, Bedürftig, 2010).

Current parameters are not equilibrium time development dependent one moment time parameters, e. g. the usual Newtonian coordinates or Lagrange equilibrium generalized coordinates, one moment time description space like parameter q and one moment time momentum p from Hamiltonian equations, with usual one moment time derivatives d/dt*[p] = -1. ∂/∂q*[U(q)], U acquired, “potential”, energy, and when assuming mass m invariant, (Goldstein, 1950), (Arnold, 1989). Current parameters are introduced in comment B, to describe 2-dim. surface parts in 3-dim. space.

A surface part implies a one moment time place in space and parametrization describes the 2-dim. domain.

The concept of parametrization with object rulers, i. e. current parameters, seems independent of the time development and metric for the overall cosmology, however due to calibration connects to it. For this necessarily the emission wave propagation surface relevant event time interval ∆t should somehow be related to the calibration event, which is unlikely, since these different events are not simultaneous necessarily. With independent current parameters, i. e. current parameters that do not follow equilibrium time development, calibration measurement event properties should possibly be ignored.

This is not to be inconsistent with the traditional measurement description, regarded are two different measurements, the object ruler measurement, and the object equilibrium time development measurement. Most of comment B is within this assumption, except par. B2. Another question is whether a cosmology described within physics should be measurement related at all, however this is part of philosophy of science.

Dimensional Domain Definition and the Fundamental Theorem for Polynomials

The definition of current parameters in this paper follows what is termed dimensional domain definition in comment B, par. B12(A3) and in comment B, par. B5 and definition 5. This definition implies domain t^^ is a multiplication of n different 1-dim. product spaces, in this case t^^ is equal to multiplication t^^ = tv. tw with n = 2, while the resulting t^^ domain is divided in domain parts for t^^ finite and t^^ infinite.

Within comment B derived is, an addition to the fundamental theorem on polynomial equations in algebra, which is also discussed in par. 2.6 and par. B12(A7). Usually this theorem is introduced mentioning the number of solutions to these equations to be at least one or less than or equal to n for a polynomial equation of degree n, (Hocking, Young, 1961), (Arnold, 1989).

Comment 3. With any nth degree polynomial equation P(n)(t^^) = 0 for t^^, apply the m dimensional domain definition, introduced in comment B, par. B12(A3), with m = 2 and t^^ = tv. tw, with special attention to t^^ with respect to infinities. This can be extrapolated to any dimension m, e. g. to include (m – 1)-dimensional tv and 1-dimensional tw, with t^^ = tv(m – 1). tw(1), for any nth order polynomial P(n)(t^^) = 0 similarly implying existence of nth order polynomials P(n)(tv) = 0 and P(n)(tw) = 0 and different solutions can be found by applying the transformations TU and I related to the surface parts ∆Pi and ∆Pu discussed in comment B, the example propagation surfaces. It depends on the parameters, related to the construction of the ∆Pi and ∆Pu, that can constrain the dimensional domain definition. Where the combination TU*I equals the identity, the single tranformations TU and I do not, and thus, to the mentioned at most n solutions, add new solutions for the same polynomial equations.

1.3. Spherical Symmetric Wave Propagation Surface

Within the time interval only description events and properties depend on a specific relevant event time interval ∆t, with the properties A[∆t1, ∆t] = ∆t1 and A[∆t, ∆t1] = ∆t1 for any time interval ∆t1, including ∆t1 = ∆t within the time interval only set, with A meaning time interval only set addition and ∆t = ‘addition zero’||∆t, according to the addition properties. Similarly M[∆t1, ∆t] = M[∆t, ∆t1] = ∆t1, with the same ∆t = ‘multiplication unit’||∆t. Relevant time interval

∆t relates to the specific description of spherical symmetric wave propagation, and to a possibly related measurement event time interval, a description assumed exemplary for any event within the time interval only description. The time interval only description approach is intended to be universal, the discussion starting with spherical symmetric star source emission propagation is considered exemplary for this. Not assumed is parallel wave propagation, however emission is assumed to have a source and a time development limit surface, i. e. surface ∆A, time interval ∆t dependent.

Wave propagation implies complementary wave particles of zero or non zero mass with group-velocity c(∆t). Finite time interval ∆t is indicator for measurement events and interaction events. The wave propagation spherical symmetric surface within the time interval description, where propagation surface energy implies simultaneous emission energy during ∆t, was discussed earlier in (Hollestelle, 2021).

Star-source emission, interpreted with wave propagation spherical symmetric surfaces, is different from parallel wave propagation. The space-like surface measure mA, not meaning the usual line element dependent metric, for wave propagation surfaces A(∆t), where ∆t indicates the relevant event time interval, is invariant with change of ∆t, the variable for equilibrium time development within the time interval only description, due to overall energy conservation.

From quantum mechanical e.m. field theory within the usual one moment time description, for instance (Sakurai, 1978), (De Wit, Smith, 1986), the time interval only description version of qm concepts only the following are included; zero or non zero mass wave particle complementarity including momentum p = h. k, the linear energy frequency relation E = h. ν, with the usual indications, and the operator ‘working to the right’ concept.

Spherical symmetric surfaces and thermodynamic properties are discussed starting from the surface concept in (Hollestelle, 2017). It’s surface includes a second, lower, layer, such that surface interaction within the second layer, decides on its properties. Within the time interval description and applying the sphere-like surface concept, one finds the time interval qm version for e.m. energy and equivalence with gravitation energy, (Hollestelle, 2021).

To assume wave propagation surfaces does not imply complementary particle paths. The discussion of wave propagation starting from the concept of paths in qm is not the subject of this paper.

The wave emission energy equals emission propagation surface energy Es related to surface A(∆t) and is equal to its metric surface measure mA. Specific Lorentz transformations TS can be defined that leave metric 2-dim. surface measures including mA invariant, rather than metric 1-dim. measures, invariant with traditional Lorentz transformations TL. TS transformations respect energy conservation during time interval equilibrium time development as it should be.

The Three Dimensional Simultaneous Propagation Volume

The time interval version for overall 3-dim. simultaneous wave emission propagation surface Ac is indicated with ∆Ac such that its measure is m∆Ac = d x m∆A, including surface normal measure d estimated equal to m∆q, considering simultaneous wave emission only. Surface normal estimate d = m∆q(∆t), introduced in (Hollestelle, 2020, 2021), equals m∆t, and depends on the ‘simply measurable’ property for ∆t, par. 1.4.

1.4. ‘Simply Measurable’ Relevant Event time Interval ∆t and Specific Lorentz Transformations TS

Time interval simultaneous emission, when measurement event dependent with time interval ∆t, is defined due to wave propagation, assuming finite group-velocity, and emitted and ‘on the way’ during relevant event time interval ∆t. Within this definition, ∆t is termed ‘simply measurable’, meaning the time interval measure m∆t can be measured within time interval ∆t itself, a property derived to be always valid in (Hollestelle, 2021, 2024). The property ‘simply measurable’ could be introduced to define why a relevant event time interval is relevant.

A reciprocal pair of one moment time commutation quantities cn and cn’ is defined: t. cn’ = cn. t and 1/t. cn = cn’. 1/t, corresponding with time interval only quantities cn(∆t) and cn’(∆t). The specific Lorentz transformation, TS(∆t) = ∆t’ is consistent with cn’(∆t) = cn(∆t’), and implies invariance for wave propagation surface energy ∆Es, which is as should be when no interaction occurs, and implies invariance for measure m∆A proportional with m∆Es for wave propagation surface ∆A, meaning m∆A’(∆t’) = m∆A(∆t), the defining property for TS in (Hollestelle, 2021), meaning TS is different from the usual Lorentz transformation TL. In terms of one moment time coordinates infinitesimal transformations TS equal linear equilibrium transformations and the metric distance measure remains preserved together with the concept of covariance, similar with the usual TL. The complete non-infintesimal TS is a equilibrium transformation quadratic in the change of one moment time description 3+1 dim. space-time equilibrium coordinates q(t) and t, (Hollestelle, 2021). Within the time interval only discription non-infinitesimals TS always are linear, due to time interval properties similar to M[∆t1, ∆t1] = ∆t1, while still preserving surface measure m∆A.

1.5. The Time Interval Only Description, Time Interval Quantities and Correspondence

Emission wave propagation energy ∆Es = #n. h. ν, with variable number #n for a multiple wave function, can depend on one moment time parameter t, variable during ∆t, for instance due to external interaction, while wave energy ∆Ew = h. ν remains invariant, a qm relation verified by e. g. photo-electric event measurements, (Millikan, 1916). The assumption of the wave function number #n implies countable properties, similar to qm properties of wave groups, whereas wave propagation time development assumes a unity of simultaneous waves with certain frequencies ν, where ν is a countable property.

A measurement related interaction, e. g. wave function collapse, includes a change for wave emission energy from non- localizable to localizable, where the time interval only description, rather than the one moment time description, can give meaning to this difference (Hollestelle, 2020, 2021) and in comment B, par. B7, discussing ‘wave-group propagation collapse’, meaning multiple wave function collapse, and the simultaneous property.

Correspondence

Time interval quantities ∆A1(∆t) relate to one moment time quantities A1(t), and correspondence, indicated with ~, is the term for relations ∆A1 ~ A1. In some cases averaging A1(t), i. e. averaging of A1 during ∆t with variable t, can define

∆A1 ~ < A1(t) >||∆t. m∆t or ∆A1 = I*||∆t [A1(t)]. M[∆ti, ∆t], however this is not the regular definition for time interval quantity correspondence. One can write time interval integration I*||∆t with ∆A1 = ꭍ||∆t dt [A1(t)]. M[∆ti, ∆t].

The definition of correspondence, from part 1 (Hollestelle, 2024), is dependent on units and zero’s from both time interval and one moment time sets. Starting with average < t >||∆t = t0, the one moment time description ‘multiplication unit’||t, where t0 ~ ∆t0 = ∆U and M[∆U, ∆t1] = M[∆t1, ∆U] = ∆t1. There is ‘multiplication zero’||∆t = ∆U0, with M[∆t1, ∆U0] = A[∆U0, ∆t1] = ∆U0. For finite relevant event time interval ∆t, ∆U ≠ ∆U0.

Correspondence from units and zero’s and, in different cases, averaging for quantities, is introduced in (Hollestelle, 2024), and depends on time interval only equilibrium MVT, par. 2.1, together with time interval multiplication M and addition A introduced in (Hollestelle, 2020, 2024). Discussion par. 7 gives an overview of set units and zero’s.

External Interaction

Assuming external interaction during time interval ∆t defined with [tb, ta], and collapse for most of the complete wave function within time interval ∆t, one derives for the wave propagation surface energy Es(ta) = #n. Es(tb) ~ (#n)^2. ∆Ew, with a multiple #n wave function with emission energy ∆Es = #n. h. ν, (Hollestelle, 2020).

With external interaction and wave function collapse, energy ∆Es is not an invariant time interval quantity. One can write

∆Es as a function of #n, for t during ∆t, however #n = #n(t) time dependent during ∆t, ∆Es(t) in this case does not belong to the time interval only set. For the original discrete #n with #n(tb) >> #n(ta) >> 1, an approximation of #n with continuous function n(t) can be assumed valid, while Es(t) = (n)^(t/ t0), with example values ta = 2. t0 and tb = 1. t0, leaving out t0 = ‘multiplication unit’||t within the one moment time description. ∆ti is multiplication inverse for ∆t with M[∆ti, ∆t] = ∆U. It follows:

∆Es(∆t) = < Es(t) >||∆t. ∆t = I*||∆t [Es(t)]. M[∆ti, ∆t] = I*||∆t [ (n)^(t/t0) ]. ∆t = [ 1/(t/t0+1). (n)^(t/t0+1) ] ||∆n. ∆t = ( 1/3. (n)^3 - 1/2. (n)^2 ). ∆t.

For ta/t0 >> tb/t0 >> 1, and with #n(tb) >> 1, there is ∆Es is quadratic in n, where n = #n, and linear in ν.

This differs from internal interaction in terms of the time interval only set, with ∆Es(ta) = ∆Es(tb), meaning time interval energy remains invariant with ∆Es = #n. h. ν, linear in ν.

A time interval only version of a qm measurement of wave function properties is described, while the stationary property remains valid, in par. 1.6 and for interaction discussing zero temperature, par. 4.2. With energy relation Es ~ ∆Es = #n. h. ν, this was discussed, among other results, to find energy density ρ = α/β. ν^2. T, with constants α and β, including frequency ν, and temperature T, in (Hollestelle, 2021), which originates from Planck’s relation, for lower frequencies. Recall frequency ν and number #n are quantities that can only be measurable during a certain finite measurement event time interval, an argument of its own to consider, finite, time intervals instead of one moment time tangents.

1.6. Theorem on Averages

The relations from par. 1.5 quadratic in #n and quadratic in ν seem compatible when considering the assumption of countable properties, one for the simultaneous wave emission propagation surface, i. e. for counting simultaneous space like occurrences, and for counting in time, i. e. counting simultaneous time like occurrences within a time interval. This is a unexpected measurement dependent approach and result, which can be applied to support the theorem on averages, an approach not mentioned in e. g. (Arnold, 1989).

The theorem on averages implies equivalence of integration over space and time within space time phase space and is usually introduced within textbooks on mathematical mechanics, e. g. (Arnold, 1989).

Space time considerations, e. g. Lorentz transformations, seem to be overall related to the development of this theorem, that appears similar to equilibrium with generalized coordinates. The time interval version for the theorem on averages has value for the discussion of many equilibrium related concepts, e. g. measurements in physics, of qm phase space, entropy change, and Noether charge.

Another argument related to the theorem on averages is the time interval description itself. Simultaneous space like and time like parameters are related by assuming time interval description equilibrium, independent from ‘one value’ one moment time coordinates and one moment time description equilibrium, such that a ‘discrete’ concept for space time is not necessary. A discussion of qm and discontinuity can be found in (Beller, 1999). Newton’s approach applies the center of weight concept which seems useful for the tangent method in ordinary one moment time 3+1 space time to reduce multiple coordinates to one value space time coordinates (Newton, 1729). The one moment time description and time interval only description are introduced together in earlier papers, correspondence to relate these descriptions is introduced in (Hollestelle, 2024).

Stationary Events

The term ‘stationary’ implies a ‘stationary event property’ where a possible interaction variance for the energy is considered to be much less, infinitesimal with respect to the energy itself, (Merzbacher, 1970). This can be applied to a measurement event including wave function reduction, introducing two different time intervals, one to describe on-going wave propagation invariance and one to describe interaction and the measurement event. A universal stationary event, also without interaction or measurement, implies invariant entropy.

The description of a stationary wave emission event and a wave function collapse measurement event for wave emission propagation within the time interval description originates from (Hollestelle, 2021). The assumption of the occurrence of stationary wave emission seems prerogative to the development of a time interval description dependent on a relevant event time interval that is ‘simply measurable’ and on simultaneity, defined in par. 1.4. This approach has some similarity with the description of qm and experiments to be divided in two parts, with a preparation procedure and a resultants collecting procedure, which is part of the operational type of interpretation of qm, relating to the EPR argument, (Roos, 1978), (Omnès, 1994), (Beller, 1999).

Descriptions of measurement events can include a description of physical phenomena and measurement items both, which is a traditional approach within physics, including for qm, e. g. the prepared system for a specific qm measurement event, (Roos, 1978), (Van Kampen, 1988). The discussion how experiment and phenomenon influence each other within or outside physics, is not the subject of this paper, neither is meant the discussion of sketches of laboratory situations which have been studied elsewhere.

Stationary Events and Time (In-)Dependence

Time independence is a problematic concept since a measurement not easily can detect time independence within a finite time interval. When a measurement implies counting occurrences, the ‘no occurrence’ result can realistically be counted only within a finite measurement event time interval, since infinite time intervals for measurements are not feasible. The change from time independence to time dependence seems contradictory to the stationary event property.

Any event, including equilibrium time development and Hamiltonian time dependence, to change due to time development into a state with H time independence, implies reaching an asymptotic stationary event. This is consistent with the interpretation of Van Kampen related to reaching a macroscopic situation from a qm situation.

1.7. Preliminary Results

The Wave Emission Propagation Surface

Within comment B the construction of the example includes time interval description surface ∆Pu, a union of surface parts ∆Pi. ∆Pu is constructed to be a covering of emission propagation surface ∆A. The ∆ indicates the time interval description for ∆Pu and ∆Pi and other quantities. The surfaces ∆Pi are described with two current parameters w and t^^ with 2-dim. domain dw x dt, which is with a combination transformation TU*I transformed to the regular square and successively by transformation TQ to a regular sphere ∆Q.

Propagation theorem 1. Any transformation TU*I is equal to the identity transformation. It’s proof is included in the main part of comment B.

This property implies TU*I provides extra degrees of freedom by iteration of its application: transformations TU*I can be applied iteratively and an extra degree of freedom can be added to the description. The specific example surface and

∆Pu are introduced originally in comment B.

Time interval surface part ∆Pi and overall surface measure m∆Pu remain invariant with any transformation TU*I, the m indicating measure. For any ∆Pu, and ∆Pu’ = TU*I*[∆Pu], there is m∆Pu’ = m∆Pu, and covering least requirement RS, necessary for ∆Pu to be a covering for wave propagation surface ∆A, remains fulfilled, meaning m∆Pu’ ≥ m∆A remains valid due to m∆Pu ≥ m∆A. When surface measure is preserved this means any TU*I equals some specific Lorentz transformation TS. Surface measure preserving TS, different from the usual Lorentz transformations TL, are defined in (Hollestelle, 2021, 2024).

Propagation theorem 2. Any transformation TU*I, i. e. equal to the identity transformation, can be related to the time reversal transformation squared and to a specific Lorentz transformation TS. This is discussed in par. 2, and in comment B, par. B12(A4).

Number of degrees of freedom. Due to the independent parameters introduced with construction of ∆Pu, the number D of degrees of freedom is D = 4, par. 2.6 and comment B. Apart from these parameters there are three ways to introduce extra degrees of freedom: when introducing current parameters due to iteration and the dimensional domain definition. The mentioned interation for TU*I applies to introduce an extra degree of freedom. The reduce the domains of the current parameters applies similarly to an extra degree of freedom, comment B, par. B3 and B8. In this way D can be considered to increase, however not to D infinite, which would imply a transition to a Newtonian description. When introducing (a-)symmetry with parameter N, i. e. the number of surface parts ∆Pi for the example propagation surface

∆Pu, and when introducing lower bound temperature ∆T_zero, both can be applied to provide an estimate for the actual number of degrees of freedom D, par. 4.3.

Gravitation Constant, Describing Energy with Wave Emission Energy or Gravitation Energy

Derived is within the time interval only description, the gravitation constant is |g| = m∆t = m∆NC, while ∆NC is the time interval Noether charge. Simultaneous qm wave emission energy and gravitation energy are interpreted to be equivalent and represent the same energy with two different interaction constants.

In comment A, eq. A1, introduced is wave propagation surface energy ∆E = ∆Eg = M[∆qi, M[m1, m2] ], equal to ∆E = ∆Es = M[c(∆t)i, M[∆m1, ∆m2] ] with c(∆t) = [∆q, ∆ti], the former energy allows for the interpretation of ∆m2 to be a gravitational mass, the latter allows to consider a finite group-velocity c(∆t) to be of wave emission origin. ∆m1 is the zero or non zero complementarity wave particle mass.

A transformation from time interval energy interpretation ∆Es to ∆Eg, due to time interval quantity definition c(∆t) = M[∆q, ∆ti], implies multiplication with a quantity, indicated with ∆Em and equal to ∆t, and since ∆t = ∆NC, with ∆Em equal to the time interval Noether charge. There is M[∆Em, ∆Es] = ∆Eg and M[∆Emi, ∆Eg] = ∆Es, with ∆Emi time interval multiplication inverse for ∆Em, which indicates ∆Em is a time interval energy due to the multiplication closure theorem derived in part 1.

The time interval description wave emission constant ∆v, with zero or non zero mass wave particle, equals ∆t, and the time interval description gravitation constant ∆g = ‘multiplication unit’||∆t from the above expressions, and trivially ∆g = M[∆Em, ∆v]. Set units including the time interval unit ‘multiplication unit’||∆t are defined and discussed in par. 7.

From multiplication factor ∆Em = ∆t it follows 2-dim. expressions [∆Egi] = [h+, ∆E], and [∆Esi] = [c(∆t), ∆E] with i = 1, 2, can be included into a 4-dim. quantity ∆E4(s, g) = [[∆Esi], [∆Egi]], the i = 1, 2, instead of indication i meaning multiplication inverse. In this way an internal transformation is enough to define the two energies ∆Es and ∆Eg to be the transformed of the other. ∆Es and ∆Eg both equal ∆E = M[∆m1, ∆m2] where h+ and c(∆t) equal ‘multiplication unit’||∆t = ∆t, derived in comment A. This allows the interchange of Es and Eg within average brackets, when the average is over the same set ∆E_set. This results in the following relations for the one moment time and time interval energy parameter set.

< Es >||∆E_set = < Eg >||∆E_set

< ∆Es >||∆t_set = < ∆Eg >||∆q_set

From 4-dim. time interval quantity ∆E4(s, g) = [∆Esi] x [∆Egi] and resultant ∆E’4(s, g) = Oij* [∆E4j] due to transition or interaction, which follow from, e. g. particle decay description, with the usual conventions for matrices ε and ẟ, and transition tensor ∆Z4 with scalar parameters L1 to L3, (Veltman, 1974). Within the time interval description one finds, from eq. 1, (indices ijkl, writing upper and lower indices all with lower indices, for reading purpose).

∆Z4ij = M[∆E’4i, Oij*[∆E4j]] = < L >. A[ m[∆Esi], m[∆Egi] ]. ẟij = < L > M[∆E4, ∆E4] =

= M[M[∆L, ∆g], M[∆E4, ∆E4]]

With interaction tensor Oij = L1. ẟij + L2. εik. ẟkl. εlj + L3. εik. εkj, there is Oij = < L >. ẟij, considering the usual symmetry properties of the matrices ε and ẟ. When Oij is the 4 x 4 matrix element for the transition with invariant trace Tr[Oij] = 4. < L >, there is ∆L = < L >||∆t. ∆t. The time interval multiplication factor ∆Em, the ‘multiplication unit’||∆t, is equal to ∆v = ∆t when ∆Em = M[∆Eg, ∆Esi] and ∆Em = M[∆g, ∆vi], the wave emission constant ∆v being to ∆Es what ∆g is to ∆Eg.

The transition parameter ∆Z4 does not have to be necessarily invariant depending on properties before and after transition, e. g. zero or non zero mass for the wave particle, meaning change of ∆m1 or ∆m2.

Another way to describe the transformation follows from the multiplication factor ∆Em = ∆t and the two energy expressions ∆Es and ∆Eg. The q(t < tb) = qb implies ∆E = ∆Es(qb) before the event, the t > ta means ∆E = ∆Eg(qa(t > ta)) after the relevant event transition time interval ∆t = [tb, ta], both equal to ∆E = ‘multiplication unit’||∆E_set = ∆t when ∆E transforms completely from ∆Es to ∆Eg. This is a transition, different from claiming equivalence, from ∆Es to ∆Eg. One finds time interval equations ∆Eg = < ∆Es >||∆t_set and ∆Es = < ∆Eg >||∆q_set different from the one moment time equations with Es and Eg. Gravitation energy ∆Eg is space like coordinates ∆q only dependent. There is ∆Eg(t < tb) = ∆Es(t > ta) = ‘multiplication zero’||∆E_set. Indication ∆t_set indicates the complete time interval only set, etc.

There is ∆q = a2. ∆t and D*||∆t*[∆q] = M[a2. D||∆t*[∆t], D||∆t*[a2]] = a2. ∆t, from par. 4.2, and from (Hollestelle, 2024). Conservation of energy is achieved for scalar a2 = ‘multiplication unit’||a_set, par. 4.2, par. 6, and ∆q = ∆t, and then a2 equals the multiplication factor ∆Emi = ∆Em from ∆Eg = M[∆Em, ∆Es]. Time interval wave emission energy

∆Es(∆t) depends on quantities h+ or h-, defined in comment A, that is ∆p and ∆q, and frequency ν, where gravitation energy ∆Eg(∆q) depends on M[m1, m2].

Gravitation Constant, Wave Emission Properties and the Wave Emission Propagation Surface

Due to the construction of example surface ∆Pu with N ≥ 1, it follows overall equation |∆g| = |m∆Pu – 1/N| = |m∆t - 1/N|, which includes a relation for m∆t and m∆q, and includes the specific case N = 1.

In this construction ∆Pi = M[∆q, ∆w], where ∆q and ∆w are averages to surface parts ∆Pi for the current parameters ∆t^^ and ∆w^^, and due to covering requirement m∆Pu ≥ m∆Q for ∆Pu, multiplication M[∆N, ∆w] is an invariant, and m∆Pi = |M[|∆qi|, ∆Ni]| and |∆q| = |A[ |∆Ni|, m∆Pi ].

One applies the relation m∆Pi = m∆Pu, included in preliminary propagation theorem 3, derived in par. 2.7. Within the time interval description, there is multiplication (M) equals addition (A), due to theorem 6, par. 3.8. When preliminary propagation theorem 3 seems contradictory recall the ∆Pi are open, disjoint surface parts of closed continuous example surface ∆Pu.

Comment 4. The gravitation constant value |∆g| equals |‘multiplication unit’||∆t | = |∆t| = |∆NC|, i. e. |∆g| is equal to the measure of the time interval Noether charge which is interesting in its own right. Necessarily m∆q = m∆t to maintain conservation of energy, par. 6.

Comment 5. Within the time interval only description, |g| = |∆t| is a direct argument for the indivisibility or unit property for gravitation constant |g|, since distributions do not exist for ∆t, comment 23.

Wave particle energy ∆Ew = M[D*||∆q [∆t], ∆m2] = h. ν when ∆m1 = ‘multiplication unit’||∆t, valid for zero mass wave particles, photons, comment A. It follows |∆q| = |M[∆m2, ∆m2i]| = |‘multiplication unit’||∆t | = |M[∆m2, ∆NCi] |= |∆m2| when N >> 1, which is an approximation for the time interval space like measure |∆q| = |∆qi|, i indicates multiplication inverse. There is relation M[∆m2i, ∆m2] = ∆NC, which can be interpreted to be universal, with ∆NC time interval Noether charge and ∆m2 a time interval gravitational mass, inferred to be similar to a type of renormalised source mass. The overall relation is, from comment B, par B12(A6).

|∆Ew| = |h. ν| ≤ |h+|. |c(∆t)i|. |∆q| = |h+|. |c(∆t)|. |m∆A/N – 1/(Z+1)|

This relation implies the specific value |h. ν| = |‘addition zero’||∆t| for certain m∆A = |multiplication unit’||∆t |, and is a possible solution when ( |1/N – 1/(Z+1)) = |‘addition zero’||∆t |.

From this it is derived |∆g| = |1/N – 1/(Z+1)|. |∆NC|, where N and Z+1 are construction parameters, comment B. A continuous domain for |∆g| can occur when N or Z both >> 1 and approximately infinite, however due to construction requirements, parameter Z has to remain finite. This does not prevent any value for |∆g| can be reached within any sufficiently small margin. The time interval measure |∆g| = |‘multiplication zero’||∆t| is reached for |(1/N – 1/(Z+1))|. |∆NC| = |‘multiplication zero’||∆t| when |1/N – 1/(Z + 1)| = |‘addition zero’||t | regardless the value of |∆NC| including |’multiplication zero’||∆t|. Within the one moment time description, which applies to N and Z, ‘addition zero’||t = ‘multiplication zero’||t. This time interval measure |∆g| does not necessarily imply the one moment time measure |g| equals |‘multiplication zero’||t| = 0. t0 since preliminary propagation theorem 3 does not apply in this case.

Non Zero and Zero Equilibrium

Introduced in par. 3 and 4 is the concept of non zero equilibrium. Non zero equilibrium can be transformed into zero equilibrium, the usual equilibrium, by introducing an un-known and yet un-interpreted time interval energy ∆dU, while the corresponding one moment time energy dU(t) is variable with one moment time parameter t, or it is an acquired energy, e. g. gravitation energy, acquired since variable with one moment time 3-dim. space like coordinate q. The un- interpreted energy can be an addition to gravitation energy ∆Eg, thereby altering equivalence with wave emission energy ∆Es, or ∆dU is a new energy independent of wave emission energy or gravitation energy, which is a different occurence. Both ways, however different, can imply an acquired mass.

Zero Temperature

The time interval description is applied to star-source wave emission clouds, for which zero and non-zero temperature, par. 3 and par. 4, is defined for the wave emission collective. Zero temperature within qm, usually is defined with ground state energy. Applying the definition from averages for time interval quantities, an estimate for a lower bound wave emission temperature and for a lower bound collective radius, for the wave emission collective, is derived in par. 4.

2.1. What Is Time Development? Time Development and the Functions of Addition and Multiplication

To describe change within the usual one moment time description, derivatives depend on addition and the method of tangents (Newton, 1729), (Goldstein, 1950), (Arnold, 1989). One moment time Lagrange equilibrium including the principle of least action implies application of one moment time space-like coordinate variations, with invariant transit space-like and time-like end-coordinates however with variations for the space-like path, where a variation means an infinitesimal change, an infinitesimal addition, to the existing coordinate value, (De Wit, Smith, 1986). These variations are not to be confused with the interval concept like time intervals in this paper. There exist several different approaches in the literature concerning coordinate variations.

Continuous and invariant time development, within the one moment time description meaning at any ‘time-lapse’ a similar ‘time-lapse’ is added, was introduced within the time interval description, (Hollestelle, 2020, 2024).

Due to the existence of unresolved difficulties with the one moment time description and the method of tangents, the time interval description is introduced to be an alternative. In a number of papers this alternative is discussed including how it relates to quantum mechanic field theory (qm) and to general relativity (GR). The following properties for time development applying the time interval only set, independent of the one moment time description, are derived in (Hollestelle, 2022, 2024). Time interval description equilibrium depends on the ‘mean speed theorem’ including the graphical method originating from medieval researchers, and in this paper termed ‘mean velocity theorem’ MVT, (Hollestelle, 2020).

To describe time development it is argued (Hollestelle, 2024), time intervals do not exist outside themselves, do not add time from outside to themselves and remain only with themselves. For the time interval only set including only finite measure intervals this means time interval addition includes the property: any finite time interval ‘addition’ the same finite time interval, remains the same finite time interval.

1 A[∆t1, ∆t1] = ∆t1, for any time interval ∆t1

∆t1 can be any time interval within the time interval only set. This confirms the interpretation of time interval only set addition with domain addition where addition of two identical domains results in the same domain. Introduced is the set addition unit ‘addition zero’||∆t = ∆t0 such that A[∆t0, ∆t1] = A[∆t1, ∆t0] = ∆t1. For sets with finite, i.e. asymmetric, time intervals, implying H time dependent, there is ∆t0 = ∆t. It is derived, (Hollestelle, 2024), that within the time interval only set, addition A is similar to multiplication M, since the generators of the set are the set elements themselves.

2.2. Time Intervals

Consistency for any time interval depends on relation eq. 2, for finite time intervals ∆t1 = [tb, ta], introduced in (Hollestelle, 2020). This relation does not define a causality relation for tb and ta. The one moment time parameters t = tb and t = ta belong to the one moment time description where time development implies tb << ta. Coordinate q is the one moment time description 3-dim. space-like coordinate q(t), part of the usual 3+1 dim. covariant space time coordinate q4 = (q, ict), similarly in par. 2.4. Due to equilibriumq(t) is dependent on one moment time parameter t, while invariant scaling quantity n_ is defined such that scalar product (n_. q) remains dimensionless, where the value | n_| can be chosen as a scaling parameter for 3-dim. one moment time space like space time coordinates q(t).

2 tb = -1. (1 – (n_. q(ta))^2)^(-1). ta

For any quantity a the indication a^(-1) indicates the multiplication inverse ai = 1/a. When ∆t1=[tb, ta] is defined from eq. 2, this implies it relates to space-like interval ∆q1 = [q(tb), q(ta)]. The one moment time parameters tb = tb(ta) and ta are defined starting from certain ‘elements’ [n] and [i] that each describe time measurements from counting time, introduced in (Hollestelle, 2020), a definition which agrees exactly with eq. 2, even though of different character. Another way tb and ta can be defined is through specific Lorentz transformations TS, with the following transformation property: instead of metric line measures, 2-dim. surface measures remain invariant, due to which TS agrees with eq. 2. From the definition of parameters being related to eq. 2, one can not conclude tb and ta belong to a discrete set, the one moment time parameter set remains continuous.

Even while eq. 2 can provide a change from one moment time description to time interval description, proceeded is directly from the time interval only set properties derived in (Hollestelle, 2024). These include correspondences, indication (~) that relate the one moment time description and time interval only description, from set units and zero’s, e. g. time interval addition zero, ‘addition zero’||∆t = ∆t, with A[∆t1, ∆t] = A[∆t, ∆t1] = ∆t1. There is correspondence ∆t ~ 0.t0 where 0.t0 + t1 = t1+ 0.t0 = t1. Within the time interval only description, MVT equilibrium assumes Hamiltonian ∆H and Lagrangian ∆L are related by the same Legendre transformation as H and L in the one moment time Lagrange equilibrium description, (Hollestelle, 2021).

2.3. Wave Propagation, Parameters, Time Interval Only Description

Assuming wave propagation group-velocity c(∆t) = M [∆q, ∆ti] invariant during ∆t, c(∆t) is a time interval quantity linear in ∆t, with M indicateing time interval multiplication. Space-like time interval quantity ∆q relates to relevant event time interval ∆t, including measurement events for deriving c(∆t).

Theorem 1. To resolve the multiplication dimension problem (MDP). Any space-like time interval quantity ∆A1 can be interpreted to be a multiplication of ∆t with a quantity of non-zero dimension, which is allowed due to the multiplication closure theorem, discussed in theorem 1b and theorem 1c, par. 3.8, and derived in (Hollestelle, 2024).

Comment 6. For instance space-like time interval ∆q1 = M[∆q1, ∆t] = a1. ∆t, since time interval only set ‘multiplication unit’||

∆t = ∆t0 = ∆t, and for ∆q1 = ∆q there is a1 = c(∆t) considering ∆t finite. Assumed is the first associative, series, multiplication property (series), M[M[∆t1, ∆t2], ∆t3]] = M[∆t1, M[∆t2, ∆t3]]. The problem of theorem 1, (MDP), can be reduced to the dimension problem t1. t0 = t0. t1 = t1 for any one moment time parameter t1, where t0 = ‘multiplication unit’||t, and ∆t0 ~ t0, (Hollestelle, 2020, 2024).

Star source wave propagation is assumed to imply a limit spherical symmetric surface A, meaning the wave propagation space like limit for waves emitted and ‘on the way’ simultaneous, from the star source within time interval ∆t, depending on ∆t and c(∆t), discussed in par. 2.4. For star source emission there remains a wave composition for emission symmetric in all directions, where qm relations energy E = h. ν and complementarity relation wave momentum p = h. k remain valid for zero and non-zero mass wave particles within the time interval description, with h constant of Planck, and k wave ‘number’. The relation wave momentum h. k with frequency ν is discussed in (Hollestelle, 2021), and in relation to the theorem on averages, in par. 1.5.

It is argued the non-localizable overall wave propagation surface energy ∆Es = #n. h. ν equals ∆NC = #n. h. ν, corresponds with the invariant one moment time Noether charge, where ∆Es equals the relevant event time interval, i. e.

∆t = ‘multiplication unit’||∆t, being non-divisible and defining simultaneous wave propagation.

Time interval quantity ∆NC refers to simultaneous wave function energy #n. ∆Ew = #n. h. ν, instead of to wave propagation surface energy ∆Ew for one wave-particle. This applies for zero mass photons, and multiple addition of photon mass remains ‘addition zero’||∆t. The time interval wave propagation energy approach for wave-particles with zero or non zero mass is discussed in comment A.

For the cosmological universe described in par. 3 in terms of the star source emission cloud, ∆Es is the relevant phenomenon in physics, a cosmological unit, related to collective simultaneous wave propagation and energy for ∆t. Time interval energy density ∆Ew = h. ν remains un-known unless through some measurement event or wave function collapse event, occuring with e. g. the photo-electric effect.

2.4. Simultaneous Events, Time Reversal, Kramer’s Equation

The property simultaneous can be defined for time interval ∆t’ related measurement events, depending on ∆t’ being ‘timely’, which is assured by ∆t itself being ‘timely’, while ∆t’ occurs during ∆t, (Hollestelle, 2020). The term ‘timely’ implies: a time interval ∆t’ is timely when it is ‘simply measurable’, meaning with a time interval measurement event result |∆t’| from one measurement event within ∆t. This resembles Einstein’s introduction of simultaneity and ‘locality’ for space-time discussed within the one moment time description, (Einstein, 1952). Simultaneity within the time interval only description differs from one moment time description space-time simultaneity, due to finite time intervals not being symmetric in one moment time parameters tb and ta, due to eq. 2, and not time translation or time reversal invariant.

Comment 7. Simultaneous measurement events for any ∆t’ and ∆t’’ are possible when ∆t’ and ∆t’’ both ‘timely’ within relevant event time interval ∆t. However ∆t itself always is ‘timely’, which is assured since ∆t is always ‘simply measurable’, i. e. measurable within one measurement event assuming finite wave group-velocity, properties discussed in (Hollestelle, 2020, 2024).

Comment 8. The possibility of free choice for one moment time q0, the space-like 3-dim. coordinate with only zero values, which (usually) is regarded equal to one moment time space-like ‘addition zero’||q for the existing cosmological universe, is an assumption related to translation invariance within infinite 3-dim. one moment time description space and to similar invariance for measurement results within physics. Translation invariance within the time interval description can be lost due to the necessity to introduce Lorentz transformation variants TS however not when only infinitesimal transformations are considered, (Hollestelle, 2020). Invariance is discussed, from the perspective of philosophy of science, in e. g. (French, Krause, 2006).

Comment 8a. Time reversal within both relativistic and non-relativistic qm being necessarily nonlinear, is discussed in (Jauch, Rohrlich, 1980). They derive e.m. field equations from the Lagrangian and the action principle, to decide on time reversal properties in terms of Lorentz transformations. Their approach is not an approach within GR.

Within the qm version of the time interval description for equilibrium MVT and time development, specific Lorentz transformations TS can be applied, different from the usual Lorentz transformations TL, preserving surface measure rather than the metric line element. Within the one moment time description of 4-dim. space time, traditional time reversal is a TL Lorentz transformation.

Time interval only description time reversal can be defined directly from parameter relation eq. 2 or from time elements [n] and [i], par. 2.2, which is problematic with respect to time reversal, or from specific Lorentz transformations TS.

Comment 8b. One moment time average ‘multiplication unit’||t = < t >||∆t = t0 ~ ∆t0 = ∆t is assumed asymmetric for finite ∆t, directly related to the asymmetry of the time interval only set for finite ∆t, due to which the result of time reversal of a time interval is not a time interval, and does not belong to the time interval only set, (Hollestelle, 2020, 2024).

Comment 8c. Time development, since time is 1-dim. where however space is 3-dim., does not allow time translation symmetry or time reversal symmetry, for any time interval ∆t, due to any of the definitions mentioned in par. 2.2. For H time dependent, equivalent to time interval ∆t being finite and asymmetric, time development from MVT equilibrium is 1- dim. and does not allow the canonical property for addition or multiplication, a property that could relate the time interval only set with time reversal symmetry. One moment time description time reversal transformation τ is meant to imply: τ*[t] = t’ = -1. t, for time reversal of one moment time parameter t to t’.

When through some process H reaches time independence, similarly time interval ∆t reaches the infinite and symmetric part of the time interval set. Time development within the time interval description with MVT equilibrium means linear and canonical infinitesimal time development equations and does not allow time reversal symmetry, (Hollestelle, 2024). Within general relativity (GR), equilibrium time development equations are quadratic, i.e. nonlinear, (Jauch, Rohrlich, 1980). Recall the specific Lorentz transformations TS are indeed quadratic in one moment time coordinates, however linear when infinitesimal.

The time interval description version of the qm e.m. wave equation is derived without the minus sign, which is as it should be, since within the time interval description multiplication inverse ∆ti = ∆t. The time interval version describes qm e.m. and gravitational interaction to be equivalent, and relies on specific time interval only set properties including property eq. 1, (Hollestelle, 2024).

Jauch and Rohrlich, following the usual qm e.m. field theory, find linear time reversal is incompatible with the e.m. field, i.e., has no invariance properties for the field variables. The qm time reversal transformation TL = τ, due to invariance for the traditional e.m. field equations, is nonlinear and follows Kramers equation, τ*[q4]. τ*[q4] = q4. q4, for one moment time description 3+1-dim. space time coordinate q4 = (q, ic. t). Meant is time reversal transformation τ, which differs from the time reversal operator. The indication . implies inner or scalar product. Within the time interval only description, Kramers equation is M[τ*[∆t1], τ*[∆t1]] = M[∆t1, ∆t1]. Solution τ*[∆t1] = -1. ∆t1 for any ∆t1, scalar multiplication with -1, does not belong to the time interval only set, due to eq. 2. All other solutions τ have the same problem, however, well defined is τ*[∆t1] = ∆t1, implying τ equals the, linear, identity transformation. This time interval description τ does not agree, following correspondence, with the traditional one moment time solution τ*[q4] = τ*[(q, ic. t)] = q4’ = (q, -1.ic. t). This is not the identity transformation, however when squared is a solution that equals identity, τ*τ*[q4] = q4 and τ*[q4]. τ*[q4] = q4. q4, (Jauch, Rohrlich, 1980).

There is D*||t*D*||t*[NC] = -1. NC, with one moment time derivative D*||t, the wave equation for the one moment time Noether charge NC. Indication * implies transformation or operator ‘working to the right’, not is meant Hermitian conjugate *. With e.m. field theory with field variables A and A’, one finds τ*τ*[A] = A and τ*τ*[A’] = -1. A’, within the one moment time description. Due to these properties a ‘super selection’ rule decomposes A, A’ overall phase space and is complete and indefinite (Jauch, Rohrlich, 1980).

Within the time interval description, the wave equation is D*||∆t*D*||∆t*[∆NC] = ∆NC, (Hollestelle, 2024). Within this description the difference for the two equivalent energies ∆Es and ∆Eg depends on the group-velocity c(∆t) = M[∆q, ∆ti], where ∆ti = ∆t, while one moment time description (dt)i = -1. dt. The wave equation can be written including I*||∆t instead of one of the D*||∆t, within the time interval only description only, due to which τ equals identity.

Comment 9. Time reversal τ depends on the multiplication inverse. When field theory allows negative energy values, within the usual one moment time description, this is not immediately clear for gravitation and Eg, where negative energy is problematic. This can be argued to depend on the difference c(∆t) and ∆q within the equations for ∆Es and ∆Eg, comment A, meaning c(∆t) = M[∆q, ∆t] invariance differs from ∆q non invariance due to ∆t dependence. The Dirac equation is linear in D*||∆t instead of quadratic and the identity argument is not valid in this case.

The ‘super selection’ rule, that does apply within the one moment time description, does not apply within the time interval description, where τ equals the identity at all phase space. This means in phase space variables are allowed to become non- selective, and this saves 4-dim. space time covariance. Inferred is from the wave equation, time reversal operator τ equals derivative D*||t and D*||∆t resp. within the one moment time and time interval description.

2.5. Wave Propagation Surfaces

An example for a spherical symmetric wave propagation surface, approximation to a realistic wave propagation surface, star source emission dependent, is discussed in comment B. Assumed is time development of wave propagation surfaces depends on relevant event time interval ∆t and wave group-velocity c(∆t). Starting from stationary state properties, a time interval description for wave propagation was introduced in (Hollestelle, 2021), where it was derived surface energy is equivalent to wave emission energy following usual qm field theory, although including zero and non zero mass wave complementary particles, and is similarly equivalent to gravitation energy.

The time interval version, with surface energy for wave propagation surfaces ∆A(∆t), requires, measures on surface ∆A do not apply the linear metric line element, even when the metric surface measure of a regular sphere usually relates to the metric radius measure squared. To describe wave emission energy with the time interval description version of qm, introduced is necessarily the requirement 2-dim. space like surface measure m∆A(∆t) to remain invariant for variable ∆t, to maintain energy conservation within time development, (Hollestelle, 2021, 2024).

2.6. Example Approximation Surface for Spherical Symmetric wave Propagation Surface ∆A Parametrization

For wave propagation surface ∆A, assumed is approximation surface ∆Pu, a union of a variable number of surface parts ∆Pi, i = 1 to N. The ∆Pi = ∆Pi(N, w, tl^^) are open, disjoint and dense parts of ∆Pu, union ∆Pu is an indecomposable medium and covering for ∆A, and ∆Pi and ∆Pu are 2-dim. time interval description space like surfaces in 3-dim. space. The ∆Pi are described with two parameters w^^ and t^^. Surfaces ∆Pi tend to be ‘linear’-like, due to a decreasing width-parameter domain measure mdw that indicates the measure of ∆Pi in one dimension, d1, however this does not imply the metric line-element for ∆Pi itself. More specific, parameters for the 2-dim. surface parts ∆Pi are: parameter w(t^^), domain limit for w^^ domain dw = [w0, w] with w assumed finite, and parameter t^^ with domain dt = [t0^^, tlim^^] with tlim^^ assumed infinite, where w = w( t^^ ) and t^^ parametrisation for ∆Pi, which could be the other way around. The parameters w and t^^ are current parameters, introduced in par. 1.5 and in the next paragraph on time development.

Time Development

For the ‘object’, one moment time description ‘surface part Pi’, parameters w(t^^) and t^^ are assumed not to depend on one moment time parameter t, not to depend on equilibrium time development, they are similar to object rulers. A ‘synchronous’ change of dw and dt implies w = w(t^^), and implies the existence of surface measure mPi preserving, specific Lorentz type of transformations TS for parameters TS*[w] and TS*[t^^], and TS*[dw] and TS*[dt], to continue to describe TS*[Pi] while mPi = mTS*[Pi] remains invariant.

Any measurement for a one moment time space like coordinate qi = (w, t^^) existing on Pi can be described with one moment time coordinate t when assuming t^^ = t^^(t), dependent on one moment time equilibrium time development for surface part Pi(t) with parameter domain dw(t) and dt^^(t). This is similar to the usual one moment time development for 3+1 dim. space time coordinates q(t) and t for the center of weight or any space like coordinate of an object, in this case qi and Pi, with Newtonian coordinates q(t) and t in space time. The qi = qi(t) follows one moment time development together with complete object Pi(t). Recall not always there will be t^^ = t^^(t), since t^^ is a current parameter, when Pi including qi remains time independent, with invariant space coordinates.

Construction of an Example

The variable number N allows for a new construction II for approximation surface Pu, introduced in Comment B. It is an example approximation for a wave propagation surface, starting from the 2-dim surface parts Pi and union Pu and a transformation applied to transform Pu to the regular sphere Q while mPu ≥ mA is the covering least requirement for Pu to be a covering for A. The regular sphere is the most simple spherical symmetric wave propagation surface due to star source emission, i. e. symmetric in all directions, and this remains even though the introduction of the qm photon wave function concept c.q. wave particle equivalence and the photo-electric effect explained with photons, (Beller, 1999).

Construction II resembles well-known construction I for which N = N0 invariant, and applies a theorem from topology for indecomposable media to argue the resultant is a covering for the wave propagation surface, (Hocking, Young, 1961). It is argued for construction II the overall number of degrees of freedom is one larger than for construction I. With construction II one can find an example approximation for spherical symmetric wave propagation surface A, depending on the identity transformation in two dim. An addition to the fundamental theorem on polynomial equations, mentioned in par. 1.2, relates to this identity transformation, and in comment B, par. B12(A7), theorem B8.

Interpretation A: Symmetry Transformation with N Changing, Interpretation B: Zero Temperature

With ∆t and A(∆t) unchanged, parameter N ≥ 1, when variable, is undecided when interdependent with domain dw and dt, meaning parameter N supports an extra degree of freedom. Least requirement for union surface Pu to be a covering for regular sphere Q(∆t) and wave propagation surface A(∆t) is, surface measure mPu ≥ mA. Time interval ∆t is left out of indications, for simplicity, in the following.

Two interpretations are mentioned in (Hollestelle, 2024). Interpretation A: when energy does not change with N, a change of N implies a symmetry transformation. For any number N ≥ 1, interdependent with dw, the union Pu remains a covering for wave propagation surface A. Time interval ∆t can not be divided into a simultaneous distribution, like is possible for space-like interval ∆q, comment 23. This symmetry possibly is not a complete symmetry, which could imply the introduction of some un-specified energy is necessary, this is discussed in par. 3, starting from the concept of closure of non zero equilibrium introducing acquired energy.

Interpretation B: implied is zero temperature for wave emission, discussed in par. 4. A star source emission cloud within the time interval description including overall and zero temperature is applied in par. 3, and par. 4.

2.7. Results: Symmetry Transformations Symmetry Transformations

Several relations are derived for time interval propagation surface ∆A and example surface ∆Pu and parts ∆Pi within construction II. These can be summarised, starting from interpretation (A).

When ∆Pu is the union of all ∆Pi, the ∆Pi can be united in pairs. It follows the union of all pairs ends with only the last pair or the last ∆Pi = ∆PN. Due to time interval property eq. 1, all unions end with the last ∆PN, or equivalently end with the first ∆P1 ‘moving’ through the union collection, cancelling the other ∆Pi. Union or addition is indicated with Σ for one moment time quantities, or A for addition for time interval quantities. For this to be valid within the one moment time description the following equations need to be valid:

Σi < wi >||∆t^^ = N

Pu = Σ[P(i=1), P(i=2)] ~ ∆Pu = A[< P(i=1) >||∆t^^, < P(i=2) >||∆t^^] = A[∆P(i=1), ∆P(i=2)]

Correspondence including one moment time quantities P(i=1) and P(i=2) follows from the expression including one moment time quantity Pi averages for time interval quantities, ∆Pi = < Pi >||∆t^^. ∆t^^ = I*||∆t^^. dt^^[Pi]. M[∆t^^i, ∆t^^], with ∆t^^i multiplication inverse for ∆t^^, from par. 1.5. One derives the former equation from this latter one, when to maintain invariance of measure m∆Pi = m(∆w(∆t^^) x ∆t^^) = mdw. mdt to define all possible domain d∆w(∆t^^) and d∆t^^, for ∆Pi. Similarly N is variable with w and tlim^^ when invariance of measure m∆Pu is regarded, such that m∆Pu = N. m∆w. mdt remains invariant.

From these equations one finds time interval description ∆Pu = ∆PN for any N, e. g. ∆Pu = ∆P2 for N =2, where ∆P1 is added to all succesive ∆Pi, untill ∆Pu = A[∆PN, A[∆P1, ∆PN-1]] = A[∆PN, ∆P1] = ∆P1 due to A[∆P1, ∆Pi] = ∆P1 for all i ≤ N.

One finds m∆Pi = mdw. mdt, where time interval current parameter ∆t^^ = η(i). ∆q(i, dt) with ∆q(i, dt) related to surface part Pi, and the same is valid for mdw = m(< w >||∆t^^. ∆t^^), i. e. the measure for domain dw. One finds time interval derivative D*||∆t[∆t^^] = η(i). D*||∆t[∆q(dt)] = η(i), with ∆q(dt) = ∆t for any surface part Pi for A(∆t), for all N, and due to this result, there is time interval measure m∆Pu = N. mPi ≈ N. mdw. mdt = N. mw. m∆t^^. m∆t^^ = m∆Pu. (< η(i) >)^2, which means < η(i) > = 1 with m∆t^^ = m∆t = 1 is a valid assumption, and m∆Pu = N. w. (m∆t^^)^2 and m∆Pi = w. (m∆t^^)^2 is of the order mA and mQ, and independent of N. These arguments result in the following preliminary propagation theorem 3.

Preliminary propagation theorem 3. The surfaces ∆Pu and ∆P(i) for all i ≤ N fulfill the covering least requirement RS for surface A(∆t). Surface measure m∆P(i) = m∆Pu ≥ mQ, required is invariance of surface measure m∆P(i=1) during addition of successive ∆P(i). The result is valid within the time interval only description.

Starting from the 2-dim. surfaces ∆Pi = ∆q x ∆w there is,

m∆Pi = M[m∆qi(∆t^^), m(< w(t^^) >||∆t^^. ∆t^^)] = A[ < qi(t^^) >||∆t^^. mdt, mdw. mdt ], the addition A occurs due to theorem 6, where N. m∆w. mdt is invariant, and mM[∆t^^, ∆t^^] = m∆t^^ = mdt.

For N = 1, there is trivially mPu = mPi, and it follows m∆q = m∆t = mM[∆m2, ∆m2i], and m∆q = m∆Pu – 1 = m(‘addition zero’||∆t). For all N the result is M[∆m2, ∆m2i] = 1/N. (m∆Pu - 1), where theorem 1b and 1c, par. 3.8, is applied.

The ∆t^^ average for q(Pi)(t^^) dependent on current parameter t^^ can be transformed into a ∆t average for q(Pi)(t), dependent on one moment time parameter t, due to calibration of t^^ to one moment time parameter t, which does not have to be simultaneous with time interval ∆t, however calibration should occur during some time interval ∆t’ for current parameters to be real object ruler. This requires to interpret t^^ and w(t^^) to be one moment time equilibrium dependent parameters during ∆t’, which is discussed in par 1.2, and par. B12(A4).

2.8. Parameters and Completeness

Not necessarily, when contruction II is applied to find an approximation example for a spherical symmetric wave propagation surface, which is a fundamental phenomenon in physics, it includes a hidden symmetry or hidden parameter. The parameters needed to describe surface parts Pi and the example construction, are defined in comment B. Hidden parameters are those that are not field variables. It is not investigated whether the parameters with construction II are complete or are sufficient to create a ‘path’ narrative for qm.

The Example Construction

Construction II includes transformation from infinite 2-parameter space to the regular square and successively to the regular sphere. The former transformation includes a singularity and requires the definition of parameter r = tlim^^/tl^^. The two space like one moment time current parameters for Pi are d1 dimension parameter w with domain dw = [w0, w], and d2 dimension parameter t^^ with domain dt = [t0^^, tlim^^]. Both the limit tl^^ and tlim^^ are infinite. By changing limit tl^^ to tlim^^ for domain dt the singularity at tl^^ can be solved, therefore parameter r is introduced, however r is not undecided. In comment B included are the covering least requirement for r and the other free parameters.

There is, ∆NC is ‘multiplication unit’||∆t and equal to ‘addition zero’||∆t due to time interval theorem 1c, par. 3.8. Interchanging the ∆Pi can be the result of specific Lorentz transformations TS, that leave surface measures invariant. Interchanging the N surface parts, with N invariant, does not change the surface wave propagation energy ∆Es, however change of N implies possibly change for parameter ∆w and other parameters and for energy ∆Es.

The symmetry during change of N not necessarily implies invariance for all parameters, and can mean a change of #n due to external interaction including wave function collapse. It is not meant the number N of surface parts ∆Pi should depend on number #n of wave particles or equals the number n of star sources within the star- source emission cloud collective.

Current Parameters

For any surface parts Pi, current parameter w = w(t^^) is a t^^-quantity. One applies correspondence with averages to define time interval description quantity ∆t^^ ~ < t^^ >||Pi(dt). mPi(dt), and ∆w ~ < w >||Pi(dt). mPi(dt), following par.

1.4. However it seems not valid averaging one moment time current parameter w during time interval ∆t, since surface part Pi being a real factual surface part is assumed to remain time independent. One can write ||Pi(dt). mPi(dt) = ||dt. Mdt = ||∆t^^. m∆t^^. Instead of averaging during ∆t one can introduce averaging for domain dw = [w0, w(t^^)], i. e. averaging during ∆t^^ which itself depends on averaging with Pi for t^^ domain D1 = [t0^^, tl^^]: the two domains dw and dt are ‘synchroneous’. Assumed is t^^ remains independent of one moment time parameter t and one writes time interval description ∆t^^ ~ t^^, and the same for ∆w ~ w. Including these comments, one derives:

Σi ∆wi = Σi < wi >||∆t^^. ∆t^^ = Σi I*||∆t^^[wi]. M[∆t^^i, ∆t^^] = Σi ‘multiplication unit’||∆t^^ = ∆N ~ Σi ((mdt. mdwi). (1/mdt. mdt)) = Σi (mdt. mdwi) = N

The average for wi, wi0 = < wi >||∆t^^ or wi0 = ‘multiplication unit’||∆t^^, can be expressed in terms of parameter r, assuming mD1 = ‘multiplication unit’||t^^. When during ∆t^^ the parameter wi remains asymmetric while wi0. w differs from wi0, and wi0. wi = wi. wi0 = wi can be defined non trivially valid, one finds wi0 = ‘multiplication unit’||∆t^^, due to the definitions for units and zero’s in (Hollestelle, 2024). The applied domain for t^^ is dt = [t0^^, tlim^^] = [t0^^, r. tl^^] with r < 1 or r > 1, where dt ≠ D1, in Comment B, to assure singularity tl^^ of the transformation of current parameter domain dw x dt to the regular sphere domain dQ is resolved.

3.1. Star Source Emission Cloud

Temperature dependent equilibrium for a set of spherical symmetric surfaces can be interpreted as a concept for the star source emission cloud collective, within the one moment time and time interval approach this was discussed in (Hollestelle, 2017, 2021).

With an emission cloud from star-sources meant is a combination of n star-sources, at one moment time space-like places qi, i = 1 to n, that are grouped with a certain one moment time space-like average, group center Q(n) = < qi >||n. To one moment time description 3 + 1 dim. space time coordinate q4 = (q, ic. t), due to one moment time equilibrium time development, define with i.c. t the usual fourth coordinate, c the velocity of light, t the one moment time parameter, and i the imaginary scalar unit. One can define a space-like measure m(Q(n) - q0) = mQ(n), towards the overall wave propagation surface A, when chosen is q0 remains at A(∆t). By definition q0 = ‘addition zero’||q for the one moment time space-like parameter q set. The prescript m indicates measure. The spherical and non-spherical spacial configurations for star- or galaxy clusters have been subject matter of extensive research within astrophysics, which is within the usual one moment time description, (De Theije, e. a., 1995), (Hollestelle, 2017b).

Star-sources are emission sources, together they are assumed to one collective source for wave propagation including collective wave propagation surface A(∆t). At surface A for any ∆t measurement place qm is chosen with m(qm – Q(n)) = mQ(n), and with m(qm – q0) infinitesimally small, i. e. place qm is infinitesimal ‘close’ to place q0 and both qm and q0 continue to remain at surface A(∆t) during time development say from ∆t = ∆t’ to ∆t = ∆t’’ (Hollestelle, 2021). This is a realistic choice since star source wave propagation surface A changes from A(∆t’) to A(∆t’’), by definition reaches the measurement apparatus at qm at A(∆t) with exactly the relevant measurement event time interval ∆t changing from ∆t’ to ∆t’’.

3.2. Inter-Dependent Quantities Q(n) and Q2(n), Zero Equilibrium

For star i defined is one moment time space-like measure mqi = m(qi - q0), i.e. place qi relative to q0, and space-like coordinate Qi = qi - Q(n), i.e. qi relative to average Q(n) = < qi >||n. The one moment time average < Qi >||n = < qi - Q(n) >||n = Q(n) – < Q(n) >||n = Q(n) – A[qn, A(n-1)]. 1/n, since by definition the average Q(n) = < qi >||n equals the one moment time quantity Q2(n) = A(n). 1/n, with A(i) = q(i) + A(i-1) ~ ∆A(i) with A(i = 1) = q(i = 1). Correspondence eqs. 3 and 4 are the result, arguing < Q(n) >||n = Q(n) since Q(n) remains independent from parameter i:

3 < Qi >||n = Q(n) - < Q(n) >||n = q0 ~ ∆Q0 = ‘addition zero’||∆q

For the time interval quantities ∆qi: the time interval average is equal to the time interval addition, applying the iterative addition ∆Q2(n) = ∆A(n), in terms of time intervals, without factor 1/n due to eq. 1, and it follows eq. 4:

4 < Qi >||n = Q(n) – Q2(n) = 0. t0 ~ ∆Q(n) - ∆A(n) = ∆U0

The one moment time, left side, difference in eq. 4 corresponds with time interval ‘addition zero’||∆q. Within par. 3.8, theorem 1, and theorem 1b and 1c, it is derived units and zero’s for the ∆q interval set equal those for the ∆t time interval set, due to which ‘addition zero’||∆q = ‘addition zero||∆t = ∆U0 corresponding with ‘addition zero’||q = 0. t0.

3.3. Equivalence of Emission Wave Energy and Gravitation Energy

The time intergral of the time interval Noether charge density is an overall 3-dim space-like quantity, and one expects

∆NC to relate to the invariant overall wave propagation surface energy ∆Es, due to conservation of energy during time development of simultaneous emission.

The time interval description relates integration to addition such that Noether charge ∆NC equals the Noether charge density in value, which implies time interval cosmological ‘volume’ ∆R3 ~ R3 which then should equal a unit, ‘multiplication unit’||∆q. This depends on time interval description eq. 1. Thus, m∆R3 equals m∆q = m∆t, measure for

∆t = ‘multiplication unit’||∆t.

Comment 10. One can write the one moment time propagation surface energy with the following equations, comment A, where support for the theorem on averages is implied by the derivative expressions.

Eg = M[1/|∆q|, M[m1, m2]] = D*||∆q*[M[m1, m2]], and Es = A[1/|c(∆t)|, M[m1, m2]] = D*||∆t*[M[m1, m2]]

where the first expression refers to time interval gravitation interpretation energy ∆Eg, the second one to time interval qm emission wave interpretation energy ∆Es = #n. ∆Ew = #n. h. ν. Two related expressions are ‘apparent emission density’ ∆m2, and ‘apparent wave density’ ∆m1 = h. ν, both in terms of density, while energy measures m∆Eg = m∆Es imply two different interpretations with the same energy value. Due to the unit value of universe volume measure m∆R3 = m∆t these time interval densities equal quantities, apparent emission mass ∆m2 and apparent wave mass ∆m1 = h. ν.

Comment 11. The existence of time interval ‘addition zero’||∆q can be assumed, independent of existence for the one moment time set ‘addition zero||q, due to theorem 1. Eq. 4 interpreted with: specific one moment time average for Qi, equal to addition of two quantities Q and Q2, corresponds with time interval wave energy ∆Ew = h. ν, a result that can be derived directly from time interval description equilibrium and time interval addition properties.

The following theorem is valid due to regarding eq. 4 to be an equilibrium equation rather than a equation with zero value result.

Theorem 2. Correspondence canonical property. Due to regarding eq. 4 to define equilibrium, and applying equilibrium theorem c, which originates from (Hollestelle, 2017), the time interval quantity corresponding with an addition of one moment time quantities is the addition of the time interval quantities corresponding with these one moment time quantities.

Comment 12. For e.m. wave propagation with ‘zero’ complementary particle mass ∆m1 = ∆m0 = ‘addition zero’||∆m, it follows wave emission energy ∆Es = A[c(∆t)i, ∆m2] = A[∆qi, ∆m2], which can be interpreted to be apparent emission density ∆m2 = #n. h. ν. Wave group-velocity c(∆t) = M[∆qi, ∆m1], with ∆m1 = ‘multiplication unit’||∆m and c(∆t)i = ∆qi, can be interpreted to be apparent wave group density ∆m1 = h. ν. The time interval mass quantities ∆m0 = ‘addition zero’||∆m equals ∆m0 = ‘multiplication unit’||∆m, which is rather similar, however not necessarily, to Newton’s equivalence relation for gravitation mass from weight, and slow mass from acceleration, (Newton, 1729).

Including a non-zero mass ∆m1 one finds ∆t = M[∆m1, ∆m2] and ∆Ew is linear in ∆qi, a linear energy interaction, not the usual quadratic field interaction, and is consistent with the surface energy proposed in (Hollestelle, 2017). This surface energy concept has its own merits within the one moment time description. A linear energy resembles an object energy change during free fall, implying gravitation and kinetic energy exchange.

Recall field energy requires some type of change to be meaningfull, the field wave. Wave energy, within the wave interpretation of qm, for free waves without interaction, depends on the frequency momentum relation described by wave density functions for dispersion free, stationary events (Hollestelle, 2021). This situation can be interpreted to imply ‘zero’ temperature, par. 4. Some of these relations, e. g. internal interaction, are discussed in Comment A, and originate from (Hollestelle, 2021).

3.4. Kinetic Energy and Acquired Energy

From Newton’s second law retained is d/dt*[p] = -1. ∂/∂q*[U], with p and q generalized coordinates within the one moment time description and acquired energy U. Acquired energy U occurs due to coordinate dependence of energy, the term acquired since an object occurs within space-time. Change of one moment time space like coordinates means a change of U explaining the usual term potential or non-mechanical energy. This is the conservative system energy description to include kinetic energy T quadratic in one moment time coordinate p for a one particle situation.

Newton’s law in this form relates to the conservation of energy. However it applies the center of mass concept, which allows the description of a mass and volume combination in 3-dim. space, which is partly independent of the mass distribution, i. e. how mass is related to multiple one moment time space-time coordinates.

Comment 13. Where simultaneous emission wave energy equals ∆Es, wave particle energy ∆Ew = h. ν remains unknown untill measurement events including wave function collapse occur. This is especially clear from the equivalence of gravitation energy and e.m. field energy, which describe the same energy ∆Es. This depends on the non distribution property for ∆t, par. 5, comment 23.

Usually and in general in any one moment time description T is considered quadratic in p, however in the time interval description ∆T quadratic in ∆p implies, due to conservation of energy and due to eq. 1, conservation of time interval ‘velocity’ ∆x ~ x, with one moment time quantity x = d/dt*[q] defined from Newton’s second law, rather than conservation of momentum ∆p = M[∆m1, ∆x], including mass ∆m1 of the wave-particle, possibly a variable. This seems to imply a change of mass ∆m1 does not change time interval kinetic energy ∆T. Velocity ∆x to be a constant of the motion, does not imply ∆x invariance during transformations of coordinate systems, however during space time coordinate transformations due to time interval equilibrium time development. The relation of Lorentz transformations, including covariance conservation, with equilibrium time development, depends on the commutator relations of time intervals, defined in (Hollestelle, 2024), discussed in comment C.

One moment time acquired energy U, acquired due to external parameters, e. g. one moment time space-like coordinate parameters q, relates to a certain time interval ∆t which allows variation of these parameters.

Not is meant one moment time wave-particle coordinates within the quantum domain, the description remains within the wave rather than the particle concept, while localizable properties are expected to exist after measurement, i. e. after wave function collapse. The time interval description of wave function collapse and measurement events was introduced in (Hollestelle, 2020). Kinetic energy ∆T = ∆Es equals the wave propagation surface energy unit ‘addition unit’||∆E, while ∆Es is non localizable. Theorem 4b, par. 3.8, can be applied such that units and zero’s for the ∆E set equal units and zero’s for the ∆t set. Within the time interval only description, gravitation energy ∆Eg is an acquired energy, depending on time interval space like quantity ∆q. Within the one moment time description the invariant total energy H0 = T + U ~ ∆H0 = A[∆T, ∆U], while ∆H0 = ‘addition zero’||∆E = ’addition zero’||∆t.

3.5. Non Zero to Zero Equilibrium with a Situation Change Energy dU

Within a yet to consider situation, indication (with-*) and introducing an unknown, and not before included, acquired energy dU* implies Lagrangian L changes to L* = L – dU* and Hamiltonian H changes to H* = H + dU*. The original situation (no-*) does not include energy dU*, i. e. dU*(no-*) = ‘addition zero’||E in the (no-*) situation, while dU* not necessarily equals ‘addition zero’||E* in the (with-*) situation. This is the one moment time description.

Introducing energy dU* adiabatically can be related to a situation change within time development, however it is a thermodynamic, macroscopic, change of event (Roos, 1978), (Van Kampen, 1988). Rather meant is, fundamentally, energy dU* is regarded occuring only, emerging only, in the (with-*) situation. This allows to interprete (with-*) to describe a new energy.

Definition 2. Non zero (no-*) to zero equilibrium (with- *) with new energy dU*. The proposed, one moment time description, interpretation is, the original (no-*) situation describes energy with non zero, open, equilibrium and acquired energy dU* can be assumed to occur due to zero, closed, equilibrium in the (with-*) situation.

Comment 14. When including a possible difference for ∆Es and ∆Eg, due to considering them independent, can contribute to acquired energy. In par. 3.3. following (Hollestelle, 2021, 2024), derived was equivalence of the two ∆Es and ∆Eg interpretations of the same energy, similar to equivalence within GR of gravitation and kinematic energy. Wave propagation surface energy ∆Es and gravitation energy ∆Eg, with m∆Eg = m∆Es, are both time interval expressions derived from the wave propagation surface description, par. 3.3. When some ‘non zero’ acquired energy ∆dU* is necessary for zero equilibrium, this can imply equivalence falls short.

Interpretation A for ∆dU*

It is a possible inference to assume gravitation energy is different from ∆Eg and similarly zero or non zero mass wave particle surface energy different from ∆Es, in this way introducing a non zero equilibrium original (no-*) situation. Non zero equilibrium in this case would mean the assumption is open to arguments, to assert wave surface energy is in principle equivalent with gravitation energy and to introduce a third and new energy dU*, or to assert the energy difference ∆Eg and ∆Es implies the existence of a third energy dU*. These two different arguments are to achieve zero equilibrium for the (with-*) situation.

A third option is to accept non zero equilibrium as a possibility in reality, described by time interval asymmetry, where the asymmetry of ∆t implies < t >||∆t = t0 ~ ∆t0 = ’multiplication unit’||∆t is different from ‘multiplication zero’||∆t = 0. t0, meaning one moment time description t0 = 0.t0 is not valid, and similarly D*||∆t[∆A1] = ‘multiplication zero’||∆t = ‘addition zero’||∆t not necessarily is valid for an invariant time interval quantity ∆A1.

Change of eq. 4 (no-*) and eq. 5 (no-*) and resultant ∆U0 and ∆Es*, i. e. change to closure (with-*) with resultant ‘addition zero’||E* = ∆U0*, means including ∆dU* within situation (with-*), while for (no-*) Lagrangian L = T - U does not close eq. 4 or eq. 5 to resultant ‘addition zero’||E = ∆U0 or ‘addition zero’||E = ∆U0*. Recall in this case (no-*) implies dU* = ‘addition zero’||E = U0 and (with-*) implies dU* differs from ‘addition zero||E* = ∆U0.

The Lagrangian L is applied to define zero or non zero equilibrium since it appears in the action principle, where for equilibrium, L is related to Q(n) – Q2(n) = ‘addition zero’||E. The application of invariance of L to derive equilibrium equations is an accepted guiding principle within the 4-dim. covariant Lorentz approach, (Sakurai, 1978).

3.6. Non Zero to Zero Equilibrium Due to Including Extra Energy dU*

To reach zero equilibrium eq. 5a one can choose Q* and Q2* to be inter-dependent energy quantities while a ‘non zero’ energy dU* is introduced. It follows from par. 3.5 the originally space like quantities Q and Q2 can be interpreted to be energy quantities. There is quantity Q0* = ‘addition zero”||E*= ‘addition zero”||t* = U0*. This is a well known solution for non-zero-resultant equations like eq. 5a without dU*, for instance when introducing e.m. gauge transformations, (De Wit, Smith, 1986), a transformation meaning to include ‘addition’ -1/+1. dU*.

5a Q*(n) – Q2*(n) + dU* = Q0* ~ A[∆U0, ∆dU*] = A[∆Es*, ∆dU*] = ∆U0* = ‘addition zero’||∆E*

From time interval eq. 1, there is ∆U0 = ∆Es and ∆Es = #n. ∆Ew = ∆Ew. Within the one moment time description situation (with-*) including closure of eq. 5, there is L* = L - dU* = ‘addition zero’||E*, and due to L = T – U = T = Es, follows eq. 6 and eq. 7. With correspondence eq. 7, the time interval description resultant is: ∆L* = A[∆T, A[∆U, ∆dU*]iv ] = ‘addition zero’||∆E*. Addition inverse is indicated with iv, different from multiplication inverse indicated with i and star source parameter i = 1 to n.

6

Q(n)* – Q2(n)* = ‘addition zero’||E* = L* = L – dU*

7

Q(n)* – Q2(n)* ~ ‘addition zero’||∆E* = ∆L* = A[∆T, A[∆U, ∆dU*]iv]

Comment 15. From eq. 4 to eq. 7 one can define equilibrium in the style of the action principle depending on variations. The variables are the Qi with i = 1 to n. Proposed is the action principle, variation of space-like integration, to mean the resultant for eq. 6 and eq. 7 remains exactly the same resultant L* = L and ∆L*= ∆L due to ‘zero’ closure remaining valid with variation. The one moment time description equilibrium resultant change should cancel, for variations to this zero equation equilibrium principle, since dU = ‘addition zero’||E and L* = L.

Applied, within eq. 7, is associative property (series property), par. 2.3. Time interval description eq. 7 depends on application of theorem 1, while time interval equilibrium depends on < t >||∆t, being non zero for H time dependent and for ∆t asymmetric and finite.

The usual one moment time description applies Lagrange equilibrium, time interval only description equilibrium is defined with the MVT theorem, while for both descriptions H and L are the Legendre transform of one another, (Hollestelle, 2020). This suggests dU is not kinetic energy, rather acquired energy. Solutions differ with description when H time dependent and ∆t asymmetric and finite, where results for the one moment time and the time interval description are the same for H time independent and ∆t symmetric and infinite, (Hollestelle, 2020).

3.7. Non Zero to Zero Equilibrium with Independent Quantities Q’ and Q2’

Within the time interval description, integration and derivatives can be related to time interval multiplication and addition. The time interval Noether charge ∆NC depends on time interval derivatives, D*||∆t, this was discussed extensively in (Hollestelle, 2024). In par. 3.5 the quantities Q and Q2 were defined being inter-dependent.

Assuming time interval equilibrium and introducing independent one moment time quantities Q’, Q2’, and Q1’ indicated with an accent, the difference Q’ – Q2’ = Q1’ corresponds with ∆Es = ∆NC to be equal to ‘multiplication unit’||∆t = ∆t, due to theorem 4 and 4b, where closure, zero equilibrium, occurs for the (accent) situation, open, non zero, equilibrium for the (no accent) situation, recall the 3rd option in par. 3.5.

For eq. 4 and eq. 5a closure occurs, with inter-dependent quantities Q, Q2 and Q – Q2 = Q1 ~ ‘addition zero’||∆t, where for eq. 5b closure occurs with independent quantities Q’ – Q2’ = Q1’ ~ ‘multiplication unit’||∆t’ = ∆dU’ = ∆NC’ and from introducing ∆dU’. When ‘addition zero’||∆t = ∆t and ‘multiplication unit’||∆t’ = ∆t’ = ∆t, this ensures the correspondence Q1 ~ ‘multiplication unit’||∆t, within eq. 5b applied with all accents.

Definition 3. Non zero to zero equilibrium (by accent) by addition energy dU’.

Introduce energy dU’ with a change of energy Es to Es’ = Es + dU’ ~ ∆Es’ = A[∆Es, ∆dU’], to reach zero equilibrium with Q1 changing to Q1’ = ‘multiplication unit’||∆t. This was assumed for eq. 4 with energy Es, the zero difference Q1 is lost in eq. 5b with independence of the Q’ and Q2’, and is regained with the extra dU’, and relates to Noether charge ∆NC to ∆NC’. For this interpretation, time interval description quantities are necessary. 5b Q’(n) – Q2’(n) = Q0’ ~ ∆U0’ = A[∆U0, ∆dU’] = A[∆Es, ∆dU’]

In eq. 4, zero equilibrium with Q(n) – Q2(n) = Q0, corresponds with time interval ‘addition zero’||∆t = ∆U0, and is, in eq. 5b, interpreted with zero equilibrium for energy ∆Es’ = M[∆Es, ∆dU’] = ∆U0’.

Interpretation A for ∆dU’

Differences in physics can be described for the (with-*) and (with-’) situations, when acquired energy ∆dU* and ∆dU’ is of un-recognised origin. With situation change is not meant a physical change, rather a change of concept of description, from open equilibrium to closed equilibrium in both cases.

This paragraph includes several concepts that depend on the time interval only description. Introduced is a equilibrium type starting from a cloud of star sources, defined from the space like coordinates of individual stars in it. Then, introduced is wave propagation dependent energy in terms of Noether charges. An equilibrium can be asymmetric, similar to the relevant event time interval ∆t, where asymmetry is included within the units and zero’s of resp. the one moment time set and the time interval only set.

Comment 16. An asymmetry within equilibrium means, following Curie’s principle, there is an asymmetry in the cause, i. e. in the energies involved, (Curie, 1894). This gives the freedom, due to the time interval only description concept of interval and asymmetry, to derive properties for an energy different from the usual energies T and U.

It is well known asymmetry, interpreted to be a variation of a symmetry transformation, can imply differences for a property like mass. Curie’s principle regarding asymmetry can be adjusted to include asymmetric time intervals, i. e. non zero symmetry, this is earlier discussed in (Hollestelle, 2020) and in Comment A.

Asymmetry is applied to describe wave propagation, regarding star source emission clouds and acquired energies, par. 3, and regarding zero temperature, par. 4. Independence for ∆Q’ and ∆Q2’ can be applied to the set of star-source time interval quantities ∆qi which remain with the average < ∆qi >||n unaltered with ∆Q(n) equal to ∆Q’(n), while there is possibly a change ∆Q2(n) towards ∆Q2’(n) = ∆A’(n).

Similar to interpreting the occurence of (with-*) situation energy ∆dU*, one can argue the difference Q’ and Q2’ to become zero equilibrium to mean gravitation energy and wave propagation surface energy should be approached

independently within the (accent-’) situation, i.e. energy ∆dU’ is an un-recognised energy to give zero equivalence to ∆Eg and ∆Es.

3.8. Theorems Related to the Time Interval Only Set

The time interval space-like ∆qi, the variation of individual star-sources, is integral with wave propagation time interval ∆t and wave group velocity c(∆t), responsible for the property simultaneous, and star-source emission from one star is considered to be simultaneous within ∆t. It is assumed ‘addition zero’||∆q = ∆q0 exists and is well defined. Several theorems for the time interval set and star-source emission cloud can be derived, while partly dependent on theorems from (Hollestelle, 2024). Theorems 1 and 2 are included in par. 2.3, and par. 3.3. They are repeated here for completeness.

Theorem 1. To resolve the multiplication dimension problem (MDP). Any space-like interval ∆q can be interpreted to be a multiplication of ∆t with a quantity a1 of non-zero dimension, which is allowed due to the multiplication closure theorem, in this paper theorem 4, and derived in paper 1, (Hollestelle, 2024). This is discussed also in comment 6.

Theorem 1b. All emission cloud properties can be described in terms of time interval only set properties and vice versa, meaning these two sets are linearly dependent. This is due to the multiplication closure theorem, theorem 4.

Within the time interval only description, to introduce a derivative to a quantity ∆y starting from a derivative to time interval ∆t, theorem 1b can be applied.

There is ∆y0 = ‘addition zero’||∆y = ‘addition zero’||∆t = ∆t, applying theorem 1b and with ∆y = D*||∆t[∆z], with time interval quantity ∆z, while for any other ∆y1(∆t) a similar equation, up to some scalar constant a1, can be derived due to theorem 4: A[∆y1, ∆y0] = ∆y1 = a1. ∆t1 = a1. A[∆t1, ∆t] = a1. A[a1i. ∆y1, ∆t] = A[∆y1, ∆t], and this implies a solution is ∆y0 = ∆t, up to some uniqueness issue.

Theorem 1c. Due to theorem 1b, there is ∆Q1 = a1. ∆t and for any quantity ∆Q1, from the set of ∆Q quantities, it follows ‘addition zero’||∆Q units equal time interval set ‘addition zero’ ||∆t units.

Theorem 3. According to (Arnold, 1989), equilibrium can be proposed to relate to invariance of some parameters where other and different parameters are variable, and in this way equilibrium can be included in and become part of Curie’s principle for symmetries (Curie, 1894), this was discussed in (Hollestelle, 2020).

Theorem 3b. Within the time interval description, from eq. 1, including space-like interval quantity ∆q, any star-source emission cloud center ‘addition’ any other emission cloud center, remains invariant and equal to ‘addition zero’||∆q.

This seems however contradictory. It seems to imply a star-cloud ‘addition’ any one star-source, remains with the same, thus invariant, star-cloud center. Recall Q(n) = < qi >||n and qi are within the one moment time description. For time interval quantities ∆Q(n) = < ∆qi >||n and ∆qi, indeed there can be some quantity ‘addition’ the same quantity with resultant the original quantity, e. g. ∆Q(n) and ∆qi, in agreement with eq. 1. This means two similar. i. e. two the same, star-source emission cloud center Q1 and Q2, can be ‘added’ to each other to result in the un-changed center Q1 = Q2, according to this argument.

Theorem 3c. Within the time interval only description a union ∆Pu of several ∆Pi corresponds with excluding overlap. A union ∆Pu of the ∆Pi does not construct overlap for the ∆Pi. The Pi are always disjoint.

For time interval quantities addition implies the overlap parts are reduced to one part, with the non-overlap parts unchanged, to result in a one multiple star-source emission cloud where double star entries are avoided. This means for time interval quantities, e. g. star-source emission cloud properties, theorem 3 is valid, even while the relativation by theorem 3b, the quantity, center in theorem 3b, have to be similar, does not apply.

Theorem 4. Multiplication closure theorem, originating from (Hollestelle, 2024): for any multiplication M[∆A2, ∆A3] = A1, where ∆Ai, i = 2, 3, are time intervals, a1 scalar, there is A1 = a1. ∆t is time interval, implying closure of multiplication within the time interval only set. Multiplication linearity theorem, equivalent to it, is introduced in the same paper.

The multiplication closure theorem, is derived within the time interval only description for multiplication, however similarly valid for addition due to theorem 6.

Theorem 4b. Due to theorem 4, there is ∆Q1 = a1. ∆t and for any quantity ∆Q1, from the set of ∆Q quantities, it follows ‘addition zero’||∆Q units equal time interval set ‘addition zero’ ||∆t units.

Theorem 5. There is an argument for the usual one moment time description definition of the TDL, describing cosmological phenomena and events at large scale, due to theorem 1. Within the time interval description, indicating with i time interval multiplication inverse, only when < f. gi > = < f >. < gi > the individual quantities remain individual, following the definition of correspondence applying averages, par. 1.4. Definition of TDL: The thermodynamical limit (TDL) includes enlarging the complete ‘system’ with its overall volume V including the number N of objects occuring, however not the average value N/V or individual mass values (Fetter, Walecka, 1971) .

Theorem 6. Time interval set theorem, originating from (Hollestelle, 2024). Within the time interval only description, addition equals multiplication in value: A[∆t1, ∆t2] = M[∆t1, ∆t2] for any ∆t1 and ∆t2, time intervals or time interval quantities. This theorem is supported by time interval derivative and integral of any time interval quantity being equal in value similarly.

Theorem 7. Time interval complementarity relation. Due to the theorem on averages there is D*||∆t[∆A1] = D*||∆q[∆A1], and due to the wave equation D*||∆t*[ D*||∆t*[∆A2] ] = ∆A2 implying ∆A2 = a1. ∆A1, where there is D*||∆t[∆A1] = ∆A1, and this means a1. ∆A1 = ∆NC and ∆NC = #n. h. ν, which seems acceptable being a way to write the time interval version of the complementarity relation, for the overall cosmology of the star source emission cloud. This suggests the usual complementarity relation, ∆Es = #n. h. ν and Ew = h. ν, is due to similar considerations. Within the time interval description structure constants and Noether charge are the same and follow both from the wave equation, disregarding the multiplication scalar, (Hollestelle, 2024).

Theorem 8. There exist arguments for the ergodic hypothesis and arguments to derive the theorem on averages, starting from theorem 3, implying a new version for it. Within GR and qm both, relations occur similar to the theorem on averages. Theorem 3 seems to be equivalent to the following theorem from thermodynamics. Thermodynamic functions and ensembles: time average at one state equals the state average over many states at equal time, (Van Kampen, 1988).

To render the theorems in this paragraph universal, depended is on the star source emission cloud properties being valuable reduction of a cosmological universe properties. Theorem 2 is included in par. 3.3, theorem 9 continues in par. 4.2 for time interval Noether charge ∆NC, theorem 4b, 10 and 11 are continued with par. 4.3 for ‘zero’ temperature, and theorem 12 in par. 5 for gravitation and entropy.

Correspondence of time interval derivative and one moment time derivative, D*||∆t[a1] ~ D*||t[a1], for one moment time invariant scalar a1, reduces to D*||∆t[a1] = D*||t[a1]. ∆t.

Due to theorem 2, difference Q(n) – Q2(n) equals one moment time Noether charge NC ~ ∆NC, i. e. corresponds with time interval Noether charge ∆NC, linear in ∆t. There is ∆NC = ∆Es = #n. h. ν, the kinetic wave propagation energy, proportional to wave propagation surface measure mA, which equals, due to eq. 1, wave propagation group-velocity c(∆t). Specific Lorentz transformation TS, preserving surface measures including mA, differs from the usual Lorentz transformations TL that preserve the metric line element, where ∆Es and ∆NC both remain TS invariant, (Hollestelle, 2024).

4.1. Zero and Non Zero Equilibrium

For a change from (no-*) situation to (with-*) dU* situation with Hamiltonian H changing to H* = H + dU*, the extra energy introduced within the description is acquired energy dU*, without change of physics. Indication * implies the (with- *) situation, not the operator * or transformation * indication.

Legendre transformation TD remains valid, within one moment time description and time interval description: ∆H = TD*[∆L] = A[ M[∆p, ∆x], ∆Liv] and similarly ∆L = TD*[∆H], which applies the symmetric associative property (series), and theorem 6, (Hollestelle, 2024). Within one moment time description for quantities H and L, and p and x(p) = d(q)/dt one finds eq. 8.

Defined is H = H0 + dH, (Hollestelle, 2020) while avoiding confusion with time interval description indication ∆: in earlier papers H = H0 + ∆H, where in the present paper, differently, correspondence H ~ ∆H applies the indication ∆. Hamiltonian part H0 is time independent and equal to one moment time description energy ‘addition zero’||t ~ ‘addition zero’||∆t due to theorem 4, and dH is the time dependent and variable part of the Hamiltonian. To derive eq. 9, theorem 6 is applied again. One finds multiplication M[∆p*, ∆x*] = ‘addition zero||∆t* effectively for any situation. Eq. 8 reaches closure, i.e. zero equilibrium, in the (with-*) situation through addition with dU*: H* = H + dU*.

8

H* = H + dU* = H0* + dH* = dH* = ‘addition zero’||t* ~ M[∆p*, ∆x*] = A[∆L*, ∆H*] = A[∆L, ∆H]

9

∆H* = A[∆H, ∆dU*] = ‘addition zero’||∆t* = A[∆dH, ∆dU*] = ∆dU*

There is M[∆p, ∆x] = M[∆p*, ∆x*] = M[∆p, ∆x]iv, and M[∆p, ∆x] remains invariant changing from (no-*) to (with-*). Traditionally the invariant energy H0 = T + U can be chosen equal to ‘addition zero’||t, due to re-scaling.

There is H0* = ‘addition zero’||t*. Within both situations (no-*) and (with-*) the one moment time units ‘zero’ are equal, ‘addition zero’||t = ‘addition zero’||t*.

Due to correspondence there is H0 = ‘addition zero’||t ~ ∆H0 = ‘addition zero’||∆t = ∆U0. From eq. 8 and with A[∆H, ∆H*] = ∆H* there is A[ A[∆L, ∆H], ∆H*] = A[∆L, A[∆H, ∆dU*] ] = A[∆L, ∆dU*] = ∆dU* and:

10

A[∆L, ∆H] = M[∆p, ∆x] = ∆L = ∆H = ‘addition zero’||∆t* = ‘addition zero’||∆t

Theorem 4b. Due to theorem 4, there is unit ∆Q1 = a1. ∆t and for any quantity ∆Q1, from the set of ∆Q quantities, it follows ‘addition zero’||∆Q units equal time interval set ‘addition zero’ ||∆t units. This theorem is also included in par. 3.8.

With eq. 8, and A[∆L, ∆H] = ‘addition zero’||∆t*, and theorem 4b, it follows eq. 11. For space-like interval ∆q and spacetime interval ∆p, eqs. 12 are valid, i.e. the canonical time interval MVT equilibrium equations (Hollestelle, 2020). The derivative of time interval momentum ∆p*, second line eq. 12, equals the time interval version of Newton’s second law, par. 3.4, for the complete acquired energies U and U* = U + dU*.

11

∆H = ‘addition zero’||∆t = A[A[ ∆L, ∆H], ∆H] = A[∆L, ‘addition zero’||∆t] = ∆L

12

D||∆t* [∆q*] = M[∆t*i, ∆q*] = ∆q*

D||∆t* [∆p*] = -1. D||∆q* [∆H*] = M[∆q*i, ∆H*] = M[c*(∆t*)i, ∆Es*]

It is not assumed the group-velocity necessarily remains un-altered in the (with-*) situation always. The group-velocity c(∆t) and wave energy ∆Es* are specific for the time interval version of wave propagation spherical symmetric surfaces. Some of these properties are applied to derive overall time interval set properties and definitions, e. g. time interval derivative D||∆t* [∆t1] for any time interval ∆t1, where the * indicates ‘working to the right’ or, similarly, indicates an operator, (Hollestelle, 2021).

4.2. Zero Temperature

Definition of Zero or Lowest Bound Temperature

The definition of zero temperature can be chosen to imply: a situation without interaction, when interaction and time dependence imply change, related to non zero temperature. A situation without interaction can imply ‘free’ or ‘plane’ waves, a stationary state with uncertainty relation |∆*E|. |∆*t| ≈ |h|, not necessarily implies zero temperature, the ∆* indicating variances. Within this definition zero interaction includes the zero, lowest bound, temperature property, a ‘frozen’ situation, where however interaction is assumed necessarily to imply action and increased energy.

The dispersion free star source emission cloud is described, instead of with plane waves, with sphere-like propagation surfaces. This makes sense for a multiple star-source emmision cloud including simultaneous wave propagation, where no interaction is meant to imply continuous and invariant simultaneous emission from the star-sources and invariant simultaneous wave emission energy for equilibrium time development, described in par. 3.1, and (Hollestelle, 2020).

Within qm, defined can be a ground state, however possibly including interaction, (Fetter, Walecka, 1971). When field wave energy and Lagrange density depend on wave type of variation, i.e. change, of field variables (Sakurai, 1978), (De Wit, Smith, 1986), the concept of zero temperature is un-interpreted, since a field plane wave assumes change, if only due to wave group-velocity and the wave frequency-momentum relation, which defines ground state energy and energy exchange which is one of the reasons for the qm field to be defined. The following depends on the time interval description version of the qm complementarity relation, introduced in (Hollestelle, 2021).

The usual introduction of ensemble temperature from the partition function and the traditional β-function inversely proportional to temperature, (Fetter, Walecka, 1971), does not apply for all zero temperature cases. Temperature refers to the emitted wave propagation, not is meant star-source temperature in this paper. Within the time interval description one can define absence of interaction to mean absence of interaction within relevant event time interval ∆t, the finite measurement event time interval. In (Hollestelle, 2021) an attempt is made to describe a measurement event including wave function reduction. A combination of time dependent and time independent properties was assumed for two subsequent time intervals which together describe the measurement event.

Wave Emission Propagation Energy and the Constant of Planck h

The simultaneous emission energy ∆Es is a multitude of wave function energy ∆E = h. ν, for wave propagation with average wave momentum p = h. k_, for k_ = 1/ν, and this ∆E relation becomes invalid, due to a singularity, for zero energy and zero frequency ν. The qm emission spectrum often assumes emission spectral lines, however instead one can similarly assume a spectrum weight function. For a continuous weight function, introduce average < k > = k_, momentum variance |∆*k|, where variance can be assumed equal to weight function width, i. e. |∆*k| = 2u, (Merzbacher, 1970). Introduced is indication * for variances, different from the * , that indicate transformations and operators, e. g. TS*[∆t], transformation TS ‘working to the right’ on ∆t, and the addition indication (with-*) from par. 3. This follows indications in (Hollestelle, 2020).

It is derived |∆*k_|. |∆*q| = h+/h from which it follows < |∆*p|^2 > = h+^2, with h the constant of Planck and h+ = +1/2 (∆*p. ∆*q + ∆*q. ∆*p), defined in comment A, with variances ∆* equal to interval measures, h+. This is valid only within the time interval description where time interval quantity ∆q equals time interval ∆t = ∆U = ‘multiplication unit’||∆t. The following is from (Hollestelle, 2020).

Assume zero temperature to imply no interaction change occuring, with time independence and zero energy. In this situation, with E = ‘addition zero’||t, the usual requirement for time independence is the stationary situation with m∆*E << mE which however does not apply, and m∆*E = mE is allowed to be assumed valid. One finds ∆*k. ∆*q = c_ = h+/h where < c_ > = 1 implies time independence including h+ = h. This result relates to the limit for qm to be applicable, while < c_ > = 0 implies time dependence. The < c_ > values are written with scalar numbers.

For E = ‘addition zero’||t this would mean < c_ > = 0 where c_ = mE/m∆*E. h+, and thus not necessarily a time independent situation with < c_ > = 1, which can still be valid however due to the presence of two zero value quantities, mE and m∆*E. This special case can be interpreted with both averages < |E| > and < |c_| > to differ from zero infinitesimaly such that h+ can be assumed approximately equal to h. This depends on the weight function for the frequency density, and is different from what one expects from one value ν frequency emission.

It follows eqs. 13, within the time interval description, energy ∆Es = ∆T is a kinetic energy and m1 wave particle mass related to kinetic energy T, where zero temperature does not imply zero energy, due to application of averages. The third part of eq. 13 includes a singularity for m1 = ‘multiplication zero’||m_set, e. g. for m1 equal to zero photon mass, which can be resolved with the time interval asymmetry average < Es >||∆t = ∆Es different from ‘multiplication zero’||E_set for finite ∆t and time dependent H, comment 13.

Comment 17. At temperature T_zero, the photon mass m1 can be different from m0 = ‘multiplication zero’||m_set, or otherwise written, temperature T_zero implies wave particles for ∆Ew = h. ν different from zero mass m1 = m0 photons.

13

< |∆*p|^2 > = h+^2 ≈ h^2

< |∆*k_|^2 > = h+^2/h^2 = c_^2 ≈ 1(scalar) m∆*Es = h+^2. 1/|∆m1| ≈ h^2. 1/|∆m1|

Equilibrium and Interaction Time Development

Comment 18. From the discussion in this paragraph one can infer that more symmetrical time intervals could imply lower temperature. Any time interval measure m∆t is no garantee for interaction to occur or not. One can infer from relevant event time interval measures m∆t’ < m∆t’’ to imply ∆t’’ is ‘more’ stationary and measurement of interaction to be ‘more’ diffuse, with reference to ∆t’. This can be related to a measure for stationary event property change.

From eq. 5b and eq. 11, and the discussion from par. 3.8, one finds eq. 14, the accent originating from par. 3.7 for interaction, for any possible star-source emission cloud with independent Q’(n) and Q2’(n), with interaction implying change different from equilibrium time development with invariant energie ∆Es.

14

∆Es = A[< ∆Q’i >||n, ∆NC’iv] = ∆t

It follows for zero temperature both < ∆p >||∆t_set and < ∆q >||∆t_set equal time interval ‘addition zero’||∆p = ‘addition zero’||∆q equal to ‘addition zero’||∆t = ∆t, due to theorem 1c, and comment 10.

Eq. 15 is a simplified version of eq. 12, and the time interval version of the equilibrium time development equations. The eq. 15 right side resultant equals ∆U when ∆t finite, ∆U0 when ∆t infinite. The time interval addition inverse for ∆t is ∆tiv = ∆t, due to which the minus sign is absent.

15

D*||∆t [∆q] = ‘multiplication unit’||∆t D*||∆t [∆p] = ‘multiplication unit’||∆t

Within eq. 12 it is possible ∆H = ∆dU remains invariant, while however derivative D*||∆q [∆dU] not necessarily equal to ‘addition zero’||∆t and in this case does not reduce to the Newtonian result.

Due to the time interval multiplication closure theorem, within theorem 4, there is ∆dU = a1. ∆t and ∆q = a2. ∆t with scalars a1 and a2, while it can be derived ∆q = ∆t equal to the time interval only set Noether charge ∆NC or structure constant (Hollestelle, 2024). In the same ref. is derived the time interval ‘derivative set rule’ for any scalar multiplication a. ∆t1, D*||∆t[a. ∆t1] = A[ a. D*||∆t[∆t1], M[D*||∆t[a], ∆t] ], the first part of eq. 16, and it follows:

16

D*||∆t [∆q] = A[ a2. D*||∆t [∆t], D*||∆t [a2] ] = A[∆q, ∆t] = ∆q

One result is, ∆q is an invariant for derivative D*||∆t and D*||∆t [D*||∆t [∆q]] = ∆q, which is the time interval description wave equation in this case with solution ∆q, where the minus sign disappears naturally, similar to eq. 15, (Hollestelle, 2024).

Theorem 9. Time interval quantity ∆q is equal to the time interval only set Noether charge. It follows ∆dU = a1. ∆t = a1.

∆q equals the same structure constant, except upon linear multiplication, and one finds D*||∆t [D*||∆t [∆dU]] = ∆dU from which follows ∆dU = ∆NC.

4.3. Zero Temperature and Star Source Emission Cloud Properties

A cosmology describing a star source emission cloud, where defined is a cloud propagation surface A ‘nearby’ all star sources, was introduced in (Hollestelle, 2021). For a ‘fuzzy’ cloud one defines the places qi for the star-sources, with qi + qbiv = ∂i and m∂i << mqb, qb the one moment time 3-dim. space like coordinate for the center of the emission cloud collective, indication iv for addition inverse, mqb = < qi >||n = < qb + ∂i >||n with i = 1 to n for n the number of star- sources. The term collective can indicate a singular or multiple cloud collective. Within this paragraph T indicates wave emission, or measurement, temperature, not kinetic energy.

There exists a transition temperature Tc, such that for T >> Tc there is |qb + q0iv| << qb infinitesimal, and for T << Tc there is |qb + q0iv| >> qb, with q0 one moment time space-like coordinate zero place, which is defined to be on surface A in par. 3.1, (Hollestelle, 2017).

The above reference is written within the one moment time description, and Tc is proportional with |qb|^4. Proceeding within the time interval description, one finds: m∆Tc is proportional with m∆t^4, where there is m∆t^4 = m∆NC, applying m∆A = m∆t and m∆q = m∆t.

This can be combined with par. 4.2: Noether charge |NC| = mA^2 ~ ∆NC = M[∆A, ∆A] = ∆A. One finds: T_cloud / T_star-source ~ < |ni| >||m. #ncloud / #nstar-source. The ni refers to the cloud (i) star source number, number #n refers to the star-source emission wave function number. The ni is to facilitate multiple clouds (i = 1, m) and originates from (Hollestelle, 2021), however is left to mean m = 1 and ni = n, i. e. one star-source emission cloud including n star- sources. When overall emission cloud zero temperature, to imply T_zero = ‘lower bound’ domain dT with T_zero = T_cloud = < |n| >||t = ‘addition zero’||t = 0. t0, is not possible to apply, one can introduce time interval quantities with averages, ∆T_zero = ∆T_cloud = < |n| >||∆t = < t >||∆t = ∆U, following theorem 4b, which can be ‘non zero’, different from ∆U0, for ∆t finite. The overall emission cloud zero temperature ∆T_zero result is, from theorem 9:

17

∆q_cloud(∆T_zero) = M[∆q, ∆q]_cloud = ∆NC_cloud = ∆t_cloud(∆T_zero)

This result is valid only within the time interval description. Applied is, h. ν_cloud = h. ν_star-source and m = 1, with n for the overall star-source emission cloud equal to the number n of star-sources.

Collective Emission Temperature T, an Estimate for Temperature Lower Bound

In par. 4.2 it was derived ∆Es = ∆t, and ∆Es can be estimated due to the virial theorem, ∆Es = ½. D. k. ∆T, with D the number of degrees of freedom for the wave propagation surface, with the radius interval ∆q_cloud = ∆t_cloud, and k the constant of Boltzmann. One finds eq. 18 with time interval zero temperature ∆T_zero for the overall emission cloud with D = 4, due to the discussion in par. 3 and comment B.

The number of degrees of freedom can vary due to transition, to include resultant parameters different from original parameters. In par. 1.6 transition is different from usual, meant is a change from wave emission energy description to gravitation energy description, rather than a change of objects.

Transition of objects can mean transition of properties, however ∆T_zero remains equal to ∆U and ∆Es_zero equal to ∆t_zero. When properties change this can occur by temperature ∆T to change to a temperature ∆T ≠ ∆T_zero, e. g. related to < n >||∆t, ni = n and |n| = n >> 0. Applied is M[c(∆t), ∆t] = c(∆t) = ∆t and M[c(∆t), ∆T] = ∆T.

18

∆Es = ½. D. k. ∆T = #n. h. ν

∆Es = ½. D. k. M[c(∆t), ∆T]

19

∆T_zero = < n >||∆t. ∆t = ‘addition zero’||∆t

Theorem 10. For the star-source emission cloud, with one collective overall propagation surface, ∆Es = ∆q, eq. 14, is proportional with ∆t, at temperature T_zero. This is a re-statement and confirmation of the constant average wave group-velocity assumption for invariant temperature, i. e. without dispersion, c(∆t) = M[∆q, ∆ti] from par. 2.3 and comment A.

Comment 19. Estimates for temperature ∆T_zero. One can estimate a collective wave emission temperature from eq. 18 and eq. 19 with D = 1; wave emission surface energy ½. k. ∆T is ca. 0.43. 10^(-4) eV, and for emission temperature ∆T, linear with ∆qi and ∆t, this means ∆T_zero = ca. 0.50 K. Any degree of freedom adds ca. 0.5 K or ca. 0.4 10^(-4) eV value of energy to T_zero. The marges to these results are not derived. Applied constants are from (Cohen, Giacomo, 1987).

For D = 4, comment B, par. B3, one finds ∆T_zero equals 2 K. With approximation n = 2 for the average number of star sources within one collective this could imply a collective of two stars, with one moment time space like coordinate q1 ≠ q2. At ∆T_zero all star sources being far enough ‘remote’ from each other and from propagation surface ∆A, theorem 3b, to be independent sources both with number n = m∆T_zero = m∆U = 1, and with ‘near’ zero mass emission wave particles. This means star sources can be independently added to the emission cloud collective, similar to the resultant of theorem 3b, and 3c, par. 3.8.

One estimates value m∆q for D =1 at ∆T_zero, with m(∆q_zero)i = 1/2. k. < |n| >||∆t. |c(∆t)|, and m∆q = 1/|∆Es|. 1/| c(∆t)|= ca. 0.78 (10^12) meter = ca. 52 AE at emission temperature T_zero, linear with 1/D. From the discussion in comment A one finds ∆q = M[h+, ∆m1] for any zero or non zero mass ∆m1 wave particle and ∆q = h+ for ∆m1 = ‘addition zero’||∆m = ∆m0 for zero mass wave particles, photons, with photon mass ∆mp. For infinite ‘remote’, ∆q = h, since in this case h+ approximates h, and infinite ‘remote’ means lowest bound m∆q = 13 AE for D = 4. Similarly the zero mass ∆m1 = ‘multiplication unit’||∆t infinite ‘remote’ wave particle, and D = 1, m∆t = 1/|h+|. 1/|c|. |∆m1|. 1/D = 0.5 10^25 sec = 1.6 10^17 year, which is a highest bound with D = 1 and zero mass ∆m1.

Theorem 11. At temperature T_zero, relation ∆NC_zero = ∆q_zero means there is a preferred T_zero time interval

∆t_zero = ∆NC_zero = 2. k. ∆T_zero, due to eq. 18 and eq. 19, where both measure m∆t_zero and m∆T_zero, due to asymmetry, differ from value zero and both finite, i.e. related to ∆U and different from ∆U0.