As a particular example of an area of global scope, the dynamical development of our overall universe is here

considered.

With theories and discussions on this subject (see [1] - [4] ), the notion of a global state, which is associated with global time, seems to be very useful or even unavoidable. The following very simplified model of the dynamical evolution of our universe (originally described in [2] ) assumes global states evolving in unique global time steps.

Let us view the laws for the evolution of the state of the world (i.e., the laws of physics), represented by a “physics-interpreter”. The physics-interpreter acts upon the state of the world. The model described here assumes the continuous repeated invocation of the physics-interpreter to realize the progression of the state of the world. The physics-interpreter continuously determines new states. The physics-interpreter acts upon an in- world to generate an out-world.

world Evolution (world W): = {

W.t = 0; W.x₁ = 0; W.x₂ = 0; W.x₃ = 0;

W. = initialState;

Δt = timestep; // must be positiv //

DO UNTIL (nonContinueState(W)) {

physicsInterpreter (W, Δt);

}

}

physicsInterpreter(W, Δt): = {

tdt = W.t + Δt ;

W = applyLawsOfPhysics(W, Δt);

discardSpacetimePointsWithTimeCoordinate(W.t < tdt);

}

with state parameters

world: = {spacetimepoint...}

spacetimepoint: = {t, x₁, x₂, x₃, ψ}

ψ: = {stateParameter₁, …, stateParameter_n}

With reference to Barbours’ work [1] , D. Zeh writes in [5] (page 168): “Barbours’ essential assumption of global states in the sense of simultaneousness seems to be a relict from his non-relativistic theory. A relativistic theory should replace the global “Now” by “Here and Now”. Although it is true that relativity theory has dropped the concept of “global simultaneousness”, this does not disallow the idea of a progression of the (global) state in conjunction with a global time.

The twin paradox example (Section 2) helps to explain that for the calculation of the (local) proper time τ for the twins, SRT/GRT allows a suitable coordinate system to be chosen. This includes the specification of the space coordinates for the points P₁, P₂, P₃, as well as the specification of the velocities at which A and B are traveling. The choice for these parameters, of course, must be such that the relations between the positions and between the velocities are in accordance with the specific example. With the example described in Section 2, the positions have been chosen to be P₁ = (0,0,0,0), P₂ = (10,0,0,0), P₃ = (5,4,0,0). This implies that the choice for the velocity of A is v_A = 0 because P₁ and P₂ have identical space coordinates (0,0,0). If, in the given example, instead of assuming A to be at rest, we assume that A is traveling with a speed v_A not = 0 such that the space-time points have coordinates P₁ = (0,0,0,0), P₂ = (10,2,0,0) and, accordingly, B is traveling from P₃ = (5,4.75,0,0) to P₂ = (10,2,0,0) (see Figure 2), then approximately the same difference in elapsed (local) times is obtained as with the original example.

τ_A = sqrt{10 ² − 2 ²} = 9.8.

τ’_B1 = sqrt{5² − 4.75²} = 1.6 and τ’_B2 = sqrt{5² − 2.75²} = 4.2; together, τ’_B1 + τ’_B2 = 5.8.²

Thus, for the calculation of a concrete example, a coordinate system (including assumptions on the velocities of the moving objects) has to be chosen. For the twin paradox example, this may be considered as the specification of a local coordinate system. Similarly, a global coordinate system can be defined that provides a choice for all possible local coordinate systems.

Definition: A global coordinate system is a space-time coordinate system that assigns coordinates

X_i = {t_i, x_i, y_i, z_i} to all space-time points of the scope considered and, in addition, selects an object within the space to which a definite velocity is assigned. (A convenient and useful choice for the velocity gauge is the determination of an object for which a constant velocity of 0 is assumed.)

“Global Time” is understood in this paper as the time coordinate of a global space-time coordinate system. Thus, within this paper, “global time” has a proper meaning only in the context of a global space-time coordinate system, and a global space-time coordinate system (including global time) requires at least one physical object with (assumed) special properties, such as uniform movement. (The inclusion of a physical object with an assumed uniform velocity distinguishes the global space-time coordinate system from the purely mathematical space-time structures, such as, for example, Minkowsky space).

Let us assume that the objective of a global time clock is to show not only hours, minutes and seconds but also months and years. Point zero for the years need not be the point in time of the big bang, but should be far

enough in the past, e.g., the origin of the cosmic background radiation.

It may be difficult to construct a clock that shows the global time continuously. The simplest solution would be a clock that is based on the local time of the cosmic object where the clock resides. The global clock would have to be updated by the time units of the local clock multiplied by a factor f_g, which depends on the velocity of the local object relative to the reference object of the global coordinate system. Whether our cosmologist is able to determine, with reasonable precision, f_g for the earth is not clear (to the author).

The question about the age of our universe presupposes the notion of a global time (i.e., without the assumption of a global time, the question does not make sense). The global time clock described in Section 3.3 would show the age of the universe according to a chosen convention. Although the development of a global time clock faces the difficulties described in the preceding sections, there seems to be agreement that the notion of (global) age of our universe is reasonable. Cosmologists seem to have at least a very rough estimate for the age of our universe.

Global states are hardly imaginable without global simultaneousness, i.e., the ability to specify which space-time points are assigned to a (global) state. Given the definition of a global coordinate system, as defined in Section 3.2, the definition of global simultaneousness is trivial: State components and events assigned to space-time points.

X_i = {t_i, x_i, y_i, z_i,} are considered to be globally simultaneous if their t_i are equal.³

The global coordinate system defined in Section 3.2 is compatible with SRT/GRT. In addition to the theoretical possibility of global states associated with global time, it must also be shown that with the dynamical evolution of the universe, global states evolve into new global states. This subject is addressed below.

The definition of a global coordinate system implies that if the state associated with global time coordinate t_i proceeds by the global time step Δgt, then the new state will have the time coordinate t_i + Δgt. From this, it follows that if (1) a state has assigned a unique (global) time coordinate t_i and (2) the state evolves with a common (global) time step Δgt and (3) the world has started with an initial state at a unique global time t₀, then a global aging of the world can be concluded.

Thus, let us discuss to what extent conditions (1), (2), and (3) are plausible. Condition (1), which assumes that a state has assigned a unique (global) time coordinate, looks quite natural. Practically all of the laws of physics that describe how a state S₀ evolves into a state S₁ assume that state S₀ exists at a unique point in time t₀ (and, similarly, state S₁ exists at a unique point in time t₁). Any deviation from this implicit assumption would make the laws of physics rather cumbersome.

Condition (3), which assumes that the initial state of our universe includes a unique global time t_i, is also the usual assumption implied with the big bang theory. Condition (2) is assumed with the model of the dynamical evolution of the world described in Section 3.1, but is possibly more controversial. With a view of the (local) wristwatch time, which leads to the twin paradox, one may argue that objects traveling at different velocities evolve with local time steps rather than with global time steps. However, this argument ignores that, at the end, when the two objects meet again, they meet at a common point of global time. This means that the same amount of global time steps Δgt is executed on all trajectories that end at an identical global time.

For the model of the dynamical evolution of the world described in Section 3.1, this means that the overall progression of the state of the world (i.e., the global aging) occurs in global time steps Δgt. When the execution is broken down to the individual local systems, the global time steps Δgt have to be filled with differing local time steps Δlt.

The alternative model, where the dynamical evolution of the world would proceed in terms of local time steps Δlt, would make the reunification of diverging trajectories very difficult or impossible.

With the physics interpreter described in Section 3.1, the statement

“discard All Space time Points With Time Coordinate (W.t < tdt);” is contained, which means that the past is discarded. This statement may be considered arbitrary; to the author’s knowledge, there is no proof or experiment that demonstrates its truth or falseness. Given that, with the known laws of physics, the prediction of a future state never depends on a past state in addition to the present state, nor is it possible to directly access a past state, the formulation “past states are lost” and the inclusion of the referenced statement appear reasonable.

The feasibility of a global coordinate system may be questioned for two reasons:

1) The definition of a global coordinate system may be theoretically correct, but the determination of a stable reference base, as required by the definition, may not be possible given what cosmologists know about the process of the evolution of our universe. There is a high chance that it is not feasible to determine a constantly stable reference base. For example, what we know (and what we do not yet know) about dark energy and an inflationary universe may result in a rather unstable reference base. However, a stable reference base is not explicitly requested by the above definition. As long as the dynamic change process is sufficiently understood, the determination of a global coordinate system may still be possible, although more complicated.

2) The definition of a global coordinate system may be feasible theoretically but difficult with the cosmological knowledge we have at present. For example, according to the above definition, the determination of a global coordinate system requires the selection of a physical object (e.g., a star) as a reference point. A suitable object would have to be, as much as possible, unaffected by dynamical developments (e.g., changed gravitational environment or changed velocity) that influenced its proper time in the past and presumably will in the future. As an example for a cosmic object with presumably well-understood uniform development, the aging of the cosmic background radiation could be a reasonable datum point (together with some assumption about the time and initial state of the formation of the cosmic background radiation).

The major goal of this paper was to show that the notions of global states and global time are compatible with special relativity theory and general relativity theory. The paper provides extensive arguments for the validity of global notions.

Although the arguments given refer mostly to SRT, they are believed to also hold true for GRT. The local time due to GRT is more complicated than the local time due to SRT (as explained with the twin paradox). Nevertheless, the arguments given for the necessity and feasibility of the notions of global time, global states, and global simultaneousness apply equally to GRT.

Much has been written (for example, [3] [5] - [8] ) in regard to the (possible) identification of a direction of time associated with the dynamical evolution of the world. Typically, the discussion starts with the statement that the laws of physics are symmetric under a reversal of the direction of time. Observations that give the impression of the existence of a direction of time are then explained as a consequence of an increase in entropy, as stated by the second law of thermodynamics. Thus, the existence of an arrow of time is established, but it is implied purely from the entropy law. In [2] , it is argued that a direction of time becomes apparent, without recourse, to the flow of entropy, based on the laws of physics, if the laws of physics are considered in the context of their overall goal, namely, describing the dynamical evolution of the world. The reasoning described in [2] is based on the simple model of the dynamical evolution of the world, as described in Section 3.1. Because both subjects, the direction of time and the possibility of a global coordinate system, are discussed in view of a model of the dynamical evolution of the world, the question arises whether the global coordinate system defined in this paper implies an arrow of time. The author’s answer is that the definition and existence of a global coordinate system does not imply an arrow of time. However, the further subjects of this paper, specifically global states, global simultaneousness and global uniform aging of our universe, have been discussed based on the referenced model of the dynamical evolution of the world, which implies an arrow of time.