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Time, Equilibrium, and General Relativity

Harmen H. Hollestelle

2020
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The International Journal of Science & Technoledge
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In this article three new ideas are introduced to support the concept of time as a time interval. These are a new definition of time coordinates, the introduction of time intervals for derivatives and the application of the "mean velocity theorem" to describe equilibrium. The time coordinates, asymmetric to the past and future, agree with an asymmetric time experience and from there the introduction of time intervals is natural. The concept of time as a time moment is basic to many theories in
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... o many theories in physics. However, with time moments one cannot easily understand change or continuity. In Hamilton's principle of least action, a time interval occurs, however it depends on virtual, not real, variations. The resulting Lagrangian equations, that do describe equilibrium effectively, depend on derivatives to time moments only. Newtonian equilibrium as well only applies derivatives to time moments. There the problem of time moments related to change already emerges. A new description of equilibrium is proposed based on time intervals, with the help of the above three concepts. The "mean velocity theorem" (paragraph 3) includes a graphical way to describe a "time" of equilibrium in the sense of a center of weight, and naturally provides the possibility to introduce time intervals and derivatives to time intervals. Also, it provides an intuitively clear understanding of symmetries and asymmetries during equilibrium. From it follow derivatives and commutation properties related to time intervals for any function of time moments t (paragraph 4). The properties of space coordinates q(t) thus derived are applied throughout the further parts of this article. In paragraph 8) introduced is the specific time interval necessary for derivatives to time intervals. Time coordinates and their properties are defined in paragraph 6) and paragraph 8). Time is assumed to depend on two elements that added together result in a one-dimensional time coordinate. One of these elements is anti-symmetric for past and future, and it counts time with positive numbers. The other one is symmetric and decisive for from when time is counted. With these definitions time coordinates do not commute and the value of a product of time dependent quantities does depend on their writing order. The derivative to time intervals and the "mean velocity theorem" are applied to derive expressions for the time dependent Hamiltonian, (paragraph 5) and (paragraph 7) and the time interval derivative of the Hamiltonian (paragraph 8). The commutation properties for q(t) derived in paragraph 4) are the basis for these results, however this Hamiltonian can be derived independently also from the equilibrium definition in terms of the generator of time transformations. A step by step transformation for time intervals prepares for how in General Relativity stationary state "local" time intervals can be integrated towards "non-local" time interval measurements (paragraph 11). Because of limits on the resulting time interval measures, this allows for a probabilistic interpretation for quantities that have these intervals as time domain. Thus, these measures are interesting in both a GR as a QM sense. The time intervals also question the time reversal symmetry of GR. As a second application the QM description of the measurement of starlight radiation energy is expressed Harmen H. Hollestelle Abstract: Considered is "time as an interval" including time from the past and from the future, in contrast to time as a moment. Equilibrium as the basis for a description of changing properties in physics is understood to depend on the "mean velocity theorem", while a "time" of equilibrium resembles a center of weight. This turns out to be a good method to derive properties for any function of time t including space coordinates q(t) and expressions for the time dependent Hamiltonian. Introduced are derivatives depending on time intervals instead of time moments and with these a new relation between the Lagrangian L and the Hamiltonian H. As an application introduced is a step by step method to integrate stationary state "local" time interval measurements to beyond "locality" in General Relativity. Because of limits on the resulting time interval measures, this allows for a probabilistic interpretation for quantities that have these intervals as time domain. Thus, these measures are interesting in both a GR as a QM sense. Another application of time interval is the discussion of the measurement of starlight radiation energy and QM wave packet collapse as an example of a time dependent Hamiltonian. Finally, a relation between starlight frequency, metric and space-and time intervals is discussed. The time intervals also question the time reversal symmetry of GR. in terms of the interval derivative of the time dependent Hamiltonian in paragraph 12). In the final paragraph (13) discussed is the relation between time interval, space interval, starlight frequency and metric tensor. Equilibrium with Time Intervals and a Time Dependent Hamiltonian Newton's laws relate applied forces and the second derivatives to time moments of the space coordinates q, for a given mass m. Equilibrium is described as the applied forces being "equal" to the changes of the velocities x, which are the first derivatives to time moments of the coordinates q [Goldstein, 1]. For a conservative system that is described with a kinetic energy T quadratic in the derivatives dq/dt, the forces F = -V/ q is derived from V, meaning all other energy. For a conservative system the kinetic energy is conserved for a closed actual path. Equilibrium based on Hamilton's principle of least action implies that the integral: I = ∫ L dt, from time t1 to t2, with L = T -V the Lagrangian, is an extremum for the actual path of motion compared to other possible paths. Otherwise said the δ variation of the integral I is zero: δ I = δ (∫ L dt|∆t1t2) = 0. This means that the integral I for the actual path is locally stationary, does not change for infinitesimal changes of the path, and thereby determines equilibrium: the total energy H0 = T + V is time and space independent and the change in T is the same as the change in -V, thus according to δ I = 0 the first order variation of both T and V with any varied path is zero. A δ variation means the considered time interval t1 to t2 remains actual and fixed while the considered, virtual or possible however not actual, path may vary from the actual path. From there one derives the Lagrangian equilibrium equations, for L = L (x = dq/dt, q), that are equivalent to those of a system in Newtonian equilibrium [Goldstein, 1], [Arnold, 2]. This is a description in terms of energy quantities like the Lagrangian and the Hamiltonian. Newtonian equilibrium is independent of δ variation considerations, however similarly applies time moment derivatives of q. The total energy H0 = T + V remains time independent for any system. Legendre transform will remain valid for the new equilibrium description in paragraph 3) and 5) including a time dependent Hamiltonian H. The Hamiltonian H can be evaluated for a certain time interval from the difference of L and the asymptotic function p.x, with the "mean velocity theorem", reconsidering the relation dL/dx = p for x(p) which is a time moment derivative relation. For a Newtonian or conservative system H reduces again to the total energy H0 as required. The "Mean Velocity Theorem" as a Basis to Describe Variation and Change and (A-) Symmetries The symmetry properties of a system tell which transformations do not change the value of the Hamiltonian. Similarly, when the value of H changes with some transformation parameter this means an asymmetry exists for some property. This agrees with the essence of Curie's (i.e. Pierre Curie) principle [Curie, 3]. Discussion of Curie's principle in relation to the Higgs mechanism can be found in [Katzir, 4] and [Earman, 5]. In qm field theories group representations of symmetries are applied to derive particle properties, and the absence of symmetries gives clues to derive differences between properties and for transitions and changes [Veltman, 6]. In this article concentrated is on time intervals and time elements and the time dependent Hamiltonian. Consider the "mean velocity theorem" [Hannam, 7] [Dijksterhuis, 8], that can be visualized with graphs. The theorem states that the area below a horizontal line is the same as the area below a sloped line, when the two lines meet and cross each other at that value at the parameter interval for which the sloped line reaches its average, "mean", value. The first line means constant velocity and the second one means varying velocity in the case of a time parameter. Because of where the two lines meet and cross each other the theorem is also called the "fixed point theorem". In Medieval age it was derived as the "mean speed theorem", with the help of graphs. The "mean velocity theorem" is in itself a way to imagine equilibrium, like the center of weight is an "average" place. The evaluation of mean velocity graphs was generalized from one dimension to higher dimensional spaces by Brouwer, who also introduced the term "fixed points theorem" [Hocking and Young, 9]. The mean velocity theorem is part of a tradition of thinking how changing properties can be described. Newton introduced derivatives, for instance to describe continuously in time the change of velocity in terms of applied forces. To relate the function L(x = dq/dt, q) = T -V to H(p, q) as a function of a new coordinate p with dL/dx = p at x(p) was a consequence when a description in terms of energies became an alternative to the description in terms of paths. A traditional derivative depends on a limiting process from a surrounding interval towards one moment in time or one space point. It remains to be interpreted what this limit means for the description of the continuity of variables that change with time or start to change with time. For a derivative to an interval instead of to one moment these difficulties do not exist. Within quantum mechanics, change is related to probability and discontinuity. Initially in qm reasoning the concept of space and time was to be disregarded in favor of abstract energy levels at least in the quantum domain. Any attempt to localize for instance with the help of paths is refuted [Beller, 10]. Energies relate to symmetries naturally: energies can remain invariant during variation of a property, while actual coordinates mostly vary in any case. This is a reason why energy quantities can be a basis for symmetry and equilibrium description. Especially when H is time dependent and describes change, or when it describes invariance as the absence of change, a time derivative depending on time intervals seems more appropriate then a time derivative depending on a time moment. Equilibrium, similarly, needs time intervals rather than a time moment to be defined properly, since it only exists where one is in equilibrium with another one. Indeed, Hamilton's principle of least action is also defined for a time interval: the time interval [t1, t2]. Arnold mentions the criterion for an equilibrium x0 of a system dx/dt = f(x): x(t) = x0 for all t is a solution of this system, i.e. f(x0) = 0, [Arnold, 2]. One can say equilibrium means a quantity exists that expresses invariance and symmetry as being the change of several other quantities. The formulation of equilibrium with the mean velocity theorem is crucial because it describes the interdependence of one moment values of a function with a certain interval average of this same function. Interval Derivatives and Intervals The definition of a comparative derivative of a quantity or function, say f(x), to an interval ∆X that includes the parameter x(t) for some specific t belonging to ∆t = [t1, t2], using " " notation to emphasize the difference with a traditional derivative to the parameter x for the specific x = x(t), is: 1) "df/dx"|∆X = < df/dx >|∆X = ∫ (df/dx) dx (1/|∆X|) Equation 1) depends on the interpretation of the relevant "mean velocity" graph as a comparison between average and slope line. This comparison is similar to an equilibrium definition for the slope line and it liberates the derivative from a one value limit to an interval in equilibrium. With < ... >|∆X is meant the average for the interval ∆X = [x(t1), x(t2)] where t is a one-dimensional parameter for simplicity. x(t) belongs to the interval ∆X and ∆X in turn should include x(t). For convenience also is defined the interval ∆Y(y(t)) = [x(t1), y(t)] for any y(t) belonging to ∆X. For y=x(t2) there is ∆Y(y) = ∆X = [x(t1), x(t2)]. Also |∆X |= |x(t2) -x(t1)| = |x(t2)| because the value of x(t1) is quite arbitrary, and one may organize that x(t1) = 0. At least x(t2) > x(t) The interval ∆X is interpreted as the domain for the function f(x). The following approximation is valid for all y belonging to ∆X: "df/dx"|∆Y = "df/dx"|∆X (y/x(t2)) and thus < df/dx >|∆Y = < df/dx >|∆X (y/x(t2)). This means that any function f allows for a linear approximation for the complete interval ∆X. A linear approximation might be positive or negative of sign depending on f(x) being increasing or decreasing. For all x belonging to ∆X and for all increasing positive f(x), this approximation means the evaluation of f(x)/x ≈ "df(x)/dx"|∆X or written as a linear equation f(x) ≈ "df/dx"|∆X x, while assumed is f(x = 0) = 0. For decreasing positive functions f(x), "df(x)/dx"|∆X ≈f(x)/x, and similarly for negative functions. For the space coordinate q(t) one finds "dq/dt"|∆t ≈ +/-q/t, for a positive, increasing respectively positive decreasing q and for ∆t = [t1, t2]. From "dq/dt"|∆t ≈ -q/t follows the approximation [1/t, q] = -2q/t and [t, q] = -2qt and 2a) "dq/dt"|∆t = 1/2 [1/t, q(t)] and this commutation bracket relation is inferred to be a valid equation for all functions and for all t belonging to ∆t, not only for q(t), valued at "equilibrium" being the equilibrium from the "mean velocity theorem" for ∆t. The following definition for a comparative derivative is inferred to be valid for any interval ∆t = [t1, t2]: 2b) "df/dt"|∆t = 1/2 [1/t, f(t)]|∆t = 1/2 (1/t1 f(t1) -f(t2) 1/t2) Writing "comparative" commutation brackets in this way suggests a similar definition with 1/2 [t, f(t)]|∆t = 1/2 (t1f(t1) -f(t2) t2), being the comparative integral of f(t). With equations 1) and 2a/b) derivatives to an interval ∆X or ∆t is defined as an alternative to traditional time moment derivatives at x = x(t) at time moment t. Equation 2b) can also be evaluated for t1 = 0 due to the linear approximation above. On the right side, still time moment functions remain. These definitions are independent of the traditional derivative and finding a function f(t) by traditional integration does not provide a solution for a comparative derivative equation immediately. However, from the above it can be argued that a positive, decreasing, function q(t) is proportional with 1/t. With the comparative derivative, and the above approximation as a comparative method, the following equation is directly derived for the Legendre transforms f and g for which g = p.x(p) -f: 3) "df/dx"|∆X = < df/dx >|∆X = 3/2 p -3 < g >|∆X 1/x(t2) Equation 3 ) does not replace the Legendre transform relation for f and g. On the contrary, it defines the comparative derivative for f(x) to an interval ∆X, while the Legendre relation g = p.x(p) -f remains intact. Thus equation 3) defines "df/dx"|∆t as a derivative to an interval while again the right side of the expression contains time moment dependent functions. This occurs because the interval ∆X and the specific time moment coordinate t are related. The progress with equation 3) is in the application of the derivative to an interval ∆X, which itself depends on the time interval ∆t=[t1, t2]. To avoid infinite regress chosen is to keep p and x(t2) as time moment parameters included in equation 3). In this way an interval does not have an interval as border. The comparative derivative definition agrees with a theorem [Arnold, 2] concerning the equal value of averages of a function for a t interval and a q interval. Following the usual identification f = L and g = H the traditional derivative of the Lagrangian dL/dx = p at x(p) while the comparative derivative "dL/dx"|∆X for interval ∆X can differ from p, because of the liberation of the derivative from a one value limit to an interval equilibrium. With equation 3) the traditional Lagrangian equilibrium equations and equilibrium itself become time interval dependent. Time Interval Averages Even for H time dependent, the Lagrangian L and the Hamiltonian H are assumed to remain the Legendre transform of each other. With f = L and g = H and x the comparative time derivative of q, and writing H = H0 + ∆H(t), to accompany equation 3) one finds:

doi:10.24940/theijst/2020/v8/i5/st2003-035
fatcat:czfikngrtjhifbltuv2mzxatzu